Risk and Return in Practice: Problems and Questions
1. In December 1995, Boise Cascade's stock had a beta of 0.95. The treasury bill rate at the
time was 5.8%, and the treasury bond rate was 6.4%. The historical risk premium of the
Standard&Poor index is 5.5%. The firm had debt outstanding of $ 1.7 billion and a market
value of equity of $ 1.5 billion; the corporate marginal tax rate was 36%.
a. Estimate the expected return on the stock for a short term investor in the company.
b. Estimate the expected return on the stock for a long term investor in the company.
c. Estimate the cost of equity for the company.
2. Boise Cascade also had debt outstanding of $ 1.7 billion and a market value of equity of $
1.5 billion; the corporate marginal tax rate was 36%. The historical risk premium of the
Standard&Poor index is 5.5%.
a. Assuming that the current beta of 0.95 for the stock is a reasonable one, estimate the
unlevered beta for the company.
b. How much of the risk in the company can be attributed to business risk and how
much to financial leverage risk?
3. Biogen Inc., as biotechnology firm, had a beta of 1.70 in 1995. It had no debt outstanding at
the end of that year.
a. Estimate the cost of equity for Biogen, if the treasury bond rate is 6.4% and the
expected yield of S&P index is 12%.
b. What effect will an increase in long term bond rates to 7.5% have on Biogen's cost
of equity?
c. How much of Biogen's risk can be attributed to business risk?
4. Genting Berhad is a Malaysian conglomerate, with holding in plantations and tourist
resorts. The beta estimated for the firm, relative to the Malaysian stock exchange, is 1.15, and
the long term government borrowing rate in Malaysia is 11.5%. The Malaysian Stock Market
risk premium is expected to 7,5%.
a. Estimate the expected return on the stock.
b. If you were an international investor, what concerns, if any, would you have about
using the beta estimated relative to the Malaysian Index? If you do, how would you
modify the beta?
5. You have just done a regression of monthly stock returns of HeavyTech Inc., a
manufacturer of heavy machinery, on monthly market returns over the last five years and
come up with the following regression:
RHeavyTech = 0.5% + 1.2 RM
The current T.Bill rate is 3% (It was 5% one year ago). The stock is currently selling for $50,
down $4 over the last year, and has paid a dividend of $2 during the last year and expects to
pay a dividend of $2.50 over the next year. The NYSE composite has gone down 8% over the
last year, with a dividend yield of 3%. The risk premium of market is supposed to be 8,5%.
a. What is the expected return on HeavyTech over the next year?
b. What would you expect HeavyTech's price to be one year from today?
c. What would you have expected HeavyTech's stock returns to be over the last year?
d. What were the actual returns on HeavyTech over the last year?
e. HeavyTech has $100 million in equity and $ 50 million in debt. It plans to issue $50
million in new equity and retire $50 million in debt. Estimate the new beta, if
HeavyTech Inc. has a tax rate of 40%.
6. Safecorp, which owns and operates grocery stores across the United States, currently has
$50 million in debt and $100 million in equity outstanding. Its stock has a beta of 1.2. It is
planning a leveraged buyout , where it will increase its debt/equity ratio of 8. If the tax rate is
40%, what will the beta of the equity in the firm be after the LBO?
7. Novell, which had a market value of equity of $2 billion and a beta of 1.50, announced that
it was acquiring WordPerfect, which had a market value of equity of $ 1 billion, and a beta of
1.30. Neither firm had any debt in its financial structure at the time of the acquisition, and the
corporate tax rate was 40%.
a. Estimate the beta for Novell after the acquisition, assuming that the entire
acquisition was financed with equity.
b. Assume that Novell had to borrow the $ 1 billion to acquire WordPerfect. Estimate
the beta after the acquisition.
8. You are analyzing the beta for Hewlett Packard and have broken down the company into
four broad business groups, with market values and betas for each group.
Business Group
Mainframes
Personal
Computers
Software
Printers
Market Value of
Equity
$ 2.0 billion
1.10
$ 2.0 billion
1.50
$ 1.0 billion
$ 3.0 billion
2.00
1.00
Beta
a. Estimate the beta for Hewlett Packard as a company. Is this beta going to be equal
to the beta estimated by regressing past returns on HP stock against a market index.
Why or Why not?
b. If the treasury bond rate is 7.5% and the market risk premium is 5,5%, estimate the
cost of equity for Hewlett Packard. Estimate the cost of equity for each division.
Which cost of equity would you use to value the printer division?
c. Assume that HP divests itself of the mainframe business and pays the cash out as a
dividend. Estimate the beta for HP after the divestiture. (HP had $ 1 billion in debt
outstanding. Hewlett Packard’s effective tax rate is 36%)
9. The following table summarizes the percentage changes in operating income, percentage
changes in revenue and betas for four pharmaceutical firms.
Firm
% Change in Revenue
PharmaCorp
SynerCorp
BioMed
Safemed
27%
25%
23%
21%
% Change in Operating
Income
25%
32%
36%
40%
Beta
1.00
1.15
1.30
1.40
a. Calculate the degree of operating leverage for each of these firms.
b. Use the operating leverage to explain why these firms have different betas.
10. A prominent beta estimation service reports the beta of Comcast Corporation, a major
cable TV operator, to be 1.45. The service claims to use weekly returns on the stock over the
prior five years and the NYSE composite as the market index to estimate betas. You replicate
the regression using weekly returns over the same period and arrive at a beta estimate of 1.60.
How would you reconcile the two estimates?
11. Battle Mountain is a mining company, which mines gold, silver and copper in mines in
South America, Africa and Australia. The beta for the stock is estimated to be 0.30. Given the
volatility in commodity prices, how would you explain the low beta?
