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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:
26. The bond has 16 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:
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 value, 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:
The annual interest deduction is the total interest divided by the maturity of the bond, so the
straight-line deduction is:
29. a. The coupon bonds have a 5.3 percent coupon which matches the 5.3 percent required return, 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.
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
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:
P: P0 = $90(PVIFA7%,10) + $1,000(PVIF7%,10) = $1,140.47
So, the capital gains yield is:
And the current yield is:
The current price of Bond D and the price of Bond D in one year is:
D: P0 = $50(PVIFA7%,10) + $1,000(PVIF7%,10) = $859.53
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:
To calculate the HPY, we need to find the interest rate that equates the price we paid for the
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 $1,300, 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 value, or:
