We need to find the annuity payment in retirement. Our retirement savings ends and 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: FVA = $700[{[1 + (.11/12) ]
360
1} / (.11/12)] = $1,963,163.82
Bond account: FVA = $300[{[1 + (.06/12) ]
360
1} / (.06/12)] = $301,354.51
So, the total amount saved at retirement is:
$1,963,163.82 + 301,354.51 = $2,264,518.33
Solving for the withdrawal amount in retirement using the PVA equation gives us:
PVA = $2,264,518.33 = $
C
[1
{1 / [1 + (.09/12)]
300
} / (.09/12)]
C
= $2,264,518.33 / 119.1616 = $19,003.763 withdrawal per month

B-78
SOLUTIONS
33.
We need to find the FV of a lump sum in one year and two years. It is important that we use the
number of months in compounding since interest is compounded monthly in this case. So:
FV in one year
= $1(1.0117)
12
= $1.15
FV in two years
= $1(1.0117)
24
= $1.32
There is also another common alternative solution. We could find the EAR, and use the number of years as
our compounding periods. So we will find the EAR first:
EAR = (1 + .0117)
12
1 = .1498 or 14.98%
Using the EAR and the number of years to find the FV, we get:
FV in one year
= $1(1.1498)
1
= $1.15
FV in two years
= $1(1.1498)
2
= $1.32
Either method is correct and acceptable. We have simply made sure that the interest compounding period is
the same as the number of periods we use to calculate the FV.
34.
Here we are finding the annuity payment necessary to achieve the same FV. The interest rate given is a 12
percent APR, with monthly deposits. We must make sure to use the number of months in the equation. So,
using the FVA equation:
Starting today:
FVA =
C
[{[1 + (.12/12) ]
480
1} / (.12/12)]
C
= $1,000,000 / 11,764.77 = $85.00
Starting in 10 years:
FVA =
C
[{[1 + (.12/12) ]
360
1} / (.12/12)]
C
= $1,000,000 / 3,494.96 = $286.13
Starting in 20 years:
FVA =
C
[{[1 + (.12/12) ]
240
1} / (.12/12)]
C
= $1,000,000 / 989.255 = $1,010.86
Notice that a deposit for half the length of time, i.e. 20 years versus 40 years, does not mean that the
annuity payment is doubled. In this example, by reducing the savings period by one-half, the deposit
necessary to achieve the same ending value is about twelve times as large.
35.
Since we are looking to quadruple our money, the PV and FV are irrelevant as long as the FV is three times
as large as the PV. The number of periods is four, the number of quarters per year. So:
FV = $3 = $1(1 +
r
)
(12/3)
r
= .3161 or 31.61%

CHAPTER 6
B-79
36.
Since we have an APR compounded monthly and an annual payment, we must first convert the interest rate
to an EAR so that the compounding period is the same as the cash flows.
EAR = [1 + (.10 / 12)]
12
1 = .104713 or 10.4713%
PVA
1
= $95,000 {[1
(1 / 1.104713)
2
] / .104713} = $163,839.09
PVA
2
= $45,000 + $70,000{[1
(1/1.104713)
2
] / .104713} = $165,723.54
You would choose the second option since it has a higher PV.
37.
We can use the present value of a growing perpetuity equation to find the value of your deposits today.
Doing so, we find:
PV =
C
{[1/(
r
g
)]
[1/(
r
g
)] × [(1 +
g
)/(1 +
r
)]
t
}
PV = $1,000,000{[1/(.08
.05)]
[1/(.08
.05)] × [(1 + .05)/(1 + .08)]
30
}
PV = $19,016,563.18
38.
Since your salary grows at 4 percent per year, your salary next year will be:
Next year s salary = $50,000 (1 + .04)
Next year s salary = $52,000
This means your deposit next year will be:
Next year s deposit = $52,000(.05)
Next year s deposit = $2,600
Since your salary grows at 4 percent, you deposit will also grow at 4 percent. We can use the present value
of a growing perpetuity equation to find the value of your deposits today. Doing so, we find:
PV =
C
{[1/(
r
g
)]
[1/(
r
g
)] × [(1 +
g
)/(1 +
r
)]
t
}
PV = $2,600{[1/(.11
.04)]
[1/(.11
.04)] × [(1 + .04)/(1 + .11)]
40
}
PV = $34,399.45

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