Ch5-Discounted Cash Flow Valuation-11E
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CHAPTER 5
DISCOUNTED CASH FLOW VALUATION
Answers to Concepts Review and Critical Thinking Questions
1.
Assuming positive cash flows and a positive interest rate, both the present and the future value will
rise.
2.
Assuming positive cash flows and a positive interest rate, the present value will fall, and the future
value will rise.
3.
It’s deceptive, but very common. The deception is particularly irritating given that such lotteries are
usually government sponsored!
4.
The most important consideration is the interest rate the lottery uses to calculate the lump sum
option. If you can earn an interest rate that is higher than you are being offered, you can create larger
annuity payments. Of course, taxes are also a consideration, as well as how badly you really need $5
million today. 5.
If the total amount of money is fixed, you want as much as possible as soon as possible. The team
(or, more accurately, the team owner) wants just the opposite.
6.
The better deal is the one with equal installments.
7.
Yes, they should. APRs generally don’t provide the relevant rate. The only advantage is that they are
easier to compute, but, with modern computing equipment, that advantage is not very important.
8.
A freshman does. The reason is that the freshman gets to use the money for much longer before
interest starts to accrue.
9.
The subsidy is the present value (on the day the loan is made) of the interest that would have accrued
up until the time it actually begins to accrue.
10.
The problem is that the subsidy makes it easier to repay the loan, not to obtain it. However, the
ability to repay the loan depends on future employment, not current need. For example, consider a
student who is currently needy, but is preparing for a career in a high-paying area (such as corporate
finance!). Should this student receive the subsidy? How about a student who is currently not needy,
but is preparing for a relatively low-paying job (such as becoming a college professor)?
CHAPTER 8 – 2
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 time line is:
0
1
2
3
4
PV
$470
$610
$735
$920
To solve this problem, we must find the PV of each cash flow and add them. To find the PV of a
lump sum, we use:
PV = FV/(1 + r
)
t
PV@10% = $470/1.10 + $610/1.10
2
+ $735/1.10
3
+ $920/1.10
4
= $2,111.99
PV@18% = $470/1.18 + $610/1.18
2
+ $735/1.18
3
+ $920/1.18
4
= $1,758.27
PV@24% = $470/1.24 + $610/1.24
2
+ $735/1.24
3
+ $920/1.24
4
= $1,550.39
2.
The time lines are:
0
1
2
3
4
5
6
7
8
PV
$5,300
$5,300
$5,300
$5,300
$5,300
$5,300
$5,300
$5,300
0
1
2
3
4
5
PV
$7,300
$7,300
$7,300
$7,300
$7,300
To find the PVA, we use the equation:
PVA = C
({1 – [1/(1 + r
)
t
]}/
r
)
At a 5 percent interest rate:
X@5%: PVA = $5,300{[1 – (1/1.05)
8
]/.05} = $34,255.03
Y@5%: PVA = $7,300{[1 – (1/1.05)
5
]/.05} = $31,605.18
And at a 15 percent interest rate:
X@15%: PVA = $5,300{[1 – (1/1.15)
8
]/.15} = $23,782.80
Y@15%: PVA = $7,300{[1 – (1/1.15)
5
]/.15} = $24,470.73
CHAPTER 8 – 3
Notice that the cash flow of X has a greater PV at a 5 percent interest rate, but a lower PV at a 15
percent interest rate. The reason is that X has greater total cash flows. At a lower interest rate, the
total cash flow is more important since the cost of waiting (the interest rate) is not as great. At a
higher interest rate, Y is more valuable since it has larger cash flows. At the higher interest rate,
these larger cash flows early are more important since the cost of waiting (the interest rate) is so
much greater. 3.
The time line is:
0
1
2
3
4
$1,075
$1,210
$1,340
$1,420
To solve this problem, we must find the FV of each cash flow and add them. To find the FV of a
lump sum, we use:
FV = PV(1 + r
)
t
FV@6% = $1,075(1.06)
3
+ $1,210(1.06)
2
+ $1,340(1.06) + $1,420 = $5,480.30
FV@13% = $1,075(1.13)
3
+ $1,210(1.13)
2
+ $1,340(1.13) + $1,420 = $6,030.36
FV@27% = $1,075(1.27)
3
+ $1,210(1.27)
2
+ $1,340(1.27) + $1,420 = $7,275.42
Notice, since we are finding the value at Year 4, the cash flow at Year 4 is added to the FV of the
other cash flows. In other words, we do not need to compound this cash flow.
4.
To find the PVA, we use the equation:
PVA = C
({1 – [1/(1 + r
)
t
]}/
r
)
0
1
…
15
PV
$3,850
$3,850
$3,850
$3,850
$3,850
$3,850
$3,850
$3,850
$3,850
PVA@15 yrs: PVA = $3,850{[1 – (1/1.06)
15
]/.06} = $37,392.16
0
1
…
40
PV
$3,850
$3,850
$3,850
$3,850
$3,850
$3,850
$3,850
$3,850
$3,850
PVA@40 yrs: PVA = $3,850{[1 – (1/1.06)
40
]/.06} = $57,928.24
0
1
…
75
PV
$3,850
$3,850
$3,850
$3,850
$3,850
$3,850
$3,850
$3,850
$3,850
PVA@75 yrs: PVA = $3,850{[1 – (1/1.06)
75
]/.06} = $63,355.02
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CHAPTER 8 – 4
To find the PV of a perpetuity, we use the equation:
PV = C
/
r
0
1
…
∞
PV
$3,850
$3,850
$3,850
$3,850
$3,850
$3,850
$3,850
$3,850
$3,850
PV = $3,850/.06 = $64,166.67
Notice that as the length of the annuity payments increases, the present value of the annuity
approaches the present value of the perpetuity. The present value of the 75-year annuity and the
present value of the perpetuity imply that the value today of all perpetuity payments beyond 75 years
is only $811.65.
5.
Here we have the PVA, the length of the annuity, and the interest rate. We want to calculate the
annuity payment. Using the PVA equation and solving for the payment in each case, we find:
0
1
2
3
4
5
6
$12,000
C
C
C
C
C
C
PVA = C
({1 – [1/(1 + r
)
t
]}/
r
)
$12,000 = C
{[1 – (1/1.11)
6
]/.11}
C
= $12,000/4.23054
C = $2,836.52
0
1
2
3
4
5
6
7
8
$19,700
C
C
C
C
C
C
C
C
PVA = C
({1 – [1/(1 + r
)
t
]}/
r
)
$19,700 = C
{[1 – (1/1.07)
8
]/.07}
C
= $19,700/5.97130
C = $3,299.11
0
1
…
15
$134,280
C
C
C
C
C
C
C
C
C
PVA = C
({1 – [1/(1 + r
)
t
]}/
r
)
$134,280 = C
{[1 – (1/1.08)
15
]/.08}
C
= $134,280/8.55948
C = $15,687.87
0
1
…
20
$300,000
C
C
C
C
C
C
C
C
C
PVA = C
({1 – [1/(1 + r
)
t
]}/
r
)
$300,000 = C
{[1 – (1/1 .06)
20
]/.06}
C
= $300,000/11.46992
CHAPTER 8 – 5
C = $26,155.37
6.
Here we need to find the present value of an annuity. Using the PVA equation, we find:
0
1
2
3
4
5
6
7
PVA
$1,750
$1,750
$1,750
$1,750
$1,750
$1,750
$1,750
PVA = C
({1 – [1/(1 + r
)
t
]}/
r
)
PVA = $1,750{[1 – (1/1.05)
7
]/.05} PVA = $10,126.15
0
1
2
3
4
5
6
7
8
9
PVA
$1,390
$1,390
$1,390
$1,390
$1,390
$1,390
$1,390
$1,390
$1,390
PVA = C
({1 – [1/(1 + r
)
t
]}/
r
)
PVA = $1,390{[1 – (1/1.10)
9
]/.10} PVA = $8,005.04
0
1
…
18
PVA
$17,500
$17,50
0
$17,500
$17,500
$17,500
$17,500
$17,500
$17,50
0
$17,500
PVA = C
({1 – [1/(1 + r
)
t
]}/
r
)
PVA = $17,500{[1 – (1/1.08)
18
]/.08} PVA = $164,008.02
0
1
…
28
PVA
$50,000
$50,00
0
$50,000
$50,000
$50,000
$50,000
$50,000
$50,00
0
$50,000
PVA = C
({1 – [1/(1 + r
)
t
]}/
r
)
PVA = $50,000{[1 – (1/1.14)
28
]/.14} PVA = $348,033.11
7.
