ABC CORPOPATION

50

3Dividend growth rate, g 5%

Cost of equity from Gordon m 11.300%

1.ABC Corp. has a stock price P0 = 50. The firm has just paid a dividend of $3 per share, and knowledgableshareholders think that this dividend will grow by a rate of 5% per year. Use the Gordon dividend model tocalculate the cost of equity of ABC.

Current stock price, P0

Current dividend, D0

<-- =D0*(1+g)/P0+g

UNHEARDOF, INC.

5Anticipated dividend growth rate, g 15%

25%

Gordon model stock price 57.5

2.Unheardof, Inc. has just paid a dividend of $5 per share. This dividend is anticipated to increase at a rate of15% per year. If the cost of equity for Unheardof is 25%, what should be the market value of a share of thecompany?

Current dividend, D0

Cost of equity, rE

<-- D0*(1+g)/(rE-g)

DISMAL.COM

Dividend, starting in 3 years 15Growth rate 20%Cost of equity 35%Value of Dismal.com 3 years from today 15

Value of Dismal.com stock 2 years from today 100Today's stock price 54.87

will be 15, and this dividend will grow at 20% per year.Thus the Gordon model applies, and the stock priceat the end of 2 years will be 15/(0.35-0.20).Today's stock price is the price 2 years from now discountedat the cost of equity.

3.Dismal.com is a producer of depressing Internet products. The company is not currently paying dividends, butits chief financial officer thinks that starting in 3 years it can pay a dividend of $15 per share, and that thisdividend will grow by 20% per year. Assuming that the cost of equity of Dismal.com is 35%, value a sharebased on the discounted dividends.

Explanation: At the end of 2 years, the next dividend

Year-end Dividendstock per Growth

Year price share rate1986 0.401987 0.50 25.00%1988 0.50 0.00%1989 0.60 20.00%1990 0.60 0.00%1991 0.30 -50.00%1992 0.30 0.00%1993 0.33 10.00%1994 0.45 36.36%1995 1.00 122.22%1996 35.00 1.40 40.00%

Compound growth, 10 yr. 13.35%Compound growth, 5 yr. 36.08%

Gordon model

35.00 <-- Ending 1996 share price

1.40 <-- 1996 dividend

10-yr. growth 17.88% <-- =B22*(1+C17)/B21+C17 5-yr. growth 41.53% <-- =B22*(1+C18)/B21+C18

CHRYSLER CORPORATION 4.Consider the following dividend and price data for Chrysler Corporation: in Rows 2-15..Use the Gordon model to calculate Chrysler's cost of equity in 1996.

P0

D0

rE

g10=g1(1+g )10

g=(g10g1 )1/10

−1

g=g2−g1g1

∗100%=(g2g1

−1)∗100%

P0=D1RE−g

RE=D1P0

+g

g10=g1(1+g )10

g=(g10g1 )1/10

−1

g=g2−g1g1

∗100%=(g2g1

−1)∗100%

P0=D1RE−g

RE=D1P0

+g

S&P 500 AND IBM

S&P 500 IBMindex price ($)

Jan-97 786.16 78.32 Jan-97

Feb-97 790.82 71.77 Feb-97

Mar-97 757.12 68.52 Mar-97Apr-97 801.34 80.13 Apr-97

May-97 848.28 86.37 May-97Jun-97 885.14 90.12 Jun-97Jul-97 954.31 105.59 Jul-97

Aug-97 899.47 101.23 Aug-97Sep-97 947.28 105.84 Sep-97Oct-97 914.62 98.35 Oct-97Nov-97 955.40 109.34 Nov-97Dec-97 970.43 104.47 Dec-97

Part a IBM beta

Part b

Classic CAPM cost of equityBenninga-Sarig cost of equity

Part cShares outstandingDebt (billion $)Market value of equity (billion $)Cost of debt

Weighted average cost of capital (WACC) Classic CAPM Benninga-Sarig

5.On the spreadsheet associated with this chapter you will find the following monthly data for IBM's stock price and the S&P 500 index during 1998:a. Use these data to calculate IBM's β.b. Suppose that at the end of 1997, the risk-free rate was 5.50 percent. Assuming that the market risk premium, E (rm)−rf = 8 percent and that the corporate tax rate TC = 40 percent, calculate IBM's cost of equity using both the classic CAPM security market line and Benninga-Sarig's tax-adjusted security market line.c. At the end of 1997, IBM had 969,015,351 shares outstanding and had $39.9 billion of debt. Assuming that IBM's cost of debt is 6.10 percent, use your calculations for the cost of equity in part b to arrive attwo estimates of IBM's weighted average cost of capital.

