Business Mathematics
Business Mathematics
Chris Kellman; Leslie Major; Don Mallory; Frank Gruen; and Amy Goldlist
BCIT
Business Mathematics by BCIT is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License,
except where otherwise noted.
Contents
Introduction 1
Acknowledgements 5
Accessibility Statement vii
Chapter 1: Business Applications of Basic Mathematics
1.1 Ratios 11
1.2 Proportions 15
1.3 Ratios With More Than Two Quantities 17
1.4 Retail Calculations 21
1.5 Rates and Currency Conversions 25
1.6 Trade Discounts 29
1.7 Cash Discounts 33
Chap 1 Knowledge Check Answer Key 41
Chapter 1 Review Questions 45
Chapter 2: Functions and Applications
2.1 Example of a Function 65
2.2 Linear Functions 69
2.3 Cost Functions 79
2.4 Equations and Functions 81
2.5 Systems of Equations 83
2.6 Profit Volume Analysis 89
Chapter 2 Knowledge Check Answer Key 97
Chapter 2 Review Questions 99
Chapter 3: Simple Interest
3.1 Example of Simple Interest 113
3.2 Exact Time 115
3.3 Calculating the Amount of Interest 123
3.4 Calculating Principal, Rate and Time 125
3.5 Calculating the Future Value 127
3.6 Time Diagrams 129
3.7 Calculating Present Value 133
3.8 Time Value of Money 139
3.9 Equations of Value 145
Solutions to Chapter 3 Knowledge Checks 151
Chapter 3 Review Questions 157
Chapter 4: Compound Interest
4.1 Introduction to Compound Interest 167
4.2 Nominal and Periodic Rates 171
4.3 Compound Interest Formula 173
4.4 Present Value 177
4.5 Multiple Cash Flows 181
4.6 Equivalent Values 185
4.7 Compound Interest with the BAII Plus 189
4.8 Equivalent and Effective Rates 197
4.9 Fractional Periods 203
4.10 Average Rates 207
Chap 4 Knowledge Check Answer Key 209
Review Questions for Chapter 4 211
Chapter 5: Annuities
5.1 Adding to a Savings Account 229
5.2 Withdrawing from a Savings Account 235
5.3 Loans and Down Payments 241
5.4 Leases and Annuities Due 249
5.5 Deferred Annuities 257
5.6 Back to Back Annuities 273
5.7 Bonds 289
5.8 Perpetuities 297
5.9 Preferred Shares 307
5.10 Mortgages 315
5.11 The 32% Rule and Bi-Weekly Payments 327
5.12 Lump Sum Payments and Refinancing Mortgages 335
Chapter 5 Review Problems 339
Chapter 6: Investment Decisions
6.1 Evaluating a Business Plan 371
6.2 Rate of Return 373
6.3 Net Present Value 375
6.4 Cash Flow on the BAII Plus 381
6.5 Internal Rate of Return 385
6.6 Payback 389
6.7 Effects of Different Lifetimes 393
6.8 Cost Comparisons 395
Chapter 6 Review Problems 397
Chapter 7: Extra Problems
Chapter 1 and 2 Review 409
Chapter 3 and 4 Review 417
Chapter 5 Review 421
Entire Course Review 427
Appendices
Appendix A: Learning Curves in the BAII Plus 443
Appendix B: EUAC and Capitalized Cost 447
Glossary 453
Formulas 461
Introduction
OBJECTIVE
This Business Mathematics courses aim to have students understand basic business problems,
interpret them in mathematical terms, and use the tools of mathematics to solve these problems.
The objective of this material is to provide mathematical theory, demonstration and practice for
first-year Business students.
A grade of C+ or better in Algebra 11 (or equivalent) is the prerequisite for this course; students
requiring extensive review or remedial work are expected to have completed a preparatory
course such as OPMT 0199, which is offered through part-time studies at BCIT.
These notes explain the applications of basic mathematics to business and industry using ratios,
functions and graphs, simple and compound interest, financial instruments and discounting,
annuities, mortgages, loans, and leases. Cash-flow analysis applying rates of return, net present
value, and payback is included.
We expect students to read the material carefully and follow the worked examples. You should
also work through the learning activities in each chapter and check the recommended solutions.
Practice problem sets with selected answers are included at the end of each chapter. You are
expected to have a pre-programmed business/financial calculator; the Texas Instruments BA II
Plus is recommended, and will be used in solving problems.
CALCULATOR INSTRUCTIONS
The recommended calculator for this course is the Texas Instruments BAII Plus. You may
purchase another calculator as long as that calculator has per-programmed financial
functions. Your calculator should be able to perform Net Present Value (NPV) and
Internal Rate of Return (IRR) cash flow analysis. Please talk to the instructor before
purchasing any calculator other than the BAII plus.
Business Mathematics 1
1
Formatting a BAII Plus:
Step To Press Display
1 Select FORMAT function [2ND] [./FORMAT] DEC= 2.00
2
Change the number of decimals to 9
(with 9 decimals it hides the zeros after the
decimal point)
[9][ENTER] DEC= 9
3
Change calculator to BEDMAS (AOS)
(gives correct order of operations)
Scroll to where it says AOS or Chn
** You want AOS
Check by entering: $3+2\times 5$
You should get 13 (not 25)
[↓]
[↓]
[↓]
[↓]
[2ND][ENTER/SET]
DEG
US-12-31-1990
US 1,000
Chn Skip step if it
shows AOS
AOS
4 To get out of the FORMAT function [2ND][CPT/
QUIT] 0
Or if you are a visual learner, here’s a handy video, courtesy of former BCIT instructor Chris
Kellman: Setting up your BAII Plus
Another option:
One or more interactive elements has been excluded from this version of
the text. You can view them online here: https://pressbooks.bccampus.ca/
businessmathematics/?p=4#video-4-1
Memory Operations with the BAII Plus
Compute:
2 Amy Goldlist
Step To Press Display
1 Sum $525 + 287 + 309$ [5][2][5][+][2][8][7][+][3][0][9][=] 1,121
2 Store the value in Memory 1 [STO][1]
3 0.4 divided by 0.6 [.][4][÷][.][6][=] 0.666666667
2 Store the value in Memory 2 [STO][2]
5 Memory 1 divided by Memory 2 [RCL][1][÷][RCL][2][=] 1681.5
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4 Amy Goldlist
Acknowledgements
BUSINESS MATHEMATICS
Notes for courses in the Mathematics of Business and Finance
Chapters 1-4 and 6 were prepared by:
Don Mallory, Frank Gruen, and other Business Mathematics instructors of the Operations
Management Faculty
Chapter 5 prepared by:
Leslie Major, with input from Myra Andrews, Alyssa Wise, Claudine Warburton, Michelle
Nakano, Neilu Rishi and the Business Mathematics instructors of the Operations Management
Faculty at the time:
Sam Choo
Ron Correll
Tim Edmunds
Judy Li
Germain Tanoh
Daniel Anvari
This book was adapted and revised as an e-book by Amy Goldlist and Leslie Major, with
revisions and edits by Myra Andrews. This was made possible with a Open BCIT Grant in
2021.
This material was originally prepared with the assistance and support of the Learning
Resources Unit.
Revision #12- December 2017 by Chris Kellman
Revision # 13 – August 2020 by Amy Goldlist
Revision # 14 – July 2021 by Amy Goldlist & Leslie Major
The cover image was adapted from
Business Mathematics 5
5
close up photo of assorted coins by Josh Appel available under the Unsplash
License
6 Amy Goldlist
Accessibility Statement
We have tried to the best of our ability to ensure that this textbook is as accessible as possible.
If there are any issues, please contact us at Business_MAth@BCIT.ca, and we will try to fix.
The online version of this text contains videos, which are not in the PDF version. You can
view these at pressbooks.bccampus.ca/businessmathematics.The videos are additional, and not
necessary for the understanding of this material. They have been auto-captioned by YouTube,
but there may be some errors.
The online version of this text includes some interactive activities. The PDF gives an address
where each activity can be found.
Business Mathematics 7
7
8 Amy Goldlist
Chapter 1: Business Applications of Basic
Mathematics
The purpose of a business is to make a profit. To find the profit a company makes in a particular
period, say in a given month, you must start with all the money that comes into the company
as a result of its operations. Then from that money, which is called the revenue, subtract all the
expenses paid to earn it.
Thus you have
Revenue
– Expenses
= Profit
The profit is found on the famous “bottom line.” In accounting it is usually called the net
income.
Example 1.0
Suppose that a company’s revenue is earned entirely from sales and that it had expenses of
$24,000 in a period during which it had sales of $32,000.
Then,
Revenue $32,000
– Expenses $24,000
= Profit $8,000
REVENUE: $32,000
Expenses Profit
$24,000 $8,000
When you study accounting, you will find that it takes considerable effort to find the correct
values for the expenses that apply to the revenues for a period.
Business Mathematics 9
9
10 Amy Goldlist
1.1 Ratios
This chapter uses ratios to analyze basic business concepts such as profits, expenses and
revenues.
Key Takeaways
If only two quantities are involved, they are referred to as amount and base.
A ratio expresses the relative sizes of quantities. If only two quantities are involved, they are
referred to as amount and base.
A frequently used ratio is the ratio of profit to sales. In this case the base is sales and the amount
is profit.
Thus you have
This ratio is also written as
Key Takeaways
It is common to express a pure ratio in percent.
Since both the amount and base are in dollars, this is a pure ratio or fraction, and it is common
in business to express it in percent.
Percent is a ratio in which the base is 100. Thus, for the above example,
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where the symbol % signifies that the amount 25 is to be divided by 100.
A ratio of 30% is thus:
Percent is used so much in business analysis that you must become completely familiar with
its meaning and be able to change back and forth between percent and decimals.
The profit-to-sales ratio expressed this way is called the percent net margin. It is used to
compare the profitability of different companies or branches of companies.
Example 1.1.1
Consider a company that has three branches, A, B and C, operating in small, medium-sized and
large towns. Suppose the branches have the following performances during a month:
A B C Total
Sales $12,500 $46,500 $63,200 $122,200
Expenses $9,550 $38,750 $48,850 $97,150
Profit $2,950 $7,750 $14,350 $25,050
From these raw figures it is not easy to assess the performance of the branches. It is clear that
C’s profit is greater than B’s, and that B’s is greater than A’s, but you would expect that to occur
because of the size of their markets and the volume of their sales. If you take the percent net
margin, however, you get the following:
For A:
For all branches and the company total:
A B C TOTAL
Percent Net Margin 23.6% 16.7% 22.7% 20.5%
From those percentages, you can see that, in a sense, branch A has the best performance (for its
size), and that branch B is farthest out of place. Although there may be good reasons for this,
the ratios provide a reasonable place to start looking for improvements.
12 Amy Goldlist
Examining profits on graphs illustrates the situation. First look at the graph of the dollar values
of the profits (Figure 1.1).
Figure 1.1: Profits in Dollars
Then look at the graph of profits as a percent of sales (Figure 1.2 below).
Figure 1.2: Profits as a Percent of Sales
On the last graph you can see that branch B is noticeably lower.
Ratios are frequently used in business analysis. Two examples:
Turnover ratio -This is the value (at cost) of goods sold, divided by the average
value of the goods held for sale (inventory). This gives the number of times the
inventory is “turned over” (completely sold and replaced).
Acid test ratio – This is the ratio of the cash resources (e.g., cash and customer
accounts owed to the company) to the short-term debts of the company. It is a
measure of the ability of a firm to pay its short-term debts.
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Knowledge Check 1.1
1. Central Clothing Store had sales of $80,000 last month.
a. If the total expenses were $71,000, what was the profit?
b. What must expenses have been if the profit had been $14,000?
2. Two consumer electronics stores, Sounders and Electronicks, are to be evaluated.
Their sales for last year were as follows:
Sounders Electronicks
Value of goods sold (at cost) $936,400 $1,245,900
On average, Sounders stock on hand cost them $184,000. Electronicks stock on hand
cost them $209,400. Compare their turnover ratios. Which had the better (higher) ratio?
Your Own Notes
Are there any notes you want to take from this section? Is there anything you’d like
to copy and paste below?
These notes are for you only (they will not be stored anywhere)
Make sure to download them at the end to use as a reference
An interactive H5P element has been excluded from this version of the
text. You can view it online here:
https://pressbooks.bccampus.ca/businessmathematics/?p=52#h5p-1
14 Amy Goldlist
1.2 Proportions
Another way to analyze the performance of branches A, B, and C described above is to take the
view that all of the percent margins should be equal – or, equivalently, that the ratio of profit to
sales should be the same for each branch.
When the ratio of quantities should be the same in all circumstances, the quantities are said to
be proportional to one another.
Example 1.2.1
Assume that the profit should be proportional to sales, and that this proportion should be that
of the company totals (20.5%). Then you could calculate the appropriate profits for branches
A, B and C individually.
For branch A:
Thus Profit =
Similarly for branches B and C, the proportional profits would be $9,532.50 and $12,956
respectively. The analysis would proceed as follows:
Profits A B C
Actual $2,950.00 $7,750.00 $14,350.00
Proportional 2,562.50 9,532.50 12,956 .00
Difference +387.50 -1,782.50 +1,394.00
Again you can see the branch B’s performance was the least effective.
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Ratios are generally used as in the above exercises – sometimes to eliminate the effect of size,
and sometimes “in reverse” to make sure the quantities are appropriate.
Knowledge Check 1.2
1. Will’s Coffee Shops expect to earn a percent net margin of 14% of sales.
a. Find the expected profit and expenses of a shop that will
have sales of $29,000 a month.
b. If a shop hopes to earn a net profit of $5,000 a month,
what should be its sales target?
2. As a guide to restaurant pricing, it is often assumed that the price of a meal
should be proportional to the cost of the food in the meal. Suppose that a
restaurant decided to make that ratio 2.8.
That is:
Use this guide to estimate the price of meals for which the food cost is (a) $3.50; (b)
$8.00; (c) $14.00.
Your Own Notes
Are there any notes you want to take from this section? Is there anything you’d like
to copy and paste below?
These notes are for you only (they will not be stored anywhere)
Make sure to download them at the end to use as a reference
An interactive H5P element has been excluded from this version of the
text. You can view it online here:
https://pressbooks.bccampus.ca/businessmathematics/?p=96#h5p-1
16 Amy Goldlist
1.3 Ratios With More Than Two Quantities
In some business situations, it is desirable to examine the relative sizes of a number of
quantities. In such cases, ratios of the quantities are used.
Consider, for example, the expenses of a merchandising company. These expenses can be
usefully broken down into two major components:
the cost of the goods sold (COGS)
all other expenses, called operating expenses.
When analyzing merchandising companies, it is customary to first deduct the cost of goods
sold from sales revenue, and to call the result the gross profit.
Operating expenses are then deducted from the gross profit to get net profit.
SALES
– COGS
GROSS PROFIT
– OPERATING EXPENSES
NET PROFIT
All of these quantities are commonly compared with the sales revenue as the base.
Example 1.3.1
Suppose that in our opening example, in which the company had sales of $32,000 in a period,
the cost of these goods was $20,000 and operating expenses were $4,000. You would arrange
the analysis as follows:
$ % Sales
Sales $32,000 100.00
– COGS 20,000 62.5
Gross Profit 12,000 37.5
– Operating Expenses 4,000 12.5
Net Profit 8,000 25.0
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This has the format of a (very) simple Income Statement.
When gross profit is compared with sales it is called percent gross margin. Net profit
compared with sales is called percent net margin.
Key Takeaways
Gross profit compared with sales is called percent gross margin; Net profit compared with
sales is called percent net margin.
Suppose now that a similar company was expected to perform with the same ratios but to have
sales of $50,000 in a period. Then each of the other entries could be calculated by using the
above ratios. For example:
So
Notice that while the ratio was stated as a percent, it was necessary to replace it with the
decimal equivalent in order to do the calculations. You should check that the other quantities
come out, as in the following table:
% Sales
Sales $50,000 100.0
COGS 31,250 62.5
Gross Profit 18,750 37.5
Operating Expenses 6,250 12.5
Net Profit 12,500 25.0
Another use of ratios is to allocate resources according to some measure of need or
performance in an organization. For example, bonuses may be given as a percent of sales;
budgeted expenses may be a percent of the previous year’s expenses.
18 Amy Goldlist
Knowledge Check 1.3
A merchandising company finds that its COGS is 65% of sales and its monthly
operating expenses are $14,000.
What would be the COGS and the gross and net profits on sales of $90,000 in a month?
What would be its percent gross margin and percent net margin?
Your Own Notes
Are there any notes you want to take from this section? Is there anything you’d like
to copy and paste below?
These notes are for you only (they will not be stored anywhere)
Make sure to download them at the end to use as a reference
An interactive H5P element has been excluded from this version of the
text. You can view it online here:
https://pressbooks.bccampus.ca/businessmathematics/?p=58#h5p-1
Business Mathematics 19
20 Amy Goldlist
1.4 Retail Calculations
Many retail calculations parallel the foregoing ratio calculations for profits and expenses.
When goods are bought for resale, a decision has to be made about the selling price of the
individual articles. The goods are said to be marked up, the amount added to the cost of the
goods being the markup.
Thus
Selling Price = Cost + Markup
or
if S is used for selling price, C for cost and M for markup.
Key Takeaway
In setting markup, it is common to apply the same fraction of the cost for marking up all goods
in the same category.
This is similar to the calculation of gross profit for a company:
COMPANY ARTICLE
SALES SELLING PRICE
– COGS – COST
GROSS PROFIT GROSS PROFIT (MARKUP)
To simplify the process of finding markup for each product, it is common to apply the same
fraction of the cost as that used for marking up all goods in the same category – for example,
sports shoes.
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This ratio is usually written as a percent and called percent markup.
For a markup ratio of 25% and goods that cost $20.00 per unit.
Key Takeaway
Gross profit: Markup ratio times cost of goods.
So
And the selling price would be
The $5.00 is also called the gross profit on the sale.
Note that, in general, markup ratios are based on cost prices and that margin ratios are based
on sales or selling price. For the above example, if you wanted a margin ratio for the product,
you would compare the $5.00 to the selling price of $25.00 and get
Since the percent markup is based on cost, which is known at the time goods are purchased,
the normal way to calculate the increase is to use the percent markup method.
Careful!
Even though this book stays with the definitions of margin and markup as above, you should be
aware that some texts and industries may use different terms (for example, “markon” instead of
“markup”, or “markup on sales” for margin) and different meanings for the terms. But the ones
used here are frequently used, and you will find them on most business calculators.
22 Amy Goldlist
Knowledge Check 1.4
1. Complete the following table.
a.
COST
$10.00
%MARKUP
35%
MARKUP
?
PRICE
?
b. $20.00 ? $9.00 ?
c. $16.00 ? ? $20.00
Your Own Notes
Are there any notes you want to take from this section? Is there anything you’d like
to copy and paste below?
These notes are for you only (they will not be stored anywhere)
Make sure to download them at the end to use as a reference
An interactive H5P element has been excluded from this version of the
text. You can view it online here:
https://pressbooks.bccampus.ca/businessmathematics/?p=104#h5p-1
Business Mathematics 23
24 Amy Goldlist
1.5 Rates and Currency Conversions
Key Takeaway
Rate: The result when you take the ratio of two different units of measurement.
When you take the ratio of two quantities that have different units of measurement (for
example, money and time), you refer to the resulting ratio as a rate. As before, using the terms
amount and the base:
A sales rate would have the amount in dollars and the base, perhaps, in days. So if sales were
$18,000 in a 10-day period, you would have:
Similarly a pay rate might be in dollars per hour and an exchange rate in dollars per British
pound.
One use of such rates is to find a new amount for a different base (assuming the rate is
constant). You would have to do this if you wanted to find the cost of a given amount of foreign
money. In this case, you would check the exchange rates, which are determined by money
markets and reported regularly in the financial pages of newspapers.
Example 1.5.1
Suppose the exchange rate is $2.10 per British pound and you need £300. Then:
£ £
So
Business Mathematics 25
25
£ £
£
£
And
\text{Amount in \$} = 300 \times \$2.10=\$630.00
Notice that if, given the same rate, you wanted to find the number of pounds you could buy
with, say $500, you could reverse the ratio:
£ £ £
Key Takeaway
Keep track of the units. You can use them as a guide for calculations.
When dealing with rates, it is critical to keep track of the units; they can guide your calculations
if you use them for arithmetic operations.
If, for example, you want Canadian dollars (CAD) for US dollars (USD), and if you know that
the exchange rate is 0.68 USD per CAD, then your rates are
And
Check this by using a simple example – say, $1,000 (Canadian). Then, by the first rate, you
would have:
26 Amy Goldlist
By the second rate, you would find:
So the rates are equivalent.
The following exchange rates were taken from a local newspaper.
Currency Rate
Euros € 1.532
Hong Kong (HKD) 0.1678
Japan (¥) 0.011648
Each rate gives the number of Canadian dollars required to purchase one unit of the other
currency.
Example 1.5.2
To purchase Euros you would use the rate 1.532 CAD per Euro. So, if you wanted 300€ at this
rate, you would need:
€ € €
To get the number of marks for, say $100.00 Canadian, you would calculate:
€ €
Knowledge Check 3.5
1. Find the Canadian dollars you require to purchase 200 units of each
currency listed above.
Business Mathematics 27
2. Find the amount of each currency you can buy for $500 Canadian
Your Own Notes
Are there any notes you want to take from this section? Is there anything you’d like
to copy and paste below?
These notes are for you only (they will not be stored anywhere)
Make sure to download them at the end to use as a reference
An interactive H5P element has been excluded from this version of the
text. You can view it online here:
https://pressbooks.bccampus.ca/businessmathematics/?p=111#h5p-1
28 Amy Goldlist
1.6 Trade Discounts
Most of the goods you purchase as a consumer are not bought directly from the manufacturer.
The goods are distributed through a marketing channel – that is, a sequence of middlemen
which move the goods to the consumer. In this section, you will examine the price calculations
for a typical channel. Consider an industry (such as clothing) which distributes goods through
wholesalers and retailers. Note that the manufacturer may sell to different levels of the channel,
perhaps to retailers near the production facilities and to wholesalers everywhere else.
Manufacturers (and wholesalers) often produce a single price list of their products for their
customers. This usually lists the prices (the list price) at the level a final consumer would pay.
Clearly wholesalers and retailers cannot pay this price because they have to mark up the price
they pay before they sell the product.
The price to be paid by these organizations is found by reducing the list price by a trade
discount. The trade discount is given by a percent of list. Suppose, for example, that a
manufacturer of men’s suits lists a suit at $300 (this is roughly what a consumer might pay).
Then a retailer buying directly from the manufacturer might get a 25% trade discount and pay:
CHAINED DISCOUNTS
If retailers are expected to pay $225 for a suit, a wholesaler has to pay less for it, because the
wholesaler’s customers are retailers. Hence the wholesaler must get an additional discount to
stay in business. This is usually given as a chained discount; which is a percentage of the price
after the first (retailer’s) discount is taken.
In the example above, this extra discount is 20%. So the wholesaler would pay the
manufacturer:
As a single equation you would have 300(1− 0.25)(1− 0.20). This discount would be described
to the wholesaler as list less 20%, 25%.
To get the final price (the net price), after all discounts, you can set up the following formula:
where:
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N = net price to be paid
L = list price
d1= discount rate 1;
d2= discount rate 2
Note that (1-d1) and (1-d2) are the fractions of the price to be paid after the discounts are taken.
Thus, in the example above:
For d1, there remains 1 – 0.25 = 0.75 = 75% to be paid.
And for d2 = 20% = 0.20, there is 1 – 0.20 = 0.80 = 80% of the remainder to be
paid.
Sometimes there are additional discounts – special sale discounts, seasonal discounts, quantity
discounts – that are treated the same way.
Check Your Knowledge 1.6
1. A wholesaler is purchasing refrigerators from a manufacturer and is offered
discounts of “list less 30%, 25%” on refrigerators with a list price of
$700.00. What price would the wholesaler pay?
2. A retailer purchased a television set from a wholesaler and paid $244.30
after getting a single discount on a set listed at $349. What single discount
did the retailer get?
Your Own Notes
Are there any notes you want to take from this section? Is there anything you’d like
to copy and paste below?
These notes are for you only (they will not be stored anywhere)
Make sure to download them at the end to use as a reference
An interactive H5P element has been excluded from this version of the
30 Amy Goldlist
text. You can view it online here:
https://pressbooks.bccampus.ca/businessmathematics/?p=127#h5p-1
Business Mathematics 31
32 Amy Goldlist
1.7 Cash Discounts
In the previous section you learned to calculate the prices businesses paid for goods. This
section examines the terms of payment allowed by sellers.
After goods are sold, the vendor sends a request for payment to the purchaser. This request
is usually on an invoice, a document which lists all the pertinent facts about the transaction,
including the following:
the items sold, with list prices
the trade discounts
conditions of delivery
other charges (e.g., freight)
the PAYMENT TERMS.
PAYMENT TERMS
Usually payment is not required immediately and a payment period (for example, 60 days) is
given. This payment period normally begins on the date of the invoice (that is, the next day is
day 1) and payment is expected by the last day of the period. The payment period allows for
goods to be partly or wholly resold by the purchasing company before payment is due, and is
thus an inducement to buy from the seller.
Key Takeaway
A cash discount encourages early payment.
Sellers have to finance the sale during the payment period and consequently some sellers
encourage payment before the end of the period by allowing a discount, called a cash discount,
for early payment. The discount also helps to ensure preferential payment to the seller by
companies that may be experiencing difficulties meeting all payment commitments.
The following are examples of terms:
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2/10, n/30 2% discount allowed if paid within 10 days, otherwise net (full) payment due within 30 days
(of the invoice date).
1/15, n/60 ROG 1% discount if paid within 15 days of the RECEIPT OF GOODS, otherwise net payment due
60 days from receipt of goods.
2/5, n/30, EOM 2% discount if paid within 5 days of the END OF THE INVOICE MONTH, otherwise full
payment due 30 days from the end of the current month.
Key Takeaway
The payment period normally begins on the date of the invoice.
Example 1.7.1
Consider the following invoice:
34 Amy Goldlist
Ajax Wholesale Ltd.
Invoice No: 16274
Date: August 15, 2020
Ship via: Hermes Delivery
Special Instructions:
Sold to: Electronicks
Ltd. None
110 B Avenue Surrey, BC
Ship to: Same
Terms: 2/5, n/30 FOB
PST Exemption #XXXXXXX
Quantity Description List Price per
Unit
Amount
15 #1472 AM-FM Radio; less
30% $17.50 $183.75
10 #1821 Telephone; less 30% 12.25 85.75
10 #2171 Answering machine;
less 30%, 10% 36.00 226.80
Subtotal 496.30
Business Mathematics 35
Delivery charge 35.00
TOTAL PAYABLE $531.30
Note that the amounts calculated have the trade discounts deducted, but do not have the cash
discounts deducted since it is not known whether or not the purchaser will pay early.
The delivery charge is listed separately and cash discounts are not usually allowed on it. This
is because the delivery in this case is not the seller’s business; it is being paid on behalf of the
purchaser.
If the invoice was paid in full by August 20, 2020, the payment would be:
Otherwise the full amount would be due on September 14, 2020.
Knowledge Check 1.7
An invoice dated September 3, 2020, for six television sets with a list price of $749.00
has trade discounts of 30% and 5%. If the payment terms are 3/10, n/30, what is the
payment required if the cash discount is taken?
PARTIAL PAYMENTS
Sometimes it is not possible to pay an entire invoice at once. In this case, we can apply the
discount to only part of an amount owing. This is called a partial payment.
36 Amy Goldlist
Example 1.7.2
S&C sports receives an invoice for $4,500 for a shipment of curling shoes. The invoice is
dated April 1 and has terms 3/10, 2/15, n/30.
a. On April 6, S&C sent in a payment of $1000. The Curling Equipment company
gives discounts for partial payments. How much does S&C owe on the invoice?
b. S&C plans to send in a second and final payment on April 11. How much should
this payment be?
For part (a), we need to see how much credit should be given for the $1,000 payment, given
that it is eligible for a 3% discount. The discount applies to the credit given (the part of the
invoice that is paid off) and not to the amount paid.
To calculate how much to pay multiply by (1 – d)
If given the amount paid divide by (1 – d) to get the credit
To see this, we can solve the equation:
Dividing both sides by (1-0.03), we have:
So, by paying the $1,000 early, S&C now owes only:
To solve (b), we multiply the amount owing by (1-0.02) to see that the final payment is:
Key Takeaways
Not all suppliers allow retailers to receive a cash discount when only part of the
invoice is paid early (partial payments).
In this text, we always assume retailers will receive a discount on the portion of
Business Mathematics 37
the invoice that was paid early.
You will notice that we did some intermediate rounding in the problem above.
Often companies will do this type of intermediate rounding when storing
amounts, but it is always important to check with the supplier, to see how they
record the numbers.
Knowledge Check 1.8
In the middle of a kitchen remodel, you decide to splurge on a new refrigerator and
range. The range was on back order, but you received the following invoice for the
fridge:
Invoice 1 (fridge): $2,500 dated April 4 (5/20, n/30)
On April 6, you sent in a payment of $760. Two weeks later, you receive the invoice
for the range. It has the same terms as the first invoice:
Invoice 2 (range): $3,250 dated April 19 (5/20, n/30)
a. If you pay off both debts at the same time, what is the last day you can pay?
b. How much will you pay?
Your Own Notes
Are there any notes you want to take from this section? Is there anything you’d like
to copy and paste below?
These notes are for you only (they will not be stored anywhere)
Make sure to download them at the end to use as a reference
An interactive H5P element has been excluded from this version of the
38 Amy Goldlist
text. You can view it online here:
https://pressbooks.bccampus.ca/businessmathematics/?p=135#h5p-1
Business Mathematics 39
40 Amy Goldlist
Chap 1 Knowledge Check Answer Key
Knowledge Check 1.1
1. a. $9,000 b. $66,000
2. Sounders: , Elecktronicks: (Elecktronicks is better)
Knowledge Check 1.2
1. a. Net Profit = 14% of .
b. , so
2 a. $9.80 b. $22.40 c. $39.20
Knowledge Check 1.3
1.a. COGS = 0.65 × $ 90,000 = $58,500
Sales $90,000
COGS 58,500
Gross Profit 31,500
Operating Expenses 14,000
Net Profit 17,500
2.
Business Mathematics 41
41
Knowledge Check 1.4
COST % MARKUP MARKUP PRICE
a. $10.00 35% $3.50 $13.50
b. $20.00 45% $9.00 $29.00
C. $16.00 25% $4.00 $20.00
Knowledge Check 1.5
1. a.
€ € €
b.
€ €
Knowledge Check 1.6
1.
2. Discount =
42 Amy Goldlist
Knowledge Check 1.7
Before Discount:
After Discount:
Knowledge Check 1.8
a. The last day you can pay off both debts is the day the fridge invoice is due: May 4.
On this day, the fridge will not get a discount, but the range will be eligible for a 5%
discount
b. The first payment of $760 is worth a credit of:
so the we owe on the fridge, and
on the range, so our final payment will be:
Business Mathematics 43
44 Amy Goldlist
Chapter 1 Review Questions
Unless otherwise stated, round the final answer to 2 decimal places.
1 A jewelry store’s sales and expenses are given below for the months of October, November
and December:
October November December
Sales $40,000 $59,000 $109,000
Expenses (all) $38,000 $50,000 $84,000
a. Find the profits for each
b. Find the percent net margin for each
2 FST Computer Stores has decided to give a total bonus of $10,000 to its outlets. The bonus
will be proportional to the monthly sales (given below) for each outlet. Find the bonus for each
outlet.
Outlet Burnaby Downtown Richmond Surrey
Sales $162,000 $119,000 $172,000 $215,000
3 A-Plus Appliances two branches reported the following results for a one week period:
Sales
East
$20,000
West
$25,000
COGS 12,000 16,000
Other Expenses 3,000 4,000
a. Find the gross and net profits for each branch.
b. Find the percent gross margin and percent net margin for each
Business Mathematics 45
45
4 Joan’s Co. knows that, nationwide, companies in its industry earn, on average, a percent gross
margin of 38% and a percent net margin of 11%. Joan’s Co. forecasts sales of $730,000 next
year. If it aims at the same profit ratio as the nationwide average, estimate Joan’s cost of goods,
gross profit and net profit.
5 Jim’s Co. sees that its competitor sells an article for $42.00. It knows that the competitor pays
$30.00 for the article.
a. What is the competitor’s markup (dollar amount)?
b. What is the competitor’s percent markup?
c. What is the competitor’s percent gross margin?
6 At one time the exchange rate for the Canadian dollar was: 1 CAD = 0.76 USD
For the Danish krone the rate was: 1 krone= 0.155 (USD)
a. Find the value of $1 USD in terms of Canadian dollars (4 decimal places).
b. Find the value of one Danish krone in terms of Canadian dollars (6 decimal places).
c. How much would it cost (in Canadian dollars) to purchase 10,000 krones (2 decimal
places)?
d. How many krones can be purchased for $500 Canadian (2 decimal places)?
7 JB Appliances received an invoice which contained the following information:
Date, June 17, 2018 Items:
10 Refrigerators; list $710 each, less 25%, 20%
5 Dryers; list $420 each, less 25%, 20%, 5% (special sale)
Freight charge $208
Payment terms 3/10 net 60
Find:
a. The last day for
b. The amount due if the invoice was paid on the last day.
c. The last day at which the cash discount
d. The amount of discount allowed on the day in (c).
46 Amy Goldlist
a. The amount to pay off the invoice on the day in (c)
8 Mac’s Wholesale buys from a distributor which allows discounts of 25%, 20% and 5%.
a. What single discount (called the single equivalent discount) would be equivalent to
the above discounts?
b. Compare the above discounts to discounts of 30%, 20%.
9 A branch of Jack’s Hardware has been told by its head office to sell goods at a percent gross
margin of 40%. What percent markup should the branch use?
10 CHMCO sells a product with discounts of 25%, 15% to its distributors. It plans to allow
a seasonal discount to make the overall discount equal to 45%. What additional (chained)
discount should CHMCO allow?
11 Wilson Co. has received an invoice dated October 17 for 5 items with list price $900 each,
and for freight of $170. The terms were:
Trade terms: list less 20%,15%.
Payment terms: 3/10, n/30.
Find:
a. The last day for payment
b. The amount due if payment is made on the last day
c. The last day for taking a cash discount
d. The amount needed to pay the invoice on the day in (c).
e. If the seller agrees to give a cash discount on a partial payment, how much credit
would be applied to the account if $1,500 were paid immediately?
12 Three companies, Alpha, Beta and Gamma, are the only suppliers to a specialized market.
Last year Alpha’s sales were $700,000, Beta’s $1,200,000 and Gamma’s $1,600,000.
a. What was the market share (percent of market) of each company? Note: retain all
decimal places calculated in (a) to solve (b) and (c).
Business Mathematics 47
b. If next year’s total market is expected to be $5,000,000 and each company’s share
remains the same, what would be the sales of each company?
c. Alpha has an aim of making its sales 75% of Beta’s. With total sales as in (b) and
Gamma’s as in (b) what would have to be the sales of Alpha and Beta if Alpha were
to make its aim?
13 Janet’s Co. has sales of $90,000, COGS of 80% of sales and operating expenses of $5,000.
a. Find the gross and net profits.
b. Find the rate of markup (based on cost).
c. Find the percent net margin.
14 Jim’s Co. has set a requirement on stock items of a turnover ratio of 2.6 per year. It is
examining three stocked items, A, B and C, which have to be bought in large amounts. As a
result of the purchasing requirements, the maximum stock for A is $1,000, for B $1,200 and
for C $2,500. If the average stock is assumed to be one-half the maximum stock, what would
be the required annual sales of each of these items?
15 Majjor Oil Co. allocates its maintenance budget to three of its branch offices on the basis
(i.e., proportional to) of total floor space. Branch I has 9,000 sq. ft., Branch Il has 20,000 sq. ft.
and Branch III has 15,000 sq. ft. If $880,000 is to be allocated, how much should each branch
receive?
16 A paint is advertised for sale with a price of $11.00 USD per gallon. Find the price in CAD/
liter if 1 CAD = 0.73 USD
Given:
1 liter = 1.0567 quarts
1 gallon = 4 quarts
17 The wellhead price of a certain grade of oil is $69.00 USD per barrel. What is the price in
CAD per liter?
1 CAD = 0.73 USD
48 Amy Goldlist
1 Barrel = 31.5 US Gallons
1 liter = 1.0567 quarts
1 US Gallon = 4 quarts
18 An invoice arrives from the Home Shopping Network. The invoice is dated May 28th, and
the invoice amount is $600. Calculate the net payment of the invoice if the te1ms are 2/10, n/
30 and the invoice is paid on June 6.
19 An invoice dated Aug. 12th for $745 containing the terms “3/15, n/60” was paid on Aug.
25th How much should the cheque be to pay this invoice?
20 An invoice dated May 4th for $1,200 has terms 3/10, 2/30, n/60. How much should be paid
on May 31st to fully pay off the invoice?
21 An invoice dated May 1st for $4,000 has terms 3/10, 2/20, n/45. It was partially paid by
a $1,000 payment on May 10th and a second payment of $1,500 on May 20th. What is the
outstanding balance after the May 20th payment?
22 A dealer purchased 10 DVD players at $300 each less discounts of 25% and 5%, and 5 TV’s
at $450 each less 30%, 10% and 2%. The invoice for these items was dated June 5th and had
terms 2/10, n/30.
a. If the dealer wishes to take advantage of the cash discount, what would be the last
day for payment?
b. What amount must be paid on the day found in part a) to fully pay it off?
c. If the dealer only sent in $2,500 on June 15th what would be the outstanding
balance?
23 An invoice for $12,000 has payment terms 3/10, 2/20, n/45. Discounts are allowed for partial
payments. The company made of payment of $4,000 9 days after the date of invoice, and a
second payment 18 days after the date of the invoice that reduced the balance owing to $2,000.
What is size of the second payment?
Business Mathematics 49
24 An invoice for $2,500 has terms 2/10, n/30. The invoice is dated April 27th and half of
the invoice amount is paid on May 1st and the rest is paid on May 20th. If the seller grants
discounts for partial payments, by how much is the balance reduced by the 1st payment? What
is the total amount paid?
25 A seller grants discounts for partial payments. An invoice for $1,200, dated May 1st, has
terms 2/10, 1/15, n/60 and $400 is paid on May 5th, $400 on May 14th and the rest is paid on
June 5th
How much is the balance reduced by the two early payments? How much is paid on
June 5th?
26 Chan Enterprises sent CK Air an invoice dated Sept. 14th for $5,400 with terms 3/10, 1.5/
20, n/45. CK Air made a payment of $2,000 on Sept 24th and a second payment on Oct 3 that
reduced the balance owing to $1,000. What was the size of the second payment?
27 You receive an invoice from AG Electronics dated Feb. 10th for:
20 TV sets at $500 per set less 20% and 15%.
30 DVD players at $200 each less 25% and 10%.
AG Electronics has payment terms 2/10, 1/20, n/45. Cash discounts are allowed for partial
payments. They make of payment of $4,900 on Feb 18th and a second payment on Feb 26th
that reduced the balance owing to $1,500. What is the size of the second payment?
28. The Harrison Lake Corporation likes to maintain a capital structure of 5:4:2 (dollar value
of debt to common shares to preferred shares). The company is considering a new project and
will need to raise $3,300,000. If the company wishes to maintain its existing capital structure,
how much debt, common stock and preferred stock should they issue?
29 The Ministry of Education allocates its budget in proportion to student enrollments. The
Ministry has $1,500,000 to allocate and the student enrollments are as follows:
High Knowledge Institute, The 12,000
Institute of Technology 15,000
School of Hard Knox 13,000
a. Allocate the budget in proportion to
b. The following year total enrollment is expected to reach 65,000. High Knowledge
50 Amy Goldlist
Institute is expected to have 3,000 more students and the ratio of students at The
Institute of Technology to Students at Hard Knocks will remain the same at 15:13.
Give the new estimate for enrollment at each
30 An engineering firm is owned by three partners (Alan, Barbara, and Chuck) with 4,000,
3,000 and 1,000 shares respectively. The company’s net income for the year is $4,500,000 and
10% of that income has been set aside as a performance bonus. The bonus will be allocated in
the following way:
$70,000 to be distributed amongst all staff excluding the partners.
Each partner will get $80,000.
The remainder is to be divided among the partners in proportion to the number of
shares held.
What was the total amount of bonus received by each partner?
31 The $300,000 yearly rent of a four-story building is to be expensed to the four departments
using it according to the number of floors occupied. Sales occupies two floors, Finance one
floor, Administration and Research and Development (R & D) share one floor equally. How
much of the rent expense is allocated to each department?
32 Last year, Reliable Securities established a sales achievement bonus fund of $10,000 to be
distributed at the year’s end among its four-person mutual fund sales force. The distribution
is to be made in the same proportion as the amounts by which each person’s sales exceed the
basic quota of $500,000. How much bonus will each salesperson receive from the fund if the
sales figures for the year were $910,000 for Alicia; $760,000 for Bob; $460,000 for Charles;
$630,000 for Diana?
33 For the past seven years the sales of Departments A, B, and C have maintained a relatively
stable ratio of 4:3:2. Department A is predicting that their sales will be $500,000 next year.
Based on their past sales ratio, what sales would be predicted for Department B?
34 If the chain discount on an item listed at $2,200 is 40%, 15%, 5%, what is the net price?
What single rate of discount is equivalent to this series of discounts?
Business Mathematics 51
35 The Ray department store is selling all summer clothes at 50% off the regular retail price
during the month of September. On Saturdays only, they are offering an additional 20% off the
already discounted price. What is the single equivalent discount rate on Saturdays?
36 A publisher sells its romance novels with chained discounts of 20% and 10%. The publisher
would like to add a third discount to bring the overall discounts to 31.6%. Find the third
discount rate.
37 If the net price of a typewriter was $425.59 after chain discounts of 25%, 10% and 3%, what
was the list price? What is the equivalent single discount rate?
38 A patio set is listed at $350. What is the size of the discount (in dollars) if the buyer is
eligible for discounts of 30%, 15% and 5%? Convert the chain discount into a single equivalent
discount rate.
39 Which offers the larger discount? Manufacturer A offering chain discounts of 30%, 20%,
5%, 2% or Manufacturer B giving 45%, 13%?
40 Island Remanufactured Wood Products Inc. sells its products at trade discounts of 30%,
10%. A competitor has been offering products at the same list prices but with trade discounts
of 25%, 20%. Island Reman wishes to beat the competitor’s prices by offering a third trade
discount. At least how big must the additional discount rate be to meet this objective?
41 Toys-R-Costly sells Barbie Computers for $840 less 20% and 15%. A competitor sells
a similar computer for $800 less 25%. What further discount (second discount) must the
competitor allow so that its net price is the same as Toys-R-Costly’s?
42 In order to stimulate sales of file cabinets, a manufacturer had to offer a temporary additional
discount to its present rate of 25%. The list price is $300. What additional temporary discount
would have to be given in order to achieve a net price of $150?
52 Amy Goldlist
43 Wadi Air Conditioners Ltd. sells one model at a list price of $500. Retailers are offered
discounts of 15% and 10%.
a. Find the net price of an air conditioner.
b. Find the single equivalent discount rate
c. Wadi would like to add a third chained discount to bring the total discount to 27%.
Calculate the rate of the third discount.
44 A company manufactures Bluetooth Speakers. The Speakers retails for $400 and are offered
to the wholesaler with chained discounts of 10% and 15%.
a. Find the net price after discount.
b. What is the single equivalent discount rate?
c. The manufacturer would like to add a third discount to bring the total discount to
30%. Calculate the rate of the third discount.
d. A competitor offers a single discount of 25%. This amounts to a discount of $87.50
off the list price. What is the list price?
45 After graduation you decide to start up a business selling spices imported from Colombia.
The main product you import, saffron, sells for 95,149.87 Pesos for a 4-ounce jar. In Canada,
you sell the product in 125 gram bags. Find the selling price in Canadian dollars per 125 gram
bag. Use the following:
1 pound = 454 grams
1 USD = 1.5644 CAD
125 grams = 1 bag
16 ounces = 1 pound
0.000427 USD = 1 COP (Columbian Peso)
46 You are taking a holiday in Britain. You are taking $750 USD with you. How many British
pounds can you buy with this money?
Rates:
$1 USD= $1.5644 CAD
1 British pound (£) = $2.1356 CAD
Business Mathematics 53
47 You are thinking of expanding your bottled water company to the US. Your company
bottles pure Fraser River water collected near the Oak Street Bridge. Your water currently sells
for $0.85 per 600 ml bottle. The water shipped to the US would be sold in quart bottles. What
price, in US dollars, should you charge for a quart of your water? (Convert the Canadian price
per 600 ml to a US price per quart).
Rates:
$1 USD= $1.5644 CAD
1 Gallon = 3.7853 Litres
4 Quarts = 1 Gallon
1,000 ml = 1 Litre
48 You missed the Back Yard Boys concert at GM Place, and have decided to drive to Seattle
to see them. After driving through customs, you stop at the Thrifty market in Blaine. You see
a quart size bottle of Duff Cola selling for $1.89 USD. You would like to compare this price to
the price of a Canadian can of Cola. Convert the price to Canadian dollars per can.
Rates:
$1 USD = $1.5644 CAD
1.0567 Quarts = 1 Litre
1 Litre = 1,000 ml
1 can = 355 ml
49 In California gasoline sells for $1.99 USD per gallon. In British Columbia gasoline is
currently selling for $0.729 CAD per liter. Convert the Canadian price per liter to US dollars
per gallon. Is the price of gasoline cheaper in California or in BC?
Rates:
$1 USD= $1.5644 CAD
1 Litre = 1.0567 Quarts
4 Quarts = 1 Gallon
50 You purchase gasoline for your gas stations from local refineries. A number of labour
54 Amy Goldlist
disruptions and increasing costs have motivated you to consider importing gasoline from the
Emirate of Abu Dhabi. Refined gasoline sells for 91.30 Dirhams per barrel. Calculate the
equivalent Canadian price in liters given:
Rates:
3 Dirhams = 1 USD
0.638 USD = 1 CAD
1 Barrel = 42 Gallons
1 Gallon = 4 Quarts
1 Quart = 0.9464 Liters
51 Fill in the Following Table:
Cost Sales Price Markup in dollars % Markup % Margin
$125 $175
$75 $95
52 A computer dealer buys NADIR brand VR headsets for $3,500 less 10%, 5% and sells them
for $3,299.
a. What is the Cost?
b. What is the rate of markup?
c. What is the percent margin?
53 An outboard motor costs the retailer $500 less chain discounts of 30%, 25% and 5%: The
company maintained a margin of 40% on all items. The motor was sold after it had been
marked down 25%.
a. What was the regular selling price?
b. What was the actual selling price (sale price)?
c. What rate of markup did they use? (based on part b)
54 A souvenir stand bought 300 hockey sweaters for $15 each. They sold 170 at the regular
selling price of $30 each but had to sell another 70 sweaters at a 35% discount (markdown)
near the end of the hockey season and the remaining sweaters were cleared by selling them
Business Mathematics 55
at a breakeven price which exactly covered the cost of the sweaters plus overhead. Overhead
(operating expenses) are roughly 25% of cost.
a. What was the selling price per sweater for the 70 sweaters sold near the end of the
season?
b. What was the selling price per sweater for those sweaters sold at the breakeven
price?
c. What was the gross profit (loss) and net profit earned from the sale of the 300
sweaters?
55 Wilma Inc. prices its products to provide a 25% margin. What rate of markup do they use?
56 You know that a TV retailer makes $50 on the sale of a certain model of television set and
the retailer has a markup policy of 20% of cost.
a. How much did this TV set cost the retailer?
b. What is the selling price of the TV set?
c. Find the percent margin.
57 A diamond ring sells for $4,500. If the rate of markup is 110%, what did the ring cost the
retailer? What is the percent margin?
58 An item that cost the dealer $350 less 35% and 12.5% carries a price tag at a markup of
150% of cost. For quick sale, the item was reduced 30%. What was the sale price?
59 The selling price of an automobile is $9,500. If the markup is 15% of cost and the operating
expenses are 2% of the selling price, how much is the gross profit (markup in $)? What is the
net profit?
60 ABC Co. prices its products to provide a 43.5% margin. What rate of markup do they use?
61 A bookstore has a policy of maintaining a margin of 20%.
56 Amy Goldlist
a. If an item costs $90, how much should it sell for?
b. If another book sells for $120, how much did it cost?
62 The regular selling price of merchandise sold in a store includes a margin of 40%. During a
sale, an item that costs the store $180 was marked down 20%. For how much was the item sold
for?
63 An item sells for $500. If the company uses a 100% rate of markup, what is the cost of the
item?
64 Scoopy-Doo Pet Foods prices its pet food to maintain a 45% margin. A 7-kg bag of Woof-
Woof dog food costs $27.50. Calculate the selling price.
65 CK Winery sells its merlot red wine for $21.00. If the rate of markup is 200%, what did the
wine cost the retailer? What is the percent margin?
66 If you knew that an appliance dealer makes $100 on the sale of a certain model of washing
machine and the dealer has a markup policy of 20% of cost, how much did this washer cost the
dealer? What is the selling price of the washer? Find the percent margin.
67 Find the cost of an item sold for $1,904 to realize a rate of markup of 40%.
68 A retailer bought an article for $78 and his rate of markup is 32%. He sold this article in a
sale after having marked it down by 30%. What is the sale price?