Here we have the FVA, the length of the annuity, and the interest rate. We want to calculate the
annuity payment. Using the FVA equation:
0
1
8
$25,600
C
C
C
C
C
C
C
C
FVA = C
{[(1 + r
)
t
– 1]/
r
}
$25,600 = C
[(1.05
8
– 1)/.05]
C
= $25,600/9.54911
C = $2,680.88
CHAPTER 8 – 6
0
1
…
40
$1,250,000
C
C
C
C
C
C
C
C
C
FVA = C
{[(1 + r
)
t
– 1]/
r
}
$1,250,000 = C
[(1.07
40
– 1)/.07]
C
= $1,250,000/199.63511
C = $6,261.42
0
1
…
25
$535,000
C
C
C
C
C
C
C
C
C
FVA = C
{[(1 + r
)
t
– 1]/
r
}
$535,000 = C
[(1.08
25
– 1)/.08]
C
= $535,000/73.10594
C = $7,318.15
0
1
…
13
$104,600
C
C
C
C
C
C
C
C
C
FVA = C
{[(1 + r
)
t
– 1]/
r
}
$104,600 = C
[(1.04
13
– 1)/.04]
C
= $104,600/16.62684
C = $6,291.03
8.
Here we need to find the future value of an annuity. Using the FVA equation, we find:
0
1
10
FVA
$2,100
$2,100
$2,100
$2,100
$2,100
$2,100
$2,100
$2,100
$2,100
$2,100
FVA = C
{[(1 + r
)
t
– 1]/
r
}
FVA = $2,100[(1.07
10
– 1)/.07] FVA = $29,014.54
0
1
…
40
FVA
$6,500
$6,500
$6,500
$6,500
$6,500
$6,500
$6,500
$6,500
$6,500
FVA = C
{[(1 + r
)
t
– 1]/
r
}
FVA = $6,500[(1.08
40
– 1)/.08] FVA = $1,683,867.37
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CHAPTER 8 – 7
0
1
9
FVA
$1,100
$1,100
$1,100
$1,100
$1,100
$1,100
$1,100
$1,100
$1,100
FVA = C
{[(1 + r
)
t
– 1]/
r
}
FVA = $1,100[(1.09
9
– 1)/.09] FVA = $14,323.14
0
1
…
30
FVA
$5,000
$5,000
$5,000
$5,000
$5,000
$5,000
$5,000
$5,000
$5,000
FVA = C
{[(1 + r
)
t
– 1]/
r
}
FVA = $5,000[(1.11
30
– 1)/.11] FVA = $995,104.39
8.
Here we need to find the FVA. The equation to find the FVA is:
FVA = C
{[(1 + r
)
t
– 1]/
r
}
0
1
…
20
FVA
$5,300
$5,300
$5,300
$5,300
$5,300
$5,300
$5,300
$5,300
$5,300
FVA for 20 years = $5,300[(1.098
20
– 1)/.098] FVA for 20 years = $296,748.26
0
1
…
40
FVA
$5,300
$5,300
$5,300
$5,300
$5,300
$5,300
$5,300
$5,300
$5,300
FVA for 40 years = $5,300[(1.098
40
– 1)/.098] FVA for 40 years = $2,221,767.13
Notice that because of exponential growth, doubling the number of periods does not merely double
the FVA.
10.
The time line is:
0
1
…
∞
PV
$30,00
0
$30,000
$30,000
$30,000
$30,000
$30,000
$30,00
0
$30,000
$30,000
CHAPTER 8 – 8
This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation:
PV = C
/
r
PV = $30,000/.056 PV = $535,714.29
11.
The time line is:
0
1
…
∞
–$525,000
$30,00
0
$30,000
$30,000
$30,000
$30,000
$30,000
$30,00
0
$30,000
$30,000
Here we need to find the interest rate that equates the perpetuity cash flows with the PV of the cash
flows. Using the PV of a perpetuity equation:
PV = C
/
r
$525,000 = $30,000/
r
We can now solve for the interest rate as follows:
r
= $30,000/$525,000 r = .0571, or 5.71%
12.
For discrete compounding, to find the EAR, we use the equation:
EAR = [1 + (APR/
m
)]
m
– 1
EAR = [1 + (.078/4)]
4
– 1
= .0803, or 8.03%
EAR = [1 + (.153/12)]
12
– 1
= .1642, or 16.42%
EAR = [1 + (.124/365)]
365
– 1 = .1320, or 13.20%
To find the EAR with continuous compounding, we use the equation:
EAR = e
q – 1
EAR = e
.114
– 1 EAR = .1208, or 12.08%
13.
Here we are given the EAR and need to find the APR. Using the equation for discrete compounding:
EAR = [1 + (APR/
m
)]
m
– 1
We can now solve for the APR. Doing so, we get:
APR = m
[(1 + EAR)
1/
m
– 1]
EAR = .142 = [1 + (APR/2)]
2
– 1
APR = 2(1.142
1/2
– 1)
= .1373, or 13.73%
EAR = .184 = [1 + (APR/12)]
12
– 1
APR = 12(1.184
1/12
– 1)
= .1701, or 17.01%
CHAPTER 8 – 9
EAR = .111 = [1 + (APR/52)]
52
– 1
APR = 52(1.111
1/52
– 1)
= .1054, or 10.54%
EAR = .089 = [1 + (APR/365)]
365
– 1
APR = 365(1.089
1/365
– 1)
= .0853, or 8.53%
14.
For discrete compounding, to find the EAR, we use the equation:
EAR = [1 + (APR/
m
)]
m
– 1
So, for each bank, the EAR is:
First National: EAR = [1 + (.138/12)]
12
– 1 = .1471, or 14.71%
First United: EAR = [1 + (.141/2)]
2
– 1 = .1460, or 14.60%
Notice that the higher APR does not necessarily result in the higher EAR. The number of
compounding periods within a year will also affect the EAR.
15.
The reported rate is the APR, so we need to convert the EAR to an APR as follows:
EAR = [1 + (APR/
m
)]
m
– 1
APR = m
[(1 + EAR)
1/
m
– 1]
APR = 365[(1.182)
1/365
– 1] APR = .1672, or 16.72%
This is deceptive because the borrower is actually paying annualized interest of 18.2 percent per
year, not the 16.72 percent reported on the loan contract.
16.
The time line is:
0
1
…
34
$5,500
FV
For this problem, we need to find the FV of a lump sum using the equation:
FV = PV(1 + r
)
t
It is important to note that compounding occurs semiannually. To account for this, we will divide the
interest rate by two (the number of compounding periods in a year), and multiply the number of
periods by two. Doing so, we get: FV = $5,500[1 + (.084/2)]
17(2)
FV = $22,277.43
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CHAPTER 8 – 10
17.
For this problem, we need to find the FV of a lump sum using the equation:
FV = PV(1 + r
)
t
It is important to note that compounding occurs daily. To account for this, we will divide the interest
rate by 365 (the number of days in a year, ignoring leap year), and multiply the number of periods by
365. Doing so, we get: 0
1
…
5(365)
$7,500
FV
FV in 5 years = $7,500[1 + (.083/365)]
5(365) FV in 5 years = $11,357.24
0
1
…
10(365)
$7,500
FV
FV in 10 years = $7,500[1 + (.083/365)]
10(365)
FV in 10 years = $17,198.27
0
1
…
20(365)
$7,500
FV
FV in 20 years = $7,500[1 + (.083/365)]
20(365)
FV in 20 years = $39,437.39
18.
The time line is:
0
1
…
10(365)
PV
$95,000
For this problem, we need to find the PV of a lump sum using the equation:
PV = FV/(1 + r
)
t
It is important to note that compounding occurs daily. To account for this, we will divide the interest
rate by 365 (the number of days in a year, ignoring leap year), and multiply the number of periods by
365. Doing so, we get: PV = $95,000/[(1 + .09/365)
10(365)
] PV = $38,628.40
CHAPTER 8 – 11
19.
The APR is the interest rate per period times the number of periods in a year. In this case, the interest
rate is 25.5 percent per month, and there are 12 months in a year, so we get:
APR = 12(25.5%) = 306% To find the EAR, we use the EAR formula:
EAR = [1 + (APR/
m
)]
m
– 1
EAR = (1 + .255)
12
– 1 EAR = 14.2660, or 1,426.60%
Notice that we didn’t need to divide the APR by the number of compounding periods per year. We
do this division to get the interest rate per period, but in this problem we are already given the
interest rate per period.
20.
The time line is:
0
1
…
60
$84,500
C
C
C
C
C
C
C
C
C
We first need to find the annuity payment. We have the PVA, the length of the annuity, and the
interest rate. Using the PVA equation:
PVA = C
({1 – [1/(1 + r
)
t
]}/
r
)
$84,500 = C
[1 – {1/[1 + (.047/12)]
60
}/(.047/12)]
Solving for the payment, we get:
C
= $84,500/53.3786
C = $1,583.03
To find the EAR, we use the EAR equation:
EAR = [1 + (APR/
m
)]
m
– 1
EAR = [1 + (.047/12)]
12
– 1 EAR = .0480, or 4.80%
21.
The time line is:
0
1
…
t
–$18,000
$500
$500
$500
$500
$500
$500
$500
$500
$500
Here we need to find the length of an annuity. We know the interest rate, the PVA, and the
payments. Using the PVA equation:
PVA = C
({1 – [1/(1 + r
)
t
]}/
r
)
$18,000 = $500{[1 – (1/1.018)
t
]/.018}
CHAPTER 8 – 12
Now we solve for t
:
1/1.018
t
= 1 – ($18,000/$500)(.018)
1/1.018
t
= .352
1.018
t = 1/.352 = 2.841 t
= ln 2.841/ln 1.018 t = 58.53 months
22.
The time line is:
0
1
–$3
$4
Here we are trying to find the interest rate when we know the PV and FV. Using the FV equation:
FV = PV(1 + r
)
$4 = $3(1 + r
) r
= 4/3 – 1 r = .3333, or 33.33% per week
The interest rate is 33.33% per week. To find the APR, we multiply this rate by the number of weeks
in a year, so:
APR = (52)33.33% APR = 1,733.33%
And using the equation to find the EAR:
EAR = [1 + (APR/
m
)]
m
– 1
EAR = [1 + .3333]
52
– 1 EAR = 3,139,165.1569, or 313,916,515.69%
23.
The time line is:
0
1
…
∞
–$260,000
$1,500
$1,500
$1,500
$1,500
$1,500
$1,500
$1,500
$1,500
$1,500
Here we need to find the interest rate that equates the perpetuity cash flows with the PV of the cash
flows. Using the PV of a perpetuity equation:
PV = C
/
r
$260,000 = $1,500/
r
We can now solve for the interest rate as follows:
r
= $1,500/$260,000 r = .0058, or .58% per month
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CHAPTER 8 – 13
The interest rate is .58% per month. To find the APR, we multiply this rate by the number of months
in a year, so:
APR = 12(.58%) = 6.92%
And using the equation to find an EAR:
EAR = [1 + (APR/
m
)]
m
– 1
EAR = [1 + .0058]
12
– 1 EAR = .0715, or 7.15%
24.
The time line is:
0
1
…
480
$475
$475
$475
$475
$475
$475
$475
$475
$475
This problem requires us to find the FVA. The equation to find the FVA is:
FVA = C
{[(1 + r
)
t
– 1]/
r
}
FVA = $475[{[1 + (.10/12)]
480 – 1}/(.10/12)] FVA = $3,003,937.80
25.
The time line is:
0
1
…
40
$5,700
$5,700
$5,700
$5,700
$5,700
$5,700
$5,700
$5,700
$5,700
In the previous problem, the cash flows are monthly and the compounding period is monthly. The
compounding periods are still monthly, but since the cash flows are annual, we need to use the EAR
to calculate the future value of annual cash flows. It is important to remember that you have to make
sure the compounding periods of the interest rate are the same as the timing of the cash flows. In this
case, we have annual cash flows, so we need the EAR since it is the true annual interest rate you will
earn. So, finding the EAR:
EAR = [1 + (APR/
m
)]
m
– 1
EAR = [1 + (.10/12)]
12
– 1 EAR = .1047, or 10.47%
Using the FVA equation, we get:
FVA = C
{[(1 + r
)
t
– 1]/
r
}
FVA = $5,700[(1.1047
40
– 1)/.1047] FVA = $2,868,732.50
26.
The time line is:
0
1
…
16
PV
$3,000
$3,000
$3,000
$3,000
$3,000
$3,000
$3,000
$3,000
$3,000
CHAPTER 8 – 14
The cash flows are an annuity with four payments per year for four years, or 16 payments. We can
use the PVA equation:
PVA = C
({1 – [1/(1 + r
)
t
]}/
r
)
PVA = $3,000{[1 – (1/1.0057)
16
]/.0057} PVA = $45,751.83
27.
The time line is:
0
1
2
3
4
PV
$815
$990
$0
$1,520
The cash flows are annual and the compounding period is quarterly, so we need to calculate the EAR
to make the interest rate comparable with the timing of the cash flows. Using the equation for the
EAR, we get:
EAR = [1 + (APR/
m
)]
m
– 1
EAR = [1 + (.085/4)]
4
– 1 EAR = .0877, or 8.77%
And now we use the EAR to find the PV of each cash flow as a lump sum and add them together:
PV = $815/1.0877 + $990/1.0877
2
+ $1,520/1.0877
4
PV = $2,671.72
28.
The time line is:
0
1
2
3
4
PV
$2,480
$0
$3,920
$2,170
Here the cash flows are annual and the given interest rate is annual, so we can use the interest rate
given. We can find the PV of each cash flow and add them together.
PV = $2,480/1.0932 + $3,920/1.0932
3
+ $2,170/1.0932
4
PV = $6,788.39
Intermediate
29.
The total interest paid by First Simple Bank is the interest rate per period times the number of
periods. In other words, the interest by First Simple Bank paid over 10 years will be:
.064(10) = .64
First Complex Bank pays compound interest, so the interest paid by this bank will be the FV factor
of $1 minus the initial investment of $1, or:
(1 + r
)
10
– 1
CHAPTER 8 – 15
Setting the two equal, we get:
(.064)(10) = (1 + r
)
10
– 1 r
= 1.64
1/10
– 1 = .0507, or 5.07%
30.
The time line is:
0
1
…
60
$65,500
C
C
C
C
C
C
C
C
C
We need to use the PVA due equation, which is:
PVA
due
= (1 + r
)PVA
Using this equation:
PVA
due
= $65,500 = [1 + (.041/12)] × C
[{1 – 1/[1 + (.041/12)
60
]}/(.041/12)]
$65,276.97 = C
{1 – [1/(1 + .041/12)
60
]}/(.041/12)]
C
= $1,205.12
Notice, to find the payment for the PVA due, we find the PV of an ordinary annuity, then compound
this amount forward one period. 31. Here we need to find the FV of a lump sum, with a changing interest rate. We must do this problem
in two parts.
After the first six months, the balance will be:
FV = $9,000[1 + (.0125/12)]
6
FV = $9,056.40 This is the balance in six months. The FV in another six months will be: FV = $9,056.40[1 + (.178/12)]
6
FV = $9,892.90
The problem asks for the interest accrued, so, to find the interest, we subtract the beginning balance from the FV. The interest accrued is:
Interest = $9,892.90 – 9,000 Interest = $892.90
32.
We will calculate the time we must wait if we deposit in the bank that pays simple interest. The
interest amount we will receive each year in this bank will be:
Interest = $90,000(.048) Interest = $4,320 per year
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CHAPTER 8 – 16
The deposit will have to increase by the difference between the amount we need by the amount we
originally deposit divided by the interest earned per year, so the number of years it will take in the
bank that pays simple interest is:
Years to wait = ($275,000 – 90,000)/$4,320 Years to wait = 42.82 years
To find the number of years it will take in the bank that pays compound interest, we can use the
future value equation for a lump sum and solve for the periods. Doing so, we find:
0
1
…
t
–$90,000
$275,000
FV = PV(1 + r
)
t
$275,000 = $90,000[1 + (.048/12)]
t
t
= 279.80 months, or 23.32 years
33.
The time line is:
0
1
…
12
–$1
FV
Here we need to find the future value of a lump sum. We need to make sure to use the correct
number of periods. So, the future value after one year will be:
FV = PV(1 + r
)
t
FV = $1(1.0121)
12
FV = $1.16
And the future value after two years will be:
0
1
…
24
–$1
FV
FV = PV(1 + r
)
t
FV = $1(1.0121)
24
FV = $1.33
34.
