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Practice Problems


Risk and Return: Diversification, Portfolio Selection and the Capital Asset Pricing Model


Short Questions:

  • What is the required rate of return on a lottery ticket according to the Capital Asset Pricing model, and why?
    Ans: The risk-free rate of return. The beta of the lottery ticket is zero, because all of the uncertainty on a lottery ticket's return is diversifiable.
  • Why does diversification reduce firm-specific risk?
    Ans: Firm specific risk is specific to that firm; this means that it is uncorrelated with firm-specific risk of other stocks.  As a result, if a portfolio contains many different stocks, some stocks have positive firm-specific shocks, while other stocks have negative firm-specific stocks, and the two will tend to cancel out, more or less.  This is why diversification reduces firm-specific risk. 
  • What are the two characteristics that a risk-free rate must have?  Illustrate your answer with reference to the 10 year Treasury bond.
    Ans: A risk-free rate must have two characteristics -- one, there must be no default risk; and two, there must be no reinvestment risk.  If we are talking about the 10-year Treasury bond, condition one is satisfied.  However, condition two is not satisfied perfectly, even if the investor has a ten-year risk horizon.  The reason for this is that the 10-year bond will have coupons that are paid before ten years are past, and they will need to be reinvested at a rate that is currently unknown.  Hence, even for an investor with a ten-year horizon, the yield on the ten-year T-bond is an imperfect measure of the risk-free rate.
  • True or false: Industry-specific risk cannot be diversified away.  Explain your answer; no credit without explanation.
    Ans: Some industry-specific risk is correlated with the market; that portion cannot be diversified away.  However, the portion that is uncorrelated with the market can be diversified away in a portfolio that is diversified across industries.
  • What is the only measure of asset risk that is considered relevant, in the CAPM?
    Ans: Beta risk.  This is because only non-diversifiable risk is relevant in the CAPM world, and the beta measures that.
  • When would you use a geometric average and when would you use an arithmetic average to estimate a historical market risk premium?
    Ans: You would use a geometric average if you were interested in a summary statistic regarding past performance.  However, if you wanted an estimate of the future one-year rate of return, an arithmetic average of past annual rates of return would be better.  Nevertheless, if you were interested in an estimate of a long-term rate of return, you would still use a geometric average. 
  • When would you use a bottom-up beta?
    Ans: You could use a bottom-up beta, if you did not have information about historical returns of the stock, as, for example, if the stock related to a new company, or if the stock had not been traded for a long time, but you felt that the underlying company was a typical one for the industry.
  • When would you use an accounting beta?
    Ans: You could use an accounting beta, if you had a sufficiently large history of accounting earnings for the company, but if you didn't have a history of stock returns.
  • How would you measure a beta for a new firm in a new industry?
    Ans: You could use scenario analysis, if the industry, itself, were new.  If you could make comparisons between this new industry and existing industries, you could use the betas of those existing industries as base estimates, as well.
  • If you know that a new firm uses more labor per unit of finished product than the average firm in the industry, how would you go about computing an adjusted beta, using the bottom-up method?
    Ans: You would use the bottom-up beta and then adjust downwards (since a higher amount of labor implies a lower operating leverage), in a way similar to the adjustment used with financial leverage that varies from company to company.
  • Two firms, A and B, are in the same industry.  Firm A has a higher financial leverage than firm B.  Which firm's stock will have a higher beta, A or B?  Why?
    Ans: All other things being the same, a firm with higher financial leverage will have a higher beta. The reason is that financial leverage increases the stock beta, for a given asset beta. Firms in the same industry will probably have the same asset beta.
  • If firm A has a higher stock beta than firm B, can you definitely conclude that firm A has higher financial leverage than firm B?  Why or why not?
    Ans: If both firms are in the same industry that may very well be true, but it is not necessarily so -- this is because not all firms in an industry have the same asset beta. Firms not in the same industry need not have the same asset beta, or the same ability to support debt; hence we cannot make any conclusions regarding a firm's financial leverage from its stock beta.
  • A firm with higher idiosyncratic risk will necessarily have a higher beta.  True or False?
    Ans: If anything, the opposite; however, logically the two are not so closely connected. Higher idiosyncratic risk is diversifiable risk, and only non-diversifiable risk contributes to beta.
  • If the beta of a stock is close to 1, then its volatility will be similar to that of the market.  True or False?
    Ans: If the beta is one, then the variance of the non-diversifiable component will be similar.  However, you still have the diversfiable component.  Hence the answer is -- no.  
  • We usually estimate the variance of an asset's return by looking at historical (past) returns.  Under what conditions may this approach not be appropriate for estimating variance for use in a risk-and-return model?
    Ans: This would not be appropriate if, for any reason, the analyst feels that the current situation of the firm is not comparable to its historical experience.  For example, if the firm has radically changed its product line, or its operating or financial strategy, historical betas may not have any relevance to current beta.  For example, IBM, according to some, has reinvented itself as an innovative firm; if this is true, then its historical beta may be more stable (closer to one) than its current beta.
  • A stock with a high return variance will have a higher beta than one with lower return variance.  True or False?
    Ans: This is not necessarily true because the higher return variance stock could have more diversifiable risk.