Solutions
Business Mathematics 57
Notes
1. October November December
a. $2,000 $9,000 $25,000
b. 5.00% 15.25% 22.94%
2. Burnaby Downtown Richmond Surrey
$2,425.15 $1,781.44 $2,574.85 $3,218.56
3. East West
a. Gross Profit $8,000 $9,000
Net Profit $5,000 $5,000
b. Gross Margin 40% 36%
Net Margin 25% 20%
4. Sales $730,000
COGS 452,000
Gross Profit 277,400
Operating Expenses 197,100
Net Profit 80,300
5. a. $12,000,
b. 40%
c. 28.57%
6. a. 3158 CAD = 1.00 USD
b. 0.203947 CAD = 1 Krone
c. 10,000 Krones =2,039.47 CAD
d. 500 CAD = 2,451.61 Krones
7. a. August 16, 2018
b. $5,665.00
c. June 27, 2018
d. $163.71
58 Amy Goldlist
e. $5501.29
8. a. 43%
b. 44%
9. 66.67%
10. 13.7255%
11. a. November 16
b. $3,230.00
c. October 27
d. $3,138.20
e. $1,546.39
12. Alpha α Beta β Gamma γ
a. 20.00% 34.2857142% 45.7142857%
b. $1,000,000 $1,714,285.71 $2,285,714.29
c. $1,163,265.31 $1,551,020.41 $2,285,714.29
13. a. Gross Profit= $18,000, Net Profit= $13,000
b. 25%
c. 14,44%
14. A: $1,300 B: $1,560 C: $3,250
15. Branch I = $180,000; Branch II = $400,000; Branch ID= $300,000
16. $3.98 CAD/liter
17. $0.79 CAD/liter
18. $588
19. $722.65
20. $1,176.00
21. $1,438.46
22. a. June 15th
b. N = $3,456.12
c. $3,526.65 - $2,551.02 = $975.63
23. $5,758.76
24. $1,275.51 is credited and a total of $2,474.49 is paid
25. $408.16 + $404.04 = $812.20 is the credit and $387.80 is still owing.
26. $2,303.07
27. $4,306.50
Business Mathematics 59
28. Debt: $1,500,000 C/S: $1,200,000 P/S: $600,000
29. a. $450,000; L: $562,500; S: $487,500
b. N: 15,000 students L: 26,786 students S:23,214 students
30. Alan: $150,000; Barbara: $132,500 Chuck: $97,500
31. Sales: $150,000; Fin: $75,000; Admin: $37,500 (same for R & D)
32. Alicia: $5,125; Bob: $3,250; Charles: 0; Diana: $1,625
33. $375,000
34. N = $1,065.90 D = 51.6%
35. 60%
36. 5%
37. L = $650.00; D = 34.525%
38. $152.16, 43.5%
39. Mfg A: Total discount= 47.864%; Mfg B: Total discount= 52.15% (B is larger)
40. more than 4.76%
41. 4.8%
42. 33.33%
43. a. N = $382.50
b. 23.5%
c. 4.575%
44. a. $306
b. 23.5%
c. $d_3 = 8.50%$
d. $350
45. $70/bag
46. £549.40
47. 0.856959 USD/quart, so $0.86 USD/quart
48. 1.11 CAD/can
49. 1.76 USD/gallon, so cheaper in Canada
50. 0.30 CAD/Liter
51. Cost Sales Price Markup in dollars % Markup % Margin
$125 $175 $50 40% 28.57%
$75 $95 $20 26.67% 21.05%
52. a. Cost = $2,992.50
b. 10.24%
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c. 9.29%
53. a. $415.63
b. $311.72
c. 25%
54. a. $19.50
b. $18.75
c. $3,090 gross profit, $1,965 net profit
55. 33.33%
56. a. C = $250
b. S = $300
c. % margin = 16.67%
57. C = $2,142.86, percent margin= 52.4%
58. $348.36
59. Markup = $1,239.13 NP= $1,049.13
60. 76.99%
61. a. $112.50
b. $96.00
62. $240.00
63. $250
64. $50
65. C = $7.00, % margin= 14/21 = 66.67%
66. C = $500, S = $600, % margin = 16.67%
67. $1,360.00
68. $72.07
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Chapter 2: Functions and Applications
Key Takeaways
The purpose of a function is to enable you to find the value of a quantity (variable) you want
from a quantity (variable) you know or can choose.
Most quantities in business (for example, profits, costs, sales) can take on a wide range of
values (profits can even be negative!). These quantities are referred to as variables, and there
are relationships between many of these variables that help you to plan and improve the
operation of a business. The relationships examined in this chapter are called functions.
The purpose of a function is to enable you to find the value of a quantity (variable) you want
from a quantity (variable) you know or can choose. The value of the quantity you want thus
depends on the value of the quantity you know. The quantity you start with (the known one) is
called the independent variable; the quantity you want is the dependent variable.
An independent variable can take on any value consistent with the situation being investigated.
A dependent variable must take on the value that is a consequence of the known value of the
independent variable.
These variables are often denoted by x for the independent variable and by y for the dependent
variable. The letters are just a convenient shorthand, which makes it easy to do algebraic
manipulation. In applications you will generally find it preferable to use letters that remind you
of what is being represented (for example, cost).
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2.1 Example of a Function
Example 2.1.1
A shirt manufacturer may decide to produce a certain number of shirts of a particular size and
style. The first task is to find out the amount of material needed.
Then,
A function must be described in such a way that you can find the value of the dependent
variable from the value of the independent variable. Three standard ways of doing this are used
in this course.
They are,
an equation, which allows you to calculate the value of the dependent variable for
each value of the independent variable.
a table, which lists the value of the dependent variable for each value of the
independent variable.
a graph, which displays the value of the dependent variable for each value of the
independent variable.
Example 2.1.2
Suppose a series of items are to be marked up by 40% of cost. The selling price (the value wanted) can
be described in terms of the cost (the value known) in the following ways:
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All of these approaches give the selling price, (dependent variable), as a function of the cost,
(independent variable). A function like this, where one variable is a multiple of another, is
called direct variation.
Note that such functions follow the algebra convention which states that when single letters are
used to stand for variables, multiplication is to be used when no other operation is given, so
that
Key Takeaways
Equations provide short descriptions of functions and enable you to find precise values of the
dependent variable.
Also, you will follow the graphing convention which requires that the independent variable be
drawn on the horizontal axis and equal the dependent variable on the vertical axis.
Each way of describing functions has its own advantages. Equations have the advantage
of providing short descriptions of functions and enabling you to find precise values of the
dependent variable when you need them. However, it may be difficult to do the calculation, or
to understand the overall behavior of the function.
Key Takeaways
Tables give values immediately, without calculation … Graphs communicate an overall
66 Amy Goldlist
understanding of the way a function behaves.
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68 Amy Goldlist
2.2 Linear Functions
Tables give the values immediately, without calculation, but they may have to be very large to
be useful. (Clearly the small table used in Example 2.1.1 is not large enough to be of real use in
a business.) Tables are still commonly used in some areas of business – for example, to find the
payments required for loans such as mortgages. They are useful because the calculations are
fairly complicated and a precise answer is required. Later chapters deal with such payments.
Normally we use spreadsheet programs such as Microsoft Excel for this type of problem.
Graphs communicate an overall understanding of the way a function behaves. Complex
functions are often referred to by graphs. The function values and equations may not be known
in detail, and the general behavior of the function may be all that is required. In economics, for
example, the major issue may be the direction (up or down) that changes in one variable (such
as price) have on another variable (such as revenue).
The rest of this chapter examines a type of function often used in industry and finance: linear
functions.
A function is linear if the graph of the function forms a straight line. You will get a straight-line
graph if the only operations on the independent variable are as follows:
1. The independent variable is multiplied by a constant. For example, ; Or
2. The independent variable is added to or subtracted from a constant. For example,
; Or
3. A combination of 1 and 2 is used. For example,
All of the above would qualify as linear functions. Note that direct variation is a special case
of linear function.
The following equations would result in non-linear functions:
Example 2.2.1
Take the cost of producing advertising posters as an example of applying a linear function.
Suppose that the management of the print shop finds that producing the posters for local
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companies costs $80.00 for artwork and set-up, and an additional $0.11 for materials and labor
for each poster printed.
The cost of production could be described by the (linear) equation
where
C is the cost of production and
x is the number of posters produced
Number Of
Posters Cost
100 $ 91.00 This table would be satisfactory
200 102.00 for costing orders to the printer
300 113.00 as long as orders were limited
400 124.00 to these sizes.
500 135.00
600 146.00
700 157.00
800 168.00
900 179.00
1,000 190 .00
To make a graph showing this function you should take as a horizontal axis the independent
variable x (number of posters), and as a vertical axis the dependent variable C (cost). Both axes
must be scaled so as to allow for the largest value of each variable (2,000 for x , $300 for C).
Plotting the points from the table above, you get, by joining them, the graph illustrated below.
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Figure 2.2.1 Graph of Cost (C) of Posters (x)
The values corresponding to a few points have been marked on the graph and dotted lines given
to show how these points relate to the axes. Note that the graph is a straight line.
All linear functions are quite simple: only two points are necessary to establish the appearance
of the function. An extra point serves as a check to see that the graph is correct.
Key Takeaways
Key fact about linear functions is the uniform rate of change, or slope of the line.
Also note that as the independent variable (number of posters) increases, the dependent
variable (cost) increases at a steady rate ($0.11 per poster).
The key fact about linear functions is this uniform rate of change, which is called the slope of
the line. If you revert to the generally used and for independent and dependent variables.
Then,
= change in y per unit change of x
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Figure 2.2.2 Rise over Run
The slope will always have the value of the coefficient (multiplier) of if y is written as a
function of x.
Thus,
Where,
a and b are constants.
a is the y-intercept ( y-value at = 0).
b is the slope of the line.
Key Takeaway
When x changes by 1, y will change by an amount b, which increases if the slope is positive
and decreases if it is negative.
Example 2.2.2
As an example, suppose you have the following linear function:
Then the slope will have the value +2. Make a table of a few values and graph the function.
Just substitute selected values of x to get values of y.
For example, if :
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x y
-2 -1
-1 1
0 3
4 11
Key Takeaways
1. When x changes by 1, y changes by 2.
2. When x changes by 4, y changes by 2 × 4 = 8.
3. Also is the value of y when .
The graph would then appear as shown in Figure 2:
Image 2.2.3: Graph of y = 3+ 2x
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Knowledge Check 2.1
1. For each of the following linear equations find the intercept and the slope.
a.
b.
c.
2. A machine that wraps boxes of candy requires 90 minutes to set up, then one-half
minute per box to do the wrapping. Use
T = total time required (in minutes)
x = number of boxes to be wrapped
a. Complete the equation for T. Note the units used.
Total time = set up time + wrapping time.
without units:
b. Complete the table for the given x-values:
x T
0
80
150
c. Plot the points from your table on the graph and join them to form a line.
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d. Estimate the time to wrap 100 bags and check by calculating it with the function.
LINEAR EQUATIONS FROM DATA
You will not always be given the equation for a linear function. Instead you will have available
some information about it and, from that data, you will have to work out the equation.
The most common case is to have two data points, and which satisfy the
equation. Then, since you are working with the slope-intercept form of linear equation, you
should first work out the slope, b, as follows:
Next, substitute one data point, say into the equation to find the intercept, a, as
follows:
which can then be solved for the intercept as it is the only unknown.
Example 2.2.3
Suppose that in the poster printing example above you were only given the facts that, to
make 1,000 posters, it would cost $190 and, to make 1,500 posters, it would cost $245. The
information could be organized by thinking of the points as being in a table or on a graph as
follows:
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Then to find the slope:
To find the y-intercept:
and write the whole equation as follows:
Knowledge Check 2.2
There are two temperature scales in North America: Fahrenheit and Celsius. They are
related by a linear equation. The temperature of boiling water is 212° Fahrenheit and
100° Celsius. The temperature of freezing water is 32° Fahrenheit and 0° Celsius.
Let,
F = temperature on the Fahrenheit scale (the y).
C = temperature on the Celsius scale (the x).
Then find the equation for F in terms of C (i.e., ) by using the following
steps:
1. Choose one point as another as in the formula,
2. Since you now know b, substitute b and one of the known points in
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and solve for a.
3. Write the final equation and find the Fahrenheit temperature equivalent to
20 degrees Celsius (approximately room temperature).
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2.3 Cost Functions
Key Takeaways
Cost functions give the cost of an activity.
The function used as an example in this chapter, which gives cost in terms of a number of
posters, has some elements typical of cost functions. Cost functions give the cost of an activity,
usually in terms of some measurement of a level of activity.
The fixed cost is the cost that must be paid no matter how low the measured level of activity
(the cost at zero). For the posters the fixed cost was the $80.00 for artwork and set-up, since
these things would have to be done even if only one poster was produced.
The variable cost is the part of the cost that changes with the level of activity. For linear cost
functions, the variable part of the cost changes directly with the level of the activity. In the
poster example, the variable cost is $0.11x. The $0.11 per poster is called the variable cost per
unit. The fact that the variable cost per unit is the same for all units makes the function linear.
Key Takeaways
Variable cost: that part of the cost that changes with the level of the activity
The fixed cost is usually denoted by F or FC.
The variable cost per unit is usually denoted by V or VC.
If you take x to be the number of units produced, then you have the following as a general
formula:
Thus, for the poster example:
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FC = $80.00
VC=$0.11 per poster
x = number of posters
Hence,
Your Own Notes
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80 Amy Goldlist
2.4 Equations and Functions
Key Takeaways
In linear equations it is always possible to solve for any variable shown in the equation
An equation always shows a relationship between variables, but the relationship is not
necessarily to be viewed as a function with independent and dependent variables. For example,
in the equation , there is no requirement that one variable be independent and the
other dependent.
In linear equations it is always possible to solve for any variable shown in the equation -that is,
to rewrite the equation with that variable by itself on one side of the equation. If this is done,
it is a convention to write the variable on the left-hand side of the equation and treat it as the
dependent variable. Thus, in the example above, you could solve for p as follows:
and treat p as a function of q. Similarly you could solve for q as follows:
and treat as a function of p.
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2.5 Systems of Equations
If a number of equations involve the same variables, they are called a system of equations.
Example 2.5.1
An agent is to purchase two products, G and H, and send them to the company’s warehouse.
He has a budget of $70,000 and truck space of 9,000 cubic meters. The product costs and sizes
are as follows:
Cost per Case Volume per Case
G $10 3 m3
H $20 2 m3
The question to be answered here is: how much of each product should be purchased if both
the budget and space are to be used up?
Key Takeaways
Limitations are called constraints. Each constraint gives an equation.
Limitations such as those for budget and space are called constraints. Each constraint gives
an equation. To find the equation from the data given, you should consider the totals for each
constraint separately and find the contribution of each variable. In this case; the variables are
the amounts of each product to be bought. Let,
g = the amount (number of cases) of G
h = the amount (number of cases) of H
Then the budget constraint equation can be obtained by considering:
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Similarly,
So,
Thus you have the system of equations
for which the graphs are given below.
Graph of G and H, Example 6
Notice that the data for the problem was chiefly in terms of rates, the costs per case and the
volume per case. These were used to get equations for totals, using the idea of,
for each equation. Notice, also, the use of the units of measurement to help keep track of the
parts of the equations.
The values required by the problem are those of the point at which both equations are satisfied
– the point on the graph at which the lines cross. This point is called the solution of the
equations. It can be estimated from the graph and also calculated from the equations.
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METHOD OF ELIMINATION
To solve the equations, using the method of elimination write each one in the form for which h
is given as a function of g.
For the budget: h = 3,500 − 0.5g
For the space: h = 4,500 − 1.5g
Next, note that at the solution, the values of h must be the same, so:
By adding to each side, you get:
Subtracting $3,500 from each side gives:
Substituting the result in one of the original equations (space):
Whence
Thus, both the budget and space allowances will be used up if 1,000 cases of G and 3,000 cases
of H are bought.
METHOD OF SUBSTITUTION
The method of substitution gives you another way of solving such equations. It consists of the
following:
Once the first equation has been solved for h, the result is
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This result would be substituted in the second equation, which would produce an equation
containing only the variable g.
Thus,
and
Substituting in the other equation:
Knowledge Check 2.3
One of the production facilities of Jones Furniture Company produces chairs from kits,
each of which contains all the parts for the chair. The facility makes two types of
chairs: a regular chair for normal inside use and a special chair for outside use. Each
chair is assembled and then painted.
There is only a limited amount of labor time for each operation: 120 hours per week for
assembly, 80 hours per week for painting. Each regular chair requires 0.5 hours for
assembly and 0.3 hours for painting. Each special chair requires 0.4 hours for assembly
and 0.4 hours for painting.
Find the number of regular and special chairs that would have to be produced each
week in order to use up both the assembly and painting time available for that week.
Let,
r = number of regular chairs produced per week
s = number of special chairs produced per week
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1. Find the equation that shows the constraint for the available assembly time:
_______ (total hours) = _______ _(hours/chair) × r (chairs) + _______
(hours/chair) × s (chairs)
2. Find the equation that shows the constraint for the available painting time:
? = ? × r + ? × s
3. Solve the equation in Question 1 for r and substitute the result in the
second equation, in Question 2, to get the value for I.
4. Substitute the result from Question 3 in one of the equations to get the
value of r.
5. Check your results by placing them in the other equation (but not the one in
Question 4) and see that they give the correct value.
6. Graph the two equations to make a visual check of your answer.
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2.6 Profit Volume Analysis
A system of equations is also used in the analysis of profits for different volumes of business.
As an example, consider cost and revenue for the poster example discussed previously.
Suppose the printing business decided that it should charge $0.17 per poster. Then it would
have a revenue function:
Together with the cost function, this forms a system of equations:
You have already seen a table for the cost function. Now prepare a short table for revenue, then
graph both functions on the same axes, as follows:
Graph of Cost (C) and Revenue (R)
To analyze this system for business purposes, you should use the relationship (from Chapter
1):
You can see from the graph that the printing business would lose money at those values of x
(the low values) for which the cost function is above the revenue function. And it would make
a profit at those values of x (the high values) where revenue exceeds the cost.
As in most systems of equations, the point at which all the equations are satisfied – the solution
of the equations – is of particular interest. In the example above, it is the place at which,
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That is, the point at which the profit is zero – the break-even point.
The break-even point can be estimated from the graph, but it can also be calculated from the
equations in the system.
To find this point, set:
So, by subtracting 0.11p from both sides, you get:
And:
So
But we cannot make a fraction of a poster! So we always round up.
x = 1,334 posters (rounded up)
This analysis shows the printing company that small orders lose money at $0.17 per poster. In
real life such companies cannot accept orders on which they lose money; they will either set a
minimum order size (say, 1,500) or charge proportionately more for small orders (for example,
$100 for the first 100 posters).
Key Takeaways
As long as the revenue and cost functions are known, you can evaluate what the
profits will be at any level of activity.
To analyze the relationship between profits and volume for an entire business or
product line, do the analysis for a time period.
90 Amy Goldlist
As long as the revenue and cost functions are known, you can evaluate what the profits will be
at any level of activity. Using the revenue function above ,
so that for, say, 2,000 posters:
The above example was for a job in a business. To analyze the relationship between profits and
volume for an entire business or product line, do the analysis for a time period, such as a month
or a year. As before, the fixed costs are those that do not depend on the level of activity. They
include:
rent
property taxes
salaries of employees who will be employed regardless of the level of activity (for
instance, managers, sales clerks)
utilities such as heat and light
The variable costs include:
the costs of materials
labor costs which are the result of the level of production (called direct labor costs)
commissions on sales
Example 2.6.1
Suppose a manufacturer of sports bags examined its accounts and found the following costs.
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Fixed Costs (Per Month)
Salaries (Office, Sales Management) $50,000.00
Depreciation of Equipment 8,000.00
Rent 6,000.00
Interest on Loans 4,000.00
Other Fixed Expenses 2,000.00
Total Fixed Cost $70,000.00
Variable Cost Per Unit Produced
Materials $4.00
Labour 3.00
Total Variable Cost $7.00 per bag
Thus the cost equation would be
where
C = cost
x = number of bags produced in a month
This manufacturer sells the bags for $12 per bag. Hence the revenue equation is
if we assume all bags produced are sold in the same month as they are produced.
The monthly profit would then be
To break even would require:
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So
x = 14,000 bags (produced and sold)
Below 14,000 bags the company would show a loss; above 14,000 bags it would show a profit.
You can find the level required for any given profit by solving for x with that profit. For
example, to earn a $10,000 profit:
thus
Knowledge Check 2.4
Suppose, in the above example, new equipment were to be bought, increasing the
depreciation cost to $12,000 and the interest on loans to $5,000, but reducing the labor
cost to $2.50 per unit. Find the following:
1. The resulting cost function.
2. The volume (number of bags) required to break even.
3. The volume required to make a profit of $10,000.
USING A BAII PLUS CALCULATOR FOR CVP
After graduation you have moved to Australia and discovered a need for bottle openers to take
the tops off “stubbies” of pop. You start a company called Pieces of Roo that will make novelty
bottle openers shaped like Kangaroo forepaws.
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You estimate fixed costs of $2,500 per month and variable costs of $3.50 per bottle opener.
The maximum capacity of your factory will be 1,000 units per month. You expect to sell the
openers for $12.00 each.
This makes your cost equation:
And your revenue equation
Use the pre-programmed functions of the BA II plus to answer the following questions:
1. Calculate the breakeven in units.
Enter the breakeven worksheet by keying 2nd BRKEVN
Enter fixed costs 2500 ENTER
Scroll down using the down arrow +
Enter variable costs 3.50 ENTER
Scroll down using the down arrow +
Enter price 12.00 Enter
Scroll down until Q appear then press CPT
The answer is 294.11 which you would round up to 295
2. Calculate the Profit at 1,000 units
Change quantity to 1,000 units 1000 ENTER
Scroll down using the down arrow+ until PFT is reached
Key CPT
The answer is $6,000
3. You would like to make a profit of $5000 – how many units should you produce?
Change PFT to 5000, 5000 ENTER
Scroll down until Q appear then press CPT
882.35 which you would round up to 883 units
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4. At maximum capacity of 1,000 units, you would like to make a profit of $7,500.
What price should you charge?
Change Q to 1,000
Change PFT to 7500
CPT P
Answer is $13.50 per unit
Your Own Notes
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Chapter 2 Knowledge Check Answer Key
Knowledge Check 2.1
1. Intercept Slope
a. 6 3
b. 4 2
c. 2 -1 (note that )
2. a.
X T
0 90
80 130
150 165
d. 140 minutes
Knowledge Check 2.2
1. Slope:
2. Using the point (0°C, 32°F): , so a = 32°F
3. F = 32 + l .8°C. So putting in C = 20, we have
Knowledge Check 2.3
1.
2.
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3.
4.
Knowledge Check 2.4
1.
2. 13,637 bags
3. 15,455 bags
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Chapter 2 Review Questions
1 Ocean Marina rents moorage space for boats. Its charge for boats between 20 and 40 feet long
is $180 plus $12 a foot.
Let x = boat length in feet C = cost of moorage:
x 20 24 28 32 36 40
C
a. Complete the above table relating length and cost.
b. State the equation describing cost as a function of length.
c. Graph cost as a function of length for 20 to 40 feet.
2 An agent has available truck shipping capacity for 62.5 tons of materials A and B. Each unit
of A weighs 2.4 tons and each unit of B weighs 1.7 tons. Find an equation that relates the
amounts of A and B that can be shipped.
3 A company finds that when the temperature is 12°C it uses 100 litres of heating oil per day.
When the temperature is 5°C, it uses 170 litres of heating oil per day.
a. Mark the above data points on a graph (allow for negative temperatures).
b. Find the equation (assumed to be linear) for oil usage in terms of temperature.
c. Find the amount of oil that would be used at -5°C. Check your result on a graph.
4 A company has a choice between two copiers, A and B. A costs $120 a month plus $0.05 per
page; B costs $250 a month plus $0.03 per page.
a. Find the cost equations for each copier.
b. Which copier would be best for 5,000 pages a month? For 10,000 copies?
c. Graph the equations on the same axes and show clearly where each copier is
cheapest.
d. Find the volumes at which the costs are equal and mark the point on your graph.
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5 Jones Stereo Company sells stereo sets for $150 each. The parts for each stereo cost $39.50
and the labor costs $53 per set. Fixed costs are $16,000 a month.
a. Find Jones’ cost and revenue functions.
b. How many must Jones produce and sell every month in order to (i) break even? (ii)
make a $6,000 profit?
6 Find the equations of the following lines.
a. Passing through (3, 10) and (I, 6).
b. With slope 1.5 and passing through (2, 5).
c. Passing through (1, 7) and falling 2 units in for each increase of 1 in x.
7 Jimms Company believes that the time required to produce its widgets is a linear function of
the number of widgets to be produced in a run. It finds that to produce a run of 600 widgets
requires 7,050 minutes, and that to produce a run of 350 widgets requires 4,300 minutes.
a. Find the equation giving time required as a function of the number of widgets.
b. How long will it take to produce 500 widgets?
c. How many widgets can one expect to produce with a run that is assigned 81 hours?
8 Deluxe Tire Company finds that to produce each additional tire it costs $21.50. The fixed
costs of the plant operation amount to $126,000 per month. The tires are sold for $58 each.
a. Find the revenue and cost functions for the operation.
b. How many tires must be produced and sold in order for Deluxe to break even in a
month?
c. How many tires must be produced and sold in order for Deluxe to earn a profit of
$40,000 in a month?
9 A company can rent a car using either of two methods of payment. Method A requires $20
per day and $0.20 per kilometer. Method B requires $35 per day and $0.12 per kilometer.
a. Which method is cheaper for 300 kilometres of use in one day?
b. At what level of use per day are the costing methods equally expensive?
100 Amy Goldlist
c. A competitor to the company charged $37 for a day’s use during which 80 km were
driven, and $43 for a day during which 120 km were driven. Assume the cost
function to be linear and find the cost, C, as a function of distance traveled, x.
10 Solve the following system of equations for x and y:
11 Solve the following system of equations for x and y:
12 A publishing company finds that, when it prices its computer books at $20 per book, it can
sell 7,000 books per month. When it prices them at $25 per book, it can sell 5,500 books per
month. The company assumes the relationship is linear.
a. Find the equation which gives the number of books that can be sold per month in
terms of the price.
b. How many books should it be able to sell at $22?
13 FG Company produces housings for a major electrical manufacturer. In a four-week period,
during which it produced 2,550 housings, its total costs were $120,000, and in a four-week
period, during which it produced 1,750 housings, its total costs were $98,000.
Assume that the costs are linear functions of the number of housings produced each four-week
period.
a. Find the equation relating total costs to the number of housings produced in a four-
week period.
b. According to your equation in (a), what are the following?
i. the fixed costs per four-week period
ii. the variable cost per housing
c. If the housings are sold for $55 each, how many must be produced and sold each
four-week period in order to break even?
d. Graph your results for costs and revenues. Identify all major areas on the graph.
Business Mathematics 101
14 A furniture company finds that to make a run of 6,200 cabinets costs $205,000 and run of
8,300 costs $267,000. Assume the relationship between cost and number of cabinets produced
is linear.
a. Find the equation which gives cost as a function of number of cabinets produced.
b. If cabinets are sold for $38 each, what is the minimum number of cabinets in a run
such that the sales revenue would pay for the cost of the run?
15 For a certain good there is a demand for 8,000 units when the price is $8/unit and a demand
for 6,200 when the price is $12/unit. Assuming demand, d , is a linear function of price, p, find
the slope of the line and the equation.
16 HJ Outdoor Company found that it cost $6,100 to make a run of 105 jackets, and $7,900 to
make a run of 150 jackets. Assume the cost is a linear function of the number of jackets made.
a. Find the cost as a function of the number of jackets made in a run, and graph it from
80 to 200 jackets.
b. Estimate the cost of a run of 180 jackets.
c. If HJ sells the jackets for $55 each, how many must be in a run in order to barely
recover the cost of making the jackets?
17 Find the equation of the line:
a. with slope 3 that contains the point (4, 1).
b. with slope -5 that contains the point (-2, 3).
c. containing the points (2, 3) and (-6, 1).
d. containing the points (12, 16) and (1, 5).
e. containing the points (-4, 5) and (-2, -3).
f. that contains the point (-4, 2) and falls 2 units in y for every one unit increase in x.
18 You use your calling card to make a telephone call to your friend who lives in Oyster River.
You receive your monthly telephone bill and two of the items are as follows:
Time of call Area Amount
5 min Oyster River $2.50
11 min Oyster River $4.60
102 Amy Goldlist
a. Define the two variables and create an equation to calculate the total cost of a call.
b. How much would a 60-minute call cost?
19 You have been asked to predict the cost of flying a Dash 7 aircraft from Vancouver to Seattle.
You are provided with the following information:
Number of Passengers Cost
12 $7,680
15 $7,725
a. Define the two variables and create an equation to calculate the total cost of a flight.
b. How much would it cost to fly the plane empty (with no passengers)?
c. Interpret the y intercept using the words of the problem.
d. How much extra does it cost to have one more passenger?
e. Interpret the slope using the words of the problem.
f. How much would it cost to fly 14 passengers?
20 B. Furniture Co. believes that the cost to produce its chairs is a linear function (straight line
relationship) of the number of chairs to be produced in a run. It finds that to produce a run of
130 chairs requires $8,200, and that a run of 250 chairs requires $13,000.
a. Find the equation giving cost, C, based on the number of chairs produced, x, in a
run.
b. What is the cost if no chairs are produced?
c. Interpret the y intercept using the words of the problem.
d. What is the extra cost to make one more chair?
e. Interpret the slope using the words of the problem.
f. How much would it cost to produce 400 chairs?
g. How many chairs can be expected to be produced with a run that is assigned
$8,600?
21 Mr. Smith wants to rent a truck for one day to make a number of deliveries. He can rent the
Business Mathematics 103
type of truck he needs from either of two companies. Company A charges $100 plus $0.40 per
kilometre. Company B charges $40 plus $0.80 per kilometre.
a. Write equations for the cost, C, of each truck in terms of x, the number of
kilometres traveled.
b. Suppose you rent a truck from Company A,
i. How much would it cost to rent the truck but not actually drive it
(0 km)?
ii. How much would it cost to drive 300 kilometres?
c. Suppose you rent a truck from Company B,
i. How much would it cost to rent the truck but not actually drive it
(0 km)?
ii. How much would it cost to drive 300 kilometres?
d. Graph the equations on the same axes and show on the graph where each company’s
deal is best. Graph from 0 to 300 km. Use a large scale, graph paper and straight
edge, and apply proper conventions.
e. Using your graph, determine the point where the costs are equal. Mark the point on
the graph you made in (d).
22 Best Buy Furniture Store manufactures and sells bedroom suites. Each suite costs $800 and
sells for $1,500. Fixed costs total $150,000.
a. Write down the cost equation and the revenue equation.
b. Find the breakeven point in units (number of suites).
23 A company has determined that a minimum of 25,000 units of their product can be sold at
a selling price of $10 per unit. However, if the selling price was reduced to $8.00 per unit, a
minimum of 35,000 units can be sold. Relevant cost data for this product is as follows:
Fixed Costs Variable Costs
$75,000.00 Labour cost per unit $4.50
Material cost per unit $1.50
a. Determine the breakeven points in units.
b. Determine the profit earned by selling the minimum quantity for each price
alternative (i.e., at both $10 and $8).
104 Amy Goldlist
24 CK Air has begun a new discount air service to Port Alberni. It costs $2,070 to fly an empty
aircraft with crew to Port Alberni. Each passenger costs an extra $12 in food and extra fuel
costs. Tickets sell for $150 for a one-way flight.
a. Write down the cost equation and the revenue equation.
b. How many passengers must the plane hold for the company to break even?
c. Calculate the profit if 17 passengers fly to Port Alberni.
d. If the company wants to make a profit of $690 per trip, how many seats must they
sell?
e. What is the contribution margin?
25 Finicky-Cat Gourmet Pet Food makes organic cat food and sells them in 2 kg bags. The
company has annual fixed costs of $150,000 and variable costs of $4 per bag. Finicky-Cat sells
each bag for $10. Production capacity is 50,000 bags per year.
a. Find the revenue and cost equations.
b. How many bags of cat food does the company have to sell to break even?
i. What are total sales at the breakeven point?
ii. What is the percent capacity at the breakeven point?
c. The company anticipates that it will be able to make and sell 40,000 bags of cat
food this year. What will it cost to produce these 40,000 bags?
d. Graph the cost equation and the revenue equation on graph paper. Graph from 0 to
50,000 bags. Make sure to label the axes.
i. Clearly identify the breakeven point on the graph.
ii. Identify the area on the graph where the company makes a profit
and where it has a loss.
e. Finicky-Cat wants to increase its selling price from the current $10 so that it could
make a profit of $150,000 from selling 40,000 bags of cat food. What price must
they charge?
26 It costs Brawn Products $8 to make a particular model of shaver. The fixed costs are
$12,000 per month. The shavers are sold to retailers for $25 less trade discounts of 33.5%,
10%.
a. Write down the revenue and cost equations.
Business Mathematics 105
b. Compute the breakeven point (in units and in sales dollars).
c. Find the profit if 2,200 shavers are sold in a month.
d. Find the profit if monthly sales (in dollars) are $37,500.
e. Brawn wants to make a profit of $5,000/month. How many shavers must be sold?
f. If the fixed cost increased to $18,000/month, how many shavers must they sell to
break even?
g. If the fixed costs remain the same (at $12,000/month) but the selling price is
reduced to $10, what would the new breakeven point be?
h. Brawn Products wants to add a third discount to reduce their selling price to $10 (as
in part g). Find the rate of the third discount.
i. The cost to make a shaver has increased by 25% from the current $8 and fixed costs
have fallen by 25% from the current $12,000/month. If Brawn wants to make a
profit of $20,000/month from the sale of 5,000 shavers, what should the new selling
price be?
27 Your company needs to lease a photocopier. It has a choice between three photocopiers A,
B, and C. Copier A charges a flat fee of $3,000 per month. Copier B charges $2,000 per month
plus $0.05 per copy; and copier C charges $1,000 per month plus $0.15 per copy.
a. Find the cost equation for each copier.
b. If the requirement is 15,000 copies per month which copier is cheapest?
c. Calculate the points of indifference based on cost (i.e., determine the number of
photocopies which will make the costs equal).
i. between copier A and B
ii. between copier A and C.
iii. between copier B and C.
d. On the same graph, graph the three cost equations over the range 0-30,000 copies.
e. Over what range of values of x (the number of copies) is it most cost-effective
(cheapest) to rent from A, B, or C?
Notes
1. a.
x 20 24 28 32 36 40
C 420 468 516 564 612 660
106 Amy Goldlist
b. [latex]C=180+12x[/latex]
2. 62.5=2.4A+1.7B
3. b. y = 220 −10x, where x is the temperature and y is oil usage; c. 270 litres
4. a. Copier A: C =$120 +$0.05x (x = number of pages; C = total cost); Copier B: C
=$250 +$0.03x b. 5,000/month: Total cost for A= $370 Total cost for B = $400
Copier A is cheaper 10,000/ month: Total cost for A= $620 Total cost for B = $550
Copier B is cheaper d. Costs are equal when x = 6,500
5. a. C =$16,000 +$92.50x (C = total costs; R = total revenue; x = number of stereos
sold); R =$150x b. Breakeven = 279 stereos (rounded up from 278.26) For a
$6,000 profit number of stereos= 383
6. a. y= 4 + 2x b. y = 2 + 1.5x c. y = 9- 2x
7. a. y=450+11x (y = time in minutes; x = number of widgets) b. 5,950 minutes or
99.17 hours c. 401 widgets
8. a. R= 58x; C = $126,000 + 21.50x; x = number of tires b. 3452.05 round up to 3453
tires c. 4547.9 round up to 4548 tires
9. a. Method B is cheaper by $9.00 b. 187.5 kilometres c. C=$25+$0.15x
10. (0.1389, 2.7222)
11. (-1, 2)
12. a. b =$13,000 −$300p; (p = price; b = number of books) b. 6,400 books
13. a. C = $49,875 + $27.50h; (C = total costs; h = number of housings) b. fixed costs=
$49,875; variable costs= $27.50 per housing c. 1,814 housings
14. a. C = $21,952 + $ 29.52x; (C = total cost; x = number of cabinets produced) b. 2,589
cabinets
15. d = 11,600 − 450p
16. a. C = $1,900 + $40x; (C = total cost; x = number of jackets made) b. $9,100 c. 127
jackets
17. a. y = −11 +3x
b. y = −7−5x
c. y = 2.5 + 0.25x
d. y = 4+x
e. y = −11 −4x
f. y =−6−2x
18. a. C =$0.75 +$ 0.35x where x = length of time of the call (in minutes); C
= total cost of the call.
b. $21.75
19. a. C = $7,500 + $15x where x = number of passengers; C = total cost to
fly the plane
b. $7,500
c. It would cost $7,500 to fly the plane empty (i.e., with no passengers).
d. $15
Business Mathematics 107
e. It costs an extra $15 for each additional passenger.
f. $7,710
20. a. C =$3,000 + $40x b. $3,000 c. The cost incurred even when no chairs are produced
i.e., the overhead or expenses, (e.g., rent, property taxes). d. $40 e. Each additional
chair made will cost $40 (the cost of labour and materials). f. $19,000 g. 140 chairs
21. a. CA=$100 +$0.40x, CB=$40+$0.80x b. and c.
# Km x Total Cost CA Total Cost CB
0 $100 $40
300 $220 $280
d. Below 150 km it is cheaper to use B; above 150 km it is cheaper to use A. e. The
costs are equal at 150 km (both cost $160)
22. a. C = $150,000 + $800x; R = $1,500x; x = the number of bedroom suites produced
and sold b. 215 suites (rounded up from 214.3 units)
23. a. 18,750 units/ 37,500 units b. $25,000 profit/ $5,000 loss
24. a. C =$2,070 +$12x; R =$150x ; x = the number of passengers on the
plane
b. 15 passengers
c. $276 profit
d. 20 passengers
e. $138/passenger
25. a. R =$10x; C =$150,000 +$4x; x = the number of bags produced and sold b. 25,000
bags of cat food, revenue= $250,000; 50% capacity c. $310,000 d.
Number of Bags Revenue Cost
0 0 $150,000
50,000 $500,000 $350,000
25,000 $250,000 $250,000
e. $11.50/bag
26. a. R =$15x, C =$12,000 + $8x, x = the number of shavers produced and
sold
b. 1,715 shavers (rounded up from 1714.28). Sales: $25,725
c. $3,400
d. $5,500
e. 2,429 shavers
108 Amy Goldlist
f. 2,572 shavers (after rounding up)
g. 6,000 shavers
h. 33.33%
i. $15.80/shaver
27. a. CA= $3,000; CB= $2,000 + $0.05x; CC= $1,000 +$0.15x; where x = the number of
copies b. A: $3000; B: $2,750; C: $3,250 so B is cheapest for 15,000 copies c.
CA=CB at 20,000 copies; CA= CC at 13,333 copies; CB = CC at 10,000 copies d.
Plot these points below. Your lines must cross at the indifference points you found
in (c). If not, your graph is wrong.
Number of copies (x) Cost A Cost B Cost C
0 $3,000 $2,000 $1,000
30,000 $3,000 $3,500 $5,500
e. A is cheapest for more than 20,000 copies, B is cheapest between
10,000-20,000, copies C is cheapest below 10,000 copies
Business Mathematics 109
110 Amy Goldlist
Chapter 3: Simple Interest
Key Takeaways
Interest is based on the amount of money borrowed, the time allotted for paying it off, and the
rate of interest.
If you were to borrow money from an individual or a financial institution such as a Bank
or Credit Union, you would expect to be charged a number of dollars for the use of this
money. This amount of compensation is called interest, and is based on the amount of money
borrowed (called the principal), the amount of time allotted for paying it off and the rate of
interest.
By the same token, if you are to deposit some money in a financial institution, you would
expect to get paid interest for allowing the financial institution to use your money. In reality,
you are loaning money to the financial institution and they in tum earn an income on this
money by either loaning it out or investing it.
The amount of simple interest is calculated by using the following relationship:
Where:
I is the amount of interest earned;
P is the amount of money (principal) borrowed or deposited;
r is the annual rate of simple interest; and
t is the time period in years
Usually, r (rate) is quoted as a percent per year (percent per annum or percent pa) and the time
(t) in years. The units of rate and time must match.
Business Mathematics 111
111
112 Amy Goldlist
3.1 Example of Simple Interest
Example 3.1.1
If the time is 48 days, then
If the time is 16 months, then
Note: When the time is given in months, you will treat one month as being equal to year.
If the time is 2 years, then:
Key Takeaways
Be very careful with rounding; be sure to input the exact fraction when using your
calculator!
Your Own Notes
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113
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114 Amy Goldlist
3.2 Exact Time
If the two dates that determine the interest period are known, it is important to calculate the
exact number of days between those two dates. To do this, we count the beginning day but not
the ending day of the period. We will always use a 365 day year, essentially ignoring leap year.
Example 3.2.1
If the interest period ranges from May 12 to May 27,
Be sure to satisfy yourself that this simple subtraction of dates actually includes the first date
but not the last.
You should also know that April, June, September and November have 30 days, February has
28 days (29 days during a leap year) and the remaining months have 31 days. Leap years
occur every four years and are determined by getting an integer (a whole number) answer when
dividing the last two digits of the year by four.
Example 3.2.2
How many days between April 16 and August 12?
Start Date: April 16 (counted)
End Date: August 12 (not counted)
Number of days:
April = (30 – 15) = 15
May = = 31
June = = 30
July = = 31
August = (12-1) = 11
TOTAL = = 118 days
Business Mathematics 115
115
Therefore,
Some calculators and computer software will calculate the days for the user. In this text we
present two methods of date counting: using your calculator, and using a table. We also highly
encourage students to take the time to learn how to use the date functionality in Microsoft
Excel, as it is well worth the effort.
Using the BA II Plus to Find Exact Dates
One or more interactive elements has been excluded from this version of
the text. You can view them online here: https://pressbooks.bccampus.ca/
businessmathematics/?p=60#oembed-1
We will use Question 16 from the Practice Problems to illustrate the exact date feature of the
BA II plus calculator.
Example 3.2.3
AW borrowed $9,000 on January 30, 2002 and agreed to pay 14% simple interest on the balance
outstanding at any time. He paid $5,000 on March 9, 2002 and $2,500 on May 25, 2002. How much
did he have to pay on June 30, 2002 in order to pay off the debt?
First, find the number of days between the debt and the focal date.
2nd Date DTl = 1.3002 Enter Enter the date of the loan
(month.day year)
DT2= 6.3002 Enter Enter the focal date
DBD= CPT 151 from debt to focal date
Next, find the number of days from the $5,000 payment to the focal date.
2nd Date DTl = 3.0902 Enter Enter the date of the first payment
DT2= 6.3002 Enter This is the focal date – you do not need to re-enter
DBD= CPT 113 days from the payment to the focal date
116 Amy Goldlist
Find the number of days from the second payment to the focal date.
2nd Date DTl = 5.2502 Enter Enter the date of the second payment
DT2= 6.300 Enter This is the focal date – you do not need to re-enter
DBD= CPT 36 days from the payment to the focal date
We will finish solving this problem later on.
Dates using a Table
If you prefer to use tables, you can use the following:
Business Mathematics 117
Day of
Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
1 1 32 60 91 121 152 182 213 244 274 305 335
2 2 33 61 92 122 153 183 214 245 275 306 336
3 3 34 62 93 123 154 184 215 246 276 307 337
4 4 35 63 94 124 155 185 216 247 277 308 338
5 5 36 64 95 125 156 186 217 248 278 309 339
6 6 37 65 96 126 157 187 218 249 279 310 340
7 7 38 66 97 127 158 188 219 250 280 311 341
8 8 39 67 98 128 159 189 220 251 281 312 342
9 9 40 68 99 129 160 190 221 252 282 313 343
10 10 41 69 100 130 161 191 222 253 283 314 344
11 11 42 70 101 131 162 192 223 254 284 315 345
12 12 43 71 102 132 163 193 224 255 285 316 346
13 13 44 72 103 133 164 194 225 256 286 317 347
14 14 45 73 104 134 165 195 226 257 287 318 348
15 15 46 74 105 135 166 196 227 258 288 319 349
16 16 47 75 106 136 167 197 228 259 289 320 350
17 17 48 76 107 137 168 198 229 260 290 321 351
18 18 49 77 108 138 169 199 230 261 291 322 352
19 19 50 78 109 139 170 200 231 262 292 323 353
20 20 51 79 110 140 171 201 232 263 293 324 354
21 21 52 80 111 141 172 202 233 264 294 325 355
22 22 53 81 112 142 173 203 234 265 295 326 356
23 23 54 82 113 143 174 204 235 266 296 327 357
24 24 55 83 114 144 175 205 236 267 297 328 358
118 Amy Goldlist
Day of
Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
25 25 56 84 115 145 176 206 237 268 298 329 359
26 26 57 85 116 146 177 207 238 269 299 330 360
27 27 58 86 117 147 178 208 239 270 300 331 361
28 28 59 87 118 148 179 209 240 271 301 332 362
29 29 88 119 149 180 210 241 272 302 333 363
30 30 89 120 150 181 211 242 273 303 334 364
31 31 90 151 212 243 304 365
Table 3-1: Days of the Year
Note carefully: For leap years (those years where the last two digits are divisible by four with
an integer result), February 29 must be included and becomes day 60. One day must then be
added to all dates following. (Examples of leap years: 2016, 2020, 2024, 2028, 2032 )
There are two occasions when extra care must be taken when using Table 3-1:
LEAP YEAR
In a leap year, if the time period includes the end of February, every date after February 28
increases its count by one.
Be careful when your calculations include a leap year or a period that covers the end of a year.
Write the day count on a diagram.
Example 3.2.4
How many days between January 15 and May 28, 2024? (2024 is a leap year.)
January 15 is the 15th day of the year from Table 3-1.
May 28 is the 148th day of the year from Table 3-1. However, since the time period includes
February 29, you must add one day to the value shown in the table.
Business Mathematics 119
Therefore, the exact number of days = (148 + 1) − 15 = 134.
WHEN THE TIME PERIOD INCLUDES THE END OF THE YEAR
Example 3.2.5
How many days between October 15, 2021 and February 13, 2022? The technique is to subtract
the first day value from 365 (to get to the end of the year) and then to add the second day value
to that difference (which moves you into a second year).
October 15 is the 288th day of the year from Table 3-1. Therefore, the number of days to the
end of the year is (365 – 288) or 77.
February 13 is the 44th day of the year from Table 3-1. Therefore, the exact number of days is
77 + 44 = 121. You can draw a time line with the exact dates shown. To assist in calculating
the exact time, get into the habit of also writing the day count on the diagram.
This will automatically lead you to an exact count of the number of days between any two
dates. To illustrate this, note:
178 – 44 = 134 days between the first two dates,
288 – 178 = 110 days between the second two dates, and
288 – 44 = 244 days between the first and last dates.
Your Own Notes
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Business Mathematics 121
122 Amy Goldlist
3.3 Calculating the Amount of Interest
We will apply the formula:
Example 3.3.1
1. How much interest must be paid on a $5,000 loan at 7.5% (per annum or pa) simple
for 32 months?
Note: It is recommended that you always state the data before solving the problem.
Therefore:
2. How much interest must be paid on a $6,500 loan at 8.5% (per annum or pa) simple
if the loan was taken out on February 12, 2023 and repaid on August 15, 2023?
Using Table 3-1:
Or using the BAII PLUS:
DT1 = 2.1223 [ENTER]
↓DT2 = 8.1523 [ENTER]
↓[CPT] DBD = 184
Business Mathematics 123
123
Therefore,
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124 Amy Goldlist
3.4 Calculating Principal, Rate and Time
If any three of the four variables in the relationship ( ) are given, you should be able to
calculate the value of the unknown variable. Starting with the basic interest equation:
You can solve for P or r or t as follows:
Since r is the rate in percent per year, the answer for t will be in years (or part thereof):
Since is the time in years, the answer for will be the annual rate of interest in decimal form.
Knowledge Check 3.1
Now try these exercises.
1. What is the interest charged to borrow $3,000 for 180 days at 6% simple
interest?
2. If $55 interest was charged for a loan at 5.5% simple interest for 125 days,
how much was borrowed?
3. How many days will it take a savings deposit of $900 to earn at least $65
interest, if the simple interest rate is 7.5%?
4. What rate of simple interest is used when a deposit of $975 earns $36.73
interest in 220 days?
Business Mathematics 125
125
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126 Amy Goldlist
3.5 Calculating the Future Value
The future value (or maturity value) is the total amount due at the end of a loan and is
calculated by adding the interest due to the original principal.
Example 3.5.1
1. If $9,600 is borrowed and $800 interest is due, what is the future value (maturity
value) of the loan?
P = $9,600
I = $800
FV = $9,600 + $800 = $10,400
2. Calculate the future value (maturity value) of a $6,500 loan at 8.5% (per annum or pa) simple
if the loan was taken out on February 12, 2023 and repaid on August 15, 2023.
First we count the days:
Using the Calculator:
DT1 = 2.1223 [ENTER]
↓DT2 = 8.1523 [ENTER]
↓[CPT] DBD = 184
Using Table 3-1:
Therefore,
And
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127
Key Takeaways
Note: Some texts refer to the Future Value of a Simple Interest Problem as S, instead of FV.
You can simplify the Future Value equation as follows:
Since and
now, substitute for I. Therefore,
Factoring out the P, you get
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128 Amy Goldlist
3.6 Time Diagrams
The time diagram, or time line, graphically represents the time value of money.
The time diagram (Figure 3.6.1) is also called the time line . It is used to graphically represent
the time value of money.
Figure 3.6.1: Time Diagram or Line
Although this diagram seems very simple and almost trivial at this point in the course, it is
very important that you learn the basics here, and that you include the diagram in all problem
solutions that involve cashflow.
Example 3.6.1
Calculate the maturity value of a 6-month loan of $5,000 at 8.25% simple interest.
Therefore:
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129
Example 3.6.2
Calculate the maturity value of a $4,200 deposit made at a simple interest rate of 6.75% pa
from March 3, 2001 to November 9, 2001.
