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We have assumed so far that there is only one systematic factor affecting stock returns. This simplifying assumption is in fact too simplistic. It is easy to think of several factors driven by the business cycle that might affect stock returns: interest rate fluctuations, inflation rates, oil prices, and so on. Presumably, exposure to any of these factors will affect a stock's risk and hence its expected return. We can derive a multifactor version of the APT to accommodate these multiple sources of risk.
Suppose that we generalize the factor model expressed in equation 11.1 to a two-factor model:
Factor 1 might be, for example, departures of GDP growth from expectations, and factor 2 might be unanticipated inflation. Each factor has a zero expected value because each measures the surprise in the systematic variable rather than the level of the variable. Similarly, the firm-specific component of unexpected return, eb also has zero expected value. Extending such a two-factor model to any number of factors is straightforward.
Establishing a multifactor APT is similar to the one-factor case. But first we must introduce the concept of a factor portfolio, which is a well-diversified portfolio constructed to have a beta of 1 on one of the factors and a beta of 0 on any other factor. This is an easy restriction to satisfy, because we have a large number of securities to choose from, and a relatively small number of factors. Factor portfolios will serve as the benchmark portfolios for a multifactor security market line.
Suppose that the two factor portfolios, called Portfolios 1 and 2, have expected returns E(r1) = 10% and E(r2) = 12%. Suppose further that the risk-free rate is 4%. The risk
CHAPTER 11 Arbitrage Pricing Theory 333
premium on the first factor portfolio becomes 10% - 4% = 6%, whereas that on the second factor portfolio is 12% - 4% = 8%.
Now consider an arbitrary well-diversified portfolio, Portfolio A, with beta on the first factor, pA1 = .5, and beta on the second factor, pA2 = .75. The multifactor APT states that the overall risk premium on this portfolio must equal the sum of the risk premiums required as compensation to investors for each source of systematic risk. The risk premium attributable to risk factor 1 should be the portfolio's exposure to factor 1, pA1, multiplied by the risk premium earned on the first factor portfolio, E(r1) - rf. Therefore, the portion of Portfolio A's risk premium that is compensation for its exposure to the first factor is pA1[E(r1) - rf] = .5(10% - 4%) = 3%, whereas the risk premium attributable to risk factor 2 is PA2[E(r2) - rf] = .75(12% - 4%) = 6%. The total risk premium on the portfolio should be 3 + 6 = 9%. Therefore, the total return on the portfolio should be 13%:
4% (risk-free rate) + 3 (risk premium for exposure to factor 1) + 6 (risk premium for exposure to factor 2)
13% (total expected return)
To see why the expected return on the portfolio must be 13%, consider the following argument. Suppose that the expected return on Portfolio A were 12% rather than 13%. This return would give rise to an arbitrage opportunity. Form a portfolio from the factor portfolios with the same betas as Portfolio A. This requires weights of .5 on the first factor portfolio, .75 on the second factor portfolio, and -.25 on the risk-free asset. This portfolio has exactly the same factor betas as Portfolio A: It has a beta of .5 on the first factor because of its .5 weight on the first factor portfolio, and a beta of .75 on the second factor.
However, in contrast to Portfolio A, which has a 12% expected return, this portfolio's expected return is (.5 X 10) + (.75 X 12) - (.25 X 4) = 13%. A long position in this portfolio and a short position in Portfolio A would yield an arbitrage profit. The total return per dollar long or short in each position would be
.13 + .5F1 + .75F2 (long position in factor portfolios) - (.12 + .5F1 + .75F2) (short position in Portfolio A)
for a positive, risk-free return on a zero net investment position.
