APT/Multi-Factor Models

 The index model introduced in Chapter 8 gave us a way of decomposing stock variability into market or systematic risk, due largely to macroeconomic events, versus firm-specific or idiosyncratic effects that can be diversified in large portfolios. In the single-index model, the return on a broad market-index portfolio summarized the impact of the macro factor. In Chapter 9 we introduced the possibility that risk premiums may also depend on correlations with extra-market risk factors, such as inflation, or changes in the parameters describing future investment opportunities: interest rates, volatility, market-risk premiums, and betas. For example, returns on an asset whose return increases when inflation increases can be used to hedge uncertainty in the future inflation rate. Its price may rise and its risk premium may fall as a result of investors’ extra demand for this asset.Risk premiums of individual securities should reflect their sensitivities to changes in extra-market risk factors just as their betas on the market index determine their risk premiums in the simple CAPM. When securities can be used to hedge these factors, the resulting hedging demands will turn the SML into a multifactor model, with each significant risk source generating an additional factor. Risk factors can be represented either by returns on these hedge portfolios (just as the index portfolio represents the market factor) or more directly by changes in the risk factors themselves, for example, changes in interest rates or inflation.Factor Models of Security ReturnsWe begin with a familiar single-factor model like the one introduced in Chapter 8. Uncertainty in asset returns has two sources: a common or macroeconomic factor and firm-specific events. By construction, the common factor has zero expected value because it measures new information concerning the macroeconomy; new information implies a revision to current expectations, and if initial expectations are rational, then such revisions should average out to zero.If we call F the deviation of the common factor from its expected value, βi the sensitivity of firm i to that factor, and ei the firm-specific disturbance, the factor model states that the actual excess return on firm i will equal its initially expected value plus a (zero expected value) random amount attributable to unanticipated economywide events, plus another (zero expected value) random amount attributable to firm-specific events.Formally, the single-factor model of excess returns is described by Equation 10.1:(10.1)where E(Ri ) is the expected excess return on stock i. Notice that if the macro factor has a value of 0 in any particular period (i.e., no macro surprises), the excess return on the security will equal its previously expected value, E(Ri ), plus the effect of firm-specific events only. The nonsystematic components of returns, the ei s, are assumed to be uncorrelated across stocks and with the factor F.Example 10.1 Factor ModelsTo illustrate the factor model, suppose that the macro factor, F, represents news about the state of the business cycle, which we will measure by the unexpected percentage change in gross domestic product (GDP). The consensus is that GDP will increase by 4% this year. Suppose also that a stock’s β value is 1.2. If GDP increases by only 3%, then the value page 309of F would be −1%, representing a 1% disappointment in actual growth versus expected growth. Given the stock’s beta value, this disappointment would translate into a return on the stock that is 1.2% lower than previously expected. This macro surprise, together with the firm-specific disturbance, ei, determines the total departure of the stock’s return from its originally expected value.Concept Check 10.1Suppose you currently expect the stock in Example 10.1 to earn a 10% rate of return. Then some macroeconomic news suggests that GDP growth will come in at 5% instead of 4%. How will you revise your estimate of the stock’s expected rate of return?The factor model’s decomposition of returns into systematic and firm-specific components is compelling, but confining systematic risk to a single factor is not. Indeed, when we motivated systematic risk as the source of risk premiums in Chapter 9, we noted that extra-market sources of risk may arise from a number of sources such as uncertainty about interest rates, inflation, and so on. The market return reflects all of these macro factors as well as the average sensitivity of firms to those factors.It stands to reason that a more explicit representation of systematic risk, allowing different stocks to exhibit different sensitivities to its various components, would constitute a useful refinement of the single-factor model. It is easy to see that models that allow for several factors—multifactor models—can provide better descriptions of security returns.Apart from their use in building models of equilibrium security pricing, multifactor models are useful in risk management applications. These models give us a simple way to measure investor exposure to various macroeconomic risks and construct portfolios to hedge those risks.Let’s start with a two-factor model. Suppose the two most important macroeconomic sources of risk are uncertainties surrounding the state of the business cycle, news of which we will again measure by unanticipated growth in GDP, and changes in interest rates. We will denote by IR any unexpected change in interest rates. The return on any stock will respond both to sources of macro risk and to its own firm-specific influences. We can write a two-factor model describing the excess return on stock i in some time period as follows:(10.2)The two macro factors on the right-hand side of the equation comprise the systematic factors in the economy. As in the single-factor model, both of these macro factors have zero expectation: They represent changes in these variables that have not already been anticipated. The coefficients of each factor in Equation 10.2 measure the sensitivity of share returns to that factor. For this reason the coefficients are sometimes called factor loadings or, equivalently, factor betas. An increase in interest rates is bad news for most firms, so we would expect interest rate betas generally to be negative. As before, ei reflects firm-specific influences.To illustrate the advantages of multifactor models, consider two firms, one a regulated electric-power utility in a mostly residential area and the other an airline. Because residential demand for electricity is not very sensitive to the business cycle, the utility has a low page 310beta on GDP. But the utility’s stock price may have a relatively high sensitivity to interest rates. Because the cash flow generated by the utility is relatively stable, its present value behaves much like that of a bond, varying inversely with interest rates. Conversely, the performance of the airline is very sensitive to economic activity but is less sensitive to interest rates. It will have a high GDP beta and a lower interest rate beta. Suppose that on a particular day, a news item suggests that the economy will expand. GDP is expected to increase, but so are interest rates. Is the “macro news” on this day good or bad? For the utility, this is bad news: Its dominant sensitivity is to rates. But for the airline, which responds more to GDP, this is good news. Clearly a one-factor or single-index model cannot capture such differential responses to varying sources of macroeconomic uncertainty.Example 10.2 Risk Assessment Using Multifactor ModelsSuppose we estimate the two-factor model in Equation 10.2 for Northeast Airlines and find the following result:This tells us that, based on currently available information, the expected excess rate of return for Northeast is 13.3%, but that for every percentage point increase in GDP beyond current expectations, the return on Northeast’s shares increases on average by 1.2%, while for every unanticipated percentage point that interest rates increase, Northeast’s shares fall on average by .3%.Factor betas can provide a framework for a hedging strategy. An investor concerned with her current exposure to one or another macro factor might offset that risk by initiating a position with equal but opposite exposure. Often, futures contracts can be used to hedge particular factor risk. We explore this application in Chapter 22.Like the index model of Chapter 8, the multifactor model is no more than a description of the factors that affect security returns. There is no “theory” in the equation. The obvious question left unanswered by Equation 10.2 is where E(R) comes from, in other words, what determines a security’s expected excess rate of return. This is where we need a theoretical model of equilibrium security returns. We therefore now turn to arbitrage pricing theory to help determine the expected value, E(R), in Equations 10.1 and 10.2.10.2 Arbitrage Pricing TheoryStephen Ross developed the arbitrage pricing theory (APT) in 1976.1 Like the CAPM, the APT predicts a security market line linking expected returns to risk, but the path it takes to the SML is quite different. Ross’s APT relies on three key propositions: (1) Security returns can be described by a factor model; (2) there are sufficient securities to diversify away idiosyncratic risk; and (3) well-functioning security markets do not allow arbitrage opportunities to persist. We begin with a simple version of Ross’s model, which assumes that only one systematic factor affects security returns. Once we understand how the model works, it will be much easier to see how it can be generalized to accommodate more than one factor.page 311 Arbitrage, Risk Arbitrage, and EquilibriumAn arbitrage opportunity arises when an investor can earn riskless profits without making a net investment. A trivial example of an arbitrage opportunity would arise if shares of a stock sold for different prices on two different exchanges. For example, suppose IBM sold for $165 on the NYSE but only $163 on NASDAQ. Then you could buy the shares on NASDAQ and simultaneously sell them on the NYSE, clearing a riskless profit of $2 per share without tying up any of your own capital. The Law of One Price states that if two assets are equivalent in all economically relevant respects, then they should have the same market price. The Law of One Price is enforced by arbitrageurs: If they observe a violation of the law, they will engage in arbitrage activity—simultaneously buying the asset where it is cheap and selling where it is expensive. In the process, they will bid up the price where it is low and force it down where it is high until the arbitrage opportunity is eliminated.Strategies that exploit violations of the Law of One Price all involve long–short positions. You buy the relatively cheap asset and sell the relatively overpriced one. The net investment, therefore, is zero. Moreover, the position is riskless. Therefore, any investor, regardless of risk aversion or wealth, will want to take an infinite position in it. Because those large positions will quickly force prices up or down until the opportunity vanishes, security prices should satisfy a “no-arbitrage condition,” that is, a condition that rules out the existence of arbitrage opportunities.The idea that market prices will move to rule out arbitrage opportunities is perhaps the most fundamental concept in capital market theory. Violation of this restriction would indicate the grossest form of market irrationality.There is an important difference between arbitrage and risk–return dominance arguments in support of equilibrium price relationships. A dominance argument holds that when an equilibrium price relationship is violated, many investors will make limited portfolio changes, depending on their degree of risk aversion. Aggregation of these limited portfolio changes is required to create a large volume of buying and selling, which in turn restores equilibrium prices. By contrast, when arbitrage opportunities exist, each investor wants to take as large a position as possible; hence, it will not take many investors to bring about the price pressures necessary to restore equilibrium. Therefore, implications for prices derived from no-arbitrage arguments are stronger than implications derived from a risk–return dominance argument.The CAPM is an example of a dominance argument, implying that all investors hold mean-variance efficient portfolios. If a security is mispriced, then investors will tilt their portfolios toward the underpriced and away from the overpriced securities. Pressure on equilibrium prices results from many investors shifting their portfolios, each by a relatively small dollar amount. The assumption that a large number of investors are mean-variance optimizers is critical. In contrast, the implication of a no-arbitrage condition is that a few investors who identify an arbitrage opportunity will mobilize large dollar amounts and quickly restore equilibrium.Practitioners often use the terms arbitrage and arbitrageurs more loosely than our strict definition. Arbitrageur often refers to a professional searching for mispriced securities in specific areas such as merger-target stocks, rather than to one who seeks strict (risk-free) arbitrage opportunities. Such activity is sometimes called risk arbitrage to distinguish it from pure arbitrage.Diversification in a Single-Factor Security MarketWe begin by considering the risk of a portfolio of stocks in a single-factor market. We first show that if a portfolio is well diversified, its firm-specific or nonfactor risk becomes page 312negligible, so that only factor (equivalently, systematic) risk remains. The excess return, RP, on an n-stock portfolio with weights wi, ∑ wi = 1, is(10.3)whereare the weighted averages of the βi and risk premiums of the n securities. The portfolio nonsystematic return (which is uncorrelated with F ) is eP = ∑ wi ei , which similarly is a weighted average of the ei of the n securities.There are two random (and uncorrelated) terms on the right-hand side of Equation 10.3, so we can separate the variance of the portfolio into its systematic and nonsystematic sources:where  is the variance of the factor F and σ2(eP) is the nonsystematic variance of the portfolio, which is given byIn deriving the nonsystematic variance of the portfolio, we depend on the fact that the firm-specific eis are uncorrelated (so all covariances across assets are zero) and, hence, the variance of the “portfolio” of nonsystematic eis is the weighted sum of the individual nonsystematic variances with the square of the investment proportions as weights.If the portfolio were equally weighted, wi = 1/n, then the nonsystematic variance would be(10.4)where the last term is the average value of nonsystematic variance across securities. In words, the nonsystematic variance of the portfolio equals the average nonsystematic variance divided by n. Therefore, as n increases, nonsystematic variance falls. This is the effect of diversification.We see this effect dramatically in the scatter diagram of Figure 10.1. The colored dots are the monthly returns over a five-year period for a single stock, Intel. The black dots are for a diversified stock mutual fund, Vanguard’s Growth & Income Fund. During the period, both the stock and the fund had virtually identical betas. But their nonsystematic risk differed considerably: The scatter of Intel’s returns fall considerably further from the regression line than the mutual fund’s, for which diversification has eliminated most residual risk.Figure 10.1 Scatter diagram for a single stock (Intel) and a diversified mutual fund (Vanguard Growth and Income). The fund exhibits much smaller scatter around the regression line.In sum, the number of securities in the portfolio has no bearing on systematic risk. Only average beta matters. In contrast, firm-specific risk becomes increasingly irrelevant as the portfolio becomes more diversified. You can see where we are going. Risk premiums should depend only on systematic risk that cannot be diversified; firm-specific risk should not command a risk premium because it is easily eliminated.Well-Diversified PortfoliosEquation 10.4 tells us that as diversification progresses, that is, as the weight in each security approaches zero, the nonsystematic variance of the portfolio approaches zero. We will define a well-diversified portfolio as one for which each weight, wi, is small enough that for practical purposes the nonsystematic variance, σ2(eP), is negligible.page 313 Concept Check 10.2A portfolio is invested in a very large number of shares (n is large). However, one-half of the portfolio is invested in stock 1, and the rest of the