SOLUTIONS TO END-OF-CHAPTER PROBLEMS

9-1 With your financial calculator, enter the following:

N = 10; I = YTM = 9%; PMT = 0.08 ? 1,000 = 80; FV = 1000; PV = VB = ?
PV = \$935.82.

Alternatively,

VB = \$80(PVIFA9%,10) + \$1,000(PVIF9%,10)
= \$80((1- 1/1.0910)/0.09) + \$1,000(1/1.0910)
= \$80(6.4177) + \$1,000(0.4224)
= \$513.42 + \$422.40 = \$935.82.

9-2 With your financial calculator, enter the following:

N = 12; PV = -850; PMT = 0.10 ? 1,000 = 100; FV = 1000; I = YTM = ?
YTM = 12.48%.

9-3 With your financial calculator, enter the following to find YTM:

N = 10 ? 2 = 20; PV = -1100; PMT = 0.08/2 ? 1,000 = 40; FV = 1000; I = YTM = ?
YTM = 3.31% ? 2 = 6.62%.

With your financial calculator, enter the following to find YTC:

N = 5 ? 2 = 10; PV = -1100; PMT = 0.08/2 ? 1,000 = 40; FV = 1050; I = YTC = ?
YTC = 3.24% ? 2 = 6.49%.

9-4 With your financial calculator, enter the following to find the current value of the bonds, so you can then calculate their current yield:

N = 7; I = YTM = 8; PMT = 0.09 ? 1,000 = 90; FV = 1000; PV = VB = ?
PV = \$1,052.06. Current yield = \$90/\$1,052.06 = 8.55%.

Alternatively,

VB = \$90(PVIFA8%,7) + \$1,000(PVIF8%,7)
= \$90((1- 1/1.087)/0.08) + \$1,000(1/1.087)
= \$90(5.2064) + \$1,000(0.5835)
= \$468.58 + \$583.50 = \$1,052.08.

Current yield = \$90/\$1,052.08 = 8.55%.

9-5 The problem asks you to find the price of a bond, given the following facts:

N = 16; I = 8.5/2 = 4.25; PMT = 45; FV = 1000.

With a financial calculator, solve for PV = \$1,028.60

9-6 a. VB = PMT(PVIFAi,n) + FV(PVIFi,n)
= PMT((1- 1/(1+in))/i) + FV(1/(1+i)n)

1. 5%: Bond L: VB = \$100(10.3797) + \$1,000(0.4810) = \$1,518.97.
Bond S: VB = (\$100 + \$1,000)(0.9524) = \$1,047.64.

2. 8%: Bond L: VB = \$100(8.5595) + \$1,000(0.3152) = \$1,171.15.
Bond S: VB = (\$100 + \$1,000)(0.9259) = \$1,018.49.

3. 12%: Bond L: VB = \$100(6.8109) + \$1,000(0.1827) = \$863.79.
Bond S: VB = (\$100 + \$1,000)(0.8929) = \$982.19.

Calculator solutions:

1. 5%: Bond L: Input N = 15, I = 5, PMT = 100, FV = 1000, PV = ?,
PV = \$1,518.98.
Bond S: Change N = 1, PV = ? PV = \$1,047.62.

2. 8%: Bond L: From Bond S inputs, change N = 15 and I = 8, PV = ?, PV = \$1,171.19.
Bond S: Change N = 1, PV = ? PV = \$1,018.52.

3. 12%: Bond L: From Bond S inputs, change N = 15 and I = 12, PV = ? PV = \$863.78.
Bond S: Change N = 1, PV = ? PV = \$982.14.

b. Think about a bond that matures in one month. Its present value is influenced primarily by the maturity value, which will be received in only one month. Even if interest rates double, the price of the bond will still be close to \$1,000. A one-year bond's value would fluctuate more than the one-month bond's value because of the difference in the timing of receipts. However, its value would still be fairly close to \$1,000 even if interest rates doubled. A long-term bond paying semiannual coupons, on the other hand, will be dominated by distant receipts, receipts which are multiplied by 1/(1 + kd/2)t, and if increases, these multipliers will decrease significantly. Another way to view this problem is from an opportunity point of view. A one-month bond can be reinvested at the new rate very quickly, and hence the opportunity to invest at this new rate is not lost; however, the long-term bond locks in subnormal returns for a long period of time.

