April 3, 2019

CHAPTER 7 B-127

23. The bond has 14 years to maturity, so the bond price equation is:

Using a spreadsheet, financial calculator, or trial and error we find:

This is the semiannual interest rate, so the YTM is:

The current yield is the annual coupon payment divided by the bond price, so:

24. a. The bond price is the present value of the cash flows from a bond. The YTM is the interest rate

used in valuing the cash flows from a bond.

b. If the coupon rate is higher than the required return on a bond, the bond will sell at a premium,

since it provides periodic income in the form of coupon payments in excess of that required by

investors on other similar bonds. If the coupon rate is lower than the required return on a bond,

the bond will sell at a discount since it provides insufficient coupon payments compared to that

required by investors on other similar bonds. For premium bonds, the coupon rate exceeds the

YTM; for discount bonds, the YTM exceeds the coupon rate, and for bonds selling at par, the

YTM is equal to the coupon rate.

c. Current yield is defined as the annual coupon payment divided by the current bond price. For

premium bonds, the current yield exceeds the YTM, for discount bonds the current yield is less

than the YTM, and for bonds selling at par value, the current yield is equal to the YTM. In all

cases, the current yield plus the expected one-period capital gains yield of the bond must be

equal to the required return.

25. The price of a zero coupon bond is the PV of the par, so:

b. In one year, the bond will have 24 years to maturity, so the price will be:

P

B-128 SOLUTIONS

The interest deduction is the price of the bond at the end of the year, minus the price at the

beginning of the year, so:

The price of the bond when it has one year left to maturity will be:

P

c. Previous IRS regulations required a straight-line calculation of interest. The total interest

received by the bondholder is:

The annual interest deduction is simply the total interest divided by the maturity of the bond, so

the straight-line deduction is:

d. The company will prefer straight-line methods when allowed because the valuable interest

deductions occur earlier in the life of the bond.

26. a. The coupon bonds have an 8% coupon which matches the 8% required return, so they will sell

at par. The number of bonds that must be sold is the amount needed divided by the bond price,

so:

b. The repayment of the coupon bond will be the par value plus the last coupon payment times the

number of bonds issued. So:

CHAPTER 7 B-129

c. The total coupon payment for the coupon bonds will be the number bonds times the coupon

Note that this is cash outflow since the company is making the interest payment.

For the zero coupon bonds, the first year interest payment is the difference in the price of the

zero at the end of the year and the beginning of the year. The price of the zeroes in one year

will be:

P

The year 1 interest deduction per bond will be this price minus the price at the beginning of the

The total cash flow for the zeroes will be the interest deduction for the year times the number of

zeroes sold, times the tax rate. The cash flow for the zeroes in year 1 will be:

Notice the cash flow for the zeroes is a cash inflow. This is because of the tax deductibility of

the imputed interest expense. That is, the company gets to write off the interest expense for the

cash flows from the coupon bond and the zero coupon bond, they will be the same. This is

because of the much larger repayment amount for the zeroes.

27. We found the maturity of a bond in Problem 22. 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 22, the number of periods can be any positive number.

28. We first need to find the real interest rate on the savings. Using the Fisher equation, the real interest

rate is:

(1 + R) = (1 + r)(1 + h)

B-130 SOLUTIONS

Now we can use the future value of an annuity equation to find the annual deposit. Doing so, we

find:

FVA = C{[(1 + r)t – 1] / r}

Challenge

29. 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:

Capital gains yield = (New price – Original price) / Original price

The current price of Bond D and the price of Bond D in one year is:

All else held constant, premium bonds pay high current income while having price depreciation as

maturity nears; discount bonds do not pay high current income but have price appreciation as

maturity nears. For either bond, the total return is still 9%, but this return is distributed differently

between current income and capital gains.

30. 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:

P

Using a spreadsheet, financial calculator, or trial and error we find:

CHAPTER 7 B-131

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:

P

To calculate the HPY, we need to find the interest rate that equates the price we paid for the

bond with the cash flows we received. The cash flows we received were $70 each year for two

years, and the price of the bond when we sold it. The equation to find our HPY is:

P

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.

31. The price of any bond (or financial instrument) is the PV of the future cash flows. Even though Bond

M makes different coupons 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:

P

Notice that for the coupon payments of $1,400, we found the PVA for the coupon payments, and

then discounted the lump sum back to today.

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:

P

32. 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:

P

2 = 0.68181P1 + 0.31819P3

P

B-132 SOLUTIONS

The call value is the difference between this implied bond value and the actual bond price. So, the

call value is:

Assuming $1,000 par value, the call value is $119.73.

33. 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. Given the high

coupon rate of the bond, it is extremely likely to be called, which means the bondholder will not

34. 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:

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)

CHAPTER 7 B-133

35. 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

APR = m[(1 + EAR)1/m – 1]

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:

B-134 SOLUTIONS

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,

we can increase the real dollar withdrawal by the inflation rate. We can increase the real withdrawal

by the effective annual inflation rate since we are only interested in the nominal amount of the last

withdrawal. So, the last withdrawal in nominal terms will be:

FV = PV(1 + r)t

Calculator Solutions

3.

Enter 10 8.75% $75 $1,000

N I/Y PV PMT FV

Solve for $918.89

4.

Enter 9 ±$934 $90 $1,000

N I/Y PV PMT FV

Solve for 10.15%

5.

Enter 13 7.5% ±$1,045 $1,000

N I/Y PV PMT FV

Solve for $80.54

Coupon rate = $80.54 / $1,000 = 8.05%