SOLUTIONS TO END-OF-CHAPTER PROBLEMS

10-1 D0 = \$1.50; g1-3 = 5%; gn = 10%; D1 through D5 = ?

D1 = D0(1 + g1) = \$1.50(1.05) = \$1.5750.
D2 = D0(1 + g1)(1 + g2) = \$1.50(1.05)2 = \$1.6538.
D3 = D0(1 + g1)(1 + g2)(1 + g3) = \$1.50(1.05)3 = \$1.7364.
D4 = D0(1 + g1)(1 + g2)(1 + g3)(1 + gn) = \$1.50(1.05)3(1.10) = \$1.9101.
D5 = D0(1 + g1)(1 + g2)(1 + g3)(1 + gn)2 = \$1.50(1.05)3(1.10)2 = \$2.1011.

10-2 D1 = \$0.50; g = 7%; ks = 15%; = ?

= = = \$6.25.

10-3 P0 = \$20; D0 = \$1.00; g = 10%; = ?; = ?

= P0(1 + g) = \$20(1.10) = \$22.

= + g = + 0.10
= + 0.10 = 15.50%. = 15.50%.

10-4 Dps = \$5.00; Vps = \$60; kps = ?

kps = = = 8.33%.

10-5 0 1 2 3
| | | |
D0 = 2.00 D1 D2 D3

Step 1: Calculate the required rate of return on the stock:

ks = kRF + (kM - kRF)b = 7.5% + (4%)1.2 = 12.3%.

Step 2: Calculate the expected dividends:

D0 = \$2.00
D1 = \$2.00(1.20) = \$2.40
D2 = \$2.00(1.20)2 = \$2.88
D3 = \$2.88(1.07) = \$3.08

Step 3: Calculate the PV of the expected dividends:

PVDiv = \$2.40/(1.123) + \$2.88/(1.123)2 = \$2.14 + \$2.28 = \$4.42.

Step 4: Calculate :

= D3/(ks - g) = \$3.08/(0.123 - 0.07) = \$58.11.

Step 5: Calculate the PV of :

PV = \$58.11/(1.123)2 = \$46.08.

Step 6: Sum the PVs to obtain the stock’s price:

= \$4.42 + \$46.08 = \$50.50.

Alternatively, using a financial calculator, input the following:

CF0 = 0, CF1 = 2.40, and CF2 = 60.99 (2.88 + 58.11) and then enter I = 12.3 to solve for NPV = \$50.50.

10-6 The problem asks you to determine the constant growth rate, given the following facts: P0 = \$80, D1 = \$4, and ks = 14%. Use the constant growth rate formula to calculate g:

= + g
0.14 = + g
g = 0.09 = 9%.

10-7 The problem asks you to determine the value of , given the following facts: D1 = \$2, b = 0.9, kRF = 5.6%, RPM = 6%, and P0 = \$25. Proceed as follows:

Step 1: Calculate the required rate of return:

ks = kRF + (kM - kRF)b = 5.6% + (6%)0.9 = 11%.

Step 2: Use the constant growth rate formula to calculate g:

= + g
0.11 = + g
g = 0.03 = 3%.

Step 3: Calculate :

= P0(1 + g)3 = \$25(1.03)3 = \$27.3182 ? \$27.32.

Alternatively, you could calculate D4 and then use the constant growth rate formula to solve for :

D4 = D1(1 + g)3 = \$2.00(1.03)3 = \$2.1855.
= \$2.1855/(0.11 - 0.03) = \$27.3188 » \$27.32.

10-8 Vps = Dps/kps; therefore, kps = Dps/Vps.

a. kps = \$8/\$60 = 13.3%.

b. kps = \$8/\$80 = 10%.

c. kps = \$8/\$100 = 8%.

d. kps = \$8/\$140 = 5.7%.

10-9 = = = = = = \$23.75.

10-10 a. ki = kRF + (kM - kRF)bi.
kC = 9% + (13% - 9%)0.4 = 10.6%. kD = 9% + (13% - 9%)-0.5 = 7%.

Note that kD is below the risk-free rate. But since this stock is like an insurance policy because it “pays off” when something bad happens (the market falls), the low return is not unreasonable.

b. In this situation, the expected rate of return is as follows:

= D1/P0 + g = \$1.50/\$25 + 4% = 10%.

However, the required rate of return is 10.6 percent. Investors will seek to sell the stock, dropping its price to the following:

= = \$22.73.

At this point, = + 4% = 10.6%, and the stock will be in equilibrium.

10-11 D0 = \$1, kS = 7% + 6% = 13%, g1 = 50%, g2 = 25%, gn = 6%.