(from table 3.1)
OR
Using the Calculator:
DT1 = 3.0301 [ENTER]
↓DT2 = 11.0901 [ENTER]
↓[CPT] DBD = 251
Therefore:
Knowledge Check 3.2
Now try these examples (be sure to draw the time diagrams).
1. How much would have to be repaid if you borrowed $4,000 for 210 days at
8% simple interest?
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2. Calculate the maturity value of $1,250 which was invested from March 10,
2022 to September 8, 2022 at 6.75% simple interest.
3. Joan Smith borrowed $2,500 from her father to start the school term. If she
agreed to fully repay him the amount borrowed plus interest to be
calculated at 3.75% simple, how much would she have to repay in exactly
two years?
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Business Mathematics 131
132 Amy Goldlist
3.7 Calculating Present Value
Key Takeaways
“Moving” money backwards in time.
Whenever interest is paid for the use of money, the value of the original principal will increase
in relation to time. This concept is known as the time value of money.
Quite often, you need to calculate the Principal (when the future or maturity value, the rate and
the time are known). You call this principal the present value. It can be found by rewriting the
relationship as follows:
Therefore, dividing both sides by , you get:
Example 3.7.1
Calculate the present value of a 6-month loan which requires a repayment of $6,500 including
interest calculated at 8.25% pa simple interest.
Therefore,
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Example 3.7.2
How much should be invested on April 6, 2001 to amount to $9,200 (FV or maturity value) on
September 19, 2001 at 8.5% simple interest?
(from table 3.1)
OR
Using the Calculator:
DT1 = 4.0601 [ENTER]
↓DT2 = 9.1901 [ENTER]
↓[CPT] DBD = 166
Therefore,
Key Takeaways
When you use your calculator, be careful when you apply the ORDER OF OPERATIONS.
Also, DO NOT ROUND OFF any intermediate values; instead use appropriate keystroke
sequences or the calculator memory.
The following is a suggested sequence of keystrokes for the BAII Plus Calculator:
134 Amy Goldlist
[166][÷][365][×][0.085][=][+][1][=][1/x][×][9200][=]
Key Takeaways
The two formulae, FV = P(1+rt) and I = Prt, interlink five variables, P, r, t, I and FV.
Depending on the context any unknown of the five variables may be calculated if you know
three of the variables. A useful way to deal with problems in this area is to write down the
“known,” and look for a formula which includes the “known” variables and the “unknown”
variable.
Example 3.7.3
How long will it take to earn $50 interest if $1,000 is deposited at 6%?
Note that $t$ will be in “years” since the interest rate is understood to be “per year.”
Now look at the same type of problem in a slightly different way:
Example 3.7.4
How long will it take $2,000 to accumulate to $2,100 if the simple interest rate is 6%?
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So:
Then:
And (given that ):
Knowledge Check 3.3
1. What rate of simple interest is used if a deposit of $2,000 amounts to
$2,210 over 1.5 years?
2. What deposit will amount to $1,871.25 over a period of 33 months if
interest is calculated at 9% simple?
136 Amy Goldlist
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138 Amy Goldlist
3.8 Time Value of Money
The value of money involves accounting/or interest over time.
INTRODUCTION USING FUTURE VALUE
The concept of VALUE of money involves accounting for interest over time.
Consider this question:
Would you rather have $5,000 now or $5,450 one year from now? Before you can answer this
question, you must know what interest rate could be earned on the $5,000 if you made that
choice. Assume 9% simple interest, and calculate the future value.
So, if you chose the $5,000 now and deposited it at 9%, in one year you would have $5,450,
which “coincidentally” is the same as your other alternative. So there is no monetary difference
between the two alternatives. The two VALUES are EQUIVALENT.
Key Takeaways
$5,000 now and $5,450 one year from now have EQUIVALENT VALUES AT 9%.
Remember that VALUE involves both interest rate and time. Try the following example
Knowledge Check/
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139
Knowledge Check 2.4
Would you rather have $10,000 now or $10,325 in 6 months time?
a. the interest rate is 7%.
b. the interest rate is 6%.
EQUIVALENT VALUES
Key Takeaways
Do not add $ values at different points in time.
Now consider this problem:
Your company has borrowed $10,000 one year ago and $15,000 today. How much does the
company owe? If you answered $25,000 you are making a common error. Dollars may NOT
be compared (added or subtracted) when they are at different points in time. They may ONLY
be compared by calculating EQUIVALENT values at the SAME POINT in time.
Example 3.8.1
Assume an interest rate of 9% and look at the problem again. (We are effectively saying that
the $10,000 borrowed one year ago has a 9% simple interest obligation.) We can find the
equivalent debt by calculating the equivalent value now of the $10,000 debt.
140 Amy Goldlist
Now the two values may be added, since they are at the same point in time.
EQUIVALENT DEBT = $10,900 + $15,000 = $25,900
If both debts are settled TODAY, the company must pay $25,900.
How much must be paid to settle both debts SIX MONTHS from now?
If both debts are “moved” individually to the 1 year 6 month point they may be added.
INTERPRETATION:
Each debt accumulates interest charges based on 9% per year, for the time the debt is
outstanding.
Debt #1: $10,000 outstanding for 1½ years accumulates $1,350 interest charge and requires
$11,350 repayment.
Debt #2: $15,000 outstanding for 6 months accumulates $675 interest charge and requires
$15,675 repayment.
The total repayment is $27,025, which includes $2,025 total interest charges, at a point 6
months from now.
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Knowledge Check 3.5
Debts of:
1. $20,000 borrowed nine months ago, and
2. $5,000 borrowed four months ago, and
3. $10,000 borrowed today
must be repaid, together with simple interest at 8%, 6 months from now. How much
must be paid to settle all the debts?
PRESENT VALUE
In a similar manner to the “forward movement” along the time line, to obtain the future value,
dollar amounts can be “moved backward” along the time line. The value of a FUTURE dollar
amount at a point EARLIER in time is called the PRESENT VALUE. Usually this present
value is calculated as the value now of an amount in the future, hence the name “Present”
value. However, this is not a necessary condition. Any value, earlier in time than the dollar
amount being considered is called “Present Value” at the point in time, and can be calculated
in the same way.
Example 3.8.2
Find the value on March 1, 2021, of a repayment of $2,000 due on August 1, 2021.
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Note: The day count number is in brackets.
To calculate the present value, a rate of interest, r, is required. The concept is that the present
value can be considered as a deposit, at a given interest rate, which would amount to the $2,000
in the given time. For example, at an interest rate of 10%:
Three interpretations of the answer are:
1. $1,919.54 is the present value of $2,000, 153 days ahead in time if the interest rate
is 10%.
2. $1,919.54 now and $2,000 in 153 days have EQUIVALENT VALVES at 10%.
3. To accumulate $2,000 in 153 days at 10%, a deposit of $1,919.54 is required.
The wording of the three interpretations is different but they really all say the same thing
monetarily.
Knowledge Check 3.6
Try these examples:
1. Find the value today of a promise to pay $5,000 1 year 7
months from now, if the simple interest rate is 9%.
2. Find the total equivalent debt today of two future debts.
a. $2,000 in 7 months, and
b. $4,000 in 13 months.
Use a simple interest rate of 7% to value the debts.
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144 Amy Goldlist
3.9 Equations of Value
Key Takeaways
All debts “moved” to date of final payment.
In many business situations, debts are not paid exactly as specified in contracts. Often a
payment is made to partially offset a debt.
When two companies regularly carry out transactions with each other, there is often a running
account arrangement, where debts and payments accrue over a period of time, with a final
payment to settle the account.
Focal Dates
When there are multiple inflows and outflows, it is common to evaluate them all at a common
date, often the date of the last payment. In this case, the time of the final payment is used as a
focal date, and all debts and payments are “moved” to the focal date, and their EQUIVALENT
VALUES calculated.
Example 3.9.1
Assume that we have two debts: $10,000 owed today and $15,000 owed one year from today.
The company makes a payment of $8,000 six months from today and will make a final payment
1.5 years from today. The interest rate is 9%.
If the company wishes to settle the outstanding debt at the 1 year 6 month point, all the dollar
amounts are “moved” to the 1 year 6 month point (the date of the final payment) where they
can be added and subtracted. This point is called the FOCAL DATE.
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First, calculate all FV. (EQUIVALENT VALUES)
Next, think about a balancing transaction at the focal date, where:
Key Takeaways
INTERPRETATION: On the FOCAL date, each debt will have accumulated interest
CHARGES.
The $10,000 debt is a debt of $11,350 on the focal date.
The $15,000 debt is a debt of $15,675 on the focal date.
The payment will have accumulated interest CREDIT, so that the $8,000 payment
is worth $8,720 on the focal date.
The total debt value on the focal date $11,350 + $15,675 = $27,025 is offset by the
value of the payment, $8,720, so that the balance owing is $27,025 − $8,720 =
$18,305. All payments and debts are “moved” to a focal date at the time of the final
payment, and a payment−debt balancing EQUATION of VALUE is set up and
solved for the unknown value.
The effect of interest is calculated every time a payment is made.
146 Amy Goldlist
Solving Equations of Value
Key Takeaways
Balancing equation with a specified focal date.
The technique of “moving” all dollar amounts to a FOCAL DATE and setting up a balancing
equation, can be used in a variety of situations where unknown values must be calculated.
There are many occasions when the focal date may be established at ANY TIME by agreement
between the creditor and the debtor.
Example 3.9.2
To settle a current debt your company has agreed to make payments of $10,000, 3 months from
now and $15,000, 9 months from now. When the first payment becomes due, the company
finds itself short of cash and enters into negotiation with the creditor. Eventually the creditor
agrees that if the company makes a partial payment of $4,000 at the 3 month point, the balance
must be paid at the 6 month point.
This is the situation:
Both debtor and creditor agree that the six month point will be the FOCAL DATE for
calculations and that the payments will be valued using 9% simple interest. The two old
payments are replaced by two new payments, with the second new payment, unknown, shown
as x.
All dollar values are compared at the focal date.
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Each dollar value is “moved” to the agreed focal date, and an EQUATION OF VALUE is set
up to balance the values of old payments and new payments at the FOCAL DATE.
The FOCAL DATE EQUATION is:
Now, let us look at a slightly more complicated set of payments. Assume that the agreement is
for two payments, one at the 3 month point and one at the 6 month point as before, but with the
second payment being exactly $15,000 more than the first payment. Now both new payments
are unknown. However, they are related. If x is the first payment, then (x +$15,000) must be
the second payment.
The EQUATION OF VALUE at the focal date is:
Value of new payments = Value of old payments
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The first new payment (x) is $4,892.43 and the second new payment is $4,892.43 + $15,000 or
$19,892.43.
Knowledge Check 3.8
Payments of $3,500 at month 4 and $5,500 at month 11 are to be replaced by a $3,000 payment
at month 6 and a final payment at month 12. The agreed simple interest rate is 8% and the
agreed focal date is the date of the second payment. Find the size of the final payment.
Knowledge Check 3.9
Mr. M owes Ms. K a payment of $10,000 that was due four months ago. He also owes her a
second payment of $9,000 due two months from today. The two parties negotiate a new
payment arrangement whereby Mr. M will repay these debts with two equal installments -one
due today and the other due six months from today . The agreed simple interest rate is 9% and
the agreed focal date is today. Find the size of the two new equal payments.
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Solutions to Chapter 3 Knowledge Checks
Knowledge Check 3.1
1. P = $3,000; r = 6% = 0.06; t =180/365 years
2. I=$55; r =5.5%=0.055; t =125/365 years
3. P = $900, I = $65, r = 7.5% = 0.075
To convert this answer to days, you must multiply by 365. To eliminate a rounding error, be
sure to use the exact value from your calculator, i.e., just multiply the above value by 365
without re-entering the displayed value .
This should now be rounded up to 352 days.
4. P = $975, I = $36.73, t =220/365 years
Knowledge Check 3.2
1. P = $4,000, r = 8% = 0.08, t = 210/365 years
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2. P = $1,250, r = 6.75% = 0.0675; years (from Table
3- 1)
3. P = $2,500, r = 3.75% = 0.0375, t = 2 years
Knowledge Check 3.3
1. P = $2,000, FV = $2,210, t = 1.5 years
Either of the two following approaches is acceptable:
Approach A:
So
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Approach B:
2. FV = $1,871.25, r = 9% = 0.09, t = 33 months =33/12 years.
Knowledge Check 3.4
a. Find FV at 7%:
Conclusion: The value of $10,000 now, in six months’ time is $10,350. Since this is $25 greater
than $10,325, you would prefer $10,000 now from a purely financial point of view.
b. Find FV at 6%:
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Conclusion: The value of $10,000 now, in six months time is $10,300. Since this is less than
$10,325, you would prefer $10,325 in six months time.
Knowledge Check 3.5
Total Debt Outstanding = $37,733.33(6 months from now)
Knowledge Check 3.6
1.
2.
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Total Equivalent Debt Now = $5,639.59
Knowledge Check 3.8
Value of “old” payments at the focal date :
Value of “replacement” payments at the focal date:
Therefore:
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Knowledge Check 3.9
Let the amount of two equal payments be x. Value of “old” payments at the focal date:
Value of “replacement” payments at the focal date:
And we can set these to be equal:
And solve, remembering to store all intermediate values in the calculator:
Therefore: The size of the two equal payments is $9,794.38.
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Chapter 3 Review Questions
Do not forget to draw your time diagrams!
1 For each principal, rate and time given below, compute the interest:
a. $2,500 at 14.2% for 1.5 years.
b. $3,200 at 8.75% for 16 months.
c. $8,300 at 11.2% for 160 days.
d. $800 at 13.6% for 212 days.
2 Calculate the interest for each of the following loans:
a. $850 at 11.5% from June 14, 2022 to October 19, 2022.
b. $2,800 at 11.25% from September 9, 20199 to March 19, 2020. (2020 was a leap
year!)
c. $4,100 at 7.5% from July 15, 2022 to September 6, 2022.
3 Complete each row in the following table:
Interest Principal Rate Time
a. ? $2,800 12% 210 days
b. $461.25 $6,000 ? 8 months
c. $ 54.00 $1,440 11.5% ? days
d. $ 81.30 ? 6.25% 205 days
4 Find the interest rate which will pay $36.40 interest on a principal of $2,140 borrowed for 69
days.
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5 If a loan of $1,900 borrowed from October 22, 2021 to December 17, 2021 resulted in $33.85
interest, what was the simple interest rate charged?
6 What principal will earn $95.20 if borrowed at 13.5% for 4 months?
7 How many days will it take for a principal of $19,200 to earn $650.00 interest at 10%?
8 What is the future value of $1,680 over 260 days at 11.25%?
9 Find the principal and the interest if a loan at 12.5% for 9 months is completely paid off by
the payment of $1,732.22 at the end of the 9 months.
10 If 9 months interest at 8.725% is $186.20, what principal was borrowed?
11 A loan at 9% was repaid by a payment of $3,710 of which $307.40 was interest. What was
the length of time (in days) of the loan?
12 If the future value of a loan for 222 days at 11.75% was $937.72, what was the principal of
the loan?
13 A loan is to be repaid in 9 months by a payment of $1,300. If interest is allowed at 13.15%,
what is the present value of the loan?
14 Payments of $5,000 due in 3 months and $6,000 due in 9 months are to be paid off with
interest allowed at 13%. How much would be required to pay off the loan today? (Use today as
the focal date.)
15 LH should have paid a loan company $2,700 3 months ago and should also pay $\1,900
158 Amy Goldlist
today. He agrees to pay $2,500 in 2 months and the rest in 6 months, and agrees to include
interest at 11%. What would be the size of his final payment? Use 6 months as the focal date .
16 AW borrowed $9,000 on January 30, 2022 and agreed to pay 14% simple interest on the
balance outstanding at any time. She paid $5,000 on March 9, 2022 and $2,500 on May 25,
2022. How much will she have to pay on June 30, 2022 in order to pay off the debt? Use June
30, 2022 as the focal date.
17 Debts of $8,000 due 8 months ago and $3,000 due in 4 months are to be paid off today with
interest at 12%. Use today as a focal date and find the size of the payment.
18 $5,000 due today is to be paid instead by payments of $2,000 in 4 months and the balance
in 9 months. Find the size of the last payment if interest is at 9% and the focal date is today.
19 Two payments of $1,200 each were due 30 and 60 days ago. They are to be paid off by two
equal payments, one in 60 days and one in 90 days. If the focal date is 90 days from today and
interest is at 12%, find the size of the payments.
20 Find the present values of the following payments if money is worth 8%:
a. $2,800 to be paid in 60 days.
b. $950 to be paid in 120 days.
c. $56,000 to be paid in 1 year.
21 You invest $1,000 for 4 years at 8% simple interest. How much interest will you earn?
22 You invest $6,000 for 2.5 years at 9% simple interest. How much interest will you earn?
23 $6,000 earns $180 in interest when invested for 30 months. What simple rate of interest is
being paid?
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24 A $1,000 savings bond earns $600 in interest over the 12 years of the investment. What
simple rate of interest is being paid?
25 You would like to earn $1,000 in interest each year. If the interest rate is 6% simple how
much money should you invest?
26 You take a 3-year loan and repay the loan and $800 in interest. How much did you borrow
if the interest rate was 10% simple?
27 You would like to save for a vacation in Edmonton. You need $4,000 for your dream
vacation. You deposit $3,000 in an account that pays 8% simple. How many months will it take
you to save for your vacation if you make no other deposits?
28 You invest $1,000 for 18 months at 8% simple interest. How much interest will you earn?
29 You take out a loan for 400 days at 10% simple interest and at the end of that time you repay
your loan plus $500 in interest. How much did you borrow?
30 You invest $8,000 on March 3rd and withdraw the money on October 4th. If the interest rate
is 9% simple, how much interest did you earn?
31 You borrow $7,000 on August 16th and agree to pay back the loan plus interest calculated at
5% simple on June 15th of the next year (not a leap year). How much interest would you pay?
32 You borrow $5,000 on June 15th and agree to pay back the loan plus interest calculated at
8% simple on March 31st of the next year (not a leap year). How much interest would you pay?
33 You put $5,000 into a savings account earning 6% simple interest.
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a. How many months will it take to for you to earn $75 of interest?
b. How many months will it take for your money to grow to $6,200?
34 You invest some money today at 4.5% simple interest for 120 days and the money grows to
$7,408. How much did you invest today?
35 You invest $12,000 today into a fund that pays 6% simple. How much money will you have
in 40 months time?
36 You borrow $6,000 to purchase a Jeep and agree to pay back all the money in 3.5 years.
How much should you pay back if the interest rate is 12% simple?
37 You need $6,000 to return to school in 8 months time. How much should you invest today
at 6% simple to achieve your goal?
38 A Freedom 35 financial planner claims you will need $1,175,000 to retire in 15 years time.
How much should you invest today at 9% simple interest to reach your retirement goal?
39 How long will it take a sum of money to double if it earns 12% simple interest? (Answer in
months)
40 You work as a real estate agent for Honest Dave’s Realty Co. located in Burnaby. You have
two debts corning due, one in six months for $5,000 and one in 12 months for $6,000. You
recently sold a couple of houses and now have some extra cash. How much must you pay today
to pay off both debts if interest is 6% simple? Use today as your focal date.
41 One of your customers has two debts outstanding, $600 is due 3 months from today and $900
was due 6 months ago. Instead, the customer would like to pay off both debts with a single
payment one year from today. Calculate the size of that payment if interest is 12% simple. Use
one year from today as the focal date.
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42 You should have made two car payments of $1,000, 6 months ago and 3 months ago. The
bank has agreed to let you repay the loan with equal payments in 3 and 6 months (from today).
Calculate the size of these payments if interest is 14% simple. Use 6 months as your focal
date.
43 You are attempting to repay your line of credit. One year ago you borrowed $5,000 and 6
months ago you borrowed $4,000. You have examined your cash flow projections and decide
to repay the line of credit with two payments in 12 and 18 months. The second payment will
be $2,000 larger than the first. Find the size of the payments using 18 months as your focal
date. Interest is 6% simple.
44You have borrowed from your line of credit. 6 months ago you borrowed $5,000 and today
you borrowed $15,000 . You plan to pay off the entire line of credit with three equal payments
at 3, 5 and 8 months (from today). Find the size of each payment if your bank charges you
9.75% simple interest? Use today as the focal date.
45 Repeat the previous problem using five months as the focal date. (By comparing the
answers to both questions you will see that it depends slightly on the focal date chosen -but
only for simple interest.)
46 You were supposed to make a payment of $3,500 three months ago and a second payment
of $6,100 five months from today. Instead you have arranged with the bank to make a payment
one month from now and a second payment, half as large, 6 months from today. Calculate
these payments if the bank charges 8.25% simple interest. Use the date of the first unknown
payment as the focal date.
47 You borrowed $1,000 on November 30th and another $1,500 on December 31st. Your
arrange with the bank to pay the entire amount on February 15th of the following year. If the
interest is 12% simple (per annum) how much must you pay on February 15th? Use February
15th as the focal date.
48 You are considering purchasing a car. The owner has offered to let you make two payments
of $4,000 each with the first payment at 6 months and the second payment at 10 months.
162 Amy Goldlist
Instead, you would like to make a payment of $4,000 in 8 months and pay the rest today. Find
the size of today’s payment if the interest rate is 6% simple. Use today as your focal date.
49 You have two debts coming due. A $1,500 debt is due in 15 months and another debt for
$1,000 is due in 33 months. Instead, you would like to repay the debts with two equal payments
at 3 and 9 months. Find the size of the equal payments if interest is calculated at 6% simple per
year. Use 9 months as your focal date.
Notes
1. a. $532.50
b. $373.33
c. $407.50
d. $63.19
2. a. $34.01
b. $165.70
c. $44.65
3. a. interest = $193.32 b. rate= 11.53125%c. time= 119.02 or 120 days d. principal
= $2,316.06
4. 8.9977% or 9%
5. 11.6121%
6. $2,115,56
7. 123.5677 or 124 days
8. $1,814.63
9. Principal= $1,583.74 Interest= $148.48
10. $2,845.46
11. 366.3898 or 367 days
12. $875.17
13. $1,183.30
14. $10,309.59
15. $2,335.58
16. $1,770.03
17. $11,524.62
18. $3264.68
19. $1,247.11
Business Mathematics 163
20. a. $2,763.66
b. $925.65
c. $51,851.85
21. $320
22. $1,350
23. 1.2%
24. 5%
25. $16,666.67
26. $2,666.67
27. 50 months
28. $120
29. $4,562.50
30. $424.11
31. $290.55
32. $316.71
33. a. 3 months b. 48 months
34. $7300 exactly
35. $14,400
36. $8,520
37. $5,769.23
38. $500,000
39. 100 months
40. $10,514.75
41. $1,716
42. $1,103.19 each
43. First payment: $4,054.19; second payment: $6,054.19
44. 7,038.53
45. $7,036.45
46. $6,426.52 and $3,213.26 in 6 months
47. $2,548.00
48. $3,846.87
49. $1,157.23
164 Amy Goldlist
Chapter 4: Compound Interest
In the previous chapter, you saw that, in simple interest calculations, the amount of interest
earned is the same for every time period – for example, the amount earned during the first
30-day period is the same as the amount earned in the second 30-day period.
In long-term loans, an increase in the amount of interest paid should follow an increase in the
loan’s value.
In long-term loans, it is felt that, since the value of the loan increases with time (because
of interest), the amount of interest should increase in later time periods. This increase is
accomplished by calculating the interest as compound interest.
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165
166 Amy Goldlist
4.1 Introduction to Compound Interest
Compound interest is the interest paid on previously earned interest as well as on the original
principal.
Example 4.1.1
Consider a loan of $1,000 for two years at 10% per year interest. If simple interest is used, the
lender would receive at the end of the loan.
But at the end of the first year, the lender could reasonably consider the value of the loan to be
and feel that the interest for the second year should be based on this amount as principal. The
lender would then receive
at the end of the second year. This is described as interest compounded annually, or converted
(changed to principal) annually.
Compounding takes effect in accounts where interest is regularly added to the balance.
While it is possible to consider the principal of a loan as continuously increasing because of
interest, the usual way to state and calculate compound interest is to follow the idea above and
increase the principal by the interest at regular intervals. The interval to be used is stated in the
rate. Commonly used intervals are given by the terms:
Annually
semi-annually (every half year = every six months)
quarterly (every three months)
monthly
biweekly (every two weeks or fortnight)
Occasionally loans are compounded weekly or daily.
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167
The interest rate in the example on the previous page would be described as 10% compounded
annually. Compounding takes effect in accounts in which the interest is added to the balance at
regular intervals.
Example 4.1.2
Imagine a trust account set up with an initial balance of $5,000 and an earning of 8%
compounded semi-annually. After two years the account would look like this:
Time Interest Balance
0 $5,000.00
6 months $200.00 $5,200.00
1 year $208.00 $5,408.00
18 months $216.32 $5,624.32
2 years $224.97 $5,849.29
The interest calculations for the account would be as follows: At 6 months:
Note: 8% compounded semi-annually means 4% every half year. At 1 year:
At 18 months:
At 2 years:
The total interest is $849.29 vs $800 at 8% simple interest.
168 Amy Goldlist
Knowledge Check 4.1
Try to perform these fundamental operations of compound interest. Complete the
following account table, using an interest rate of 16% compounded quarterly.
Time Interest Balance
0 $8,000.00
3 months $320.00 $8,320.00
6 months $332.80 ?
9 months ? $8,998.91
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170 Amy Goldlist
4.2 Nominal and Periodic Rates
Key Takeaways
Nominal rate: Percent annual rate; Periodic rate: Rate that gives percent interest each period.
When compound-interest rates are given as above, the percent annual rate is called the nominal
rate and denoted by j. To show clearly what the compounding interval is, add a subscript to
the j (jm) . This indicates the number of times per year the interest is to be compounded.
Thus, 8% compounded semi-annually would be j2 = 8%, m=2.
Thus, 18% compounded monthly would be j12 = 18%, m=12.
The periodic rate is used in most BAII Plus calculations.
The rate that gives the percent interest each period is called the periodic rate and is denoted by
i. This is the rate actually used in most formulas. The following are examples.
Example 4.2.1
1. For 8% compounded semi-annually (j2 = 8% =0.08),
2. For 18% compounded monthly (j12 = 18% =0.18),
In general, to get i from jm:
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171
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172 Amy Goldlist
4.3 Compound Interest Formula
The procedure for adding interest each period can always be used to find the future value of a
loan or deposit, but the following general formula gives the future value more directly.
where:
FV = future value of the loan
PV = present value of the loan (principal)
i = periodic interest rate
n = number of compounding periods
Figure 4.1: Cash Flow
Example 4.3.1
To see how the formula is developed, consider the $5,000 loan at 8% compounded semi-
annually for two years.
First,
The balances would be:
At 6 months:
At 1 year:
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173
At 18 months:
At 2 years:
This last calculation for the two-year balance is the general formula for FV with:
PV = $5,000
i = 0.04
n = 4 = 2× 2
In general, the values for i and n are found by:
and
Knowledge Check 4.2
Use the compound-interest formula to find the following future values:
1. The future value of a deposit of $8,000 at 16% compounded qua1terly for
nine months.
2. The future value of a loan of $1,000 for two years at 10% compounded
annually.
3. The future value of a loan of $2,500 for four years at 8% compounded
monthly.
174 Amy Goldlist
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176 Amy Goldlist
4.4 Present Value
For many investment decisions, it is necessary to find the principal, or present value, that
corresponds to a given future value.
Example 4.4.1
Consider a note that will pay $10,000 to whoever owns it three years from now. If an investor
wants to earn 10% compounded annually, what is the most he or she should pay for the note?
You have:
i=10%=0.10 per year
n=3 years
FV= $10,000
Thus, using the compound-interest formula:
This result can be checked by accumulating the money in an account, as shown in the next box.
Time Interest Balance
0 $7,513.15
1 $751.32 8,264.46
2 826.45 9,090.91
3 909.09 10,000.00
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177
FORMULA FOR PRESENT VALUE
The compound-interest formula can be rewritten to give the present value directly. Divide both
sides of the equation by
Cancel and rewrite:
Knowledge Check 4.3
Find the present values by using the formula for PV given above.
1. The present value of $5,849.29 due in two years if interest is at 8%
compounded semi-annually.
2. The present value of $8,998.91 due in nine months if interest is at 16%
compounded quarterly.
3. The principal of a loan that would amount to $50,000 in six years at 8.5%
compounded annually.
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180 Amy Goldlist
4.5 Multiple Cash Flows
Up to this point, you have considered situations that have only a single loan or deposit and a
repayment. There are, however, many cases in which there are many inflows and/or outflows
to a particular individual or company. For example, consider the company that is developing a
business and needs cash inflows (such as loans) early in its life and will balance these inflows
with outflows later.
Example 4.5.1
Suppose, for instance, that a builder plans to finance a project through a bank and will borrow
$150,000 now and $100,000 in three months, then repay $50,000 in six months and the rest in
one year’s time. Interest is to be paid on the outstanding balance at 12% compounded monthly.
Then you can find the size of the last payment (denoted by x above) by following through an
account and adding interest every month.
Cash inflows to the builder are positive.
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181
Time (months) Interest Cash Flow Balance
0 0.00 $150,000.00 $150,000.00
1 1,500.00 0.00 151,500.00
2 1,515.00 0.00 153,015.00
3 1,530.15 100,000.00 254,545.15
4 2,545.45 0.00 257,090.60
5 2,570.91 0.00 259,661.51
6 2,596.62 -50,000.00 212,258.12
7 2,122.58 0.00 214,380.70
8 2,143.81 0.00 216,524.51
9 2,165.25 0.00 218,689.76
10 2,186.90 0.00 220,876.65
11 2,208.77 0.00 223,085.42
12 2,230.85 0.00 225,316.27
A shorter way to check the account is to accumulate the future value on the outstanding
balance only at the time of cash flows. For the above account, then, the result is as follows:
Time
(months) Interest Cash Flow Balance
0 $150,000 $150,000
3 100,000.00 254,545.15
6 -50,000 212,258.12
12 -225,316.27 0
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Business Mathematics 183
184 Amy Goldlist
4.6 Equivalent Values
We say that the payments of $50,000 and $225,316.27 at the given times are equivalent to the
loans of $150,000 and $100,000 at their given times.
It is often the case that there are two sets of flows. (In the builder’s case mentioned previously,
the $150,000 and $100,000 inflows form one set; the $50,000 and $225,316.27 outflows form
the other.)
Two sets of cash flows are equivalent in value whenever they have the same value after interest
is allowed for.
If you proceed through an account with two sets of equivalent cash flows – one viewed as
inflows to the account, the other as outflows – the final balance must be $0.
Key Takeaways
In an equation of value, the values of inflows, after interest, are equal to the values of outflows.
Evaluating payments by following through an account is fundamental to understanding
equivalence, but this method is difficult to apply to more complex cash flows where the cash
flow to be determined occurs earlier, or where there are several cash flows to be determined. In
such cases, the following equation of value method is preferred.
An equation of value make the values of inflows equal to the values of outflows after interest
is allowed for – that is, after allowing for the time value of money. This is accomplished by
finding the value of every cash flow at some fixed date, the Focal Date. The results obtained
will be the same as those found by calculating the values by an account, as shown above.
The compound-interest formula,
is in itself a simple equation of value, with the focal date at the nth period
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Example 4.6.1
The builder’s situation discussed above could be analyzed by an equation of value and
corresponding cash-flow diagram, as follows:
For a focal date of one year, the equation is:
Through the example above, you will appreciate that the equation of value allows for the same
interest as the account method does. This is true no matter what focal date is chosen – which is
not the case in Simple Interest.
Since the time of the last flow was chosen as the focal date, all flows are evaluated by future
value. Any cash flow that precedes the focal date is evaluated by finding its future value. Any
cash flow that follows the focal date is evaluated by finding its present value.
To see how that works, study the next example, which uses a focal date of three months for the
problem above.
Example 4.6.2
186 Amy Goldlist
This is the same answer as that obtained before, but stating and solving the equation are slightly
more complex. Choosing the best focal date is a matter of convenience.
Problems may have several unknown cash flows, but it must be possible to state them in terms
of one variable, since there is only one equation.
Example 4.6.3
As an example, suppose, in the builder’s case, the loans were to be repaid with two equal-sized
payments: one at six months, the other at one year.
The equation is simplest if you choose one year as the focal date.
Then:
Business Mathematics 187
To check that this is correct, enter the values in an account as shown in the box:
Time (months) Interest Cash Flow Balance
0 $150,000 $150,000
3 100,000.00 254,545.15
6 -135,042.23 127,215.89
12 -135,042.23 0
It is easy to check results by an account, but it would be difficult to solve such a problem by an
account. And in this case an equation would have to be solved at some point.
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188 Amy Goldlist
4.7 Compound Interest with the BAII Plus
Using Financial Calculator Functions
The financial calculator recommended for this course is the BAII
Plus. Both this and other financial calculators have built-in
compound-interest functions. It is possible to do almost all of the
course calculations to the same accuracy without these functions, but
the process is much faster if they are available.
The functions you will use in this chapter are controlled by the
following keys:
P/Y and C/Y N I/Y PV PMT FV
How many times do we
compound per year?(m)
Number of
periods
Nominal
Interest Rate,
jm
Present
Value
0
(for
now)
Future Value
(One of PV and FV
is negative!)
Key Takeaways
Financial Calculators should have built-in compound-interest functions.
In the same row is the PMT key which you will use in the next chapter. For this chapter, the
PMT value should be set at 0. It’s always best practice to set it to 0 each and every time!
Example 4.7.1
Invest $100 at j2 =6% for 4 years. N = 2× 4 = 8 periods.
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189
Step To Press Display
1
Clear previous saved values
(except P/Y and C/Y)
[2ND] [CLR
TVM]
2 Enter N=8 periods [N] [8]
N = 8
3 Enter nominal interest rate, I/Y = 6%. (Annual interest rate in
percentage) [I/Y][6] I/Y = 6
4 Select P/Y and C/Y worksheet [2ND] [P/Y]
5
Set number of payments per year, P/Y = 2
[ENTER] [2] P/Y = 2
6
Set Number of compounding periods per year, C/Y=2
(By default, C/Y is set as the same as P/Y)
[↓] [2]
[ENTER] C/Y = 2
7 Return to standard calculator mode [2ND] [QUIT] 0
190 Amy Goldlist
Step To Press Display
8
Enter present value, PV =100
[1][0][0][±][PV] PV = 100
9 Enter periodic payment, PMT =0 [0][PMT] PMT = 0
10
Compute future value, FV
(positive value for inflow)
[CPT][FV] FV =
-126.6770081
We write this as:
P/Y C/Y N I/Y PV PMT FV
4 4×2=8 6 +100 0 CPT: -125.6770
Leaving some spaces for Annuities, in Chapter 5.
Example 4.7.2
To illustrate the use of the financial calculator, suppose you want to obtain the future value of
a $5,000 loan at 8% compounded semi-annually for two years.
Business Mathematics 191
P/Y C/Y N I/Y PV PMT FV
2 2×2=8 8 5,000 0 CPT: -5,849.29
You will see the answer, $5,849.29, which was obtained earlier in the chapter by an account
and by the formula. Note that the answer appears as a negative value on the calculator. This is
because the calculator performs an equation of value in the form of:
Hence it must make either inflows or outflows negative. (Since PV was made positive, it must
make FV negative.)
From now on, you will normally indicate the procedure for solving problems – especially if
they are likely to be done with computer functions – by listing the available values of the
variables and what is required.
The answer would be negative on the calculator, but this will be mentioned only if confusion
may arise from the answer.
With the calculator functions, any one of the functions N, I/Y, PV, or FV can be found from the
others. How this is done is illustrated in the next example, which uses some previous problems.
The calculator assumes each problem has a cash outflow (entered as a negative) and a cash
inflow (entered as a positive). For simplicity, we will always show PV as positive, and FV as
negative.
Example 4.7.3
You borrow $1,000 and agree to repay the loan with a single payment in 2 years. How much
should you pay if interest is charged at 8% compounded quarterly?
P/Y C/Y N I/Y PV PMT FV
4 4×2=8 8 1,000 0 CPT: -1,171.66
To look at values entered in your calculator, just press [RCL] and then the value you want to
check, e.g., [RCL] [N] should show 8.
192 Amy Goldlist
Example 4.7.4
If an invested $8,000 results in a future value of $8,998.91 in nine months, what is the interest
rate compounded quarterly?
You have:
P/Y C/Y N I/Y PV PMT FV
4 4× 9/12 =3 CPT 8,000 0 -8,998.91
Answer: 16% compounded quarterly.
Alternatively, you could solve the algebra problem:
Which simplifies to:
But this is a much tougher problem!
Example 4.7.5
If $150,000 is invested at 12% compounded monthly and results in a future value of
$169,023.75, for how long must it have been invested?
P/Y C/Y N I/Y PV PMT FV
12 CPT 12 150,000 0 -169,023.75
Answer: 11.9999973 or 12 months.
Business Mathematics 193
Alternatively, we could solve the algebra problem:
Which simplifies, using logarithms to:
In general, the calculator is a very good option – you do not need to use logarithms, and can
solve much faster.
Knowledge Check 4.5
1. Find the future value of a loan of $12,000 for 16 months at 15%
compounded monthly. In doing this, you should write down the values
entered into the TVM:
P/Y C/Y N I/Y PV PMT FV
3. How much must be invested at 11% quarterly to get $9,500 in two years?
P/Y C/Y N I/Y PV PMT FV
4. If a bank deposit of $80,000 amounts to $84,934.22 after gaining interest
compounded monthly for one year, what was the nominal rate per month?
P/Y C/Y N I/Y PV PMT FV
194 Amy Goldlist
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196 Amy Goldlist
4.8 Equivalent and Effective Rates
Interest rates are EQUIVALENT if they provide the same amount of interest on any loan.
Consider the two rates:
1. 20.5% compounded semi-annually.
2. 20% compounded quarterly.
If you take any principal and any length of time, you will find that the two rates always result
in exactly the same future value – hence the same interest. This is because they are related by
the algebraic expression:
To check the equivalence, consider the following example.
Example 4.8.1
Suppose $50,000 is invested for seven years at the interest rates noted above. Find the future
value of the $50,000 for each interest rate.
You have:
Key Takeaway
If two rates produce the same result for any principal and time, the rates will do so for any
values.
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197
For rate a: 20.5% compounded semi-annually
n = 7×2 = 14 half-years
I =
PV =$50,0000
FV = ?
Answer: $196,006.46 for rate b
That exactly the same future value is obtained for both rates bears out the claim that the rates
are equivalent. In fact, if two rates produce
the same result for any (non-zero) principal and time, then the rates will do so for any values.
Hence they are equivalent.
You can use calculator functions to find equivalent rates fairly easily, but first we will use
the Future Value formula. You can use any size of investment and any length of time, but to
illustrate this in the next example, $1 for one year is used.
Example 4.8.2
Suppose you are given the 20% compounded quarterly rate mentioned above and are asked to
find the equivalent rate compounded
semi-annually. You are given:
n = 1× 4 = 4
PV = $1
FV = ?
198 Amy Goldlist
Using the Future Value Formula, we have;
(Leave this answer in the calculator.)
For the new rate, the only thing that will be different (aside from i) is that it is to be
compounded only twice in the year.
So we have
n = 1× 2 =2
i = ?
PV = $1
FV = $1.21550625
Using the Future Value Formula, we have;
So the nominal rate would be j2 = 10.25% × 2 = 20.5%.
EFFECTIVE RATES
The equivalent rate compounded annually for a given compound interest rate.
Compound-interest rates are compared by finding for each rate the equivalent rate compounded
annually. For a given compound-interest rate, the equivalent rate compounded annually is
called its effective rate.
Business Mathematics 199
Example 4.8.3
Find Effective rates for the following:
a. 20.5% compounded semi-annually: j2 = 0.205 so i = 0.1025
To find the effective rate, we can just evaluate the Future Value equation for n = 1 year:
This is trivial to solve: j1 = i = 21.550625%. In fact, the interest earned on $1 invested for a
year is the equivalent rate!
b. 20% compounded quarterly.
n = 1 × 4 quarters, i = 20 ÷ 4 = 5%, PV = $1, FV =?
So we can see that the effective rate is also j1 = 21.550625%. You can see, then, that each
rate was equivalent to 21.550625% compounded annually – which also shows that they were
equivalent, since they were both equivalent to the same effective rate.
EFFECTIVE RATES WITH THE BAII PLUS
Example 4.8.4
200 Amy Goldlist
A bank offers a certificate that pays a nominal interest rate of 15%
with quarterly compounding. What is the annual effective interest
rate?
Step To Press Display
1 Select Interest Conversion worksheet [2ND] [ICONV] NOM = 0
2 Enter nominal interest rate, NOM = 15 [1][5][ENTER] NOM = 15
3 Enter number of compounding periods per year, C/Y = 4 [↓] [4] [ENTER] C/Y = 4
4 Compute annual effective rate, EFF [↓][CPT] EFF = 15.8650415
Example 4.8.5
Try the following: ↓
Find the interest rate, compounded quarterly, that is equivalent to 15% compounded monthly.
[2ND][ICONV]
NOM = 15 [ENTER]
↑C/Y = 12 [ENTER]
↑EFF = [CPT] 16.07545177
↓C/Y = 4 [ENTER]
↓NOM = [CPT] 15.18828125
Thus, the following three rates are all equivalent:
j12 = 15% ⇔ j1=16.07545177% ⇔ j4= 15.18828125%
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202 Amy Goldlist
4.9 Fractional Periods
Occasionally it may be necessary to deal with compound interest for a fraction of a period, for
example taking money out of a savings account after two weeks. In these cases, it is important
to understand what the policy of the bank or lender is.
For a savings account that pays interest monthly you may still receive interest for the two
weeks (at the end of the month) if they pay interest on the daily balance but pay it monthly.
In other words, some banks calculate interest based on the equivalent daily rate (j365) , but
pay monthly. The interest rate paid depends on the total daily closing balance. Interest rate is
applied to the entire balance, calculated daily, and paid monthly.
However, for GICs if you cash them in early you may get no interest or if it is a redeemable
GIC you will get interest for the time that you had it. So if you have a 3-year cashable GIC
that pays interest annually and you cash it in after 10 months it is unclear whether they will use
simple interest or fractional exponents. We reached out to a bank to ask – the head office of
TD Bank has stated that they had no idea, they said the “computer calculates it”.
For such cases, unless otherwise stated, use the compound-interest formula:
with n having a fractional part. The following example justifies this procedure.
Example
Suppose that $1,000 was borrowed at an interest rate of 10% compounded annually, and
originally was scheduled to be repaid after two years. Instead, it was decided to repay the loan
after 1.5 years.
1. Both parties have agreed to have interest calculated for the partial period. At
that time, by the compound-interest formula, the amount to be repaid would be:
. Now suppose this
amount were to be reinvested at the same rate for the remaining half year. In this
case, the money accumulated would be: ,
which is exactly what would have resulted from the original two year loan:
2. Both parties have not agreed to have interest calculated for the partial period.
In this case, we would round down, so $n = 1$, and we receive no interest for the
Business Mathematics 203
203
final year. . In this case, we get a lot less
interest!
In fact, part 1 gives us a general property of compound interest: If the balance is found at
any time and reinvested at the same rate, nothing changes. You may find that some financial
institutions prefer to deal with the fractional period by assuming that simple interest is paid for
that portion. At one time this method was popular because of the difficulty of performing the
calculations for the formula without calculators or computers. It also meant that the institutions
would receive more money since, for a partial period, simple interest is slightly higher than
compound interest.
Key Takeaways
General property of compound interest: If the balance is found at any time and reinvested at the
same rate, nothing changes.
Knowledge Check 4.4
A loan for $6,000 will be taken out for four years at 14% compounded semi-annually.
However, it is decided that the money should be repaid after three years and two
months.
a. Find the accumulated amount to be repaid.
b. Check to see that reinvesting this amount for the remaining 10 months
would produce the same amount as the original four-year loan.
204 Amy Goldlist
Your Own Notes
Are there any notes you want to take from this section? Is there anything you’d like
to copy and paste below?
These notes are for you only (they will not be stored anywhere)
Make sure to download them at the end to use as a reference
An interactive H5P element has been excluded from this version of the
text. You can view it online here:
https://pressbooks.bccampus.ca/businessmathematics/?p=560#h5p-1
Business Mathematics 205
206 Amy Goldlist
4.10 Average Rates
Often, interest rates change over the life of an investment. In this case, we often try to find the
average rate. To illustrate, here is an example:
Example 4.10.1
7 years ago, Warren’s grandmother gave each of her grandchildren different amounts of money.
Warren invested $4,000 of his money with CJC investments. For the first year, CJC put his
money in a GIC which paid an effective rate of 5.9%. For the next 2 years, CJC moved the
money into Canada Savings Bonds, which paid out a semi-annual rate of 7.8%. To capitalize
on the markets, CJC moved the money into a mutual fund, where it earned an average monthly
interest rate of 4.8% for three years. For the final year, the money was put it into stocks which
paid out 5.6% compounded quarterly. How much money does Warren have today?
To solve this, we will take the Present value, and find its value at each point:
YEAR P/Y C/Y N I/Y PV PMT FV
1 1 1 5.9 4,000 0 CPT: -4236
2-3 2 2 ×2 = 4 7.8 4236 0 CPT: -4936.488637…
4-6 12 12×
3=36 4.8 4936.488637… 0 CPT: -5699.43497…
7 4 4 5.6 -5699.43497… 0 CPT: 6025.36864…
We can also look at this algebraically:
So Warren has $6,025.37.
Now we can work out what Warren’s average effective rate is. We take the PV and FV, and
use the calculator to find out what effective rate will turn $4,000 to $6025.37 in 7 years. Make
sure to use your memory buttons to store all of the decimal points.
P/Y C/Y N I/Y PV PMT FV
1 7 CPT 4,000 0 -6025.36864..
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207
We get an effective rate of j1 = 6.027290102%. This is Warren’s average rate.
Once we have the effective average rate, we can use it to solve other problems:
Example 4.10.2
Amy also invested with CJC investments, who invested her money in the same manner. If Amy now
has $8150, how much did she invest 7 years ago with CJC?
To solve this, we can use the average rate already computed:
P/Y C/Y N I/Y PV PMT FV
1 7 6.027290102 CPT 0 -8,150
Which tells us that Amy invested $5,410.46, 7 years ago.
Your Own Notes
Are there any notes you want to take from this section? Is there anything you’d like
to copy and paste below?
These notes are for you only (they will not be stored anywhere)
Make sure to download them at the end to use as a reference
An interactive H5P element has been excluded from this version of the
text. You can view it online here:
https://pressbooks.bccampus.ca/businessmathematics/?p=585#h5p-1
208 Amy Goldlist
Chap 4 Knowledge Check Answer Key
Knowledge Check 4.1
Time Interest Balance
0 $8,000.00
3 months $320.00 $8,320.00
6 months $332.80 $8,652.80
9 months $346.11 $8,998.91
Knowledge Check 4.2
1. $8,000(1.04)3 = $8,998.91
2. $1,000(1.10)2 = $1,1210
3. $2,500(1.0066666667)48 = $3,439.17
Knowledge Check 4.3
1.
2.
3.
Business Mathematics 209
209
Knowledge Check 4.4
a. With i = 0.07, n = 3 × 2, FV = $9,209.76
b. n=(l0+12)× 2, FV=$10,309.11,
when PV = $6,000, n = 4 × 2, FV = $10,309.12
The difference is due to rounding off the FV in (a).
Knowledge Check 4.5
1. $14,638.67
B/E P/Y C/Y N I/Y PV PMT FV
END 12 16 15 12,000 0 CPT
2. $7,646,61
B/E P/Y C/Y N I/Y PV PMT FV
END 4 16 15 CPT 0 -9,500
3. 6% nominal
B/E P/Y C/Y N I/Y PV PMT FV
END 12 12 CPT 80,000 0 -84,934.22
210 Amy Goldlist
Review Questions for Chapter 4
1 Complete the following table, assuming an interest rate of 10% compounded quarterly.
Time Interest Balance
0 0 $8,000.00 (PV)
3 months
6 months
$200.00
8,405.00
9 months
12 months 8,830.50 (FV)
2 Make a table with the same headings as previous, but for a loan of $12,000 for 5 months at
15% compounded monthly.
3 Write the following nominal rates in the “jm” notation and find the corresponding periodic
rates, i (include the period with these). The first line is completed as an example.
Nominal Rate Frequency Nominal Rate, jm Periodic rate, i
8% semi-annually j2= 0.08 i =0.04
15% monthly ? ?
10% quarterly ? ?
9% annually ? ?
10.4% weekly ? ?
4 Write each of the following rates as nominal rates and complete the table.
Business Mathematics 211
211
Nominal Rate Compounding Frequency Nominal
Rate, jm Periodic Rate, i
? ? jl = 7% ?
? ? j4 = 9% ?
? ? j12 = 6% ?
? Quarterly ?
? Monthly ?
? Semi-annually ?
5 Complete the following table, using the compound-interest formula to calculate the future
value of each loan.
PV Interest Rate Length i n FV
$11,500 9% compounded quarterly 2 years ? ? ?
$7,400 6% compounded annually 4 years ? ? ?
$14,000 8.6% compounded monthly 8 months ? ? ?
6 Government compound-interest savings bonds have the interest compounded every year.
Suppose that a $1,000 bond paid interest at 8.5% compounded annually and was kept for three
years. Find the value (Future Value) of the bond at the end of the three years by:
a. Showing the interest and balance each year.
b. Using the compound-interest formula to get FV at the end of two years.
7 A loan of $24,000 for two years is to carry interest at 14% compounded semi-annually. Use
the compound-interest formula to find the value of the loan at the end of two years.