9-7 a. VB = =
= PMT((1- 1/(1+kdn))/kd) + FV(1/(1+kd)n).
M = \$1,000. INT = 0.09(\$1,000) = \$90.

1. \$829 =
\$829= \$90((1- 1/(1+kd4))/kd) + \$1,000(1/(1+kd)4).

The YTM can be found by trial-and-error. If the YTM was 9 percent, the bond value would be its maturity value. Since the bond sells at a discount, the YTM must be greater than 9 percent. Let's try 10 percent.

At 10%, VB = \$90(3.1699) + \$1,000(0.6830) = \$285.29 + \$683.00
= \$968.29.

\$968.29 > \$829.00; therefore, the bond's YTM is greater than 10 percent.

Try 15 percent.

At 15%, VB = \$90(2.8550) + \$1,000(0.5718) = \$256.95 + \$571.80
= \$828.75.

Therefore, the bond's YTM is approximately 15 percent.

2. \$1,104 = .

The bond is selling at a premium; therefore, the YTM must be below 9 percent. Try 6 percent.

At 6%, VB = \$90(3.4651) + \$1,000(0.7921) = \$311.86 + \$792.10
= \$1,103.96.

Therefore, when the bond is selling for \$1,104, its YTM is approximately 6 percent.

Calculator solution:

1. Input N = 4, PV = -829, PMT = 90, FV = 1000, I = ? I = 14.99%.

2. Change PV = -1104, I = ? I = 6.00%.

b. Yes. At a price of \$829, the yield to maturity, 15 percent, is greater than your required rate of return of 12 percent. If your required rate of return were 12 percent, you should be willing to buy the bond at any price below \$908.86 (using the tables) and \$908.88 (using a calculator).

9-8 \$1,000 =
\$1,000 = \$140((1- 1/(1+kd6))/kd) + \$1,090(1/(1+kd)6).

Try 18 percent:

PV18% = \$140(3.4976) + \$1,090(0.3704) = \$489.66 + \$403.74 = \$893.40.
18 percent is too high.

Try 15 percent:

PV15% = \$140(3.7845) + \$1,090(0.4323) = \$529.83 + \$471.21 = \$1,001.04.

15 percent is slightly low.

The rate of return is approximately 15.03 percent, found with a calculator using the following inputs.

N = 6; PV = -1000; PMT = 140; FV = 1090; I = ? Solve for I = 15.03%.

9-8 a. \$1,100 = .

Using a financial calculator, input the following:

N = 20, PV = -1100, PMT = 60, FV = 1000, and solve for I = 5.1849%.

However, this is a periodic rate. The nominal annual rate = 5.1849%(2) = 10.3699% ? 10.37%.

b. The current yield = \$120/\$1,100 = 10.91%.

c. YTM = Current Yield + Capital Gains (Loss) Yield
10.37% = 10.91% + Capital Loss Yield
-0.54% = Capital Loss Yield.

d. \$1,100 = .

Using a financial calculator, input the following:

N = 8, PV = -1100, PMT = 60, FV = 1060, and solve for I = 5.0748%.

However, this is a periodic rate. The nominal annual rate = 5.0748%(2) = 10.1495% ? 10.15%.

9-10 The problem asks you to solve for the YTM, given the following facts:

N = 5, PMT = 80, and FV = 1000. In order to solve for I we need PV.

However, you are also given that the current yield is equal to 8.21%. Given this information, we can find PV.

Current yield = Annual interest/Current price
0.0821 = \$80/PV
PV = \$80/0.0821 = \$974.42.

Now, solve for the YTM with a financial calculator:

N = 5, PV = -974.42, PMT = 80, and FV = 1000. Solve for I = YTM = 8.65%.

9-11 The problem asks you to solve for the current yield, given the following facts: N = 14, I = 10.5883/2 = 5.2942, PV = -1020, and FV = 1000. In order to solve for the current yield we need to find PMT. With a financial calculator, we find PMT = \$55.00. However, because the bond is a semiannual coupon bond this amount needs to be multiplied by 2 to obtain the annual interest payment: \$55.00(2) = \$110.00. Finally, find the current yield as follows:

Current yield = Annual interest/Current Price = \$110/\$1,020 = 10.78%.

