Chapter 4
Answers to Concepts Review and Critical Thinking Questions
1.
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.
2.
Future values grow (assuming a positive rate of return); present values shrink.
3.
The future value rises (assuming a positive rate of return); the present value falls.
4.
It depends. The large deposit will have a larger future value for some period, but after time, the
smaller deposit with the larger interest rate will eventually become larger. The length of time for the
smaller deposit to overtake the larger deposit depends on the amount deposited in each account and
the interest rates.
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. GMAC gets to use the $500 immediately. If GMAC uses
it wisely, it will be worth more than $10,000 in thirty years.
7.
Oddly enough, it actually makes it more desirable since GMAC only has the right to pay the full
$10,000 before it is due. This is an example of a “call” feature. Such features are discussed 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 $10,000? 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, and therefore has a lower rate of return.
10. The price would be higher because, as time passes, the price of the security will tend to rise toward
$10,000. This rise is just a reflection of the time value of money. As time passes, the time until
receipt of the $10,000 grows shorter, and the present value rises. In 2006, the price will probably be
higher for the same reason. We cannot be sure, however, because interest rates could be much
higher, or GMAC’s financial position could deteriorate. Either event would tend to depress the
security’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:
$6,000 × .07 = $420
So, after 10 years, you will have:
$420 × 10 = $4,200 in interest.
The total balance will be $6,000 + 4,200 = $10,200
With compound interest, we use the future value formula:
FV = PV(1 +r)t
FV = $6,000(1.07)10 = $11,802.91
The difference is:
$11,802.91 – 10,200 = $1,602.91
2.
To find the FV of a lump sum, we use:
FV = PV(1 + r)t
FV = $3,150(1.18)3
FV = $7,810(1.06)10
FV = $89,305(1.12)17
FV = $227,382(1.05)22
3.
= $ 5,175.55
= $ 13,986.52
= $613,171.78
= $665,151.63
To find the PV of a lump sum, we use:
PV = FV / (1 + r)t
PV = $15,451 / (1.04)9
PV = $51,557 / (1.12)4
PV = $886,073 / (1.22)16
PV = $550,164 / (1.20)21
= $10,855.67
= $32,765.41
= $36,788.51
= $11,958.76
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 = $307 = $221(1 + r)5
r = ($307 / $221)1/5 – 1
r = .0679 or 6.79%
FV = $761 = $425(1 + r)7
r = ($761 / $425)1/7 – 1
r = .0868 or 8.68%
FV = $136,771 = $25,000(1 + r)18
r = ($136,771 / $25,000)1/18 – 1
r = .0990 or 9.90%
FV = $255,810 = $40,200(1 + r)16
r = ($255,810 / $40,200)1/16 – 1
r = .1226 or 12.26%
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)
FV = $1,105 = $250 (1.06)t
t = ln($1,105 / $250) / ln 1.06
t = 25.50 years
FV = $3,860 = $1,941(1.08)t
t = ln($3,860 / $1,941) / ln 1.05
t = 14.09 years
FV = $387,120 = $21,320(1.14)t
t = ln($387,120 / $21,320) / ln 1.14
t = 22.13 years
FV = $198,212 = $32,500(1.29)t
t = ln($198,212 / $32,500) / ln 1.29
t = 7.10 years
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 – 1
r = ($280,000 / $39,000)1/18 – 1
r = .1157 or 11.57%
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
t = 10.24 years
The length of time to quadruple your money is:
FV = $4 = $1(1.07)t
t = ln 4 / ln 1.07
t = 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 slight 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 = ($17,825 / $1)1/124 – 1