8 A “junk” bond was supposed to pay interest at 25% paid annually. Unfortunately, no
payments were made for the last seven years, so the interest was allowed to compound at 25%
compounded annually.
a. Find the value at the end of each of the seven years for a bond with a principal of
$1,000 at the start.
212 Amy Goldlist
b. Also find the value the loan would have had at the end of each year if the interest
had been simple interest at 25% annually. Note the difference caused by
compounding.
9 Find the present values of each of the amounts below, filling in the rest of the table when you
do so.
PV Interest Rate Length n FV
? 6.5% compounded semi-annually 2 years ? $11,364.76
? 16% compounded monthly 3.5 years ? $25,000.00
? I0% compounded quarterly 1 year ? $30,000.00
Check your last result by assuming the PV was invested for one year in an account paying 10%
compounded quarterly and adding on the interest each period as in Problem 1.
10 According to the terms of his uncle’s will, Tom Jones is to receive $50,000 2.5 years from
now. Tom would like to borrow as much as he can now and pay it off with his inheritance. Use
the compound-interest formula to find out how much he can borrow if the interest rate is as
follows:
a. 10% compounded monthly.
b. 9% compounded monthly.
c. 8% compounded monthly.
11 Ann Lee has a lease on a government property which must be paid by a lump sum of $6,000
every year. The next payment is due in 10 months from now, and Ann plans to invest enough
money in an account at her bank so that the amount in the account in 10 months will cover the
$6,000 payment. If the interest rate is 8.5% compounded monthly, how much should she place
in the account?
12 Ajax Company is borrowing $100,000 now and has agreed to pay 11% compounded
annually on the outstanding balance at all times. Ajax plans to pay $30,000 at the end of the
first year of the loan and $35,000 at the end of the second year. The remaining debt is to be
completely paid off by a single payment at the end of the third year. Draw a cash-flow diagram
and find the amount that should be paid at the end of the third year by:
Business Mathematics 213
a. Finding the interest and balance due at the end of each year.
b. Using an equation of value at a focal date (e.g., at year 3).
13 AC Holdings has taken over a company with a debt that was to be paid by a payment of
$85,000 one year from now. Instead, AC has agreed with the holder of the debt to pay it off
early and be allowed 12% compounded quarterly for early payment. It plans to pay $40,000
now and the rest in six months.
a. Draw a cash-flow diagram and find the amount that should be paid in 6 months.
b. Repeat (a) but assume the payments now and in 6 months are to be equal in size.
14 North Credit Union advertises that it will pay 7% compounded annually on money on
deposit for periods longer than one year and that the money may be taken out, with interest, at
any time after the first year. A depositor placed $20,000 on deposit at the above rate for four
years, but decided to withdraw it after 2.5 years.
a. How much should the depositor receive?
b. Show that, if this amount were deposited at the same rate for the remaining 1.5
years, the result would be the same as if it had never been withdrawn.
15 A loan of $17,000 for three years resulted in a future value of $21,560.11. Use your
calculator functions to find the nominal interest rate if the loan was compounded:
a. quarterly
b. semi-annually.
16 Use your calculator functions to find the FV of a principal of $1.00 invested for one year
at 16% compounded quarterly, and leave the results in your calculator. Then find the nominal
monthly rate that will give the same FV – the rate compounded monthly that is equivalent to
16% compounded quarterly. Check your answer by finding the future value of a principal of
$10,000 invested for two years at each rate. You should get the same answer in each case.
17 Complete the following table of equivalent nominal rates, giving answers in percent to
six decimals. Each row will contain equivalent rates.
214 Amy Goldlist
Effective Rate, j1 j2 j4 j12
? ? ? j12 = 16%
? ? j4 = 12% ?
? j2 = 9% ? ?
18 Use your results in Row 1 in the previous problem to find the future value of a loan of
$4,000 for 18 months by doing the compounding:
a. monthly.
b. Quarterly.
c. Semi-annually
d. Annually
19 Your first child was born this year and you decide to save for her education. You deposit
$2,000 into an RESP (registered education savings plan) that pays 4.5% compounded annually
(j1=0.045). How much will your child have in 18 years? Use both the formula and the TVM
Keys.
20 You borrow $5,000 from a private loan company and agree to pay it back with interest
calculated at 10% compounded quarterly. How much will you owe in 30 months? Use both the
formula and the TVM Keys.
21 You are expecting a tax refund of $3,000 six months from now. You take your T4 slips to
H&P Square Tax preparation service and they agree to give you the money now if you sign
over the refund to them. How much money will you receive today if interest is calculated at j12
= 9$%. Use both the formula and the TVM Keys.
22 You go to purchase a brand new motorcycle and the dealer quotes you a price of $14,000 to
be paid as a single payment in 30 months. You would like to pay cash today for the motorcycle.
How much should you offer if interest is calculated at 8% compounded semi annually? Use the
formula and the TVM Keys.
Business Mathematics 215
23 You have $75,000 and decide to purchase a 30 year Government of Canada strip bond with
an interest rate of 7.0% compounded semi-annually. How much will the bond be worth in 30
years? (i.e., what is the maturity value of the strip bond?)
24 Financial planners predict you will need one million dollars to retire.
a. You would like to buy a 30-year strip bond with an interest rate of 7% compounded
semi-annually that has a maturity value of one million dollars. How much will you
need to pay for this strip bond today?
b. Unfortunately, you only have $75,000 available to buy a bond. You are considering
buying a junk bond (risky bond) because you know that the interest rates are much.
higher than for a Government of Canada bond. What nominal interest rate,
compounded semi-annually, do you require so that the bond will have a maturity
value of $1,000,000 in 30 years?
c. Your friend convinces you that junk bonds are too risky to be used as a retirement
investment. If instead, you buy the Government of Canada bond (with your
$75,000) that pays 7% compounded semi-annually, how many years will it take you
to reach your goal of having $1,000,000?
25 If an investment grows from $10,000 to $16,000 in 27 months, what was the nominal rate
of interest, compounded quarterly?
26 If an investment grows from $4,000 to $6,000 in 48 months, what was the nominal rate of
interest:
a. compounded monthly?
b. compounded quarterly?
c. compounded semi-annually?
d. compounded annually?
27 You would like to save to return to school. You deposit $4,000 into a GIC that pays . You
have decided to return to school when your savings grows to at least $6,000. If you make no
more contributions, how many years will it take you to reach your goal?
28 How many years will it take $300.00 to accumulate to $425.29 at 7% compounded monthly?
216 Amy Goldlist
29 An investment of $1,500.00 made 27 months ago is now worth $1753.48. What nominal rate
of interest, compounded quarterly, did this investment earn?
30 You have always wanted to go to Orlando, Florida. You estimate you will need $6,000 for
your trip. You deposit your tax refund of $5,016.10 into an account that pays 12% compounded
monthly. How many years will it take to reach $6,000?
31 You put $5,000 into a 3-year term deposit that pays interest at 6.3% compounded quarterly.
After the end of the three years you renew the term deposit plus accumulated interest at 7.2%
compounded semi annually for an additional three years.
a. How much money will you have at the end of the six years?
b. How much interest did you earn?
32 You borrowed some money from the Honest Shark Finance Co. You made only one
payment of $25,292.49 at the end of 5 years to pay off the loan. The interest rate was 18.5%
compounded semi annually for the first 2 years and 19.6% compounded quarterly for the
remaining years.
a. How much did you borrow?
b. How much interest did you pay?
33 You are going to purchase a car and are given two options to pay.
You can pay $18,000 in cash today, or
$5,000 after one year, and a second payment of $15,000 in 2 years (from today).
Which option is better? Your answer should be stated in terms of today’s dollars. The
prevailing interest rate is 7% compounded monthly (j12 = 7%)?
34 You have a line of credit that charges interest at j12 = 8%. You borrowed $4,000 six months
ago and $2,000 two months ago. You would like to repay the loan with a single payment in 6
months time. Calculate the size of the payment. Use 6 months as the focal date.
Business Mathematics 217
35 Repeat the above question with you making two equal payments at six and twelve months.
Find the size of the equal payments. Use 6 months as the focal date. Note: To save time use
some of the values you calculated above since the focal date is the same.
36 You purchased a machine for your plant and the contract calls for equal payments of $12,000
in 12 months and 24 months. Your cash flow is better than you projected so you would like to
repay the loan early with a single payment today. Calculate the size of the payment if interest
is 9% compounded semi annually. Use today as the focal date.
37 Repeat the above question with you making two equal payments, one today and one in six
months time. Find the size of the equal payments. Use today as the focal date. Note: To save
time use some of the values you calculated above since the focal date is the same.
38 You were supposed to pay $5,000 today. Up until yesterday you had the money but you lost
it all gambling at the Great American Casino. You arrange with the bank to defer the payment.
You will pay $2,000 at the end of 18 months and 3 equal payments at the end of 24 months, 30
months and 36 months . The interest rate is 10% compounded quarterly (j4 = 10%).
a. Find the size of the equal payments. Use 30 months as the focal date.
b. How much interest did you pay as a result of gambling and losing the $5,000?
39 You borrowed $12,000 2.5 years ago. You agreed to repay the loan with one payment 15
months from today, and a second payment, $3,000 larger than the first, 27 months from today.
Find the size of each payment if money is worth 11% compounded quarterly? Use 27 months
as the focal date.
40 You have $10,000 to invest today. How much would you have (to the nearest $100) at the
end of 30 years if your money earns:
a. 12% interest, compounded monthly?
b. 12% simple interest?
218 Amy Goldlist
41 Your Credit Card charges you a nominal interest rate of 18.6% per year. If they compound
interest monthly, what effective rate of interest do they charge?
42 You are offered two options for your mortgage.
A Canadian bank offers you a rate of j2 = 8.40%
A US bank offers you j12 = 8.30%.
Which rate is better? Convert both rates to effective rates to compare them.
43 What is the effective rate of interest earned on an investment of $10,652,952,497,853.65 that
earns 15% compounded quarterly?
44 What nominal rate, compounded semi-annually, is equivalent to 8% compounded quarterly?
45 What nominal rate, compounded quarterly, is equivalent to 8% compounded monthly?
46 What is the effective interest rate of 8% compounded monthly?
47 Canada Premium Bonds are a new kind of Canada Savings Bond. They were available for
sale until November 1, 2022. They proposed to pay the following interest rates, compounded
annually.
Year 1 2.50%
Year2 3.00%
Year3 4.00%
Year4 4.85%
Year 5 6.00%
What effective interest rate would a Canada Premium Bond purchaser average over the five
years? Hint: use the formula to find the FV. Do not round the FV.
Business Mathematics 219
48 What nominal rate, compounded monthly, is equivalent to 10% compounded semi-annually?
49 What is the effective interest rate of 20% compounded quarterly?
50 You have $5,000 saved. You are considering two investments:
Canada Savings Bonds (CSB) which pay 5.25% in the first year, 6% in the second
year, and 6.75% in the third year, compounded semi-annually.
A 3-year “Bond-Beater” Guaranteed Investment Certificate (GIC) offered by a bank
that pays 5.75%, 6.5%, and 7.25% compounded annually in the three successive
years.
a. How much would you have at the end of 3 years if· you bought a $5,000 CSB and
how much if you bought a $5,000 GIC?
b. Find the average effective interest rate earned for the entire 3-year period for both
the CSB and the GIC.
51 You invest $6,000 in a mutual fund. Your investment of $6,000 earns the following
returns.
Year Return
Year 1 j2 = 10%
Year 2 j12 = 6%
Year 3 j4 = 8%
Year 4 j1 = 7%
a. How much would your $6,000 investment be worth at the end of the fourth year?
b. How much did you earn (in dollars) during those four years?
c. In the fifth year the mutual fund loses money and the value of your investment
decreases by $400. Calculate the average annual rate of return, compounded semi-
annually, for the five years you have held your investment.
d. A friend has invested in a different mutual fund and says she doubled her money in
seven years. What effective interest rate did she average over the seven years?
220 Amy Goldlist
52 A Canada Savings Bond pays j2 = 5% in the first year, j2 = 8% in the second year, and
j2 = 10% in the third year. What nominal interest rate, compounded quarterly, would provide
the same return? Do not round the FV.
53 An investment earned 12%, compounded quarterly, for two years and 10% compounded
annually for the next three years. Calculate the average annual rate of return, compounded
monthly, for the five years. Do not round the FV.
54 An investment earned 20%, 15%, -10%, 25%, and -5% in 5 successive years. What average
annual rate of return, compounded annually, was earned for the entire 5-year period? Hint: use
the formula to find the FV. Do not round the FV.
Notes
1. Time Interest Balances:
0m $8,000.00
3m $200.00 $8,200.00
6m $205.00 $8,405.00
9m $210.13 $8,615.13
12m $215.37 $8,830.50
2. Time Interest Balances:
0m $12,000.00
lm $150.00 $12,150.00
2m $151.88 $12,301.88
3m $153.77 $12,455.65
4m $155.70 $12,611.35
5m $157.64 $12,768.99
Business Mathematics 221
3. Nominal Rate Frequency Nominal Rate, jm Periodic rate, i
8% semi-annually j2 = 0.08 i =0.04
15% monthly j12 = 0.15 i = 0.0125
10% quarterly j4 = 0.10 i= 0.025
9% annually j1 = 0.09 i= 0.09
10.4% weekly j52 = 0.104 i= 0.002
4. Nominal Rate Compounding Frequency Nominal Rate, jm Periodic Rate, i
7% annually jl = 7% i= 0.07
9% quarterly j4 = 9% i = 0.0225
6% monthly j12 = 6% i = 0.005
12% quarterly j4 = 12% i = 0.03
24% monthly j12 = 24% i = 0.02
17% semi-annually j2 = 17% i = 0.085
5. PV Interest Rate Length i n FV
$11,500 9% quarterly 2 years 0.0225 8 $13,740.56
$7,400 6% annually 4 years 0.06 4 $9,342.33
$14,000 8.6% monthly 8 months 0.007166 8 $14,823.09
6. Year Interest Balances
0 $1,000.00
1 $85.00 $1,085.00
2 $92.23 $1,177.23
3 $100.06 $1,277.29
7. $31,459.10
222 Amy Goldlist
8. a. Year Balance
0 $1,000.00
1 $1,250.00
2 $1,562.50
3 $1,953.13
4 $2,441.41
5 $3,051.76
6 $3,814.70
7 $4,768.38
b. Year Balance
0 $1,000.00
1 $1,250.00
2 $1,500.00
3 $1,750.00
4 $2,000.00
5 $2,250.00
6 $2,500.00
7 $2,750.00
9. PV Interest Rate Length n FV
$11,500 6.5% compounded semi-annually 2 years 8 $11,364.76
$7,400 16% compounded monthly 3.5 years 4 $25,000.00
$14,000 I0% compounded quarterly 1 year 8 $30,000.00
10. a. $38,980.40
b. $39,959.34
c. $40,963.72
11. $5,591.10
Business Mathematics 223
12. Year Balance
1 $81,000.00
2 $54,910.00
3 $60,950.10
b. $100,000 (1.11)3 = $30,000 (1.11)2 + $35,000 (1.11) + x x = $60,950.10
13. a. $37,684.65 b. $38,876.54
14. 1. $23,685.88
2. FV = $ 23,685(1.07)1.5 =$ 20,000(1.07)4 =$ 26,215.92
15. a. 8% compounded quarterly
b. 8.08% compounded semi-annually
16. 16% compounded quarterly = 15.7913% compounded monthly
17. Effective Rate, j1 j2 j4 j12
17.227080% 16.214281% 16.542910% 16%
12.550881% 12.180000% 12% 11.881961%
9.202500% 9% 8.900966% 8.835748%
18. a. $5,076.94
b. $5,076.94
c. $5,076.94
d. $5,076.94 (using a fractional period )
19. $4,416.96
20. $6,400.42
21. $2,868.47
22. $11,506.98
23. $590,856.82
24. a. $126,934.31
b. 8.82331%
c. 37.65 years
25. j4 = 21.44411%
26. a. j12 = 10.17956%
b. j4 = 10.26616%
c. j2 = 10.39790%
224 Amy Goldlist
d. j1 = 10.66819%
27. 5.5 years
28. 5.0 years
29. 7.0000%
30. 1.5 years
31. $7,457.11 (the interest earned is $2,457.11)
32. $10,000 (the interest paid is $15,292.49)
33. In today's dollars the 2nd option is worth $17,708.60 which is less than $18,000 so
the second option is cheaper
34. $6,441.19
35. $3,284.78
36. $21,051.50
37. $10,757.36
38. a. Payment= $1,396.46
b. the interest paid is $1,189.38
39. First payment: $8,083.05; second payment $11,083.05
40. a. $359,500 b. $46,000
41. j1 = 20.2705% (effective means compounded annually)
42. Canada: j1= 8.57640%; US: j1 = 8.62314%, Canadian bank is the better deal
43. j1 = 15.86504%
44. j2 = 8.08%
45. j4 = 8.05345%
46. j1 = 8.29995%
47. j1 = 4.062377967%, not 4.07%
48. j12 = 9.79782%
49. 21.550625%
50. a. CSB: $5,970.10 GIC: $6,039.45
b. CSB: j1= 6.08904%; GIC: j1 = 6.49825%
51. a. $8,134.05
b. $2,134.05
c. j2 = 5.14246%
d. j1 = 10.40895137%
52. j4 = 7.58458%
53. j12 = 10.49364%
54. 8.08142% compounded annually
Business Mathematics 225
226 Amy Goldlist
Chapter 5: Annuities
An annuity is a series of payments, with one payment per period for a given number of periods.
The variables in an annuity are:
N: the number of periods, in the term (equal to the number of payments)
I/Y: the nominal interest rate
PMT: the periodic payment
PV: the present value of the annuity
FV: the future value of the annuity
Annuities can be used for:
amortizing a loan, where the loan value is the present value of the annuity. The
variables are N, I/Y, PMT and PV.
accumulating an amount, where the amount is the future value of the annuity. The
variables are N, I/Y, PMT and FV.
Annuities can be solved in many ways:
by formula
by calculating interest period by period
by focal date methods
by calculator program, which is what we will concentrate on.
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5.1 Adding to a Savings Account
Learning Outcomes
Calculate the amount saved and interest earned when regular deposits are made.
You make a series of equal-sized payments (deposits), at regular intervals over
the course of a fixed time period. When you make these regular deposits, you are
adding to the money in the account and building up a positive account balance.
Therefore, both PV (your initial deposit) and PMT (your subsequent regular
deposits) must have the same sign (positive). The future value (FV) must have
the opposite sign (negative)1:
PV Interest PMT FV
Initial Deposit + % Gain + Regular Deposits = Final Withdrawal
0 or + + + −
To make sense of FV (your final withdrawal) being negative, you could imagine that we are
closing the account at the end of the investment period. When we close it, we withdraw all of
the funds, making the final/future value opposite in sign from the regular deposits (PMT) and
the initial amount deposited (PV).
See the sections below for key formulas, tips and examples related to calculations when adding
to a savings account.
CALCULATING AN AMOUNT SAVED (FV)
Let us start by calculating the ending account balance (FV) when regular deposits (PMT) are
made into an account. For this scenario, it is possible that there is an initial balance (PV) or no
money in the account (PV = 0). Let’s see an example where you start saving for your retirement
and we’ll examine both scenarios — when you have no money in the account at the start and
when you have an initial balance.
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EXAMPLE 5.1.1— SAVING FOR RETIREMENT
For the next ten years, you deposit $400 each month into a retirement savings account. You
deposit the first payment one month from now. The account will pay 4.5%, compounded
monthly. You start with $0 in the account at the start of the ten years. How much money will
you have in the account at the end of ten years?
B/E P/Y C/Y N I/Y PV PMT FV
END 12 12 10×12=120 4.5 0 +400 CPT -60,479.23
Why is B/E set to END? This is because the first deposit is being made in one month, at the
end of the first payment interval. When regular payments or deposits are made at the end of
the payment interval, we call this an ordinary annuity. We do not need to do anything in the
calculator (it is set to END by default). We will talk more about changing this setting in the
later sections.
Why are P/Y and C/Y equal to 12? P/Y equals the number of payments or deposits per year.
C/Y equals the number of times the interest rate compounds per year and the interest. Since
both the compounding and deposits are monthly, both P/Y and C/Y equal 12.
Why does N equal 120? N equals 120 because you will make monthly deposits for 10 years.
This will give you a total of 10×12 deposits.
Note: N = The total number of payments or deposits = Number of years × P/Y
Why does PV equal 0? PV is set to 0 because there is no money in the account at the start. PV
is the initial (starting) balance in the account.
Finally, why does PMT equal +400? PMT equals to +400 because we make regular deposits of
$400 into our savings account. We treat all deposits as positive because it is money going into
a savings account2.
Now we can calculate the amount saved at the end of 10 years (FV). Notice that the BAII Plus
will give us a negative value for the future value. This minus sign (negative sign) indicates that
the future value (FV) is going in the opposite direction of the deposits (PMT and PV). Don’t
include the minus sign (negative sign) in your final answer.
Conclusion: You will have $60,479.23 in your savings account at the end of 10 years.
Check Your Knowledge 5.1.1
Redo the above example but instead, start with a balance of $1,000 in the account. What will be
the ending balance in the account? Drag the values that you would enter into the BAII Plus
230 Amy Goldlist
Calculator Keys in the exercise below.
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Conclusion: You will have $62,046.22 in your savings account at the end of 10 years. Note that
the initial deposit of $1,000 makes your ending balance $1,566.99 larger.
CALCULATING THE INTEREST EARNED
For all investments, loans, etc., we can consider the following formula to be true when
calculating the interest earned or charged on our investment or loan:
Interest = Money Out – Money In = $ OUT – $ IN
In the case of regular deposits into a savings account, in this textbook, we will consider the
initial deposit, PV (if there is one) to be “money in” since we are depositing this money INTO
the account.3 We consider the regular deposits, PMT, to be “money in” since they are also
deposits going INTO the savings account. Finally, we consider FV to be “money out” since it
is being withdrawn from the account when we close it:
PV Interest PMT FV
Initial Deposit + % Gain + Regular
Deposits
= Final
Withdrawal
$ IN $ IN $ IN $ OUT
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This gives us the following equation for the interest earned:
Check Your Knowledge 5.1.2
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KEY TAKEAWAYS
Key Takeaways: Regular Deposits into Savings Accounts
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YOUR OWN NOTES
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THE FOOTNOTES
Business Mathematics 233
Notes
1. Other texts might treat PV and PMT as negative and FV as positive. What is most
important is that we are aware that PV and PMT must have the same sign and FV
must be opposite in sign.
2. Some texts might treat deposits into a savings account as negative. If they do, the
initial deposit, PV (if non-zero) will also be negative and FV (the ending balance)
will be positive.
3. Other texts or instructors might treat signs differently than we do here. What is most
important is that we are aware that money flows in two different directions (in and
out). It could be possible to consider the regular deposits as negative and the
ending balance (FV) as positive. You will still get the correct answer because the
deposits and final balance are opposite in sign.
234 Amy Goldlist
5.2 Withdrawing from a Savings Account
Learning Outcomes
Calculate the payment size and interest earned for regular withdrawals from a savings
account.
It is also possible to make regular withdrawals from a savings account. When
you make these regular withdrawals, you’re deducting from the money you’ve
built up in the account. At the end of the annuity (when the account is closed),
all remaining funds (FV) will need to be withdrawn. If nothing is specified for a
remaining $ amount, assume it is zero.
PV Interest PMT FV
Initial
Deposit + % Gain = Regular
Withdrawals
+ Final
Withdrawal
+ + − 0 or −
A common example where these type of regular withdrawals are made is a retirement fund
where the retiree withdraws a certain amount of money every month to pay their bills or an
education account where a student withdraws money twice a year to pay their tuition.
See the sections below for key formulas, tips and examples related to calculations when
withdrawing from a savings account.
CALCULATING THE PAYMENT SIZE
Let us start by calculating the size of your regular withdrawals (PMT) from an education fund.
It is possible that there is an ending balance (FV) or nothing leftover at the end (FV = 0).
We will examine both scenarios — when you have no money leftover and when you have an
ending balance.
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EXAMPLE 5.2.1
Your have $20,000 in your savings account to pay for your post-secondary education. You plan
on making semi-annual withdrawals from the account for three years while you attend BCIT.
The first withdrawal will be in six months. The savings account will earn 4%, compounded
monthly. What will be the size of your semi-annual withdrawals?
B/E P/Y C/Y N I/Y PV PMT FV
END 2 12 3×2=6 4 +20,000 CPT −3,572.53 0
Why is B/E set to END? This is because the first withdrawal is being made in six months, at
the end of the first payment interval. Semi-annual payments are payments that occur twice
per year or once every six months. Again, we do not need to anything in the calculator (it is set
to END by default).
Why does P/Y equal 2? The payments are semi-annual, or twice per year.
Why does C/Y equal 12? The account earns 4% compounded monthly (12 times per year).
Note that P/Y and C/Y are not equal in this example. When P/Y ≠ C/Y, we call this a general
annuity.
Why does N equal 6? N equals six because you will make semi-annual withdrawals for 3 years.
This will give you a total of 3×2 withdrawals. Remember that N = number of years × P/Y =
total number of withdrawals.
Why does PV equal +20,000? The initial balance (or deposit) will be equal to PV and in this
example, that initial balance equals $20,000. In this text, we will treat that initial balance (or
deposit) as positive.
Finally, why does FV equal 0? Since all of the money will be withdrawn, FV = 0.
Now we can calculate the size of your withdrawals (PMT). Notice that the BAII Plus will
give us a negative value for the payments. This minus sign (negative sign) indicates that the
payments (PMT) are reducing the balance in the account until it eventually gets to zero (FV =
0). We will drop this minus sign (negative sign) for our final answer.
Conclusion: You will be able to withdraw $3,572.53 from your savings account every six
months.
Check Your Knowledge 5.2.1
236 Amy Goldlist
What if you wanted to have $3,000 remaining at the end of three years to travel after
you graduate? Redo the above example but instead, have an ending balance of $3,000
remaining in the account. What will be the size of your regular withdrawals? Drag in
the values that you would enter in your calculator.
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Conclusion: You will be able to withdraw $3,097.16 from your savings account every
six months. Note that PMT and FV have the same sign (both negative). You can
consider them both as withdrawals. We are withdrawing $3,097.16 every six months
and withdrawing $3,000 from the account at the end of three years.
CALCULATING THE INTEREST EARNED
For all investments, again, interest is the difference between money in and money out.
In the case of annuities with regular withdrawals, we consider the initial deposit, PV, to be
money in ($ IN) because this money is being deposited into the account. We consider the
regular withdrawals, PMT, to be money out ($ OUT) because they are being withdrawn from
the account. Finally, we consider FV (if there is any balance remaining at the end) to be money
out ($ OUT) because we assume that the final amount (if it’s not 0) will be withdrawn from the
account when the account is closed.
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PV Interest PMT FV
Initial
Deposit + % Gain = Regular
Withdrawals
+ Final
Withdrawal
$ IN $ IN $ OUT $ OUT
This gives us the following equation for the interest earned:
Check Your Knowledge 5.2.2
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KEY TAKEAWAYS
Key Takeaways: Regular Withdrawals from Saving Accounts
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YOUR OWN NOTES
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Business Mathematics 239
240 Amy Goldlist
5.3 Loans and Down Payments
Learning Outcomes
Calculate the duration, selling price and cost of financing for loans with down
payments.
The first type of debt annuity we will examine is the (loan) — an annuity where
we borrow an initial amount of money (PV) and repay the loan with a series of
equal-sized payments (PMT), at regular intervals, over the course of a fixed time
period. At the end, we owe nothing (FV = 0).
PV Interest PMT FV
Amount Borrowed + %Charged = Regular Payments + 0
+ + −
Note: PV and PMT have opposite signs. To better understand this: PV is the initial amount
we receive (loan amount) and PMT is the repayments of that loan that we must pay back after
receiving the loan. The interest adds to the amount owed (we are charged interest each period
on what we owe).
See the sections below for key formulas, tips and examples related to loans and down
payments.
THE TIME REQUIRED TO REPAY A LOAN & KEY QUESTIONS
For loans, you can be asked to calculate the time, rate, initial amount borrowed or size of the
loan payments. In the example below, we will calculate the amount of time required for Zhang
Min to repay her line of credit that she took out to pay for school.
EXAMPLE 5.3.1
Today, Zhang Min borrowed $10,000 from her line of credit. Her line of credit charges 3.75%,
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241
compounded monthly. She can afford to pay $300 per month on her line of credit with the first
payment one month from today. How many years will it take her to repay her line of credit?
Before we determine what to enter into your BAII Plus, let’s ask a few important questions:
Key Questions: Loans
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This gives us the following values in the BAII Plus:
B/E P/Y C/Y N I/Y PV PMT FV
END 12 12 CPT 35.255 3.75 +10,000 −300 0
Zhang Min will make 35 full-sized payments and a smaller final payment at the end of 36
months.
To calculate the number of years, we use the following formula:
Conclusion: It will take 3 years for Zhang Min to repay her line of credit.
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DOWN PAYMENTS & KEY QUESTIONS FOR UNKNOWN SELLING PRICES
A down payment is a lump-sum payment made before you take out the loan. Down payments
are often required when taking out a car loan or mortgage. The down payment will save money
in interest charges over the duration of the loan. This is because it reduces the amount borrowed
(PV):
If we are asked to calculate the selling price when we know the value of the down payment
and amount borrowed (PV), we can rework the above equation to solve for the selling price:
EXAMPLE 5.3.2
Raj wants to buy an All Wheel Drive Tesla Model S. He can take out a 5-year loan with Tesla
Lending. He must make a $10,000 down payment followed by monthly payments of $2,074/
month with the first payment one month after the car is purchased. Tesla Lending charges Raj
3.75% effective on the loan. What is the selling price of the car?
Before we determine what to enter into your BAII Plus, let’s ask a few important questions:
Key Questions: Loans with Known Down Payments and Unknown Selling Prices
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From the above answers and the values given in the problem, enter the following in the BAII
Plus:
B/E P/Y C/Y N I/Y PV PMT FV
END 12 1 5×12=60 3.75 CPT 113,484.44 −2074 0
Because PV equals to the amount borrowed, we know that Raj will borrow $113,484.44. We
still need to calculate the selling price. In order to do this, we use the following formula:
Conclusion: The selling price is $123,484.44 for the All Wheel Drive Tesla Model S.
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COST OF FINANCING ON LOANS & PRACTICE EXERCISE
We call the interest charged on loans the cost of financing. The same interest formula is used
as before:
We consider the amount borrowed, PV, to be “money in” since we are receiving this money at
the start of the loan. We consider the regular payments, PMT to be “money out.” To calculate
the total amount paid from the regular payments, calculate PMT×N since we will make N
payments of size PMT. Finally, FV equals zero because nothing is owed at the end of a loan.
PV Interest PMT FV
Amount Borrowed + %Charged = Regular Payments + 0
$ IN $ IN $ OUT
This gives us the following equation for cost of financing for loans:
Notice that the down payment is not used in the above formula. Only the loan payment size
and amount borrowed are used to calculate the cost of financing for a loan.
Check Your Knowledge 5.3.2
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Check Your Knowledge 5.3.3
Raj feels like he can’t afford the monthly payments for the 5-year loan (see Example
2). Instead, he takes out an 8-year loan. He will still make the $10,000 down payment
followed by monthly payments with the first payment one month after the car is
purchased. Tesla charges 4.25% effective on the 8-year car loan. How much extra
interest will Raj pay if he takes out the 8-year loan instead of the 5-year loan (from
Example 2)?
First we calculate the size of Raj’s new payments. Drag in the values in the correct
calculator keys:
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Because PMT = -1,392.25, we know that Raj will need to pay $1,392.25 per month if
he takes out the 8-year loan. Use this payment size to calculate the amount of interest
(cost of financing) on the 8-year loan:
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Business Mathematics 247
Conclusion: Raj will pay an extra $9,216 in interest if he takes out the 8-year loan
instead of the 5-year loan.
YOUR OWN NOTES
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248 Amy Goldlist
5.4 Leases and Annuities Due
Learning Outcomes
Calculate the rate and cost of financing for car leases and understand annuities due.
Leasing a car (or vehicle) is much like renting a vehicle. The lessee (person leasing the car)
makes a series of equal-sized payments (lease payments), at regular intervals over the course
of a fixed time period (lease period). These payments will be smaller than loan payments. This
is because at the end of the lease, the lessee either must pay an additional amount called the
residual value to purchase the car or they must return the car to the car dealership. To calculate
any of the above values, we enter the following in for each value:
PV Interest PMT FV
Amount Leased + %Charged = Lease
Payments + Residual Value
+ + − −
Leases are examples of annuities due. We will examine this term in the section below.
See the sections below for key formulas, tips and examples related to leases and annuities due.
ORDINARY ANNUITIES VS ANNUITIES DUE
When calculating an ordinary annuity (an annuity where the payments occur at the end of
each payment interval), you do not likely need to change anything in your calculator. By
default, your calculator will be set to END mode (which is the setting you need for ordinary
annuities).
It is also possible that payments occur at the beginning of each interval. In this case, you
need to “tell” your calculator that the payments are occurring at the beginning of the payment
interval. You do this by setting the calculator to BGN mode. This type of annuity is called an
annuity due.
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HOW DO WE SWITCH BETWEEN BGN AND END ON THE BAII PLUS?
How Do We Set the BAII Plus to BGN?
Hit [2ND] followed by [PMT] on your calculator (this gets you into the BGN/END
menu).
You will see END displayed on the screen. To change this to BGN, click on [2ND]
and [ENTER]
You will now see BGN displayed on the screen as well as in the top right of the
screen.
Hit [2ND] followed by [CPT] to exit the BGN/END menu.
Be careful! Ordinary annuities are more common that annuities due so once you
have completed your annuity due question, turn off BGN (ie: set your calculator
back to END).
How Do We Set the BAII Plus to END?
Hit [2ND] followed by [PMT] on your calculator.
You will see BGN displayed on the screen. To change this to END, click on [2ND]
and [ENTER].
You will now see END displayed on the screen and you will see BGN disappear
from the top right of the calculator screen.
Hit [2ND] followed by [CPT] to exit the BGN/END menu.
Now, you can practice these steps in the exercise below.
Check Your Knowledge 5.4.1
Drag the keys onto the calculator screen in the correct order to switch from END to
BGN.
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Drag the keys onto the calculator screen in the correct order to switch from BGN to
END.
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Did you notice that the same keys are used to switch from END to BGN as BGN to
END?
CALCULATING THE RATE CHARGED & DOWN PAYMENTS ON LEASES
When leasing a car, it can be required to make a down payment. If a down payment is made,
we deduct that from the selling price to determine the amount leased:
If we instead need the selling price, we re-shuffle the above equation to give:
Let us now look at Raj’s lease. He is hoping to lease a Tesla Model X.
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EXAMPLE 5.4.1
Raj can lease a Tesla Model X car for three years. From the Tesla website, he finds the
following pricing information:
Selling Price: $110,880
Term: 36 months
Required Down Payment: $7,500
Monthly payment size: $1,702, the first monthly payment is due the day the car is
purchased
Amount owed (residual value) in 3 years to purchase the Model X: $50,568.84
What effective interest rate is Tesla charging on the 3-year lease?
Check Your Knowledge 5.4.2
First, calculate the amount leased (PV):
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Next, enter the values into the BAII Plus:
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Conclusion: Tesla is charging 3.75% effective on their 3-year lease.
COST OF FINANCING FOR LEASES
We also call the interest charged on leases the cost of financing. The same interest formula is
used:
In the case of car leases, we consider the selling price of the car, PV to be “money in.” We
consider the regular payments (PMT) and the residual value (FV) to be ‘money out’:
PV Interest PMT FV
Amount Leased + %Charged = Lease
Payments + Residual Value
$ IN $ IN $ OUT $ OUT
This gives us the following equation for cost of financing for leases:
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Check Your Knowledge 5.4.3
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KEY TAKEAWAYS
Key Takeaways: Car Leases and Annuities Due
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256 Amy Goldlist
5.5 Deferred Annuities
Learning Outcomes
Calculate deferred withdrawals from a retirement fund and deferred payments on a
debt.
To understand deferred annuities, let us first go back and examine the definition of an annuity.
An annuity is “a series of equal-sized payments, at regular intervals, over a fixed period of
time.” What then does it mean to defer this annuity? It means to delay (or defer) the regular
payments for a period of time.
Because there is a deferral period and a payment (annuity) period, there are actually two parts
to the problem:
Part 1: Deferral Period
There are no payments in part 1 (PMT1 = 0).
The only money being added to the initial balance (PV1) is the interest being earned
(or charged).
The ending balance from the deferral period (FV1) equals the starting balance for
the annuity (PV2).
Part 2: Annuity with Regular Payments
Payments are being made or withdrawn (PMT2).
Assume the final future value (FV2) equals 0 unless told otherwise.
See the sections below for key formulas, tips and examples related to deferred annuities
calculations.
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257
EXAMPLES OF DEFERRED ANNUITIES
The most common example of a deferred annuity is a retirement fund where the investor is not
yet ready to retire. They defer their withdrawals (payments) until they retire. In the mean time,
the fund earns interest. The fund continues to earn interest as the investor withdraws money
from the fund.
Other possible examples of deferred annuities are student loans1 and in-store credit cards that
offer “pay nothing for 18 months” promotions on their in-store credit cards. These are both less
than perfect examples of deferred annuities in that they both come with exceptions.
In the case of the student loan, the student defers repaying the loan until after they have finished
school2. What is special about this case is that the Government does not charge them any
interest during the deferment. If the loan is granted by a bank or other financial institution, then
the student gets charged interest during the deferral period.
In the case of “pay nothing for 18 months” promotion, the credit card holder defers payments
(possibly waits for 18 months to start making payments) on the card. These cards and
promotions come with a lot of fine print and exceptions. The credit card example becomes like
a deferred annuity if the credit card holder fails to pay off the entire balance on the card before
the 18 months is up. They get back-charged interest on the amount borrowed. If, after this
happens, they set up a payment schedule to repay the amount owing on the card with regular,
equal-sized payments, then the credit card example becomes a deferred annuity example.
SAVING FOR RETIREMENT EXAMPLES
If you search the web for ‘deferred annuity’ — the savings deferred annuity related to planning
for retirement will come up in at least the top 5 searches. It is the most common type of
deferred annuity. When people plan for their retirement, they start saving before they retire.
They usually do not need to withdrawn money from a retirement fund until they retire. In other
words, they defer the withdrawals until retirement. Let us look at an example of this to better
understand this type of deferred annuity.
Example 5.5.1: Deferred Withdrawals from a Retirement Fund with BGN Off
Ann and Sue sell their recreation property. They make $245,000 on the sale and deposit it
immediately into a retirement fund. The fund earns 2.45%, compounded monthly, for the entire
time. Exactly ten years after the sale, Ann retires. They now want to start supplementing their
income with payments from the retirement fund. They want the payments to start one month
after Ann retires. They want these payments to last for 5 years. How much can they withdraw
per month from the retirement fund?
Let us first organize this information into a time diagram:
258 Amy Goldlist
The deferral period (part 1) occurs first.
The deposit of $245,000 occurs at the very beginning of this deferral period.
No payments nor withdrawals are made for the first 10 years (only interest is added
for these 10 years).
After the 10 years is complete, part 2, the annuity, begins.
One month after the start of part 2 (annuity), the first withdrawal occurs.
These withdrawals repeat for 5 years until no money is left in the annuity.
The tricky part with these types of problems with two parts is determining which part to start
with. Start where the “known” money is. In this example, we know that Ann and Sue deposit
$245,000 immediately (PV1 = 245,000). For this reason, we will start with part 1 because PV1
is known.
Part 1: Let us now organize the information for Part 1 into a BAII Plus table:
B/E P/Y C/Y N1 I/Y PV1 PMT1 FV1
–– 12 12 10×12=120 2.45 +245,000 0 CPT -312,939.05
Since there are no payments, B/E doesn’t matter (that is why ‘––‘ is set for B/E).
Because the interest rate is 2.45% compounding monthly, I/Y = 2.45 and C/Y = 12.
Because there are no payments, match P/Y to C/Y (P/Y = 12).
The deferral period lasts 10 years, so N1 = 10×12=120.
Remember, the $245,000 deposit occurs at the very beginning so PV1 = +245,000.
There are no payments for the deferral period so PMT1 = 0.
CPT FV1 = −312,939.05. Ann and Sue’s initial deposit grows to $312,939.05 after
10 years due to the interest earned on the deposit.
The $312,939.05 (FV1) will become the starting value for the annuity in part 2
(PV2).
Part 2: Let us now organize the information for Part 2 into a BAII Plus table:
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B/E P/Y C/Y N2 I/Y PV2 PMT2 FV2
END 12 12 5×12=60 2.45 +312,939.05 CPT −5,546.95 0
The payments start one month after Ann retires (in one payment interval), so B/E =
END.
Because the interest rate is still 2.45% compounding monthly, I/Y = 2.45 and C/Y =
12.
Ann and Sue receive monthly payments so P/Y = 12.
They want these payments to last 5 years so, N2 = 5×12=60.
Use the (positive) value of FV1 for PV2. So, PV2 = +312,939.05.
There is no money left at the end of 5 years so FV2 = 0.
Compute PMT2 to get the size of Ann and Sue’s monthly withdrawals.
Conclusion: Ann and Sue can withdraw $5,546.95 per month from their retirement fund.
Example 5.5.2: Deferred Withdrawals from a Retirement Fund with BGN On
What would change in Example 1 if Ann and Sue made the first withdrawal exactly 10 years
after the sale of their property? Ie: what if their first withdrawal occurred the day that Ann
retired?
Let us first look at the timeline for this problem:
Part 1 is the same as in Example 1, so we do not need to redo those calculations.
Part 2 has changed. The first withdrawal (PMT2) now occurs exactly at the start
(beginning) of that annuity.
This means we set the calculator to BGN when calculating the annuity payments in
part 2 (PMT2):
B/E P/Y C/Y N2 I/Y PV2 PMT2 FV2
END 12 12 5×12=60 2.45 +312,939.05 CPT −5,535.64 0
260 Amy Goldlist
Conclusions:
The amount withdrawn by Ann and Sue decreases slightly from $5,546.95 to
$5,535.64 when they make the first withdrawal on the day that Ann retires.
Withdrawing the first payment one month earlier means that Ann and Sue will miss
out on the interest that would have been earned on that payment.
This is because the money in the annuity continues to earn interest as the
withdrawals are being made.
As a result, Sue and Ann withdraw $11.31 less per month if they make the first
withdrawal exactly 10 years after the sale of their property.
INTEREST EARNED WHEN SAVING FOR RETIREMENT
Again we use same interest formula:
We need to be careful when calculating money in and money out for deferred annuities.
What money was deposited in the account? Only the initial deposit (PV1) is
considered money in ($ IN).
What money do we take out (withdraw)? Only the regular withdrawals (PMT2) and
final withdrawal (FV2) in part 2 are considered to be money out ($ OUT).
Because FV1 does not get withdrawn but instead becomes the starting balance for
the annuity (PV2), it is not included in our interest calculation. The money is neither
being deposited nor withdrawn.
PV1 Interest1 PMT1 FV1 PV2 Interest2 PMT2 FV2
Initial
Deposit
+ %
Earned + $0
=Ending
Balance
Part 1
=Starting
Balance
Part 2
+ %
Earned
= Regular
Withdrawals
+ Final
Withdrawal
$ IN $ IN –– –– –– $ IN $ OUT $ OUT
This gives us the following equation for interest earned:
Business Mathematics 261
Check Your Knowledge 5.5.0
Calculate the amount of interest Ann and Sue will earn in on their retirement savings
plan from Example 2:
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DEFERRING PAYMENTS ON A DEBT EXAMPLE & KEY TAKEAWAYS
It is quite common to defer payments when purchasing items, especially in retail. A retailer
purchases their goods from a supplier. If cash flow is an issue for the retailer, they can often
choose to defer the payments on the shipment until a later date. It is also common when
consumers purchase items using “in-store” credit cards. Some stores have “Pay Nothing”
promotions for their customers. If a customer purchases goods from the store on the in-store
credit card, they can choose to make no payments on their purchase for a certain amount of
time (6 months, 12 months, 18 months,…).
In the above cases, the retailer or customer owes the value of the amount of goods they
purchased. They make no payments for a certain amount of time (they defer their payments).
262 Amy Goldlist
After the deferral period has passed, they make payments to repay the value of the goods plus
the interest charged during the deferral period. See the timeline describing this process:
Part 1: The initial amount owing gathers interest. There are no payments in part 1 (the
deferral period). The only money being added to the balance is the interest being charged.
This problem is a compound interest problem (Chapter 4):
Part 2: Payments are now being made on the balance owing. This is the ‘annuity’ part of the
problem.
Example 5.5.3: Deferred Repayment of Money Owed with BGN ON
Luis is renovating his kitchen. He takes advantage of a promotion offered by Home Depot:
“Pay nothing for 18 months.” He purchases $7,500 of materials on his Home Depot card.
Home Depot charges him 28.8% compounded monthly. 18 months after the purchase, Luis
makes his first of 6 monthly payments to pay off the credit card. What is the size of Luis’s
monthly payments?
Let us first look at the timeline for this problem:
From there, fill in the exercise below to figure out the size of Luis’ payments.
Check Your Knowledge for Example 3
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To determine what to enter in the BAII Plus for Part 1, let’s ask a few important
questions:
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Next, enter the values for Part 1 into the BAII Plus:
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264 Amy Goldlist
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Conclusion: Luis will owe $11,493.71 at the end of the deferral period (end of the 18
months). This amount will become the starting balance (PV2) in part 2.
Next, let’s ask some key questions on what to enter in the BAII Plus for Part 2:
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Next, enter the values for Part 2 into the BAII Plus:
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Conclusion: Luis will need to make six monthly payments of $2,030.97 to pay off the
credit card.
COST OF FINANCING WHEN DEFERRING PAYMENTS ON A DEBT
Again we use same interest formula:
We need to be careful when calculating money in and money out for deferred annuities.
What is the money in ($ IN)? Only the initial amount borrowed (PV1) is considered
money in ($ IN).
What is the money out ($ OUT)? Only the regular payments (PMT2) are considered
to be money out ($ OUT):
You should note that because FV1 is not a payment but instead becomes the starting
balance for the annuity (PV2), it is not included in our interest calculation.
266 Amy Goldlist
PV1 Interest1 PMT1 FV1 PV2 Interest2 PMT2 FV2
Initial
Deposit
+ %
Earned + $0
=Ending
Balance
Part 1
=Starting
Balance
Part 2
+ %
Earned
= Regular
Payments + $0
$ IN $ IN –– –– –– $ IN $ OUT ––
This gives us the following equation for cost of financing:
Check Your Knowledge 5.5.3
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Business Mathematics 267
DEFERRED REPAYMENT WITH THE INITIAL AMOUNT OWED UNKNOWN
It is possible that the initial amount borrowed (PV1) for a deferred annuity is unknown.
In the case, we must know the payment size in part 2 (PMT2). For this reason, we start with
part 2 (the annuity) in these cases. See Jessa’s example below where she knows the size of the
monthly payments required to finance a bedroom set at Ikea.
Example 5.5.4: Deferred Repayment with the Initial Amount Owed Unknown
Jessa is shopping at Ikea. She sees a sign for a beautiful bedroom furniture set. The sign reads
the following. “Make no payments for 6 months followed by 12 easy payments of $129.99 for
this entire bedroom set.” Jessa has just moved into her new apartment and needs new bedroom
furniture but she’s a bit tight on cash right now. She is interested in the furniture and curious
about how much the actual bedroom set costs. She reads the fine-print on the sign that states
that Ikea charges an effective interest rate of 28.8% on financed purchases. What is the cost of
financing that Jessa will pay if she purchases the bedroom set this way?
Let us first look at the timeline for this problem:
Check Your Knowledge 5.5.4
Let’s find the price of the bedroom furniture set by breaking down this question into
pieces. First, determine which “part” to start with:
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Next, drag and drop the correct values for Part 2 into the BAII Plus table:
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Jessa will owe $1,373.71 when she starts making regular payments. Use this amount
(and make it negative) for FV1.
Next, drag and drop the correct values for Part 1 into the BAII Plus table:
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The price of the furniture set is $1,191.51. Use this amount to calculate the cost of
financing that Jessa will pay on her Ikea purchase:
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Conclusion: Jess will pay $368.37 in interest on her Ikea purchase.
270 Amy Goldlist
YOUR OWN NOTES
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THE FOOTNOTES
Notes
1. A student loan is a special case of a deferred annuity in that if the student receives a
government-sponsored student loan, they do not need to pay any interest while
they are in school.
2. If the Government of Canada has given out the student loan, then it is a special case
of a deferred annuity. The student does indeed defer their payments until after they
have finished school (they can wait up to 6 months after they are done school to
start their payments).
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272 Amy Goldlist
5.6 Back to Back Annuities
Learning Outcomes
Calculate the initial deposit value or payment sizes in back-to-back annuities.
What are back-to-back annuities? They are a series of equal-sized, regular deposits (payments)
over a fixed period of time (annuity 1) followed by a series of equal-size regular withdrawals
for a fixed time period (annuity 2). In both cases, the balance in the account will be earning
interest during the deposits and withdrawals. See the diagram below:
Notice that, unless there is a special deposit or withdrawal between the two annuities, the
ending balance of the first annuity (FV1) becomes the starting value of annuity 2 (PV2) with
one caution: we need to be careful of signs. FV1 will be negative. PV2 will be positive (they
should be opposite in sign). We will talk more about this later during our first example.
See the sections below for key formulas, tips and examples related to back-to-back annuities
calculations.
EXAMPLES OF BACK-TO-BACK ANNUITIES
It is common when saving for retirement, or for a child’s education to save up by making
regular deposits into an RRSP (registered retirement savings plan) or an RESP (registered
education savings plan). Often, after making these regular deposits, the retiree (or student)
starts making regular withdrawals from the account upon retirement (or upon starting school
for the student).
THE SIGNS OF PV, PMT & FV FOR BACK-TO-BACK ANNUITIES
When calculating deposit or withdrawal amounts for back-to-back annuities, it is important to
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be careful of the signs of each of the values (for PV, PMT and FV). Let’s examine the signs
below:
Annuity 1: The initial balance (PV1) is considered positive. This balance gathers interest. The
subsequent payments (PMT1) add to the existing balance in the account and are therefore also
positive. At the end of the annuity, we consider the future value (FV1) as the amount we would
need to withdraw from the account to close the account. For this reason, the future value (FV1)
is recorded as negative:
PV1 Interest1 PMT1 FV1
Initial Deposit + % Gain + Regular
Deposits = Ending Balance
0 or + + + −
Annuity 2: The initial balance (PV2), in most cases, is the amount of money saved up in
annuity 1 (FV1). There is one exception to this rule – when there is a lump-sum deposited or
withdrawn between the end of Annuity1 and the start of Annuity2. We will see examples of this
later in this section.
PV2 Interest2 PMT2 FV2
Initial Deposit + % Gain = Regular Withdrawals + Final
Withdrawal
+ + − 0 or −
For annuity 2, both the regular withdrawals (PMT2) and the final ending balance (FV2) deduct
from the balance in the account. They should both be negative:
Putting this all together gives:
PV1 Interest1 PMT1 FV1 PV2 Interest2 PMT2 FV2
Initial
Deposit
+ %
Earned
+
Regular
Deposits
=Ending
Balance
Part 1
=Starting
Balance
Part 2
+ %
Earned
= Regular
Withdrawals
+ Final
Withdrawal
+ + + − + + − −
Notice that, we use the ending balance of annuity 1 (FV1) becomes the starting balance of
annuity 2 (PV2) except we change the sign. FV1 should be negative and PV2 should be positive
(they should be opposite in sign). We will talk more about this later during our first example.
274 Amy Goldlist
DETERMINING THE SIZE OF THE REGULAR DEPOSITS (PMT1)
Some people plan for their retirement by deciding on the size of the withdrawals they would
like to receive upon retirement (PMT2). They then back-calculate the size of the deposits
(PMT1) they will need to make to achieve their retirement goals.
Let’s have a look at Raj’s retirement plan in the next example. Raj is very wise and starts saving
when he turns 25!
Example 5.6.1
Today is Raj’s 25th birthday, and he has opened an account to start his retirement savings
with an initial deposit of $1,000. He plans to make regular deposits into the account on a
monthly basis, with the first deposit today. He estimated that, the retirement account will earn
an average interest rate of 6% compounded annually. At age 65 he will turn his retirement
saving into an annuity paying 4% compounded annually and he will be able to withdraw $4,000
per month for 30 years with the first withdrawal occurring on his 65th birthday. How much
does Raj’s monthly deposit need to be in order to meet his retirement goals?
Let us first organize this information into a time diagram:
Next, we need to determine where to start. For back-to-back annuities, always start where
the known payment is. In this example, we know the size of Raj’s withdrawals during his
retirement (PMT2), therefore, we will “start” with part 2.
Let us now fill in the BAII Plus table for Part 2:
B/E P/Y C/Y N2 I/Y PV2 PMT2 FV2
BGN 12 1 30×12=360 4 CPT +847,893.56 −4,000 0
Because the withdrawals start right on Raj’s 65th birthday, BGN is on.
Raj’s makes monthly withdrawals but the interest compounds annually so P/Y = 12
and C/Y = 1. Be careful when entering a different value into P/Y than C/Y in your
BAII Plus.
Raj wants to withdraw $4,000 per month for 30 years so N2 = 30×12 and PMT2 =
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−4,000 (remember to make withdrawals negative).
We assume there is nothing left in the account at the end of 30 years (FV2 = 0)
because we are not told otherwise.
The present value (PV2) becomes $847,893.56. This means that Raj will need
$847,893.56 in his account when he retires in order to withdraw $4,000 per month
for 30 years.
The present value from the second annuity will become the future value for the first
annuity but we change its sign
Enter FV1 as negative. Ie: PV2 = −FV1. This is because FV1 is will be considered as
the final withdrawal when ending annuity1.
We can now calculate the size of Raj’s monthly deposits (PMT1). Let us fill in the BAII Plus
table for Part 1:
B/E P/Y C/Y N1 I/Y PV1 PMT1 FV1
BGN 12 1 40×12=480 6 +1,000 CPT +436.95 −847,893.56
Because the deposits start today, BGN is on for part 1.
Again, Raj makes monthly payments and the interest is compounded annually, so
P/Y = 12 and C/Y = 1.
Because Raj makes monthly deposits for 40 years, N1 = 40×12 = 480.
Finally, be careful of the signs for PV and FV. They must be opposite in sign.
We will make PV1 positive (as it is an initial deposit) and FV1 negative (we treat it
as a withdrawal at the end).
Conclusion: Raj needs to deposit $436.95 per month for the next 40 years to achieve his
retirement goals.
INTEREST EARNED ON BACK-TO-BACK ANNUITIES
Again we use the usual interest formula:
We need to be careful when calculating money in and money out for deferred annuities.
All deposits are considered money in ($ IN).
All withdrawals are both money out ($ OUT).
Do not include FV1 nor PV2 in the $ IN or $ OUT calculations.
276 Amy Goldlist
Because FV1 does not get withdrawn but instead becomes the starting balance for
the annuity (PV2), it is not considered money out.
Similarly, because PV2 does not get deposited but instead is actually the ending
balance from annuity1 (FV1), it is not considered money in.
PV1 Interest1 PMT1 FV1 PV2 Interest2 PMT2 FV2
Initial
Deposit
+ %
Earned
+
Regular
Deposits
=Ending
Balance
Part 1
=Starting
Balance
Part 2
+ %
Earned
= Regular
Withdrawals
+ Final
Withdrawal
$ IN $ IN $ IN –– –– $ IN $ OUT $ OUT
This gives us the following equation for interest earned:
Example 5.6.2
How much interest will Raj earn over the 70 years that his money is invested in Example 1a?
The Money Out, in this case, is the amount that Raj withdraws during his retirement:
The Money In is the amount Raj deposits into the account:
Now take the difference between the money out and the money in (notice that neither FV1 nor
PV2 are included in the $ OUT nor $ IN calculations):
Conclusion: Raj will earn $1,229,264 in interest over the 70 years that his money is invested!
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SWITCHING FROM BGN TO END
Let us examine Raj’s retirement example (Example 1a) once again. In Example 1a, BGN
was turned on for both annuity1 and annuity2. This was because Raj’s first deposit was made
immediately (at the start of annuity1) and his first withdrawal was made exactly on this 65th
birthday (at the start of annuity2). Let us now look at an example where BGN is turned off (ie:
the calculator is set to END).
Example 5.6.3
What would change if Raj withdrew his first retirement payment of $4,000 (PMT2) two months
after his last deposit (last PMT1)? How much would Raj need to deposit into the retirement
fund each month (PMT1) in this case?
Because Raj made his deposits into the saving account (annuity1) at the beginning of each
month then the last deposit would go into the account one month before his 65th birthday.
His first withdrawal occurs two months after this last deposit. That means the first withdrawal
occurs one month after his 65th birthday, which would be the end of the first payment interval
for annuity2. That means we set BGN to off (END) for annuity2. Let us look at the new
timeline for this question:
Notice that turning BGN ‘off’ (setting the calculator to END) will change the value of PV2 for
annuity2. Let us start by re-calculating the value of PV2:
B/E P/Y C/Y N2 I/Y PV2 PMT2 FV2
END 12 1 30×12=360 4 CPT +845,126.83 −4,000 0
Then, let’s use the value for PV2 in FV1 but make FV1 negative. Don’t forget to turn BGN on
for this calculation:
B/E P/Y C/Y N1 I/Y PV1 PMT1 FV1
BGN 12 1 40×12=480 6 +1,000 CPT +435.50 −845,126.83
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Conclusion: Raj will need to deposit $435.50 per month into his retirement fund.
Check Your Knowledge for Example 1c
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LUMP SUM DEPOSITS
It is possible to either deposit or withdrawal money between annuities. Let us again examine
Raj’s retirement example and see what happens if Raj deposits additional money into his
retirement savings account when he turns 65.
Example 5.6.4
Raj anticipates downsizing (selling his house and buying a smaller property) when he turns 65.
He thinks he can deposit $100,000 from the sale of his property into his retirement savings plan
when he turns 65. How much are Raj’s new monthly deposits into his savings plan (annuity 1)
with this extra deposit into the retirement fund? Use the values from part c) and add the extra
$100,000 deposit when Raj turns 65.
Let us first look at the timeline for this question:
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Let us next examine the BAII Plus Table for Annuity2.
Annuity 2 (Regular Withdrawals):
B/E P/Y C/Y N2 I/Y PV2 PMT2 FV2
END 12 1 30×12=360 4 CPT +845,126.83 −4,000 0
Notice that nothing changes for Annuity2 between examples 1c and 1d (PV2 does not change).
Where example 1d differs is in Annuity1.
All together, Raj needs $845,126.83 saved up at the start of annuity2 (PV2). He receives an
extra $100,000 from the sale of his property at that time. This means he only needs to save
$745,126.83 during Annuity1. This will be the value of FV1:
We can see, above, that FV1 must equal $745,126.83. Let us write out the formal equation to
solve for FV1when there is a lump-sum deposit between annuities:
Now that we know FV1, we can calculate the new value for PMT1.
Annuity 1 (Regular Deposits)
B/E P/Y C/Y N1 I/Y PV1 PMT1 FV1
BGN 12 1 40×12=480 6 +1,000 CPT +383.34 −745,126.83
Conclusion: Raj will need to deposit $383.34 per month into his retirement fund.
Check Your Knowledge for Example 1d
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INTEREST EARNED WITH LUMP SUM DEPOSITS
When calculating interest earned on back-to-back annuity problems with lump-sum payments,
there is an additional value we include in the money in ($ IN) — the lump sum deposit:
The money out ($ OUT) does not change. Taking the difference between the money out and
money in gives the following equation for interest earned on back-to-back annuities with lump-
sum deposits:
Let us now look at Raj’s retirement example again to see an example of this calculation.
Example 5.6.5
How much interest will Raj earn in over the 70 years of his retirement investment in Example
1d?
The $ IN will now include the additional $100,000 deposit as well as the initial deposit (PV1)
and the regular monthly deposits (PMT1) into the savings account (annuity1):
The $ OUT will not change:
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Taking the difference between $ OUT and $ IN gives:
Conclusion: Raj will earn $1,154,996.80 in interest over the 70 years that his money is
invested.
Check Your Knowledge for Example 1e
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REGISTERED EDUCATION SAVINGS PLAN (RESP)
It is common when saving for a child’s education to save up by making regular deposits into
an RESP (registered education savings plan). There is a grant called the Canada Education
Savings Grant (CESG) that adds 20% to these regular deposits (up to a maximum of $500 per
year). Let us examine how this grant works for Sofia’s RESP in the example below.
Example 5.6.6
Dmitry and Elena just had a baby girl, Sofia. They set up an RESP for Sofia. Given below are
the terms:
They deposit $625 every quarter into the RESP
The first deposit occurs 3 months after Sofia is born
The last deposit occurs on Sofia’s 18th birthday
The deposits receive the CESG grant — 20% is added to each deposit by the
government
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Sofia makes her first of 8 semi-annual withdrawals on her 18th birthday to attend
university
The education fund earns 4.25% compounded monthly for the entire time
What is the size of Sofia’s semi-annual withdrawals?
Before we determine what to enter on the time diagram & BAII Plus, let’s ask a few important
questions:
Key Questions to Get Started on Example 2
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Let us gather all of this information into a timeline:
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Always start where the known payment (PMT) is. We know that $750 per quarter will be
deposited into the RESP (= PMT1). Therefore, “start” with Part 1.
Annuity 1 (Regular Deposits):
B/E P/Y C/Y N1 I/Y PV1 PMT1 FV1
END 4 12 18×4=72 4.25 0 +750 CPT −80,614.76
Enter the value for FV1 into PV2. Don’t forget to make PV2 positive. We can now calculate
PMT2:
Annuity 2 (Regular Withdrawals):
B/E P/Y C/Y N2 I/Y PV2 PMT2 FV2
BGN 2 12 4×2=8 4.25 +80,614.76 CPT −10,840.65 0
Conclusion: Sofia can withdraw $10,840.65 every 6 months to pay for university 1.
LUMP SUM WITHDRAWALS
Let us now examine the scenario where a lump-sum withdrawal occurs between annuities.
We will see what happens if Sofia makes a lump-sum withdrawal on her 18th birthday in the
example below:
Example 5.6.7
Redo Example 2 but add the following: on Sofia’s 18th birthday, she purchases a car for
$6,000. Because she will be using the car to get to and from university, she is able to withdraw
$6,000 from her RESP fund. What will be the size of her new semi-annual withdrawals if
everything else remains the same on Sofia’s RESP account?
With the new $6,000 withdrawal when Sofia turns 18, the timeline will now become:
The value of FV1 will remain the same:
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Annuity 1 (Regular Deposits):
B/E P/Y C/Y N1 I/Y PV1 PMT1 FV1
END 4 12 18×4=72 4.25 0 +750 CPT −80,614.76
Now, let’s use the value of FV1 to calculate PV2.
We can now enter $74,614.76 in for PV2 and calculate the new value for PMT2.
Annuity 2 (Regular Withdrawals):
B/E P/Y C/Y N2 I/Y PV2 PMT2 FV2
BGN 2 12 4×2=8 4.25 +74,614.76 CPT −10,033.80 0
Conclusion: Sofia can now withdraw $10,033.80 every 6 months from her RESP fund.
INTEREST EARNED WITH LUMP SUM WITHDRAWALS
When calculating interest earned on back-to-back annuity problems with lump-sum
withdrawals, there is an additional value we include in the money out ($ OUT) — the lump
sum withdrawal:
The money in ($ IN) does not change. Taking the difference between the money out and money
in gives the following equation for interest earned:
Let us now look at Sofia’s RESP example again to see an example of this calculation.
Example 5.6.8
How much interest does Sofia’s RESP account earn in Example 2b over the 22 years?
Let us first calculate the Money In to the RESP account over the 22 years:
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The Money Out will include the $6,000 withdrawal:
Taking the difference between the Money Out and In gives:
Sofia’s RESP will earn $32,270.40 in interest over the 22 years that the money is invested.
FINAL WITHDRAWALS
Let us finish this (very long) section with one final topic — money left over at the end. If there
is any money over at the end of a back-to-back annuity, this amount will be entered into FV2
and we make it negative (it is treated as a final withdrawal). Let us look again at Sofia’s RESP
example and assume Sofia wants money left over at the end.
Example 5.6.9
Redo Example 2b but assume Sofia wants $30,000 left in her RESP after she graduates from
university to help pay for her MBA. Calculate the size of her semi-annual withdrawals in this
case.
With the $30,000 left at the end, the timeline now becomes:
All of Annuity1 will remain the same as well as PV2 (the same as in Example 2b). We will start
our calculations with Annuity2. Almost all of the values are the same except FV2: enter the
ending balance amount ($30,000) into FV2 and make it negative, Then calculate Sofia’s new
withdrawal size (PMT2):
Annuity 2 (Regular Withdrawals):
B/E P/Y C/Y N2 I/Y PV2 PMT2 FV2
BGN 2 12 4×2=8 4.25 +74,614.76 CPT −6,629.23 −30,000
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Conclusion: Sofia can withdraw $6,629.33 every 6 months and still have $30,000 left over at
the end.
YOUR OWN NOTES
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THE FOOTNOTES
Notes
1. The current maximum allowable withdrawal amount on RESP's is $5,000 in the first
quarter while attending post-secondary. Let's assume that this maximum allowable
amount will increase in 18 years once Sofia attends university.
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5.7 Bonds
Learning Outcomes
Calculate the coupon payment size, market value or gain/loss for bonds.
If a corporation or government is looking to raise money (capital), they can issue bonds. The
issuer (the company or government) must pay the bond holder (owner of the bond) a series
of equal-sized regular interest payments for a fixed period of time. The size of these payments
is determined by the agreed-upon interest rate (coupon rate). At the end of the fixed period
of time, or maturity date, the issuer must repay the bond holder the principal (the amount of
money the bond issuer borrowed).
We consider the bond is a debt owed by the issuer to the bond holder. The amount owed never
increases because the issuer pays the interest owed each period to the holder in the form of a
coupon payment. This is why the final amount owed by the issuer to the bond holder (the
face value) will be the principal (amount borrowed).
There are several different types of calculations for bonds — see the sections below for the key
formulas, tips and examples related to bond calculations.
CALCULATING COUPON PAYMENTS FOR BONDS
The coupon payment (PMT) is the interest earned in one period (6 months):
where the principal is the initial purchase price of the bond (when the bond is purchased from
the issuer) and i is the periodic rate. The periodic rate is the interest rate for one period.
Example 5.7.1
Amir purchased a $3,000 bond that has a 10-year term (will mature in 10 years). The bond has
a coupon rate of 5%, compounded semi-annually. What is the size of the semi-annual coupon
payment?
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289
Check your Knowledge for Example 1
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CALCULATING THE FAIR MARKET VALUE OF A BOND
Often, the bond holder can sell the bond (transfer the bond) to a secondary bond holder
before the end of the term. Because interest rates change throughout the term of the bond,
the bond can be worth different amounts at different times throughout its term. The amount
the purchaser (secondary bond holder) is willing to pay is called the fair market value of the
bond. The fair market value is determined by the current interest rate, the size of the coupon
payment, the number of years remaining in the term (number of years before the maturity date)
and the face value of the bond.
We will use the BAII Plus to calculate the fair market value of a bond:
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PV Interest PMT FV
Fair Market
Value
+ % Market
Rate
= Coupon
Payments + Face Value
− − + +
Bonds are the ONLY type of investment in this text where we will have a negative value for PV
and a positive value for FV. To make sense of this, imagine the bond from the perspective of
the bond holder. The bond holder initially ‘lends’ money to the bond issuer when purchasing
a bond from the issuer. We consider this amount lent (PV) to be negative because this is the
amount the bond holder must outlay to purchase the bond.
Each period, the issuer will owe the bond holder interest on that loan. The issuer pays this
interest (coupon payment) to the holder each period. The issuer also repays the face value of
the bond (FV) at the end to the holder1. For this reason, we consider PMT and FV as positive
because they are money that the bond holder receives.
Example 5.7.2
Francis purchases a $4,000 bond that has a 30-year term. The bond has a coupon rate of 5.25%,
compounded semi-annually and a semi-annual coupon payment of $65. If Francis chooses to
sell the bond after 10 years when the current interest rate is 3%, compounded semi-annually,
what is the fair market value of the bond?
Check your Knowledge for Example 2
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Conclusion: Francis can sell the bond for $4,149.58 in 10 years.
CALCULATING THE GAIN OR LOSS WHEN SELLING A BOND
If the interest rate (coupon rate) on a bond is above the market rate then the bond increases in
value (sells at a premium). If the interest rate (coupon rate) on a bond is below the market
rate, then the bond goes down in value (sells at a discount). The gain (or loss) is calculated by:
Let us use this formula to determine Francis’ gain or loss in Example 2.
Example 5.7.3: Selling at a Premium
What is Francis’ gain or loss on the sale of his bond in Example 2?
Check your Knowledge for Example 2b
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Conclusion: Francis will make a $149.58 gain when he sells the bond in 10 years.
Example 5.7.4 — Selling at a Discount
Redo Example 2 with an interest rate (market rate) of 6% when Francis chooses to sell the bond
in 10 years. What is Francis’ gain or loss on the sale of the bond?
Check your Knowledge for Example 2c
First, we need to calculate the fair market value of the bond in 10 years:
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The fair market value of the bond will be $2,728.69 when Francis sells the bond in 10
years. We can use this value to now calculate Francis’ gain or loss:
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Conclusion: Francis will lose $1,271.31 on the sale of the bond.
KEY TAKEAWAYS FOR BONDS
Key Takeaways for Bonds
Coupon Payment (PMT) = PV × i = Principal × Periodic Rate
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Periodic Rate = Nominal Rate ÷ 2 (Bonds are always semi-annual)
For bonds, PV will be negative and PMT and FV are positive.
Bonds are the ONLY type of investment in this text where PV will be
negative.
Bonds are the ONLY type of investment in this text where FV will be
positive.
If the market rate rises above the coupon rate, the bond decreases in value.
If the market rate drops below the coupon rate, the bond increases in value.
The gain or loss on the sale of bonds is given by:
YOUR OWN NOTES
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THE FOOTNOTES
Notes
1. Information thanks to https://www.investopedia.com/terms/b/bond.asp ;
https://en.wikipedia.org/wiki/Bond_(finance) and https://www.investopedia.com/
articles/investing/062813/why-companies-issue-bonds.asp
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5.8 Perpetuities
Learning Outcomes
Calculate the payment sizes or present values for regular and deferred perpetuities.
A perpetuity is like a bond, but with no fixed term (no fixed maturity date). If a corporation
issues a perpetuity to an investor, the perpetuity will continue making payments to this investor
indefinitely1. Common examples of perpetuities are scholarship funds, perpetual trusts and
war bonds, like the British Treasury Bonds issued for World War 12. Finally, some business
investments can also be treated like perpetuities3.
There are several different types of perpetuities — see the sections below for the key formulas,
tips and examples related to perpetuity calculations.
PAYMENTS FOR ORDINARY PERPETUITIES
If the payments for a perpetuity are withdrawn at the end of each interval, we call this an
ordinary perpetuity. The payment (PMT) for an ordinary perpetuity can be given by the
following formula:
where PV is the initial value of the perpetuity and i is the periodic rate (the rate per period).
We can also use the BAII Plus to calculate the PMT. There are a few ‘tricks’ to know when
using the BAII Plus:
B/E = “END” for ordinary perpetuities
N = 1000 × P/Y. Use total # years = 1,000 for perpetuities4.
PV = the initial balance of the perpetuity.
FV = 0. The funds are never withdrawn. The amount withdrawn equals 0 for this
reason.
CPT PMT. Calculate the value of the payment (PMT).
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Example 5.8.1
Sasha, a wealthy BCIT Alumnus, just donated $100,000 to create a scholarship at BCIT for
first year business students. The scholarship will be awarded semi-annually and the fund will
earn 5% compounded semi-annually. The first scholarship will be awarded in 6 months. What
will be the size of the semi-annual scholarships awarded?
Using the formula:
Using the BAII Plus:
B/E P/Y C/Y N I/Y PV PMT FV
END 2 2 1000×2=2000 5 +100,000 CPT −2,500 0
Conclusion: BCIT can pay out a scholarship of $2,500 every 6 months.
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THE PRESENT VALUE FOR ORDINARY PERPETUITIES
There are also two ways to calculate the present value of an ordinary annuity. We can use the
following formula:
or we can, again, use the BAII Plus and use the same tricks as when calculating the payment
(PMT).
Let us look again at Sasha’s example but instead, let us figure out the amount Sasha needs to
donate.
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Example 5.8.2
How much does Sasha (the wealthy BCIT alumnus) need to donate if he wants BCIT to give
out $3,000 semi-annual scholarships with the first scholarship awarded in 6 months. Assume
the fund still earns 5% compounded semi-annually.
Using the formula:
Using the BAII Plus:
B/E P/Y C/Y N I/Y PV PMT FV
END 2 2 1000×2=2000 5 CPT +120,000 −3,000 0
Conclusion: Sasha will need to donate $120,000 to the scholarship fund at BCIT.
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THE PRESENT VALUE FOR PERPETUITUES DUE
It is possible to have a perpetuity that gives out the first payment immediately. We call this
a perpetuity due. An example of this is a scholarship fund where the first scholarship is
awarded as soon as the fund has been created. 5
If payments are withdrawn from a perpetuity fund at the start of each payment interval
(perpetuity due), there will need to be slightly more in the perpetuity fund to account for the
initial balance in the account dropping in value right at the start of the perpetuity. That “slightly
more” amount will be the value of the payment6:
Let us now revisit Sasha’s scholarship example and determine the size of Sasha’s donation if
the scholarship fund is a perpetuity due.
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Example 5.8.3
Suppose that Sasha would like to donate enough money such that the semi-annual scholarships
are still $3,000 but the first scholarship will be awarded immediately. Assume the scholarship
fund still earns 5% compounded semi-annually. How much more will Sasha need to donate?
Using the formula and i = 0.025 and PMT = 3,000 gives:
Using the BAII Calculator gives:
B/E P/Y C/Y N I/Y PV PMT FV
BGN 2 2 1000×2=2000 5 CPT +123,000 −3,000 0
Sasha will need to donate $123,000 if the first scholarship is awarded immediately. Taking
the difference between the required donation amount donated for Example 2 (an ordinary
perpetuity) and Example 3 (a perpetuity due) gives:
Conclusion: Sasha will need to donate $3,000 more if the first scholarship is awarded today
instead of 6 months from now. It is no coincidence that the difference is equal to the payment
size.
THE PRESENT VALUE FOR DEFERRED PERPETUITUES
It is possible to defer the payments received from a perpetuity. A first example is if a non-profit
or charity receives a donation to cover future costs for the organization7. Another example of
a perpetuity is a business decision where there is an initial amount invested to start the business
(PV1) and then it takes several years for the business to start earning income8.
When calculating payment sizes for deferred perpetuities or the initial amount needed to be
deposited at the start of the perpetuity, it is important to remember that there are actually two
parts to the deferred perpetuity problem:
Part 1: The initial balance gathers interest. There are no payments nor withdrawals in part 1
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(the deferral period). The only money being added to the balance is the interest being earned
(or charged). This problem is a compound interest problem (Chapter 4):
Part 2: Payments are now being withdrawn. These payments exactly equal to the interest
earned on the current balance in the perpetuity account. The starting balance for the perpetuity
(PV2) equals to the ending balance from the deferral period (FV1) .
Example 5.8.4
What if Sasha’s scholarship fund (from Examples 1 to 3) doesn’t give out its first scholarship
for one year? How much would Sasha need to donate if the semi-annual scholarships are still
$3,000 and the fund still earns 5% compounded semi-annually?
Let us first draw out the timeline for this problem:
Because we know the payments for part 2, start there. We can either use the formula or the
BAII Plus to do the calculations for this problem. Let us start by using the BAII Plus:
Part 2: Perpetuity (using the BAII Plus)
Let us ‘start’ part 2 exactly when the first scholarship is awarded. For this reason, B/E will be
set to BGN:
B/E P/Y C/Y N2 I/Y PV2 PMT2 FV2
BGN 2 2 1000×2=2000 5 CPT +123,000 −3,000 0
Notice that PV2 equals to the amount Sasha would need to donate in Example 3. We will now
enter that value into FV1 and calculate the amount donated initially (PV1).
Part 1: Deferral Period (using the BAII Plus)
There are no payments nor withdrawals for part 1 so enter ‘––’ for B/E. Also, because the
deferral period lasts for 1 year, N1 =1×2 = 2.
B/E P/Y C/Y N1 I/Y PV1 PMT1 FV1
–– 2 2 1×2=2 5 CPT +117,073.17 0 −123,000
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Conclusion: Sasha will only need to donate $117,073.17 if the first scholarship is awarded in
one year.
Method 2 — Using Formulas:
We will again start with part 2. Let us rewrite the timeline, slightly differently, however:
Notice that we will only run the deferral period for 6 months and start the perpetuity (part 2) 6
months before the first scholarship is withdrawn. This makes the calculation easier for part 2:
Part 2: Perpetuity (using Formulas)
We now have ‘started’ part 2 one payment period earlier. This now makes the perpetuity in part
2 an ordinary annuity. We use the following formula to calculate its present value:
There will need to be $120,000 in the scholarship fund in 6 months. We can find the present
value of this amount using the compound interest formula from Chapter 4.
Part 1: Deferral Period (using Formulas)
Conclusion: Again, we find that Sasha will need to donate $117,073.17 now if the first
scholarship is awarded in 1 year.
PAYMENTS FOR DEFERRED PERPETUITIES
It is possible to use formulas to calculate the payment size for deferred perpetuities. We will
however, just use the BAII Plus method for this section.
Let us examine a similar donation example. In this example, Ezra, a wealthy philanthropist will
make a large donation to help out a medical center that will not open for several years.
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Example 5.8.5
Ezra, a wealthy philanthropist, donates $1,000,000 to the Moshi Medical Diagnostics Center,
located in Moshi, Tanzania. The Center is due to open in 3 years. The Center will use the
donated funds to cover their annual equipment and testing costs. They will invest the funds at
4.25%, effective, into a perpetuity. They will withdraw the first perpetuity payment in exactly
3 years when the center opens. Calculate the size of the annual withdrawals.
Let us first draw out a timeline for this problem:
Next, determine which part to start with. In this case, because we know the initial deposit
(PV1), start with part 1 (the deferral period).
Part 1 (the deferral period):
B/E P/Y C/Y N1 I/Y PV1 PMT1 FV1
–– 1 1 3×1=3 4.25 +1,000,000 0 CPT −1,132,995.52
It does not matter what we enter in B/E above because there are no payments. Let us now enter
the ending value for the deferral period (FV1) into the starting value for the perpetuity (PV2).
Part 2 (the perpetuity):
B/E P/Y C/Y N2 I/Y PV2 PMT2 FV1
BGN 1 1 1000×1=1000 4.25 +1,132,995.52 CPT −46,189.27 0
Notice that BGN is on in part 2. This is because we started part 2 (the perpetuity) right when
the first payment was withdrawn.
Conclusion: The Moshi Medical Center will receive $46,189.27 annually to cover equipment
and testing costs starting in 3 years when the center opens.
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KEY TAKEAWAYS FOR PERPETUITIES
Key Takeaways for Perpetuities
When entering perpetuities into the BAII Plus:
FV is always equal to 0 for perpetuities.
We always use 1,000 years for perpetuities.
This gives N = 1000 × P/Y
It does not matter what sign we use for PV or PMT (as we are computing
the other one)9
When using formulas:
For ordinary perpetuities,
For perpetuities due,
For deferred perpetuities:
Run the deferral period until the first payment occurs
Turn BGN on for part 2
Set PV2 = FV1
Set FV2 = 0.
YOUR OWN NOTES
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to copy and paste below?
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304 Amy Goldlist
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THE FOOTNOTES
Notes
1. Information thanks to https://languages.oup.com/google-dictionary-en/ ;
https://www.investopedia.com/terms/p/perpetuity.asp ; https://en.wikipedia.org/
wiki/War_bond; https://www.advisor.ca/tax/estate-planning/design-a-dynasty-with-
perpetual-trusts/; https://www.investopedia.com/terms/t/trust-fund.asp;
https://thephilanthropist.ca/original-pdfs/Philanthropist-21-3-370.pdf;
2. Historically, governments issued bonds to raise money to meet the growing costs of
war. Some of these bonds lasted in perpetuity.
3. Suppose a company invests a certain initial amount of money with the hope/
expectation that they will earn a return on their investment in the form of regular
income from the business (possibly for an indefinite amount of time). If we
assume the returns last for an indefinite amount of time, we treat this problem like
a perpetuity as well.
4. A perpetuity, technically, lasts forever. In the calculator, the closest to 'forever' we
should enter for a number of years is 1,000. This number of years will work with
all types of calculations in the TVM keys.
5. The same principle applies as in ordinary perpetuities, the balance in the scholarship
fund never increases nor decreases in value. This is because after a payment has
been withdrawn from the account, interest is earned on the remaining balance in
the scholarship fund. The payments are calculated such that the after the interest
has been earned, the balance in the account increases back to its original value
(PV).
6. We, again, need to be careful if the number of payments per year (P/Y) is not equal to
the number of compounding periods per year (C/Y). If that is the case, we need to
calculate the equivalent interest rate with the number of compounding periods
equal to the number of payment intervals for the perpetuity.
7. They deposit the money once it’s received but do not need to start withdrawing from
the account until the non-profit opens. If we assume that the non-profit will remain
open indefinitely, then we assume that they will need withdrawals to cover their
costs for an undetermined amount of time as well. We treat this problem as a
perpetuity.
8. In that case, the repayments of the initial investment are delayed (deferred) for a
certain number of years. If we assume that there is no fixed end date to the
Business Mathematics 305
business, we treat this problem as a perpetuity where the returns continue on
indefinitely.
9. If ever both were being entered into the BAII plus - we would need to use opposite
signs for PV and PMT
306 Amy Goldlist
5.9 Preferred Shares
Learning Outcomes
Calculate the fair market value and gain (or loss) when purchasing preferred shares.
When a company is looking to raise funds, they can issue preferred shares. A preferred share
has no maturity date (no expiration date or fixed term) but will stop regular payments if the
company stops making a profit or goes out of business. Because we usually have no way of
knowing when a business will end or stop making profits, we will treat preferred shares like
perpetuities and assume the profits (and therefore regular payments) will continue indefinitely
(forever).
Let us first understand some of the terminology:
Dividend: The regular payment (PMT) paid by the issuer of the preferred share.
The value of this dividend remains fixed and does not change through the lifetime
of the share.
Price: The selling price of the share (PV). This is the fair market value (ie: what
you are willing to pay for the share).
Market Rate: The current interest rate (I/Y).
See the sections below for the key formulas, tips and examples related to calculating values for
preferred shares.
CALCULATING FAIR MARKET VALUES USING THE FORMULAS
To calculate any values for preferred shares, you can use the same formulas as for perpetuities.
If the dividend is awarded at the end of the each payment interval, and if you would like to
know the fair market value of the share, use the following formula:
If you would like to know the fair market value of the preferred share and the dividends are
awarded at the start of each payment interval, use the following formula:
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307
Let us now look at Kiana’s purchase of preferred shares and determine how much she should
be willing to pay to purchase the preferred shares.
Example 5.9.1
Kiana purchases 100 Koodo preferred shares. Koodo pays a dividend of $0.80 every quarter on
their preferred shares. The next dividend will be paid out in 3 months. If Kiana wants to earn
a minimum of 5% compounded quarterly on her investment, how much should she be willing
to pay for the 100 shares?
Let us first calculate the periodic rate (i) using the nominal rate (j4 = 2.75%):
Because the first payment will be paid out at the end of the first interval (in 3 months), we use
the following formula to calculate PV (the price per share):
Inputting i = 0.0125 and PMT = $0.80 into the formula gives the following price per share:
Kiana should be willing to pay $64 per share. To calculate the price she should pay for all 100
shares, multiply by the 100 shares:
Conclusion: Kiana should pay $6,400 for the 100 Koodo preferred shares.
Example 5.9.2
Let us again look at Kiana’s purchase of 100 Koodo preferred shares (from Example 1). What
if the first dividend were paid immediately instead of in 3 months? How much should Kiana
be willing to pay for the 100 shares?
Because the first payment will be paid out immediately, we use the following formula to
calculate PV (the price per share):
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Inputting i = 0.0125 and PMT = $0.80 into the formula gives:
Kiana should be willing to pay $64.80 per share. To calculate the price she should pay for all
100 shares, multiply by the 100 shares:
Conclusion: Kiana should pay $6,480 for the 100 Koodo preferred shares.
CALCULATING FAIR MARKET VALUES USING THE BAII PLUS
We can also use your BAII Plus calculator to calculate the fair market value of preferred shares.
We use the same ‘tricks’ as for perpetuities:
N = 1000×P/Y. Use 1000 years when calculating the number of payments (N)1.
I/Y = Nominal Interest Rate (the current interest rate).
FV = 0. You can never take the principal out. You only ever get paid the regular
dividends.
PMT = Regular Dividend Amount. We will enter this amount as positive because it
is the amount the share holder receives2.
CPT PV. Calculate the fair market value of the share (PV).
Example 5.9.3
Calculate Kiana’s price per share for shares from Example 1 using the BAII Plus:
B/E P/Y C/Y N I/Y PV PMT FV
END 4 4 1000×4=4000 5 CPT −64 +0.80 0
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text. You can view it online here:
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Conclusion: Kiana should pay $64 per share the Koodo preferred shares (the same prices as in
Example 1).
Example 5.9.4
Calculate Kiana’s price per share for shares from Example 2 using the BAII Plus:
B/E P/Y C/Y N I/Y PV PMT FV
BGN 4 4 1000×4=4000 5 CPT −64.80 +0.80 0
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Conclusion: Kiana should pay $64.80 per share the Koodo preferred shares (the same prices
as in Example 2).
CALCULATING THE GAIN OR LOSS FROM PREFERRED SHARES
We calculate the gain or loss from the sale of preferred ways in the same way as for bonds:
When the selling price is higher than the purchase price, we gain money on the sale of the
shares. When the selling price is lower than the purchase price, we loose money on the sale.
Example 5.9.5
Kiana paid $64.00 per share when she bought 100 Koodo shares. Kiana would like the sell the
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shares several years later when interest rates have risen to 6.4%. Assume the next dividend of
$0.80 will be paid in 3 months. Calculate Kiana’s gain (or loss) on the sale of the 100 shares.
Check Your Knowledge for Example 5.9.5
We first need to calculate the fair market value of the shares. Which value do we
calculate do this?
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Which formula could we use to calculate the fair market value for this question?
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Determine what to enter into the BAII Plus to calculate this value:
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Business Mathematics 311
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Kiana can sell each share for $50 when she goes to sell them several years later. We
can use this price to calculate the gain or loss on Kiana’s 100 shares:
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An interactive H5P element has been excluded from this version
of the text. You can view it online here:
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Conclusion: Kiana will lose $1,400 on the sale of her 100 Koodo shares.
KEY TAKEAWAYS FOR PREFERRED SHARES
Key Takeways for Preferred Shares
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Calculating the price per share in the BAII Plus:
FV = 0. FV is always equal to 0.
N = 1000 × P/Y. Always use N = 1,000 years × P/Y.
PMT = Dividend value. Enter PMT as positive3.
CPT PV. Fair Market Value = PV. 4
Price per Share formulas:
Payments at end of interval:
Payments at beginning of interval:
Gain or loss per share:
Total gain or loss:
YOUR OWN NOTES
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THE FOOTNOTES
Business Mathematics 313
Notes
1. Again, this is because the payments continue on forever and 1000 years is the closest
to forever that we can input in the BAII Plus
2. It does not actually matter what sign we input for PMT because we will be
calculating PV each time. If ever we were inputting both PV and PMT into the
BAII Plus, we would need to make sure they are opposite in sign.
3. It does not matter what sign we use for PMT (as we are computing the PV).
4. If ever both were being entered into the BAII plus - we should enter PMT as positive
and PV as negative or at least use opposite signs for PV and PMT.
314 Amy Goldlist
5.10 Mortgages
Learning Outcomes
Calculate the payment size, balance, principal repaid or interest charged during a
mortgage and understand how to renew a mortgage.
It is often necessary to take out a mortgage when purchasing a home or
property. A mortgage is usually a long-term (roughly 25 year), general (P/Y
≠ C/Y), ordinary annuity (PMTs at END of interval). Buyers can choose a
fixed or variable rate mortgage. A fixed rate mortgage means that the
interest rate charged remains fixed for the mortgage term. A variable rate
mortgage means that the interest rate varies throughout the mortgage term.
In general, the buyer (mortgage holder) is required to pay down some of the
balance owing (principal) on the mortgage in addition to the interest charges
each interval.
HISTORY OF HOW MORTGAGES EVOLVED OVER THE YEARS
In the 1900’s, home buyers who took out a mortgage only paid the interest owed each month.
The buyer would save up while making these interest-only payments and fully repay the
mortgage when they had enough saved.1
Major world events (like the first world war & great depression) changed this practice. Large
numbers of people were unable to pay their mortgages. Because of this, buyers were then
required to pay some of the balance owing (principal) each payment interval in addition to
the interest owed each month. To further protect lenders, in 1946, the “Canada Mortgage
and Housing Corporation(CMHC)” was created. The CMHC insures buyers’ mortgages if the
buyer puts down less than a minimum % down payment.2 This protects lenders if a buyer
defaults on their mortgage (can’t pay their mortgage).
Before the 1970/80’s, buyers would be guaranteed a fixed interest rate for the entire duration
of their mortgage. This changed after the interest rate inflation in the 1970’s and 80’s (interest
rates hit a record high of 21.46%). Banks who had previously locked in interest rates with
buyers were missing out on thousands of potential dollars in interest when buyer were locked
in at rates as low as 6.9% and when the current interest rate was 21.46%. After the interest rate
inflation of the 1970’s and 1980’s, banks created “mortgage terms.”
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315
MORTGAGE TERMS
A mortgage term is the length of time your mortgage agreement and interest rate will be in
effect.3 Mortgage terms are, most often, 5 years in length but can vary anywhere from 6
months to 10 years in length. If the buyer chooses a fixed-rate mortgage, they are guaranteed
a fixed interest rate for the duration of the mortgage term. After the term is ‘up’ (after the time
period defined by the term has passed), the buyer negotiates a new interest rate with the lender
(bank). The buyer renews their mortgage.
Basically, a new mortgage is drawn up at the end of each term when the buyer renews their
mortgage. The buyer can pay off part (or all) of the balance owing with a lump-sum payment.
The buyer can move their mortgage to another bank. The buyer is charged interest at the new
interest rate and pays this interest on the current balance (principal) owing on the mortgage.
If each term is 5 years in length, a buyer will renew their mortgage roughly five times on a
25-year mortgage. The full length of the mortgage (ex: 25 years) is the amortization period.
CONSTRUCTING AMORTIZATION TABLES
Amortization is “the process of paying off debt over time with regular installments of interest
and principal sufficient to repay the loan in full by its maturity date.4”
We can create an amortization table (or schedule) to show the amount of principal and interest
that make up each payment. The table also shows the balance owing after each payment. The
table can run until the loan (or mortgage) if fully paid off or to the end of the term.
Let us now look at an example of an amortization schedule for Kerry — she just bought a car
and borrowed some money from her line of credit to do so.
Example 5.10.1
Kerry purchases a new Toyota Rav4. She borrows $16,000 from her line of credit for the
purchase. She is charged 3% compounded monthly on the line of credit. Kerry wants to repay
her line of credit with 6 monthly payments. The first payment will be in one month. Construct
an amortization schedule and use it to answer how much interest she will pay in total and what
the size of her final payment will be. Assume Kerry owed nothing on her line of credit before
she borrowed the $16,000.
In order to construct the amortization schedule, we need to determine the size of Kerry’s
monthly payments:
B/E P/Y C/Y N I/Y PV PMT FV
END 12 12 6 3 +16,000 CPT −2,690.05 0
316 Amy Goldlist
Let’s round Kerry’s payment up to the next dollar5 which means that Kerry will pay $2,691 per
month.
Because we round up Kerry’s payments, this means that she will overpay by roughly $0.95 per
month. This makes her final payment slightly smaller. Let us now construct the amortization
schedule to determine the size of this final payment as well as the interest paid and balance
outstanding each month.
Line of Credit Amortization Table
Payment
#
Payment
(PMT) Interest Paid (INT) Principal Repaid (PRN) Ending Balance (BAL)
1 $2,691 $16,000×0.0025 =$40 $2,691−$40 =$2,651 $16,000−$2,651 = $13,349
2 $2,691 $13,349×0.0025 =$33.37 $2,691−$33.37 =$2,657.63 $13,349−$2,657.63 =
$10,691.37
3 $2,691 $10,691.37×0.0025
=$26.73 $2,691−$26.73 =$2,664.27 $10,691.37−$2,664.27
=$8,027.10
4 $2,691 $8,027.10×0.0025 =$20.07 $2,691−$20.07 =$2,670.93 $8,027.10−$2,670.93
=$5,356.17
5 $2,691 $5,356.17×0.0025 = $13.39 $2,691−$13.39 =$2,677.61 $5,356.17−$2,677.61
=$2,678.56
6 $2,6916 $2,678.56×0.0025 =$6.70 $2,691−$6.70 =$2,684.30 $2,678.56−$2,684.30
=−$5.74
Key Takeaways for the Above Amortization Table
Payment Amount = PMT = $2,691 for this example
Interest Paid (INT) = Previous Ending Balance × i
Where i = periodic rate = jm ⁄m = 0.03⁄12 = 0.0025
Principal Repaid (PRN) = Payment Amount − Interest Paid
Ending Balance (BAL) = Previous Ending Balance − Principal Repaid
Final Payment = Payment (PMT) + Final Ending Balance = $2,691+(−$5.74) =
$2,685.26
Add up all of the interest paid to calculate the total interest paid:
Total Interest = $40 + $33.37 + $26.73 + $20.07 + $13.39 + $6.70 = $140.26
You could also construct this table in Excel (click here to download the Excel file).
Conclusion: Kerry will pay $140.26 in interest total and make a final payment of $2,685.26 to
pay off her line of credit.
Business Mathematics 317
AMORTIZATION USING AMRT IN THE BAII PLUS
It can take a long time to construct an amortization table, especially for long-term loans. You
can instead construct the table in Excel or use the BAII Plus’ AMRT (amortization) menu. In
order to access the AMRT menu in your BAII Plus, you need to hit 2ND PMT after you
have already entered all values in the TVM keys and have rounded up your payment (PMT)
and re-entered it into your calculator as a negative value. These steps are written out below:
Steps to Using the AMRT Menu in Your BAII Plus
1. Compute the missing value in the TVM keys (ex: PMT)
2. Input the rounded up payment value. Then make it negative and re-
enter it using: + | − PMT
3. Hit 2ND PMT to enter the AMRT menu
4. Enter in a value for P1 and hit ENTER ↓
5. Enter in a value for P2 and hit ENTER ↓
6. Scroll through the AMRT menu using ↑ ↓
In the above steps, we did not explain what P1 and P2 mean. Let’s now
make sense of these values as well as the rest of the values given by
the AMRT menu.
Understanding the Values in the AMRT Menu:
P1: Starting payment in period in question
P2: Ending payment in period in question
BAL: Outstanding Balance at end of period in question
PRN: Principal Repaid during period in question
INT: Interest Paid during period in question
We have not yet clearly defined what the “period in question” means. This is because this
period can vary. For monthly payments, the period in question is the starting and ending month
numbers. Some examples are given below for P1 and P2 values.
Examples of “Periods in Question” for Monthly Payments
1. The first year of a mortgage ⇒ P1=1, P2=12 (there are 12 months in the first year)
2. The first month of a mortgage ⇒ P1=1, P2=1 (period is just month 1)
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3. The third year of a mortgage ⇒ P1=25, P2=36 (months 25 to 36 are in the third
year)
4. The third month of a mortgage ⇒ P1=3, P2=3 (just period 3)
5. The first three years of a mortgage ⇒ P1=1, P2=36 (payments 1 to 3×12=36)
Example 5.10.2
Maksim purchases an apartment in New Westminster. He pays $600,000 less a 20% down
payment. He takes out a 25-year mortgage with a 5-year term. He is charged 2.95%,
compounded semi-annually. He makes monthly payments with the first payment in one month.
How much interest will he pay during the first 5-year term? Assume that his payments are
rounded up to the next dollar.
Step 0: Determine the amount Maksim borrows (PV):
Step 1: Determine the size of Maksim’s monthly mortgage payments:
B/E P/Y C/Y N I/Y PV PMT FV
END 12 2 25×12=300 2.95 +480,000 CPT −2,259.28 0
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Step 2: Round up the payment and re-enter as a negative value: 2260 + | − PMT
Step 3: Access the AMRT menu: 2ND PMT
Step 4: Input P1: 1 ENTER ↓
Step 5: Input P2: 60 ENTER ↓
Step 6: Scroll down to INT: ↓ ↓ ↓
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Conclusion: Maksim will pay $65,436.89 in interest during the first 5 years of his mortgage.
Wondering why we used P1=1 and P2=60? There are 60 months in the first 5 years, starting
with month one and ending with month 60. Also see the table in the section below below.
P1 & P2 Values for First 5 years of Monthly Payments
We want to calculate the interest for the first 5 years of the mortgage (the first term). The first
month in this time-period is month 1 (the start of the mortgage). The last month will be month
60 (=5×12). See the table below for the month numbers for the first 5 years.
Year P1 P2 Month Numbers (P1 to P2)
1 1 12 The first year contains months 1 to 12
2 13 24 The 2nd year contains months 13 to 24
3 25 36 The 3rd year contains months 25 to 36
4 37 48 The 4th year contains months 37 to 48
5 49 60 The 5th year contains months 49 to 60
Example 5.10.3 — How much of the first mortgage payment will be interest?
Step 0—2: Make sure the values from Example 2 entered into the TVM keys (N, I/Y, PV, PMT,
FV) in your BAII Plus.
Step 3: Access the AMRT menu: 2ND PMT
Step 4: Input P1 (the first payment ‘starts’ in month 1): 1 ENTER ↓
Step 5: Input P2 (the first payment ‘ends’ in month 1): 1 ENTER ↓
Step 6: Scroll down to INT: ↓ ↓ ↓
Conclusion: Maksim will pay $1,172.82 in interest during the first month of his mortgage.
Example 5.10.4— How much will the balance owing be reduced by with the first
payment?
In this case, we want to calculate the principal repaid (this is the amount we reduce the balance
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owing by with each payment that we make). Make sure all values from Example 2b are still in
your calculator (Steps 0 to 5). Just scroll back up ↑ to PRN.
Conclusion: Maksim will reduce his balance owing by $1,087.19 with his first mortgage
payment.
Example 5.10.5— How much interest does Maksim repay in the first year?
There are 12 months of payments in the first year, starting at payment 1:
P1 = 1
P2 = 12
INT = $13,896.99
Conclusion: Maksim will pay $13,896.99 in interest in the first year.
Example 5.10.6 — How much principal does Maksim repay in the fifth year?
The fifth year starts at month 49 and ends at month 60:
P1 = 49
P2 = 60
PRN = $14,866.29
Conclusion: Maksim will reduce his balance owing by $14,866.29 in the fifth year.
