It is important to distinguish the multifactor APT from the multi-index CAPM. In the lat- ter, the factors are derived from a multiperiod consideration of a stream of consumption as well as randomly evolving investment opportunities pertaining to the distributions of rates of return. Hence, the hedge index portfolios must be derived from considerations of the utility of consumption, nontraded assets, and changes in investment opportunities.
A multi-index CAPM therefore will inherit its risk factors from sources of risk that a broad group of investors deem important enough to hedge. If hedging demands are com- mon to many investors, the prices of securities with desirable hedging characteristics will be bid up and their expected return reduced. This process requires a multifactor model to explain expected returns, where each factor arises from a particular hedging motive. Risk sources that are “priced” in market equilibrium (that is, are sufficiently important to result in detectable risk premiums) presumably will be systematic sources of uncertainty that affect investors broadly.
In contrast, the APT is largely silent on where to look for priced sources of risk. This lack of guidance is problematic, but by the same token, it accommodates a less structured search for relevant risk factors. These may reflect the concerns of a broader set of inves- tors, including institutions such as endowment or pension funds that may be concerned about exposures to risks that would not be obvious from an examination of individual consumption/investment decisions.
CONCEPT CHECK
6
Consider the following regression results for stock X.
rX 2% 1.2 (percentage change in oil prices )
a. If I live in Louisiana, where the local economy is heavily dependent on oil industry profits, does stock X represent a useful asset to hedge my overall economic well-being?
b. What if I live in Massachusetts, where most individuals and firms are energy consumers?
c. If energy consumers are far more numerous than energy producers, will high oil-beta stocks have higher or lower expected rates of return in market equilibrium than low oil-beta stocks?
1. Multifactor models seek to improve the explanatory power of single-factor models by explicitly accounting for the various systematic components of security risk. These models use indicators intended to capture a wide range of macroeconomic risk factors.
2. Once we allow for multiple risk factors, we conclude that the security market line also ought to be multidimensional, with exposure to each risk factor contributing to the total risk premium of the security.
3. A (risk-free) arbitrage opportunity arises when two or more security prices enable investors to construct a zero net investment portfolio that will yield a sure profit. The presence of arbitrage opportunities will generate a large volume of trades that puts pressure on security prices. This pressure will continue until prices reach levels that preclude such arbitrage.
4. When securities are priced so that there are no risk-free arbitrage opportunities, we say that they satisfy the no-arbitrage condition. Price relationships that satisfy the no-arbitrage condition are important because we expect them to hold in real-world markets.
SUMMARY
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5. Portfolios are called “well-diversified” if they include a large number of securities and the invest- ment proportion in each is sufficiently small. The proportion of a security in a well-diversified portfolio is small enough so that for all practical purposes a reasonable change in that security’s rate of return will have a negligible effect on the portfolio’s rate of return.
6. In a single-factor security market, all well-diversified portfolios have to satisfy the expected return–beta relationship of the CAPM to satisfy the no-arbitrage condition. If all well-diversified portfolios satisfy the expected return–beta relationship, then all but a small number of securities also must satisfy this relationship.
7. The APT does not require the restrictive assumptions of the CAPM and its (unobservable) market portfolio. The price of this generality is that the APT does not guarantee this relationship for all securities at all times.
8. A multifactor APT generalizes the single-factor model to accommodate several sources of sys- tematic risk. The multidimensional security market line predicts that exposure to each risk factor contributes to the security’s total risk premium by an amount equal to the factor beta times the risk premium of the factor portfolio that tracks that source of risk.
9. A multifactor extension of the single-factor CAPM, the ICAPM, is a model of the risk–return trade-off that predicts the same multidimensional security market line as the APT. The ICAPM suggests that priced risk factors will be those sources of risk that lead to significant hedging demand by a substantial fraction of investors.
Related Web sites for this chapter are available at www.
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single-factor model multifactor model factor sensitivity factor loading
factor beta
arbitrage pricing theory arbitrage
Law of One Price
risk arbitrage
well-diversified portfolio factor portfolio
KEY TERMS
1. Suppose that two factors have been identified for the U.S. economy: the growth rate of industrial production, IP, and the inflation rate, IR. IP is expected to be 3%, and IR 5%. A stock with a beta of 1 on IP and .5 on IR currently is expected to provide a rate of return of 12%. If industrial production actually grows by 5%, while the inflation rate turns out to be 8%, what is your revised estimate of the expected rate of return on the stock?
2. The APT itself does not provide guidance concerning the factors that one might expect to deter- mine risk premiums. How should researchers decide which factors to investigate? Why, for exam- ple, is industrial production a reasonable factor to test for a risk premium?
3. If the APT is to be a useful theory, the number of systematic factors in the economy must be small. Why?
4. Suppose that there are two independent economic factors, F 1 and F 2 . The risk-free rate is 6%, and all stocks have independent firm-specific components with a standard deviation of 45%. The following are well-diversified portfolios:
Portfolio Beta on F 1 Beta on F 2 Expected Return
A 1.5 2.0 31%
B 2.2 0.2 27%
What is the expected return–beta relationship in this economy?
5. Consider the following data for a one-factor economy. All portfolios are well diversified.
Portfolio E ( r ) Beta
A 12% 1.2
F 6% 0.0
PROBLEM SETS
i. Basic
ii. Intermediate
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Suppose that another portfolio, portfolio E, is well diversified with a beta of .6 and expected return of 8%. Would an arbitrage opportunity exist? If so, what would be the arbitrage strategy?
6. Assume that both portfolios A and B are well diversified, that E ( r A ) ⫽ 12%, and E ( r B ) ⫽ 9%.
If the economy has only one factor, and  A ⫽ 1.2, whereas  B ⫽ .8, what must be the risk-free rate?
7. Assume that stock market returns have the market index as a common factor, and that all stocks in the economy have a beta of 1 on the market index. Firm-specific returns all have a standard deviation of 30%.
Suppose that an analyst studies 20 stocks, and finds that one-half have an alpha of ⫹ 2%, and the other half an alpha of –2%. Suppose the analyst buys $1 million of an equally weighted portfolio of the positive alpha stocks, and shorts $1 million of an equally weighted portfolio of the negative alpha stocks.
a. What is the expected profit (in dollars) and standard deviation of the analyst’s profit?
b. How does your answer change if the analyst examines 50 stocks instead of 20 stocks?
100 stocks?
8. Assume that security returns are generated by the single-index model, Ri5ai1biRM1ei
where R i is the excess return for security i and R M is the market’s excess return. The risk-free rate is 2%. Suppose also that there are three securities A, B, and C, characterized by the follow- ing data:
Security  i E ( R i ) ( e i )
A 0.8 10% 25%
B 1.0 12% 10%
C 1.2 14% 20%
a. If M ⫽ 20%, calculate the variance of returns of securities A, B, and C.
b. Now assume that there are an infinite number of assets with return characteristics identi- cal to those of A, B, and C, respectively. If one forms a well-diversified portfolio of type A securities, what will be the mean and variance of the portfolio’s excess returns? What about portfolios composed only of type B or C stocks?
c. Is there an arbitrage opportunity in this market? What is it? Analyze the opportunity graphically.
9. The SML relationship states that the expected risk premium on a security in a one-factor model must be directly proportional to the security’s beta. Suppose that this were not the case. For example, suppose that expected return rises more than proportionately with beta as in the figure below.
B
C
A E(r)
β
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a. How could you construct an arbitrage portfolio? ( Hint: Consider combinations of portfolios A and B, and compare the resultant portfolio to C. )
b. Some researchers have examined the relationship between average returns on diversified portfolios and the and 2 of those portfolios. What should they have discovered about the effect of 2 on portfolio return?
10. Consider the following multifactor (APT) model of security returns for a particular stock.
Factor Factor Beta Factor Risk Premium
Inflation 1.2 6%
Industrial production 0.5 8
Oil prices 0.3 3
a. If T-bills currently offer a 6% yield, find the expected rate of return on this stock if the mar- ket views the stock as fairly priced.
b. Suppose that the market expected the values for the three macro factors given in column 1 below, but that the actual values turn out as given in column 2. Calculate the revised expecta- tions for the rate of return on the stock once the “surprises” become known.
Factor Expected Rate of Change Actual Rate of Change
Inflation 5% 4%
Industrial production 3 6
Oil prices 2 0
11. Suppose that the market can be described by the following three sources of systematic risk with associated risk premiums.
Factor Risk Premium
Industrial production ( I ) 6%
Interest rates ( R ) 2
Consumer confidence ( C ) 4
The return on a particular stock is generated according to the following equation:
r515%11.0I1.5R1.75C1e
Find the equilibrium rate of return on this stock using the APT. The T-bill rate is 6%. Is the stock over- or underpriced? Explain.
12. As a finance intern at Pork Products, Jennifer Wainwright’s assignment is to come up with fresh insights concerning the firm’s cost of capital. She decides that this would be a good opportunity to try out the new material on the APT that she learned last semester. She decides that three promising factors would be (i) the return on a broad-based index such as the S&P 500; (ii) the level of interest rates, as represented by the yield to maturity on 10-year Treasury bonds; and (iii) the price of hogs, which are particularly important to her firm. Her plan is to find the beta of Pork Products against each of these factors by using a multiple regression and to estimate the risk premium associated with each exposure factor. Comment on Jennifer’s choice of factors.
Which are most promising with respect to the likely impact on her firm’s cost of capital? Can you suggest improvements to her specification?
Use the following information to Answer Problems 13 – 16 :
Orb Trust (Orb) has historically leaned toward a passive management style of its portfolios. The only model that Orb’s senior management has promoted in the past is the capital asset pricing model (CAPM). Now Orb’s management has asked one of its analysts, Kevin McCracken, CFA, to investi- gate the use of the arbitrage pricing theory (APT) model.
McCracken believes that a two-factor APT model is adequate, where the factors are the sensitiv- ity to changes in real GDP and changes in inflation. McCracken has concluded that the factor risk
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premium for real GDP is 8% while the factor risk premium for inflation is 2%. He estimates for Orb’s High Growth Fund that the sensitivities to these two factors are 1.25 and 1.5, respectively.
Using his APT results, he computes the expected return of the fund. For comparison purposes, he then uses fundamental analysis to also compute the expected return of Orb’s High Growth Fund.
McCracken finds that the two estimates of the Orb High Growth Fund’s expected return are equal.
McCracken asks a fellow analyst, Sue Kwon, to provide an estimate of the expected return of Orb’s Large Cap Fund based on fundamental analysis. Kwon, who manages the fund, says that the expected return is 8.5% above the risk-free rate. McCracken then applies the APT model to the Large Cap Fund. He finds that the sensitivities to real GDP and inflation are .75 and 1.25, respectively.
McCracken’s manager at Orb, Jay Stiles, asks McCracken to compose a portfolio that has a unit sensitivity to real GDP growth but is not affected by inflation. McCracken is confident in his APT estimates for the High Growth Fund and the Large Cap Fund. He then computes the sensitivities for a third fund, Orb’s Utility Fund, which has sensitivities equal to 1.0 and 2.0, respectively. McCracken will use his APT results for these three funds to accomplish the task of creating a portfolio with a unit exposure to real GDP and no exposure to inflation. He calls the fund the “GDP Fund.” Stiles says such a GDP Fund would be good for clients who are retirees who live off the steady income of their investments. McCracken says that the fund would be a good choice if upcoming supply side macroeconomic policies of the government are successful.
13. According to the APT, if the risk-free rate is 4%, what should be McCracken’s estimate of the expected return of Orb’s High Growth Fund?
14. With respect to McCracken’s APT model estimate of Orb’s Large Cap Fund and the informa- tion Kwon provides, is an arbitrage opportunity available?
15. The GDP Fund composed from the other three funds would have a weight in Utility Fund equal to ( a ) 2.2; ( b ) 3.2; or ( c ) .3.
16. With respect to the comments of Stiles and McCracken concerning for whom the GDP Fund would be appropriate:
a. McCracken was correct and Stiles was wrong.
b. Both were correct.
c. Stiles was correct and McCracken was wrong.
17. Assume a universe of n (large) securities for which the largest residual variance is of an order not larger than nsM2. Construct as many different weighting schemes as you can that generate well-diversified portfolios.
18. Derive a more general (than the numerical example in the chapter) demonstration of the APT security market line:
a. For a single-factor market.
b. For a multifactor market.
19. Small firms will have relatively high loadings (high betas) on the SMB (small minus big) factor.
a. Explain why.
b. Now suppose two unrelated small firms merge. Each will be operated as an independent unit of the merged company. Would you expect the stock market behavior of the merged firm to differ from that of a portfolio of the two previously independent firms? How does the merger affect market capitalization? What is the prediction of the Fama-French model for the risk premium on the combined firm? Do we see here a flaw in the FF model?
iii. Challenge
1. Jeffrey Bruner, CFA, uses the capital asset pricing model (CAPM) to help identify mispriced securities. A consultant suggests Bruner use arbitrage pricing theory (APT) instead. In compar- ing CAPM and APT, the consultant made the following arguments:
a. Both the CAPM and APT require a mean-variance efficient market portfolio.
b. Neither the CAPM nor APT assumes normally distributed security returns.
c. The CAPM assumes that one specific factor explains security returns but APT does not.
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State whether each of the consultant’s arguments is correct or incorrect. Indicate, for each incor- rect argument, why the argument is incorrect.
2. Assume that both X and Y are well-diversified portfolios and the risk-free rate is 8%.
Portfolio Expected Return Beta
X 16% 1.00
Y 12 0.25
In this situation you would conclude that portfolios X and Y:
a. Are in equilibrium.
b. Offer an arbitrage opportunity.
c. Are both underpriced.
d. Are both fairly priced.
