38. To answer this question, we need to find the monthly interest rate, which is the APR divided by 12.
We also must be careful to use the real interest rate. The Fisher equation uses the effective annual
rate, so, the real effective annual interest rates, and the monthly interest rates for each account are:
Stock account:
APR = m[(1 + EAR)1/m – 1]
Monthly rate = APR / 12
Bond account:
(1 + R) = (1 + r)(1 + h)
APR = m[(1 + EAR)1/m – 1]
Monthly rate = APR / 12
Now we can find the future value of the retirement account in real terms. The future value of each
account will be:
Stock account:
FVA = C {(1 + r)t – 1] / r}
Bond account:
FVA = C {(1 + r)t – 1] / r}
The total future value of the retirement account will be the sum of the two accounts, or:
CHAPTER 7 – 2
Now we need to find the monthly interest rate in retirement. We can use the same procedure that we
used to find the monthly interest rates for the stock and bond accounts, so:
(1 + R) = (1 + r)(1 + h)
APR = m[(1 + EAR)1/m – 1]
Monthly rate = APR / 12
Now we can find the real monthly withdrawal in retirement. Using the present value of an annuity
equation and solving for the payment, we find:
PVA = C({1 – [1/(1 + r)]t } / r )
This is the real dollar amount of the monthly withdrawals. The nominal monthly withdrawals will
increase by the inflation rate each month. To find the nominal dollar amount of the last withdrawal,
FV = PV(1 + r)t
Calculator Solutions
3.
Enter 23 4.7% €58 €1,000
N I/Y PV PMT FV
4.
Enter 18 ±¥91,530 ¥3,400 ¥100,000
N I/Y PV PMT FV
5.
Enter 8 5.9% ±$948 $1,000
N I/Y PV PMT FV
CHAPTER 7 – 3
6.
Enter 28 4.5% / 2 $41 / 2 $1,000
N I/Y PV PMT FV
7.
Enter 36 ±$1,050 $53 / 2 $1,000
N I/Y PV PMT FV
8.
Enter 29 5.3% / 2 ±$965 $1,000
N I/Y PV PMT FV
9.
Enter 34 4.9% / 2 ±$10,000
N I/Y PV PMT FV
10.
Enter 26 3.8% / 2 ±$49 / 2 ±$2,000
N I/Y PV PMT FV
11.
Enter 32 3.9% / 2 ±$185 / 2 ±$5,000
N I/Y PV PMT FV
18. Bond X
P0
Enter 26 7% / 2 $85 / 2 $1,000
N I/Y PV PMT FV
P1
Enter 24 7% / 2 $85 / 2 $1,000
N I/Y PV PMT FV
P3
Enter 20 7% / 2 $85 / 2 $1,000
N I/Y PV PMT FV
CHAPTER 7 – 4
P8
Enter 10 7% / 2 $85 / 2 $1,000
N I/Y PV PMT FV
P12
Enter 2 7% / 2 $85 / 2 $1,000
N I/Y PV PMT FV
Bond Y
P0
Enter 26 8.5% / 2 $70 / 2 $1,000
N I/Y PV PMT FV
P1
Enter 24 8.5% / 2 $70 / 2 $1,000
N I/Y PV PMT FV
P3
Enter 20 8.5% / 2 $70 / 2 $1,000
N I/Y PV PMT FV
P8
Enter 10 8.5% / 2 $70 / 2 $1,000
N I/Y PV PMT FV
P12
Enter 2 8.5% / 2 $70 / 2 $1,000
N I/Y PV PMT FV
19. If both bonds sell at par, the initial YTM on both bonds is the coupon rate, 6.5 percent. If the YTM
suddenly rises to 8.5 percent:
PSam
Enter 6 8.5% / 2 $65 / 2 $1,000
N I/Y PV PMT FV
PDave
Enter 40 8.5% / 2 $65 / 2 $1,000
N I/Y PV PMT FV
CHAPTER 7 – 5
If the YTM suddenly falls to 4.5 percent:
PSam
Enter 6 4.5% / 2 $65 / 2 $1,000
N I/Y PV PMT FV
PDave
Enter 40 4.5% / 2 $65 / 2 $1,000
N I/Y PV PMT FV
All else the same, the longer the maturity of a bond, the greater is its price sensitivity to changes
in interest rates.
20. Initially, at a YTM of 6 percent, the prices of the two bonds are:
PJ
Enter 38 6% / 2 $30 / 2 $1,000
N I/Y PV PMT FV
PK
Enter 38 6% / 2 $90 / 2 $1,000
N I/Y PV PMT FV
If the YTM rises from 6 percent to 8 percent:
PJ
Enter 38 8%2 $30 / 2 $1,000
N I/Y PV PMT FV
PK
Enter 38 8% / 2 $90 / 2 $1,000
N I/Y PV PMT FV
If the YTM declines from 6 percent to 4 percent:
PJ
Enter 38 4% / 2 $30 / 2 $1,000
N I/Y PV PMT FV
CHAPTER 7 – 6
PK
Enter 38 4% / 2 $90 / 2 $1,000
N I/Y PV PMT FV
All else the same, the lower the coupon rate on a bond, the greater is its price sensitivity to
changes in interest rates.