0 1 2 3 4
| | | | |
1.50 1.875 1.9875
1.327 + 28.393 = 1.9875/(0.13 - 0.06)
= 30.268
23.704
\$25.03

10-12 Calculate the dividend stream and place them on a time line. Also, calculate the price of the stock at the end of the supernormal growth period, and include it, along with the dividend to be paid at t = 5, as CF5. Then, enter the cash flows as shown on the time line into the cash flow register, enter the required rate of return as I = 15, and then find the value of the stock using the NPV calculation. Be sure to enter CF0 = 0, or else your answer will be incorrect.

D0 = 0; D1 = 0, D2 = 0, D3 = 1.00
D4 = 1.00(1.5) = 1.5; D5 = 1.00(1.5)2 = 2.25; D6 = 1.00(1.5)2(1.08) = \$2.43.
= ?

0 1 2 3 4 5 6
| | | | | | |
1.00 1.50 2.25
0.66 34.71
0.86 36.96
18.38
\$19.89 =

= D6/(ks - g) = 2.43/(0.15 - 0.08) = 34.71. This is the price of the stock at the end of Year 5.

CF0 = 0; CF1-2 = 0; CF3 = 1.0; CF4 = 1.5; CF5 = 36.96; I = 15%.

With these cash flows in the CFLO register, press NPV to get the value of the stock today: NPV = \$19.89.

10-13 a. Vps = = = \$125.

b. Vps = = \$83.33.

10-14 0 1 2 3 4
| | | | |
D0 = 2.00 D1 D2 D3 D4

a. D1 = \$2(1.05) = \$2.10. D2 = \$2(1.05)2 = \$2.21. D3 = \$2(1.05)3 = \$2.32.

b. PV = \$2.10(0.8929) + \$2.21(0.7972) + \$2.32(0.7118) = \$5.29.

Calculator solution: Input 0, 2.10, 2.21, and 2.32 into the cash flow register, input I = 12, PV = ? PV = \$5.29.

c. \$34.73(0.7118) = \$24.72.

Calculator solution: Input 0, 0, 0, and 34.73 into the cash flow register, I = 12, PV = ? PV = \$24.72.

d. \$24.72 + \$5.29 = \$30.01 = Maximum price you should pay for the stock.

e. = = = = \$30.00.

f. The value of the stock is not dependent upon the holding period. The value calculated in Parts a through d is the value for a 3-year holding period. It is equal to the value calculated in Part e except for a small rounding error. Any other holding period would produce the same value of ; that is, = \$30.00.

10-15 a. g = \$1.1449/\$1.07 - 1.0 = 7%.

Calculator solution: Input N = 1, PV = -1.07, PMT = 0, FV = 1.1449,
I = ? I = 7.00%.

b. \$1.07/\$21.40 = 5%.

c. = D1/P0 + g = \$1.07/\$21.40 + 7% = 5% + 7% = 12%.

10-16 a. 1. = = \$9.50.

2. = \$2/0.15 = \$13.33.

3. = = = \$21.00.

4. = = = \$44.00.

b. 1. = \$2.30/0 = Undefined.

2. = \$2.40/(-0.05) = -\$48, which is nonsense.

These results show that the formula does not make sense if the required rate of return is equal to or less than the expected growth rate.

c. No.

10-17 The answer depends on when one works the problem. We used the June 16, 2000, Wall Street Journal:

a. \$33? to \$61.

b. Current dividend = \$0.88. Dividend yield = \$0.88/\$33.9375 ? 2.6%. You might want to use an expected dividend yield of (\$0.88)(1 + g)/\$33.9375, with g estimated somehow.

c. The \$33.9375 close was up 5/8 from the previous day’s close.

d. The return on the stock consists of a dividend yield of about 2.6 percent plus some capital gains yield. We would expect the total rate of return on stock to be in the 10 to 12 percent range.

10-18 a. End of Year: 01 02 03 04 05 06 07
| | | | | | |

D0 = 1.75 D1 D2 D3 D4 D5 D6

Dt = D0(1 + g)t
D2002 = \$1.75(1.15)1 = \$2.01.
D2003 = \$1.75(1.15)2 = \$1.75(1.3225) = \$2.31.
D2004 = \$1.75(1.15)3 = \$1.75(1.5209) = \$2.66.
D2005 = \$1.75(1.15)4 = \$1.75(1.7490) = \$3.06.
D2006 = \$1.75(1.15)5 = \$1.75(2.0114) = \$3.52.

b. Step 1
PV of dividends = .
PV D2002 = \$2.01(PVIF12%,1) = \$2.01(0.8929) = \$1.79
PV D2003 = \$2.31(PVIF12%,2) = \$2.31(0.7972) = \$1.84
PV D2004 = \$2.66(PVIF12%,3) = \$2.66(0.7118) = \$1.89
PV D2005 = \$3.06(PVIF12%,4) = \$3.06(0.6355) = \$1.94
PV D2006 = \$3.52(PVIF12%,5) = \$3.52(0.5674) = \$2.00
PV of dividends = \$9.46

Step 2

= = = = = \$52.80.