r = .0821 or 8.21%
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)
FV = $140,000 = $30,000(1.042)t
t = ln($140,000 / $30,000) / ln 1.042
t = 37.44 years
10. To find the PV of a lump sum, we use:
PV = FV / (1 + r)t
PV = $800,000,000 / (1.07)20
PV = $206,735,202
11. To find the PV of a lump sum, we use:
PV = FV / (1 + r)t
PV = $2,000,000 / (1.13)80
PV = $113.44
12. To find the FV of a lump sum, we use:
FV = PV(1 + r)t
FV = $50(1.053)106
FV = $11,922.78
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 = ($1,125,000 / $150)1/109 – 1
r = .0853 or 8.53%
To find what the check will be in 2040, we use the FV of a lump sum, so:
FV = PV(1 + r)t
FV = $1,125,000(1.0853)36
FV = $21,428,376.58
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 = ($400,000 / $0.01)1/212 – 1
r = .0861 or 8.61%
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
r = –.0446 or –4.46%
Intermediate
16. a. 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 = ($10,000 / $500)1 /30 – 1
r = .1050 or 10.50%
b. Using the FV formula and solving for the interest rate, we get:
r = (FV / PV)1 / t – 1
r = ($6,998.79 / $500)1 /23 – 1
r = .1216 or 12.16%
c. Using the FV formula and solving for the interest rate, we get:
r = (FV / PV)1 / t – 1
r = ($10,000 / $6,998.79)1 /7 – 1
r = .0523 or 5.23%
17. To find the PV of a lump sum, we use:
PV = FV / (1 + r)t
PV = $140,000 / (1.1075)10
PV = $50,430.20
18. To find the FV of a lump sum, we use:
FV = PV(1 + r)t
FV = $4,000(1.12)45
FV = $655,950.42
If you wait 10 years, the value of your deposit at your retirement will be:
FV = $4,000(1.12)35
FV = $211,198.48
Better start early!
19. Even though we need to calculate the value in eight years, we will only have the money for six years,
so we need to use six years as the number of periods. To find the FV of a lump sum, we use:
FV = PV(1 + r)t
FV = $13,000(1.08)6
FV = $20,629.37
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
$140,000 = $30,000(1.09)t
t = ln($140,000 / $30,000) / ln 1.09
t = 17.88 years
From now, you’ll wait 2 + 17.88 = 19.88 years
21. To find the FV of a lump sum, we use:
FV = PV(1 + r)t
In Regency Bank, you will have:
FV = $9,000(1.01)120
FV = $29,703.48
And in King Bank, you will have:
FV = $9,000(1.12)10
FV = $27,952.63
22. 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. We also need to be careful about the number of periods. Since the length of the
compounding is six months and we have 24 months, there are four compounding periods. 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 = ($3 / $1)4 – 1
r = .3161 or 31.61%
23. 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
$2,500 = $1,300(1.004)t
t = ln($2,500 / $1,300) / ln 1.004
t = 163.81 months
24. To find the PV of a lump sum, we use:
PV = FV / (1 + r)t
PV = $50,000 / (1.0055)108
PV = $27,650.69
25. To find the PV of a lump sum, we use:
PV = FV / (1 + r)t
So, if you can earn 11 percent, you will need to invest:
PV = $1,000,000 / (1.11)45
PV = $9,129.90
And if you can earn 5 percent, you will need to invest:
PV = $1,000,000 / (1.05)45
PV = $111,296.51
Calculator Solutions
1.
Enter
10
N
7%
I/Y
±$6,000
PV
PMT
FV
$11,802.91
Solve for
$5,802.92 – 10($420) = $1,620.91
2.
Enter
3
N
18%
I/Y
±$3,150
PV
PMT
FV
$5,175.55
10
N
6%
I/Y
±$7,810
PV
PMT
FV
$13,986.52
17
N
12%
I/Y
±$89,305
PV
PMT
FV
$613,171.78
Solve for
Enter
Solve for
Enter
Solve for
Enter
22
N
5%
I/Y
9
N
4%
I/Y
4
N
12%
I/Y
16
N
22%
I/Y
21
N
20%
I/Y
±$227,382
PV
PMT
Solve for
3.
Enter
Solve for
Enter
Solve for
Enter
Solve for
Enter
Solve for
4.
