Simon Benninga. Financial Modelling 3-rd edition

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88 Chapter 2

5. The current stock price of TransContinental Airways is $65 per share. TCA currently

pays an annual per-share dividend of $3. Over the past ﬁ ve years this dividend has

grown annually at a rate of 23 percent. A respected analyst assumes that the current

growth rate of dividends will hold up for the next ﬁ ve years, after which dividend

growth will slow to 5 percent annually. Use the twostagegordon function to compute

the cost of equity.

6. 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 ten years will be 15 percent per year. After ten years you think that the com-

pany’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?

7. The cash ﬂ ow to equity holders for Merck, Inc. during the years 1991–2000 is given

in the following chart. Compute Merck’s cost of equity r

at year-end 2000 on the

assumption that the equity cash ﬂ ow growth will be 20.15 percent for the following

four years, followed by a long-term growth rate of 6 percent.

Notes:

• The numbers are taken from Merck’s ﬁ nancial statements for 1991–2000.

When employee stock options are exercised, the company gets a tax shield

on the beneﬁ t accruing to the option holders. This beneﬁ t, measured as

Stock price

at option exercise

Option

exercise price

−

, is computed in column E.

• At the end of 2000, Merck had 2,307,599,179 shares and the stock price was 93.625

per share.

18. To do this problem you will have to copy the formula from the spreadsheet fm3_

chapter02.xls to your answer spreadsheet. See the ﬁ le “Adding Getformula to your

spreadsheet” on the disk that comes with Financial Modeling for details.

89 Calculating the Cost of Capital

AB C D E F G

Year

Purchase

of treasury

stock

Cash

dividends

Proceeds

from

exercise of

stock

options

Tax Benefits

from exercise

of stock

options

Cash

flow to

equity

holders

1991 184 893 -48 -32 997

<-- =SUM(B3:E3)

1992 863 1,064 -52 -37 1,838

1993 371 1,174 -83 -31 1,431

1994 705 1,434 -139 -61 1,939

1995 1,571 1,540 -264 -119 2,727

1996 2,493 1,729 -442 -207 3,573

1997 2,573 2,040 -413 -333 3,867

1998 3,626 2,253 -490 -488 4,900

1999 3,582 2,590 -323 -401 5,447

2000 3,545 2,798 -641 -501 5,202

Growth 38.91% 13.53% 20.15%

<-- =(F12/F3)^(1/9)-1

MERCK & CO. 1991 - 2000

ll figures in $millions

MERCK--Cash Dividends versus Total Equity Cash Flow, 1991-

2000

500

1,000

1,500

2,000

2,500

3,000

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

Cash dividends

1,000

2,000

3,000

4,000

5,000

6,000

Total equity cf

Cash dividends Cash flow to equity holders

8. Consider a company that has b

equity

= 1.5 and b

debt

= 0.4. Suppose that the risk-free

rate of interest is 6 percent, the expected return on the market E(r

) = 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 tax-adjusted CAPM.

90 Chapter 2

9. On the spreadsheet fm3_problems02.xls you will ﬁ nd the following monthly data for

Cisco’s stock price and the S&P 500 index. Compute the equation r

CSCO,t

= a

CSCO

SP,t

and include the R

and t-statistics for the equation and its coefﬁ cients.

ABC

Date S&P 500 CSCO

3-Jul-02 911.62 13.19

1-Aug-02 916.07 13.82

3-Sep-02 815.28 10.48

1-Oct-02 885.76 11.18

1-Nov-02 936.31 14.92

2-Dec-02 879.82 13.10

2-Jan-03 855.70 13.37

3-Feb-03 841.15 13.98

3-Mar-03 848.18 12.98

1-Apr-03 916.92 15.00

1-May-03 963.59 16.41

2-Jun-03 974.50 16.79

1-Jul-03 990.31 19.49

1-Aug-03 1008.01 19.14

2-Sep-03 995.97 19.59

1-Oct-03 1050.71 20.93

3-Nov-03 1058.20 22.70

1-Dec-03 1111.92 24.23

2-Jan-04 1131.13 25.71

2-Feb-04 1144.94 23.16

CISCO (CSCO) AND S&P

500 PRICES

July 2002 - June 2007

10. 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 percent is selling for $95.

You consider the probability that the company will actually survive to pay off the

bond 80 percent. With 20 percent 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?

19. To do this problem you will have to copy the functions tintercept and tslope from the

spreadsheet fm3_chapter02.xls to your answer spreadsheet. See previous footnote.