9-12 The bond is selling at a large premium, which means that its coupon rate is much higher than the going rate of interest. Therefore, the bond is likely to be called--it is more likely to be called than to remain outstanding until it matures. Thus, it will probably provide a return equal to the YTC rather than the YTM. So, there is no point in calculating the YTM--just calculate the YTC. Enter these values:

N = 10, PV = -1353.54, PMT = 70, FV = 1050, and then solve for I.

The periodic rate is 3.24 percent, so the nominal YTC is 2 x 3.24% = 6.47%. This would be close to the going rate, and it is about what the firm would have to pay on new bonds.

9-13 a. The bonds now have an 8-year, or a 16-semiannual period, maturity, and their value is calculated as follows:

VB = = \$50(12.5611) + \$1,000(0.6232)
= \$628.06 + \$623.20 = \$1,251.26.

Calculator solution: Input N = 16, I = 3, PMT = 50, FV = 1000,
PV = ? PV = \$1,251.22.

b. VB = \$50(10.1059) + \$1,000(0.3936) = \$505.30 + \$393.60 = \$898.90.

Calculator solution: Change inputs from Part a to I = 6, PV = ?
PV = \$898.94.

c. The price of the bond will decline toward \$1,000, hitting \$1,000 (plus accrued interest) at the maturity date 8 years (16 six-month periods) hence.

9-14 The answer depends on when one works the problem. We used the June 2, 1998, Wall Street Journal:

a. AT&T's 8.625%, 2031 bonds had a 7.9 percent current yield. The bonds sold at a premium, 109.875% of par, so the coupon interest rate would have to be set lower than 8.625% for the bonds to sell at par. If we assume the bonds aren’t callable, we can do a rough calculation of their YTM. Using a financial calculator, we input the following values: N = 34 x 2 = 68, PV = 1.09875 x -1,000 = -1098.75, PMT = 0.8625/2 x 1,000 = 86.25/2 = 43.125, FV = 1000, and then solve for YTM = kd = 3.8968% x 2 = 7.79% ? 7.8%.
Thus, AT&T would have to set a rate of 7.8 percent on new long-term bonds.

b. The return on AT&T's bonds is the current yield of 7.9 percent, less a small capital loss in 2031. The total return is about 7.8 percent.

9-15 a. The original yield to maturity was 3.4 percent. This can be demon- strated by showing that a value of 3.4 percent for kd solves this equation:

\$1,000 =

Calculator solution: Input N = 30, PV = -1000, PMT = 34, FV = 1000,
I = ? I = 3.40%.

b. In February 1985, the bond had a remaining life of 17 years. Thus, its value is calculated as follows:

VB = \$650 =
= .

Trying PVIFA and PVIF for 7 percent, we obtain the following:

VB = \$34(9.7632) + \$1,000(0.3166) = \$648.55 ? \$650.

Therefore, kd ? 7%.

Solving for kd using a financial calculator gives 6.98 percent. Input N = 17, PV = -650, PMT = 34, FV = 1000, I = ? I = 6.98%.

c. In February 2000, the bonds have a remaining life of 2 years. Thus, their value is calculated as follows:

VB = = \$934.91.

Calculator solution: Input N = 2, I = 7, PMT = 34, FV = 1000, PV = ? PV = \$934.91.

d. Just before maturity, the bond has a value of \$1,000 (plus accrued interest, which should not concern students at this point).

e. The price of the bonds will rise. There is a built-in capital gain; thus, for discount bonds kd = Interest yield + Capital gains yield. Of course, if interest rates rise, part of this built-in gain can be offset for holding periods less than the years to maturity.

f. 1. In February 1985 the current yield = \$34/\$650 = 5.23%.

2. In February 2000 the current yield = \$34/\$934.91 = 3.64%.
kd = Total yield = Capital gains yield + Current yield.

Here kd = 7% as solved in Part b of the problem, so the capital gains yield in February 1982 was 7.0% - 5.23% = 1.77%, and the capital gains yield in February 1997 was 7.0% - 3.64% = 3.36%. Alternatively, the capital gains yield could have been calculated by finding the price with 16 years and 1 year remaining and using the formula:

In February 1985 the capital gains yield = (\$661.51 - \$650)/\$650 = 1.77%.