Example 5.10.7 — How much Maksim still owe at the end of five years?
If we want the balance owing, it only matters what is entered into P2. This is because BAL
(balance) is the amount owing at the end of payment P2. For this reason, we can just leave the
values from Example 2e in the calculator and scroll to BAL:
P1 = 49
P2 = 60
BAL = $409,836.89
Conclusion: Maksim will owe $409,836.89 at the end of five years.
RENEWING MORTGAGES
At the end of a mortgage term, the mortgage holder renews their mortgage (or refinances the
mortgage if they borrow more money).
Business Mathematics 321
When the mortgage holder renews their mortgage, their terms and interest rate will most likely
be changed. Use this new rate and use the number of years remaining to determine the size of
the new mortgage payments.
Example 5.10.8
Let us assume Maksim has made 5 years of mortgage payments and it is now time for
Maksim’s to renew his mortgage. Maksim renews his mortgage for another 5 years at 3.5%
compounded semi-annually. What is the size of Maksim’s new mortgage payments? Round up
to the next dollar.
B/E P/Y C/Y N I/Y PV PMT FV
END 12 2 20×12=240 3.5 +409,836.89 CPT −2,371.57 0
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Conclusion: Maksim will pay $2,372 per month (round the payments up).
CALCULATING THE FINAL PAYMENT
The final payment using the BAII Plus AMRT method can be calculated the same way as when
we using the amortization table (see Example 1). The only differences are that we enter the
final payment number into P2 and scroll down to BAL to determine the final balance owing.
We then use the same formula as before:
Note: the ending balance will be negative. That means that when we add that negative number
to the regular payment size, the payment size drops in value.
Example 5.10.9
Let us continue on with Example 5.10.9. Let us assume Maksim continues to pay 3.5%
compounded semi-annually for the entire 20 years remaining in his mortgage. What will be the
size of Maksim’s final payment if this were true?
322 Amy Goldlist
Step 1: Make sure the values from Example 3 entered into the TVM keys.
Step 2: Round up the payment and re-enter as a negative value: 2372 + | − PMT
Step 3: Access the AMRT menu: 2ND PMT
Step 4: Input P1 (input any value up to 240): 1 ENTER ↓
Step 5: Input P2 (final payment is the 240th payment): 240 ENTER ↓
Step 6: Scroll down to BAL: ↓
Step 7: Calculate the final payment size using BAL=−147.20:
Conclusion: Maksim will make a final payment of $2,224.80 at the end of 20 years.
CALCULATING THE INTEREST CHARGED USING THE FORMULA
There are two ways to calculate the total interest charged on a mortgage. We can calculate
the difference between the money out and money in or we can use the AMRT menu and read
off the INT values. Let us first step through taking the difference between money out and in
calculation:
To determine $ OUT, where be sure to add up ALL payments and be careful of the final
payment — it is often smaller than the rest:
To determine $ IN, total all money borrowed. If the mortgage holder borrows more money at
some point during the mortgage (if they refinance), then also include that amount in the $ IN
calculation:
Let us now determine how much interest Maksim will be charged on this mortgage.
Example 5.10.10
Let us continue on with this example. Let us assume Maksim continues to pay 3.5%
compounded semi-annually for the entire 20 years remaining in his mortgage. If this is true,
how much interest does Maksim pay in total on his mortgage?
Business Mathematics 323
Let us first calculate the money out ($ OUT):
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Next, let’s determine the money in ($ IN):
Taking the difference between the money out and in gives:
Conclusion: Maksim will be charged $224,732.80 in interest over the 25 years.
CALCULATING THE INTEREST CHARGED USING THE AMRT MENU
Let us now step through how to use the AMRT menu results to calculate the total interest
charged on a mortgage.
For each term in the mortgage (or for each period where the interest rate is fixed), calculate the
total interest paid during that term by doing the following:
P1 = 1
P2 = Final Payment # in term
INT = Total interest paid during that term
Redo this calculation for each term in the mortgage and add up all of these values to determine
the total interest paid over the entire mortgage. Let us revisit Maksim’s mortgage a final time
in this section to understand how to perform this calculation.
324 Amy Goldlist
Example 5.10.11 — Calculate the total interest charged using the AMRT menu
We have already calculated the total interest charged for the first term in Maksim’s mortgage.
We found that Maksim paid $65,436.89 in interest during the first 5 years of his mortgage.
To determine the amount of interest Maksim will pay in the remaining 20 years7, let us look
back at Example 4 and do the following:
1. Make sure the values from Example 4 are still in the TVM keys (they should be)
2. Access the AMRT menu: 2ND PMT
3. Make sure P1=1 (it should still be 1)
4. Make sure P2=240 (it should still be 240)
5. Scroll down to INT: ↓ ↓ ↓ = $159,295.91
Use this interest amount as well as the $65,436.89 to determine the total interest charged:
Total Interest Charged = $65,436.89 + $159,295.91 = $224,732.80
Conclusion: Maksim will be charged $224,732.80 in interest over the 25 years (the same
amount as calculated in Example 5).
YOUR OWN NOTES
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THE FOOTNOTES
Business Mathematics 325
Notes
1. Information thanks to the Financial Services Commission of Ontario:
https://www.fsco.gov.on.ca/en/mortgage/Pages/history.aspx
2. Currently, the CMHC insures mortgages with “mortgage default insurance” if the less
than a 20% down payment is made at the start of the mortgage.
3. Information thanks to https://www.canada.ca/en/financial-consumer-agency/services/
financial-toolkit/mortgages/mortgages-2/6.html
4. https://www.investopedia.com/terms/a/amortization.asp
5. For mortgages and loans, banks round mortgage payments up to the next dollar or the
next cent. The final mortgage (or loan) payment is smaller as a result of the regular
overpayments.
6. This will not be the actual size of the final payment. The final payment will actually
be equal to this value minus the overpayment (final ending balance)
=$2,691−$5.74 = $2,685.25
7. Normally, Maksim would renew his mortgage every 5 years. In this case, the interest
rate would change every five years and there would roughly 5 different interest rate
calculations (one per term). For Maksim's mortgage example, we assumed Maksim
would be charged 3.5% for the remaining 20 years of his mortgage. Although this
is unlikely, it simplified our calculations (this section is already incredibly long and
the calculations would be similar every time Maksim renews).
326 Amy Goldlist
5.11 The 32% Rule and Bi-Weekly Payments
Learning Outcomes
Use the 32% rule to determine the maximum mortgage amount. Also, calculate the size
of accelerated biweekly mortgage payments or the savings in interest.
There are two more topics to examine to conclude our section on Mortgages. The first is the
32% rule, which states that no more than 32% of your gross monthly can go towards housing
costs. The second (and final topic) will be how to calculate accelerated bi-weekly payments
and savings when choosing to make accelerated bi-weekly payments.
THE 32% RULE
Your housing costs shouldn’t be more than 32% of your gross monthly income.1 Housing costs
include your mortgage payment, property taxes, heating costs and half of your strata (condo)
fees.2
PMT + Prop Taxes + Heat Costs + 0.5 × Strata Fees ≤ 0.32(Gross Monthly Income)
We can rework this formula to determine the maximum allowable monthly mortgage payment:
Let us now use this rule to determine how much Brenda and Huong will be allowed to borrow
when they go to purchase an apartment in Central Surrey.
Example 5.11.1
Huong and Brenda are looking to purchase a 2-bedroom apartment in Central Surrey. The
sisters’ combined gross income is $150,000. The property taxes on the 2 bedroom apartment
are $3,600/year. The average heating cost is $43/month. The strata fee for the apartment is
$500/month. What is the largest mortgage payment Brenda and Huong would be allowed to
make per month?
Business Mathematics 327
327
Let us use the 32% rule to determine Huong and Brenda’s maximum allowable mortgage
payment. To use the rule, we need all housing expenses monthly. Let us calculate the equivalent
monthly amounts for Huong and Brenda’s income and their potential property taxes:
This gives the following for the maximum allowable mortgage payment:
Conclusion: Huong and Brenda can pay no more than $3,407 per month on their mortgage
payment.
Example 5.11.2
Huong and Brenda have $100,000 saved up for a down payment on an apartment. The current
interest rate is 2.4% compounded semi-annually for a fixed rate 25 year mortgage with a 5-year
term. What is the most expensive apartment Brenda and Huong would be allowed to purchase?
Let us use the maximum allowable monthly mortgage payment (PMT) from Example 1 to
calculate the maximum Huong and Brenda are allowed to borrow (PV):
B/E P/Y C/Y N I/Y PV PMT FV
END 12 2 25×12=300 2.4 CPT +769,071.85 −3,407 0
Huong and Brenda can borrow $769,071.85. To determine price of the most expensive
apartment, include the down payment:
Conclusion: Huong and Brenda can buy an apartment costing up to $869,071.85.
328 Amy Goldlist
BI-WEEKLY & ACCELERATED BI-WEEKLY PAYMENTS
A bi-weekly payment is a payment that occurs once every 2 weeks. Bi-weekly payments
are popular because people are commonly paid bi-weekly. There are 26 bi-weekly payment
periods in a year.
When paying bi-weekly, people can choose the accelerated bi-weekly option. When they
choose an accelerated bi-weekly mortgage, they pay half of the normal monthly payment every
two weeks. When making this choice, they end up making 2 extra bi-weekly payments per
year. This happens because for two of the twelve months in the year, the borrower will receive
three paychecks during that month. This happens in months where their payday lands on the
following days3:
The 1st, 15th and 29th of the month
The 2nd, 16th and 30th of the month
Those two extra bi-weekly payments equal to one extra full-sized monthly mortgage payment.
This overpayment by one full-sized mortgage payment per year will lead to savings in interest
paid on the mortgage and the time required to pay off the mortgage.
Let us look again to Huong and Brenda and their mortgage options when they purchase an
apartment in Central Surrey.
Example 5.11.3
Huong and Brenda have found the perfect apartment! It costs $850,000 and they will make a
$100,000 down payment. Now they just need to decide whether to make monthly mortgage
payments or accelerated bi-weekly mortgage payments.
Help them decide by determining the size of the mortgage payments for both options as well
as the potential time and money saved if Huong and Brenda make accelerated bi-weekly
payments (if they can afford it)!
Let us start by calculating the size of Huong and Brenda’s monthly mortgage payments.
Example 5.11.3a – The Size of Huong and Brenda’s Monthly Payments
Step 0: Determine the amount Huong and Brenda will borrow:
Business Mathematics 329
Step 1: Determine the size of the regular monthly mortgage payments (assume Huong and
Brenda will still have a 25-year mortgage and be charged 2.4%, compounded semi-annually):
B/E P/Y C/Y N I/Y PV PMT FV
END 12 2 25×12=300 2.4 +750,000 CPT
−3,322.51 0
Conclusion: If Huong and Brenda make monthly payments, they will pay $3,323 per month.
Example 5.11.3b – Total payments Per Year (Monthly)
Let us now calculate how much Huong and Brenda will make in mortgage payments per year:
Conclusion: Huong and Brenda will make $39,876 in mortgage payments each year if they
pay monthly.4.
Example 5.11.3c – Total Interest Paid with Monthly Payments
Now that we have the size of Huong and Brenda’s monthly payments, we can calculate the
interest they will pay over the 25 years. We will assume, to avoid making this example too
long, that the interest rate will remain fixed at 2.4% for the entire 25 years.
Step 1: Make sure all values from example 3a are still in your BAII Plus (TVM keys).
Step 2: Round up the payment and re-enter as a negative value: 3323 + | − PMT
Step 3: Access the AMRT menu: 2ND PMT
Step 4: Input P1: 1 ENTER ↓
Step 5: Input P2: 300 ENTER ↓
Step 6: Scroll down ↓ ↓ ↓ to INT: −246,699.76
Conclusion: Huong and Brenda will pay $246,699.76 in interest if they choose to make
monthly payments.
330 Amy Goldlist
Example 5.11.3d – Time-Savings with Accelerated Bi-Weekly
When making accelerated bi-weekly payments, we divide the regular monthly payments in
half:
We then enter this payment amount into the BAII Plus:
B/E P/Y C/Y N I/Y PV PMT FV
END 12 2 CPT 583.157 2.4 +750,000 −1,661.50 0
An interactive H5P element has been excluded from this version of the
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It will take them 583 full-sized and one smaller payment to pay off the mortgage with
accelerated bi-weekly payments. This will still be 584 payments (even though the last one is
smaller, it still counts as a payment). Let us calculate how many years that is:
It will take them 22.46 years to pay of their mortgage with accelerated bi-weekly payments.
Let’s compare that to the time required to pay off the mortgage with monthly payments:
Conclusion: Huong and Brenda will save 2.54 years if they choose to make bi-weekly
payments.
Example 5.11.3e – Additional Amount Paid per Year with Accelerated Bi-Weekly
Let us now calculate how much Huong and Brenda will make in bi-weekly payments per year:
Business Mathematics 331
Compare this amount to the total paid with monthly payments to determine how much more
they will pay per year if they make accelerated bi-weekly payments:
Conclusion: Huong and Brenda pay an extra $3,233 in mortgage payments if they make
accelerated bi-weekly payments. Notice that this is exactly equal to one monthly payment.
Example 5.11.3f — Savings in Interest with Accelerated Bi-Weekly
Steps 0-2: Check that all values for Example 3d are still in your BAII Plus.
Step 3: Access the AMRT menu: 2ND PMT
Step 4: Input P1: 1 ENTER ↓
Step 5: Input P2: 584 ENTER ↓ (There are 584 bi-weekly payments in total).
Step 6: Scroll down ↓ ↓ ↓ INT: −218,915.65
Huong and Brenda will pay $218,915.65 in interest if they make accelerated bi-weekly
payments. We can use this amount to figure out their savings in interest with this payment plan:
Conclusion: Huong and Brenda will save $27,784.11 in interest if they pay with accelerated
bi-weekly payments instead of monthly payments.
Example 5.11.3g — What Should Huong and Brenda Decide?
If Huong and Brenda can afford to pay the extra $3,233 per year, then they should choose the
accelerated bi-weekly payment option and save 2.54 years and $27,784.11 in interest when
repaying their mortgage!
YOUR OWN NOTES
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to copy and paste below?
332 Amy Goldlist
These notes are for you only (they will not be stored anywhere)
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THE FOOTNOTES
Notes
1. This rule is also called the GDS - Gross Debt Service Ratio.
2. There is also the Total Debt Service Ratio (TDS) that lenders use when borrowers
have other loans. No more than 40% of a borrower's income can go towards their
mortgage payment, property taxes, heating costs, half of their strata fees and other
debt payments. They will take the minimum payment generated by the GDS (32%)
and TDS rules.
3. Their payday could also land on the 3rd, 17th and 31st of the month. This is more rare
because only 7 months in the year have 31 days and February only has 28 days,
normally. In general, there will be two months in the year where they have three
paychecks.
4. The final year is the only exception. They will pay slighly less in the final year
because the final payment in that year will be smaller.
Business Mathematics 333
334 Amy Goldlist
5.12 Lump Sum Payments and Refinancing Mortgages
Learning Outcomes
Calculate the extra amount borrowed when refinancing a mortgage or the reduced
payment size when renewing a mortgage and making a lump sum payment that drops
the balance owing.
It is possible to borrow more money or pay off part of your mortgage when you go to renew
your mortgage1. Let us first examine making a lump sum payment upon renewal (ie: making
an additional payment when renewing your mortgage).
LUMP SUM PAYMENTS
After a mortgage term is up and before a borrower renews their mortgage, they can choose to
pay down some of the balance owing with a lump sum payment. This is like the initial down
payment but part-way through the mortgage. It is often advisable to make these payments, if
you can afford them, as they will save you a lot of interest if there are many years left in your
mortgage.
Example 5.12.1
The Frasers term is up on their mortgage. They owe $523,324.15 on their at the end of the
term. They have $125,000 saved up to pay down on their mortgage before they renew the
mortgage. They will renew for another 15 years. They negotiate an interest rate of 2.88%,
compounded semi-annually. What is the size of their new monthly mortgage payments?
First, let us determine the amount they will borrow (PV):
Next, let’s input the all values into the BAII Plus and calculate the size of their payments
(PMT):
Business Mathematics 335
335
B/E P/Y C/Y N I/Y PV PMT FV
END 12 2 15×12=180 2.88 +398,324,15 CPT
−2,724.56 0
The Frasers will pay $2,724.56 per month on their mortgage.
INCREASING THE SIZE OF A MORTGAGE UPON RENEWAL (REFINANCING)
After a mortgage term is up, a borrower can choose to increase the size of their new mortgage2
by borrowing additional money against the mortgage when they renew. Unless necessary, it is
often not advisable to increase the size a mortgage. This is because a borrower will pay a lot
of interest on the extra money borrowed if there are many years left on the mortgage.
Example 5.12.2
Craig and Joel’s mortgage term is up. For the last 5 years, their mortgage payments have been
$3,904/month. Interest rates have dropped since they first bought their place. They are hoping
to borrow some money to renovate their kitchen while still keeping their mortgage payments
the same (at $3,904/month). How much can they borrow to renovate if they owe $523,324.15
and have 15 years left on their mortgage? Their new interest rate will be 1.99%, compounded
semi-annually.
If we want to know how much Craig and Joel can borrow, we need to calculate the PV (amount
owed) using the $3,904 payment and using the 15 years remaining to calculate N:
B/E P/Y C/Y N I/Y PV PMT FV
END 12 2 15×12=180 1.99 CPT +607,464.81 −3,904 0
Take the difference between the amount they can borrow (PV) and the original amount they
would have owed to determine how much extra Craig and Joel can borrow:
Conclusion: Craig and Joel can borrow $84,140.66 to renovate their kitchen3.
YOUR OWN NOTES
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to copy and paste below?
336 Amy Goldlist
These notes are for you only (they will not be stored anywhere)
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text. You can view it online here:
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THE FOOTNOTES
Notes
1. If you borrow more money when your mortgage term is up, this is called refinancing
(instead of renewing).
2. Up to 80% of the value of the house can be financed (borrowed).
3. Provided that the new value of their mortgage does not exceed 80% of the assessed
value of their house. Their house would need to worth at least $759,331.02 for the
bank to lend them up to $84,140.66 for their renovation.
Business Mathematics 337
338 Amy Goldlist
Chapter 5 Review Problems
1 Calculate the future value of the ordinary annuity in each part:
Size of Payment Term of the
Annuity
Nominal
Interest Rate
Payment and
Conversion Period FV
a. $2,100 10 years 9.50% 6 months ?
b. $4.25 12 years 9.00% 1 day ?
c. $750 8 years 10.00% 1 month ?
d. $3,500 27 months 12.00% 3 months ?
Note: b. might explain how much an “average” smoker could save in 12 years (assuming a
constant price for cigarettes and a fixed interest rate).
2 $500.00 is deposited at the end of every six months for nine years in an account paying 10.0%
compounded semi-annually. Calculate the accumulated value of the deposits.
3 Calculate the amount of interest included in the accumulated value of $600.00 deposits made
at the end of each month for 5 years. The interest rate is 13.5% compounded monthly.
4 Bill Holden is preparing retirement plans for his employees. He requires each employee to
deposit $265.00 at the end of each month for 9 years. The interest rate is 8.75% compounded
monthly.
a. How much money will be in each employee’s account at the end of 9 years?
b. How much will each employee have actually contributed?
c. How much of the amount will be interest?
Business Mathematics 339
339
5 Corinne Smith made $2,750 deposits every 6 months into a registered retirement savings
plan paying 11.25% compounded semi-annually. Just after making the 16th deposit, the interest
rate changed to 10.00% compounded quarterly. If neither deposits nor withdrawals were made
during the next five years how much would Ms. Smith then have in her account?
6 Calculate the present value of the ordinary annuity.
Size of Payment Term of the
Annuity
Interest
Rate Payment and Conversion Period PV
a. $2,100 10 years 9.50% 6 months ?
b. $4.25 12 years 9.00% 1 day ?
c. $750
8 years 10.00% 1 month
?
d. $3,500 27 months 12.00% 3 months ?
7 You wish to take two years off work to attend school and also wish to receive $950.00 at the
end of every month for the 2 years. If you were able to deposit money into an account paying
10.00% compounded monthly:
a. How much should be deposited when you take the time off?
b. How much interest will you receive in the two years?
8 A laptop was bought by paying $750 down and an installment contract with payments of $85
at the end of each month for 2.5 years. If the interest was calculated at 16.9% compounded
monthly:
a. What was the equivalent cash price of the laptop?
b. How much was the cost of the financing?
340 Amy Goldlist
9 Peter Van Dusen opened a trust account to fund his son’s education. The account paid 10.25%
compounded quarterly. His son is expected to require four years of quarterly payments of
$2,000 with the first payment occurring 10 years 3 months from today . How much must Mr.
Van Dusen deposit now so that his son will be able to receive the 4 years of payments? This is
an example of a deferred annuity.
10 To purchase a new trawler-type yacht for chartering, Henry Skipper signed an agreement to
borrow the entire amount and to make payments of $2,750 at the end of every month for seven
years.
a. What was the purchase price of the yacht if money was worth 15.5% compounded
monthly?
b. In his third year of operation, an economic downturn caused Mr. Skipper to miss
payments 25 and 26. What payment was required at the time that payment 27 was
due in order to bring the contract up to date?
c. Upon receipt of payment 27, the mortgage company wished to invoke a contractual
clause and cancel the mortgage. How much (in addition to the payment calculated
in part b above) would Mr. Skipper have to pay in order to fully pay out the
mortgage?
11 Calculate the payment for each annuity:
Future Value Present
Value
Interest
Rate
Payment and Conversion
Period Time PMT
a. $0 $17,750 10.5% 1 quarter 4 years, 6
months ?
b. $12,000 $0 16.5% 6 months 20 years ?
c. $6,500 $0 8.4% 3 months 9 years, 3
months ?
d. $0 $12,500 10.5% 1 month 8 years ?
12 A used Corvette can be bought for $15,000 cash or for equal payments at the end of each
quarter for 5 years. Calculate the size of the quarterly payments at 10% compounded quarterly.
Business Mathematics 341
13 A gaming computer priced at $4,600 can be purchased for $1,900 down and the balance
paid by 36 equal monthly payments at 13.8% compounded monthly. Calculate the size of the
monthly payments.
14 Calculate the term of each annuity:
Future Value Present Value Size of
Payment Interest Rate
Payment and
Conversion
Period
Term (N)
a. $0 $8,500 $675 10.75% 3 months ?
b. $23,500 $0 $491.25 8.25% 1 month ?
c. $0 $962 $100 10.00% 1 week ?
d. $85,000 $0 $2,500 13.00% 6 months ?
15 Charlie Horseshoe invested their $250,000 lottery winnings in a term deposit paying 8%
compounded monthly for 10 years. For how long can $3,500 be withdrawn from the account at
the end of each month starting at the end of the term deposit? Does your answer make sense?
If not, why not?
16 $1,000 is deposited at the end of each month for 5 years. Find the nominal rate of interest
compounded monthly at which the deposits will accumulate to $80,000. (Answer in % using 2
places of decimals.)
17 A camper van can be purchased for $27,500 plus 14% taxes and fees. The dealer will finance
the balance owing after the payment of the sales tax and a down payment of 20% of the price
(without the taxes and fees). Payments will be $330.30 at the end of each month for 7 years.
What nominal rate of interest compounded monthly is being charged?
18 A computer valued at $2,800 can be bought for 25% down and monthly payments of $84.60
for 2.5 years. What effective rate of interest is being charged?
19 A loan of $30,000 is to be repaid with monthly payments over a period of 20 years. Calculate
342 Amy Goldlist
the total savings in interest if the loan is financed at 12.25% compounded monthly rather than
13% compounded monthly.
20 Judith Leisure-Lee is able to set aside $1,500 every 3 months from her income. She plans
to buy a studio condominium at Hemlock Valley Ski Area when she has accumulated at least
$50,000. How long will it take her if she can invest her savings at 9.5% converted quarterly?
(State the answer in months.)
21 A firm believer in technological education wishes to provide an educational institution with
$6,000 bursaries to be awarded at the end of each year for the next 10 years. If the institution
can invest money at 8.75% effective, how much should the philanthropist donate, one year
prior to the first award, to set up the fund for the 10 bursaries?
22 An agreement for sale contract carries payments of $4,500 at the end of every six months
for 10 years. How much should you be willing to pay for the contract if you require a return of
12.25% compounded semi-annually on your money?
23 Samuel Hardy received $48,650 as a severance settlement when his position was terminated.
Harvey had just celebrated his 41st birthday. He immediately and prudently invested the money
in an account paying a guaranteed 9.6% compounded semi-annually until his 60th birthday last
year. At that time, he converted the existing balance into an ordinary annuity paying $3,750 per
month with interest at 10% compounded monthly. For how long will the annuity run until all
the funds have been paid out?
24 Judith Leisure-Lee purchased a small studio condominium at Whistler for $115,000. She
paid $40,000 down and agreed to make equal payments at the end of every month for 25 years.
The interest rate was 13.25% compounded monthly.
a. What size payment is Judith making each month?
b. After 10 years of payments, how much will Judith still owe?
c. How much will she have paid, in total, over the 25 years?
d. How much total interest will she have paid after 25 years of payments?
Business Mathematics 343
25 A well-used car, priced at $1,250 was sold on “easy terms” for a down payment of $250 and
$50 per month for two years. What effective interest rate is being charged?
26 Dogwood Holdings financed a factory expansion by borrowing $325,000 at 10%
compounded semi-annually for 10 years. Payments are to be made at the end of every six
months.
a. Calculate the size of the payment.
b. How much of the 4th payment is interest?
c. Calculate the outstanding balance after the 4th payment.
d. Construct an amortization schedule for the first 4 payments.
27 Save-On-Auto Parts borrowed $120,000 to purchase a fleet of seven vans. They intend to
repay by making monthly payments of $2,400. Interest is at 16% compounded monthly.
a. How many full payments will Save-On-Auto Parts have to make?
b. Calculate the size of the final payment, to be made one month after the last full
$2,400 payment, which will fully amortize the debt.
c. How much interest is included in the 40th payment?
d. What percentage of the loan will have been repaid by the first 48 monthly
payments?
e. How much total interest will be paid?
28 A New York lottery offers a choice to the winner of $1,000,000 cash or $8,500 per month
for 15 years. Which alternative should the winner select if money is worth:
a. 6.0% compounded monthly?
b. 6.25% compounded monthly?
29 Find the present value and future value for an annuity whose periodic payments of $1,000.00
are made at the beginning of every quarter for 7 years. The rate of interest is 13% compounded
quarterly.
344 Amy Goldlist
30 Find the present value and future value for an annuity whose periodic payments of $3,000
are made at the beginning of each six months for 15 years. The rate of interest is 8.5%
compounded semi-annually.
31 Amby Dextrous bought a grand piano, and agreed to a series of monthly payments of $225
for 4 years starting the date of sale. The rate of interest is 13% compounded monthly.
a. Calculate the cash price.
b. How much will be paid over the term of its financing?
c. How much of the total payments will be interest?
32 Find the cash value of a three year service contract for monthly payments of $1,600 paid at
the beginning of each month. The value of money is 14% compounded monthly.
33 If an annuity with a present value of $50,000 has periodic payments at the beginning of
each quarter for 10 years, find the size of the periodic payment. The rate of interest is 12%
compounded quarterly.
34 An annuity whose present value is $50,000 is extinguished by payments of $650.00 made at
the beginning of each month for 12 years.
a. Find the nominal rate of interest compounded monthly.
b. Calculate the effective rate.
35 Payments of $200 were made into a stock ownership plan at the beginning of each quarter
for 15 years. They now have a net value of $50,000. What has been the nominal rate of return
on the investment?
36 $100 is deposited into a retirement plan at the beginning of every month for 20 years. One
month after the last deposit, money is withdrawn in equal monthly payments for 15 years. If
interest is 8.5% compounded monthly, find the size of the monthly withdrawals.
Business Mathematics 345
37 If $75 is deposited into an education fund at the beginning of each month for 18 years and
one month after the final deposit monthly withdrawals of $310.56 a month are made until the
fund is exhausted, find the term of the annuity of withdrawals. Interest is 7.5% compounded
monthly. (Try this problem both as an annuity due and an ordinary annuity.)
38 Find the present value of a deferred annuity whose periodic payment is $550 at the beginning
of each year for 20 years, with the first payment following a two year period of deferment. The
interest rate is 3.6% compounded annually.
39 Find the present value of a deferred annuity whose periodic payment of $360.00 is made at
the end of every semi-annual period for 18 years after a deferment period of six months. The
interest rate is 10% compounded semi-annually.
40 A deferred annuity has a present value of $15,000 and periodic payments are made at the
beginning of each quarter for 10 years after a deferral period of 8 years. The rate of interest is
6% compounded quarterly. Find the size of the periodic payment.
41 Tri-City Holdings borrows $500,000 to fund the expansion of the firm into its eastern market
region. If the loan is to be repaid by making equal payments at the end of each quarter for
8 years beginning after a deferral period of 2 years, find the size of the periodic payments.
Interest rate is 11% compounded quarterly.
42 Find the present value of a perpetuity whose periodic payments of $4,000 are made at the
end of each quarter. Interest is at 6.8% compounded quarterly.
43 An institute lecturer position in Math of Finance is being funded by a perpetual fund . The
fund earns interest at I0% compounded annually and is to pay $60,000 at the end of each year,
with the first payment two years from the date of the fund being set up. Find the size of initial
funding required.
44 You want to have accumulated $4,000 for your European trip four years from now. If
interest is 6.4% compounded quarterly, find the size of quarterly deposits required for 4 years
if deposits are made:
346 Amy Goldlist
a. at the beginning of each quarter.
b. at the end of each quarter.
45 A three-year car lease has a present value of $16,600. If money is worth 13.5% compounded
monthly, find the equivalent monthly lease payments, payable in advance for 3 years.
46 A Caribbean holiday tour package may be financed by making monthly payments of $300
at the beginning of each month for 2 years. Interest is 15% compounded monthly.
a. Find the purchase price now.
b. What will be the total paid in installments over the term?
c. How much interest will be paid over the term of the financing?
d.
47 A building will produce net monthly incomes of $2,000 at the beginning of each month
indefinitely. What is the maximum purchase price if one can get 14% compounded monthly on
one’s money?
48 $400 is deposited into a retirement fund at the end of each quarter for 10 years and interest
is paid at a rate of 9.6% compounded quarterly.
a. Find the accumulated balance at the end of the 10 years.
b. How much of that balance is interest?
49 A recreational property is purchased for $54,000 with a down payment of 10% and the
balance secured by a mortgage, amortized by equal monthly payments over 20 years. Interest
is 16% compounded monthly.
a. Find the size of the monthly payments.
b. Find the balance after 5 years.
c. How much will have been paid for the property in total over the 20 year term of the
financing?
d. How much of the total payments will represent interest?
Business Mathematics 347
50 An agreement for sale contract carries payments of $4,500 at the end of every six months
for 10 years. How much should you be willing to pay for the contract if you require a return of
12.25% compounded monthly on your money?
51 John Leisure purchased a small studio condominium at Whistler for $115,000. He paid
$40,000 down and agreed to make equal payments at the end of every month for 25 years. The
interest rate was 8.75% compounded semi-annually. (see #36)
a. What size payment is John making each month?
b. After 10 years of payments, how much will John still owe?
c. How much will he have paid, in total, over the 25 years?
d. How much total interest will he have paid after 25 years of payments?
Simple Ordinary Annuities, Annuities Due, General Annuities
52 Starting today, Mrs. Robinson will put $500 into her RRSP every month for 20 years. If her
RRSP earns 6% compounded monthly, how much will she earn in interest over the 20 years?
53 Mrs. Watson wants to save $52,450 for a down payment on a house. She will save $2,000
per quarter, starting today.
a. If her invested funds earn 6% compounded quarterly, how long will it take her to
reach her goal?
b. How much interest will she earn during that time?
54 You have just graduated from BCIT. You owe $15,238.98 in student loans. You will be
charged 7% interest compounded monthly. You can afford to make monthly payments of $300
starting today.
a. How long (in years) will it take you to repay your student loans?
b. How much interest will you have paid?
55 How long will it take to save $100,000 if you start saving$1,597 every 3 months, starting
today? Assume an interest rate of 6%, compounded quarterly.
348 Amy Goldlist
56 You borrow $50,000 from the bank to consolidate your credit card debt and student loans.
The bank charges you 12% interest, compounded monthly. The first payment is one month
from now.
a. How long will it take to pay off the loan if you pay$504.25 per month?
b. What is the cost of financing?
c. How long will it take to pay off the loan if you pay only $500 per month?
57 Mr. Eskanderian contributes $1,000 into his RRSP at the end of every quarter for 10 years.
If his RRSP earns 10% compounded quarterly, how much interest will he earn in the 10 years?
Deferred Annuities
58 Judith transfers $25,000 into an RRSP today. She plans to let the RRSP accumulate earnings
at the rate of 8.75% compounded annually for exactly 10 years and then immediately purchase
a 15-year annuity. The first withdrawal will start 3 months after she purchases the annuity.
The annuity earns 9% compounded quarterly. What size of payment will she receive every 3
months?
59 Katherine has recently received an inheritance. She wants to set aside part of the inheritance
to put it into an RRSP to save for her retirement. She anticipates that she will need to receive
$1,200 per month for 15 years with the first withdrawal starting exactly 10 years from today.
The invested funds will earn 7% compounded monthly for the entire 25 years. What amount
must she contribute to her RRSP today?
60 Barry wants to set up an annuity that will pay him $3,000 per month for 20 years beginning
when he turns 65 years of age. If his current age is 50 years and the invested funds will earn
6.5% compounded monthly, what amount must he invest today?
61 Samuel recently inherited money from his grandfather’s estate. He wants to purchase an
annuity that will pay $5,000 every 3 months between age 60 (when he plans to retire) and age
65 (when his permanent pension will begin). The first withdrawal is to be 3 months after he
reaches 60, and the last is to be on his 65th birthday. If Sam is currently 50.5 years old, and the
invested funds will earn 6% compounded quarterly, what amount must he invest today?
Business Mathematics 349
62 It is time to start saving for your retirement. You are 40 years old and want to retire when
you tum 60. You will deposit $1,000 into an RRSP at the beginning of every month for 20
years. You will then use the accumulated funds to purchase a 15-year annuity with the first
withdrawal one month after your 60th birthday. Assume that the RRSP earns 7% compounded
monthly, and the funds invested in the annuity earn 5% compounded monthly.
a. Find the size of the monthly withdrawals.
b. You decide that you will need at least $5,000 per month to live on when you retire
at age 60. How much extra money must you contribute to your RRSP each month
so you can withdraw $5,000 every month for 15 years?
63 Starting today, Giselle Lafleur will deposit $200 in her RRSP each month for 20 years. One
month after the last deposit, she will withdraw the money in equal monthly withdrawals
for 10 years.
a. Find the size of the monthly withdrawals if the invested funds earn j12 = 9%.
b. How much interest will Miss Lafleur earn over the next 30 years?
64 You have just celebrated your 20th birthday. You want to retire when you tum 65. Starting
today , you are going to make monthly contributions to your RRSP, so that when you retire you
can withdraw $2,250 per month for 15 years. The first withdrawal is made when you turn 65.
You anticipate the invested funds will earn 7% compounded monthly.
a. How much money must you contribute each month to your RRSP to achieve your
goal?
b. How much interest did you earn during the entire time?
65 Jean is thinking about retiring in five years. He would like to have $50,000 in an account
when he retires. He decides to make monthly deposits (at the beginning of each month) in his
local credit union where he can earn 7.0% compounded quarterly.
a. What would be the required monthly deposit to accumulate $50,000 in five years?
b. How much interest does John earn over the 5 years?
66 Six years from now, when you tum 55, you are planning to retire. You want to set aside
350 Amy Goldlist
some money today so you can receive $2,500 at the end of every quarter for 15 years with the
first withdrawal 3 months after you tum 55. The invested funds earn 9% compounded semi-
annually.
a. What amount must you invest today?
b. How much interest will you earn during the 21 years?
Perpetuities and General Annuities
67 You have become wealthy beyond your wildest dreams and would like to create a
scholarship at BCIT. You would like to give $2,000 per year to a student studying business
math. You would like your scholarship to continue in perpetuity.
a. How much money should you set aside today if the first payment is in one year and
the interest rate is 8% compounded annually?
b. How much should you set aside if the first scholarship is today?
68 You take out a loan to buy a black Honda S2000 convertible sports car with leather interior .
The bank requires that you put $9,000 down followed by payments of $925 at the end of every
month for 5 years.
a. If the bank charges you 12% interest compounded monthly, what is the selling price
of your car?
b. If the bank charges you 12% interest compounded semi annually, how much would
your monthly payments be? Note: You borrow the same amount as found in part
(a).
69 Mr. Bean borrows $7,500 today. He will repay the loan with quarterly payments at the end
of every three months for three years. The interest rate charged is 9% compounded monthly.
Find the size of the payment.
70 A corporation donates $11,000 to Langara. The funds are invested at 10% compounded
annually per year. The interest is paid out each year as a scholarship. How much will be paid
out each year if the first scholarship is paid immediately after the donation is received? Note:
It’s easier to use the calculator.
Business Mathematics 351
71 You have just won the Set for Eternity Lottery; the lottery will pay you and your descendants
$5,000 per month forever with the first payment in one month . Instead of receiving $5,000 per
month forever you would like to receive the cash now. What is the cash value of the prize if the
interest rate is j12 = 6%?
72 The TRIUMF research lab at UBC recently received a donation from a private individual.
The funds were matched two-to-one by the government. Each year they plan to pay out
a scholarship of $8,596.59 as they anticipate earning 11% compounded quarterly on the
invested funds. What amount did the private individual donate? (The first scholarship will be
one year later). Round answer to the nearest dollar.
73 You have decided to purchase preferred shares of Plutonium Fuel Cells Inc. that pays a semi-
annual dividend of $1.25 per share.
1. What would you be willing to pay per share if you want to earn at least 5%
compounded semi-annually on your investment and the next dividend is to be paid
in 6 months?
2. You are short of cash and need to sell your shares of Plutonium Fuel Cells Inc.
Unfortunately, interest rates have risen to 8% compounded semi-annually. Calculate
your gain or loss per share if the next dividend is due in six months – and the
dividend per share remains the same.
74 You purchase 500 preferred shares of Yardmucks Coffee that pays a quarterly dividend
of $0.26 per share. The next dividend is due in 3 months. Current interest rates are 8%
compounded quarterly.
a. What would you be willing to pay for the 500 shares?
b. If interest rates drop to 6.5% compounded quarterly, how much would you expect to
gain or lose if you sell all of your shares? Note: the next dividend is due in 3 months
and the dividend per share is still $0.26.
75 The Winfall lottery offers you two choices for its grand prize. Either a cash prize of
$1,500,000 today or $7,000 per month forever (with the first installment one month from now).
a. Which choice should you select if interest is 6% compounded monthly?
b. You choose the $1,500,000 today and deposit it into an account earning 6%
compounded monthly. How much could you withdraw each month forever?
Assume the first withdrawal is one month from now.
352 Amy Goldlist
76 You borrow $50,000 and agree to make monthly payments for 15 years starting one month
later. Calculate the size of your payments if the interest rate is 9% effective.
77 You borrow $20,000 today from a moneylender called, The Money Branch. As a result of
your bad credit The Money Branch charges you an interest rate of 28.8%. You will repay the
loan with equal monthly payments over four years with the first payment one month from now.
a. Find the size of the monthly payment if the interest is: (i)28.8% compounded
monthly or (ii) 28.8% compounded semi-annually.
b. How much less interest would you pay if they compound the interest semi-annually,
instead of monthly?
78 The University of Edmonton received a donation from a wealthy individual. Some of the
donated money will be set aside to create a scholarship fund that will pay out $10,000 at the end
of every 6 months, in perpetuity. If the invested funds can earn 8% compounded semi-annually,
instead of 5% compounded semi-annually, how much less money must they set aside today to
pay out a $10,000 scholarship?
79 You purchase 200 preferred shares of Black Bear Brewing that pays a dividend of $1 per
share every 3 months. Current interest rates are 12% compounded monthly. The next dividend
is due in 3 months.
a. How much would you be willing to pay for the 200 shares?
b. If interest rates fall to 8% compounded quarterly, how much would you expect to
gain or lose if you sell all of your shares? Note: the next dividend is due in 3
months and the dividend per share is unchanged.
Mortgages
80 You take out a $200,000 mortgage:
a. What is the periodic interest rate per month if the rate is
i. 12% compounded monthly?
ii. 12% compounded semi-annually?
Business Mathematics 353
b. If you borrow $200,000, how much interest is paid in the first month? Use both
rates and compare your answers.
81 A debt of $1,200 is repaid by monthly payments of $350 at the end of each month.
The interest rate is 12% compounded monthly. Construct the complete amortization schedule
without using the AMRT keys. Then go back and verify all the answers using the AMRT and
Pl/P2 keys .
Period
Payment
PMT
Interest
INT
Principal Paid
PRN
Balance Owing
BAL
0
1
2
3
4
TOTALS
82 You purchase a house in Surrey for $220,000 and put 25% down so you could receive a
conventional mortgage. You decide to get your mortgage at Superstore since they give you free
groceries each year as an incentive. A 5-year term mortgage (i.e., the interest rate is fixed for
5 years) is negotiated with Simplii Financial in which the balance is amortized over 20 years
(repaid with equal payments over 20 years) at 6.70% interest compounded semi-annually.
a. Calculate your monthly payment. The bank rounds up the payment to the next
dollar.
b. COMP n to verify that n is slightly smaller than 240. If the value of n = -100 then
you forgot to make the payment negative. If the value of n is exactly 240 then you
forgot to re-enter the payment.
c. Find out how the first payment is distributed between interest and principal.
Compare this to the results for the 60th payment.
d. How much interest did you pay in the first year? By how much was the balance
outstanding reduced in the first year?
e. What percent of the original mortgage was paid off in the first two years?
f. How much principal was paid off in the first five years? How much interest did you
pay in the first five years?
354 Amy Goldlist
g. How much interest did you pay in the fifth year of the mortgage? How much
principal was repaid in the fifth year?
h. How much will you still owe after you have made five years of payments?
i. Calculate the value of the final payment assuming that the interest rate never
changes during the 20 years.
83 The Archibald’s are eligible for a Canada Mortgage and Housing Corp. insured mortgage
allowing them to qualify for a mortgage of up to 95% of the selling price of the house. They are
also subject to the 30% rule: no more than 30% of their gross income can go towards paying
the mortgage and property taxes.
a. What is the maximum mortgage they qualify for if their gross monthly income is
$5,000 and they want to amortize the mortgage over 25 years? Assume that the
property taxes on the house are $1,800 per year (after the home owner’s grant).
Current mortgage rates are 6.80% compounded semi-annually. Round answer to
the nearest $100.
b. The Archibald’s take out a mortgage for $195,000 with Citizen’s Bank amortized
over 25 years at 6.8% interest compounded semi-annually for a 5-year term. What
is the size of the monthly mortgage payment(round up to the next dollar)?
c. How much interest did they pay in the first five years of the mortgage?
d. How much money would they still owe on this mortgage after five years of
payments?
e. When they renew their mortgage in five years time, mortgage rates have fallen to
6.0 % compounded semi annually for a five-year term. They have saved $15,000
and will use it to reduce the size of the mortgage. Find the size of their new monthly
payments assuming the balance outstanding is amortized over the remaining time.
The bank rounds up the payment to the next dollar.
f. What is the size of the final payment of the renewed mortgage assuming that the
interest rate does not change during the remaining 20 years?
84 Banks normally use the 30% rule: no more than 30% of your gross income can go towards
paying your mortgage and property taxes.
a. If your gross monthly income is $4,000 per month and your property taxes are
$2,400 per year (after the $470 home owners grant), what is the largest mortgage a
bank would authorize if the mortgage is amortized over 25 years and the rate of
interest is 7.45% compounded semi-annually. Round answer to the nearest $100.
(Assume monthly payments).
b. You purchase a fixer-upper house in downtown Mission for $164,000. Your down
Business Mathematics 355
payment is 25% and you negotiate a first mortgage for the balance. The mortgage is
at 7.45%, compounded semi-annually, amortized over 25 years for a 3-year term.
How large a monthly payment is required? The bank rounds payment up to the next
dollar.
c. How much interest did you pay in the first three years of the mortgage?
d. What percent of the original mortgage have you paid off in the first three years?
e. How much interest did you pay in the third year only?
f. In three years the term of your mortgage is up and you wish to renew. Interest rates
have increased to 9.25% compounded semi-annually for a three-year term
mortgage. At this time you make a lump-sum payment of $10,000 to reduce the size
of your mortgage. Calculate the size of the new monthly payments assuming the
balance outstanding is amortized over the remaining time. Round up to the next
dollar.
g. What is the size of the final payment of the renewed mortgage assuming that the
interest rate does not change during the remaining 22 years?
85 Suppose you take out a mortgage amortized over 25 years for $179,940 at j2 = 12.5%.
a. Find the size of the monthly payment. Round payment to nearest penny.
b. How much time and money would you save if you make 26 bi-weekly payments
(equal to half of the monthly payment) instead of twelve monthly payments each
year? Assume that the interest rate never changes during the 25 years.
86 In March 2012 the Beckers purchased a house in Delta for $512,500. They made a down
payment of exactly 20% and took out a mortgage with TD Canada Trust for the balance at an
interest rate of 7.5% compounded semi-annually, for a 5-year term, amortized over 25 years.
a. What is the size of the monthly payment? The bank rounds the payment up to the
next dollar.
b. How much of the 30th payment was interest?
c. How much interest did the Beckers pay in the 3rd year of the mortgage?
d. Today, (March 2017), they have decided to increase the size of their mortgage and
use the money for house renovations. How much extra money can they borrow if
they want to keep the same monthly payment as before but still pay off the
mortgage by March 2037 (twenty years from now)? The interest rate has fallen to
5.1%, compounded semi-annually for a five-year term.
e. The Beckers increase their mortgage to $450,000 and amortize it over 20 years at
5.1%, compounded semi annually. What is the size of their monthly payment? (The
356 Amy Goldlist
bank rounds up the payment to the next dollar.)
f. What is the size of the final payment assuming that the interest rate stays the same
over the remaining twenty years?
87 The Blacks are considering purchasing a three bedroom townhouse in the Killarney area
of Vancouver. Their gross monthly income is $12,000 per month. The property taxes on the
townhouse are $3,000 per year, and they also have to pay a monthly maintenance fee of $150
per month for the upkeep on the townhouse complex. Banks normally use the 30% rule: no
more than 30% of your gross income can go towards paying your mortgage, property taxes,
and monthly maintenance fees. What is the largest mortgage a bank would authorize if the
mortgage is amortized over 25 years and the rate of interest is 5.6%, compounded semi-
annually? (Assume monthly payments.)
88 Five years ago the Smiths purchased a home in North Vancouver for $650,000. They made
a down payment of exactly 20% and mortgaged the balance with Westminster Savings Credit
Union. The interest rate was 5.6%, compounded semi-annually, for a 5-year term, amortized
over 25 years.
a. Calculate the size of the monthly payment required. The credit union rounds the
payment up to the next dollar.
b. What percentage of the original mortgage was paid off in the first 3 years?
c. How much interest did the Smiths pay in the fourth year of the mortgage?
d. The Smiths, after having made 5 years of payments, made a lump-sum payment to
reduce the outstanding balance to $450,000. What was the amount of the lump-sum
payment?
e. After making the lump sum payment, the Smiths renew their mortgage for another
5-year term, amortized over the remaining time, at 6.8%, compounded semi-
annually. Calculate the new monthly payment. Round the payment up to the next
dollar.
f. Assuming the interest rate remains the same over the remaining time, what is the
size of the final payment?
89 In October 2012 the Reids obtained a mortgage for $380,000 at 6.8% interest compounded
semi-annually, for a 5-year term, amortized over 25 years. (The bank rounds the payment up to
the next dollar.) Today, (October 2017), they have decided to increase the size of their mortgage
and use the money for house renovations. How much more money can they borrow if they want
to keep the same monthly payment as before but still pay off the mortgage by October 2037
Business Mathematics 357
(twenty years from now)? The interest rate has fallen to 4.3%, compounded semi-annually for
a five-year term.
90 You purchase a new car. The dealer offers you terms of 20% down and the remainder
financed over five years at an interest rate of 8% compounded monthly.
a. Find the size of your monthly payment if your first payment is due
at the end of the month and the price of the car was $33,906.25
including GST, PST, documentation fee, and government
environmental levies.
b. What is the cost of financing (how much interest will you pay
over the life of the loan)?
91 You would like to save for your retirement by making monthly deposits of $200 into
an account. What nominal interest rate of interest, compounded monthly, must you earn to
accumulate $1,000,000 in thirty years? Assume that the first payment will be at the end of the
month.
92 You are considering quitting smoking due to the high cost of a pack of cigarettes. You smoke
1 pack a day at a cost of $7.50. If you put the $7.50 you would have spent on cigarettes, into
a savings account earning 5.75% interest compounded daily, how much would you have in the
bank at the end of ten years?
93 Upon graduation, you have a student loan of $15,000. The most you can afford to pay is
$550 per month. How long will it take you to repay the loan with payments of $550 per month
starting in one month if the interest rate is 6% compounded monthly?
94 Kent sold his car to Carolyn for $1,000 down and monthly payments of $120.03 at the end
of every month for 3.5 years. The interest rate charged is 12%, compounded monthly. What
was the selling price of the car?
95 Rajinder bought a car with $5,000 down followed by equal monthly payments of $783.41 at
the end of every month for 2 years at 16% compounded monthly.
a. What is the selling price of the car?
358 Amy Goldlist
b. What is the cost of financing?
96 Manpreet paid $14,000 to buy a used car. He made a down payment followed by equal
monthly payments of $249.50 at the end of every month for 4 years at 8% compounded
monthly.
a. What is the size of the down payment?
b. What is the cost of financing?