The time line is:
0
1
…
31
–£440
£60
£60
£60
£60
£60
£60
£60
£60
£60
CHAPTER 8 – 17
Here we are given the PVA, number of periods, and the amount of the annuity. We need to solve for
the interest rate. Even though the currency is pounds and not dollars, we can still use the same time
value equations. Using the PVA equation:
PVA = C
({1 – [1/(1 + r
)
t
]}/
r
)
£440 = £60[{1 – [1/(1 + r
)
31
]}/
r
]
To find the interest rate, we need to solve this equation on a financial calculator, using a spreadsheet,
or by trial and error. If you use trial and error, remember that increasing the interest rate decreases
the PVA, and decreasing the interest rate increases the PVA. Using a spreadsheet, we find:
r
= 13.36%
Not bad for an English Literature major!
35.
Here we need to compare two cash flows, so we will find the value today of both sets of cash flows.
We need to make sure to use the monthly cash flows since the salary is paid monthly. Doing so, we
find:
0
1
…
24
$7,500
$7,500
$7,500
$7,500
$7,500
$7,500
$7,500
$7,500
$7,500
PVA
1
= $90,000/12({1 – 1/[1 + (.07/12)
24
]}/(.07/12)) PVA
1
= $167,513.24
0
1
…
24
$20,000
$6,417
$6,417
$6,417
$6,417
$6,417
$6,417
$6,417
$6,417
$6,417
PVA
2
= $20,000 + $77,000/12({1 – 1/[1 + (.07/12)
24
]}/(.07/12)) PVA
2
= $163,316.89
You should choose the first option since it has a higher present value.
36.
The time line is:
0
1
…
20
PVA
$25,000
$25,00
0
$25,000
$25,000
$25,000
$25,000
$25,00
0
$25,000
$25,000
The cash flows are an annuity, so we can use the present value of an annuity equation. Doing so, we
find:
PVA = C
({1 – [1/(1 + r
)
t
]}/
r
)
PVA = $25,000({1 – [1/(1.09)
20
]}/.09) PVA = $228,213.64
CHAPTER 8 – 18
37.
The investment we should choose is the investment with the higher rate of return. We will use the
future value equation to find the interest rate for each option. Doing so, we find the return for
Investment G is:
0
6
–$25,000
$60,000
FV = PV(1 + r
)
t
$60,000 = $25,000(1 + r
)
6
r
= ($60,000/$25,000)
1/6
– 1 r = .1571, or 15.71%
And the return for Investment H is:
0
9
–$25,000
$92,000
FV = PV(1 + r
)
t
$92,000 = $25,000(1 + r
)
9
r
= ($92,000/$25,000)
1/9
– 1 r = .1558, or 15.58%
So, we should choose Investment H since it has a higher return.
38.
The time line is:
0
1
…
13
$10,000
$10,00
0
$10,000
$10,000
$10,000
$10,000
$10,00
0
$10,000
$10,000
The relationships between the present value of an annuity and the interest rate are:
PVA falls as r
increases, and PVA rises as r
decreases.
FVA rises as r
increases, and FVA falls as r
decreases.
The present values of $10,000 per year for 13 years at the various interest rates given are:
PVA@10% = $10,000{[1 – (1/1.10)
13
]/.10} = $71,033.56
PVA@5% = $10,000{[1 – (1/1.05)
13
]/.05} = $93,935.73
PVA@15% = $10,000{[1 – (1/1.15)
13
]/.15} = $55,831.47
39.
The time line is:
0
1
…
t
–$15,000
$225
$225
$225
$225
$225
$225
$225
$225
$225
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CHAPTER 8 – 19
Here we are given the FVA, the interest rate, and the amount of the annuity. We need to solve for the
number of payments. Using the FVA equation:
FVA = $15,000 = $225[{[1 + (.065/12)]
t
– 1}/(.065/12)]
Solving for t
, we get:
1.00542
t
= 1 + ($15,000/$225)(.065/12) t
= ln 1.36111/ln 1.00542 t = 57.07 payments
40.
The time line is:
0
1
…
60
–$95,000
$1,850
$1,850
$1,850
$1,850
$1,850
$1,850
$1,850
$1,850
$1,850
Here we are given the PVA, number of periods, and the amount of the annuity. We need to solve for
the interest rate. Using the PVA equation:
PVA = $95,000 = $1,850[{1 – [1/(1 + r
)
60
]}/
r
]
To find the interest rate, we need to solve this equation on a financial calculator, using a spreadsheet,
or by trial and error. If you use trial and error, remember that increasing the interest rate lowers the
PVA, and decreasing the interest rate increases the PVA. Using a spreadsheet, we find:
r
= .525%
The APR is the periodic interest rate times the number of periods in the year, so:
APR = 12(.525%) = 6.30%
This is the monthly interest rate. To find the APR with a monthly interest rate, we multiply the
monthly rate by 12, so the APR is:
APR = .00525 × 12
APR = .0630, or 6.30%
41.
The time line is:
0
1
2
3
4
5
PV
$39,344,970
$42,492,56
8
$45,640,165
$48,787,763
$51,935,36
0
To solve this problem, we must find the PV of each cash flow and add them. To find the PV of a
lump sum, we use:
PV = FV/(1 + r
)
t
CHAPTER 8 – 20
PV = $39,344,970/1.11 + $42,492,568/1.11
2
+ $45,640,165/1.11
3 + $48,787,763/1.11
4
+ $51,935,360/1.11
5
PV = $166,264,654.66
CHAPTER 8 – 21
42.
The time line is: 0
1
2
3
4
$600,000 $8,000,000
$8,000000
$12,000,00
0
$12,00000
0
To solve this problem, we must find the PV of each cash flow and add them. To find the PV of a
lump sum, we use:
PV = FV/(1 + r
)
t
PV = $600,000 + $8,000,000/1.11 + $8,000,000/1.11
2
+ $12,000,000/1.11
3
+ $12,000,000/1.11
4
PV = $30,979,254.94
43.
Here we are finding the interest rate for an annuity cash flow.
We are given the PVA, the number of
periods, and the amount of the annuity. We should also note that the PV of the annuity is the amount
borrowed, not the purchase price, since we are making a down payment on the warehouse. The
amount borrowed is:
Amount borrowed = .80($2,600,000) = $2,080,000
The time line is:
0
1
…
360
–$2,080,000
$14,200
$14,200
$14,200
$14,200
$14,200
$14,200
$14,200
$14,200
$14,200
Using the PVA equation:
PVA = $2,080,000 = $14,200[{1 – [1/(1 + r
)
360
]}/
r
]
To find the interest rate, we need to solve this equation on a financial calculator, using a spreadsheet,
or by trial and error. If you use trial and error, remember that increasing the interest rate lowers the
PVA, and decreasing the interest rate increases the PVA. Using a spreadsheet, we find:
r
= .605%
The APR is the monthly interest rate times the number of months in the year, so:
APR = 12(.605%) APR = 7.26%
And the EAR is:
EAR = (1 + .00605)
12
– 1 EAR = .0750, or 7.50%
44.
Here we have two cash flow streams that will be combined in the future. In essence, we have three
time lines. We will start with the time lines for the savings period, which are:
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CHAPTER 8 – 22
Bond account:
0
1
2
3
4
5
6
7
8
9
10
$60,000
$9,000
$9,000
$9,000
$9,000
$9,000
$9,000
$9,000
$9,000
$9,000
$9,000
Stock account:
0
10
$230,000
To find the withdrawal amount, we need to know the present value, as well as the interest rate and
periods, which are given. The present value of the retirement account is the future value of the stock
and bond account. We need to find the future value of each account and add the future values
together. For the bond account, the future value is the value of the current savings plus the value of
the annual deposits. So, the future value of the bond account will be:
FV = C
{[(1 + r
)
t
– 1]/
r
} + PV(1 + r
)
t
FV = $9,000{[(1 + .06)
10
– 1]/.06} + $60,000(1 + .06)
10
FV = $226,078.02
The total value of the stock account at retirement will be the future value of a lump sum, so:
FV = PV(1 + r
)
t
FV = $230,000(1 + .105)
10
FV = $624,238.59
The total value of the account at retirement will be:
Total value at retirement = $226,078.02 + 624,238.59
Total value at retirement = $850,316.61
So, at retirement, we have:
0
1
…
25
–$850,316.61
C
C
C
C
C
C
C
C
C
This amount is the present value of the annual withdrawals. Now we can use the present value of an
annuity equation to find the annuity amount. Doing so, we find the annual withdrawal will be:
PVA = C
({1 – [1/(1 + r
)
t
]}/
r
)
$850,316.01 = C
[{1 – [1/(1 + .053)
25
]}/.053]
C
= $62,158.80
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CHAPTER 8 – 23
45.
The time line is:
0
1
…
60
$465
$465
$465
$465
$465
$465
$465
$465
$465
Here we are given the PVA for an annuity due, number of periods, and the amount of the annuity.