21.
Enter 36 ±$1,068 $64 / 2 $1,000
N I/Y PV PMT FV
Enter 5.79% 2
NOM EFF C/Y
22. The company should set the coupon rate on its new bonds equal to the required return; the required
return can be observed in the market by finding the YTM on outstanding bonds of the company.
Enter 40 ±$1,083 $70 / 2 $1,000
N I/Y PV PMT FV
25. Current yield = .0755 = $80 / P0 ; P0 = $80 / .0755 = $1,059.60
Enter 7.2% ±$1,059.60 $80 $1,000
N I/Y PV PMT FV
26.
Enter 26 ±$1,089.60 $57 / 2 $1,000
N I/Y PV PMT FV
28.
a. Po
Enter 50 5.8% / 2 $1,000
N I/Y PV PMT FV
b. P1
Enter 48 5.8% / 2 $1,000
N I/Y PV PMT FV
CHAPTER 7 – 7
P24
Enter 2 5.8% / 2% $1,000
N I/Y PV PMT FV
Solve for $944.43
29. a. The coupon bonds have a 6% coupon rate, which matches the 6% required return, so they will
sell at par; number of bonds = $47,000,000 / $1,000 = 47,000.
For the zeroes:
Enter 40 6% / 2 $1,000
N I/Y PV PMT FV
b. Coupon bonds: repayment = 47,000($1,030) = $48,410,000
Zeroes:
Enter 38 6% / 2 $1,000
N I/Y PV PMT FV
Solve for $325.23
During the life of the bond, the zero generates cash inflows to the firm in the form of the
interest tax shield of debt.
32.
Bond P
P0
Enter 10 7% $100 $1,000
N I/Y PV PMT FV
P1
Enter 9 7% $100 $1,000
N I/Y PV PMT FV
CHAPTER 7 – 8
Bond D
P0
Enter 10 7% $40 $1,000
N I/Y PV PMT FV
P1
Enter 9 7% $40 $1,000
N I/Y PV PMT FV
All else held constant, premium bonds pay high current income while having price depreciation
33.
a.
Enter 17 ±$1,060 $70 $1,000
N I/Y PV PMT FV
This is the rate of return you expect to earn on your investment when you purchase the bond.
b.
Enter 15 5.41% $70 $1,000
N I/Y PV PMT FV
The HPY is:
Enter 2 ±$1,060 $70 $1,160.52
N I/Y PV PMT FV
The realized HPY is greater than the expected YTM when the bond was bought because interest
CHAPTER 7 – 9
34.
PM
CFo$0
C01 $0
F01 12
C02 $1,100
F02 16
PN
Enter 40 5.6% / 2 $20,000
N I/Y PV PMT FV
37. To find the present value, we need to find the real weekly interest rate. To find the real return, we
need to use the effective annual rates in the Fisher equation. So, we find the real EAR is:
Now, to find the weekly interest rate, we need to find the APR.
Enter 3.47% 52
NOM EFF C/Y
Now we can find the present value of the cost of the roses. The real cash flows are an ordinary
annuity, discounted at the real interest rate. So, the present value of the cost of the roses is:
Enter 30 × 52 3.41% / 52 $7
N I/Y PV PMT FV
38. To answer this question, we need to find the monthly interest rate, which is the APR divided by 12.
We also must be careful to use the real interest rate. The Fisher equation uses the effective annual
rate, so, the real effective annual interest rates, and the monthly interest rates for each account are:
Stock account:
CHAPTER 7 – 10
Now, to find the APR:
Enter 6.73% 12
NOM EFF C/Y
Bond account:
Enter 2.88% 12
NOM EFF C/Y
Now we can find the future value of the retirement account in real terms. The future value of each
account will be:
Stock account:
Enter 30 × 12 6.53% / 12 $800
N I/Y PV PMT FV
Bond account:
Enter 30 × 12 2.85% / 12 $400
N I/Y PV PMT FV
The total future value of the retirement account will be the sum of the two accounts, or:
Now we need to find the monthly interest rate in retirement. We can use the same procedure that we
used to find the monthly interest rates for the stock and bond accounts, so:
(1 + R) = (1 + r)(1 + h)
Enter 4.81% 12
NOM EFF C/Y
CHAPTER 7 – 11
Now we can find the real monthly withdrawal in retirement. Using the present value of an annuity
equation and solving for the payment, we find:
Enter 25 × 12 4.70% / 12 $1,117,492.64
N I/Y PV PMT FV
This is the real dollar amount of the monthly withdrawals. The nominal monthly withdrawals will
Enter 30 + 25 4% $6,342.06
N I/Y PV PMT FV