12. You have collected returns on AnaDone Corporation (AD Corp.), a large diversified
manufacturing firm, and the NYSE index for five years:
Year
1981
1982
1983
1984
1985
AD Corp
10%
5%
-5%
20%
-5%
NYSE
5%
15%
8%
12%
-5%
a. Estimate the intercept (alpha) and slope (beta) of the regression.
b. If you bought stock in AD Corp. today how much would you expect to make as a
return over the next year? [The six-month T.Bill rate is 6%] The market risk premium
is supposed to be 8,5%.
c. Looking back over the last five years, how would you evaluate AD's performance
relative to the market? (The riskfree rate during the period was also 6% on an annual
basis)
d. Assume now that you are an undiversified investor and that you have all of your
money invested in AD Corporation. What would be a good measure of the risk that
you are taking on? How much of this risk would you be able to eliminate if you
diversify?
e. AD is planning to sell off one of its divisions. The division under consideration has
assets which comprise half of the book value of AD Corporation, and 20% of the
market value. Its beta is twice the average beta for AD Corp (before divestment).
What will the beta of AD Corporation be after divesting this division?
13. You run a regression of monthly returns of Mapco Inc, an oil and gas producing firm, on
the S&P 500 index and come up with the following output for the period 1991 to 1995.
Intercept of the regression = 0.06%
X-coefficient of the regression = 0.46
Standard error of X-coefficient = 0.20
R squared = 5%
There are 20 million shares outstanding, and the current market price is $ 2. The firm has $ 20
million in debt outstanding. (The firm has a tax rate of 36%)
a. What would an investor in Mapco's stock require as a return, if the T.Bond rate is
6% and the market risk premium is 5,5%?
b. What proportion of this firm's risk is diversifiable?
c. Assume now that Mapco has three divisions, of equal size (in market value terms).
It plans to divest itself of one of the divisions for $ 20 million in cash and acquire
another for $ 50 million (It will borrow $ 30 million to complete this acquisition). The
division it is divesting is in a business line where the average unlevered beta is 0.20,
and the division it is acquiring is in a business line where the average unlevered beta is
0.80. What will the beta of Mapco be after this acquisition?
14. You have just run a regression of monthly returns of American Airlines (AMR) against
the S&P 500 over the last five years. You have misplaced some of the output and are trying to
derive it from what you have.
a. You know the R squared of the regression is 0.36, and that your stock has a variance
of 67%. The market variance is 12%. What is the beta of AMR?
b. You also remember that AMR was not a very good investment during the period of
the regression and that it did worse than expected (after adjusting for risk) by 0.39 % a
month for the five years of the regression. During this period, the average riskfree rate
was 4.84%. What was the intercept on the regression?
c. You are comparing AMR Inc. to another firm which also has an R squared of 0.48.
Will the two firms have the same beta? If not, why not?
15. You have run a regression of monthly returns on Amgen, a large biotechnology firm,
against monthly returns on the S&P 500 index, and come up with the following output ñ
Rstock = 3.28% + 1.65 RMarket R2= 0.20
The current one-year treasury bill rate is 4.8% and the current thirty-year bond rate is 6.4%,
and the market risk premium is 8,5%. The firm has 265 million shares outstanding, selling for
$ 30 per share.
i. What is the expected return on this stock over the next year?
ii. Would your expected return estimate change if the purpose was to get a discount
rate to analyze a thirty-year capital budgeting project?
iii. An analyst has estimated, correctly, that the stock did 51.10% better than expected,
annually, during the period of the regression. Can you estimate the annualized riskfree
rate that she used for her estimate?
iv. The firm has a debt/equity ratio of 3%, and faces a tax rate of 40%. It is planning to
issue $2 billion in new debt and acquire a new business for that amount, with the same
risk level as the firm's existing business. What will the beta be after the acquisition?
16. You have just run a regression of monthly returns on MAD Inc., a newspaper and
magazine publisher, against returns on the S&P 500, and arrived at the following result ñ
RMAD = - 0.05% + 1.20 RS&P
The regression has an R-squared of 22%. The current T.Bill rate is 5.5% and the current
T.Bond rate is 6.5%. The riskfree rate during the period of the regression was 6%.. Answer
the following questions relating to the regression ñ
a. Based upon the intercept, you can conclude that the stock did
A. 0.05% worse than expected on a monthly basis, during the regression.
B. 0.05% better than expected on a monthly basis during the period of the
regression
C. 1.25% better than expected on a monthly basis during the period of the
regression.
D. 1.25% worse than expected on a monthly basis during the period of the
regression.
E. None of the above. (1 point)
b. You now realize that MAD Inc went through a major restructuring at the end of last
month (which was the last month of your regression), and made the following changes
The firm sold off its magazine division, which had an unlevered beta of 0.6, for
$ 20 million.
o It borrowed an additional $ 20 million, and bought back stock worth $ 40
million.
o
After the sale of the division and the share repurchase, MAD Inc. had $ 40 million in debt and
$ 120 million in equity outstanding.
If the firm's tax rate is 40%, re-estimate the beta, after these changes.
17. Time Warner Inc., the entertainment conglomerate, has a beta of 1.61. Part of the reason
for the high beta is the debt left over from the leveraged buyout of Time by Warner in 1989,
which amounted to $10 billion in 1995. The market value of equity at Time Warner in 1995
was also $ 10 billion. The marginal tax rate was 40%.
a. Estimate the unlevered beta for Time Warner.
b. Estimate the effect of reducing the debt ratio by 10% each year for the next two
years on the beta of the stock.
18. Chrysler, the automotive manufacturer, had a beta of 1.05 in 1995. It had $ 13 billion in
debt outstanding in that year, and 355 million shares trading at $ 50 per share. The firm had a
cash balance of $ 8 billion at the end of 1995. The marginal tax rate was 36%.
a. Estimate the unlevered beta of the firm.
b. Estimate the effect of paying out a special dividend of $ 5 billion on this unlevered
beta.
c. Estimate the beta for Chrysler after the special dividend.
