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35. To calculate this, we need to set up an equation with the callable bond equal to a weighted average of
the noncallable bonds. We will invest X percent of our money in the first noncallable bond, which
means our investment in Bond 3 (the other noncallable bond) will be (1 – X). The equation is:
C2 = C1 X + C3(1 – X)
So, we invest about 68 percent of our money in Bond 1, and about 32 percent in Bond 3. This
combination of bonds should have the same value as the callable bond, excluding the value of the
call. So:
P2= .68182P1 + .31819P3
The call value is the difference between this implied bond value and the actual bond price. So, the
call value is:
36. In general, this is not likely to happen, although it can (and did). The reason this bond has a negative
YTM is that it is a callable U.S. Treasury bond. Market participants know this information. Given
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:
(1 + R) = (1 + r)(1 + h)
Now, to find the weekly interest rate, we need to find the APR. Using the equation for discrete
compounding:
EAR = [1 + (APR/m)]m – 1
We can solve for the APR. Doing so, we get:
APR = m[(1 + EAR)1/m – 1]
So, the weekly interest rate is:
Weekly rate = APR/52
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:
PVA = C({1 – [1/(1 + r)t] }/r)
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:
(1 + R) = (1 + r)(1 + h)
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:
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.
4.
N I/Y PV PMT FV
5.
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6.
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7.
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8.
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9.
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10.
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11.
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18. Bond X
P0
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P1
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P3
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P8
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P12
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Bond Y
P0
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P1
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P3
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P8
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P12
N I/Y PV PMT FV
19. If both bonds sell at par, the initial YTM on both bonds is the coupon rate, 7.3 percent. If the YTM
suddenly rises to 9.3 percent:
PSam
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PDave
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If the YTM suddenly falls to 5.3 percent:
PSam
N I/Y PV PMT FV
Solve for $1,054.81
PDave
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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
N I/Y PV PMT FV
PK
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If the YTM rises from 6 percent to 8 percent:
PJ
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PK
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If the YTM declines from 6 percent to 4 percent:
PJ
N I/Y PV PMT FV
Solve for $893.59
PK
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Solve for $1,532.03
All else the same, the lower the coupon rate on a bond, the greater is its price sensitivity to
changes in interest rates.
21.
N I/Y PV PMT FV
Solve for 3.480%
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.
N I/Y PV PMT FV
Solve for 2.660%
Enter 7.2% ±$1,059.60 $80 $1,000
N I/Y PV PMT FV
26.
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Solve for 2.443%
28.
a. Po
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b. P1
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P24
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29.
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32.
Bond P
P0
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P1
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Bond D
P0
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P1
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33.
a.
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b.
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The HPY is:
N I/Y PV PMT FV
34.
PM
CFo$0
C01 $0
F01 12
C02 $900
F02 16
C03 $1,300
F03 11
C04 $21,300
F04 1
PN
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.
NOM EFF C/Y
Now we can find the present value of the cost of the flowers. The real cash flows are an ordinary
annuity, discounted at the real interest rate. So, the present value of the cost of the flowers is:
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:
Now, to find the APR:
NOM EFF C/Y
Bond account:
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:
N I/Y PV PMT FV
Bond account:
N I/Y PV PMT FV
Solve for $227,089.04
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:
NOM EFF C/Y
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:
N I/Y PV PMT FV
This is the real dollar amount of the monthly withdrawals. The nominal monthly withdrawals will
N I/Y PV PMT FV