Enter
5
N
Solve for
Enter
7
N
Solve for
Enter
18
N
Solve for
Enter
Solve for
16
N
I/Y
6.79%
I/Y
8.68%
I/Y
9.90%
I/Y
12.26%
PV
–$10,855.67
PV
–$32,765.41
PV
–$36,788.51
PV
–$11,958.76
±$221
PV
±$761
PV
±$25,000
PV
±$40,200
PV
FV
$665,151.63
PMT
$15,451
FV
PMT
$51,557
FV
PMT
$886,073
FV
PMT
$550,164
FV
PMT
$307
FV
PMT
$425
FV
PMT
$136,771
FV
PMT
$255,810
FV
5.
Enter
Solve for
N
25.50
Enter
Solve for
N
14.09
Enter
Solve for
N
22.13
Enter
Solve for
6.
Enter
N
7.10
18
N
Solve for
7.
Enter
Solve for
N
10.24
Enter
Solve for
8.
Enter
Solve for
N
20.49
20
N
6%
I/Y
±$250
PV
5%
I/Y
PMT
$1,105
FV
±$1,941
PV
PMT
$3,860
FV
14%
I/Y
±$21,320
PV
PMT
$387,120
FV
29%
I/Y
±$32,500
PV
PMT
$198,212
FV
±$39,000
PV
PMT
$280,000
FV
7%
I/Y
±$1
PV
PMT
$2
FV
7%
I/Y
±$1
PV
PMT
$4
FV
±$10,000
PV
PMT
$45,000
FV
I/Y
11.57%
I/Y
7.81%
9.
Enter
Solve for
10.
Enter
N
37.44
4.2%
I/Y
20
N
7%
I/Y
80
N
13%
I/Y
106
N
5.30%
I/Y
Solve for
11.
Enter
Solve for
12.
Enter
±$30,000
PV
PV
–$206,735,202
PV
–$113.44
±$50
PV
PMT
$140,000
FV
PMT
$800,000,000
FV
PMT
$2,000,000
FV
PMT
Solve for
13.
Enter
109
N
Solve for
Enter
36
N
I/Y
8.53%
8.53%
I/Y
±$150
PV
PMT
±$1,125,000
PV
PMT
±$0.01
PV
PMT
$400,000
FV
±$12,377,500
PV
PMT
$10,311,500
FV
±$500
PV
PMT
$10,000
FV
Solve for
14.
Enter
212
N
Solve for
15.
Enter
4
N
Solve for
16. a.
Enter
Solve for
30
N
I/Y
8.61%
I/Y
–4.46%
I/Y
10.50%
FV
$11,922.78
$1,125,000
FV
FV
$21,428,376.58
b.
Enter
23
N
Solve for
c.
Enter
7
N
Solve for
17.
Enter
I/Y
12.16%
I/Y
5.23%
±$6,998.79
PV
PMT
$6,998.79
FV
PMT
$10,000
FV
PMT
$140,000
FV
10
N
10.75%
I/Y
45
N
12%
I/Y
±$4,000
PV
PMT
FV
$655,950.42
35
N
12%
I/Y
±$4,000
PV
PMT
FV
$211,198.48
6
N
8%
I/Y
$13,000
PV
PMT
FV
$20,629.37
9%
I/Y
±$30,000
PV
PMT
Solve for
18.
Enter
±$500
PV
PV
–$50,430.20
Solve for
Enter
Solve for
19.
Enter
Solve for
20.
Enter
Solve for
N
17.88
$140,000
FV
You must wait 2 + 17.88 = 19.88 years.
21.
Enter
120
N
1%
I/Y
±$9,000
PV
PMT
FV
$29,703.48
10
N
12%
I/Y
±$9,000
PV
PMT
FV
$27,952.63
Solve for
Enter
Solve for
22.
Enter
4
N
Solve for
23.
Enter
Solve for
24.
Enter
N
163.81
I/Y
31.61%
.4%
I/Y
108
N
.55%
I/Y
45
N
11%
I/Y
45
N
6%
I/Y
Solve for
25.
Enter
Solve for
Enter
Solve for
±$1
PV
±$1,300
PV
PV
–$26,844.00
PV
–$9,129.90
PV
–$111,296.51
PMT
$3
FV
PMT
$2,500
FV
PMT
$50,000
FV
PMT
$1,000,000
FV
PMT
$1,000,000
FV