19. You are trying to estimate the beta of a private firm that manufactures home appliances.
You have managed to obtain betas for publicly traded firms that also manufacture home
appliances.
Firm
Black & Decker
Fedders Corp.
Maytag Corp.
National Presto
Whirlpool
Beta
1.40
1.20
1.20
0.70
1.50
Debt
$ 2,500
$5
$ 540
$8
$ 2900
MV of Equity
$ 3,000
$ 200
$ 2250
$ 300
$ 4000
The private firm has a debt equity ratio of 25%, and faces a tax rate of 40%. The publicly
traded firms all have marginal tax rates of 40%, as well.
a. Estimate the beta for the private firm.
b. What concerns, if any, would you have about using betas of comparable firms?
20. As the result of stockholder pressure, RJR Nabisco is considering spinning off its food
division. You have been asked to estimate the beta for the division, and decide to do so by
obtaining the beta of comparable publicly traded firms. The average beta of comparable
publicly traded firms is 0.95, and the average debt/equity ratio of these firms is 35%. The
division is expected to have a debt ratio of 25%. The marginal corporate tax rate is 36%.
a. What is the beta for the division?
b. Would it make any difference if you knew that RJR Nabisco had a much higher
fixed cost structure than the comparable firms used here?
21. Southwestern Bell, a phone company, is considering expanding its operations into the
media business. The beta for the company at the end of 1995 was 0.90, and the debt/equity
ratio was 1. The media business is expected to be 30% of the overall firm value in 1999, and
the average beta of comparable firms is 1.20; the average debt/equity ratio for these firms is
50%. The marginal corporate tax rate is 36%.
a. Estimate the beta for Southwestern Bell in 1999, assuming that it maintains its
current debt/equity ratio.
b. Estimate the beta for Southwestern Bell in 1999, assuming that it decides to finance
its media operations with a debt/equity ratio of 50%.
22. The chief financial officer of Adobe Systems, a growing software manufacturing firm, has
approached you for some advice regarding the beta of his company. He subscribes to a service
which estimates Adobe System's beta each year, and he has noticed that the beta estimates
have gone down every year since 1991 - 2.35 in 1991 to 1.40 in 1995. He would like the
answers to the following questions ñ
a. Is this decline in beta unusual for a growing firm?
b. Why would the beta decline over time?
c. Is the beta likely to keep decreasing over time?
23. You are analyzing Tiffany's, an upscale retailer, and find that the regression estimate of
the firm's beta is 0.75; the standard error for the beta estimate is 0.50. You also note that the
average unlevered beta of comparable specialty retailing firms is 1.15.
a. If Tiffany's has a debt/equity ratio of 20%, estimate the beta for the company based
upon comparable firms. (The tax rate is 40%)
b. Estimate a range for the beta from the regression.
c. How would you reconcile the two estimates? Which one would you use in your
analysis?
Solutions
Problem 1
a. Expected Return to Short-term Investor = 5.8% + 0.95 (8.5%) = 13.88%
(I am using the historical premium of 8.5% to estimate expected returns)
b. Expected Return to Long-term Investor = 6.4% + 0.95 (5.5%) = 11.63%
c. I would use the expected return of 11.63% as the cost of equity
Problem 2
a. Unlevered Beta = 0.95 / (1 + (1- 0.36) (1700/1500)) = 0.55
b. The beta of 0.95 can be broken down into business risk (0.55) and financial risk (0.40).
Problem 3
a. Cost of Equity = 6.40% + 1.70 (5.5%) = 15.75%
b. If the long term bond rate rises to 7.5%, the cost of equity will rise by 1.1%.
c. Since the firm had no debt, all of the risk can be attributed to business risk.
Problem 4
a. Expected Return = 11.5% + 1.15 (7.50%) = 20.13%
{I am using a premium of 7.50% for Malaysian stocks to reflect its higher risk.}
b. This beta measures risk relative to a Malaysian index. For an international investor an more
appropriate beta may be the one estimated relative to a global index.
Problem 5
a. Expected Return = 3% + 1.2 (8.5%) = 13.20%
b. Expected Price Appreciation = 13.20% - ($ 2.50 / $ 50) = 8.20%
Expected Price one year from today = $ 50 (1.082) = 54.10 $
c. Expected Returns over last year = 5% + 1.20 (-5% - 5%) = -7.00%
Returns on Market = -8% + 3% = -5%
d. Actual Returns over last year = (50-54+2)/54 = -3.70%
e. Unlevered Beta = 1.20 / (1+ (1-.4) (50/100)) = 0.923
If the firm issues $ 50 million in equity and retires debt, its beta will drop to 0.923
Problem 6
Unlevered Beta = 1.20 / (1 + (1-0.4) (50/100)) = 0.923076923
New Beta = 0.923 (1 + (1-0.4) (8)) = 5.35
Problem 7
a. Unlevered Beta for Novell = 1.50 ! Firm has no debt
Unlevered Beta for WordPerfect = 1.30 ! Firm has no debt
Unlevered Beta for Combined Firm = 1.50 (2/(2+1)) + 1.30 (1/(2+1)) = 1.43
This would be the beta of the combined firm if the deal is all-equity.
b. If the deal is financed with debt,
New Debt/Equity Ratio = 1/2 = 0.5
New Beta = 1.43 (1 + (1-.4) (0.5)) = 1.86
Problem 8
a. Beta for Hewlett Packard = 1.10 (2/8) + 1.50 (2/8) + 2.00 (1/8) + 1.00 (3/8) = 1.275
This beta may not be equal to the regression estimate of beta, because both of these are
estimated with error
b. Cost of Equity = 7.5% + 1.275 (5.5%) = 14.51%
Mainframes Cost of Equity = 7.5% + 1.10 (5.5%) = 13.55%
Personal Computers Cost of Equity = 7.5% + 1.5(5.5%) = 15.75%
Software Cost of Equity = 7.5% + 2 (5.5%) = 18.50%
Printer Division's Cost of Equity = 7.5% + 1 (5.5%) = 13.00%
To value the printer division, I would use a 13.00% cost of equity.