This is the price of the stock 5 years from now. The PV of this price, discounted back 5 years, is as follows:

PV of = \$52.80(PVIF12%,5) = \$52.80(0.5674) = \$29.96.

Step 3

The price of the stock today is as follows:

= PV dividends Years 2002-2006 + PV of
= \$9.46 + \$29.96 = \$39.42.

This problem could also be solved by substituting the proper values into the following equation:

=

Calculator solution: Input 0, 2.01, 2.31, 2.66, 3.06, 56.32 (3.52 + 52.80) into the cash flow register, input I = 12, PV = ? PV = \$39.43.

c. 2002
D1/P0 = \$2.01/\$39.42 = 5.10%
Capital gains yield = 6.90*
Expected total return = 12.00%

2007
D6/P5 = \$3.70/\$52.80 = 7.00%
Capital gains yield = 5.00
Expected total return = 12.00%

*We know that k is 12 percent, and the dividend yield is 5.10 percent; therefore, the capital gains yield must be 6.90 percent.

The main points to note here are as follows:

1. The total yield is always 12 percent (except for rounding errors).

2. The capital gains yield starts relatively high, then declines as the supernormal growth period approaches its end. The dividend yield rises.

3. After 12/31/06, the stock will grow at a 5 percent rate. The dividend yield will equal 7 percent, the capital gains yield will equal 5 percent, and the total return will be 12 percent.

d. People in high income tax brackets will be more inclined to purchase “growth” stocks to take the capital gains and thus delay the payment of taxes until a later date. The firm’s stock is “mature” at the end of 2003.

e. Since the firm’s supernormal and normal growth rates are lower, the dividends and, hence, the present value of the stock price will be lower. The total return from the stock will still be 12 percent, but the dividend yield will be larger and the capital gains yield will be smaller than they were with the original growth rates. This result occurs because we assume the same last dividend but a much lower current stock price.

f. As the required return increases, the price of the stock goes down, but both the capital gains and dividend yields increase initially. Of course, the long-term capital gains yield is still 4 percent, so the long-term dividend yield is 10 percent.

10-19 a. Part 1. Graphical representation of the problem:

Supernormal Normal
growth growth
0 1 2 3 ?
| | | | |
D0 D1 (D2 + ) D3 D?
PVD1
PVD2
PV
P0

D1 = D0(1 + gs) = \$1.6(1.20) = \$1.92.
D2 = D0(1 + gs)2 = \$1.60(1.20)2 = \$2.304.

= = = = \$61.06.

= PV(D1) + PV(D2) + PV( )
=
= \$1.92(0.9091) + \$2.304(0.8264) + \$61.06(0.8264) = \$54.11.

Calculator solution: Input 0, 1.92, 63.364(2.304 + 61.06) into the cash flow register, input I = 10, PV = ? PV = \$54.11.

Part 2.

Expected dividend yield: D1/P0 = \$1.92/\$54.11 = 3.55%.

Capital gains yield: First, find which equals the sum of the present values of D2 and , discounted for one year.

= D2(PVIF10%, 1) + (PVIF10%, 1) = = \$57.60.

Calculator solution: Input 0, 63.364(2.304 + 61.06) into the cash flow register, input I = 10, PV = ? PV = \$57.60.

Second, find the capital gains yield:

= = 6.45%.

Dividend yield = 3.55%
Capital gains yield = 6.45
10.00% = ks.

b. Due to the longer period of supernormal growth, the value of the stock will be higher for each year. Although the total return will remain the same, ks = 10%, the distribution between dividend yield and capital gains yield will differ: The dividend yield will start off lower and the capital gains yield will start off higher for the 5-year supernormal growth condition, relative to the 2-year supernormal growth state. The dividend yield will increase and the capital gains yield will decline over the 5-year period until dividend yield = 4% and capital gains yield = 6%.

c. Throughout the supernormal growth period, the total yield will be 10 percent, but the dividend yield is relatively low during the early years of the supernormal growth period and the capital gains yield is relatively high. As we near the end of the supernormal growth period, the capital gains yield declines and the dividend yield rises. After the supernormal growth period has ended, the capital gains yield will equal gn = 6%. The total yield must equal ks = 10%, so the dividend yield must equal 10% - 6% = 4%.

d. Some investors need cash dividends (retired people) while others would prefer growth. Also, investors must pay taxes each year on the dividends received during the year, while taxes on capital gains can be delayed until the gain is actually realized.

