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24. 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:
Current yield = .0842 = $90/P0
P0 = $90/.0842 = $1,068.88
Now that we have the price of the bond, the bond price equation is:
The bond has 8 years to maturity.
25. The bond has 11 years to maturity, so the bond price equation is:
P = $1,053.12 = $36.20(PVIFAR%,22) + $1,000(PVIFR%,22)
Using a spreadsheet, financial calculator, or trial and error we find:
26. We found the maturity of a bond in Problem 24. However, in this case, the maturity is indeterminate.
27. 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:
Year 1 interest deduction = $278.37 – 263.92
Year 1 interest deduction = $14.44
The price of the bond when it has one year left to maturity will be:
c. Previous IRS regulations required a straight-line calculation of interest. The total interest
received by the bondholder is:
Total interest = $1,000 – 263.92
Total interest = $736.08
d. The company will prefer straight-line methods when allowed because the valuable interest
deductions occur earlier in the life of the bond.
28. 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:
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
payment. For the cash flow of the coupon bonds, we need to account for the tax deductibility of
the interest payments. To do this, we will multiply the total coupon payment times one minus
the tax rate. So:
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 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
During the life of the bond, the zero generates cash inflows to the firm in the form of the
interest tax shield of debt. We should note an important point here: If you find the PV of the
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:
P: P0 = $85(PVIFA7%,10) + $1,000(PVIF7%,10) = $1,105.35
P1 = $85(PVIFA7%,9) + $1,000(PVIF7%,9) = $1,097.73
The current price of Bond D and the price of Bond D in one year is:
D: P0 = $55(PVIFA7%,10) + $1,000(PVIF7%,10) = $894.65
P1 = $55(PVIFA7%,9) + $1,000(PVIF7%,9) = $902.27
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
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:
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
bond with the cash flows we received. The cash flows we received were $49 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:
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:
Notice that for the coupon payments of $800 and $1,000, we found the PVA for the coupon
payments, and then discounted the lump sums back to today.
32. 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
33. 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:
34. 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:
APR = m[(1 + EAR)1/m – 1]
APR = 12[(1 + .0769)1/12 – 1]
APR = .0743, or 7.43%
APR = m[(1 + EAR)1/m – 1]
APR = 12[(1 + .0288)1/12 – 1]
APR = .0285, or 2.85%
Now we can find the future value of the retirement account in real terms. The future value of each
account will be:
Stock account:
Bond account:
The total future value of the retirement account will be the sum of the two accounts, or:
Account value = $1,196,731.96 + 170,316.78
Account value = $1,367,048.74
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:
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:
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
35. 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 will grow at 9.18 percent, and the costs will grow at 2 percent, so the savings each year
for the next five years will be:
Year Revenue Costs Savings
1 $305,704.00 $127,500.00 $178,204.00
2 333,767.63 130,050.00 203,717.63
Now we can find the value of each year’s savings using the future value of a lump sum equation, so:
FV = PV(1 + r)t
Year Future Value
1 $178,204.00(1 + .09)4 = $251,549.49
He will spend $500,000 on a luxury boat, so the value of his account will be:
Calculator Solutions
1.
a.
Enter 30 3% $1,000
N I/Y PV PMT FV
Solve for $411.99
b.
c.
Enter 30 5% $1,000
N I/Y PV PMT FV
Solve for $231.38
2.
a.
b.
Enter 40 4.5% $35 $1,000
N I/Y PV PMT FV
Solve for $815.98
c.
3.
4.
Enter 23 3.65% ±$1,080 $1,000
N I/Y PV PMT FV
5.
Enter 15 3.90% €45 €1,000
6.
Enter 21 ±¥106,000 ¥2,800 ¥100,000