c. Assuming that the leverage is equally distributed across the divisions,
Division
Beta Unlevered Beta Value of Equity Debt Firm Value
Mainframes 1,10 1,019
2,00
0,25 2,25
PCs
1,50 1,389
2,00
0,25 2,25
Sofware
2,00 1,852
1,00
0,125 1,13
Printers
1,00 0,926
3,00
0,375 3,38
Unlevered Beta = 1.389 (2.25/6.75) + 1.852 (1.125/6.75) + 0.926 (3.375/6.75) = 1.235
New Debt/Equity Ratio = 1/5.75
New Levered Beta = 1.235 (1 + (1-.36) (1/5.75)) = 1.37
Problem 9
Firm
%Revenue %OI Operating leverage Beta
PharmaCorp 27%
25% 0,93
1,00
SynerCorp
25%
32% 1,28
1,15
BioMed
23%
36% 1,57
1,30
Safemed
21%
40% 1,90
1,40
b. Firms with high operating leverage also have high betas.
Problem 10
Beta estimation services adjust betas towards one. It is possible that this adjustment is the
reason for the difference between the regression beta (1.60) and the reported beta of 1.45.
Problem 11
The beta reflects market risk and is estimated relative to a stock index. To the extent that
commodity prices and stock prices are not highly positively correlated the low betas reflect
the low market risk inherent in these stocks. I would expect these firms to have substantial
firm-specific risk.
Problem 12
Year AD
1981 10%
1982 5%
1983 -5%
1984 20%
1985 5%
NYSE
5%
15%
-8%
12%
-5%
a. Regression Results
Returns on AD = 0.047 + 0.60 ( NYSE)
The intercept is 4.7%; the beta is 0.60.
b. Expected Return over next year = 6% + 0.60 (8.5%) = 11.10%
c. Intercept = 4.70%
Riskfree Rate (1-Beta) = 6% (1-0.60) = 2.40%
Intercept - Riskfree Rate (1-Beta) = 4.7% - 2.4% = 2.30%
On an annual basis, the stock did 2.3% better than expected.
d. If you were an undiversified investor, you would be exposed to all risk in AD; this can be
measured in terms of the standard deviation in AD returns (9.08%).
The R-squared of this regression is 45%; this suggests that 55% is diversifiable risk.
e. Beta for divested division = 2 (0.60) = 1.20
1.20 (.2) + X (0.8) = 0.6
Solve for X,
X = 0.36/.8 = 0.45
This is the beta after the divestiture, assuming that the cash is paid out and that the leverage is
unaffected.
Problem 13
a. Expected Return = 6% + 0.46 (5.5%) = 8.53%
b. Proportion of the firm's risk that is diversifiable = 1 - R squared = 95%
c. Existing Debt/Equity Ratio = 20/(20*2) = 50.00%
Unlevered Beta for the firm = 0.46/(1+(1-.36) (.5)) = 0.35
Unlevered Beta of Firm without divested division
0.20 (20/60) + X (40/60) = 0.35
Solving for X,
X = (0.35 - 0.0667)/(40/60) = 0.42
New Unlevered Beta with new division,
0.42 (40/90) + 0.80 (50/90) = 0.63
New Debt/Equity Ratio = (20 + 30)/40 = 1.25
New Levered Beta = 0.63 (1 + (1-.36) (1.25)) = 1.13
Problem 14
a. R squared = β 2 * σ m2 / σ i2
Beta2 = (0.36*0.67)/0.12 = 2.01
Beta = 1.42
b. Jensen's alpha = Intercept - Riskfree Rate (1-Beta)
-0.39% = Intercept - 0.39% (1-1.42)
Monthly Riskfree Rate = (1.0484)1/12 - 1 = 0.39%
Solving for the intercept,
Intercept = -0.39% + 4,84% (1-1.42) = -2,41%
c. The two firms might not have the same beta, because they might have different total
variances.
Problem 15
a. Expected Return on the Stock over next year = 4.8% + 1.65 (8.5%) = 18.83%
b. Yes, i would use the long term bond rate as my riskless rate.
Expected Return on the Stock = 6.4% + 1.65 (5.5%) = 15.48%
c. Monthly Jensen's Alpha = (1.51)1/12 - 1 = 3.49%
3.49% = 3.28% - Riskfree Rate (1-1.65)
Solving for the riskless rate,
Monthly Riskless Rate = 0.21%/(0.65) = 0.32%
Annualized Riskless Rate = (1.0032)12 - 1 = 3.91%
d. Unlevered Beta = 1.65 / (1 + (1-.4)(.03)) = 1.62
Market Value of Equity = 265 * $ 30 = 7,950.00 $
Existing Debt = 0.03 (7950) = 238.50 $
New Debt = $ 238.50 + $ 2000 = 2,238.50 $
New Levered Beta = 1.62 (1 + (1-.4) (2238.5/7950) = 1.89
Problem 16
a. Intercept - Riskfree Rate (1-Beta) = -0.05% - 0.49% (1-1.20) = 0.05%
Monthly Riskfree Rate = 1.06(1/12) - 1 = 0.49%
The stock did 0.05% better than expected during the period of the regression.
