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How can one estimate the opportunity cost of capital for an "average-risk" project?
Over the past 73 years the return on the Standard & Poor's Composite Index of common stocks has averaged almost 9.4 percent a year higher than the return on safe Treasury bills. This is the risk premium that investors have received for taking on the risk of investing in stocks. Long-term bonds have offered a higher return than Treasury bills but less than stocks.
If the risk premium in the past is a guide to the future, we can estimate the expected return on the market today by adding that 9.4 percent expected risk premium to today's interest rate on Treasury bills. This would be the opportunity cost of capital for an average-risk project, that is, one with the same risk as a typical share of common stock.
How is the standard deviation of returns for individual common stocks or for a stock portfolio calculated?
The spread of outcomes on different investments is commonly measured by the variance or standard deviation of the possible outcomes. The variance is the average of the squared deviations around the average outcome, and the standard deviation is the square root of the variance. The standard deviation of the returns on a market portfolio of common stocks has averaged about 20 percent a year.
Why does diversification reduce risk?
The standard deviation of returns is generally higher on individual stocks than it is on the market. Because individual stocks do not move in exact lockstep, much of their risk can be diversified away. By spreading your portfolio across many investments you smooth out the risk of your overall position. The risk that can be eliminated through diversification is known as unique risk.
What is the difference between unique risk, which can be diversified away, and market risk, which cannot?
Even if you hold a well-diversified portfolio, you will not eliminate all risk. You will still be exposed to macroeconomic changes that affect most stocks and the overall stock market. These macro risks combine to create market risk—that is, the risk that the market as a whole will slump.
Stocks are not all equally risky. But what do we mean by a "high-risk stock"? We don't mean a stock that is risky if held in isolation; we mean a stock that makes an above-average contribution to the risk of a diversified portfolio. In other words, investors don't need to worry much about the risk that they can diversify away; they do need to worry about risk that can't be diversified. This depends on the stock's sensitivity to macroeconomic conditions.
Related web Links
www.financialengines.com Some good introductory material on risk, return, and inflation www.stern.nyu.edu/~adamodar/ This New York University site contains some historical data on market risk and return
Dow Jones Industrial Average Standard & Poor's Composite Index maturity premium risk premium variance standard deviation diversification unique risk market risk
Brealey-Myers: I III. Risk I 9. Introduction to Risk, I I © The McGraw-Hill
Fundamentals of Corporate Return, and the Opportunity Companies, 2001
Finance, Third Edition Cost of Capital chapter 9 Introduction to Risk, Return, and the Opportunity Cost of Capital 283
QUIZ 1. Rate of Return. A stock is selling today for $40 per share. At the end of the year, it pays a dividend of $2 per share and sells for $44. What is the total rate of return on the stock? What are the dividend yield and capital gains yield?
2. Rate of Return. Return to problem 1. Suppose the year-end stock price after the dividend is paid is $36. What are the dividend yield and capital gains yield in this case? Why is the dividend yield unaffected?
3. Real versus Nominal Returns. You purchase 100 shares of stock for $40 a share. The stock pays a $2 per share dividend at year-end. What is the rate of return on your investment for these end-of-year stock prices? What is your real (inflation-adjusted) rate of return? Assume an inflation rate of 5 percent.
4. Real versus Nominal Returns. The Costaguanan stock market provided a rate of return of 95 percent. The inflation rate in Costaguana during the year was 80 percent. In the United States, in contrast, the stock market return was only 14 percent, but the inflation rate was only 3 percent. Which country's stock market provided the higher real rate of return?
5. Real versus Nominal Returns. The inflation rate in the United States between 1950 and 1998 averaged 4.4 percent. What was the average real rate of return on Treasury bills, Treasury bonds, and common stocks in that period? Use the data in Self-Test 9.2.
6. Real versus Nominal Returns. Do you think it is possible for risk-free Treasury bills to offer a negative nominal interest rate? Might they offer a negative real expected rate of return?
7. Market Indexes. The accompanying table shows the complete history of stock prices on the Polish stock exchange for 9 weeks in 1991. At that time only five stocks were traded. Construct two stock market indexes, one using weights as calculated in the Dow Jones Industrial Average, the other using weights as calculated in the Standard & Poor's Composite Index.
Prices (in zlotys) for the first 9 weeks'trading on the Warsaw Stock Exchange, beginning in April 1991. There was one trading session per week Only five stocks were listed in the first 9 weeks.