We need to solve for the interest rate. Using the PVA equation:
PVA
due
= C
[{1 – [1/(1 + r
)]
t
}/ r
](1 + r
)
$24,500 = $465[{1 – [1/(1 + r
)]
60
}/
r
](1 + r
)
To find the interest rate, we need to solve this equation on a financial calculator, using a spreadsheet,
or by trial and error. If you use trial and error, remember that increasing the interest rate decreases
the PVA, and decreasing the interest rate increases the PVA. Using a spreadsheet, we find:
r
= .00452, or .452%
This is the monthly interest rate. To find the APR with a monthly interest rate, we multiply the
monthly rate by 12, so the APR is:
APR = .00452 × 12
APR = .0542, or 5.42%
46.
a.
If the payments are in the form of an ordinary annuity, the present value will be:
0
1
2
3
4
5
$14,500
$14,500
$14,500
$14,500
$14,500
PVA = C
({1 – [1/(1 + r
)
t
]}/
r
)
PVA = $14,500[{1 – [1/(1 + .071)
5
]}/ .071]
PVA = $59,294.01
If the payments are an annuity due, the present value will be:
0
1
2
3
4
5
$14,500
$14,500
$14,500
$14,500
$14,500
PVA
due
= (1 + r
)PVA
PVA
due
= (1 + .071)$59,294.01
PVA
due
= $63,503.89
b.
We can find the future value of the ordinary annuity as:
FVA = C
{[(1 + r
)
t
– 1]/
r
}
FVA = $14,500{[(1 + .071)
5
– 1]/.071}
FVA = $83,552.26
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CHAPTER 8 – 24
If the payments are an annuity due, the future value will be:
FVA
due
= (1 + r
)FVA
FVA
due
= (1 + .071)$83,552.26
FVA
due
= $89,484.47
c.
Assuming a positive interest rate, the present value of an annuity due will always be larger than
the present value of an ordinary annuity. Each cash flow in an annuity due is received one
period earlier, which means there is one period less to discount each cash flow. Assuming a
positive interest rate, the future value of an annuity due will always be higher than the future
value of an ordinary annuity. Since each cash flow is made one period sooner, each cash flow
receives one extra period of compounding.
47.
Here we need to find the difference between the present value of an annuity and the present value of
a perpetuity. The annuity time line is:
0
1
…
30
PVA
$11,500
$11,500
$11,500
$11,500
$11,500
$11,500
$11,500
$11,50
0
$11,500
PVA = C
({1 – [1/(1 + r
)
t
]}/
r
)
PVA = $11,500{[1 – (1/1.043)
30
]/.043} PVA = $191,810.81
And the present value of the perpetuity is:
0
1
…
∞
PV
$11,500
$11,50
0
$11,500
$11,500
$11,500
$11,500
$11,50
0
$11,500
$11,500
PVP = C
/
r
PVP = $11,500/.043
PVP = $267,441.86
So, the difference in the present values is:
Difference = $267,441.86 – 191,810.81
Difference = $75,631.05
There is another common way to answer this question. We need to recognize that the difference in
the cash flows is a perpetuity of $11,500 beginning 31 years from now. We can find the present
value of this perpetuity and the solution will be the difference in the cash flows. So, we can find the
present value of this perpetuity as:
PVP = C
/
r
PVP = $11,500/.043
PVP = $267,441.86
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CHAPTER 8 – 25
This is the present value 30 years from now, one period before the first cash flows. We can now find
the present value of this lump sum as:
PV = FV/(1 + r
)
t
PV = $267,441.86/(1 + .043)
30
PV = $75,631.05
This is the same answer we calculated before.
48.
The time line is:
0
1
…
18
19
…
30
$8,250
$8,250
$8,250
$8,250
Here we need to find the present value of an annuity at several different times. The annuity has
semiannual payments, so we need the semiannual interest rate. The semiannual interest rate is:
Semiannual rate = .08/2 Semiannual rate = .04
Now, we can use the present value of an annuity equation. Doing so, we get:
PVA = C
({1 – [1/(1 + r
)
t
]}/
r
)
PVA = $8,250{[1 – (1/1.04)
12
]/.04} PVA = $77,426.86
This is the present value one period before the first payment. The first payment occurs nine and one-
half years from now, so this is the value of the annuity nine years from now. Since the interest rate is
semiannual, we must also be careful to use the number of semiannual periods. The value of the
annuity five years from now is:
PV = FV/(1 + r
)
t
PV = $77,426.86/(1 + .04)
8
PV = $56,575.05
And the value of the annuity three years from now is:
PV = FV/(1 + r
)
t
PV = $77,426.86/(1 + .04)
12
PV = $48,360.59
And the value of the annuity today is:
PV = FV/(1 + r
)
t
PV = $77,426.86/(1 + .04)
18
PV = $38,220.07
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CHAPTER 8 – 26
49.
The time line is:
0
1
2
3
4
5
6
…
20
PV
$5,100
$5,100
$5,100
$5,100
We want to find the value of the cash flows today, so we will find the PV of the annuity, and then
bring the lump sum PV back to today. The annuity has 15 payments, so the PV of the annuity is: PVA = $5,100{[1 – (1/1.079
15
)]/.079} PVA = $43,921.16
Since this is an ordinary annuity equation, this is the PV one period before the first payment, so this
is the PV at t
= 5. To find the value today, we find the PV of this lump sum. The value today is:
PV = $43,921.16/1.079
5
PV = $30,030.78
50.
The time line is:
0
1
…
180
PV
$1,750
$1,750
$1,750
$1,750
$1,750
$1,750
$1,750
$1,750
$1,750
This question is asking for the present value of an annuity, but the interest rate changes during the
life of the annuity. We need to find the present value of the cash flows for the last eight years first.
The PV of these cash flows is:
PVA
2
= $1,750[{1 – 1/[1 + (.06/12)]
96
}/(.06/12)] PVA
2
= $133,166.63
Note that this is the PV of this annuity exactly seven years from today. Now we can discount this
lump sum to today. The value of this cash flow today is:
PV = $133,166.63/[1 + (.09/12)]
84
PV = $71,090.38
Now we need to find the PV of the annuity for the first seven years. The value of these cash flows
today is:
PVA
1
= $1,750[{1 – 1/[1 + (.09/12)]
84
}/(.09/12)] PVA
1
= $108,769.44
The value of the cash flows today is the sum of these two cash flows, so:
PV = $71,090.38 + 108,769.44 PV = $179,859.81
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CHAPTER 8 – 27
51.
The time line is:
0
1
…
156
$1,600
$1,600
$1,600
$1,600
$1,600
$1,600
$1,600
$1,600
$1,600
Here we are trying to find the dollar amount invested today that will equal the FVA with a known
interest rate and payments. First we need to determine how much we would have in the annuity
account. Finding the FV of the annuity, we get:
FVA = $1,600[{[ 1 + (.078/12)]
156
– 1}/(.078/12)] FVA = $430,172.31
Now we have:
0
1
…
13
PV
$430,172.31
So, we need to find the PV of a lump sum that will give us the same FV. Using the FV of a lump
sum with continuous compounding, we get: PV = FV/(1 + r
)
t
PV = $430,313.02/(1 + .09)
13
PV = $140,313.02
52.
The time line is:
0
1
…
7
…
14
15
…
∞
PV
$7,300
$7,300
$7,300
$7,300
To find the value of the perpetuity at t
= 14, we first need to use the PV of a perpetuity equation.
Using this equation, we find:
PV = $7,300/.046 PV = $158,695.65 0
1
…
7
…
14
PV
$158,695.65
Remember that the PV of perpetuity (and annuity) equations give the PV one period before the first
payment, so, this is the value of the perpetuity at t
= 14. To find the value at t
= 7, we find the PV of
this lump sum as:
PV = $158,695.65/1.046
7
PV = $115,835.92
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CHAPTER 8 – 28
53.
The time line is:
0
1
…
12
–$20,000 $1,973.33
$1,973.3
3
$1,973.33
$1,973.33
$1,973.33
$1,973.33
$1,973.33
$1,973.3
3
$1,973.33
To find the APR and EAR, we need to use the actual cash flows of the loan. In other words, the
interest rate quoted in the problem is only relevant to determine the total interest under the terms
given. The interest rate for the cash flows of the loan is:
PVA = $20,000 = $1,973.33{(1 – [1/(1 + r
)
12
])/
r
}
Again, we cannot solve this equation for r
, so we need to solve this equation on a financial
calculator, using a spreadsheet, or by trial and error. Using a spreadsheet, we find:
r
= 2.699% per month
So the APR that would legally have to be quoted is:
APR = 12(2.699%) APR = 32.39% And the EAR is:
EAR = 1.02699
12
– 1 EAR = .3766, or 37.66%
54.