Risk-free, rf

Market risk-premium, E(rm)-rf

Corporate tax rate, TC

S&P 500 IBMreturn return

0.01 -0.08 <= Return=(P2-P1)/P1 or P2/P1-1-0.04 -0.050.06 0.170.06 0.080.04 0.040.08 0.17

-0.06 -0.040.05 0.05

-0.03 -0.070.04 0.110.02 -0.04

1.66 <= COVAR(G6:G16;H6:H16)/VARP(G6:G16)

5.50%

8%

40%

18.74%20.18%

969,015,35139.9

101.2346.10%

Weighted average cost of capital (WACC)14.48%15.51%

5.On the spreadsheet associated with this chapter you will find the following monthly data for IBM's stock price . Use these data to calculate IBM's β.b. Suppose that at the end of 1997,

the risk-free rate was 5.50 percent. Assuming that the market risk premium, E (rm)−rf = 8 percent and that the corporate tax rate TC = 40 percent, calculate IBM's cost of equity using both the classic CAPM security market

. At the end of 1997, IBM had 969,015,351 shares outstanding and had $39.9 billion of debt. Assuming that IBM's cost of debt is 6.10 percent, use your

COST OF EQUITY

Use DATA-TABLE functionCurrent stock price 50Current dividend 5 21.00%Anticipated dividend growth rate 10% 0% 10.0%Implied cost of equity 21.00% 3% 13.3%

6% 16.6%9% 19.9%

12% 23.2%15% 26.5%18% 29.8%21% 33.1%24% 36.4%

6. A firm has a current stock price of $50 and has just paid a dividend of $5 per share.

a. Assuming that investors in the firm anticipate a dividend growth rate of 10 percent, what is the firm's

cost of equity?b. Draw a graph showing the relation between the cost of

equity and the anticipated dividend growth rate.

0% 5% 10% 15% 20% 25% 30%0.0%

10.0%

20.0%

30.0%

40.0%

Cost of Equity

Use DATA-TABLE function

ABC Corp ("supernormal" growth)

Current dividend 3.00 Dividend valuation

15% PV of years 1-10

10%Cost of equity 12% Share value

Year1 3.45 <-- =B3*(1+$B$4)2 3.97 <-- =B9*(1+$B$4)3 4.56 <-- =B10*(1+$B$4)4 5.25 <-- =B11*(1+$B$4)5 6.036 6.947 7.988 9.189 10.55 <-- =B16*(1+$B$4)

10 12.14 <-- =B17*(1+$B$4)11 13.35 <-- =B18*(1+$B$5)12 14.6913 16.1514 17.7715 19.5516 21.50

7. Exercise on supernormal growth: ABC Corporation has just paid a dividend of $3 per share. You—an experienced analyst—feel quite sure that the growth rate of the company's dividends over the next 10 years will be 15 percent per year. After 10 years you think that the company's dividend growth rate will slow to the industry average, which is about 5 percent per year. If the cost of equity for ABC is 12 percent, what is the value today of one share of the company?

Growth rate g1, years 1-10 ("supernormal")

Growth rate g2, years 11 - ¥ PV of years 11 - ¥

34.79 <-- =NPV(B6,B9:B18)

214.92 <-- =B18*(1+B5)/(B6-B5)/(1+B6)^10249.72 <-- =SUM(E4:E5)

7. Exercise on supernormal growth: ABC Corporation has just paid a dividend of $3 per share. You—an experienced analyst—feel quite sure that the growth rate of the company's dividends over the next 10 years will be 15 percent per year. After 10 years you think that the company's dividend growth rate will slow to the industry average, which is about 5 percent per year. If the cost of equity for ABC is 12 percent, what is the value today of