97 For $42,000 an individual can purchase a 5-year annuity from Continental Life and receive
monthly payments of $871.85 for 5 years with the first payment one month from now. What
effective rate of interest does this investment earn?
Deferred Perpetuities
98 A scholarship fund is to be set up. The fund will pay out a scholarship of $20,000 every year
with first scholarship paid out two years after the fund is set up. Find the size of the donation
needed. Assume the fund will earn 10% effective.
99 A bursary fund for BCIT honor students is to be funded by a perpetual fund. The fund earns
interest at 10% compounded annually and is to pay $30,000 each year, with the first payment
four years after the fund is set up. Find the size of the initial funding that is required.
100 An alumnus wants to donate a sum of money to his Alma Mater that will provide a
scholarship of $750.00 every six (6) months in perpetuity. If money can be invested at 6%
compounded semi-annually and the first $750.00 is to be awarded at the end of one year, how
much must he donate to the school today?
101 You are considering purchasing shares of New Wave Technology Corp. The company
has stated that they will pay dividends of $0.72 per share every three (3) months with the
first dividend paid exactly four (4) years from today. If the current interest rates are 8%
compounded quarterly, what would you be willing to pay for one share today?
Business Mathematics 359
Using All 5 Keys
102 You just celebrated your 40th birthday. You dream about retiring when you tum 55. You
currently have $80,000 accumulated in your retirement plan. You decide to make deposits each
month into a retirement plan for exactly 15 years, starting today. You want to purchase an
annuity which will pay you $6,000 per month for l 0 year, with the first withdrawal starting
one month after your 55th birthday. The retirement plan and the annuity earn 6% compounded
monthly.
a. How much must you deposit each month into your retirement plan?
b. How much interest will you earn over the ENTIRE 24 years?
103 Barney just celebrated his 40th birthday . He currently has $52,034 accumulated in his
retirement plan and he plans to continue making equal monthly deposits into a savings account
for 15 years, starting today. Two months after his last deposit, he intends to withdraw $4,000
per month for his living expenses for a period of 5 years. The invested funds earn 6%,
compounded monthly, for the entire 20 years.
a. Find the size of the monthly deposits.
b. How much interest will he earn over the entire 20 years?
104 You have $50,000 in your RRSP today. For the next 12.5 years you will contribute $500
per month into your RRSP with the first deposit made one month from now. How much will
you have in your RRSP at the end of 12.5 years if you earn 6.9598% compounded monthly on
your RRSP?
105 You take out a loan for $50,000 and make monthly payments of $500 with the first payment
made one month from now. If the interest rate on the loan is 6.9598% compounded monthly,
how long (in months) will it take you to pay off the loan?
106 Starting today, you will contribute $695.09 per month into an RRSP for a period for 5 years.
You will then use the accumulated funds to purchase an annuity with monthly payments paid
out over 12.5 years. What will be the size of the monthly withdrawals if the first withdrawal
is made 2 months after the last deposit? Assume you earn 6.9598% compounded monthly the
entire time.
360 Amy Goldlist
107 What amount will be in an RRSP after 20 years if monthly contributions of $300 are made
for the first 15 years and then contributions of $600 per month are made for the subsequent 5
years? The first deposit is made one month from now and the funds invested in the RRSP earn
7% compounded monthly.
108 Starting today, you contribute $1,200 every 3 months into your RRSP for five years. The
interest rate was 10% compounded quarterly for the first 2 years and 9% compounded quarterly
for the last 3 years. How much will you have in your RRSP at the end of 5 years?
109 Herb has made contributions of $2,000 to his RRSP at the end of every 6 months for the
past 8 years. The RRSP has earned 9.5% compounded semi-annually. Today, he moved the
funds to a different play, paying 8% compounded quarterly. He will now contribute $1,500 at
the end of every 3 months. How much will he have in his RRSP 7 years from now?
110 The Alumni Association of BCIT would like to set up a scholarship that will pay out
$2,000 every 6 months forever. The funds will be deposited in an account which earns 8.16%
effective. The first scholarship will be awarded 2 years after the funds are deposited. How
much money do they need to set aside today?
111 You plan to contribute $900 every 6 months into your retirement plan for a period of
15 years. How much INTEREST would you earn over the 15 years if your RRSP earns 7%
compounded monthly? The first contribution is made 6 months from now.
112 Barney just celebrated his 25th birthday. Starting today he will make contributions every
month into his retirement plan for a period of 30 years. How much must Barney contribute
every month to his retirement plan so he can withdraw $5,000 per month for a period of 20
years with the first withdrawal starting two months after his last deposit? Assume Barney earns
8.5%, compounded quarterly, the entire time.
113 Marika has already accumulated $18,000 in her RRSP . If she contributes $2,000 at the end
of every 6 months for the next 10 years, and $300 per month for the subsequent 5 years, what
amount will she have in her plan at the end of the 15 years? Assume that her plan earns 9%,
compounded semi annually, for the first 10 years and 9% compounded monthly for the next 5
years.
Business Mathematics 361
114 How much larger will the value of an RRSP be at the end of 25 years if the contributor
makes month-end contributions of $300 instead of year-end contributions of $3,600? In both
cases the RRSP earns 8.5%, compounded semi-annually.
115 What will be the amount in an RRSP after 25 years if contributions of $2,000 are made
at the beginning of each year for the first 10 years and contributions of $4,000 are made at
the beginning of each year for the subsequent 15 years? Assume that the RRSP earns 8%,
compounded quarterly.
Notes
1. a. $67,631.83
b. $33,511.96
c. $109,635.81
d. $35,556.87
2. $14,066.19
3. $15,021.08
4. a. $43,307.02
b. $28,620.00
c. $14,687.02
5. $112,179.12
6. a. $26,734.40
b. $11,382.03
c. $49,426.12
d. $27,251.38
7. a. $20,587.31
b. $2,212.69
8. a. $2,818.12 b. $481.88
9. $9,443.94 if you assume the first withdrawal occurs 10 years and three (3) months
from today or $9,685.94 if you assume that the first withdrawal occurs at the 10
year point.
10. a. $140,461.30
b. $8,357.02
c. $110,460.68
11. a. $1,250.01
362 Amy Goldlist
b. $43.36
c. $117.93
d. $193.00
12. $962.21
13. $92.02
14. n = 15.58 → 16 quarters n = 41.5 → 42 months[ n = 9.72→ 10 weeks n = 18.51→ 19
semi-annual periods
15. Payments are indefinite because the withdrawals are less than the interest
accumulated for the period.
16. 11.2269% compounded monthly
17. 6.8381% compounded monthly
18. 15.2222% compounded monthly = 16.3305% effective
19. $3,816.00 total savings
20. n = 24.84→ 25 quarters or 75 months
21. $38,933.32
22. $51,094.94
23. 123 months at $3,750 and one (1) month at a smaller amount
24. a. $860.03
b. $67,097.29
c. $257,998 with the last payment= $849.03
d. $257.998 − 75,000 = $182,998
25. 19.7469% effective
26. a. $26,078.85 (rounded up)
b. $14,700.73
c. $282,636.43
Pmt# Amount Interest Principal Balance
0 $325,000.00
1 $26,078.85 $16,250.00 $9,828.85 315,171.15
2 26,078.85 15,758.56 10,320.29 304,850.85
3 26,078.85 15,242.54 10,836.31 294,014.55
4 26,078.85 14,700.73 11,378.12 282,636.43
27. a. 82 full payments and one (1) smaller payment
b. $2,266.47
Business Mathematics 363
c. $1,058.99
d. 44.4%
e. $79,066.47
28. a. PV of monthly payments= $1,007,279.88 which is more than the $1
million cash payment. Choose the payments.
b. PV of monthly payments= $991,342.82 which is less than the $1
million cash payment. Choose the $1 million cash payment.
29. PV = $18,794.90; FV = $46,021.60
30. PV = $52,476.38; FV = $182,913.49
31. a. $8,477.78
b. $10,800.00
c. $2,322.22
32. $47,360.41
33. a. $2,100.12
b. $10,800.00
c. $2,322.22
34. a. 11.9930%,
b. compounded monthly = 12.6747% effective
35. 16.1008% compounded quarterly
36. $617.43
37. n = 185.9986 → 186 monthly payments (15 years and 6 months)
38. $7,477.38 assuming that the first payment occurs at year two
39. $5,673.21 assuming that the first payment occurs at 12 months
40. $795.49 with first payment at year eight
41. $29,439.84 assuming that there are eight years of payments (for example, n = 32)
42. $235,294.12
43. $545,454.55
44. a. $217.86
b. $221.35
45. $557.06
46. a. $6,264.61
b. $7,200.00
c. $935.39
47. $173,428.57
48. a. $26,370.83
b. $10,370.83
49. a. $676.16 (rounded up)
364 Amy Goldlist
b. $46,036.44
c. $162,261.81 with the last payment= $659.57
d. $162,261.81 − $48,600 = $113,661.81
50. $50,447.62
51. a. $608.72 (rounded up)
b. $61,465.87
c. $182,606.20 d.
d. $107,606.20
52. $112,175.55
53. $5.5 years, $8,450 interest earned
54. a. 5 years
b. $2,761.02
55. 11 years
56. a. 40 years and $192,040 of interest
b. b. You never will since the payment only covers the interest.
57. $27,402.55
58. $1,766.18
59. $66,820.18
60. $152,996.91
61. $48,752.43
62. a. $4,143.48
b. $1,206.71−$1,000 = $206.71 per month
63. a. $1,692.10
b. $1,692.10 ×120 − $200 × 240 = $155,052
64. a. $66/month
b. $2,250 × 180− $66 ×540 = $369,360
65. a. $695.09
b. $50,000 − $695.09 × 60 = $8,294.60
66. a. $48,559.18
b. $2,500 × 60 − $48,559.18 = $101,440.82
67. a. $25,000
b. $27,000
68. a. $50,583.41
b. $918.93 per month
69. $720.87
70. $1,000
Business Mathematics 365
71. $1,000,000
72. $25,000
73. a. $50.00/share
b. a loss of $18.75/share
74. a. $6,500
b. Gain of $1,500.
75. a. The PV of the $7,000 perpetual payment is $1,400,000, so $1,500,000
today is better
b. $7,500 per month forever
76. $496.74 per month
77. a. i. $706.23/month ii. $688.03/month
b. $873.60
78. $150,000 less
79. a. $6,600
b. $3,400 gain
80. a. i. 1% per month ii. 0.975879417% per month
b. $2,000 for the 1st month 1% per month and only $1951.76 using
0.975879417% per month.
81.
Period Payment PMT Interest
INT
Principal
Paid-PRN
Balance
Owing-BAL
0 $1,200.00
1 $350 $12.00 $338.00 862.00
2 350 8.62 341.38 520.62
3 350 5.21 344.79 175.83
4 175.83 + 1.76 =
$177.59 1.76 175.83 0
TOTAL $27.59 $1,200.00
Final payment is $350−$172.42 = $177.58 using the calculator (slight difference
due to rounding).
82. a. 1240.74 → $1241.00/month
b. n = 239.8969885
c. 1st month: $332.35 principal, $908.65 interest 60th month: $459.53
principal, $781.47 interest
d. Principal paid off= $4,111.26: Interest is $10,780.74
366 Amy Goldlist
e. e. $8,502.60/ $165,000 = 5.153%
f. Principal paid off= $23,554.39: Interest is $50,905.61
g. Principal paid off= $5,351.30: Interest is $9,540.70
h. Would need $141,445.61 to pay it all off
i. $1,113.48
83. a. $196,200 rounded
b. $1,341.82 → $1,342/month
c. $62,591.48
d. $177,071.48
e. $1,154.254 - $1,155/month
f. $813.47
84. a. 137,283.97..., $137,300 (rounded)
b. $895.95 ..., $896/month
c. $26,478.31
d. $5,777.69/$123,000 = 4.697%
e. $8,683.64
f. 939.527 ..., $940/month
g. The balance outstanding is $107,222.31
h. $545.58
85. a. $1,920.00 per month
b. 447.8742287/26 = 17.2259 years, so save 25 years - 17.23 years =
7.77 years The interest savings is: $1,920 × 300- $960 ×
447.8742287 = $146,041
86. a. $3,000 (n = 299.8201542)
b. $2,430.53
c. $29,143.82
d. $77,273.24 ($452,806.23 −$375,532.99)
e. $2,982
f. $2,737.26
87. $519,290.79, $519,300
88. a.
a. $3,205 (n = 299.8729184)
b. 6.056% ($31,492.35/$520,000)
c. $26,748.44
d. $14,419.50
e. $3,410
Business Mathematics 367
f. $3,297.03
89. $76,785.12 ($421,860.24- $345,075.12)
90. a. $550
b. $5875
91. 13.58942%
92. $36,994.34
93. 30 months
94. $5,100
95. a. $21,000
b. $2,801.84
96. a. $3,780
b. $1,756
97. j2 = 9.38064%
98. $181,818.18
99. $225,394.44
100. $24,271.84
101. $26.75/share
102. a. $1,177.37
b. $428,073.40
103. a) $271.00/month b) $139,186 53
104. $238,086.11
105. 150 months or 12.5 years
106. $500/month
107. $177,755.87
108. $30,723.99
109. $136,300.67
110. $44,449.82
111. $19,855.69
112. $352.39/month
113. $188,830.07
114. Month-end: $302,244.75 Year-end: $290,846.96 $11,397.79 more
115. $223,904.53
368 Amy Goldlist
Chapter 6: Investment Decisions
An investment consists of spending money or other resources (e.g., time) with the aim of
receiving benefits later.
INVESTMENT → BENEFITS
(now) (later)
Depositing money in an interest-bearing account is an investment. The purchase of a bond
or treasury bill is an investment. Also, company projects such as the purchase and installation
of equipment, the development of new products and increasing inventory held for sale are
investments.
In this chapter we consider the problem of deciding whether or not an investment is worthwhile
and deciding which investments are preferable to others.
We view the financial aspect of an investment as being entirely determined by the cash flows
associated with it by how much money is spent and received and when it is spent and received.
To (substantially) simplify the analysis we will omit tax considerations and view the expected
cash flows of projects as being certain. Our methods and calculations will be similar to cases
with taxes and risks included, but will be less complex.
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370 Amy Goldlist
6.1 Evaluating a Business Plan
Example 6.1.1
Suppose a business plan would require an investment of $20,000 now and would result in cash
flows of $13,000 in one year and $8,000 in two years.
Then, if you ignore (just for a moment!) the time value of money, you can see that:
showing a positive profit.
Key Takeaways
A positive profit is a minimum requirement for a successful investment. No company
would choose investments which they expect to result in a loss.
Beyond this the question is: How can we decide whether this investment is preferable to
others?
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371
Suppose that an alternative investment was to place the money in a trust account which would
pay 6% effective over the two-year period, and from which withdrawals could be made at any
time.
Then, if you try to match the sequence of cash flows you would have:
Time Interest Deposit Withdrawal Balance
0 $20,000.00 $0.00 $20,000.00
1 $1,200.00 13,000.00 8,200.00
2 492.00 8,000.00 692.00
The final balance of $692.00 shows that in the account we could have withdrawn a total
payment of $8,692 at the end of the second year, which is more than the original investment
would allow; hence, the trust company account would be a better investment. The first
investment made money, but not fast enough to earn 6% effective.
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372 Amy Goldlist
6.2 Rate of Return
The rate at which profit is earned, the rate of return, is the key factor in investments. It is
described and used in exactly the same way as a compound interest rate. Usually it is stated
as an annual rate of return (a rate compounded annually, an effective rate). As in the usual
calculation of compound interest, the rate of return is applied to the outstanding balance (the
unrecovered part of the investment). Once the investment has been recovered, all additional
inflows represent profit beyond the minimum requirements.
We need a way to evaluate investments so that we can choose the best ones. The major
methods of evaluation involve discounting cash flows to find their present values (values at the
beginning of the investment).
There are several approaches to dealing with the rate of return. The first approach we shall
consider is to set a minimum rate, called the minimum acceptable rate of return (MARR), or
minimum return on investment (ROI).
If a company sets such a rate, then investments must meet or exceed this rate of return before
the company will fund them. Consequently, the company’s resources will be directed to only
those projects which provide adequate returns.
The setting of the minimum acceptable rate of return is a management decision and depends
on a number of factors, including the company’s historical earnings rate and the cost of capital
raised by the company.
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Business Mathematics 373
373
374 Amy Goldlist
6.3 Net Present Value
Suppose that we set a minimum rate of return of 6% effective and apply it to our first example.
Then, the future inflows would have present values of:
And
So $12,264.15 is the amount it is worth investing now to receive $13,000.00 in one year (and
earn 6%), and $7,119.97 is the amount it is worth investing now to receive $8,000.00 in two
years (and earn 6% compounded annually).
Thus, to earn a rate of return of 6% per year, it would be worth paying a total of:
But the investment costs $20,000, which is $615.88 more than it should if 6% is to be earned!
Consequently, the investment should not be made if 6% is the minimum acceptable rate.
The discounting procedure above finds the net present value (NPV) of investments. This dollar
amount is the difference between what “should” be worth investing and what actually has to be
invested. So the inflows (benefits) are discounted to see the maximum amount worth investing,
and the outflows (costs) are discounted to see what actually is being invested.
Using PV for present value:
Example 6.3.1
A small manufacturing business will cost $200,000 paid immediately and other expenses
incurred are expected to result in a net negative cash flow of $40,000 by the end of the first
year. It is then expected to earn annual positive net cash flows of $50,000 starting at the end of
the second year and continuing each year until the end of the fifth year. It will be sold at the
end of the last year (year five) for $250,000.
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375
If the investors have set a minimum rate of return of 20%, should they invest in the business?
To evaluate this investment by NPV we find for outflows:
PV of $200,000 is $200,000.00
PV of $40,000 (yr 1) is $33,333.33
Total PV Outflows $233,333.33
PV of $50,000 (yr 2) is $34,722.22
PV of $50,000 (yr 3) is $28,935.19
PV of $50,000 (yr 4) is $24,112.65
PV of $300,000 (yr 5) is $120,563.27
Total PV Inflows $208,333.33
So that:
The negative NPV shows that the investment did not earn the required rate of return.
Note that the net outflow of $40,000 in year 1 was also discounted to its present value
of $33,333.33 to make the evaluation. You could assume that if $33,333.33 were invested
elsewhere it would earn 20% and be available as $40,000 at year 1.
To summarize decisions based on NPV:
If, for a given required rate of return, an investment has NPV > 0 (i.e., a positive net
present value), then the investment earns more than the required rate.
If NPV < 0 (i.e., a negative net present value), then the investment earns less than
the required rate.
If NPV = 0, then the investment earns exactly the required rate.
376 Amy Goldlist
Be careful to note that a negative NPV does not necessarily imply that the investment resulted
in a loss. It may be that the investment earned a profit, but did not do so fast enough to make
the required rate of return. This was the case in the example above, which would have had a
profit of $210,000, but would not have made the profit at a rate of 20% effective.
Knowledge Check 6.1
To check the result of the last example, complete the following “account” calculation at
20% effective.
Time Interest Deposit Withdrawal Balance
0 $200,000 $200,000
1 $40,000 40,000 ?
2 ? 50,000 ?
3 ? 50,000 ?
4 ? 50,000 ?
5 ? 300,000 ?
You should find a final balance of $62,208 which shows that to earn 20% an extra
$62,208 would be required at the end of the fifth year. This again shows that the
original investment did not earn 20%.
Question: How is the required $62,208 related to the NPV of -$25,000?
Knowledge Check 6.2
Business Mathematics 377
Find the present value of each cash flow in the above problem using a minimum acceptable rate
of return of 15% effective, and use the NPV to determine whether or not the investment would
be acceptable using this lower rate.
Now consider the problem of choosing between two different investments, which
might both be acceptable when considered separately.Suppose the investors in Example
2, who are considering the manufacturing business, have also the choice of purchasing
a share in an existing business, which is described below. If they have set a minimum
rate of return of 15%, how can we determine which investment is preferable?
Example 6.3.2
The share of the existing business would cost $225,000 cash and is expected to produce cash
inflows of $15,000 a year for five years, then to be sold for $425,000.
For the share investment we have the following cash diagram:
To evaluate this investment by NPV at 15% effective, we have for the only outflow a present
value of $225,000, and for the inflows:
Time (year) Inflow PV
1 $15,000 $13,043.48
2 15,000 11,342.15
3 15,000 9,862.74
4 15,000 8,576.30
5 440,000 218,757.76
Total PV Inflows $261,582.43
Then the net present value is:
378 Amy Goldlist
In Knowledge Check 6.2 you should have found that at 15% the NPV of the manufacturing
investment is $13,641.07; hence, by this criterion the purchase of the share would be more
profitable.
Knowledge Check 6.3
Use the net present value criterion to evaluate the following investments:
OPTION A: A small restaurant business in leased premises can be purchased for
$30,000. It is expected to produce cash inflows from operations of $12,000 a year (at
the end of each year) for 4 years. At the end of the 4 years the lease will expire and the
equipment will be sold for $14,000.
OPTION B: A similar restaurant with its own premises can be purchased for $92,000.
It would produce net cash inflows of $21,000 at the end of each year for 5 years and
would be sold for $95,000 at that time.
If the prospective investors set a minimum rate of return of 20%, should they purchase
a restaurant, and if so, which one would be the best deal?
You should attempt the problem by discounting each cash flow and finding the NPV that
way. In the following setup for OPTION A, we have assumed that outflows will be treated as
negative. This is a standard practice and allows a straightforward total to get NPV.
Business Mathematics 379
Time (yr) Cash Flow PV
0 -$30,000 ?
1 12,000 ?
2 12,000 ?
3 12,000 ?
4 26,000 ?
(12,000 + 14,000)
NPV = $7,816.36
For OPTION B you should get an NPV of $8,981.22.
On the basis of NPV both restaurants are acceptable investments, but the second restaurant
(OPTION B) would be the most beneficial if only one is to be purchased.
Calculating the present value of the inflows can always be done by discounting each flow
separately, but in the case of equal flows it can also be done by an annuity. For example, in
OPTION A the first three cash flows of $12,000 can be treated as an annuity by setting:
and finding PV. Then find the present value of the final $26,000 at year 4. You should get a
total of $37,813.36, which agrees with the result above.
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380 Amy Goldlist
6.4 Cash Flow on the BAII Plus
Example 6.4.1
You invested $1,000 and received $300 for the first 2 years at the end of each year. At the end of the
3rd year, you sold your investment for $800. The minimum Annual Rate of Return (MARR) is 14%.
The NPV is:
But we will solve using the BAII Plus Calculator:
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381
Step To Press Display
1 Select Cash Flow worksheet [CF] CFo= (previous entered
value)
2 Clear previous work [2ND] [CE|C]
CFo= 0
3 Enter initial investment, negative for
outflows. [1][0][0][0][+\-][ENTER] CFo = – 1,000
4 Enter 1st cash flow [↓][3][0][0][ENTER] C01= 300
5
Enter frequency of 1st cash flow
(frequency =2)
[↓][2][ENTER] F01= 2
6 Enter 2nd cash flow [↓][8][0][0][ENTER] C02= 800
7 Enter frequency of 2nd cash flow [↓][1][ENTER] F02= 1
8 To find NPV [NPV] I = 0
9 Enter the discount rate [1][4][ENTER] I = 14
10 Compute NPV [↓][CPT] NPV = 33.97536624
11 To find IRR [IRR] IRR = 0
12 Compute IRR [CPT] IRR = 15.69595604
We will write this as:
382 Amy Goldlist
CF0 = -1000
C01 = 300 F01 = 2
C02 = 800 F02 = 1
Your Own Notes
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Business Mathematics 383
384 Amy Goldlist
6.5 Internal Rate of Return
When NPV is positive we know that the project being examined earns more than the required
rate. When NPV is negative we know that the project being examined earns less than the
required rate.
Only when NPV is zero does the project actually earn exactly the required rate. In this special
case we can view the inflows as paying back our investment plus interest at the required rate.
The internal rate of return (IRR) is the rate at which this happens, i.e., the rate for which:
Consider the manufacturing company in Example 6.3.1. In that case at a 20% effective rate of
return the present value of inflows was
$208,333.33 and the present value of outflows was $233,333.33 so that:
We also found in Learning Activity #2 that the NPV at 15% was $13,641.07.
If the required rate is changed to 10% effective, the present value of inflows becomes
$299,315.12 and the present value of outflows becomes $236,363.64 so that:
Similarly we can establish the following values:
Required Rate 16% 17%
NPV $5,156.73 -$2,927.98
It should be clear that the rate which would make NPV = 0 is between 16% and 17% effective.
From the graph below we can see that this value, the internal rate of return of this project, is
about 16.6%.
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385
The internal rate of return can be obtained from business calculators. In the case of the
BAII Plus, place the cash flows in the calculator as for the NPV, remembering to clear all
([2ND][CF][2ND][CE|C]) before entering:
CF0 = -200,000
C01 = -40,000 F01 = 1
C02 = 50,000 F02 = 3
C03 = 300,000 F03 = 1
and then pressing [IRR][CPT]
You should see the answer, IRR= 16.63227%
Note that this is a solution to the equation:
Which is quite difficult to solve by hand.
The answer in your calculator will always be a periodic rate for whatever period was used for
the flows. Since the cash flows were yearly, the rate is an effective rate in this case. Since we
are working with estimates, it is not uncommon to work exclusively with annual rates, as we
do in this textbook.
386 Amy Goldlist
Your Own Notes
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Business Mathematics 387
388 Amy Goldlist
6.6 Payback
The payback period of an investment is the length of time it takes to recover the initial
investment. The shorter the time needed, the better the investment by this criterion. Unless
stated otherwise, the time will be calculated using the undiscounted cash flows.
Example 6.6.1
Suppose that an investment of $20,000 will produce cash inflows of $2,000 in the first year and
$6,000 annually for many years after that. Then the payback period of this investment is four
years, since by accumulating the inflows you can obtain the following table:
Time (yr) Cash Flow Balance
0 -$20,000 -$20,000
1 2,000 -18,000
2 6,000 -12,000
3 6,000 -6,000
4 6,000 0
The payback is at four years.
Example 6.6.2
What if the final cash flow in year four was $10,000? Then we would have the following:
Time (yr) Cash Flow Balance
0 -$20,000 -$20,000
1 2,000 -18,000
2 6,000 -12,000
3 6,000 -6,000
4 10,000 4,000
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389
In this case, convention says that we treat all inflows and outflows as taking place uniformly
throughout the year, so payback is:
Note that no allowance is made for the exact timing of flows (if the inflows were $5,000 a
year the same result would have been obtained) or for the size of cash flows after the payback
period. Also, there is no mention of a rate of return, although a short payback period usually
indicates a high rate of return.
If the investment is not paid back exactly at the end of a year, and, if the cash flows could
reasonably be assumed to occur uniformly throughout the year, then you can estimate the
payback period by adding the fraction of the last year needed to recover the investment. This
is an approximation, but it is a useful tool.
Suppose that in Example 6.8 above the investment required had been $21,500. Then the
payback period would have been extended to 4.25 years since the additional $1,500 would have
to be earned in the 5th year and would require 1,500/6,000 = 0.25 years, so that the payback
period would be 4 + 0.25 = 4.25 years.
If the cash flows were given on a monthly basis there would be no need to estimate a part of a
month – the time could be given to the end of the first month at which the recovery is complete.
The payback period is one of the simplest measures of investment effectiveness. Its benefits
are that it is easily understood, easily calculated and that it provides some assessment of the
risk to which a company is exposed in an investment. Investments which require a long period
before the investment is recovered are generally felt to carry a higher risk, since predictions of
cash flows are more likely to be in error if they are for times far in the future.
Knowledge Check 6.8
Complete the tables and calculations below to obtain the payback periods for the
restaurant OPTIONS A and B given in Example 2.
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OPTION A ($30,000 invested at time 0)
Time Inflow Balance
0 -$30,0000
1 12,000
2 12,000
3 12,000
Time needed beyond end of second year:
Amount needed = ?
Amount in year three ?
Your Own Notes
Are there any notes you want to take from this section? Is there anything you’d like
to copy and paste below?
These notes are for you only (they will not be stored anywhere)
Make sure to download them at the end to use as a reference
An interactive H5P element has been excluded from this version of the
text. You can view it online here:
https://pressbooks.bccampus.ca/businessmathematics/?p=845#h5p-1
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392 Amy Goldlist
6.7 Effects of Different Lifetimes
When we evaluate two options using the NPV criterion, we need to be mindful of the different
lifetimes. A car with an NPV of $25,000 seems like a better investment than a car with an
NPV of $30,000, but if the second car will last 10 years, but the first only five, that will change
the calculation.
There are methods of evaluating investments with differing lifespans, such as the Equivalent
Uniform Annualized Cost (EUAC) or Capitalized Cost, which give an average cost per year.
We cover these optional topics in Appendix B.
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394 Amy Goldlist
6.8 Cost Comparisons
All of the methods presented in this chapter try to answer the question: Should we invest?
If there was a clear answer to this question that we could teach you, the world of business
would be much simpler. Every time we need to make a decision, we will have ot evaluate the
criteria given here, as well as looking at the risks and rewards. The concept of risk will be
covered more thoroughly in Business Statistics.
Here are the tools we have covered:
1. PROFIT (Revenue – Costs) – if we do not end up with a profit, than it is useless to
evaluate the NPV or IRR.
2. PAYBACK: How long until we get our original investment back? When we
choose, a shorter payback is preferred.
3. NET PRESENT VALUE (NPV): When we evaluate the Profit with interest, is it
positive?
4. INTERNAL RATE OF RETURN (IRR): What is our actual rate of return?
Your Own Notes
Are there any notes you want to take from this section? Is there anything you’d like
to copy and paste below?
These notes are for you only (they will not be stored anywhere)
Make sure to download them at the end to use as a reference
An interactive H5P element has been excluded from this version of the
text. You can view it online here:
https://pressbooks.bccampus.ca/businessmathematics/?p=841#h5p-1
Business Mathematics 395
395
396 Amy Goldlist
Chapter 6 Review Problems
1 For an investment of $15,000 now, a company can receive $9,250 one year from now, and
$9,200 two years from now. Check its earnings rate on this deal by comparing it to an account
using the following rates. For each part, fill out the following balance sheet
Time (year) Interest Deposit Withdrawal Balance
0 $15,000 $15,000
1 ? $9,250 ?
2 ? 9,200 ?
a. 10% effective
b. 20% effective
c. 15% effective
2 For the investment in the previous problem , find the NPV at each of:
a. 10% effective
b. 20% effective
c. 15% effective.
Relate each value to the results in (a), (b) and (c)
3 A finance company has an opportunity to purchase three promissory notes from a broker for
a total of $140,000 cash. The first note would pay $20,000 in one year, the second would pay
$70,000 in two years and the third $80,000 in three years. The finance company has set for
itself a minimum acceptable rate of return of 10% effective.
a. Evaluate the net present value of the investment at the company’s MARR.
b. State your conclusion about the acceptability of the profitability of this deal.
4 A project is to cost $60,000 immediately and to produce net cash inflows of $20,000 at the
end of the first year, $30,000 at the end of the second year and $25,000 at the end of the third
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year. At the end of the third year the business will be sold for $5,000. The company aims at a
rate of return of 15% effective.
a. Show whether or not the company will achieve its objective.
b. Find the value of the bonus paid at the start which would cause the project to earn
exactly 20% effective.
5 Samuels Co. plans to start a repair business. It would provide net returns of $35,000 at the
end of the first year, and $50,000 at the end of each of the next three years. The equipment
would then be sold for $11,000 and the business terminated. Samuels aims at a rate of return of
15%. How much should Samuels be willing to invest (now)?
6 Williams Co. has the chance to start a transportation company in the north. It would require
$108,000 to start the business and it would provide net returns of $35,000 at the end of the first
year, and $40,000 at the end of each of the next two years. The equipment would then be sold
for $9,000 and the business terminated. Williams aims at a rate of return of 12%.
a. Find the NPV at 12% effective, and the IRR. Should Williams Co. start this
company?
b. If the government wanted to subsidize Williams so that it would earn 15%
compounded annually, what subsidy paid at the end of the three years would be
required?
7 PA Lumber Co. is bidding on the right to cut lumber from a private forest for 10 years. The
amount of lumber permitted to be cut each year would allow PA to produce a net cash flow
(excluding cost of cutting rights) of $120,000 a year. If PA aims at a rate of return of 12.5%,
how much could it afford to bid as a lump sum for the cutting rights?
8 Iffi Co. plans to install insulation in a building which it plans to use for six years. The
insulation and its installation will cost $193,000 and will result in savings of $37,000 a year
and will increase the residual value of the building by $90,000. Find the internal rate of return
of the planned installation of the insulation.
9 Tessa Quaid has purchased a 10-year video rental franchise for $12,000. She will have to
invest another $22,000 in the store immediately. She expects returns of $8,000 a year at the end
398 Amy Goldlist
of each year. Will TQ make a rate of return of at least 15%? What will be the internal rate of
return?
10 TW Cable Co. plans to begin operation in a large town and to expand later to a nearby small
town. TW estimates its net cash flows to be:
Time (year) Cash Flow
0 -$600,000
1 100,000
2 100,000
3 -50,000 (expansion)
4-9 150,000
10 700,000 (includes sale of business)
Find the net present value at 17% and the internal rate of return.
11 A food concession at an airport has a five year life and costs $130,000. Initial investments in
equipment, training and inventory amount to $75,000. The concession’s operation is expected
to produce net cash inflows of $50,000 a year and the residual value of equipment and the
ending inventory are expected to total $30,000.
a. Find the NPV at 15% effective.
b. Find the internal rate of return.
12 The owners of a small business are considering three offers from potential purchasers:
Offer #1: $100,000 cash.
Offer #2: $80,000 now and $10,000 at the end of each year for five years.
Offer #3: $25,000 now and $25,000 at the end of each year for five years.
Which offer is most attractive if money is worth 9% to the owners?
13 A “10%” Government bond is to pay interest of $100 per year for nine years. Find the
price investors should be willing to pay for the bond if they want to earn:
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a. 15% effective.
b. 10% effective.
c. 5% effective.
14 The Titan Package Co. finds that a $95,000 investment in automated packaging equipment
will save it $20,000 a year for the next 10 years. The equipment will have no value at the end
of the 10 years. If the company aims at a rate of return of 15% effective should it purchase the
equipment? Why/why not?
15 The Fast Food Co. can expand into either of two locations. Location A will need an
expenditure of $130,000 now and will result in a yearly net cash inflow of $35,000 (at the end
of each year) for 10 years and no other benefits. Location B will need expenditures of $150,000
now and $50,000 in one year. It will then produce a yearly net cash flow of $45,000 a year for
the following 12 years (all at the end of each year). In location B there would also be no other
benefits. Which location gives the higher net present value at a required rate of return of 16%
effective per year?
16 You are considering a home-based business. You would like to open a kennel. You estimate
that your start-up costs will be $250,000 this year and you will need to spend another $10,000
next year. You expect revenues of $35,000 per year for the first two years and $42,000 for the
next three years. Your expenses are thought to be $6,000 per year. At the end of five years you
plan to sell the kennel and anticipate receiving $255,000. You want to earn a minimum of 14%
compounded annually on your investment. All revenue occurs at the end of the year and all
expenses are paid at the beginning of the year.
a. Calculate the payback for this project.
b. Calculate the net present value for this project. Should you open a kennel?
c. If you require only 12% effective should you open a kennel? Why or why not?
d. How much should the start-up costs be reduced by so that you would be willing to
open a kennel? Use a MARR of 14%.
e. Your estimate of the kennel’s selling price may be too low. You think you may be
able to sell the kennel for more than $255,000. What is the minimum selling price
you must have so that you would be willing to open the kennel? Assume the start-
up costs remain unchanged. Round to the nearest $. Use a MARR of 14%.
f. Calculate the internal rate of return (IRR) for this project. Should you open a kennel
if you want to earn at least 14% compounded annually on your investment?
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17 You are considering purchasing a home-based business that uses AI to create customized
meal plans. You think that you can buy the business from the current owner for $120,000. You
expect revenues of $40,000 per year. Your expenses are thought to be $20,000 per year for the
first 2 years and $10,000 per year for the last 3 years. At the end of five years you plan to sell
the business, retire and move to Las Vegas. You anticipate you could sell it for $130,000. You
want to earn at least 16% compounded annually on your investment?
a. Calculate the payback for this project.
b. Find the NPV. Should you buy the business?
c. How high could the purchase price of the business be and still make it worthwhile
for you to buy the AI business?
d. You think you might be overly optimistic in your estimate of the selling price of the
business. What is the lowest selling price that you can accept and still be willing to
buy the business? Round to the nearest $.
e. Calculate the internal rate of return (IRR) for this investment. Should you buy the
business?
f. Your accountant tells you that your projected annual revenue is too high. What is
the maximum annual decrease in revenue you could withstand and still have this
investment be worthwhile?
18 You want to start up a business in a mall selling CFL souvenirs. You plan to spend $80,000
in start up costs. You anticipate that you will break even in the first year and second year,
make $20,000 the third, and make $20,000 each year after that. You plan to sell the business
for $100,000 at the end of the sixth year. You want to earn at least 15% effective on your
investment.
a. What is the net present value of your business plan?
b. What is the internal rate of return?
c. Should you undertake the business plan? Why or why not?
d. You think you might be overly optimistic in your estimate of the selling price of the
business. What is the lowest selling price that you can accept and still undertake the
project? Round to the nearest dollar.
19 You are a restaurant owner and are considering expanding your business by buying the
recently vacated store next door. The purchase price is $50,000. You will also need to spend
an extra $35,000 in re-modeling. The larger dining room is expected to generate net profits of
$12,000 a year for the first 4 years and $15,000 a year for the next 6 years. At the end of 10
years, you will retire. You expect to be able to sell the restaurant for $70,000. You want to earn
a minimum of 20% on your investment.
Business Mathematics 401
a. What is the net present value?
b. What is the Internal Rate of Return?
c. Should you expand the restaurant? Why or why not?
d. You hear news reports that real estate prices may rise over the next few years and
that this is especially true for commercial real estate. You now think you can sell the
restaurant for more than $70,000. What is the minimum selling price you must have
so that you would be willing to expand the restaurant? Round to the nearest $.
20 You have decided to start a business selling vitamin supplements. You estimate that you will
have $200,000 in start-up costs. Annual expenses are expected to be $30,000 and revenue is
expected to be $50,000 per year for the first 2 years and $75,000 in each of the next 3 years.
You plan to sell the business at the end of 5 years for an estimated $250,000. You want to earn
at least a 15% rate of return on your investment (MARR is 15%).
a. Should you invest? What is the IRR?
b. What is the NPV? Should you invest?
c. Your accountant tells you that your estimated selling price is too high. What is the
minimum selling price you could accept and still have this investment be
worthwhile?
d. One of the supplements you were going to sell has recently been banned. This will
cause a substantial decrease in your revenue. What is the maximum annual decrease
in revenue you could withstand and still have this investment be worthwhile?
21 You are operations manager for Flatugas, a small natural gas producer in the Peace River
Region of British Columbia. You are considering investing in a new technology that captures
more of the gas from the wellhead. The technology will cost $1,000,000 and operating costs
are estimated at $25,000 per year payable at the beginning of each year. The technology should
result in $250,000 in extra natural gas revenue per year and have the added benefit of reducing
greenhouse gas emissions. The technology is expected to last for 5 years when it could be
sold back to the manufacturer for a guaranteed price of $249,908.70. The company has a 15%
MARR and requires a payback period of 4 years or less.
a. What is the payback period of the investment? Should you invest?
b. Calculate the IRR for the investment and explain how you would use IRR to make a
decision on whether to invest or not.
c. Calculate the NPV. Should you invest?
d. The director of finance reviews your analysis and tells you that a MARR of 13% is
more appropriate for the project. He also informs you that the company should be
able to sell clean air credits under the Paris agreement. What annual revenue would
402 Amy Goldlist
be required from the credits to make this a worthwhile investment at a 13% MARR?
(Assume 5 equal payments at the end of the year).
22 The Blue Sky Resort plans to install a new chair lift to serve a new ski area. Construction
of the lift is estimated to require an immediate outlay of $190,000. In addition, the company
must spend $30,000 today and $30,000 per year for the next 3 years to clear and groom the
new area. The life of the lift is estimated to be 11 years with a salvage value of $80,000. Profits
from the lift (excluding the cost of grooming and clearing) are expected to be $40,000 per year
for the first five years, and $70,000 per year for the next six years. The company wants to earn
a minimum of 14% effective on the investment. All revenue occurs at the end of the year and
all expenses are paid at the beginning of the year.
a. Determine the net present value. Should they go ahead with the project? Why or
why not?
b. Determine the internal rate of return. Should they go ahead with the project? Why
or why not?
c. Determine the payback.
d. You hear on the news that we might be in a recession for the next few years. You
now wonder if your salvage value estimate of $80,000 is too high. What is the
lowest salvage value you can accept and still undertake the project? Round to the
nearest $.
23 A project is to cost $60,000 immediately and to produce net cash flows of $20,000 at the
end of the first year, $30,000 at the end of the second year, and $25,000 at the end of the third
year. At the end of the third year the business will be sold for $5,000.
a. If the company requires 15% effective, will it be able to achieve its goal?
b. Find the value of the bonus paid at the start which would cause the project to earn
exactly 20% effective.
24 Due to a restricted capital budget, a company can undertake only one of the following 3-year
projects. Both require an initial investment of $650,000 and will have no significant salvage
value at the end. Project X is anticipated to have annual profits of $400,000, $300,000, and
$200,000 in successive years, whereas Project Y’s only profit, $1.05 million, comes at the end
of year three.
a. Calculate the IRR of each project. On the basis of their IRRs, which project should
be selected?
b. On the basis of NPV, which project should be selected if the firm wants to earn at
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least 14% effective on their investment?
c. On the basis of NPV, which project should be selected if the firm wants to earn at
least 11% effective on their investment?
Notes
1. a. Final Balance = -$1225, so this investment earns more than 10%
effective
b. Final Balance = $1300, so this investment earns less than 20%
effective
c. Final Balance = $0, so this investment earns exactly 15% effective
2. a. NPV = $1,012.40
b. NPV = -$902.78
c. NPV = $0
3. NPV = -$3,861.76. The deal does not earn 10% effective MARR.
4. a. NPV = -$198.90 at 15% effective. The company does not achieve its
objective.
b. The NPV of the cash flows using i = 20% effective is -$5,138.88. To
earn 20%, the NPV must = $0. Therefore the size of the bonus must
be $5,138.89 to have a net project cost $54,861.11 instead of
$60,000.
5. NPV = $135,994.73
6. a. NPV = -$9,985.01 at 12% effective and IRR = 6.8305% for this three
year project.
b. To earn 15% effective, NPV must = $0. Without the subsidy, the NPV
at 15% is -$15,101.18. This is the PV of the required subsidy at the
end of the third year. Therefore, the subsidy = future value of this
amount at 15% in three years or $22,967.01.
7. NPV = $664,371.70. This is the amount PA can afford to bid.
8. IRR = 12.7118%
9. NPV at 15% effective = $6,150.15. TQ will make more than 15% effective. IRR =
19.6004%
10. NPV = $9,076.27; IRR = 17.29597% effective.
11. NPV = $22,476.94 IRR = 10.5635%
12. NPVs are $100,000, $118,896.51 and $122,241.28 respectively. Offer #3 is the most
attractive one at 9% effective.
13. A 10% bond implies that if the bond is purchased for $1,000, it will generate $100
404 Amy Goldlist
annual interest, and the bondholder will receive the $1,000 bond investment back
at the end of the ninth year. To solve this problem, find the PV of nine annual cash
flows of $100 and the $1,000 paid at the end of the ninth year using the target
interest rate. a. $761.42 b. $1,000.00 c. $1,355.39
14. NPV = $5,375.37 Yes, the company should purchase the equipment.
15. NPV for A is $39,162.96 NPV for B is $8,508.47. Location A has the higher NPV at
16% effective.
16. a. 4+136/297 years = 4.46 years
b. -$17,152.42 ? No, because the NPV is negative.
c. The NPV is +$1,111.42 which is above zero so you should open a
kennel.
d. $17,152.42
e. minimum selling price is $255,000 + $33,025.52 = $288,026
f. 12.12%, No because the IRR is below the minimum acceptable rate of
return of 14%.
17. a. 4+30/170 = 4.176 years
b. $16,263.94 Yes buy the business because the NPV is positive (earning
more than 16% per year).
c. Purchase price could increase by up to $16,263.94 to $136,264.
d. Lowest selling price you could accept is $95,840.
e. 19.45%/year. This exceeds the required rate of 16%/year so buy
business.
f. Up to a $4,967.16/year decrease in revenue.
18. a. NPV = +$6408 .24
b. IRR = 16.73%
c. Yes, undertake the business plan because the NPV is positive and the
IRR is 16.73% which exceeds the minimum required return of 15%.
d. $100,000-$14,822.65=$85,177 is the minimum selling price for the
business.
19. a. -$18,573.73
b. 14.77%
c. No, because the NPV is negative and the IRR = 14.77% < MARR of
20%.
d. $70,000 + $115,004 = $185,004 is the minimum selling price.
20. a. 17.25% > 15% MARR so yes invest.
b. +$19,413.74. Yes, since the NPV is positive.
c. $210,952 (rounded to nearest dollar)
d. $5,791.42/year
21. a. 4.25 years so no, it is longer than 4 years.
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b. 9.95% is below 15% so no, do not invest.
c. -$134,086.89 which is below zero, therefore do not invest
d. $24,000/year
22.
NPV =+$7,979.31; yes, undertake the project because the NPV is
positive
IRR = 14.58% exceeds the MARR of 14%
6.57 years
$80,000 -\33,722.42 =$46,277.58 is the minimum salvage value.
23.
NPV = -$198.90; no, the company does not achieve its objective.
The bonus must be $5,138.89 to make the NPV equal zero
24. 1. Project X: 20.8% Project Y: 17.3% Project X has the higher IRR and
should be selected.
2. NPV: Project X: $66,712 Project Y: $58,720 Project X has the
higher NPV and should be selected.
3. NPV: Project X: $100,085 Project Y: $117,751 Project Y now has
the higher NPV and should be selected.
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Chapter 7: Extra Problems
The following groups of questions may be useful in studying for tests.
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Chapter 1 and 2 Review
1 Gems Inc., an upscale jewelry store, purchased a diamond ring for $2,500 less 40% and 5%.
The store’s average per unit operating expenses (overhead) is 30% of cost. The “regular selling
price” of the ring is established so that if the ring is sold in a “20% off sale” the net profit at the
reduced price will be 20% of cost.
a. What is the reduced price of a ring in a “20% off’ sale?
b. What is the “regular price” of the ring?
c. What is the net profit if the ring is sold at the “regular price”?
2 Samsong Inc., a TV manufacturer lists its deluxe models for $450 each, less trade discounts
of 15% and 8%. A retailer wants to make a net profit of 10% of the selling price. If expenses
are 15% of the selling price, at what price must he sell the TV?
3 The Alfacenturi Corp received an invoice dated July 12th for $5,400 with terms 3/10, 1.5/20,
n/45. ABC made a payment of $2,000 on July 20th, and a second payment on July 31st that
reduced the balance owing to $1,000. Find the size of the second payment.
4 An item that normally sells for $550 is put on sale for $357.50. Find the rate of markdown.
5 Radio Shock Electronics Co. makes televisions. Radio Shock sold a number of TV sets to a
wholesaler at $399.90 per set, after discount rates of 15%, 9% and 6%.
a. What is the list price of a TV set?
b. What is the size of the 3rd discount (in dollars)?
6 A retailer has a policy of maintaining a margin of 60% on all items.
a. What is their rate of markup?
b. Another company maintains a 60% rate of markup on all items. What is their
percent margin?
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409
7 A company purchases a line of basketball sneakers for $60 from Niko Inc. and sells them for
$120.
a. What is the markup in dollars?
b. What is the percent margin?
c. What is the rate of markup?
8 A pair of radically shaped skis costs the retailer $600 less chain discounts of 50%, 30% and
10%. The retailer maintained a 65% margin on all items. Since spring was fast approaching the
retailer drastically reduced the selling price of the skis by 60%.
a. What was the regular selling price?
b. What is the sale price?
c. What is the rate of markup? (use the sale price not the regular price)
9 What is the cost of an item that sells for $80 if the rate of markup is 60%?
10 You know that a retailer makes $450 on the sale of a snowboard and the retailer has a rate of
markup of 60%.
a. What is the selling price of a snowboard?
b. What is the percent margin?
11 Filters-R-Us makes plastic coffee filters . It costs the company $6,500 to make 2,000 filters
and $8,000 to make 4,000 filters. Assume the relationship between cost and the number of
coffee filters produced is linear.
a. Find an equation that determines the cost, based on the number of filters produced.
b. How much would it cost to produce 3,000 coffee filters?
12 Dick, Ed and Fran formed a partnership. The partnership agreement requires them to provide
capital when and as required by the partnership in the ratio of 7:9:8, respectively.
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a. If the total required initial investment was $96,000, how much did each contribute?
b. One year later, Dick’s share of another injection of capital was $10,500.
1. What is the total investment made by the partnership including the
initial investment?
2. What is Ed’s share of the entire investment?
13 At the Mac’s store in Burnaby a 500 ml bottle of sparkling water sells for 1.19CAD. Convert
this price to US dollars per gallon.
Rates:
1 CAD= 0.7125 USD
1.0567 Quarts= 1 Litre
1 Gallon = 4 Quarts
1,000 ml= 1 Litre
14 A diamond ring cost a jeweler $4,200. He requires a margin of 45%.
a. At what price should he sell the ring?
b. What rate of markup did he realize?