3. A zero-investment portfolio with a positive alpha could arise if:
a. The expected return of the portfolio equals zero.
b. The capital market line is tangent to the opportunity set.
c. The Law of One Price remains unviolated.
d. A risk-free arbitrage opportunity exists.
4. According to the theory of arbitrage:
a. High-beta stocks are consistently overpriced.
b. Low-beta stocks are consistently overpriced.
c. Positive alpha investment opportunities will quickly disappear.
d. Rational investors will pursue arbitrage consistent with their risk tolerance.
5. The arbitrage pricing theory (APT) differs from the single-factor capital asset pricing model (CAPM) because the APT:
a. Places more emphasis on market risk.
b. Minimizes the importance of diversification.
c. Recognizes multiple unsystematic risk factors.
d. Recognizes multiple systematic risk factors.
6. An investor takes as large a position as possible when an equilibrium price relationship is vio- lated. This is an example of:
a. A dominance argument.
b. The mean-variance efficient frontier.
c. Arbitrage activity.
d. The capital asset pricing model.
7. The feature of arbitrage pricing theory (APT) that offers the greatest potential advantage over the simple CAPM is the:
a. Identification of anticipated changes in production, inflation, and term structure of interest rates as key factors explaining the risk–return relationship.
b. Superior measurement of the risk-free rate of return over historical time periods.
c. Variability of coefficients of sensitivity to the APT factors for a given asset over time.
d. Use of several factors instead of a single market index to explain the risk–return relationship.
8. In contrast to the capital asset pricing model, arbitrage pricing theory:
a. Requires that markets be in equilibrium.
b. Uses risk premiums based on micro variables.
c. Specifies the number and identifies specific factors that determine expected returns.
d. Does not require the restrictive assumptions concerning the market portfolio.
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Unanticipated Inflation
One of the factors in the APT model specified by Chen, Roll, and Ross is the percent change in unanticipated inflation. Who gains and who loses when inflation change? Go to http://
hussmanfunds.com/rsi/infsurprises.htm to see a graph Inflation Surprise Index and Economists’ Inflation Forecasts.
E-INVEST- MENTS EXERCISES
SOLUTIONS TO CONCEPT CHECKS
1. The GDP beta is 1.2 and GDP growth is 1% better than previously expected. So you will increase your forecast for the stock return by 1.2 1% 1.2%. The revised forecast is for an 11.2%
return.
2. With these lower risk premiums, the expected return on the stock will be lower:
E(r)54%11.234%1(2.3)3(22%)59.4%
3. a. This portfolio is not well diversified. The weight on the first security does not decline as n increases. Regardless of how much diversification there is in the rest of the portfolio, you will not shed the firm-specific risk of this security.
b. This portfolio is well diversified. Even though some stocks have three times the weight of other stocks (1.5/ n versus .5/ n ), the weight on all stocks approaches zero as n increases. The impact of any individual stock’s firm-specific risk will approach zero as n becomes ever larger.
4. The SML says that the expected return on the portfolio should be 4% (1⁄3)(10 4) 6%. The return actually expected is only 5%, implying that the stock is overpriced and that there is an arbitrage opportunity. Buy $1 of a portfolio that is 2⁄3 invested in T-bills and 1⁄3 in the market. The return on this portfolio is 2⁄3 r f 1⁄3 r M 2⁄3 4% 1⁄3 r M . Sell $1 of portfolio G. The net return on the combined position is:
$1 [2⁄3 .04 1⁄3 r M ] Buy portfolio invested 2⁄3 in T-bills and 1⁄3 in the market index.
$1 [.05 1⁄3( r M .10)] Sell $1 in portfolio G, with expected return of 5% and beta of 1⁄3 on surprise in market return.
$1 .01 Total
The profit per dollar invested is risk-free and precisely equal to the deviation of expected return from the SML.
5. The equilibrium return is E ( r ) r f P 1 [ E ( r 1 ) r f ] P 2 [ E ( r 2 ) r f ]. Using the data in Example 10.5 :
E(r)541.23(1024)11.43(1224)516.4%
6. a. For Louisiana residents, the stock is not a hedge. When their economy does poorly (low energy prices), the stock also does poorly, thereby aggravating their problems.
b. For Massachusetts residents, the stock is a hedge. When energy prices increase, the stock will provide greater wealth with which to purchase energy.
c. If energy consumers (who are willing to bid up the price of the stock for its hedge value) dominate the economy, then high oil-beta stocks will have lower expected rates of return than would be predicted by the simple CAPM.