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We noted earlier the research by Fama and French showing that stocks of small firms and those with a high book-to-market ratio have provided above-average returns. This could simply be a coincidence. But there is also evidence that these factors are related to company profitability and therefore may be picking up risk factors that are left out of the simple CAPM.28
If investors do demand an extra return for taking on exposure to these factors, then we have a measure of the expected return that looks very much like arbitrage pricing theory:
r rf b market (rmarket factor ) + bsize (rsize factor ) + bbook-to-market (rbook-to-market factor)
This is commonly known as the Fama-French three-factor model. Using it to estimate expected returns is exactly the same as applying the arbitrage pricing theory. Here's an example.29
Step 1: Identify the Factors Fama and French have already identified the three factors that appear to determine expected returns. The returns on each of these factors are
27This estimate rests on risk premiums actually earned from 1978 to 1990, an unusually rewarding period for common stock investors. Estimates based on long-run market risk premiums would be lower. See E. J. Elton, M. J. Gruber, and J. Mei, "Cost of Capital Using Arbitrage Pricing Theory: A Case Study of Nine New York Utilities," Financial Markets, Institutions, and Instruments 3 (August 1994), pp. 46-73.
28E. F. Fama and K. R. French, "Size and Book-to-Market Factors in Earnings and Returns," Journal of Finance 50 (1995), pp. 131-155.
29The example is taken from E. F. Fama and K. R. French, "Industry Costs of Equity," Journal of Financial Economics 43 (1997), pp. 153-193. Fama and French emphasize the imprecision involved in using either the CAPM or an APT-style model to estimate the returns that investors expect.
CHAPTER 8 Risk and Return 209
Factor |
Measured by |
Market factor Size factor Book-to-market factor |
Return on market index minus risk-free interest rate Return on small-firm stocks less return on large-firm stocks Return on high book-to-market-ratio stocks less return on low book-to-market-ratio stocks |
Step 2: Estimate the Risk Premium for Each Factor Here we need to rely on history. Fama and French find that between 1963 and 1994 the return on the market factor averaged about 5.2 percent per year, the difference between the return on small and large capitalization stocks was about 3.2 percent a year, while the difference between the annual return on stocks with high and low book-to-market ratios averaged 5.4 percent.30
Step 3: Estimate the Factor Sensitivities Some stocks are more sensitive than others to fluctuations in the returns on the three factors. Look, for example, at the first three columns of numbers in Table 8.5, which show some estimates by Fama and French of factor sensitivities for different industry groups. You can see, for example, that an increase of 1 percent in the return on the book-to-market factor reduces the return on computer stocks by .49 percent but increases the return on utility stocks by .38 percent.31
Factor Sensitivities |
CAPM | ||||
Expected Risk |
Expected Risk | ||||
^market |
bsize |
b bbook-to-market |
Premium* |
Premium | |
Aircraft |
1.15 |
.51 |
.00 |
7.54% |
6.43% |
Banks |
1.13 |
.13 |
.35 |
8.08 |
5.55 |
Chemicals |
1.13 |
-.03 |
.17 |
6.58 |
5.57 |
Computers |
.90 |
.17 |
-.49 |
2.49 |
5.29 |
Construction |
1.21 |
.21 |
-.09 |
6.42 |
6.52 |
Food |
.88 |
-.07 |
-.03 |
4.09 |
4.44 |
Petroleum & gas |
.96 |
-.35 |
.21 |
4.93 |
4.32 |
Pharmaceuticals |
.84 |
-.25 |
-.63 |
.09 |
4.71 |
Tobacco |
.86 |
-.04 |
.24 |
5.56 |
4.08 |
Utilities |
.79 |
-.20 |
.38 |
5.41 |
3.39 |
TABLE 8.5 | |||||
Estimates of industry risk premiums |
using the Fama- |
French three-factor |
model and the CAPM. |
*The expected risk premium equals the factor sensitivities multiplied by the factor risk premiums, that is, (bmarket X 5.2 ) + (bslze X 3.2 ) + (bbook-to-market X 5.4).