b. Debt before restructuring = $ 40 million - $ 20 million = 20.00 $
Equity before restructuring = $ 120 + $ 40= 160.00 $
Debt/Equity Ratio before restructuring = 20/160 = 12.50%
Unlevered Beta before restructuring = 1.20 / (1+(1-.4)(.125)) = 1.12
Unlevered Beta after the divisional sale,
0.6(20/180) + X (160/180) = 1.12
Solving for X,
Unlevered Beta after divisional sale = 1.19
New Debt/Equity Ratio after restructuring = 40/120 = 0.333333333
New Beta = 1.19 (1 + (1-.4) (0.33)) = 1.43
Problem 17
a. Unlevered Beta = 1.61 / (1+(1-.4) (10/10)) = 1.01
b. Estimated Debt Ratios
Year Debt Ratio D/E
Beta
1
40%
0,67
1,41
2
30%
0,42871 1,27
Problem 18
a. Unlevered Beta = 1.05 / (1 + (1-.36)(13000/(355*50)) = 0.71
b. With special dividend,
Debt/Equity Ratio = 13000/(355*50-5000) = 1.02
Assuming that the beta of cash is 0,
Value of Firm before dividend = 13000 + 355*50 = 30,750.00 $
0 (8000/30750) + X (22750/30750) = 0.71
Solving for X,
X = 0.96
The unlevered beta after the special dividend will be 0.96.
c. New Beta after the special dividend = 0.96 (1+(1-.36)(1.02) = 1.59
Problem 19
Company
Beta Debt Equity D/E
Black&Decker 1,40 2500 3000
83,33%
Fedders
1,20 5
200
2,50%
Maytag
1,20 540 2250
24,00%
National Presto 0,70 8
300
2,67%
Whirlpool
1,50 2900 4000
72,50%
Average
1,20
37,00%
Unlevered beta = 1.20 / (1+(1-.4)(.37)) = 0.98
a. Beta for private firm = 0.98 (1+ (1-.4) (.25)) = 1.13
b. The firms might not be directly comparable in terms of business mix. Furthermore, the risk
of a firm may be affected by its size; if the private firm is much smaller, this beta might not
adequately reflect its risk. Finally, this beta reflects only market risk; the private firm may
care about firm-specific risk as well.
Problem 20
a. Unlevered Beta for Food Business = 0.95/ (1 + (1-.36) (.35)) = 0.78
Beta for Food Division = 0.78 (1 + (1-.36) (.25)) = 0.90
b. Yes. The higher fixed cost structure would lead me to use a higher unlevered beta for
Nabisco.
Problem 21
a. Unlevered Beta of phone business = 0.90 / (1+0.64*1) = 0.55
Unlevered Beta of media business = 1.20 / (1 + 0.64*.5) = 0.91
Unlevered Beta in 1999 = 0.7 (0.55) + 0.3 (0.91) = 0.66
Beta in 1999 = 0.66 (1 + (1-.36) (1.00)) = 1.08
b. If it finances its media operations with a 50% D/E ratio,
Overall Debt Ratio = 1 (.7) + .5 (.3) = 0.85
Beta with this debt ratio = 0.66 (1 + 0.64 (0.85)) = 1.02
Problem 22
a. No, it is not unusual. As firms grow and become larger, they generally become more
diversified and less risky.
b. I would expect it to continue since Adobe still has a beta well above one.
c. I would expect it to stop as the beta approaches one.
Problem 23
a. Unlevered Beta of comparable firms = 1.15
Beta for Tiffany's based on comparable firms = 1.15 (1+0.6*.2) = 1.29
b. Range for beta from regression
With one standard error : 0.25 - 1.25
c. The regression estimate is very noisy. It is entirely possible that both of these estimates are
from the same distribution; I would trust the "comparable firm" estimate more.
Discussion Issues and Derivations
1. A Derivation of the Capital Asset Pricing Model
I. Establish the Objects of Choice: Mean versus Variance
Theme: Investors are risk averse. They measure reward using expected return and risk
using variance.
Underlying assumptions: The mean-variance assumption can hold only if (a) all
investors have quadratic utility function or (b) returns are normally distributed.
Implication: Portfolio A with higher expected return and the same variance as
portfolio B will be preferred to B
II. Benefits of Diversification
For any desired level of risk (s) there exists a portfolio of several assets which yields a
higher expected return than any individual security
E(Rp) = Σ wi E(Ri)
σ2p = Σ Σ wi wj Covij
Efficient portfolios: maximize returns for any level of risk.
Implications: (a) Everybody should diversify (b) Investors should try to identify and
hold efficient portfolios (c) This method has very heavy computational requirements.
III. The Single Index Model: The Logical Limit of Diversification
Assumptions: (a) Riskfree lending and borrowing (b) Markets which are frictionless there are no transactions costs (c)Homogeneous expectations
Implications: (1) The risky portfolio than when combined with the riskless asset
maximizes returns is the market portfolio. (2) Everybody holds some combination of
the market portfolio and the risky asset. How much of each is held will be a function
of the investor's risk aversion.(3) Since all investors hold the same market portfolio it
must contain all assets in the economy in proportion to their value.
IV. The Risk of an Individual Asset
Step 1: Individuals diversify and hold portfolios
Step 2: The risk of a security is the risk it adds to the portfolio
Step 3: Everybody holds the market portfolio
Step 4: The risk of a security is the risk that it adds to the market portfolio.
Step 5: The covariance between an asset "i" and the market portfolio (Covim) is a
measure of this added risk. The higher the covariance the higher the risk.
Step 6: This measure can be standardized by dividing by the market variance. b =
Covim/ σ2m.
• Variants of the Capital Asset Pricing Model
I. No Riskless Asset
Basis: If no riskless asset exists investors can use a portfolio of risky assets which is
uncorrelated with the market portfolio instead as the riskless asset. This portfolio is called the
zero-beta portfolio.
Properties of the Zero-beta portfolio
(1) Of all the the zero-beta portfolios this has the minimum variance
(2) The separation principle applies here with the two portfolios, the market portfolio and the
zero-beta portfolio, i.e. all investors hold combinations of the two.
(3) The expected return on any security can be expressed as a linear function of its beta.
E(Ri) = E(Rz) + β (E(Rm) - E(Rz))
where E(Rz) is the expected return on a zero beta portfolio
II. Riskless Lending but no Riskless Borrowing
Basis:
(a) There is a piecewise linear relationship between expected return and beta for efficient
portfolios.
(b) Efficient portfolios with the riskfree asset lie along the segment RfT and those containing
only risky assets lies along the segment TMC.
III. Existence of Non-Marketable Assets (such as Human Capital)
The separation principle still holds but,
(a) Investors hold different portfolios of risky assets depending upon the portfolios of nonmarketable assets that they possess.
(b) The market price of risk includes the variance of the market and the covariance between
the market portfolio and the portfolio of non-marketable assets
IV. Existence of Taxes
Model: The model considers differential taxes on dividends and capital gains in a one-period
context where investors maximize their one-period returns. The final model for expected
return has a dividend component
E(Ri) = a + βi (E(Rm) - Rf) + c (di - Rf)
where di = Dividend yield on asset i
Rf = After-tax riskfree rate
V. Existence of Heterogeneous Expectations and Information
Model: To get strong conclusions we have to assume that all investors have a certain class of
utility functions (Constant Absolute risk aversion) and complete markets (At least as many
independent securities as states).
• Testing the CAPM: Issues and Discussion
Issue 1: The CAPM can never be tested because the market portfolio can never be observed
Central to the CAPM is the concept of a market portfolio which includes every asset in the
economy. To test the CAPM therefore one has to observe and be able to measure this efficient
market portfolio. If one cannot do so one cannot test the CAPM. One cannot use of an
inefficient portfolio like the S&P 500 or the NYSE 2000 or even every stock in the economy
to estimate betas and test for linearity (like all the studies have done) because
(a) The betas measured against an inefficient portfolio are meaningless measures and cannot
be used to accept or reject the CAPM which is really a theory about betas measured against
the efficient market portfolio
(b) For every inefficient portfolio there exists a set of betas which will satisfy the linearity
condition.
Issue 2: The CAPM is difficult to test on individual assets
The noisiness in beta estimates and the fact that the CAPM yields expected returns for
individual assets over the long term makes it difficult to test the CAPM by trying to relate
expected returns on individual assets (such as stocks) to their betas. What most tests of the
CAPM do instead is to look at portfolios of stocks, based upon betas, and then compare these
betas to expected returns in the next time period.
• More on Factor Analysis and the Arbitrage Pricing Model
Central to applying the arbitrage pricing model is the use of a factor analysis. In a typical
factor analysis, we begin with pricing data on a large number of assets over very long time
periods. In the factor analysis, we look for factors that seem to move prices on large numbers
of assets in unison. To prevent factors from being double counted, we ensure that the factors
that emerge are independent of each other. While all of this occurs behind the screen of the
factor analysis, what emerges as output from the analysis includes:
(a) the number of common factors that appeared to affect asset prices over the period for
which the data is available
(b) the betas of each asset relative to each factor, again using the same data
(c) the "risk premiums" associated with each factor
These factor betas and factor premiums are then used, in conjunction with a riskfree rate to
get an expected return for an asset.
• Estimating the Macro Economic Factors in a Multi-Factor Model
Once the number of factors have been identified in an arbitrage pricing model, the time series
behavior of each factor can be derived from the factor analysis. The search then begins for
macro economic factors that exhibit the same time series behavior. Once macro economic
factors have been matched up with the unnamed factors in the factor analysis, the betas of
each asset are re-estimated against the identified macro economic factors. The beta estimation
may be done by running a multiple regression of stock returns (for each stock) against
changes in macro economic variables (such as interest rates, inflation rates and GNP growth)
over time. The coefficients on these regressions yield the betas, and risk premiums can be
estimated also from the historical data.
• Building a Regression Model
Generally, regression models begin with the cross sectional differences in returns across
stocks at any point in time, and try to explain these differences using differences on
measurable financial characteristics of the firms issuing these assets. As an example, Fama
and French, in their much quoted study, used differences in market capitalization and price to
book ratios to explain differences in returns across stocks.
The more difficult question is deciding which financial variables to use in explaining returns.
The best place to start is to look at the empirical evidence that has been accumulated over
time on market efficiency and the CAPM. This evidence suggests that
- Low market capitalization stocks seem to earn higher returns, on average, than high market
capitalization stocks
- Low PE, PBV and PS ratio stocks seem to earn higher returns, on average, than high PE,
PBV and PS ratio stocks
- High dividend yield stocks seem to earn higher returns, on average, than low dividend yield
stocks
While the initial regression may include all of these variables, many of these variables tend to
be correlated with each other. Thus, low PE stocks tend to also be low PBV ratio stocks which
pay high dividends. In the interests of efficiency (and to prevent problems in the regression
from independent variables being correlated with each other), it makes sense to use the
measure that is most highly correlated with returns and drop the others. Thus, the use of price
to book value ratios by Fama and French.
• Why not use bond betas to arrive at the cost of debt?
Given that we use stock betas to arrive at expected returns for stocks, the question may arise
as to why we do not use bond betas to get expected returns for bonds. The reason lies in the
absence or presence of symmetry in returns for each of these asset classes. Stocks, which have
potentially unlimited upside potential as well as significant downside potential, have much
more symmetric returns than bonds. Thus, they tend to fit in much more cleanly into the
mean-variance framework than do bonds.
Corporate bonds have some upside potential, but it is limited by the fact that bonds can at best
become default-free. Thus, the upside potential for a AA rated bond is fairly limited.
Consequently, the risk measure that we have to use has to be a downside risk measure, which
is what default risk and ratings measure. Clearly, the lower the rating of a bond, the greater
the upside potential, and thus, the greater the likelihood that we can estimate bond betas and
expected returns on them. For a junk bond, for instance, it may be possible to estimate a beta
like a stock beta and get an expected return from it.
• Credit Scores as Alternatives to Bond Ratings
Bond ratings are a tool that we use to measure default risk and arrive at a cost of debt.
Lenders (such as banks) have historically used credit scores as a measure of default risk,
especially when lending to individuals and private businessess. A credit score is derived by
measuring how a borrower scores on a variety of measures, which over time have been
correlated with default risk.
Morningstar
Operating Risk
In investing, as in life, risk comes in many guises. A journal entry I wrote in December,
which looked at some commonalties among great managers, touched briefly on the idea of
separating a company's operating risk from its stock's price risk. I find that breaking down risk
along those lines is a particularly helpful way of looking at stocks, so this journal entry takes a
closer look at the concept.
So what are operating risk and price risk? Operating risk is the risk to the company as a
business. That includes anything that might adversely affect the company's market or its
profitability (such as volatile raw-materials costs or rising labor costs). A high debt load
compared with industry or market averages would also make for higher operating risk because
it would magnify the bottom-line effects of a drop in demand. Basically, anything about the
business that makes its earnings less certain or more unstable qualifies as operating risk.
Price risk, on the other hand, has more to do with the stock than the business. There are
different ways to look for price risk, but the most common is probably looking at a stock's
price multiples, such as P/E or price/book, against that of the industry, the market, or any
other index that will yield a meaningful comparison. A stock whose multiples are
comparatively high carries more price risk than one whose multiples are lower.
When I look at a stock, I try and figure out how its operating risk balances its price risk. To
think why that might be important, imagine a company in a turnaround situation--that is, one
that carries a fair amount of operating risk. If the stock is expensive, I probably wouldn't be
interested because much of the potential improvement in earnings is already figured into the
price. And if those earnings don't materialize, the chances are good that the stock will tank.
But if the stock is cheap, I might be inclined to take a chance on those theoretical, risky future
earnings. The potential upside is great if the company comes through, and downside risk--or
the price risk--is comparatively limited.
I usually visualize the balance between operating and price risk as a four-box matrix, and I try
and fit stocks into one of the four categories: low operating risk and high price risk, high
operating risk and low price risk, low operating risk and low price risk, and high operating
risk and high price risk.
Low operating risk and high price risk The leading stocks in today's market mostly fit into
this box. These are the Coca-Colas, the Gillettes, the Microsofts--companies that have
advantages such as leading market position, brand-name franchises, and superior profitability
that enable them to post reliable earnings and growth, year in and year out, quarter after
quarter. Smaller or less well-known companies with good managements or strong market
niches can also end up here. These companies--though they occasionally post lower-thanexpected growth or weak earnings--have delivered over the long haul.
And the market has noticed. Clorox, for example, now commands 31 times its trailing
earnings, a P/E that's 50% higher than its average P/E over the past five years. In other words,
the market has raised the premium on its future earnings, in large part because the company
has been so reliable in the past. If these companies don't meet or exceed those high
expectations, these stocks could be in for a long period of sub-par returns until earnings catch
up with the prices. And any glitch in earnings--and every company has glitches, no matter
how steady it is over all--could send these stocks into a tailspin.
High operating risk and low price risk Cyclicals--with their reliably unreliable earnings-tend to crowd into this corner, though other companies, with less-than-ideal markets or
mediocre managements, fit as well. In large part, these stocks can make up for their lack of
earnings stability--a la the turnaround example above--or their choppy growth, with their low
price risk.
Low operating risk and low price risk These stocks are the Holy Grail of value investors.
Warren Buffett's stocks fit into this category when he bought them, before they moved into
the low operating risk, high price risk box. The big problem with this box is finding stocks
that fit into it. As you've probably realized by now, most stocks tend to congregate in the two
boxes above. Since the market is more or less efficient, it tends to exact a higher price for
higher quality and a lower price for lower quality.
High operating risk and high price risk This playground is pretty much off limits to all but
die-hard growth investors. Most often, hot start-ups and small tech plays end up here. These
are companies whose businesses are new, or are so small that they carry a lot of operating
risk. But they are also usually in popular or glamorous industries, which means that the
market has put a high price tag on those uncertain earnings. In order for their high-priced
stocks to do well, these companies need turn in some convincing evidence that they could be
the next Microsoft.
What's important is really the exercise of considering which of the above categories is more
appropriate to the stock and why. Because once you have a handle on that, you can compare
the stock to your investment style and decide if it fits. A stock that's risky because its
operations are unstable is a different kind of investment than one that's risky because it's
expensive. An investor who might consider one might not consider the other. It's a case of
investor know thyself and thy stock.
Questions:
1. This article draws a distinction between operating risk and price risk. Given the
distinction in chapter 3 between diversifiable and non-diversfiable risk, what does
operating risk measure? What does price risk measure?
2. The article then goes on to categorize firms based upon operating risk and price risk.
Two categories include firms with high (low) operating risk and low (high) price risk.
How would you explain the mismatch? Why might a firm with high (low) operating
risk have low (high) price risk?
3. If you were an investor who is not diversified, which of the four groups would you
invest in? Why or why not?
4. If you are an investor who is diversified, is this a useful categorization? If not, how
would you modify it?
Morningstar
Market Risk & Time
While the stock and bond markets can be risky in the short run, time has a moderating effect
on market risk. The longer you hold a stock or bond investment, the lower your chances of
losing money, and the greater the odds of earning a return close to the long-term average.
For example, a one-year investment in stocks has historically produced returns ranging from
+53.9% to -43.3%. Over ten-year periods, however, returns have varied from -0.9% per year
for the worst ten years to +20.1% per year for the best ten years.
Holding Period
1 Year
5 Years
10 Years
Range Of Returns on Stocks: 1926 to 1997
Best Return
Worst Return
+53.9 %
-43.3 %
+23.9 %
-12.5 %
+20.1 %
- 0.9 %
15 Years
20 Years
25 Years
+18.2 %
+16.9 %
+14.7 %
+ 0.6 %
+ 3.1 %
+ 5.9 %
As you can see, risk can be substantial over short periods. But over longer horizons, the
chance of losing money is substantially reduced.