Stock
Week |
Tonsil (Electronics) 1,500* |
Prochnik (Garments) 1,500* |
Krosno (Glass) 2,200* |
Exbud (Construction) 1,000* |
Kable (Electronics) 1,000* |
1 |
85 |
56 |
59.5 |
149 |
80 |
2 |
76.5 |
51 |
53.5 |
164 |
80 |
3 |
69 |
46 |
49 |
180 |
80 |
4 |
62.5 |
41.5 |
47 |
198 |
79.5 |
5 |
56.5 |
38 |
51.5 |
217 |
80 |
6 |
56 |
41.5 |
56.5 |
196 |
80 |
7 |
61.5 |
45.5 |
62 |
177 |
80 |
8 |
67.5 |
50 |
60 |
160 |
80.5 |
9 |
61 |
45.5 |
54 |
160 |
72.5 |
* Number of shares outstanding.
Source: We are indebted to Professor Mary M. Cutler for providing these data. Stock Market History.
* Number of shares outstanding.
Source: We are indebted to Professor Mary M. Cutler for providing these data. Stock Market History.
a. What was the average rate of return on large U.S. common stocks from 1926 to 1998?
b. What was the average risk premium on large stocks?
c. What was the standard deviation of returns on the S&P 500 portfolio?
Brealey-Myers: III. Risk 9. Introduction to Risk, © The McGraw-Hill
Fundamentals of Corporate Return, and the Opportunity Companies, 2001
Finance, Third Edition Cost of Capital
284 PART THREE Risk
10. Market Indexes. In 1990, the Dow Jones Industrial Average was at a level of about 2,600. In early 2000, it was about 10,000. Would you expect the Dow in 2000 to be more or less likely to move up or down by more than 40 points in a day than in 1990? Does this mean the market was riskier in 2000 than it was in 1990?
11. Maturity Premiums. Investments in long-term government bonds produced a negative average return during the period 1977-1981. How should we interpret this? Did bond investors in 1977 expect to earn a negative maturity premium? What do these 5 years' bond returns tell us about the normal future maturity premium?
12. Risk Premiums. What will happen to the opportunity cost of capital if investors suddenly become especially conservative and less willing to bear investment risk?
13. Risk Premiums and Discount Rates. You believe that a stock with the same market risk as the S&P 500 will sell at year-end at a price of $50. The stock will pay a dividend at year-end of $2. What price will you be willing to pay for the stock today? Hint: Start by checking today's 1-year Treasury rates.
14. Scenario Analysis. The common stock of Leaning Tower of Pita, Inc., a restaurant chain, will generate the following payoffs to investors next year:
Dividend Stock Price
Normal economy 2.00 100
The company goes out of business if a recession hits. Calculate the expected rate of return and standard deviation of return to Leaning Tower of Pita shareholders. Assume for simplicity that the three possible states of the economy are equally likely. The stock is selling today for $90.
15. Portfolio Risk. Who would view the stock of Leaning Tower of Pita (see problem 14) as a risk-reducing investment—the owner of a gambling casino or a successful bankruptcy lawyer? Explain.
16. Scenario Analysis. The common stock of Escapist Films sells for $25 a share and offers the following payoffs next year:
Dividend Stock Price
Boom 0 $18
Normal economy $1.00 26
Recession 3.00 34
9. Risk Premiums. Here are stock market and Treasury bill returns between 1994 and 1998:
Year |
S&P Return |
T-Bill Return |
1994 |
1.31 |
3.90 |
1995 |
37.43 |
5.60 |
1996 |
23.07 |
5.21 |
1997 |
33.36 |
5.26 |
1998 |
28.58 |
4.86 |
a. What was the risk premium on the S&P 500 in each year?
b. What was the average risk premium?
c. What was the standard deviation of the risk premium?
Calculate the expected return and standard deviation of Escapist. All three scenarios are equally likely. Then calculate the expected return and standard deviation of a portfolio half
Brealey-Myers: Fundamentals of Corporate Finance, Third Edition
III. Risk
9. Introduction to Risk, Return, and the Opportunity Cost of Capital
© The McGraw-Hill Companies, 2001
chapter 9 Introduction to Risk, Return, and the Opportunity Cost of Capital 285
invested in Escapist and half in Leaning Tower of Pita (from problem 14). Show that the portfolio standard deviation is lower than either stock's. Explain why this happens. 17. Scenario Analysis. Consider the following scenario analysis:
Rate of Return
Scenario Probability Stocks Bonds
Recession .20 -5% +14% Normal economy .60 +15 +8 Boom_.20_+25_+4_
a. Is it reasonable to assume that Treasury bonds will provide higher returns in recessions than in booms?
b. Calculate the expected rate of return and standard deviation for each investment.
c. Which investment would you prefer?
18. Portfolio Analysis. Use the data in the previous problem and consider a portfolio with weights of .60 in stocks and .40 in bonds.
a. What is the rate of return on the portfolio in each scenario?
b. What is the expected rate of return and standard deviation of the portfolio?
c. Would you prefer to invest in the portfolio, in stocks only, or in bonds only?