15 You are planning to open a neighbourhood ice cream shop and are doing some financial
analysis. You can lease the shop for $4,300 for one month, salaries will cost $12,100 per month,
hydro and miscellaneous expenses will be $450 per month. You can sell ice cream cones for
$4.75 each. Each ice cream cone costs you $2.05.
a. Write down the revenue and cost equations.
b. Determine the number of ice cream cones you must sell in a month to break even.
What are your total sales (in dollars) at the breakeven point?
c. You’d like to make a profit of $4,000 in one month. Determine the number of ice
cream cones that you would need to sell.
d. Find the profit/loss if monthly sales (in dollars) are $14,250.
16 Liquidation Electronics sells an article for $1,020.00 less 25% and 15%. A competitor
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carries the same article for $927.00 less 25%. What further discount must the competitor allow
so that its net price is the same as the discount store’s?
17 Find the equation of the line that passes through the points (-2, 5) and (3, 2).
18 The license to operate a taxicab costs $2,000 per month. Insurance is $250 per month and
maintenance averages $225 per month. Fuel costs $0.20 per kilometre. Taximeter rates are
regulated and set at $0.45 per kilometre.
a. Determine the breakeven point for operating a taxicab.
b. What are the units for x, and what are the units for the slope?
c. What will be the profit/loss if 4,900 km are driven?
d. How many kilometres must be driven to make a profit of $2,500?
19 Forever-Green Press prints its elegant advertising calendars in November and sells them for
$9.60 each. The Company’s production capacity for this run is 60,000 calendars at a variable
cost of $3.80 each. The fixed costs (including artwork and licensing fees) allocated to this
production run are $232,000. Assume that the company can sell all the calendars it produces.
a. What is the breakeven point in (a) units; (b) sales; dollars; (c) percent of capacity?
b. How many calendars must be printed and sold if they want to make a profit of
$87,000?
c. If the plant operates at full capacity and November’s entire calendar production run
is sold, what profit (or loss) will be made?
d. Graph the cost and revenue equations (same graph). Mark the Breakeven Point on
the graph.
e. If the variable cost increases by 25%, the fixed costs increase to $240,000, what
new selling price should they charge if they want to earn a profit of $50,000, from
the sale of 40,000 calendars?
20 Plutonium fuel rods are sold at a list price of $5,000 with chained discounts of 15% and
25%.
a. Calculate the net price of a fuel rod.
b. Calculate the single equivalent discount rate.
412 Amy Goldlist
c. The company selling the fuel rods would like to add a third discount to bring the
overall discount rate to 40%. How large should the third discount be?
d. A competing firm offers a single discount of 35%. This amounts to a discount of
$1,680 off the list price. What is the list price?
21 Two long distance phone companies offer the following quotes:
PLAN A: $22 per month plus $0.04 per minute.
PLAN B: $15 per month plus $0.08 per minute.
Determine the point of indifference (i.e., the number of minutes per month where the costs are
equal).
22 Compute the single discount equivalent to the discount series 40%, 10%, 5%.
23 An invoice for $3,200.00, dated April 20, has terms 2/10, n/30. What payment must be made
on April 30th to reduce the debt to $1,200.00?
24 On May 30th, Restorative Services Ltd. received an invoice for $2,600, terms 1.5/10, n/
30. On June 9, Restorative Services made a partial payment of $1,379.00 on the invoice. How
much is still owing after the payment?
25 Albert, Bob and Chuck form a partnership and agree that half of the annual profit be
distributed in proportion to the total number of hours worked in the business during the year
and the other half in the ratio 4:3:2, respectively. The hours of work for Albert, Bob and Chuck
were 300, 200 and 100, respectively. How much of the $180,000 profit should Chuck receive?
26 The regular selling price of merchandise sold in a store includes a margin of 60%. During a
sale, an item that cost the store $250 was marked down 40%. For how much was the item sold?
27 An item that cost the dealer $500 less 20%, 15%, carries a price tag at a markup of 150% of
cost. For quick sale, the item was reduced 30%. What was the sale price?
Business Mathematics 413
28 An appliance shop reduces the price of an appliance for quick sale from $1,560.00 to
$1,195.00. Compute the rate of markdown.
29 Log-it-All Forest Products sells Grade-A sheets of plywood at trade discounts of 25%, 3%.
A competitor has been selling the plywood at the same list prices but with trade discounts of
20%, 12.5%. Log-it-All wants to beat the competitor’s prices by offering a third trade discount.
At least how big must the additional discount rate be to meet this objective?
30 What amount must be remitted if the following invoices, all with terms 4/10, 2/30, n/60, is
paid on May 10?
$850.00 less 20%, 10% dated March 21
$960.00 less 30%, 16.67% dated April 10
$1040.00 less 33.33%, 25%, 5% dated April 30
Notes
1. a. $2,137.50
b. $2,671.88
c. $819.38
2. $469.20
3. $2,303.07
4. 35%
5. a. $550
b. $25.53
6. a. 150%
b. 37.5%
7. a. $60
b. 50%
c. 100%
8. a. $540
b. $216
c. 14.3%
9. $50
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10. a. $1,200
b. 37.5%
11. a. C = $5,000 + $0.75x, x = the number of filters produced.
b. $7,250
12. a. Dick: $28,000, Ed: $36,000, Fran: $32,000
b. $132,000 in total; Ed $49,500
13. 6.42 USD per Gallon
14. a. $7,636.36
b. 81.82%
15. a. R = $4.75x, C = $16,850 + $2.05x where x = the number of ice cream
cones.
b. 6241 ice cream cones, $29,644.75
c. 7,723 ice cream cones
d. loss of $8,750
16. 6.47% or 6.5%
17. y=3.8 - 0.6x
18. a. 9,900 km
b. x = number of km driven, slope= $/km
c. -$1,250
d. 19,900 km
19. a. 40,000 calendars; $384,000; 66.7% of capacity
b. 55,000 calendars
c. $116,000
d. $12/calendar
20. a. $3,187.50
b. 36.25%
c. 5.8824% (or 5.88)
d. $4,800
21. 175 minutes
22. 48.7%
23. $1,960
24. $1,200
25. $35,000
26. $375
27. $595
28. 23.4%
Business Mathematics 415
29. slightly more than 3.78%
30. $1,635.04
416 Amy Goldlist
Chapter 3 and 4 Review
1 Sam agrees to make a payment of $1,000.00 in 3 months at 8% simple interest. How much
money is Sam borrowing today?
2 Alixe borrowed $5,000 today at 12% simple interest. How much will she owe the bank when
she pays off the debt in 180 days?
3 Myrna borrowed $5,000 from the bank some time ago at 9% simple interest. She paid off the
loan today with one payment of $5,675 . How many months ago did she borrow the $5,000?
(Hint: it’s easier to use I = Prt.)
4 You would like to save to purchase a Kia automobile. You deposit $3,500 into a GIC that
pays 8% interest, compounded quarterly. You need to save at least $5,000 to buy the car. If you
make no more contributions, how many years will it take you to reach your goal?
5 Lindsay made an investment of $1,500.00 made 42 months ago. It is now worth $2,110.65.
What nominal rate of interest, compounded semi-annually, did this investment earn?
6 Answer the following
a. What nominal rate, compounded semi-annually, is equivalent to 9% compounded
quarterly?
b. What nominal rate, compounded monthly, is equivalent to 9% compounded
quarterly?
c. What nominal rate, compounded quarterly, is equivalent to 12% compounded
annually?
7 A debt of $6,000 due today and $8,000 due 2.5 years from today is to be paid off with two
payments. The first payment is to be made six months from today, and a second payment,
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$5,000 larger than the first, 15 months from today. What should the two payments be if money
is worth 9%, compounded quarterly? Use 15 months as the focal date.
8 Pavel was supposed to pay $10,000 today. Instead, he arranged with the bank to pay $3,000
12 months from today, followed by two equal payments in 18 months and in 30 months from
today. The interest rate is 8% compounded quarterly. Calculate the size of each payment? Use
30 months as the focal date.
9 Anna borrowed $3,000 from her line of credit 8 months ago and a further $5,000 3 months
ago . She arranged with the bank to pay off the line of credit with two payments. The first
payment, 6 months from today, will be twice as large as the second payment made one year
from today. The bank charges you 9% interest, compounded monthly. Use 6 months as the
focal date.
a. Find the size of each payment.
b. How much interest will she pay?
10 A debt of $15,000 is due today. Instead, the borrower agrees to make one payment of $5,000
in 6 months and 2 equal payments in 3 months and 12 months from today, with simple interest
charged at 6%. What is the size of each payment? Use 6 months as the focal date.
11 Matt is going to purchase a car and is given two options to pay.
He can pay $25,000 in cash today, or
$12,000 after one year, and a second payment of $15,000 after 18 months.
Which option is better? Your answer should be stated in terms of today’s dollars. The interest
rate is 7% compounded semi annually?
12 Mortel Inc. borrows some money today. In 10 months it must pay back $854,962.11. How
much did Mortel borrow today if it is being charged 8% interest, compounded monthly?
13 An investment of $5,000 made 27 months ago is now worth $7,756.64. What nominal rate
of interest, compounded quarterly, did the investment earn?
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14 Julia is seeking an investment and has narrowed her search to three different funds.
Fund A offers a return of 12.2% compounded quarterly.
Fund B offers a rate of 12.4% compounded semi- annually.
Fund Coffers a rate of 12% compounded monthly.
Convert all rates to effective rates and explain which rate is the best investment.
15. Serena invests $3,000 in a bond fund today. Her investment earns 4% compounded semi-
annually in the first year, 6% compounded quarterly in the second year and 8% compounded
annually in the third year.
a. How much would the $3,000 investment be worth at the end of the third year?
b. If, in the fourth year the bond fund loses money and the value of her investment
falls by $200, what average annual rate of return, compounded quarterly, did she
earn for the four years that she held the investment?
c. A friend has invested in a different bond fund and says she tripled her money in
eight years. What nominal rate of interest, compounded semi-annually, did she
earn?
d. At 9% compounded quarterly, how long would it take you to triple your money?
16 Mike purchased a two-year term deposit that pays interest at 5.7% compounded quarterly.
At the end of the two years he renews the term deposit plus accumulated interest at 6.2%
compounded semi-annually for an additional three years. At the end of the five years he has
$6,724.84? How much did Mike initially pay for the term deposit?
17 Yolanda has a debt that comes due on November 26th and another that comes due on April
6th of the following year. She would like to celebrate Valentines Day by paying off the debts
with a single payment on February 14th. How much should she pay if interest is 5% simple and
each debt was for $1,000? (Use February 14th as the focal date.)
18 Davis borrowed $5,000, and 18 months later repaid the loan with a single payment of
$5,600. Calculate the simple interest rate.
Business Mathematics 419
19 Your investment of $8,500 grew to $8,893.09 at an interest rate of 4% simple. How many
days was it invested?
20 How much money do you need to deposit to earn $850 in interest in 18 months with interest
at 6% compounded quarterly?
Notes
1. $980.39
2. $5,295.89
3. 18 months
4. 4.5 years
5. j2=10.0000%
6. a. j2 = 9.1012%
b. j12=8.9333%
c. j4=11.4949%
7. First payment: $4,284.02; second payment: $9,284.02
8. $4,231.33 is paid at 18 and 30 months
9. a. First payment: $5,871.56; second payment $2,935.78. b. $807.34
10. $5,262.17
11. In today ' s dollars, second option is worth $24,731.27< $25,000 , second option is
cheaper.
12. $800,000
13. j4 = 20.0000%
14. A: 12.7696%, B: 12.7844%, C: 12.6825% , B is the best investment.
15. a. $3,577.75
b. j4= 2.9759%
c. j2= 14.2151%
d. 12.34 years
16. $5,000
17. $2,004.02
18. 8.0%
19. 422 days
20. $9,096.43
420 Amy Goldlist
Chapter 5 Review
1 What will be the amount in an RRSP at the end of 20 years if monthly contributions of $500
are made at the end of each month and the RRSP earns 6% compounded monthly for the first
15 years and 7.5% compounded monthly for the remaining 5 years?
2 You have $4,000 in your savings account today. You will deposit $500 per month starting one
month from now.
a. How much will you have in your savings account at the end of 5 years if you earn
6% compounded monthly?
b. How much interest would you earn?
3 You have just purchased a preferred share for $50.00. The company pays a dividend of $1.00
every quarter. One year later, when you sell the share, the interest rate is 10% compounded
quarterly. How much will you gain or lose? (The next dividend is due in three months.)
4 You purchase a car and finance $10,000. The loan is to be repaid with monthly payments of
$200 made at the end of the month for 5 years. What effective rate of interest is being charged?
5. You begin a savings plan. Starting now and for ten years, you deposit $100 per month
into an account that pays = 6%. After 10 years, how much interest will you have earned?
6. Upon graduation, you have a student loan of $15,000. The most you can afford to pay is
$550 per month. How long will it take you to repay the loan with payments of $550 per month
starting in one month if the interest rate is 6% compounded monthly?
7 You purchase a car for $25,000 and make a 20% down payment. Interest is charged at 6%
compounded monthly, and you will make 6 years of monthly payments. Find the size of the
monthly payment if your first payment is in one month. (Round payment up to next cent.)
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421
8 You have decided to take a passive approach to fitness and order the ‘Ab-Cruncher’ from
the shopping network. The Ab-Cruncher can be purchased for four easy monthly payments of
$19.95 starting in one month. If interest is charged at 112 = 12%, find the cost of financing.
9 You are 25-years-old and want to retire when you turn 55. Starting today you will deposit
$200 into an RRSP every month for 30 years. You will then use the accumulated funds to
purchase a 10-year annuity with the first withdrawal one month after your 55th birthday.
Assume that the RRSP and the funds invested in the annuity earns 6% compounded monthly.
a. Find the size of the monthly withdrawals.
b. You decide that you will need $3,000 per month to live on when you retire at age
55. How much extra money must you contribute to your RRSP every month, so you
can withdraw $3,000 every month for 10 years?
10 You contribute $2,500 into your RRSP at the end of every quarter for 5 years. If your RRSP
earns 8% compounded quarterly, how much interest will you earn in the 5 years?
11 You borrow $50,000 and agree to make monthly payments for 15 years with the first
payment one month from now. Calculate the size of your monthly payments if the interest rate
is 9% effective.
12 Paula, a BCIT business student, borrowed $5,000 from her parents to cover expenses and
agreed to repay the debt with 20 equal monthly payments with the first payment due in 2 years.
The interest rate is 12% compounded monthly.
a. How large is each payment?
b. How much interest did Paula pay her parents?
13 Barton Fink bought a second hand car with $1,000 down and signed a contract agreeing to
pay $299 every month for 3 years at 9% compounded monthly. The first payment is made one
month later.
a. What is the cash price of the car?
b. How much interest will Barton have paid over the 3 years?
422 Amy Goldlist
14 A debt of $5,000 is to be paid off by making payments of $502.31 at the end of every 3
months. If interest is 12% compounded quarterly, how long will it take to pay off this debt?
15 You need to set aside $2,500 for your vacation in Moose Jaw to celebrate graduation from
BCIT 21 months from today. If your bank pays 9% compounded quarterly, how much must
you deposit into your account at the end of every quarter to have $2,500 in 21 months?
16 A home entertainment centre may be purchased for $2,299 or by making payments of $199
to be made at the end of every month for 12 months.
a. What nominal rate of interest, compounded monthly, is being charged?
b. What is the effective rate of interest rate charged?
17 Fred Rock will deposit $500 in his RRSP every month for 15 years starting today. One
month after the last deposit, he will withdraw the money in equal monthly withdrawal for 10
years.
a. Find the size of the monthly withdrawals if the invested funds earn 7.0%
compounded monthly.
b. How much interest will Fred earn over the 25 years?
18 Sam Shepherd wants to borrow some money from the Federal Business Development Bank
to open a doggy daycare business. The terms of the loan specify that no payments are required
in the first year of the business. Starting in the second year, payments of $1,000 at the end of
every three months are required for a period of five years (i.e., until the end of the sixth year of
the business). If interest is 8% compounded quarterly, how much money is he borrowing?
19 Sam N Ella has just turned 50 years old today. He has decided to start saving for his
retirement. He plans to retire when he turns 65. He will set aside $1,500 per quarter, starting
today, into a money market account that earns 4% compounded monthly.
a. How much will Sam have on the day he retires?
Business Mathematics 423
b. How much interest will he have earned?
20 Judith Fortune recently won $500,000 from a hospital lottery. She has decided to put some
of her winnings into an RRSP. Miss Fortune is 55 years old and plans to retire at age 65. She
would like to receive $2,000 per month for twenty years starting on her 65th birthday.
a. If she can earn 6%, compounded monthly, how much of her winnings should she
put into her RRSP today?
b. Miss Fortune has changed her mind and has decided to put half of her winnings into
an RRSP instead. How much extra will she receive per month when she retires?
21A dining room suite sells for $4,999. It may be purchased from Junk-O Furniture by making a
down payment, followed by monthly payments of $245 for a year-and-half. (The first payment
is one month later.) Interest is j12 = 12%. Find the size of the down payment.
22An alumnus wants to donate a sum of money to his Alma Mater that will provide a
scholarship of $750.00 every 6 months in perpetuity. If money can be invested at 6%
compounded semi-annually and the first $750.00 is to be awarded at the end of six months how
much must he donate to the school today?
23 You have decided to purchase preferred shares of No-Vision Eye Treatment Centre that pays
a semi-annual dividend of $1.75 per share. Current interest rates are 7% compounded semi-
annually and the next dividend is due in six months.
a. What should you be willing to pay per share?
b. Two years later the interest rates drop to 5% compounded semi-annually. You have
decided to sell your shares. Calculate your gain or loss per share if the next dividend
is due in six months – and the dividend per share remains the same.
24 You purchase 250 preferred shares of Silent Witness Security Systems Inc. that pays a
dividend of $2.25 per share every 3 months. Current interest rates j4 = 6%.
a. What would you be willing to pay for the 250 shares?
b. If interest rates rise to 8% compounded quarterly, how much would you expect to
gain or lose if you sell all of your shares? Note: the next dividend is due in 3
months and the dividend per share is still $2.25.
424 Amy Goldlist
Notes
1. $247,586.15
2. a. a. $40,280.42
b. b. $6,280.42
3. Loss of $10
4. j1 = 7.6777%
5. $4,469.87
6. 30 months
7. $331.46
8. $1.96
9. a. $2,241.59 per month
b. $267.67−$200=$67.67 /month
10. $10,743.42
11. $496.74
12. a. $348.33
b. $1,966.60
13. a. $10,402.59
b. $1,361.41
14. 3 years
15. $337.75
16. .
a. 7.0708%
b. 7.3045%
17. a. $1,840.10
b. $1,840.10× 120- $500×180 = $130,812
18. $15,106.20
19. a. $123,866.46
b. $33,866.46
20. .
a. a. $154,203.50
b. b. $3,242.47 - 2,000 = $1,242.47
21. $981.42
22. $25,000
Business Mathematics 425
23. .
a. $50/share
b. Gain of $20/share
24. .
a. $37,500
b. $9,375 (loss)
426 Amy Goldlist
Entire Course Review
1 The executor of Sam Jackson’s estate is to divide $880,000 between three charities, the
United Way, Heart Fund and Red Cross in the ratio of 6:3:2 respectively. How many dollars
will each charity receive?
2 A publisher sells its romance novels with chained discounts of 10% and 20%.
a. Find the single equivalent discount rate.
b. The publisher would like to add a third chained discount to bring the overall
discount to 31.6%. Find the third chained discount rate.
3 An invoice with terms 2/10, 1/20, n/45 for $3,000 dated November l was partially paid by a
$1,470 payment on November 10th and a second payment of $495 on November 20th. What is
the outstanding balance after the November 20th payment?
4 Black Bear Cupcakes received an invoice dated November 28th with terms 1.5/10, n/30 for
$8,050. On December 6th, Black Bear mailed a cheque for $4,925 in partial payment of the
invoice. What is the outstanding balance after the December 6th payment?
5 Chemco Co. sells its products at trade discounts of 25%, 10%. A competitor has been offering
products at the same list prices but with trade discounts of 30%, 10%. Chemco wants to beat
the competitor’s prices by offering a third trade discount. At least how big must the additional
discount rate be to meet this objective?
6 What single discount is equivalent to a chain discount of 15%, 10%?
7 If we want a single equivalent discount of 19.25% by using chain discounts of 15% and $x$,
what must the second discount rate be?
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427
8 A furniture manufacturer offers his clients discount rates of 22.5% and 10%.
a. What single rate is equivalent to this series of rates?
b. If a client of this manufacturer paid $1,395 for a sofa, at what price was the sofa
listed?
9 Find the selling price of an item bought for $1,050.00 if the margin is 30%?
10 If a company maintains a margin of 20% on all items it sells, what is the rate of markup?
11 The regular selling price of an item sold in a store includes a margin of 60%. During a sale,
an item that cost the store $240 was marked down 20%. For how much was the item sold?
12. The net price of an article is $612 after discounts of 15% and 10%. What was the list
price?
13 An item that cost the dealer $850 less 35%, 20% carries a price tag at a markup of 25% of
cost. For quick sale, the item was reduced 25%. What was the sale price?
14 Find the cost of an item sold for $2,800 to realize a rate of markup of 40%.
15 An article cost $300 and sold for $450. What was the rate of markup?
16 The markup on an item is $50. If the margin was 40%, what was the cost?
17 The price of an item is reduced for quick sale from $950 to $760. Compute the rate of
markdown .
428 Amy Goldlist
18 You are taking a holiday in Britain. You are taking 850 US dollars with you. How many
British pounds can you buy with this money?
Rates:
1 USD= 1.38358 CAD
1 British pound(£) =$2.21824 CAD
19. A manufacturer of major appliances provides the following information about the
operations of the refrigeration division:
Fixed costs per period are $26,880
Variable costs per unit are $360
Selling price per unit is $640, and
Capacity is 150 units.
Compute the break-even point:
a. in units.
b. as a percent of capacity.
c. in dollars.
20 Find the cost-equation for shipping one ton of a product if we know that it costs $500 to
ship it a distance of1,000 kilometres and $750 to ship it 2,000 kilometres. Assume that the
relationship between cost of shipping and distance shipped is linear. What would it cost us to
ship one ton of the product 1,500 kilometres? Check your answer by putting the two points
back into the equation.
21 The High Tech computer shop assembles and sells computers. After reviewing their
accounting data, the following was determined.
Fixed costs: $100,000 per year
Variable costs $1,450 per computer
Selling price per computer: $2,450
Capacity is 1,000 computers per year
a. How many computers must it sell per year to break even?
Business Mathematics 429
b. What is the total yearly sales (revenue) required to break even?
c. Determine the computer shop’s yearly profit if it sells 250 computers per year?
d. How many computers would need to be sold to make a $25,000 profit?
e. If the variable costs increase to $1,600 per computer, what should you increase the
selling price to if you want to earn a profit of $25,000 from the sale of 125
computers?
22 A small production company has 3 choices when renting a mobile crane for short periods:
Choice A:It can pay $200 per hour used.
Choice B: Pay an annual fee of $5,000 and pay only $100 per hour used.
Choice C: Pay an annual fee of $12,000 and pay only $50 per hour used.
a. Write the cost functions for each alternative.
b. Calculate the points of indifference (where the costs are equal).
i. between alternative A and B
ii. between alternative B and C
c. Determine the range of hourly usage where the three alternatives would be
preferred.
23 You take out a loan for $10,000. The most you can afford to pay is $425 per month at the end
of every month. If the interest rate on the loan is 7% compounded monthly, how many months
will it take you to pay off the loan? (No decimal answers!)
24 Kate made $600 monthly deposits at the end of every month for 5 years into an RRSP paying
8% compounded monthly. Immediately after the end of the 5th year the rate is reduced to 6%
compounded quarterly.
a. If neither deposits nor withdrawals were made during the next 10 years, how much
would Kate have in her account at the end of 15 years?
b. How much interest did Kate earn over the entire 15 year period of time?
25 You have just turned 30 years old. Starting today, you will make monthly contributions of
430 Amy Goldlist
$600 into a RRSP for 25 years. One month after your 55th birthday you will begin taking
monthly withdrawals with your last withdrawal on your 70th birthday.
a. Find the size of these withdrawals if the interest rate is 6.5% compounded monthly.
b. How much interest did you earn over the entire 45 years?
26 Barney wants to start saving for his retirement. Starting today, he will make monthly
deposits to his RRSP for 20 years. One month after his last deposit, he wants to withdraw
$4,000 per month for 10 years. Assume the invested funds earn 6% compounded monthly for
the entire time.
a. How much must Barney deposit into his RRSP per month to achieve his goal?
b. How much interest will Barney earn over the 30 years?
27 If you agree to make a payment of $1,500 in 3 months at 6% simple interest, how much
money are you borrowing today?
28 You borrow $6,000 on November 15, 2021 at 8% simple interest. How much do you owe
the bank when you pay off the debt on March 15, 2022? What is the cost of financing?
29 You borrowed $3,000 from the bank some time ago at 6% simple interest. You paid off the
loan today with one payment of $3,360. How many months ago did you borrow the $3,000?
30 A debt of $6,000 was due 6 months ago and a debt of $14,000 is due 5 months from today.
Instead, the borrower agrees to make 2 equal payments, to be made 3 months, and 8 months
from today, with simple interest allowed at 12%. What is the size of the payments? Use 5
months as the focal date.
31 You would like to save to return to school. You deposit $4,000 into a GIC that pays j4 =
7.44%. You have decided to return to school when your savings grow to at least $6,000. If you
make no more contributions, how many years will it take you to reach your goal?
Business Mathematics 431
32 An investment of $2,000 made 30 months ago is now worth $2,676.45. What nominal rate
of interest, compounded semi annually, did the investment earn?
33 What nominal rate, compounded semi-annually, is equivalent to 12% compounded
quarterly?
34 Debts of $800 due today and $900 due in 27 months are to be repaid with 2 equal payments,
1 year and 2 years from today. If the interest rate is 9% compounded quarterly, find the size of
the payments. Use 2 years as the focal date.
35 You are searching for a mutual fund.
Fund 1 offers a rate of return of 6.4% compounded quarterly.
Fund 2 offers a rate of return of 6.6% compounded semi-annually.
Convert all rates to effective rates. Which fund is the better investment?
36 You purchase a new car. The dealer offers you terms of 20% down and the remainder
financed over three years at an interest rate of 9% compounded quarterly. The cost of the car
is $21,640.79.
a. Find the size of your monthly payment if your first payment is due one month after
you purchase the car.
b. What is the cost of financing (i.e., how much interest will you pay)?
37 A TV set may be purchased on these terms: a down payment of 20% of the selling price is to
be made on the date of purchase, followed by 18 monthly payments of $40 each, with the first
payment one month after the date of purchase. If the interest rate charged is 10% compounded
monthly, what is the cash price (selling price) of the TV set?
38 A used car that sells for $15,000 may be purchased by making payments of $500 per month
for 3 years with the first payment due the day the car is purchased.
a. What nominal rate of interest, compounded quarterly, is charged?
432 Amy Goldlist
b. What effective rate of interest is charged?
39 You take out a loan for $12,000 with quarterly payments of $806.59 at the end of each
quarter. If the interest rate on the loan is 12% compounded quarterly, how many years will it
take you to pay off the loan?
40 John will purchase an annuity that will pay him $5,000 per quarter for 10 years beginning
when he turns 60 years of age. If John’s current age is 45 years and the invested funds will earn
7.0% compounded quarterly, what amount must he invest in the annuity today so he can collect
$5,000 per quarter for 10 years with the first payment on his 60th birthday?
41 A bursary fund for BCIT honour students is to be funded by a perpetual fund. The fund earns
interest at 8% compounded semi-annually and is to pay $2,000 every six months, with the first
scholarship paid in six months. Find the size of the initial funding that is required.
42 A friend has invested in a bond fund and says she doubled her money in five years. What
rate of interest, compounded semi annually, did she earn?
43 A person can buy a piece of land for $130,000 now or $60,000 now and $100,000 in 5 years.
Which option is better if money can be invested at:
a. 6% compounded quarterly?
b. 10% compounded quarterly?
44 Meryl just turned 50 years old. She put $50,002 into her RRSP today. She will leave the
money in her RRSP until her 65th birthday. She will then purchase a 10-year annuity with the
first withdrawal one month later. Use
a. What will be the size of the monthly withdrawals?
b. How much interest did Meryl earn over the 25 years?
c. Meryl has determined that she requires a larger monthly payment than that found in
part (a). She would like to receive $2,000 per month. How much additional money
must she transfer to her RRSP today so she can collect $2,000 per month instead?
Business Mathematics 433
45 You invest $4,000 in a mutual fund. Your investment of $4,000 earns the following returns.
Year Return
Year 1 j1 = 9%
Year 2 j2=12%
Year 3 j4 = 10%
What average nominal rate of return, compounded semi annually did you earn?
46 You purchase a new car. The dealer requires that you put $6,000 down followed by monthly
payments of $999 over four years. (The first payment is made one month after you buy the
car). The interest rate is 9.9% effective.
a. What is the cash price (selling price) of the car?
b. What is the cost of financing?
47 You contribute $2,500 into your RRSP at the end of every quarter for 5 years. If your RRSP
earns 8% compounded quarterly, how much interest will you earn in the 5 years?
48 A computer that sells for $3,999 may be purchased by making a down payment , plus a series
of month-end payments of $225 for one and a half years. If the interest rate is 9% compounded
monthly, what is the size of the down payment?
49 You are buying a house for $260,000 with a down payment of 20%. The interest rate is 8.5%
compounded semi-annually. The mortgage is amortized over 25 years for a 3-year term.
a. Calculate the size of the monthly payment. The lender’s policy is to round payments
up to the next whole dollar.
b. How many payments would be required?
c. How much interest will you pay in the first 3 years?
d. How much interest would you pay in the third year only?
e. You make an extra lump-sum payment of $40,000 at the end of 3 years. What is the
outstanding balance after this lump-sum payment?
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f. When you go to renew your mortgage at the end of
g. 3 years, the rates have fallen to only 5.5% compounded semi-annually for a 3-year
term. Find the size of the new payment.
h. The payment determined in part (f) is much smaller than you thought due to the
lower rate and lump sum payment. You have decided to increase the monthly
payment so that you pay off the remaining balance in only 12 years instead of 22
years. Find the size of the new payment. Round up to the next dollar.
i. Find the size of the final payment.
50 A summer cottage, valued at $120,000, may be purchased by paying a $20,000 down
payment and financing the balance with a mortgage at 9% compounded semi-annually and
monthly payments for 15 years.
a. Find the monthly payment. Round up to the next dollar .
b. How much of the 60th payment pays interest and how much goes toward the
principal?
c. After the 100th payment, how much of the original mortgage is still left to be paid?
d. After making 115 payments, what percent of the original debt will have been paid
off?
51 You are contemplating purchasing a business selling computer software over the Internet.
You estimate that the purchase price would be $40,000.
Your expenses should be $10,000 per year.
You expect annual revenues to be $15,000 for the first two years
and $20,000 each year after that.
You plan to sell the business at the end of six years and estimate
you will get $55,000.
Your MARR is 15% effective.
Assume all expenses are paid at the beginning of the year and
revenues are received at the end of the year. Time diagram is
required.
a. What is the IRR? Should you purchase the business? Why or why not?
b. Calculate the NPV. Should you purchase the business? Why or why not?
c. What is the highest purchase price you could pay and still be willing to buy the
business?
Business Mathematics 435
d. What is the lowest selling price you could tolerate and still be willing to undertake
the business? Round to the nearest dollar.
e. You accountant advises you that your annual revenue projections are too high. What
is the maximum annual decrease in revenue you could withstand and still have this
investment be worthwhile?
52 A food concession at an airport has a 7 year life and costs $200,000. Renovations will
cost you another $50,000. The concession’s operation is expected to produce net incomes of
$60,000 a year. The salvage value of the equipment and ending inventory are expected to total
$40,000. Your MARR is 20%.
a. Find the IRR. Would you buy this concession? Why or why not?
b. Find the NPV. Would you buy this concession? Why or why not?
c. By how much does the purchase price of the concession have to fall to make the
investment worthwhile? What is the new price?
d. Your accountant tells you your salvage estimate is too low. What is the minimum
salvage value you require to make the investment worthwhile?
e. The Canadian government has decided to offer a one-time subsidy to encourage the
creation of concession stands at the airport. The subsidy is received one year later.
How large of a subsidy would you require to make the investment worthwhile?
f. Your accountant advises you that your projected net incomes are too low. What is
the minimum annual increase in revenue you require to make the investment
worthwhile?
Notes
1. United: $480,000; Heart: $240,000; Red Cross: $160,000.
2. a. 28% b. 5%
3. $1,000
4. $3,050
5. 6.67%
6. 23.5%
7. 5%
8. .
a. 30.25%
b. $2,000
9. $1,500
10. 25%
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11. $480
12. $800
13. $414.38 is the sale price, the price before the 25% markdown was $552.50.
14. $2,000
15. 50%
16. $75
17. 20%
18. £530.17 (GBP)
19. a. in units, 96
b. as a percent of capacity, 64%
c. in dollars, $61,440.
20. C = $250 + $0.25x where x is the number of kilometers driven, $625.
21. a. 100 computers
b. $245,000
c. $150,000
d. 125 computers
e. $2,600
22. a. CA=$200x; CB =$5,000 +$100x; CC=$12,000 +$50x, x = the number
of hours
b. i. 50 hours; ii. 140 hours
c. < 50 hours use A; 50-140 hours use B; > 140 hours use C.
23. 26 months (The final payment will be smaller.)
24. a. $79,973.02
b. $43,973.02
25. .
a. $3,935.10
b. $3,935.10× 180 - $600× 300 = $528,318
26. .
a. $779.79/month
b. $4,000×120 − $779.79× 240 = $292,850
27. $1,477.83
28. $6,157.81 and $157.81 is the interest
29. 24 months
30. $10,377.35
31. 5.5 years
32. 12.0%
Business Mathematics 437
33. 12.18%
34. $877.20
35. 6.5552%, 6.7089%, Fund two is highest.
36. a. $550;
b. $2,487.37
37. $832.54
38. a. a. i/month = 1.083423742%, j4 = 13.1425%
b. j1 = 13.8045%
39. 5 years
40. $51,370.97 today
41. $50,000
42. 14.35%
43. a. Financing more costly because $134,247 > $130,000
b. Financing cheaper because $121,027 < $130,000.
44. a. $1,654
b. $148,478
c. $10,459.89
45. j2=10.40678%
46. a. $45,781.31;
b. $8,170.69
47. $10,743.42
48. $223.68
49. a. 1654.356115 → $1,655/month
b. 300 payments (299 full-sized payments and 1 smaller final payment.
(n = 299.6072073)
c. $51,140.25
d. $16,809.59
e. $159,560.25
f. 1037.575056→ $1,038/month
g. $1511.061987→ $1,512/month
h. $1,322.06
50. .
a. $1,004.52 → $1,005/month
b. $414.21 principal repaid, $590.79 interest
c. $60,495.03
d. 48.345%
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51. a. 18.62% > MARR of 15% so yes, buy the business.
b. $7817.58 > 0 so yes, buy the business.
c. up to $47,817.58
d. $55,000- $18,083 = $36,917
e. decrease of $2,065.69/year
52. a. IRR= 16.83% < 20% MARR so no
b. No, NPV=-$22,561.23, NEG.
c. falls by $22,561 to $177,439
d. $120,841 ($80,841 increase)
e. $27,073.47
f. $6,259/year
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Appendices
These are optional topics, which are not always covered.
Business Mathematics 441
441
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Appendix A: Learning Curves in the BAII Plus
The BAII Plus calculator can be used for more than just Business Math. For example, we use
power functions to understand learning Curves.
Example A.1
In manufacturing a certain part, we find that the initial part takes 176 hours to manufacture,
and that there is a 90% learning curve.
a. Find a power function that shows the time to create the nth item as a function of n.
b. How much time does it take to manufacture the 144th item?
c. How many items to we need to manufacture before we are down to 50 hours per
part?
We are looking for a function of the form:
or using your calculator’s built in notation
We can find this using the Data worksheet on the calculator, by fitting a power function using
the data points given: (1, 176) and (2, 176×0.9)
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443
Step To Press Display
1 Select Data Worksheet [2ND] [7/DATA] X01=0
2 Clear old Data [ 2ND] [CE|C] X01=0
3 Enter two data points [1] [ENTER] X01=1
↓176 [ENTER] Y01=176
↓2 [ENTER] X02=2
↓176[×].9 [=][ENTER] Y02= 158.4
5 Enter STAT mode [2ND] [8/DATA] LIN
6 Get into PWR mode
[2ND] [ENTER]
[2ND][ENTER]
[2ND[ENTER]
PWR
7 Check that you entered data correctly [↓] n=2
8 Find the Power Function ↓↓↓↓↓↓↓ a= 176
↓ b= -0.152003093
9 Find the time for item 144 ↓↓ 144 [ENTER] X’ = 144
↓ [CPT] Y’ = 82.68655388
10 Find the number of the item that takes 50 hours 50 [ENTER] Y’= 50
↑ [CPT] X’ = 3940.961945
a. We have found a and b in the calculator, so we know that we have:
b. The 144th item takes 82.69 hours to manufacture
c. It will take around 50 hours to manufacture the 3941st item.
Your Own Notes
Are there any notes you want to take from this section? Is there anything you’d like
to copy and paste below?
These notes are for you only (they will not be stored anywhere)
Make sure to download them at the end to use as a reference
444 Amy Goldlist
An interactive H5P element has been excluded from this version of the
text. You can view it online here:
https://pressbooks.bccampus.ca/businessmathematics/?p=1180#h5p-1
Business Mathematics 445
446 Amy Goldlist
Appendix B: EUAC and Capitalized Cost
COST COMPARISONS
Some investment decisions involve the purchase of equipment which provides benefits that
clearly outweigh the costs. For example, a sprinkler system may be required before a building
can be used, a salesman may need an automobile, an office may need a copier. In such cases
the issue is really that of obtaining the equipment at the lowest cost. Consequently, you need
only examine those areas in which alternatives differ.
Example B.1
As an example, suppose that an office needs a heavy duty copier and has to choose between two
alternatives, Model I and Model II. Both models will do a satisfactory job. The characteristics
and costs of the two models follow:
Useful Life
Model I
3 years
Model II
3 years
Price $8,500 $10,000
Operating Cost $2,000/yr $1,500/yr
Disposal Value $1,500 $2,000
Assume the operating cost is paid at the beginning of each year.
As with our earlier work, we must allow for the time value of money – amounts paid early are
more expensive than amounts paid later since the funds to make later payments are assumed to
be invested until they are needed. We will assume the company expects a rate of return of 12%
effective.
All amounts should be discounted to find the Net Present Value of the cash flows.
For Model I, we have:
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447
TIME Cash Flow PV
0 -$10,500 -$10,500.00
1 -2,000 – 1,785.71
2 -2,000 -1,594.39
3 1,500 1,067.67
NPV = -$12,812.43
For Model II, we have:
TIME Cash Flow PV
0 -$12,000 -$12,000.00
1 -1,500 -1,339.29
2 -1,500 -1,195.79
3 2,000 1,423.56
NPV = -$13,111.52
These net present values represent the amount of money the company would have to set aside
as a lump sum, which, with its earnings, would provide the payments necessary for the copiers.
Clearly in this case Model I would be the cheapest.
If, however, no allowance was made for the time of payments and the amounts were simply
totaled (equivalent to a zero rate of return), then the costs of the copiers would appear exactly
the same.
EFFECTS OF DIFFERENT LIFETIMES
In the copier example above, the two copiers were expected to last for the same time. If
alternatives are expected to last for different times, their costs can be compared by putting them
on a per year or per month basis. To allow for the time value of money in such cases, you can
use the annuity payments which would have as a present value the NPV of the costs. If this is
448 Amy Goldlist
done on an annual basis, the annuity payment is called the Equivalent Uniform Annual Cost
(EUAC).
Example B.2
Suppose that there had been a third alternative copier with the following characteristics:
MODEL III
Useful Life
Model I
3 years
Model II
3 years
Model III
4 years
Price $8,500 $10,000 $11,000
Operating Cost $2,000/yr $1,500/yr $2,000/ yr
Disposal Value $1,500 $2,000 $1,500
This one would have to be compared with Model I (the cheaper of the other two) and the
question would be: Is the extra initial cost of Model III justified by the longer useful life?
To answer this, start by finding the NPV of Model III costs.
Time (year) Cash Flow PV
0 -$13,000 -$13,000.00
1 -2,000 -1,785.71
2 -2,000 -1,594.39
3 -2,000 -1,423.56
4 1,500 953.28
NPV -$16,850.38
The NPV cannot be simply compared to that of Model I since the NPV is for a different time
of use. Instead, we place it on an annual basis by finding the equivalent yearly annuity for each
model.
Business Mathematics 449
For Model I:
from which you will find the payment is -$5,334.44.
This is the Equivalent Uniform Annual Cost (EUAC), i.e., the annual payment equivalent in
value to the set of payments required for the given asset (in this case Model I).
For Model III:
from which the payment= EUAC = – $5,547.72.
Hence, this alternative would be more expensive than Model I.
CAPITALIZED COST
Another approach to dealing with unequal lives is to view the equipment as being replaced at
the end of each cycle with an identical piece of equipment with identical costs. Then the NPV
of all of the cash flows is found. Such an NPV is called the Capitalized Cost.
For Model I above this can be found by using the EUAC and the perpetuity formula:
Similarly for Model III:
again showing that Model I is cheapest.
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The capitalized cost method of comparison is particularly suited to constructions such as
bridges which are often viewed as never to be replaced, simply to be built and maintained.
Example B.3
Suppose a small bridge is to be built for $4,000,000 and to be maintained indefinitely for
payments of $35,000 a year, starting at the end of the first year. Then the capitalized cost at a
rate of return of 10% effective would be:
this being the amount of money set aside now which, with earnings, would make all payments
for the bridge.
Your Own Notes
Are there any notes you want to take from this section? Is there anything you’d like
to copy and paste below?
These notes are for you only (they will not be stored anywhere)
Make sure to download them at the end to use as a reference
An interactive H5P element has been excluded from this version of the
text. You can view it online here:
https://pressbooks.bccampus.ca/businessmathematics/?p=1182#h5p-1
Business Mathematics 451
452 Amy Goldlist
Glossary
––
Double dash (double en dash) is used in this textbook to denote 'N/A' or nothing needs to
be entered.
2ND
The [2ND] function key (near the top left on the BAII Plus) allows the user to access the
secondary menus. Example: [2ND][PMT] accesses the BGN menu.
accelerated bi-weekly
Take the monthly mortgage payment (P/Y=12), and divide by 2, than make this payment
bi-weekly. (P/Y=26). This amounts to one extra monthly payment per year, which speeds
up your mortgage repayment.
Acid-test ratio
Ratio of cash resources to the company's short-term debts.
amortization
The process of paying off debt through regular principal and interest payments over time.
amortization period
The total length of time until a loan is fully repaid.
amortization table
A table (or schedule) detailing the amount of principal and interest paid during each
payment as well as the balance owing after each payment.
AMRT
Amortization menu in the BAII Plus. Hit 2ND PMT to access this menu.
annuity
A series of equal-sized payments, at regular intervals, over a fixed period of time.
Business Mathematics 453
453
annuity due
An annuity with deposits or payments at the beginning of the payment period.
BGN
The setting in the BAII Plus to turn on when payments occur at the beginning of the
payment interval.
bond holder
Canada Education Savings Grant (CESG)
The CESG provides 20% of the Registered Education Savings Plan (RESP) contributions
of up to $2,500. That means the CESG can add a maximum of $500 to an RESP each year.
Compound interest
interest paid on previously earned interest as well as on the original principal.
contribution margin
Markup in dollars for one item
cost of financing
The amount of interest we must repay on the loan.
Cost of the Goods Sold (COGS)
Total Variable expenses, or all the costs that go into making your item.
coupon payment
The regular (usually semi-annual) payment from a coupon bond.
coupon rate
In a bond, this is the initial interest rate used to calculate the coupon payment
CPT
Compute key on BAII Plus. Computes selected value.
defaults
To default on a loan is to fail to make the payments. This can lead to fines, legal
procedures, or items being repossessed.
454 Amy Goldlist
deferred annuities
An annuity where the regular payments are delayed for a period of time.
Dividend
The regular payment (PMT) paid by the issuer of the preferred share.
down payment
A lump-sum payment made before you take out a loan.
effective rate
The equivalent rate compunded annually
END
The setting in the BAII Plus to make the payments occur at the end of the payment interval.
ENTER
The [ENTER] key (top left on the BAII Plus) is used to set values within the calculator's
menus.
Equivalent
This is a mathematical term, meaning that two things are the same in the ways we want
them to. In the case of interest rates, two rates are equivalent if an investment at each rate
gives the same Future Value after one year.
Equivalent Uniform Annual Cost (EUAC)
The Payment of an Ordinary Annuity with PV = NPV
Expense
A cost, either fixed or variable, Money Out
face value
In a bond, this is the original purchase price, as well as final payment for the bond.
fair market value
How much an item would sell for on the open market - normally, this is the PV according
to today's interest rates for similar investments.
Business Mathematics 455
fair market value of a bond
The amount the purchaser (secondary bond holder) is willing to pay for a bond.
fixed cost
Also can be known as the Operations Costs or Operating Expenses. Stay the same no
matter how may units are sold.
fixed interest rate
An interest rate which remains constant through the entire term, instead of fluctuating
based on market conditions.
fixed rate mortgage
A mortgage where the interest rate charged remains fixed for the duration of the mortgage
term.
FOCAL DATE
The date at which we make the calcualtions.... (FIX)
general annuity
An Annuity where P/Y ≠ C/Y
Gross Profit (GP)
Revenue minus Cost of Goods (or Variable expenses) only. The Operating expenses or
Fixed Costs are not considered.
interest
Money earned on an investment, or paid on a loan.
internal rate of return (IRR)
The rate for which the NPV = 0
issuer
lease payments
Equal sized payments for a lease.
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lease period
Duration of a lease (fixed time period).
lessee
Person or company holding the lease. Example: person who is leasing a car.
loan
An annuity where we borrow an initial amount of money (PV) and we repay the loan with
a series of equal-sized payments (PMT), at regular intervals, over the course of a fixed
time period
Market Rate
The current interest rate.
markup
maturity date
The termination or ending date for which a loan, bond, or any amount borrowed must be
paid back in full.
method of substitution
A method of solving a system of equations.
minimum acceptable rate of return (MARR)
mortgage term
The length of time your mortgage agreement and interest rate will be in effect.
Net Present Value (NPV)
The sum of all Inflows, minus all Outflows, adjusted for time.
Nominal rate
Percent annual rate in compound interest.
Operating expenses
Fixed Costs
Business Mathematics 457
ordinary annuity
An annuity where the payments occur at the end of each payment interval.
ordinary perpetuity
A perpetuity where the first payment comes at the end of the first period
partial payment
percent gross margin
Percent Markup
The ratio of Profit over Cost, (usually Gross Profit).
percent net margin
Periodic rate
Periodic Compound interest rate
perpetuity
An annuity that has no end or an annuity with regular cash flows that continue forever.
perpetuity due
A perpetuity where the first payment is at the beginning of the first period.
PMT
The 'payment' key within the TVM (time value of money keys) in the BAII Plus
Calculator.
preferred share
A preferred share has no maturity date (no expiration date or fixed term) but will stop
regular payments if the company stops making a profit of goes out of business. Because
we have no way of knowing when this might happen, we treat preferred shares like
perpetuities and assume the profits will continue infinitely (forever).
Price
The selling price, fair market value or the amount the purchaser is willing to pay.
458 Amy Goldlist
principal
The original amount of money invested or borrowed.
Profit
The difference between the revenues (sales) and expenses (cost of goods and operating
costs).
renew
A new mortgage is drawn up at the end of each term when a mortgage holder (buyer)
renews their mortgage.
residual value
Size of final lease payment required. Example: final payment required to own a car at the
end of the car lease.
RESP
Registered Education Savings Plan
return on investment (ROI)
revenue function
RRSP
Registered Retirement Savings Plan
secondary bond holder
A bond holder who has purchased a bond(s) from either the original bond holder or another
secondary bond holder.
Sells at a Discount
Sells for a price lower than the original purchase price.
Sells at a Premium
Sells for a higher selling price than the original purchase price.
Semi-annual payments
Payments that occur twice per year or once every six months.
Business Mathematics 459
simple annuity
An annuity where the payment frequency (P/Y) is equal to the compounding frequency
(C/Y).
simple interest
Interest earned without any compounding, that is interest paid only on the principal.
system of equations
A number of equations, involving the same variables
term of a bond
The amount of time between when the bond is issued and when the bond matures.
Turnover Ratio
The value of goods sold divided by the average value of goods in stock
TVM keys
Time Value of Money keys. These are the N, I/Y, PV, PMT and FV keys on the BAII Plus.
variable cost
Costs that vary based on how many items are made and sold.
variable rate
variable rate mortgage
the interest rate varies throughout the mortgage term
460 Amy Goldlist
Formulas
Chapter 1: Mathematics of Merchandising
Profit = Sales – Costs
Chapter 2: Functions
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461
Chapter 3: Simple Interest
Chapter 4: Compound Interest
Chapter 5: Annuities
Ordinary Perpetuities:
Perpetuities Due
To switch to payments at the beginning of the interval:
Press 2ND BGN (above the PMT key). The display should show END.
Press 2ND SET (above the ENTER key). The display should show BGN.
462 Amy Goldlist
Press CE/C (bottom left corner) or 2ND QUIT (top left corner).
To go back to payments at the END of the interval:
Press 2ND BGN (above the PMT key) The display should show BGN.
Press 2ND SET (above the ENTER key). The display should show END.
Press CE/C (bottom left corner) or 2ND QUIT (top left corner)
Chapter 6: Investment Decisions
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