In February 2000 the capital gains yield = (\$966.36 - \$934.91)/ \$934.91 = 3.36%.

The total yield at both dates was 7 percent.

9-16 a. Yield to maturity (YTM):

With a financial calculator, input N = 28, PV = -1165.75, PMT = 95, FV = 1000, I = ? I = kd = YTM = 8.00%. With the formulas, proceed as follows:

\$1,165.75 =
= \$95((1- 1/(1+kd28))/kd) + \$1,000(1/(1+kd)28).

Try 10 percent:

Is \$1,165.75 = \$95(PVIFA10%,28) + \$1,000(PVIF10%,28)?
= \$95(9.3066) + \$1,000(0.0693) = \$953.43 < \$1,165.75.

Try 9 percent:

Is \$1,165.75 = \$95(PVIFA9%,28) + \$1,000(PVIF9%,28)?
= \$95(10.1161) + \$1,000(0.0895) = \$1,050.53 < \$1,165.75.

Try 8 percent:

Is \$1,165.75 = \$95(PVIFA8%,28) + \$1,000(PVIF8%,28)?
= \$95(11.0511) + \$1,000(0.1159) = \$1,165.75 = \$1,165.75.

Therefore, YTM = 8%.

Yield to call (YTC):

With a calculator, input N = 3, PV = -1165.75, PMT = 95, FV = 1090,
I = ? I = kd = YTC = 6.11%. With the formulas, proceed as follows:

\$1,165.75 =
= \$95((1- 1/(1+kd28))/kd) + \$1,090(1/(1+kd)28).

Try 7 percent:

Is \$1,165.75 = \$95(PVIFA7%,3) + \$1,090(PVIF7%,3)?
= \$95(2.6243) + \$1,090(0.8163) = \$1,139.08 < \$1,165.75.

Try 6 percent:

Is \$1,165.75 = \$95(PVIFA6%,3) + \$1,090(PVIF6%,3)?
= \$95(2.6730) + \$1,090(0.8396) = \$1,169.10 > \$1,165.75.

Try 6.1 percent:

Is \$1,165.75 = \$95(PVIFA6.1%,3) + \$1,090(PVIF6.1%,3)?
= \$95(2.6681) + \$1,090(0.8372) = \$1,166.02 ? \$1,165.75.

Therefore, YTC ? 6.1%.

b. Knowledgeable investors would expect the return to be closer to 6.1 percent than to 8 percent. If interest rates remain substantially lower than 9.5 percent, the company can be expected to call the issue at the call date and to refund it with an issue having a coupon rate lower than 9.5 percent.

c. If the bond had sold at a discount, this would imply that current interest rates are above the coupon rate. Therefore, the company would not call the bonds, so the YTM would be more relevant than the YTC.
9-17

9-18 a.

b.

9-19 The detailed solution for the problem is available both on the instructor’s resource CD-ROM (in the file Solution for Ch 09-19 Build a Model.xls) and on the instructor’s side of the Harcourt College Publishers’ web site, http://www.harcourtcollege.com/finance/theory10e.

9-20 a. The price of the bond has risen to \$1,200 which is higher than the 2002 price, so interest rates must have fallen below their 2002 level.
Since interest rates have fallen, an investor should expect to receive the yield to call and would price the bond so that the YTC provided him or her with the current market required return. In this case, the YTC is approximately 3.41 percent, so the YTC has fallen from 6.1 percent in 2002 to 3.41 percent in 2003.

INPUT DATA:
Bond's par value \$1,000.00 Original Maturity 30
Coupon rate 9.50% Years remaining 27
Call price 109% Years until callable 2
Current bond price \$1,200.00

KEY OUTPUT:
Yield to maturity: 7.72%
Yield to call: 3.41%

b. Since the price of the bond has fallen to \$800, interest rates must have risen sharply. In this case, the bond will not be called, so investors should expect to receive the 12.02 percent yield to maturity.

INPUT DATA:
Bond's par value \$1,000.00 Original Maturity 30
Coupon rate 9.50% Years remaining 27
Call price 109% Years until callable 2
Current bond price \$800.00

KEY OUTPUT:
Yield to maturity: 12.02%
Yield to call: 27.79%