  • Marginal Investor
  • Free Cash Flow to Equity
  • hurdle rate
  • cost of capital
  • cost of equity
  • cost of debt
  • implied market risk premium
  • historical risk premium
  • geometric average
  • arithmetic average
  • beta
  • Jensen's alpha
  • market portfolio
  • operating leverage
  • fundamental beta
  • asset beta
  • unlevered beta
  • levered beta
  • equity beta
  • bottom-up beta
  • accounting beta
  • synthetic rating
  • preferred stock
  • convertible bond
  • hybrid security
  • standard error of the beta estimate

Problem 1. (solution) A. Assume that investors' horizons are exactly 1 year. Hence assume that the CAPM describes asset returns over 1 year holding periods. Suppose a 10 year zero-coupon sells for $295 and is known to have a beta of 0.15. Suppose further that the expected rate of return on the market is 25% per annum. What do you expect a 9 year bond to cost next year if the current yield on a 1 year bond is 5%?

The following data apply to questions B through D:

Hennessey & Associates manages a $30 million equity portfolio for the multimanager Wilstead Pension Fund. Jason Jones, financial vice president of Wilstead, noted that Hennessy had rather consistently achieved the best record among the Wilstead's six equity managers. Performance of the Hennessy portfolio had been clearly superior to that of the S&P 500 in 4 of the past 5 years. In the one less favorable year, the shortfall was trivial.

Hennessy is a "bottom-up" manager. The firm largely avoids any attempt to "time the market." It also focuses on selection of individual stocks, rather than the weighting of favored industries.

There is no apparent conformity of style among the six equity managers. The five managers, other than Hennessy, manage portfolios aggregating $250 million made of more than 150 individual issues.

Jones is convinced that Hennessy is able to apply superior skill to stock selection, but the favorable results are limited by the high degree of diversification in the portfolio. Over the years, the portfolio generally held 40 to 50 stocks, with about 2% to 3% of total funds committed to each issue. The reason Hennessy seemed to do well most years was because the firm was able to identify each year 10 or 12 issues which registered particularly large gains.

Based on this overview, Jones outlined the following plan to the Wilstead pension committee:

"Let's tell Hennessy to limit the portfolio to no more than 20 stocks. Hennessy will double the commitments to the stocks that it really favors, and eliminate the remainder. Except for this one new restriction, Hennessy should be free to manage the portfolio exactly as before."

All the members of the pension committee generally supported Jones' proposal, because all agreed that Hennessy had seemed to demonstrate superior skill in selecting stocks. Yet, the proposal was a considerable departure from previous practice, and several committee members raise questions.

Respond to each of these questions, using no more than 5 lines each:

B. Answer the following:

a. Will the limitation of 20 stocks likely increase or decrease the risk of the portfolio? Explain.

b. Is there any way Hennessy could reduce the number of issues from 40 to 20 without significantly affecting risk? Explain.

C. One committee member was particularly enthusiastic concerning Jones' proposal. He suggested that Hennessy's performance might benefit further from reduction in the number of issues to 10. If the reduction to 20 could be expected to be advantageous, explain why reduction to 10 might be less likely to be advantageous. (Assume that Wilstead will evaluate the Hennessy portfolio independently of the other portfolios in the fund.)

D. Another committee member suggested that, rather than evaluate each managed portfolio independently of other portfolios, it might be better to consider the effects of a change in the Hennessy portfolio on the total fund. Explain how this broader point of view could affect the committee decision to limit the holdings in the Hennessy portfolio to either 10 or 20 issues.