CALCULATING WACC

1.5

0.4

6%

15%

40%

Capital structurePercentage of equity 40%Percentage of debt 60%

Cost of equityClassic CAPM 19.50%Benninga-Sarig 20.70%

Cost of debtClassic CAPM 9.60%Benninga-Sarig 10.56%

Weighted average cost of capital (WACC)Classic CAPM 11.26%Benninga-Sarig 12.08%

8. Consider a company that has βequity = 1.5 and βdebt = 0.4. Suppose that the risk-free rate of interest is 6 percent, the expected return on the market E(rm) is 15 percent and the corporate tax rate is 40 percent. If the company has 40 percent equity and 60 percent debt in its capital structure, calculate its weighted average cost of capital using both the classic CAPM and the Benninga-Sarig tax-adjusted CAPM.

bequity

bdebt

Risk-free, rf

E(rm)

Corporate tax rate TC

WACC=EE+D

∗rE+DE+D

∗rD∗(1−TC )WACC=EE+D

∗rE+DE+D

∗rD∗(1−TC )WACC=

EE+D

∗rE+DE+D

∗rD∗(1−TC )

RISKY BOND PROBLEM

Face value of bond 100Coupon rate 22%Non-default probability 80%Default probability 20%Payoff in default (% of face value) 40%

Market price today 95Expected payoff in one year 105.6Expected return 11.16%

9. You are considering buying the bonds of a very risky company. A bond with a $100 face value, a one-year maturity, and a coupon rate of 22% is selling for $95. You consider the probability that the company will actually survive to pay off the bond 80%. With 20% probability, you think that the company will default, in which case you think that you will be able to recover $40. What is the expected return on the bond?

NORMAL AMERICA, INC.

Dec. 31 Dec. 15stock dividend SP 500

Year price per share return1986 33.001987 30.69 2.50 1987 4.7%1988 35.38 2.50 1988 16.2%1989 42.25 3.00 1989 31.4%1990 34.38 3.00 1990 -3.3%1991 36.25 1.60 1991 30.2%1992 32.25 1.40 1992 7.4%1993 43.00 0.80 1993 9.9%1994 42.13 0.80 1994 1.2%1995 52.88 1.10 1995 37.4%1996 55.75 1.60 1996 22.9%

Dec. 31 Dec. 15stock dividend SP 500

Year price per share return1986 33.001987 30.69 2.50 0.57% 1987 4.7%1988 35.38 2.50 23.44% 1988 16.2%1989 42.25 3.00 27.90% 1989 31.4%1990 34.38 3.00 -11.54% 1990 -3.3%1991 36.25 1.60 10.11% 1991 30.2%1992 32.25 1.40 -7.17% 1992 7.4%1993 43.00 0.80 35.81% 1993 9.9%1994 42.13 0.80 -0.17% 1994 1.2%1995 52.88 1.10 28.13% 1995 37.4%1996 55.75 1.60 8.46% 1996 22.9%

Beta 0.7458

It is January 1, 1997. Normal America, Inc. (NA) has paid a year-end dividend in each of the last 10 years, as shown by the following table.rows 3-16.Calculate NA's β with respect to the SP500.