91 Calculating the Cost of Capital

11. It is January 1, 1997. Normal America, Inc. (NA) has paid a year-end dividend in

each of the last ten years, as shown by the following table:

ABCDE

Year

Dec. 31

stock price

Dec. 15

dividend per

S&P 500

return

1986 33.00

1987 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%

NORMAL AMERICA, INC.

a. Calculate NA’s β with respect to the S&P 500.

b. Suppose that the Treasury bill rate is 5.5 percent and that the expected return

on the market is E(r

) = 13 percent. If the corporate tax rate T

= 35 percent,

calculate NA’s cost of equity using both the classic CAPM the tax-adjusted

model.

c. Assume that NA’s cost of debt is 8 percent. If the company is ﬁ nanced by 1/3

equity and 2/3 debt, what is its weighted average cost of capital using each of the

two CAPM models?

12. The ﬁ le fm3_problems03.xls with Financial Modeling gives data (see following) for

monthly returns on the Vanguard Index 500 fund for December 2000–June 2007.

Assuming that this fund proxies for the market as a whole compute the following:

a. E(r

) based on the whole sample of returns

b. E(r

) based on the last two years of monthly returns

92 Chapter 2

ABC D

Date

Vanguard

Index 500

Return

1-Dec-00 110.75

2-Jan-01 114.68 3.49% <-- =LN(B4/B3)

1-Feb-01 104.2 -9.58% <-- =LN(B5/B4)

1-Mar-01 97.32 -6.83% <-- =LN(B6/B5)

2-Apr-01 104.88 7.48% <-- =LN(B7/B6)

1-May-01 105.57 0.66%

1-Jun-01 102.98 -2.48%

2-Jul-01 101.96 -1.00%

RETURN DATA FOR VANGUARD

INDEX 500 FUND

December 2000 - June 2007

13. At the end of June 2007, the price-earnings ratio of the S&P 500 was 17.5. Assume

that the index proxies for the market, that it has a 50 percent dividend payout ratio,

and that dividends are expected to grow at 7 percent. Compute E(r

Appendix 1: Why Is b a Good Measure of Risk? Portfolio b versus Individual

Stock b

Although b may not be a very good measure of the riskiness of an indi-

vidual stock, the average b is a very good measure of the riskiness of a

diversiﬁ ed portfolio. This point is illustrated in this appendix. However,

before we ﬁ re into the illustration, we want to stress the meaning of the

previous sentence:

If portfolio b is a good measure of portfolio risk, then—for the holder

of a diversiﬁ ed portfolio (and this includes most investors)—the

individual share b is a good measure of the risk of a share, when this

share is ultimately held in a diversiﬁ ed portfolio.

To illustrate, consider the following data—prices for the components

of the Dow-Jones 30 Industrials (DJ30) over the 60 months between

April 2002 and April 2007 (some are not shown):

93 Calculating the Cost of Capital

Date

Alcoa

American

International

AIG

American

Express

AXP

Boeing

Citigroup

Caterpillar

CAT

DuPont

Disney

DIS

General

Electric

General

Motors

Home Depot

Honeywell

HON

4/2/2002 30.54 67.13 34.27 40.92 34.28 24.70 37.72 22.04 27.56 50.54 43.85 32.59

5/1/2002 31.54 65.04 35.52 39.27 34.33 23.64 39.29 21.79 27.20 49.33 39.42 35.00

6/3/2002 29.89 66.31 30.35 41.44 30.81 22.14 37.92 17.97 25.53 42.43 34.77 31.46

7/1/2002 24.53 62.12 29.53 38.23 26.66 20.38 35.80 16.86 28.30 36.95 29.24 28.89

8/1/2002 22.75 61.03 30.20 34.29 28.07 19.90 34.72 14.91 26.50 38.43 31.18 26.90

9/3/2002 17.50 53.20 26.11 31.57 25.41 16.97 31.07 14.40 21.82 31.24 24.75 19.45

10/1/2002 20.00 60.84 30.53 27.52 31.83 18.79 35.53 15.88 22.35 26.70 27.38 21.50

11/1/2002 23.32 63.37 32.68 31.67 33.49 22.96 38.75 18.85 24.00 32.35 25.03 23.55

12/2/2002 20.79 56.31 29.74 30.68 30.31 21.03 36.82 15.70 21.72 30.03 22.83 21.73