E. Consider the following table, which gives a security analyst's expected return on two stocks for two particular market returns:

Market Return

Aggressive Stock

Defensive Stock







  1. What are the betas of the two stocks?
  2. What is the expected rate of return on each stock if the market return is equally likely to be 5% or 20%? For the purpose of this answer, do not assume that the CAPM holds.
  3. If the T-bill rate is 8% and the market return is equally likely to be 5% or 20%, draw the SML for this economy clearly marking the axes, the intercepts and the slope.
  4. Plot the two securities on the SML graph. If we define the alpha of a security as the excess of the actual expected return over the CAPM required return, what are the alphas of each?
  5. What hurdle rate should be used by the management of the aggressive firm for a project with the risk characteristics of the defensive firm's stock?

Note: SML stands for Security Market Line. This line is typically drawn with the expected return on assets on the y-axis, and the asset beta on the x-axis.

The expected return should be equal to the required rate of return, but this equality will hold only if the market is in equilibrium.

Problem 2. (Fall 1999) Read the following article and answer the following questions:

 AHP, Warner-Lambert Discuss Merger --- Transaction for $65 Billion Could Spur Consolidation Among Drug Companies
The Wall Street Journal - 11/03/1999
By Robert Langreth and Steven Lipin

American Home Products Corp. and Warner-Lambert Co. are in talks to merge in a $65 billion deal that would unite two of the largest pharmaceutical companies in the world, according to people familiar with the situation.

Such a deal -- which would be the largest drug merger in history and one of the largest transactions ever -- could trigger a new wave of consolidation in what remains a relatively fragmented industry, despite some major deals in recent years.

Indeed, combining Warner-Lambert, of Morris Plains, N.J., with American Home, located just down the road in Madison, would bring together the companies behind such household names as Advil, Anacin and Chap Stick (American Home) and Dentyne gum and Certs breath mints (Warner-Lambert). American Home also makes Premarin, a fast-selling drug for menopause and osteoporosis, and Warner-Lambert manufactures blockbuster cholesterol drug Lipitor.

An announcement could come as soon as tomorrow, though, as is often the case with delicate merger talks, the discussions could fall through at the last minute.

Spokesmen for American Home and Warner-Lambert declined to comment.

The talks come at a time when American Home's stock has been hit by a series of product and legal-related setbacks. American Home recently took a $4.75 billion charge to settle thousands of lawsuits related to the diet drugs Redux and Pondimin. American Home also is restructuring its struggling Cyanamid agricultural unit.

While its share price has perked up a bit in recent weeks, American Home has been concerned about the possibility of a hostile overture by a rival drug company, according to people close to the company. It recently adopted a "poison pill" shareholder-rights plan.

In New York Stock Exchange trading at 4 p.m., American Home shares fell 43.75 cents to $50.375. Warner-Lambert shares fell $1.5625 to $78.4375, also on the Big Board.

American Home is poised to launch some promising new drugs, including a vaccine for pneumonia in children, but some of its strongest existing sellers, like Premarin, are aging and a deal with Warner-Lambert would greatly expand its portfolio of medicines.  As for Warner-Lambert, Wall Street analysts have expressed concerns about whether it has enough potentially profitable new drugs in its pipeline, though profit growth has been robust because of Lipitor's success.

American Home has looked for a merger partner in the recent past. Just last year, it announced a deal to merge with Monsanto Co. and held merger talks with SmithKline Beecham PLC. Both sets of talks fell through, partly because of issues about who would run the combined firm.

American Home's chairman, John Stafford, 62 years old, is expected to be chairman of the combined firm, and WarnerLambert's chairman, Lodewijk J.R. de Vink, who is nearly a decade younger, is likely to be chief executive, these people said. Under a scenario being discussed, the board of the combined company would be split evenly between the two sides.

American Home and Warner Lambert shareholders each are expected to end up with about 50% of the stock of the combined company. Under the structure being considered, AHP shares would be used to acquire the shares of Warner-Lambert.

The headquarters is expected to be at American Home's base in Madison.                

