Chapter 05 - Introduction to Valuation: The Time Value of Money
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 it’s positive); the present value falls.
5.
It would appear to be both deceptive and unethical to run such an ad without a disclaimer or
explanation.
6.
It’s a reflection of the time value of money. TMCC gets to use the $24,099. If TMCC uses it wisely,
it will be worth more than $100,000 in 30 years.
7.
This will probably make the security less desirable. TMCC will only repurchase the security prior to
maturity if it is to its advantage, i.e., interest rates decline. Given the drop in interest rates needed to
make this viable for TMCC, it is unlikely the company will repurchase the security. This is an
example of a “call” feature. Such features are discussed at length in a later chapter.
8.
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 $100,000? Thus, our answer does depend on who is making the promise to
repay.
9.
The Treasury security would have a somewhat higher price because the Treasury is the strongest of
all borrowers.
10. The price would be higher because, as time passes, the price of the security will tend to rise toward
$100,000. This rise is just a reflection of the time value of money. As time passes, the time until
receipt of the $100,000 grows shorter, and the present value rises. In 2019, the price will probably be
higher for the same reason. We cannot be sure, however, because interest rates could be much
higher, or TMCC’s financial position could deteriorate. Either event would tend to depress the
security’s price.
5-1
Chapter 05 - Introduction to Valuation: The Time Value of Money
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:
$64,000 × .07 = $4,480
So after 9 years you will have:
$4,480 × 9 = $40,320 in interest.
The total balance will be $64,000 + 40,320 = $104,320
With compound interest we use the future value formula:
FV = PV(1 + r)t
FV = $64,000(1.07)9 = $117,661.39
The difference is:
$117,661.39 – 104,320 = $13,341.39
2.
To find the FV of a lump sum, we use:
FV = PV(1 + r)t
FV = $2,250(1.13)11
FV = $8,752(1.09)7
FV = $76,355(1.12)14
FV = $183,796(1.06)8
3.
= $ 8,630.69
= $ 15,999.00
= $373,155.46
= $292,942.90
To find the PV of a lump sum, we use:
PV = FV / (1 + r)t
PV = $15,451 / (1.07)13
PV = $51,557 / (1.13)4
PV = $886,073 / (1.14)29
PV = $550,164 / (1.09)40
= $ 6,411.62
= $31,620.87
= $19,825.71
= $17,515.89
5-2
Chapter 05 - Introduction to Valuation: The Time Value of Money
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 – 1
FV = $297 = $240(1 + r)4;
FV = $1,080 = $360(1 + r)18;
FV = $185,382 = $39,000(1 + r)19;
FV = $531,618 = $38,261(1 + r)25;
5.
r = ($297 / $240)1/4 – 1
r = ($1,080 / $360)1/18 – 1
r = ($185,382 / $39,000)1/19 – 1
r = ($531,618 / $38,261)1/25 – 1
= 0.0547, or 5.47%
= 0.1472, or 14.72%
= 0.0855, or 8.55%
= 0.1110, or 11.10%
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)
FV = $1,389 = $560(1.09)t;
FV = $1,821 = $810(1.10)t;
FV = $289,715 = $18,400(1.17)t;
FV = $430,258 = $21,500(1.15)t;
6.
t = ln($1,389/ $560) / ln 1.09
t = ln($1,821/ $810) / ln 1.10
t = ln($289,715 / $18,400) / ln 1.17
t = ln($430,258 / $21,500) / ln 1.15
= 10.54 years
= 8.50 years
= 17.56 years
= 21.44 years
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 – 1
r = ($300,000 / $65,000)1/18 – 1 = .0887, or 8.87%
5-3
Chapter 05 - Introduction to Valuation: The Time Value of Money
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(1.065)t
t = ln 2 / ln 1.065 = 11.01 years
The length of time to quadruple your money is:
FV = $4 = $1(1.065)t
t = ln 4 / ln 1.065 = 22.01 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 – 1
r = ($283,400 / $200,300)1/10 – 1 = .0353, or 3.53%
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:
FV = PV(1 + r)t
Solving for t, we get:
t = ln(FV / PV) / ln(1 + r)
t = ln ($190,000 / $40,000) / ln 1.048 = 33.23 years
10. To find the PV of a lump sum, we use:
PV = FV / (1 + r)t
PV = $575,000,000 / (1.068)20 = $154,256,257.63
5-4
Chapter 05 - Introduction to Valuation: The Time Value of Money
11. To find the PV of a lump sum, we use:
PV = FV / (1 + r)t
PV = $1,000,000 / (1.09)80 = $1,013.63
12. To find the FV of a lump sum, we use:
FV = PV(1 + r)t
FV = $50(1.041)108 = $3,833.97
13. 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 – 1
r = ($1,350,000 / $150)1/115 – 1 = .0824, or 8.24%
To find the FV of the first prize in 2040, we use:
FV = PV(1 + r)t
FV = $1,350,000(1.0824)30 = $14,516,947.05
14. 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 – 1
r = ($125,000 / $1)1/115 – 1 = .1074, or 10.74%
15. 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 – 1
r = ($10,311,500 / $12,377,500)1/4 – 1 = – 4.46%
Notice that the interest rate is negative. This occurs when the FV is less than the PV.