1/2/2003 18.04 52.68 29.89 29.38 29.75 20.39 32.89 16.85 20.64 29.60 19.86 22.13

2/3/2003 18.86 47.98 28.25 25.77 28.88 21.79 32.16 16.42 21.62 27.90 22.29 20.89

3/3/2003 17.82 48.18 27.96 23.43 29.84 22.81 34.08 16.39 22.92 27.78 23.21 19.50

4/1/2003 21.23 56.46 31.93 25.51 34.00 24.54 37.30 17.97 26.47 29.78 26.81 21.54

5/1/2003 22.79 56.39 35.14 28.85 35.71 24.33 37.25 18.92 25.80 29.60 30.96 24.10

2/1/2007 33.41 67.11 56.72 87.26 50.37 64.43 50.73 34.25 34.91 31.90 39.36 46.40

3/1/2007 33.90 67.22 56.25 88.91 51.34 67.03 49.43 34.43 35.36 30.64 36.74 46.06

4/2/2007 34.59 67.23 55.96 90.50 51.57 67.63 49.32 34.91 35.02 31.90 38.02 47.26

PRICE DATA FOR DOW-JONES INDUSTRIAL 30 COMPONENTS, APRIL 2002 - APRIL 2007

In the next spreadsheet, we compute, for each of the components of

the DJ30, the equations r

= a

+ b

SP,t

. We indicate the r-squared and

the t-statistics of the intercept a and the slope b for each equation:

AB C DEFGHIJKLM

Alcoa

American

International

AIG

American

Express

AXP

Boeing

Citigroup

Caterpillar

CAT

DuPont

Disney

DIS

General

Electric

General

Motors

Home

Depot

Noneywell

HON

Alpha

-0.0074 -0.0058 0.0020 0.0097 0.0009 0.0100 -0.0007 0.0017 0.0003 -0.0140 -0.0098 -0.0018

Beta

1.9446 1.1873 1.2615 0.7304 1.2086 1.3915 1.0488 1.2131 0.7474 1.2860 1.5274 1.6276

R-squared

0.5646 0.3906 0.6987 0.1442 0.5050 0.4328 0.4114 0.3699 0.2533 0.1869 0.4791 0.6100

T-Alpha

-0.9545 -0.8548 0.5325 1.1917 0.1653 1.3765 -0.1147 0.2410 0.0583 -1.1417 -1.3563 -0.2964

T-Beta

8.6724 6.0965 11.5982 3.1260 7.6922 6.6532 6.3675 5.8350 4.4352 3.6511 7.3042 9.5241

Date

5/1/2002 3.22% -3.16% 3.58% -4.12% 0.15% -4.39% 4.08% -1.14% -1.31% -2.42% -10.65% 7.13%

6/3/2002 -5.37% 1.93% -15.73% 5.38% -10.82% -6.56% -3.55% -19.27% -6.34% -15.07% -12.55% -10.66%

7/1/2002 -19.76% -6.53% -2.74% -8.06% -14.47% -8.28% -5.75% -6.38% 10.30% -13.83% -17.32% -8.52%

RETURN DATA, ALPHA, BETA, R-SQUARED

For Dow-Jones 30 Industrials, April 2002 - April 2007

=tintercept(B16:B75,$AH$16:$AH$75)

=tslope(B16:B75,$AH$16:$AH$75)

=INTERCEPT(B16:B75,$AH$16:$AH$75)

=SLOPE(B16:B75,$AH$16:$AH$75)

=RSQ(B16:B75,$AH$16:$AH$75)

Here are the averages of the r-squared and the t-statistics when done

on an individual basis:

CBA

Average alpha -0.0005 <-- =AVERAGE('Page 87, bottom'!B7:AE7)

Average beta 1.1421 <-- =AVERAGE('Page 87, bottom'!B8:AE8)

Average r-squared 0.3508 <-- =AVERAGE('Page 87, bottom'!B9:AE9)

Average t-statistic, intercept 0.0052 <-- =AVERAGE('Page 87, bottom'!B10:AE10)

Average t-statistic, slope 5.7859 <-- =AVERAGE('Page 87, bottom'!B11:AE11)

REGRESSING THE DJ30 COMPONENTS ON THE S&P500

Averages

94 Chapter 2

If, however, we regress the returns of a portfolio of the Dow-Jones

components on the S&P 500, we ﬁ nd a dramatic increase in the R

AB C

Alpha -0.0020

<-- {=INTERCEPT(MMULT('Page 87, bottom'!B16:AE75,'Page 88,

bottom'!B9:B38),'Page 87, bottom'!AH16:AH75)}

Beta 1.1747

<-- {=SLOPE(MMULT('Page 87, bottom'!B16:AE75,'Page 88, bottom'!B9:B38),'Page

87, bottom'!AH16:AH75)}

R-squared 0.8191

<-- {=RSQ(MMULT('Page 87, bottom'!B16:AE75,'Page 88, bottom'!B9:B38),'Page 87,

bottom'!AH16:AH75)}

T-Alpha -0.7772

<-- {=tintercept(MMULT('Page 87, bottom'!B16:AE75,'Page 88, bottom'!B9:B38),'Page

