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21. The current yield is:
Current yield = Annual coupon payment / Price
The bond price equation for this bond is:
Using a spreadsheet, financial calculator, or trial and error we find:
This is the semiannual interest rate, so the YTM is:
The effective annual yield is the same as the EAR, so using the EAR equation from the previous
chapter:
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 the outstanding bonds of the company.
So, the YTM on the bonds currently sold in the market is:
Using a spreadsheet, financial calculator, or trial and error we find:
This is the semiannual interest rate, so the YTM is:
23. 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
And we calculate the clean price as:
CHAPTER 27 - 2
Clean price = Dirty price – Accrued interest
24. 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
And we calculate the dirty price as:
Dirty price = Clean price + Accrued interest
25. 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:
Now that we have the price of the bond, the bond price equation is:
We can solve this equation for t as follows:
The bond has 11.06 years to maturity.
26. The bond has 13 years to maturity, so the bond price equation is:
Using a spreadsheet, financial calculator, or trial and error we find:
CHAPTER 27 - 3
This is the semiannual interest rate, so the YTM is:
The current yield is the annual coupon payment divided by the bond price, so:
27. a. The bond price is the present value of the cash flows from a bond. The YTM is the interest rate
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
c. Current yield is defined as the annual coupon payment divided by the current bond price. For
28. 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:
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:
CHAPTER 27 - 4
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
29. a. The coupon bonds have a 6 percent coupon which matches the 6 percent 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:
The number of zero coupon bonds to sell would be:
b. The repayment of the coupon bond will be the par value plus the last coupon payment times the
number of bonds issued. So:
The repayment of the zero coupon bond will be the par value times the number of bonds issued,
so:
c. The total coupon payment for the coupon bonds will be the number of bonds times the coupon
Note that this is a cash outflow since the company is making the interest payment.
CHAPTER 27 - 5
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:
The Year 1 interest deduction per bond will be this price minus the price at the beginning of the
year, which we found in part b, so:
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
year even though the company did not have a cash flow for the interest expense. This reduces
the company’s tax liability, which is a cash inflow.
During the life of the bond, the zero generates cash inflows to the firm in the form of the
30. We found the maturity of a bond in Problem 25. However, in this case, the maturity is indeterminate.
31. 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)
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
CHAPTER 27 - 6
32. 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 are:
So, the capital gains yield is:
Capital gains yield = (New price – Original price) / Original price
And the current yield is:
The current price of Bond D and the price of Bond D in one year is:
D: P0 = $40(PVIFA7%, 10) + $1,000(PVIF7%, 10) = $789.29
So, the capital gains yield is:
And the current yield is:
All else held constant, premium bonds pay high current income while having price depreciation as
33. 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:
Using a spreadsheet, financial calculator, or trial and error we find:
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:
CHAPTER 27 - 7
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:
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.
34. The price of any bond (or financial instrument) is the PV of the future cash flows. Even though Bond
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:
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:
CHAPTER 27 - 8
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. Given the high
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. 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)