To generalize this argument, note that the factor exposure of any portfolio, P, is given by its betas, pP1 and pP2. A competing portfolio formed from factor portfolios with weights pP1 in the first factor portfolio, pP2 in the second factor portfolio, and 1 - pP1 - pP2 in T-bills will have betas equal to those of Portfolio P and expected return of
E(rP) = pwE(r!) + ppiE(ri) + (1 - pP1 - pP2)rf = rf + pw[E(r!) - rf] + ppi[E(ri) - rf]
Hence any well-diversified portfolio with betas pP1 and pP2 must have the return given in equation 11.6 if arbitrage opportunities are to be precluded. If you compare equations 11.3 and 11.6, you will see that equation 11.6 is simply a generalization of the one-factor SML.
Finally, the extension of the multifactor SML of equation 11.6 to individual assets is precisely the same as for the one-factor APT. Equation 11.6 cannot be satisfied by every well-diversified portfolio unless it is satisfied by virtually every security taken individually. This establishes a multifactor version of the APT. Hence the fair rate of return on any stock with p1 = .5 and p2 = .75 is 13%. Equation 11.6 thus represents the multifactor SML for an economy with multiple sources of risk.
PART III Equilibrium in Capital Markets
CONCEPT CHECK ^ QUESTION 5
PART III Equilibrium in Capital Markets
CONCEPT CHECK ^ QUESTION 5
One shortcoming of the multifactor APT is that it gives no guidance concerning the determination of the risk premiums on the factor portfolios. In contrast, the CAPM implies that the risk premium on the market is determined by the market's variance and the average degree of risk aversion across investors. As it turns out, the CAPM also has a multi-factor generalization, sometimes called the intertemporal (ICAPM). This model provides some guidance concerning the risk premiums on the factor portfolios. Moreover, recent theoretical research has demonstrated that one may estimate an expected return-beta relationship even if the true factors or factor portfolios cannot be identified.
1. 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.
2. Rational investors will want to take infinitely large positions in arbitrage portfolios regardless of their degree of risk aversion.
3. The presence of arbitrage opportunities and the resulting large volume of trades will create pressure on security prices. This pressure will continue until prices reach levels that preclude arbitrage. Only a few investors need to become aware of arbitrage opportunities to trigger this process because of the large volume of trades in which they will engage.
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.
5. Portfolios are called "well-diversified" if they include a large number of securities and the investment 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 security market line to satisfy the no-arbitrage condition.
7. If all well-diversified portfolios satisfy the expected return-beta relationship, then all but a small number of securities also must satisfy this relationship.
8. The assumption of a single-factor security market made in the simple version of the APT, together with the no-arbitrage condition, implies the same expected return-beta relationship as does the CAPM, yet it 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.
9. A multifactor APT generalizes the single-factor model to accommodate several sources of systematic risk.
KEY TERMS arbitrage risk arbitrage well-diversified portfolio zero investment portfolio arbitrage pricing theory factor portfolio
PROBLEMS 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. Suppose that there are two independent economic factors, F1 and F2. 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, |
Beta on Fz |
Expected Return |
A |
1.5 |
2.0 |
31 |
B |
2.2 |
-0.2 |
27 |
What is the expected return-beta relationship in this economy? 3. Consider the following data for a one-factor economy. All portfolios are well diversified.
Portfolio |
m |
Beta |
A |
12% |
1.2 |
6% |
0.0 |
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?
4. The following is a scenario for three stocks constructed by the security analysts of Pf Inc.
Stock |
Price ($) |
Recession |
Average |
Boom |
A |
10 |
-15 |
20 |
30 |
B |
15 |
25 |
10 |
-10 |
C |
50 |
12 |
15 |
12 |
a. Construct an arbitrage portfolio using these stocks.
b. How might these prices change when equilibrium is restored? Give an example where a change in Stock C's price is sufficient to restore equilibrium, assuming that the dollar payoffs to Stock C remain the same.
5. Assume that both Portfolios A and B are well diversified, that E(rA) = 12%, and E(rB) = 9%. If the economy has only one factor, and pA = 1.2, whereas pB = .8, what must be the risk-free rate?
6. 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?
7. Assume that security returns are generated by the single-index model,
where Ri is the excess return for security i and RM 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 following data:
Security | |||
A |
0.8 |
10% |
25% |
B |
1.0 |
12% |
10% |
C |
1.2 |
14% |
20% |
a. If aM = 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 identical 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.
8. 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 | |||
mC | |||
A | |||
CHAPTER 11 Arbitrage Pricing Theory a. How could you construct an arbitrage portfolio? (Hint: Consider combinations of Portfolios A and B, and compare the resultant portfolio to C.)
b. We will see in Chapter 13 that some researchers have examined the relationship between average return on diversified portfolios and the p and p2 of those portfolios. What should they have discovered about the effect of p2 on portfolio return?
9. If the APT is to be a useful theory, the number of systematic factors in the economy must be small. Why?
10. The APT itself does not provide guidance concerning the factors that one might expect to determine risk premiums. How should researchers decide which factors to investigate? Why, for example, is industrial production a reasonable factor to test for a risk premium?
11. 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 market 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 expectations 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 |
12. Suppose that the market can be described by the following three sources of systematic risk with associated risk premiums.
Factor |
Risk Premium |
Industrial production (t) |
6% |
Interest rates (R) |
2 |
Consumer confidence (C) |
4 |
The return on a particular stock is generated according to the following equation:
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.
Bodie-Kane-Marcus: Investments, Fifth Edition
III. Equilibrium In Capital Markets
11. Arbitrage Pricing Theory
© The McGraw-Hill Companies, 2001
PART III Equilibrium in Capital Markets
13. 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 |
V |
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.
14. 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.
15. 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.
16. 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.
17. An investor takes as large a position as possible when an equilibrium price relationship is violated. This is an example of:
a. A dominance argument.
b. The mean-variance efficient frontier.
c. Arbitrage activity.
d. The capital asset pricing model.
18. 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.
19. 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.
1. The least profitable scenario currently yields a profit of $10,000 and gross proceeds from the equally weighted portfolio of $700,000. As the price of Dreck falls, less of the equally weighted portfolio can be purchased from the proceeds of the short sale. When Dreck's price falls by more than a factor of 10,000/700,000, arbitrage no longer will be feasible, because the profits in the worst state will be driven below zero.
To see this, suppose that Dreck's price falls to $10 X (1 - 1/70). The short sale of 300,000 shares now yields $2,957,142, which allows dollar investments of only $985,714 in each of the other shares. In the high real interest rate-low inflation scenario, profits will be driven to zero:
Stock |
Dollar Investment |
Rate of Return |
Dollar Return |
Apex |
$ 985,714 |
.20 |
$197,143 |
Bull |
985,714 |
.70 |
690,000 |
Crush |
985,714 |
-.20 |
-197,143 |
Dreck |
-2,957,142 |
.23 |
-690,000 |
Total |
0 |
0 |
At any price for Dreck stock below $10 X (1 - 1/70) = $9.857, profits are negative, which means this arbitrage opportunity is eliminated. Note: $9.857 is not the equilibrium price of Dreck. It is simply the upper bound on Dreck's price that rules out the simple arbitrage opportunity. c2(eP) = a2(e; )/n. Therefore, c(eP) = Va2(e)/n a. V900/10 = 9.49%
We conclude that nonsystematic volatility can be driven to arbitrarily low levels in well-diversified portfolios.
A portfolio consisting of two-thirds of Portfolio A and one-third of the risk-free asset will have the same beta as Portfolio E, but an expected return of (1/3 X 4) + (% X 10) = 8%, less than that of Portfolio E. Therefore, one can earn arbitrage profits by shorting the combination of Portfolio A and the safe asset, and buying Portfolio E. a. For Portfolio P,
For Portfolio Q,
b. The equally weighted portfolio has an expected return of 12.5% and a beta of .75. K = (12.5 - 5)/.75 = 10. 5. Using equation 11.6, the expected return is
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