87, bottom'!AH16:AH75)}

T-Beta 16.2082

<-- {=tslope(MMULT('Page 87, bottom'!B16:AE75,'Page 88, bottom'!B9:B38),'Page 87,

bottom'!AH16:AH75)}

0.1000

AIG 0.0000

AXP 0.0000

BA 0.0000

C 0.1000

CAT 0.0000

DD 0.1000

DIS 0.1000

GE 0.0000

GM 0.0000

HD 0.0000

HON 0.0000

HPQ 0.0000

IBM 0.0000

INTC 0.1000

JNJ 0.0000

JPM 0.0000

KO 0.1000

MCD 0.0000

MMM 0.1000

MO 0.0000

MRK 0.1000

MSFT 0.0000

PFE 0.0000

PG 0.0000

T 0.0000

UTX 0.0000

VZ 0.1000

WMT 0.0000

XOM 0.1000

Sum of portfolio weights 1.00

<-- =SUM(B9:B38)

Portfolio weights

REGRESSING PORTFOLIOS OF THE DJ30 COMPONENTS ON THE S&P500

By changing the portfolio weights, you can see that this increase is

characteristic of almost all well-diversiﬁ ed portfolios (even those not

including all of the DJ30 components):

95 Calculating the Cost of Capital

The meaning of this result is that, when we invest in large diversiﬁ ed

portfolios, almost all of the risk is due to the individual assets’ bs.

Appendix 2: Getting Data from the Internet

All the data used in this chapter were retrieved from Yahoo. This appen-

dix provides a brief description of how they were gotten. Keep in mind

that since the Internet is a very lively place, it may be that by the time

you read this book, some of the technical details and addresses will have

changed.

AB C

Alpha -0.0005

<-- {=INTERCEPT(MMULT('Page 87, bottom'!B16:AE75,'Page 89'!B9:B38),'Page 87,

bottom'!AH16:AH75)}

Beta 1.1421

<-- {=SLOPE(MMULT('Page 87, bottom'!B16:AE75,'Page 89'!B9:B38),'Page 87,

bottom'!AH16:AH75)}

R-squared 0.9460

<-- {=RSQ(MMULT('Page 87, bottom'!B16:AE75,'Page 89'!B9:B38),'Page 87,

bottom'!AH16:AH75)}

T-Alpha -0.3669

<-- {=tintercept(MMULT('Page 87, bottom'!B16:AE75,'Page 89'!B9:B38),'Page 87,

bottom'!AH16:AH75)}

T-Beta 31.8910

<-- {=tslope(MMULT('Page 87, bottom'!B16:AE75,'Page 89'!B9:B38),'Page 87,

bottom'!AH16:AH75)}

0.0333

AIG 0.0333

AXP 0.0333

BA 0.0333

C 0.0333

CAT 0.0333

DD 0.0333

DIS 0.0333

GE 0.0333

GM 0.0333

HD 0.0333

HON 0.0333

HPQ 0.0333

IBM 0.0333

INTC 0.0333

JNJ 0.0333

JPM 0.0333

KO 0.0333

MCD 0.0333

MMM 0.0333

MO 0.0333

MRK 0.0333

MSFT 0.0333

PFE 0.0333

PG 0.0333

T 0.0333

UTX 0.0333

VZ 0.0333

WMT 0.0333

XOM 0.0333

Sum of portfolio weights 1.00

<-- =SUM(B9:B38)

Portfolio weights

REGRESSING PORTFOLIOS OF THE DJ30 COMPONENTS ON THE S&P500

96 Chapter 2

1. Yahoo’s ﬁ nancial information is most conveniently accessed by

clicking on the Finance link on the Yahoo home page.

2. Putting the stock symbol in the Get Quotes box takes you to informa-

tion about a particular company. Here we have asked for information on

General Motors (stock symbol, GM):

97 Calculating the Cost of Capital

3. In this book we most commonly use Historical Prices to download

price and dividend information about the stock and Key Statistics to

access a wealth of information about the company.

4. Key Statistics gives much information about General Motors. Of

particular interest to readers of this chapter is the computation of

beta, based on 60 months of monthly return data versus the S&P 500

index: