CHAPTER 5

INTRODUCTION TO VALUATION: THE TIME VALUE

OF MONEY

Answers to Concepts Review and Critical Thinking Questions

1. The four parts are the present value (PV), the future value (FV), the discount rate (r), and the life of

the investment (t).

2. Compounding refers to the growth of a dollar amount through time via reinvestment of interest

earned. It is also the process of determining the future value of an investment. Discounting is the

process of determining the value today of an amount to be received in the future.

3. Future values grow (assuming a positive rate of return); present values shrink.

4. The future value rises (assuming itâ€™s positive); the present value falls.

5. Itâ€™s a reflection of the time value of money. ScotiaMcLeod gets to use the $29.19 immediately. If

Scotia uses it wisely, it will be worth more than $100 in twenty years.

6. The key considerations would be: (1) Is the rate of return implicit in the offer attractive relative to

other, similar risk investments? and (2) How risky is the investment; i.e., how certain are we that we

will actually get the $10,000? Thus, our answer does depend on who is making the promise to repay.

7. The Government of Canada security would have a somewhat higher price because the Government of

Canada is the strongest of all borrowers.

8. The price would be higher because, as time passes, the price of the security will tend to rise toward

$100. This rise is just a reflection of the time value of money. As time passes, the time until receipt of

the $100 grows shorter, and the present value rises. In 2007, the price will probably be higher for the

same reason. We cannot be sure, however, because interest rates could be much higher, or Canadaâ€™s

financial position could deteriorate. Either event would tend to depress the securityâ€™s price.

Solutions to Questions and Problems

NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple

steps. Due to space and readability constraints, when these intermediate steps are included in this

solutions manual, rounding may appear to have occurred. However, the final answer for each problem is

found without rounding during any step in the problem.

Basic

1. The simple interest per year is:

$5,000 Ã— .07 = $350

So after 10 years you will have:

$350 Ã— 10 = $3,500 in interest.

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The total balance will be $5,000 + 3,500 = $8,500

With compound interest we use the future value formula:

FV = PV(1 +r)t

FV = $5,000(1.07)10 = $9,835.76

The difference is:

$9,835.76 â€“ 8,500 = $1,335.76

2. To find the FV of a lump sum, we use:

FV = PV(1 + r)t

FV = $2,250(1.10)19 = $ 13,760.80

FV = $9,310(1.08)13 = $ 25,319.70

FV = $76,355(1.22)4= $169,151.87

FV = $183,796(1.07)8= $315,795.75

3. To find the PV of a lump sum, we use:

PV = FV / (1 + r)t

PV = $15,451 / (1.05)6= $11,529.77

PV = $51,557 / (1.11)9= $20,154.91

PV = $886,073 / (1.16)23 = $29,169.95

PV = $550,164 / (1.19)18 = $24,024.09

4. To answer this question, we can use either the FV or the PV formula. Both will give the same answer

since they are the inverse of each other. We will use the FV formula, that is:

FV = PV(1 + r)t

Solving for r, we get:

r = (FV / PV)1 / t â€“ 1

FV = $307 = $265(1 + r)2;r = ($307 / $265)1/2 â€“ 1 = 7.63%

FV = $896 = $360(1 + r)9;r = ($896 / $360)1/9 â€“ 1 = 10.66%

FV = $162,181 = $39,000(1 + r)15; r = ($162,181 / $39,000)1/15 â€“ 1 = 9.97%

FV = $483,500 = $46,523(1 + r)30;r = ($483,500 / $46,523)1/30 â€“ 1= 8.12%

5. To answer this question, we can use either the FV or the PV formula. Both will give the same answer

since they are the inverse of each other. We will use the FV formula, that is:

FV = PV(1 + r)t

Solving for t, we get:

t = ln(FV / PV) / ln(1 + r)

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FV = $1,284 = $625(1.08)t;t = ln($1,284/ $625) / ln 1.08 = 9.36 yrs

FV = $4,341 = $810(1.07)t;t = ln($4,341/ $810) / ln 1.07 = 24.81 yrs

FV = $402,662 = $18,400(1.21)t;t = ln($402,662 / $18,400) / ln 1.21 = 16.19 yrs

FV = $173,439 = $21,500(1.29)t;t = ln($173,439 / $21,500) / ln 1.29 = 8.20 yrs

6. To answer this question, we can use either the FV or the PV formula. Both will give the same answer

since they are the inverse of each other. We will use the FV formula, that is:

FV = PV(1 + r)t

Solving for r, we get:

r = (FV / PV)1 / t â€“ 1

r = ($80,000 / $15,000)1/18 â€“ 1 = 9.75%

7. To find the length of time for money to double, triple, etc., the present value and future value are

irrelevant as long as the future value is twice the present value for doubling, three times as large for

tripling, etc. To answer this question, we can use either the FV or the PV formula. Both will give the

same answer since they are the inverse of each other. We will use the FV formula, that is:

FV = PV(1 + r)t

Solving for t, we get:

t = ln(FV / PV) / ln(1 + r)

The length of time to double your money is:

FV = $2 = $1(1.07)t

t = ln 2 / ln 1.07 = 10.24 years

The length of time to quadruple your money is:

FV = $4 = $1(1.07)t

t = ln 4 / ln 1.07 = 20.49 years

Notice that the length of time to quadruple your money is twice as long as the time needed to double

your money (the difference in these answers is due to rounding). This is an important concept of time

value of money.

8. To answer this question, we can use either the FV or the PV formula. Both will give the same answer

since they are the inverse of each other. We will use the FV formula, that is:

FV = PV(1 + r)t

Solving for r, we get:

r = (FV / PV)1 / t â€“ 1

r = ($28,835 / $21,608)1/5 â€“ 1 = 5.94%

9. To answer this question, we can use either the FV or the PV formula. Both will give the same answer

since they are the inverse of each other. We will use the FV formula, that is:

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