-10% -5% 0% 5% 10% 15% 20% 25% 30% 35% 40%

-20%

-10%

0%

10%

20%

30%

40%

f(x) = 0.745841613473167 x − 0.00230812298812698R² = 0.406956272125524

Normal America Returns vs SP500

SP500

No

rmal

Am

eric

a

-10% -5% 0% 5% 10% 15% 20% 25% 30% 35% 40%

-20%

-10%

0%

10%

20%

30%

40%

f(x) = 0.745841613473167 x − 0.00230812298812698R² = 0.406956272125524

Normal America Returns vs SP500

SP500

No

rmal

Am

eric

a

JOHNSON & JOHNSON, CASH FLOW TO EQUITY, 1995-2005

Year ending Dividends Stock issues

31-Dec-95 827,000,000 322,000,000 -112,000,000 1,037,000,000 <- SUM(B3:D3)=31-Dec-96 974,000,000 412,000,000 -149,000,000 1,237,000,000 19.29% <-=E4/E3-131-Dec-97 1,137,000,000 628,000,000 -225,000,000 1,540,000,000 24.49%31-Dec-98 1,305,000,000 930,000,000 -269,000,000 1,966,000,000 27.66%31-Dec-99 1,479,000,000 840,000,000 -221,000,000 2,098,000,000 6.71%31-Dec-00 1,724,000,000 973,000,000 -387,000,000 2,310,000,000 10.10%31-Dec-01 2,047,000,000 2,570,000,000 -514,000,000 4,103,000,000 77.62%31-Dec-02 2,381,000,000 6,538,000,000 -390,000,000 8,529,000,000 107.87%31-Dec-03 2,746,000,000 1,183,000,000 -311,000,000 3,618,000,000 -57.58%31-Dec-04 3,251,000,000 1,384,000,000 -642,000,000 3,993,000,000 10.36%31-Dec-05 3,793,000,000 1,717,000,000 -696,000,000 4,814,000,000 20.56%

End 2005 equity dataStock price 61.07Number of shares 3,119,842,000Equity value 190,528,750,940 <- =B16*B17

Equity cash flow, end 2005 4,814,000,000 <- =E13Future growth rate Based on 10-year growth rate 16.59% <- =(E13/E3)^(1/10)-1 Based on 5-year growth rate 15.82% <- =(E13/E8)^(1/5)-1

Based on 10-year growth rate 19.54% <- =B20*(1+B22)/B18+B22 Based on 5-year growth rate 18.75% <- =B20*(1+B23)/B18+B23

For x-axes on graph y-axes 35,064 8,529,000,000 38,717 -696,000,000

Repurchase of common stock

Cash flow to equity

Year on year growth

Gordon cost of equity, rE

Note: Stock issues are proceeds from issuance of stock options (most to employees)

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005-1,000,000,000

0

1,000,000,000

2,000,000,000

3,000,000,000

4,000,000,000

5,000,000,000

6,000,000,000

7,000,000,000

8,000,000,000

9,000,000,000 JNJ, CASH FLOW TO EQUITY, 1995-2005

Total dividends

Stock repurchases

Stock issues

Cash flow to equity

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005-1,000,000,000

0

1,000,000,000

2,000,000,000

3,000,000,000

4,000,000,000

5,000,000,000

6,000,000,000

7,000,000,000

8,000,000,000

9,000,000,000 JNJ, CASH FLOW TO EQUITY, 1995-2005

Total dividends

Stock repurchases

Stock issues

Cash flow to equity

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005-1,000,000,000

0

1,000,000,000

2,000,000,000

3,000,000,000

4,000,000,000

5,000,000,000

6,000,000,000

7,000,000,000

8,000,000,000

9,000,000,000 JNJ, CASH FLOW TO EQUITY, 1995-2005

Total dividends

Stock repurchases

Stock issues

Cash flow to equity

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005-1,000,000,000

0

1,000,000,000

2,000,000,000

3,000,000,000

4,000,000,000

5,000,000,000

6,000,000,000

7,000,000,000

8,000,000,000

9,000,000,000 JNJ, CASH FLOW TO EQUITY, 1995-2005

Total dividends

Stock repurchases

Stock issues

Cash flow to equity

APPENDIX 1

APPENDIX 2

When we combine the 23 shares into an equally weighted portfolio, the portfolio β is 0.944, which is equal to theaverage beta of the component securities.[12] However, the portfolio's R2 = 61.44 percent, which is much larger thanaverage R2 of the component securities. For a large, well-diversified portfolio, the portfolio R2 approaches 1.The meaning of this number is that—when we invest in large diversified portfolios—almost all of the risk is due to the individual assets' βs.

When we combine the 23 shares into an equally weighted portfolio, the portfolio β is 0.944, which is equal to theaverage beta of the component securities.[12] However, the portfolio's R2 = 61.44 percent, which is much larger thanaverage R2 of the component securities. For a large, well-diversified portfolio, the portfolio R2 approaches 1.The meaning of this number is that—when we invest in large diversified portfolios—almost all of the risk is due to the