10-20 a. ks = kRF + (kM - kRF)b = 11% + (14% - 11%)1.5 = 15.5%.
= D1/(ks - g) = \$2.25/(0.155 - 0.05) = \$21.43.

b. ks = 9% + (12% - 9%)1.5 = 13.5%. = \$2.25/(0.135 - 0.05) = \$26.47.

c. ks = 9% + (11% - 9%)1.5 = 12.0%. = \$2.25/(0.12 - 0.05) = \$32.14.

d. New data given: kRF = 9%; kM = 11%; g = 6%, b = 1.3.

ks = kRF + (kM - kRF)b = 9% + (11% - 9%)1.3 = 11.6%.
= D1/(ks - g) = \$2.27/(0.116 - 0.06) = \$40.54.

10-21 a. Old ks = kRF + (kM - kRF)b = 9% + (3%)1.2 = 12.6%.
New ks = 9% + (3%)0.9 = 11.7%.

Old price: = = = = \$38.21.

New price: = = \$31.34.

Since the new price is lower than the old price, the expansion in consumer products should be rejected. The decrease in risk is not sufficient to offset the decline in profitability and the reduced growth rate.

b. POld = \$38.21. PNew = .
Solving for ks we have the following:

\$38.21 =
\$2.10 = \$38.21(ks) - \$1.9105
\$4.0105 = \$38.21(ks)
ks = 0.10496.

Solving for b:

10.496% = 9% + 3%(b)
1.496% = 3%(b)
b = 0.49865.

Check: ks = 9% + (3%)0.49865 = 10.4960%.

= = \$38.21.

Therefore, only if management’s analysis concludes that risk can be lowered to b = 0.49865, or approximately 0.5, should the new policy be put into effect.

10-22 The detailed solution for the problem is available both on the instructor’s resource CD-ROM (in the file Solution for Ch 10-22 Build a Model.xls) and on the instructor’s side of the Harcourt College Publishers’ web site, http://www.harcourtcollege.com/finance/theory10e.

10-23 a. Supernormal growth rate = 12%; normal growth rate = 4%

INPUT DATA: KEY OUTPUT:
Supernormal growth 12.00% Current price (P0) \$31.50
Normal growth rate 4.00% Price at 12/31/2006 \$40.09
Req. rate of return 12.00% Dividend yield 2002 6.22%
Last dividend (D0) \$1.75 “ “ 2007 8.00%
Supernormal period 5 years Cap. gains yield 2002 5.78%
“ “ “ 2007 4.00%
Total return both yrs. 12.00%

MODEL-GENERATED DATA:
Expected dividends: PV of dividends:
2002 \$1.96 1999 \$1.75
2003 2.20 2000 1.75
2004 2.46 2001 1.75
2005 2.75 2002 1.75
2006 3.08 2003 1.75

Stock price Stock price
at 12/31/2006: \$40.09 at 1/1/2002: \$31.50

Yields in 2007: Yields in 2002:
Dividend 8.00% Dividend 6.22%
Capital Gain 4.00% Capital Gain 5.78%
Total 12.00% Total 12.00%

b. 1. k = 13 percent

INPUT DATA: KEY OUTPUT:
Supernormal growth 12.00% Current price (P0) \$27.86
Normal growth rate 4.00% Price at 12/31/2006 \$35.64
Req. rate of return 13.00% Dividend yield 2002 7.03%
Last dividend (D0) \$1.75 “ “ 2007 9.00%
Supernormal period 5 years Cap. gains yield 2002 5.97%
“ “ “ 2007 4.00%
Total return both yrs. 13.00%

2. k = 15 percent

INPUT DATA: KEY OUTPUT:
Supernormal growth 12.00% Current price (P0) \$22.59
Normal growth rate 4.00% Price at 12/31/2006 \$29.16
Req. rate of return 15.00% Dividend yield 2002 8.68%
Last dividend (D0) \$1.75 “ “ 2007 11.00%
Supernormal period 5 years Cap. gains yield 2002 6.32%
“ “ “ 2007 4.00%
Total return both yrs 15.00%

3. k = 20 percent

INPUT DATA: KEY OUTPUT:
Supernormal growth 12.00% Current price (P0) \$15.20
Normal growth rate 4.00% Price at 12/31/2006 \$20.05
Req. rate of return 20.00% Dividend yield 2002 12.89%
Last dividend (D0) \$1.75 “ “ 2007 16.00%
Supernormal period 5 years Cap. gains yield 2002 7.11%
“ “ “ 2007 4.00%
Total return both yrs 20.00%