American Home Products

Warner Lambert

-- Headquarters: Madison, N.J.
-- CEO: John R. Stafford
-- 1998 Revenue: $13.5 billion
-- 1998 Net income: $2.5 billion
-- Market cap: $65.8 billion
-- Business: Pharmaceutical and health-care products; vaccines and other biotechnology products; agricultural- and animal-health products
-- Some brands: Advil, Anacin, Centrum, Chap Stick, Preparation H, Premarin

-- Headquarters: Morris Plains, N.J.
-- CEO: Lodewijk J.R. de Vink
-- 1998 Revenue: $10.2 billion
-- 1998 Net income: $1.3 billion
-- Market cap: $67.0 billion
-- Business: Pharmaceutical and health-care products
-- Some brands: Lipitor, Schick, Halls, Listerine, Dilantin, Trident, Sudafed, Benadryl, Certs, Lubriderm, Neurontin

 Here's some additional information from Yahoo (http://biz.yahoo.com/p/a/ahp.html) and (http://biz.yahoo.com/p/w/wla.html).


American Home Products

Warner Lambert

Stock Beta



Long-Term Debt/Equity Ratio



a)       (20 points) What are the unlevered betas of American Home Products (AHP)and Warner-Lambert (WLA)?
b)      (20 points) Estimate the beta of the stock of the merged company, assuming that the merged company does not change any operating policies.  Assume a tax rate of 40%.  Ignore the minor differences in the sizes of the two companies.
c)      (5 points; bonus) How do you think the beta of the merged company will actually change, following the merger?  Explain your answer.
d)      (20 points) Estimate the one-year expected return on the two stocks.  You have access to the following additional information:

  1. The current yield on 3-month T-bills is 5.103% (Source: http://www.bloomberg.com/markets/iyc.html)

  2. The arithmetic average return on the US stock market in excess of the 3-month T-bill rate, computed over the period 1926-1990 is 8.41%.

  3.  The arithmetic average return on the US stock market in excess of the yield on the 30-year Treasury bond, computed over the period 1926-1990 is 7.24%

  4.  The geometric average return on the US stock market in excess of the 3-month T-bill rate, computed over the period 1926-1990 is 6.41%.

  5. The geometric average return on the US stock market in excess of the yield on the 30-year Treasury bond, computed over the period 1926-1990 is 5.50%.  (Source for the data in items b. through e. above: Damodaran, Corporate Finance: Theory and Practice, p. 126)

e) (5 points; bonus) The actual average returns on AHP (using data for the last five years from http://chart.yahoo.com) is 30.25% p.a. and that on WLA is 42.61%.  Compare these numbers to your answers from d) above.  How would you explain the discrepancy, if any?

f)   (10 points) What do you think the correlation coefficient between the stock returns of the two companies would be?  Provide a numerical estimate and justify your answer.

g)    (20 points) Use your estimate in f) above to compute the variance of returns on a portfolio consisting of $10,000 invested in American Health Products and $20,000 in Warner-Lambert (assuming that the merger does not go through).  The following additional information is available:  The standard deviation of returns on WLA stock is 93.32% per year, computed using stock-split and dividend adjusted return data from Yahoo (http://chart.yahoo.com) for the last five years.  The same number for AHP is 93.76%.  (If you have not been able to answer part e), you can use any arbitrary figure for the correlation, other than zero.)

h)      (10 points) Compute the expected return on the portfolio in g) above, using your computations from d) above.  

Problem 3. (Spring 1999): Read the following WSJ article from March 15, 1999 and answer this question, using no more than two sides of a page.  Rambling answers will be penalized.

  • Why should earnings-protection insurance not make sense from a shareholder’s point of view?

  • Bonus: (no more than half of one side): What arguments can you marshal for the idea that firms should hedge against return (not earnings) volatility?

Manager's Journal: Reducing Risk Doesn't Pay Off

Reliance National announced recently that it will soon begin offering "earnings-protection insurance." That  is, it will write insurance policies to cover companies against lower-than-expected earnings resulting from  uncontrollable events ranging from a product flop, to a supplier that goes bust, to a key customer who  defects.

The idea behind earnings insurance, like other forms of insurance, is to reduce risk. Earnings insurance is  only the latest strategy available to risk-averse managers; in the past they have formed conglomerates and  employed derivatives for the same purpose. On the surface, such strategies would seem to make sense.  After all, stockholders dislike risk. In the stock market, the essence of risk is the volatility of returns.  Thus, it would seem that if a company can smooth out its earnings through various forms of risk  management, it will keep the stockholders happy and the stock price high. Why didn't somebody think of  this before? There's got to be a catch.

Indeed there is. Stockholders don't care if earnings are hurt by uncontrollable events. At least diversified  stockholders don't care. And because it is irrational not to diversify, we don't really need to worry about  stockholders who don't diversify. A diversified stockholder -- one who has spread his money across 20 or  more different stocks -- has effectively eliminated the risk that goes with investing in any individual  company, without any sacrifice of return. For every company that underperforms compared with  expectations, there will almost certainly be one that overperforms.

If a rational investor has already avoided company-specific risk through diversification, how would he  view the news that a company in which he owns stock has bought an insurance policy covering the same  risk? Not favorably. From the point of view of the diversified shareholder, earnings insurance is a waste  of money.

This is not to say that there will not be takers for such insurance. Management may think that the  stockholders care. The problem is that a manager whose compensation is tied to earnings may have an  ulterior motive. It is obvious why he might see earnings insurance as a good idea. Fortunately for  shareholders, few managers have their incentive compensation tied to earnings these days. Most receive  stock options and therefore should not be much interested in any financial gimmick that might adversely  affect stock price.

Earnings insurance has other problems, too. For one, smart stockholders would not view the proceeds of  an insurance policy as the dollar-for-dollar equivalent of the same amount of earnings. Earnings provide  shareholders not only with money, but also with information about how well a company is managed. If a  disaster happens once, it can happen again. Stockholders will perceive more risk even if the insurance  company ponies up the difference. The price of the stock will drop, insurance or no insurance.

What's more, earnings insurance will not cover the one risk that stockholders might like to see covered. It  won't pay if the shortfall is due to fraud or other wrongdoing by management. This is a risk that can't be  diversified away. Fraud always hurts; there are no gainers to balance out the losers. That's why the courts  waive the business-judgment rule, which ordinarily protects management from judicial second-guessing, if  there is self-dealing or a similar wrong.

Earnings insurance as a way of hedging risk is part of a trend that goes back decades. The primary  motivation for the conglomerate mergers of the 1960s and 1970s was earnings management. The idea was  that assembling an array of different companies under one umbrella would yield a smooth earnings stream  at the holding-company level. But stockholders soon discovered that they could diversify much more  cheaply and easily by adjusting the stocks in their portfolio or buying shares in a mutual fund.

Conglomerate stocks thus fell into disfavor. Why buy prepackaged diversification in the form of a  conglomerate, in which the CEO has no substantive focus, when you can roll your own portfolio and  adjust the mix of business with a simple phone call to your broker? In the end, the conglomerate mergers  of the '60s and '70s became the targets of the bust-up takeovers of the '80s. Nor did the trend end then:  Last week's breakup of RJR Nabisco came after its stock had languished for a decade since the 1988  merger.

More recently, companies have turned to various derivative instruments to manage risk. Aside from the  well-publicized losses of Procter & Gamble, Gibson Greetings and others from derivatives gone wrong,  what do derivatives do for stockholders even when they work as intended? Not much. As far as a  diversified stockholder is concerned, company-specific risks from interest rate fluctuations, currency  translation, or volatile commodity prices all come out in the wash. Some companies win, some lose. Only  the average matters.

A few companies seem to recognize that stockholders dislike risk management. Homestake Mining, for  example, generally does not attempt to hedge with gold futures. Rather it allows the risk of changing gold  prices to pass through to its investors who may hedge either by owning a stock of a gold user or through  options.

First conglomerate mergers, then derivatives -- earnings insurance is more of the same. Smart  stockholders won't like it. Sell short.   


Problem 4: (Spring 1999) Here are the prices and betas, as of March 17, 1999 for three stocks that I own: 


Price per share

Betas (as reported by Yahoo Finance -- http://biz.yahoo.com/p/)

Coca Cola Co (KO)



Ascend Communications (ASND)



MBIA  Inc. (MBI)



 a.       If I have 100 shares of each company in my portfolio, compute the portfolio proportions for each stock. 

You have the following additional information on these stocks’ standard deviations (in percentages, on diagonal) and correlation coefficients (off-diagonal numbers):

















 b.      Compute the standard deviation of returns on your portfolio.

c.       If the expected return on the market portfolio is 15%, and the yield on the one-year T-bill is 5%, what is the required rate of return on your portfolio?

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