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20. 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. So,
the YTM on the bonds currently sold in the market is:
P = $1,121.80 = $32(PVIFAR%,40) + $1,000(PVIFR%,40)
Using a spreadsheet, financial calculator, or trial and error we find:
YTM = 5.40%
21. Accrued interest is the coupon payment for the period times the fraction of the period that has passed
since the last coupon payment. Since we have a semiannual coupon bond, the coupon payment per six
months is one-half of the annual coupon payment. There are two months until the next coupon
payment, so four months have passed since the last coupon payment. The accrued interest for the bond
is:
Clean price = $922.33
22. Accrued interest is the coupon payment for the period times the fraction of the period that has passed
since the last coupon payment. Since we have a semiannual coupon bond, the coupon payment per six
months is one-half of the annual coupon payment. There are four months until the next coupon
payment, so two months have passed since the last coupon payment. The accrued interest for the bond
is:
Dirty price = $1,072.67
23. To find the number of years to maturity for the bond, we need to find the price of the bond. Since we
already have the coupon rate, we can use the bond price equation, and solve for the number of years
to maturity. We are given the current yield of the bond, so we can calculate the price as:
P0 = $63/.0695
P0 = $906.47
Now that we have the price of the bond, the bond price equation is:
24. The bond has 13 years to maturity, so the bond price equation is:
P = $1,043.55 = $27(PVIFAR%,26) + $1,000(PVIFR%,26)
Using a spreadsheet, financial calculator, or trial and error we find:
R = 2.471%
This is the semiannual interest rate, so the YTM is:
25. We found the maturity of a bond in Problem 20. However, in this case, the maturity is indeterminate.
A bond selling at par can have any length of maturity. In other words, when we solve the bond pricing
equation as we did in Problem 20, the number of periods can be any positive number.
Challenge
26. To find the capital gains yield and the current yield, we need to find the price of the bond. The current
price of Bond P and the price of Bond P in one year is:
The capital gains yield is:
The current price of Bond D and the price of Bond D in one year is:
D: P0 = $56(PVIFA7%,8) + $1,000(PVIF7%,8) = $916.40
P1 = $56(PVIFA7%,7) + $1,000(PVIF7%,7) = $924.55
All else held constant, premium bonds pay a high current income while having price depreciation as
maturity nears; discount bonds pay a lower current income but have price appreciation as maturity
nears. For either bond, the total return is still 7 percent, but this return is distributed differently between
current income and capital gains.
27. a. The rate of return you expect to earn if you purchase a bond and hold it until maturity is the YTM.
The bond price equation for this bond is:
b. To find our HPY, we need to find the price of the bond in two years. The price of the bond in two
years, at the new interest rate, will be:
Solving for R, we get:
The realized HPY is greater than the expected YTM when the bond was bought because interest
rates dropped by 1 percent; bond prices rise when yields fall.
28. The price of any bond (or financial instrument) is the PV of the future cash flows. Even though Bond
M makes different coupon payments, to find the price of the bond, we just find the PV of the cash
flows. The PV of the cash flows for Bond M is:
Bond N is a zero coupon bond with a $20,000 par value; therefore, the price of the bond is the PV of
the par, or:
29. 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:
So, we invest about 28 percent of our money in Bond 1, and about 72 percent in Bond 3. This
combination of bonds should have the same value as the callable bond, excluding the value of the call.
So:
The call value is the difference between this implied bond value and the actual bond price. So, the call
30. In general, this is not likely to happen, although it can (and did). The reason that this bond has a
negative YTM is that it is a callable U.S. Treasury bond. Market participants know this. Given the
high coupon rate of the bond, it is extremely likely to be called, which means the bondholder will not
31. To find the real annual increase in salary, we need to find the nominal growth rate and the annual
inflation rate. So, the nominal growth rate in salary was:
$97,500 = $22,400(1 + R)30
And the annual inflation rate was:
1,021.39 = 415.23(1 + h)30
Using the Fisher effect, we find the real annual increase in salary was:
32. 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. Using the equation for discrete
compounding:
EAR = [1 + (APR/m)]m – 1
We can solve for the APR. Doing so, we get:
So, the weekly interest rate is:
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:
33. 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)
1 + .12 = (1 + r)(1 + .04)
r = .0769, or 7.69%
APR = m[(1 + EAR)1/m – 1]
APR = 12[(1 + .0769)1/12 – 1]
APR = .0743, or 7.43%
Monthly rate = APR/12
Monthly rate = .0743/12
Monthly rate = .0062, or .62%
Bond account:
(1 + R) = (1 + r)(1 + h)
FVA = $325{[(1 + .0024)360 – 1]/.0024]}
FVA = $184,509.85
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:
34. In this problem, we need to calculate the future value of the annual savings after the five years of
operations. The savings are the revenues minus the costs, or:
Savings = Revenue – Costs
Since the annual fee and the number of members are increasing, we need to calculate the effective
growth rate for revenues, which is:
The revenue for the current year is the number of members times the annual fee, or:
b.
Enter
34
5%
$1,000
N
I/Y
PV
PMT
FV
Solve for
$190.35
