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Answers to End–of–Chapter Problems
B–21
Chapter 5: The Time Value of Money
5.2
To find the FV of a lump sum, we use:
FV = PV(1 + r)
t
a. FV = $1,000(1.05)
10
= $1,628.89
b. FV = $1,000(1.07)
10
= $1,967.15
c. FV = $1,000(1.05)
20
= $2,653.30
d. Because interest compounds on the interest already earned, the future value in
part c is
more than twice the future value in part a. With compound interest,
future values grow
exponentially.
5.4
To find the future value with continuous compounding, we use the equation:
FV = PV e
rt
a.
FV = $1,000 e
.12(5)
= $1,822.12
b.
FV = $1,000 e
.10(3)
= $1,349.86
c.
FV = $1,000 e
.05(10)
= $1,648.72
d.
FV = $1,000 e
.07(8)
= $1,750.67
5.6.
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 = 0.081 = [1 + (APR / 2)]
2
– 1
APR = 2[(1.081)
1/2
– 1] = 0.0794 or 7.94%
EAR = 0.076 = [1 + (APR / 12)]
12
– 1
APR = 12[(1.076)
1/12
– 1] = 0.0735 or 7.35%
EAR = 0.168 = [1 + (APR / 52)]
52
– 1
APR = 52[(1.168)
1/52
– 1] = 0.1555 or 15.55%
Solving the continuous compounding EAR equation:
EAR = e
q
– 1
We get:
APR = ln(1 + EAR)
APR = ln(1 + 0.262)
APR = 0.2327 or 23.27%
5.8
Here, we are trying to find the interest rate when we know the PV and FV. Using the FV equation:
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View Full DocumentAnswers to End–of–Chapter Problems
B–22
FV = PV(1 +
r
)
$4 = $3(1 +
r
)
r = 4/3 – 1 = 0.3333 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% = 1,733.33%
And using the equation to find the EAR:
EAR = [1 + (APR /
m
)]
m
– 1
EAR = [1 + 0.3333]
52
– 1 = 3,135,086.84 or 313,508,684.1%
Friendly’s operations are not legal since its rate is extremely high, compared to the legal charge of
60% per annum.
5.10
We need to find the annuity payment in retirement. Our retirement savings ends at the same time
the retirement withdrawals begin, so the PV of the retirement withdrawals will be the FV of the
retirement savings. So, we find the FV of the stock account and the FV of the bond account and add
the two FVs.
Stock account: FV = $700[{[1 + (0.11/12) ]
360
– 1} / (0.11/12)] = $1,963,163.82
Bond account: FV = $300[{[1 + (0.07/12) ]
360
– 1} / (0.07/12)] = $365,991.30
So, the total amount saved at retirement is:
$1,963,163.82 + 365,991.30 = $2,329,155.11
Solving for the withdrawal amount in retirement using the PV equation gives us:
PV = $2,329,155.11 = C[1 – {1 / [1 + (0.09/12)]
300
} / (0.09/12)]
C = $2,329,155.11 / 119.1616 = $19,546.19 withdrawal per month
5.12
Here, we need to find the interest rate for two possible investments. Each investment is a lump sum,
so:
G:
PV = $50,000 = $85,000 / (1 + r)
5
(1 + r)
5
= $85,000 / $50,000
r = (1.70)
1/5
– 1 = 0.1120 or 11.20%
H:
PV = $50,000 = $175,000 / (1 + r)
11
(1 + r)
11
= $175,000 / $50,000
r = (3.50)
1/11
– 1 = 0.1206 or 12.06%
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 Spring '08
 Wood

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