19. Risk Premium. If the stock market return in 2004 turns out to be -20 percent, what will happen to our estimate of the "normal" risk premium? Does this make sense?
20. Diversification. In which of the following situations would you get the largest reduction in risk by spreading your portfolio across two stocks?
a. The stock returns vary with each other.
b. The stock returns are independent.
c. The stock returns vary against each other.
21. Market Risk. Which firms of each pair would you expect to have greater market risk:
a. General Steel or General Food Supplies.
b. Club Med or General Cinemas.
22. Risk and Return. A stock will provide a rate of return of either -20 percent or +30 percent.
a. If both possibilities are equally likely, calculate the expected return and standard deviation.
b. If Treasury bills yield 5 percent, and investors believe that the stock offers a satisfactory expected return, what must the market risk of the stock be?
23. Unique versus Market Risk. Sassafras Oil is staking all its remaining capital on wildcat exploration off the Côte d'Huile. There is a 10 percent chance of discovering a field with reserves of 50 million barrels. If it finds oil, it will immediately sell the reserves to Big Oil, at a price depending on the state of the economy. Thus the possible payoffs are as follows:
Value of Reserves, Value of Reserves, Value of per Barrel 50 Million Barrels Dryholes
Boom
Normal economy Recession
$200,000,000 $250,000,000 $300,000,000
Is Sassafras Oil a risky investment for a diversified investor in the stock market—compared, say, to the stock of Leaning Tower of Pita, described in problem 14? Explain.
Brealey-Myers: I III. Risk I 9. Introduction to Risk, I I © The McGraw-Hill
Fundamentals of Corporate Return, and the Opportunity Companies, 2001
Finance, Third Edition Cost of Capital
286 PART THREE Risk
Solutions to
Self-Test
Questions
The bond price at the end of the year is $1,050. Therefore, the capital gain on each bond is $1,050 - 1,020 = $30. Your dollar return is the sum of the income from the bond, $80, plus the capital gain, $30, or $110. The rate of return is
Income plus capital gain =
Original price 1,020
Real rate of return is
1 + nominal return 1 + inflation rate
1.108 1.04
9.2 The risk premium on stocks is the average return in excess of Treasury bills. This was 14.7 - 5.2 = 9.5%. The maturity premium is the average return on Treasury bonds minus the return on Treasury bills. It was 6.4 - 5.2 = 1.2%.
+70% |
+60% |
3,600 |
+10 |
0 |
0 |
+10 |
0 |
0 |
-50 |
-60 |
3,600 |
Variance = average of squared deviations = 7,200/4 = 1,800
Standard deviation = square root of variance = V 1,800 = 42.4, or about 42%
Variance = average of squared deviations = 7,200/4 = 1,800
Standard deviation = square root of variance = V 1,800 = 42.4, or about 42%
9.4 The standard deviation should decrease because there is now a lower probability of the more extreme outcomes. The expected rate of return on the auto stock is now
The variance is
[.3 x (-8 - 5)2] + [.4 x (5 - 5)2] + [.3 x (18 - 5)2] = 101.4
The standard deviation is = 10.07 percent, which is lower than the value assuming equal probabilities of each scenario.
9.5 The gold mining stock's returns are more highly correlated with the silver mining company than with a car company. As a result, the automotive firm will offer a greater diversification benefit. The power of diversification is lowest when rates of return are highly correlated, performing well or poorly in tandem. Shifting the portfolio from one such firm to another has little impact on overall risk.
9.6 The success of this project depends on the experiment. Success does not depend on the performance of the overall economy. The experiment creates a diversifiable risk. A portfolio of many stocks will embody "bets" on many such unique risks. Some bets will work out and some will fail. Because the outcomes of these risks do not depend on common factors, such as the overall state of the economy, the risks will tend to cancel out in a well-diversified portfolio.
9.7 a. The luxury restaurant will be more sensitive to the state of the economy because expense account meals will be curtailed in a recession. Burger Queen meals should be relatively recession-proof.
b. The paint company that sells to the auto producers will be more sensitive to the state of the economy. In a downturn, auto sales fall dramatically as consumers stretch the lives of their cars. In contrast, in a recession, more people "do it themselves," which makes paint sales through small stores more stable and less sensitive to the economy.
Brealey-Myers: Fundamentals of Corporate Finance, Third Edition
III. Risk
10. Risk, Return, and Capital Budgeting
© The McGraw-Hill Companies, 2001
Brealey-Myers: Fundamentals of Corporate Finance, Third Edition
III. Risk
10. Risk, Return, and Capital Budgeting
© The McGraw-Hill Companies, 2001
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