The same principle applies to bonds, though bonds are less risky than stocks. For long-term
bonds, it takes ten years before returns are consistently positive; for shorter-maturity bonds,
about three years.
Of course, it should be noted that past performance in the stock and bond markets does not
necessarily predict future performance.
Questions:
1. Do you think that stocks become less risky, relative to bonds, as your time horizon is
lengthened? If so, why? If not, why not?
2. What are the implications of this article for the measures of risk and risk premiums
used in the risk and return models developed in chapter 3?
The Wall Street Journal Interactive Edition -- January 23, 1998
Investors Must Recall Risk, Investing's Four-Letter Word
By KAREN HUBE Staff Reporter of THE WALL STREET JOURNAL
What four letter word should pop into mind when the stock market takes a harrowing nose
dive? No, not those. R-I-S-K. [Media] Risk is the potential for realizing low returns or even
losing money, possibly preventing you from meeting important objectives, like sending your
kids to the college of their choice or having the retirement lifestyle you crave. But many
financial advisers and other experts say that these days investors aren't taking the idea of risk
as seriously as they should, and they are overexposing themselves to stocks. "The market has
been so good for years that investors no longer believe there's risk in investing," says Gary
Schatsky, a financial adviser in New York. "And when the market drops hundreds of points
and rebounds immediately, that belief is confirmed." The danger is that when the market
declines and stays down for months -- as some analysts predict it eventually will -- investors
won't be able to meet their short-term financial goals. Or, they will panic and sell their
investments as their shares are declining in value, which is the worst possible time. So before
the market goes down and stays down, be sure that you understand your tolerance for risk and
that your portfolio is designed to match it.
Assessing your risk tolerance, however, can be tricky. You must consider not only how much
risk you can afford to take but also how much risk you can stand to take. What you can afford
depends mainly on your time horizon -- how long before you will need the money. If your
investments are targeted for your child's college tuition in three years, for instance, your
financial ability to take on risk is low because you may not have time to recover if the value
of your portfolio declines. But if you have 10 years before your child heads to college, you
can afford to take more risks because you would have plenty of time to ride out dips in the
market. Determining how much risk you can stand -- your temperamental tolerance for risk -is more difficult. It isn't quantifiable. "A variety of behavioral factors come into play," says
Richard Bernstein, director of quantitative research at Merrill Lynch & Co. in New York. "If
my broker asks me if I want high-risk or low-risk securities, I may say high risk because I
don't want to look wimpy." Similarly, some people will gloss over the less impressive details
of their investing histories, says Ronald Roge, a financial adviser in Bohemia, N.Y. He
routinely asks to see copies of his clients' tax returns to get a reliable account of gains or
losses. "The aim is always to find the fine line between greed and fear," he says. To that end,
many financial advisers, brokerage firms and mutual-fund companies have created risk
quizzes to help people determine whether they are conservative, moderate or aggressive
investors. Some firms that offer such quizzes include Merrill Lynch, T. Rowe Price
Associates Inc., Baltimore, Zurich Group Inc.'s Scudder Kemper Investments Inc. of New
York, and Vanguard Group in Malvern, Pa. "The typical investor may not have ever
experienced a negative turn in the stock market. They need to be prepared for that," says
Robert Benish, vice president of education programs for Scudder, whose questionnaire is part
of a broader investing-education program. "We want to help them understand what risk means
to them." Typically, risk questionnaires include seven to 10 questions about a person's
investing experience, financial security and tendency to make risky or conservative choices.
Some of these risk tests, such as one created by Vanguard (www.vanguard.com), can be
found on-line. The benefit of the questionnaires is that they are an objective resource people
can use to get at least a rough idea of their risk tolerance. "It's impossible for someone to
assess their risk tolerance alone," says Mr. Bernstein. "I may say I don't like risk, yet will take
more risk than the average person." Many experts warn, however, that the questionnaires
should be used simply as a first step to assessing risk tolerance. "They are not precise," says
Ron Meier, a certified public accountant who teaches investing-related courses at the College
for Financial Planning in Denver, a unit of Apollo Group Inc., Phoenix. "They are good for
leading discussions but not for coming up with a final risk score."
The second step, many experts agree, is to ask yourself some difficult questions, such as: How
much you can stand to lose over the long term? "Most people can stand to lose a heck of a lot
temporarily," says Mr. Schatsky. The real acid test, he says, is how much of your portfolio's
value you can stand to lose over months or years. Rather than using percentages, think in
dollar terms. "When you convert percentages to figures, often you see very different
psychological effects," says Richard Wagener, a financial adviser in Columbia, Md. The idea
of a 20% decline on, say, a $150,000 portfolio is more abstract than a $30,000 loss, he says.
Financial adviser Glen Clemans of Portland, Ore., says he cites the 1972-1974 bear market to
help his clients realize how much they can lose. "I tell my clients to imagine they had put
$100,000 in the S? 500 in November 1972," says Mr. Clemans. He then explains that after the
first year, the investment would have been worth $88,000. "I watch to see how they flinch.
Then I tell them that the next year, their investment would have been worth $67,000, and ask
them 'So now what would you do?' " As it turns out, most people rank as middle-of-the-road
risk-takers, say Mr. Clemans and other advisers. "Only about 10% to 15% of my clients are
aggressive," says Mr. Roge. For some of the biggest risk-takers, particularly those who say
they like to gamble, Mr. Roge recommends creating an "action account" that includes no
more than 10% of their assets. Money in the account can be used to speculate in the stock
market. "Some investors just love the excitement of getting a stock pick from a brother-in-
law, even though they may lose money," says Mr. Roge. "People who need that kind of action
should have an outlet for it."
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