Source: E. F. Fama and K. R. French, "Industry Costs of Equity," Journal of Financial Economics 43 (1997), pp. 153-193.
*The expected risk premium equals the factor sensitivities multiplied by the factor risk premiums, that is, (bmarket X 5.2 ) + (bslze X 3.2 ) + (bbook-to-market X 5.4).
Source: E. F. Fama and K. R. French, "Industry Costs of Equity," Journal of Financial Economics 43 (1997), pp. 153-193.
30We saw earlier that over the longer period 1928-2000 the average annual difference between the returns on small and large capitalization stocks was 3.1 percent. The difference between the returns on stocks with high and low book-to-market ratios was 4.4 percent.
31A 1 percent return on the book-to-market factor means that stocks with a high book-to-market ratio provide a 1 percent higher return than those with a low ratio.
Once you have an estimate of the factor sensitivities, it is a simple matter to multiply each of them by the expected factor return and add up the results. For example, the fourth column of numbers shows that the expected risk premium on computer stocks is r — rf = (.90 X 5.2) + (.17 X 3.2) — (.49 X 5.4) = 2.49 percent. Compare this figure with the risk premium estimated using the capital asset pricing model (the final column of Table 8.5). The three-factor model provides a substantially lower estimate of the risk premium for computer stocks than the CAPM. Why? Largely because computer stocks have a low exposure (—.49) to the book-to-market factor.
yuMMARy
The basic principles of portfolio selection boil down to a commonsense statement that investors try to increase the expected return on their portfolios and to reduce the standard deviation of that return. A portfolio that gives the highest expected return for a given standard deviation, or the lowest standard deviation for a given expected return, is known as an efficient portfolio. To work out which portfolios are efficient, an investor must be able to state the expected return and standard deviation of each stock and the degree of correlation between each pair of stocks.
Investors who are restricted to holding common stocks should choose efficient portfolios that suit their attitudes to risk. But investors who can also borrow and lend at the risk-free rate of interest should choose the best common stock portfolio regardless of their attitudes to risk. Having done that, they can then set the risk of their overall portfolio by deciding what proportion of their money they are willing to invest in stocks. The best efficient portfolio offers the highest ratio of forecasted risk premium to portfolio standard deviation.
For an investor who has only the same opportunities and information as everybody else, the best stock portfolio is the same as the best stock portfolio for other investors. In other words, he or she should invest in a mixture of the market portfolio and a risk-free loan (i.e., borrowing or lending).
A stock's marginal contribution to portfolio risk is measured by its sensitivity to changes in the value of the portfolio. The marginal contribution of a stock to the risk of the market portfolio is measured by beta. That is the fundamental idea behind the capital asset pricing model (CAPM), which concludes that each security's expected risk premium should increase in proportion to its beta:
Expected risk premium r — rf beta X market risk premium p(rm — rf)
The capital asset pricing theory is the best-known model of risk and return. It is plausible and widely used but far from perfect. Actual returns are related to beta over the long run, but the relationship is not as strong as the CAPM predicts, and other factors seem to explain returns better since the mid-1960s. Stocks of small companies, and stocks with high book values relative to market prices, appear to have risks not captured by the CAPM.
The CAPM has also been criticized for its strong simplifying assumptions. A new theory called the consumption capital asset pricing model suggests that security risk reflects the sensitivity of returns to changes in investors' consumption.
CHAPTER 8 Risk and Return 211
This theory calls for a consumption beta rather than a beta relative to the market portfolio.
The arbitrage pricing theory offers an alternative theory of risk and return. It states that the expected risk premium on a stock should depend on the stock's exposure to several pervasive macroeconomic factors that affect stock returns:
Expected risk premium = b1(rfactor 1 - rf) + b2(rfactor 2 - tf) + •••
Here b's represent the individual security's sensitivities to the factors, and rfactor - f is the risk premium demanded by investors who are exposed to this factor.
Arbitrage pricing theory does not say what these factors are. It asks for economists to hunt for unknown game with their statistical tool kits. The hunters have returned with several candidates, including unanticipated changes in
• The level of industrial activity.
• The spread between short- and long-term interest rates. Fama and French have suggested three different factors:
• The return on the market portfolio less the risk-free rate of interest.
• The difference between the return on small- and large-firm stocks.
• The difference between the return on stocks with high book-to-market ratios and stocks with low book-to-market ratios.
In the Fama-French three-factor model, the expected return on each stock depends on its exposure to these three factors.
Each of these different models of risk and return has its fan club. However, all financial economists agree on two basic ideas: (1) Investors require extra expected return for taking on risk, and (2) they appear to be concerned predominantly with the risk that they cannot eliminate by diversification.
The pioneering article on portfolio selection is:
H. M. Markowitz: "Portfolio Selection," Journal of Finance, 7:77-91 (March 1952).
There are a number of textbooks on portfolio selection which explain both Markowitz's original theory and some ingenious simplified versions. See, for example:
E. J. Elton and M. J. Gruber: Modern Portfolio Theory and Investment Analysis, 5th ed., John Wiley & Sons, New York, 1995.
Of the three pioneering articles on the capital asset pricing model, Jack Treynor's has never been published. The other two articles are:
W. F. Sharpe: "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk," Journal of Finance, 19:425-442 (September 1964).
J. Lintner: "The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets," Review of Economics and Statistics, 47:13-37 (February 1965).
The subsequent literature on the capital asset pricing model is enormous. The following book provides a collection of some of the more important articles plus a very useful survey by Jensen:
M. C. Jensen (ed.): Studies in the Theory of Capital Markets, Frederick A. Praeger, Inc., New York, 1972.
FURTHER READING
PART II Risk
The two most important early tests of the capital asset pricing model are:
E. F. Fama and J. D. MacBeth: "Risk, Return and Equilibrium: Empirical Tests," Journal of Political Economy, 81:607-636 (May 1973).
F. Black, M. C. Jensen, and M. Scholes: "The Capital Asset Pricing Model: Some Empirical Tests," in M. C. Jensen (ed.), Studies in the Theory of Capital Markets, Frederick A. Praeger, Inc., New York, 1972.
For a critique of empirical tests of the capital asset pricing model, see:
R. Roll: "A Critique of the Asset Pricing Theory's Tests; Part I: On Past and Potential Testability of the Theory," Journal of Financial Economics, 4:129-176 (March 1977).
Much of the recent controversy about the performance of the capital asset pricing model was prompted by Fama and French's paper. The paper by Black takes issue with Fama and French and updates the Black, Jensen, and Scholes test of the model:
E. F. Fama and K. R. French: "The Cross-Section of Expected Stock Returns," Journal of Finance, 47:427-465 (June 1992).
F. Black, "Beta and Return," Journal of Portfolio Management, 20:8-18 (Fall 1993).
Breeden's 1979 article describes the consumption asset pricing model, and the Breeden, Gibbons, and Litzenberger paper tests the model and compares it with the standard CAPM:
D. T. Breeden: "An Intertemporal Asset Pricing Model with Stochastic Consumption and Investment Opportunities," Journal of Financial Economics, 7:265-296 (September 1979).
D. T. Breeden, M. R. Gibbons, and R. H. Litzenberger: "Empirical Tests of the Consumption-Oriented CAPM," Journal of Finance, 44:231-262 (June 1989).
Arbitrage pricing theory is described in Ross's 1976 paper.
S. A. Ross: "The Arbitrage Theory of Capital Asset Pricing," Journal of Economic Theory, 13:341-360 (December 1976).
The most accessible recent implementation of APT is:
E. J. Elton, M. J. Gruber, and J. Mei, "Cost of Capital Using Arbitrage Pricing Theory: A Case Study of Nine New York Utilities," Financial Markets, Institutions, and Instruments, 3:46-73 (August 1994).
For an application of the Fama-French three-factor model, see:
E. F. Fama and K. R. French, "Industry Costs of Equity," Journal of Financial Economics, 43:153-193 (February 1997).
QUIZ
1. Here are returns and standard deviations for four investments.
Return |
Standard Deviation | |
Treasury bills |
6% |
0% |
Stock P |
10 |
14 |
Stock Q |
14.5 |
2B |
Stock R |
21.0 |
26 |
Calculate the standard deviations of the following portfolios.
a. 50 percent in Treasury bills, 50 percent in stock P.
b. 50 percent each in Q and R, assuming the shares have
• perfect positive correlation
• perfect negative correlation
• no correlation
CHAPTER 8 Risk and Return
FIGURE 8.13
See Quiz Question 3.
FIGURE 8.13
See Quiz Question 3.
c. Plot a figure like Figure 8.4 for Q and R, assuming a correlation coefficient of .5.
d. Stock Q has a lower return than R but a higher standard deviation. Does that mean that Q's price is too high or that R's price is too low?
2. For each of the following pairs of investments, state which would always be preferred by a rational investor (assuming that these are the only investments available to the investor):
a. Portfolio A r = 18 percent a = 20 percent Portfolio B r = 14 percent a = 20 percent b. Portfolio C r = 15 percent a = 18 percent Portfolio D r = 13 percent a = 8 percent c. Portfolio E r = 14 percent a = 16 percent Portfolio F r = 14 percent a = 10 percent
3. Figures 8.13a and 8.13b purport to show the range of attainable combinations of expected return and standard deviation.
a. Which diagram is incorrectly drawn and why?
b. Which is the efficient set of portfolios?
c. If rf is the rate of interest, mark with an X the optimal stock portfolio.
4. a. Plot the following risky portfolios on a graph:
Portfolio | ||||||||
A |
B |
C |
D |
E |
F |
G |
H | |
Expected return (r), % |
10 |
12.5 |
15 |
16 |
17 |
18 |
18 |
20 |
Standard deviation (ct), % |
23 |
21 |
25 |
29 |
29 |
32 |
35 |
45 |
b. Five of these portfolios are efficient, and three are not. Which are inefficient ones?
c. Suppose you can also borrow and lend at an interest rate of 12 percent. Which of the above portfolios is best?
d. Suppose you are prepared to tolerate a standard deviation of 25 percent. What is the maximum expected return that you can achieve if you cannot borrow or lend?
e. What is your optimal strategy if you can borrow or lend at 12 percent and are prepared to tolerate a standard deviation of 25 percent? What is the maximum expected return that you can achieve?
5. How could an investor identify the best of a set of efficient portfolios of common stocks? What does "best" mean? Assume the investor can borrow or lend at the risk-free interest rate.
6. Suppose that the Treasury bill rate is 4 percent and the expected return on the market is 10 percent. Use the betas in Table 8.2.
a. Calculate the expected return from McDonald's.
b. Find the highest expected return that is offered by one of these stocks.
c. Find the lowest expected return that is offered by one of these stocks.
d. Would Dell offer a higher or lower expected return if the interest rate was 6 rather than 4 percent? Assume that the expected market return stays at 10 percent.
e. Would Exxon Mobil offer a higher or lower expected return if the interest rate was 6 percent?
7. True or false?
a. The CAPM implies that if you could find an investment with a negative beta, its expected return would be less than the interest rate.
b. The expected return on an investment with a beta of 2.0 is twice as high as the expected return on the market.
c. If a stock lies below the security market line, it is undervalued.
8. The CAPM has great theoretical, intuitive, and practical appeal. Nevertheless, many financial managers believe "beta is dead." Why?
9. Write out the APT equation for the expected rate of return on a risky stock.
10. Consider a three-factor APT model. The factors and associated risk premiums are
Factor |
Risk Premium | |
Change |
in GNP |
5% |
Change |
in energy prices |
-1 |
Change |
in long-term interest rates |
+2 |
Calculate expected rates of return on the following stocks. The risk-free interest rate is
7 percent.
a. A stock whose return is uncorrelated with all three factors.
b. A stock with average exposure to each factor (i.e., with b = 1 for each).
c. A pure-play energy stock with high exposure to the energy factor (b = 2) but zero exposure to the other two factors.
d. An aluminum company stock with average sensitivity to changes in interest rates and GNP, but negative exposure of b = —1.5 to the energy factor. (The aluminum company is energy-intensive and suffers when energy prices rise.)
Fama and French have proposed a three-factor model for expected returns. What are the three factors?
PRACTICE
QUESTIONS
1. True or false? Explain or qualify as necessary.
a. Investors demand higher expected rates of return on stocks with more variable rates of return.
b. The CAPM predicts that a security with a beta of 0 will offer a zero expected return.
c. An investor who puts $10,000 in Treasury bills and $20,000 in the market portfolio will have a beta of 2.0.
Brealey-Meyers: Principles of Corporate Finance, Seventh Edition
II. Risk
8. Risk and Return
© The McGraw-Hill Companies, 2003
CHAPTER 8 Risk and Return d. Investors demand higher expected rates of return from stocks with returns that are highly exposed to macroeconomic changes.
e. Investors demand higher expected rates of return from stocks with returns that are very sensitive to fluctuations in the stock market.
2. Look back at the calculation for Coca-Cola and Reebok in Section 8.1. Recalculate the expected portfolio return and standard deviation for different values of x1 and x2, assuming the correlation coefficient p12 = 0. Plot the range of possible combinations of expected return and standard deviation as in Figure 8.4. Repeat the problem for p12 = +1 and for p12 = —1.
3. Mark Harrywitz proposes to invest in two shares, X and Y. He expects a return of 12 percent from X and 8 percent from Y. The standard deviation of returns is 8 percent for X and 5 percent for Y. The correlation coefficient between the returns is .2.
a. Compute the expected return and standard deviation of the following portfolios:
Portfolio |
Percentage in X |
Percentage in Y |
1 |
50 |
50 |
2 |
25 |
75 |
3 |
75 |
25 |
b. Sketch the set of portfolios composed of X and Y.
c. Suppose that Mr. Harrywitz can also borrow or lend at an interest rate of 5 percent. Show on your sketch how this alters his opportunities. Given that he can borrow or lend, what proportions of the common stock portfolio should be invested in X and Y?
4. M. Grandet has invested 60 percent of his money in share A and the remainder in share B. He assesses their prospects as follows:
A |
B | ||
Expected return (%) |
15 |
20 | |
Standard deviation (%) |
20 |
22 | |
Correlation between returns |
.5 |
a. What are the expected return and standard deviation of returns on his portfolio?
b. How would your answer change if the correlation coefficient was 0 or —.5?
c. Is M. Grandet's portfolio better or worse than one invested entirely in share A, or is it not possible to say?
5. Download "Monthly Adjusted Prices" for General Motors (GM) and Harley Davidson (HDI) from the Standard & Poor's Market Insight website (www.mhhe.com/ edumarketinsight). Use the Excel function SLOPE to calculate beta for each company. (See Practice Question 7.13 for details.)
a. Suppose the S&P 500 index falls unexpectedly by 5 percent. By how much would you expect GM or HDI to fall?
b. Which is the riskier company for the well-diversified investor? How much riskier?
c. Suppose the Treasury bill rate is 4 percent and the expected return on the S&P 500 is 11 percent. Use the CAPM to forecast the expected rate of return on each stock.
PART II Risk
Download the "Monthly Adjusted Prices" spreadsheets for Boeing and Pfizer from the Standard & Poor's Market Insight website (www.mhhe.com/edumarketinsight).
a. Calculate the annual standard deviation for each company, using the most recent three years of monthly returns. Use the Excel function STDEV. Multiply by the square root of 12 to convert to annual units.
b. Use the Excel function CORREL to calculate the correlation coefficient between the stocks' monthly returns.
c. Use the CAPM to estimate expected rates of return. Calculate betas, or use the most recent beta reported under "Monthly Valuation Data" on the Market Insight website. Use the current Treasury bill rate and a reasonable estimate of the market risk premium.
d. Construct a graph like Figure 8.5. What combination of Boeing and Pfizer has the lowest portfolio risk? What is the expected return for this minimum-risk portfolio?
The Treasury bill rate is 4 percent, and the expected return on the market portfolio is 12 percent. On the basis of the capital asset pricing model:
a. Draw a graph similar to Figure 8.7 showing how the expected return varies with beta.
b. What is the risk premium on the market?
c. What is the required return on an investment with a beta of 1.5?
d. If an investment with a beta of .8 offers an expected return of 9.8 percent, does it have a positive NPV?
e. If the market expects a return of 11.2 percent from stock X, what is its beta?
Most of the companies in Table 8.2 are covered in the Standard & Poor's Market Insight website (www.mhhe.com/edumarketinsight). For those that are covered, use the Excel SLOPE function to recalculate betas from the monthly returns on the "Monthly Adjusted Prices" spreadsheets. Use as many monthly returns as available, up to a maximum of 60 months. Recalculate expected rates of return from the CAPM formula, using a current risk-free rate and a market risk premium of 8 percent. How have the expected returns changed from the figures reported in Table 8.2? Go to the Standard & Poor's Market Insight website (www.mhhe.com/edumarket insight), and find a low-risk income stock—Exxon Mobil or Kellogg might be good candidates. Estimate the company's beta to confirm that it is well below 1.0. Use monthly rates of return for the most recent three years. For the same period, estimate the annual standard deviation for the stock, the standard deviation for the S&P 500, and the correlation coefficient between returns on the stock and the S&P 500. (The Excel functions are given in Practice Questions above.) Forecast the expected rate of return for the stock, assuming the CAPM holds, with a market return of 12 percent and a risk-free rate of 5 percent.
a. Plot a graph like Figure 8.5 showing the combinations of risk and return from a portfolio invested in your low-risk stock and in the market. Vary the fraction invested in the stock from zero to 100 percent.
b. Suppose you can borrow or lend at 5 percent. Would you invest in some combination of your low-risk stock and the market? Or would you simply invest in the market? Explain.
c. Suppose you forecast a return on the stock that is 5 percentage points higher than the CAPM return used in part (a). Redo parts (a) and (b) with this higher forecasted return.
d. Find a high-beta stock and redo parts (a), (b), and (c).
Percival Hygiene has $10 million invested in long-term corporate bonds. This bond portfolio's expected annual rate of return is 9 percent, and the annual standard deviation is 10 percent.
Amanda Reckonwith, Percival's financial adviser, recommends that Percival consider investing in an index fund which closely tracks the Standard and Poor's 500 in-
Brealey-Meyers: Principles of Corporate Finance, Seventh Edition
II. Risk
8. Risk and Return
© The McGraw-Hill Companies, 2003
CHAPTER 8 Risk and Return dex. The index has an expected return of 14 percent, and its standard deviation is 16 percent.
a. Suppose Percival puts all his money in a combination of the index fund and Treasury bills. Can he thereby improve his expected rate of return without changing the risk of his portfolio? The Treasury bill yield is 6 percent.
b. Could Percival do even better by investing equal amounts in the corporate bond portfolio and the index fund? The correlation between the bond portfolio and the index fund is + .1.
11. "There may be some truth in these CAPM and APT theories, but last year some stocks did much better than these theories predicted, and other stocks did much worse." Is this a valid criticism?
12. True or false?
a. Stocks of small companies have done better than predicted by the CAPM.
b. Stocks with high ratios of book value to market price have done better than predicted by the CAPM.
c. On average, stock returns have been positively related to beta.
13. Some true or false questions about the APT:
a. The APT factors cannot reflect diversifiable risks.
b. The market rate of return cannot be an APT factor.
c. Each APT factor must have a positive risk premium associated with it; otherwise the model is inconsistent.
d. There is no theory that specifically identifies the APT factors.
e. The APT model could be true but not very useful, for example, if the relevant factors change unpredictably.