5-5
Chapter 05 - Introduction to Valuation: The Time Value of Money
Intermediate
16. 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 – 1
a. PV = $100,000 / (1 + r)30 = $24,099
r = ($100,000 / $24,099)1/30 – 1 = .0486, or 4.86%
b. PV = $42,380 / (1 + r)11 = $24,099
r = ($42,380 / $24,099)1/11 – 1 = .0527, or 5.27%
c. PV = $100,000 / (1 + r)19 = $42,380
r = ($100,000 / $42,380)1/19 – 1 = .0462, or 4.62%
17. To find the PV of a lump sum, we use:
PV = FV / (1 + r)t
PV = $190,000 / (1.12)9 = $68,515.90
18. To find the FV of a lump sum, we use:
FV = PV(1 + r)t
FV = $5,000(1.11)45 = $547,651.21
FV = $5,000(1.11)35 = $192,874.26
Better start early!
19. We need to find the FV of a lump sum. However, the money will only be invested for six years, so
the number of periods is six.
FV = PV(1 + r)t
FV = $15,000(1.071)6 = $22,637.48
5-6
Chapter 05 - Introduction to Valuation: The Time Value of Money
20. 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)
t = ln($85,000 / $15,000) / ln(1.11) = 16.62
So, the money must be invested for 16.62 years. However, you will not receive the money for
another two years. From now, you’ll wait:
2 years + 16.62 years = 18.62 years
Calculator Solutions
1.
Enter
9
N
7%
I/Y
$64,000
PV
PMT
FV
$117,661.39
Solve for
$117,661.39 – 104,320 = $13,341.39
2.
Enter
11
N
13%
I/Y
$2,250
PV
PMT
FV
$8,630.69
7
N
9%
I/Y
$8,752
PV
PMT
FV
$15,999.00
14
N
12%
I/Y
$76,355
PV
PMT
FV
$373,155.46
8
N
6%
I/Y
$183,796
PV
PMT
FV
$292,942.90
13
N
7%
I/Y
Solve for
Enter
Solve for
Enter
Solve for
Enter
Solve for
3.
Enter
Solve for
PV
$6,411.62
5-7
PMT
$15,451
FV
Chapter 05 - Introduction to Valuation: The Time Value of Money
Enter
4
N
13%
I/Y
PV
$31,620.87
PMT
$51,557
FV
29
N
14%
I/Y
PV
$19,825.71
PMT
$886,073
FV
40
N
9%
I/Y
PV
$17,515.89
PMT
$550,164
FV
$240
PV
PMT
$297
FV
$360
PV
PMT
$1,080
FV
$39,000
PV
PMT
$185,382
FV
$38,261
PV
PMT
$531,618
FV
9%
I/Y
$560
PV
PMT
$1,389
FV
10%
I/Y
$810
PV
PMT
$1,821
FV
17%
I/Y
$18,400
PV
PMT
$289,715
FV
Solve for
Enter
Solve for
Enter
Solve for
4.
Enter
4
N
Solve for
Enter
18
N
Solve for
Enter
19
N
Solve for
Enter
25
N
Solve for
5.
Enter
Solve for
N
10.54
Enter
Solve for
N
8.50
Enter
Solve for
N
17.56
I/Y
5.47%
I/Y
6.29%
I/Y
8.55%
I/Y
11.10%
5-8
Chapter 05 - Introduction to Valuation: The Time Value of Money
Enter
Solve for
6.
Enter
N
21.44
18
N
Solve for
7.
Enter
Solve for
N
11.01
Enter
Solve for
8.
Enter
N
22.01
10
N
Solve for
9.
Enter
Solve for
10.
Enter
N
33.23
$21,500
PV
PMT
$430,258
FV
$65,000
PV
PMT
$300,000
FV
6.5%
I/Y
$1
PV
PMT
$2
FV
6.5%
I/Y
$1
PV
PMT
$4
FV
$200,300
PV
PMT
$283,400
FV
$40,000
PV
PMT
$190,000
FV
PV
$154,256,257.63
PMT
$575,000,000
FV
PV
$1,013.63
PMT
$1,000,000
FV
15%
I/Y
I/Y
8.87%
I/Y
3.53%
4.80%
I/Y
20
N
6.8%
I/Y
80
N
9%
I/Y
108
N
4.10%
I/Y
Solve for
11.
Enter
Solve for
12.
Enter
$50
PV
Solve for
5-9
PMT
FV
$3,833.97
Chapter 05 - Introduction to Valuation: The Time Value of Money
13.
Enter
115
N
Solve for
Enter
30
N
I/Y
8.24%
8.24%
I/Y
$150
PV
PMT
$1,350,000
PV
PMT
$1
PV
PMT
±$125,000
FV
$12,377,500
PV
PMT
$10,311,500
FV
$24,099
PV
PMT
$100,000
FV
$24,099
PV
PMT
$42,380
FV
$42,380
PV
PMT
$100,000
FV
PMT
$190,000
FV
Solve for
14.
Enter
115
N
Solve for
15.
Enter
4
N
Solve for
16. a.
Enter
30
N
Solve for
16. b.
Enter
11
N
Solve for
16. c.
Enter
19
N
Solve for
17.
Enter
I/Y
10.74%
I/Y
–4.46%
I/Y
4.86%
I/Y
5.27%
I/Y
4.62%
12%
I/Y
45
N
11%
I/Y
$5,000
PV
PMT
FV
$547,651.21
35
N
11%
I/Y
$5,000
PV
PMT
FV
$192,874.26
PV
$68,515.90
Solve for
Enter
FV
$14,516,947.05
9
N
Solve for
18.
Enter
$1,350,000
FV
Solve for
5-10
Chapter 05 - Introduction to Valuation: The Time Value of Money
19.
Enter
6
N
7.10%
I/Y
$15,000
PV
11%
I/Y
$15,000
PV
PMT
Solve for
20.
Enter
Solve for
N
16.62
From now, you’ll wait 2 + 16.62 = 18.62 years
5-11
PMT
FV
$22,637.48
$85,000
FV