The time line is:
0
1
2
3
4
5
FV
$20,000
$27,000
$38,000
To solve this problem, we must find the FV of each cash flow and add them. To find the FV of a
lump sum, we use:
FV = PV(1 + r
)
t
FV = $20,000(1.057)
3
+ $27,000(1.057)
2
+ $38,000 FV = $91,784.37
Notice, since we are finding the value at Year 5, the cash flow at Year 5 is added to the FV of the
other cash flows. In other words, we do not need to compound this cash flow. To find the value in
Year 10, we need to find the future value of this lump sum. Doing so, we find:
FV = PV(1 + r
)
t
FV = $91,784.37(1.057)
5
FV = $121,099.86
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CHAPTER 8 – 29
55.
The payment for a loan repaid with equal payments is the annuity payment with the loan value as the
PV of the annuity. So, the loan payment will be:
PVA = $58,500 = C
{[1 – 1/(1 + .06)
5
]/.06}
C
= $13,887.69
The interest payment is the beginning balance times the interest rate for the period, and the principal
payment is the total payment minus the interest payment. The ending balance is the beginning
balance minus the principal payment. The ending balance for a period is the beginning balance for
the next period. The amortization table for an equal payment is:
Year
Beginning
Balance
Total
Payment
Interest
Payment
Principal
Payment
Ending Balance
1
$58,500.00
$13,887.69
$3,510.00
$10,377.69
$48,122.31
2
48,122.31
13,887.69
2,887.34
11,000.35
37,121.96
3
37,121.96
13,887.69
2,227.32
11,660.37
25,461.59
4
25,461.59
13,887.69
1,527.70
12,359.99
13,101.59
5
13,101.59
13,887.69
786.10
13,101.59
0
In the third year, $2,227.32 of interest is paid. Total interest over life of the loan = $3,510 + 2,887.34 + 2,227.32 + 1,527.70 + 786.10 Total interest over life of the loan = $10,938.45
56.
This amortization table calls for equal principal payments of $11,700 per year. The interest payment
is the beginning balance times the interest rate for the period, and the total payment is the principal
payment plus the interest payment. The ending balance for a period is the beginning balance for the
next period. The amortization table for an equal principal reduction is:
Year
Beginning
Balance
Total
Payment
Interest
Payment
Principal
Payment
Ending Balance
1
$58,500
$15,210
$3,510
$11,700
$46,800
2
46,800
14,508
2,808
11,700
35,100
3
35,100
13,806
2,106
11,700
23,400
4
23,400
13,104
1,404
11,700
11,700
5
11,700
12,402
702
11,700
0
In the third year, $2,106 of interest is paid. Total interest over life of the loan = $3,510 + 2,808 + 2,106 + 1,404 + 702
Total interest over life of the loan = $10,530
Notice that the total payments for the equal principal reduction loan are lower. This is because more
principal is repaid early in the loan, which reduces the total interest expense over the life of the loan.
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CHAPTER 8 – 30
Challenge
57.
The time line is:
0
1
$17,080
–
$20,000
To find the APR and EAR, we need to use the actual cash flows of the loan. In other words, the
interest rate quoted in the problem is only relevant to determine the total interest under the terms
given. The cash flows of the loan are the $20,000 you must repay in one year, and the $17,080 you
borrow today. The interest rate of the loan is:
$20,000 = $17,080(1 + r
)
r
= $20,000/$17,080 – 1 r = .1710, or 17.10%
Because of the discount, you only get the use of $17,080, and the interest you pay on that amount is
17.10%, not 14.6%.
58.
The time line is:
–24
–23
…
–12
–11
…
0
1
…
60
$3,583.33
$3,583.33
$3,833.3
3
$3,833.33
$4,250
$4,250
$4,250
$150,000
$20,000
Here we have cash flows that would have occurred in the past and cash flows that will occur in the
future. We need to bring both cash flows to today. Before we calculate the value of the cash flows
today, we must adjust the interest rate so we have the effective monthly interest rate. Finding the
APR with monthly compounding and dividing by 12 will give us the effective monthly rate. The
APR with monthly compounding is:
APR = 12(1.059
1/12
– 1)
APR = .0575, or 5.75%
To find the value today of the back pay from two years ago, we will find the FV of the annuity, and
then find the FV of the lump sum. Doing so gives us:
FVA = ($43,000/12)[{[1 + (.0575/12)]
12
– 1}/(.0575/12)] FVA = $44,150.76
FV = $44,150.76(1.059) FV = $46,755.65
Notice we found the FV of the annuity with the effective monthly rate, and then found the FV of the
lump sum with the EAR. Alternatively, we could have found the FV of the lump sum with the
effective monthly rate as long as we used 12 periods. The answer would be the same either way.
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CHAPTER 8 – 31
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CHAPTER 8 – 32
Now, we need to find the value today of last year’s back pay: FVA = ($46,000/12)[{[1 + (.0575/12)]
12
– 1}/(.0575/12)] FVA = $47,231.04
Next, we find the value today of the five years’ future salary:
PVA = ($51,000/12){[{1 – {1/[1 + (.0575/12)]
12(5)
}]/(.0575/12)}
PVA = $221,181.17
The value today of the jury award is the sum of salaries, plus the compensation for pain and
suffering, and court costs. The award should be for the amount of:
Award = $46,755.65 + 47,231.04 + 221,181.17 + 150,000 + 20,000 Award = $485,167.86
As the plaintiff, you would prefer a lower interest rate. In this problem, we are calculating both the
PV and FV of annuities. A lower interest rate will decrease the FVA, but increase the PVA. So, by a
lower interest rate, we are lowering the value of the back pay. But, we are also increasing the PV of
the future salary. Since the future salary is larger and has a longer time, this is the more important
cash flow to the plaintiff.
59.
To find the interest rate of a loan, we need to look at the cash flows of the loan. Since this loan is in
the form of a lump sum, the amount you will repay is the FV of the principal amount, which will be: Loan repayment amount = $10,000(1.097) Loan repayment amount = $10,970
The amount you will receive today is the principal amount of the loan times one minus the points. Amount received = $10,000(1 – .02) Amount received = $9,800
The time line is:
0
1
$9,800
–$10,970
Now, we find the interest rate for this PV and FV.
$10,970 = $9,800(1 + r
) r
= $10,970/$9,800 – 1 r = .1194, or 11.94%
60.
We need to find the FV of the premiums to compare with the cash payment promised at age 65. We
have to find the value of the premiums at Year 6 first since the interest rate changes at that time. So:
FV
1
= $800(1.10)
5
= $1,288.41
FV
2
= $800(1.10)
4
= $1,171.28
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CHAPTER 8 – 33
FV
3
= $900(1.10)
3
= $1,197.90
FV
4
= $900(1.10)
2
= $1,089.00
FV
5
= $1,000(1.10)
1
= $1,100.00
Value at Year 6 = $1,288.41 + 1,171.28 + 1,197.90 + 1,089.00 + 1,100.00 + 1,000 Value at Year 6 = $6,846.59
Finding the FV of this lump sum at the child’s 65
th
birthday:
FV = $6,846.59(1.07)
59
= $370,780.66
The policy is not worth buying; the future value of the deposits is $370,780.66, but the policy
contract will pay off $350,000. The premiums are worth $20,780.66 more than the policy payoff.
Note, we could also compare the PV of the two cash flows. The PV of the premiums is:
PV = $800/1.10 + $800/1.10
2
+ $900/1.10
3
+ $900/1.10
4
+ $1,000/1.10
5
+ $1,000/1.10
6
PV = $3,864.72
And the value today of the $350,000 at age 65 is:
PV = $350,000/1.07
59
= $6,462.87
PV = $6,462.87/1.10
6
= $3,648.12
The premiums still have the higher cash flow. At time zero, the difference is $216.60. Whenever you
are comparing two or more cash flow streams, the cash flow with the highest value at one time will
have the highest value at any other time.
Here is a question for you: Suppose you invest $216.60, the difference in the cash flows at time zero,
for six years at 10 percent interest, and then for 59 years at a 7 percent interest rate. How much will
it be worth? Without doing calculations, you know it will be worth $20,780.66, the difference in the
cash flows at Time 65! Calculator Solutions
1.
CF
0
$0
CF
0
$0
CF
0
$0
C01
$470
C01
$470
C01
$470
F01
1
F01
1
F01
1
C02
$610
C02
$610
C02
$610
F02
1
F02
1
F02
1
C03
$735
C03
$735
C03
$735
F03
1
F03
1
F03
1
C04
$920
C04
$920
C04
$920
F04
1
F04
1
F04
1
I = 10%
I = 18%
I = 24%
NPV CPT
NPV CPT
NPV CPT
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CHAPTER 8 – 34
$2,111.99
$1,758.27
$1,550.39
2.
Enter
8
5%
$5,300
N
I/Y
PV
PMT
FV
Solve for
$34,255.03
Enter
5
5%
$7,300
N
I/Y
PV
PMT
FV
Solve for
$31,605.18
Enter
8
15%
$5,300
N
I/Y
PV
PMT
FV
Solve for
$23,782.80
Enter
5
15%
$7,300
N
I/Y
PV
PMT
FV
Solve for
$24,470.73
3.
Enter
3
6%
$1,075
N
I/Y
PV
PMT
FV
Solve for
$1,280.34
Enter
2
6%
$1,210
N
I/Y
PV
PMT
FV
Solve for
$1,359.56
Enter
1
6%
$1,340
N
I/Y
PV
PMT
FV
Solve for
$1,420.40
FV = $1,280.34 + 1,359.56 + 1,420.40 + 1,420 = $5,480.30
Enter
3
13%
$1,075
N
I/Y
PV
PMT
FV
Solve for
$1,551.11
Enter
2
13%
$1,210
N
I/Y
PV
PMT
FV
Solve for
$1,545.05
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CHAPTER 8 – 35
Enter
1
13%
$1,340
N
I/Y
PV
PMT
FV
Solve for
$1,514.20
FV = $1,551.11 + 1,545.05 + 1,514.20 + 1,420 = $6,030.36
Enter
3
27%
$1,075
N
I/Y
PV
PMT
FV
Solve for
$2,202.01
Enter
2
27%
$1,210
N
I/Y
PV
PMT
FV
Solve for
$1,951.61
Enter
1
27%
$1,340
N
I/Y
PV
PMT
FV
Solve for
$1,701.80
FV = $2,202.01 + 1,951.61 + 1,701.80 + 1,420 = $7,275.42
4.
Enter
15
6%
$3,850
N
I/Y
PV
PMT
FV
Solve for
$37,392.16
Enter
40
6%
$3,850
N
I/Y
PV
PMT
FV
Solve for
$57,928.24
Enter
75
6%
$3,850
N
I/Y
PV
PMT
FV
Solve for
$63,355.02
5.
Enter
6
11%
$12,000
N
I/Y
PV
PMT
FV
Solve for
$2,836.52
Enter
8
7%
$19,700
N
I/Y
PV
PMT
FV
Solve for
$3,299.11
Enter
15
8%
$134,280
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CHAPTER 8 – 36
N
I/Y
PV
PMT
FV
Solve for
$15,687.87
Enter
20
6%
$300,000
N
I/Y
PV
PMT
FV
Solve for
$26,155.37
6.
Enter
7
5%
$1,750
N
I/Y
PV
PMT
FV
Solve for
$10,126.15
Enter
9
10%
$1,390
N
I/Y
PV
PMT
FV
Solve for
$8,005.04
Enter
18
8%
$17,500
N
I/Y
PV
PMT
FV
Solve for
$164,008.02
Enter
28
14%
$50,000
N
I/Y
PV
PMT
FV
Solve for
$348,033.11
7.
Enter
8
5%
$25,600
N
I/Y
PV
PMT
FV
Solve for
$2,680.88
Enter
40
7%
$1,250,000
N
I/Y
PV
PMT
FV
Solve for
$6,261.42
Enter
25
8%
$535,000
N
I/Y
PV
PMT
FV
Solve for
$7,318.15
Enter
13
4%
$104,600
N
I/Y
PV
PMT
FV
Solve for
$6,291.03
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CHAPTER 8 – 37
8.
Enter
10
7%
$2,100
N
I/Y
PV
PMT
FV
Solve for
$29,014.54
Enter
40
8%
$6,500
N
I/Y
PV
PMT
FV
Solve for
$1,683,867.37
Enter
9
9%
$1,100
N
I/Y
PV
PMT
FV
Solve for
$14,323.14
Enter
30
11%
$5,000
N
I/Y
PV
PMT
FV
Solve for
$995,104.39
9.
Enter
20
9.8%
$5,300
N
I/Y
PV
PMT
FV
Solve for
$296,748.26
Enter
40
9.8%
$5,300
N
I/Y
PV
PMT
FV
Solve for
$2,221,767.13
12.
Enter
7.8%
4
NOM
EFF
C/Y
Solve for
8.03%
Enter
15.3%
12
NOM
EFF
C/Y
Solve for
16.42%
Enter
12.4%
365
NOM
EFF
C/Y
Solve for
13.20%
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CHAPTER 8 – 38
Enter
11.4%
2
NOM
EFF
C/Y
Solve for
12.08%
13.
Enter
14.2%
2
NOM
EFF
C/Y
Solve for
13.73%
Enter
18.4%
12
NOM
EFF
C/Y
Solve for
17.01%
Enter
11.1%
52
NOM
EFF
C/Y
Solve for
10.54%
14.
Enter
13.8%
12
NOM
EFF
C/Y
Solve for
14.71%
Enter
14.1%
2
NOM
EFF
C/Y
Solve for
14.60%
15.
Enter
18.2%
365
NOM
EFF
C/Y
Solve for
16.72%
16.
Enter
17 × 2
8.4%/2
$5,500
N
I/Y
PV
PMT
FV
Solve for
$22,277.43
17.
Enter
5
365
8.3%/365
$7,500
N
I/Y
PV
PMT
FV
Solve for
$11,357.24
Enter
10
365
8.3%/365
$7,500
N
I/Y
PV
PMT
FV
Solve for
$17,198.27
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CHAPTER 8 – 39
Enter
20
365
8.3%/365
$7,500
N
I/Y
PV
PMT
FV
Solve for
$39,437.39
18.
Enter
10
365
9%/365
$95,000
N
I/Y
PV
PMT
FV
Solve for
$38,628.40
19.
Enter
306%
12
NOM
EFF
C/Y
Solve for
1,426.60%
20.
Enter
60
4.7%/12
$84,500
N
I/Y
PV
PMT
FV
Solve for
$1,583.03
Enter
4.7%
12
NOM
EFF
C/Y
Solve for
4.80%
21.
Enter
1.8%
$18,000
$500
N
I/Y
PV
PMT
FV
Solve for
58.53
22.
Enter
1,733.33%
52
NOM
EFF
C/Y
Solve for
313,916,515.69%
23.
Enter
6.92%
12
NOM
EFF
C/Y
Solve for
7.15%
24.
Enter
40
12
10%/12
$475
N
I/Y
PV
PMT
FV
Solve for
$3,003,937.80
25.
Enter
10%
12
NOM
EFF
C/Y
Solve for
10.47%
Enter
40
10.47%
$5,700
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CHAPTER 8 – 40
N
I/Y
PV
PMT
FV
Solve for
$2,868,732.50
26.
Enter
4
4
.57%
$3,000
N
I/Y
PV
PMT
FV
Solve for
$45,751.83
27.
Enter
8.5%
4
NOM
EFF
C/Y
Solve for
8.77%
CF
0
$0
C01
$815
F01
1
C02
$990
F02
1
C03
$0
F03
1
C04
$1,520
F04
1
I = 8.77%
NPV CPT
$2,671.72
28.
CF
0
$0
C01
$2,480
F01
1
C02
$0
F02
1
C03
$3,920
F03
1
C04
$2,170
F04
1
I = 9.32%
NPV CPT
$6,788.39
29. First Simple: $100(.064) = $6.40; 10-year investment = $100 + 10($6.40) = $164
Enter
10
±$100
$164
N
I/Y
PV
PMT
FV
Solve for
5.07%
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CHAPTER 8 – 41
30.
2
nd
BGN 2
nd
SET
Enter
60
4.10%/12
$65,500
N
I/Y
PV
PMT
FV
Solve for
$1,205.12
31.
Enter
6
1.25%/12
$9,000
N
I/Y
PV
PMT
FV
Solve for
$9,056.40
Enter
6
17.8%/12
$9,056.40
N
I/Y
PV
PMT
FV
Solve for
$9,892.90
Interest = $9,892.90 – 9,000 Interest = $892.90
32.
First: $90,000(.048) = $4,320 per year
($275,000 – 90,000)/$4,320 = 42.82 years
Second:
Enter
4.8%/12
$90,000
$275,000
N
I/Y
PV
PMT
FV
Solve for
279.80
279.80/12 = 23.32 years
33.
Enter
12
1.21%
$1
N
I/Y
PV
PMT
FV
Solve for
$1.16
Enter
24
1.21%
$1
N
I/Y
PV
PMT
FV
Solve for
$1.33
34.
Enter
31
±£440
£60
N
I/Y
PV
PMT
FV
Solve for
13.36%
35.
Enter
2 × 12
7%/12
$90,000/12
N
I/Y
PV
PMT
FV
Solve for
$167,513.24
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CHAPTER 8 – 42
Enter
2 × 12
7%/12
$77,000/12
N
I/Y
PV
PMT
FV
Solve for
$143,316.89
$143,316.89 + 20,000 = $163,316.89
36.
Enter
20
9%
$25,000
N
I/Y
PV
PMT
FV
Solve for
$228,213.64
37.
Enter
6
±$25,000
$60,000
N
I/Y
PV
PMT
FV
Solve for
15.71%
Enter
9
±$25,000
$92,000
N
I/Y
PV
PMT
FV
Solve for
15.58%
38.
Enter
13
10%
$10,000
N
I/Y
PV
PMT
FV
Solve for
$71,033.56
Enter
13
5%
$10,000
N
I/Y
PV
PMT
FV
Solve for
$93,935.73
Enter
13
15%
$10,000
N
I/Y
PV
PMT
FV
Solve for
$55,831.47
39.
Enter
6.5%/12
$225
$15,000
N
I/Y
PV
PMT
FV
Solve for
57.07
40.
Enter
60
$95,000
$1,850
N
I/Y
PV
PMT
FV
Solve for
.525%
.525%
12 = 6.30%
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CHAPTER 8 – 43
42.
CF
0
0
CF
0
$600,000
C01
$39,344,970
C01
$8,000,000
F01
1
F01
2
C02
$42,492,568
C02
$12,000,000
F02
1
F02
2
C03
$45,640,165
C03
F03
3
F03
C04
$48,787,763
C04
F04
1
F04
C05
$51,935,360
C05
F05
1
F05
I = 11%
I = 11%
NPV CPT
NPV CPT
$166,264,654.66
$30,979,254.94
43.
Enter
30
12
.80($2,600,000)
±$14,200
N
I/Y
PV
PMT
FV
Solve for
.605%
APR = .605%(12) = 7.26%
Enter
7.26%
12
NOM
EFF
C/Y
Solve for
7.50%
44.
Future value of bond account:
Enter
10
6%
$60,000
$9,000
N
I/Y
PV
PMT
FV
Solve for
$226,078.02
Future value of stock account:
Enter
10
10.5%
$230,000
N
I/Y
PV
PMT
FV
Solve for
$624,238.59
Future value of retirement account:
FV = $226,078.02 + 624,238.59
FV = $850,316.61
Annual withdrawal amount:
Enter
25
5.3%
$850,316.61
N
I/Y
PV
PMT
FV
Solve for
$62,158.80
45.
2
nd
BGN 2
nd
SET
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CHAPTER 8 – 44
Enter
60
$24,500
$465
N
I/Y
PV
PMT
FV
Solve for
.452%
APR = .452%(12) = 5.42%
46.
a.
Enter
5
7.1%
$14,500
N
I/Y
PV
PMT
FV
Solve for
$59,294.01
2
nd
BGN 2
nd
SET
Enter
5
7.1%
$14,500
N
I/Y
PV
PMT
FV
Solve for
$63,503.89
b.
Enter
5
7.1%
$14,500
N
I/Y
PV
PMT
FV
Solve for
$83,552.26
2
nd
BGN 2
nd
SET
Enter
5
7.1%
$14,500
N
I/Y
PV
PMT
FV
Solve for
$89,484.47
47.
Present value of annuity:
Enter
30
4.3%
$11,500
N
I/Y
PV
PMT
FV
Solve for
$191,810.81
And the present value of the perpetuity is:
PVP = C
/
r
PVP = $11,500/.043
PVP = $267,441.86
So the difference in the present values is:
Difference = $267,441.86 – 191,810.81
Difference = $75,631.05
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CHAPTER 8 – 45
48.
Value at t
= 9
Enter
12
8%/2
$8,250
N
I/Y
PV
PMT
FV
Solve for
$77,426.86
Value at t
= 5
Enter
4
2 8%/2
$77,426.86
N
I/Y
PV
PMT
FV
Solve for
$56,575.05
Value at t
= 3
Enter
6
2 8%/2
$77,426.86
N
I/Y
PV
PMT
FV
Solve for
$48,360.59
Value today
Enter
9
2 8%/2
$77,426.86
N
I/Y
PV
PMT
FV
Solve for
$38,220.07
49.
Enter
15
7.9%
$5,100
N
I/Y
PV
PMT
FV
Solve for
$43,921.16
Enter
5
7.9%
$43,921.16
N
I/Y
PV
PMT
FV
Solve for
$30,030.78
50.
Enter
96
6%/12
$1,750
N
I/Y
PV
PMT
FV
Solve for
$133,166.63
Enter
84
9%/12
$133,166.63
N
I/Y
PV
PMT
FV
Solve for
$71,090.38
Enter
84
9%/12
$1,750
N
I/Y
PV
PMT
FV
Solve for
$108,769.44
$71,090.38 + 108,769.44 = $179,859.81
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CHAPTER 8 – 46
51.
Enter
13 × 12
7.8%/12
$1,600
N
I/Y
PV
PMT
FV
Solve for
$430,172.31
Enter
13
9%
$430,172.31
N
I/Y
PV
PMT
FV
Solve for
$140,313.02
52.
PV@ Time = 14: $7,300/.046 = $158,695.65
Enter
7
4.6%
$158,695.65
N
I/Y
PV
PMT
FV
Solve for
$115,835.92
53.
Enter
12
$20,000
$1,973.33
N
I/Y
PV
PMT
FV
Solve for
2.699%
APR = 2.699%
12 = 32.39%
Enter
32.39%
12
NOM
EFF
C/Y
Solve for
37.66%
CF
0
$0
C01
$0
F01
1
C02
$20,000
F02
1
C03
$27,000
F03
1
C04
$0
F04
1
C05
$38,000
F05
1
I = 5.7%
NPV NFV
$91,784.37
Enter
5
5.7%
±$91,784.37
N
I/Y
PV
PMT
FV
Solve for
$121,099.86
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CHAPTER 8 – 47
57.
Enter
1
$17,080
$20,000
N
I/Y
PV
PMT
FV
Solve for
17.10%
58.
Enter
5.9%
12
NOM
EFF
C/Y
Solve for
5.75%
Enter
12
5.75%/12
$43,000/12
N
I/Y
PV
PMT
FV
Solve for
$44,150.76
Enter
1
5.9%
$44,150.76
N
I/Y
PV
PMT
FV
Solve for
$46,755.65
Enter
12
5.75%/12
$46,000/12
N
I/Y
PV
PMT
FV
Solve for
$47,231.04
Enter
60
5.75%/12
$51,000/12
N
I/Y
PV
PMT
FV
Solve for
$221,181.17
Award = $46,755.65 + 47,231.04 + 221,181.17 + 150,000 + 20,000 = $485,167.86
59.
Enter
1
$9,800
$10,970
N
I/Y
PV
PMT
FV
Solve for
11.94%
60.
Value at Year 6:
Enter
5
10%
$800
N
I/Y
PV
PMT
FV
Solve for
$1,288.41
Enter
4
10%
$800
N
I/Y
PV
PMT
FV
Solve for
$1,171.28
Enter
3
10%
$900
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CHAPTER 8 – 48
N
I/Y
PV
PMT
FV
Solve for
$1,197.90
Enter
2
10%
$900
N
I/Y
PV
PMT
FV
Solve for
$1,089.00
Enter
1
10%
$1,000
N
I/Y
PV
PMT
FV
Solve for
$1,100.00
So, at Year 6, the value is: $1,288.41 + 1,171.28 + 1,197.90 + 1,089.00 + 1,100.00
+ 1,000 = $6,846.59
At Year 65, the value is:
Enter
59
7%
$6,846.59
N
I/Y
PV
PMT
FV
Solve for
$370,780.66
The policy is not worth buying; the future value of the deposits is $370,780.66 but the policy contract will pay off $350,000.
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