14. Consider the following simplified APT model (compare Tables 8.3 and 8.4):
Expected Risk | |
Factor |
Premium |
Market |
6.4% |
Interest rate |
-.6 |
Yield spread |
5.1 |
Calculate the expected return for the following stocks. Assume rf = 5 percent.
Factor Risk Exposures | |||
Market |
Interest Rate |
Yield Spread | |
Stock |
(bi) |
(b2) |
(ba) |
P |
1.0 |
-2.0 |
-.2 |
P2 |
1.2 |
0 |
.3 |
P3 |
.3 |
.5 |
1.0 |
15. Look again at Practice Question 14. Consider a portfolio with equal investments in stocks P, P2, and P3.
a. What are the factor risk exposures for the portfolio?
b. What is the portfolio's expected return?
16. The following table shows the sensitivity of four stocks to the three Fama-French factors in the five years to 2001. Estimate the expected return on each stock assuming that the interest rate is 3.5 percent, the expected risk premium on the market is 8.8 percent,
PART II Risk the expected risk premium on the size factor is 3.1 percent, and the expected risk premium on the book-to-market factor is 4.4 percent. (These were the realized premia from 1928-2000.)
Factor |
Factor Sensitivities | |||
Coca-Cola |
Exxon Mobil |
Pfizer |
Reebok | |
Market |
.82 |
.50 |
.66 |
1.17 |
Size* |
-.29 |
.04 |
-.56 |
.73 |
Book-to-markett |
.24 |
.27 |
-.07 |
1.14 |
*Return on small-firm stocks less return on large-firm stocks.
+Return on high book-to-market-ratio stocks less return on low book-to-market-ratio stocks.
*Return on small-firm stocks less return on large-firm stocks.
+Return on high book-to-market-ratio stocks less return on low book-to-market-ratio stocks.
1. In footnote 4 we noted that the minimum-risk portfolio contained an investment of 21.4 percent in Reebok and 78.6 in Coca-Cola. Prove it. Hint: You need a little calculus to do so.
2. Look again at the set of efficient portfolios that we calculated in Section 8.1.
a. If the interest rate is 10 percent, which of the four efficient portfolios should you hold?
b. What is the beta of each holding relative to that portfolio? Hint: Remember that if a portfolio is efficient, the expected risk premium on each holding must be proportional to the beta of the stock relative to that portfolio.
c. How would your answers to (a) and (b) change if the interest rate was 5 percent?
3. "Suppose you could forecast the behavior of APT factors, such as industrial production, interest rates, etc. You could then identify stocks' sensitivities to these factors, pick the right stocks, and make lots of money." Is this a good argument favoring the APT? Explain why or why not.
4. The following question illustrates the APT. Imagine that there are only two pervasive macroeconomic factors. Investments X, Y, and Z have the following sensitivities to these two factors:
Investment b1 b2
We assume that the expected risk premium is 4 percent on factor 1 and 8 percent on factor 2. Treasury bills obviously offer zero risk premium.
a. According to the APT, what is the risk premium on each of the three stocks?
b. Suppose you buy $200 of X and $50 of Y and sell $150 of Z. What is the sensitivity of your portfolio to each of the two factors? What is the expected risk premium?
c. Suppose you buy $80 of X and $60 of Y and sell $40 of Z. What is the sensitivity of your portfolio to each of the two factors? What is the expected risk premium?
CHAPTER 8 Risk and Return
d. Finally, suppose you buy $160 of X and $20 of Y and sell $80 of Z. What is your portfolio's sensitivity now to each of the two factors? And what is the expected risk premium?
e. Suggest two possible ways that you could construct a fund that has a sensitivity of .5 to factor 1 only. Now compare the risk premiums on each of these two investments.
f. Suppose that the APT did not hold and that X offered a risk premium of 8 percent, Y offered a premium of 14 percent, and Z offered a premium of 16 percent. Devise an investment that has zero sensitivity to each factor and that has a positive risk premium.
CHAPTER NINE
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