portfolio is equally divided among the other n − 1 shares. Is this portfolio well diversified?Another portfolio also is invested in the same n shares, where n is very large. Instead of equally weighting with portfolio weights of 1/n in each stock, the weights in half the securities are 1.5/n while the weights in the other shares are .5/n. Is this portfolio well diversified?Because the expected value of eP for any well-diversified portfolio is zero, and its variance also is effectively zero, any realized value of eP will be virtually zero. Rewriting Equation 10.1, we conclude that, for a well-diversified portfolio, for all practical purposesThe solid line in Figure 10.2, Panel A, plots the excess return of a well-diversified portfolio A with E(RA) = 10% and βA = 1 for various realizations of the systematic factor. The expected return of portfolio A is 10%; this is where the solid line crosses the vertical axis. At this point, the systematic factor is zero, implying no macro surprises. If the macro page 314factor is positive, the portfolio’s return exceeds its expected value; if it is negative, the portfolio’s return falls short of its mean. The excess return on the portfolio is thereforeCompare Panel A in Figure 10.2 with Panel B, which is a similar graph for a single stock (S) with βS = 1. The undiversified stock is subject to nonsystematic risk, which is seen in a scatter of points around the line. The well-diversified portfolio’s return, in contrast, is determined completely by the systematic factor.Figure 10.2 Excess returns as a function of the systematic factor: Panel A, Well-diversified portfolio A; Panel B, Single stock (S).The Security Market Line of the APTNonsystematic risk across firms cancels out in well-diversified portfolios, and one would not expect investors to be rewarded for bearing risk that can be eliminated through diversification. Therefore, only the systematic or factor risk of a portfolio of securities should be related to its expected returns. This is the basis of the security market line that we are now ready to derive.First, we show that all well-diversified portfolios with the same beta must have the same expected return. Figure 10.3 plots the returns on two such portfolios, A and B, both with betas of 1, but with differing expected returns: E(rA) = 10% and E(rB) = 8%. Could portfolios A and B coexist with the return pattern depicted? Clearly not: No matter what the systematic factor turns out to be, portfolio A outperforms portfolio B, leading to an arbitrage opportunity.Figure 10.3 Returns as a function of the systematic factor: An arbitrage opportunityIf you sell short $1 million of B and buy $1 million of A, a zero-net-investment strategy, you would have a riskless payoff of $20,000, as follows:(.10 + 1.0 × F) × $1 million from long position in A− (.08 + 1.0 × F) × $1 million from short position in B .02 × $1 million = $20,000 net proceedspage 315 Your profit is risk-free because the factor risk cancels out across the long and short positions. Moreover, the strategy requires zero-net-investment. You (and others) will pursue it on an infinitely large scale until the resulting pressure on security prices forces the return discrepancy between the two portfolios to disappear. We conclude that such arbitrage activity ensures that well-diversified portfolios with equal betas will have equal expected returns.What about portfolios with different betas? Their risk premiums must be proportional to beta. To see why, consider Figure 10.4. Suppose that the risk-free rate is 4% and that a well-diversified portfolio, C, with a beta of .5, has an expected return of 6%. Portfolio C plots below the line from the risk-free asset to portfolio A. Consider, therefore, a new portfolio, D, composed of half of portfolio A and half of the risk-free asset. Portfolio D’s beta will be (.5 × 0 + .5 × 1.0) = .5, and its expected return will be (.5 × 4 + .5 × 10) = 7%. Now portfolio D has an equal beta but a greater expected return than portfolio C. From our analysis of Figure 10.3 in the previous paragraph we know that this constitutes an arbitrage opportunity. We conclude that, to preclude arbitrage opportunities, the expected return on all well-diversified portfolios must lie on the straight line from the risk-free asset in Figure 10.4.Figure 10.4 An arbitrage opportunityNotice in Figure 10.4 that risk premiums are indeed proportional to portfolio betas. The risk premium is depicted by the vertical arrow, which measures the distance between the risk-free rate and the expected return on the portfolio. As in the simple CAPM, the risk premium is zero for β = 0 and rises in direct proportion to β.Figure 10.4 relates the risk premium on well-diversified portfolios to their betas against the macro factor. As a final step, we would like a security market line that relates the portfolio risk premium to its beta against a market index rather than an unspecified macro factor.Fortunately, this last step is easy to justify. This is because all well-diversified portfolios are perfectly correlated with the macro factor. (Again, look at Figure 10.2, Panel A, which shows that the scatter plot for any well-diversified portfolio lies precisely on the straight line.) Therefore, if a market index portfolio is well diversified, its return will perfectly reflect the value of the macro factor. This means that betas measured against the market page 316index are just as informative about relative levels of systematic risk as are betas measured against the macro factor.Therefore, we can write the excess return on a well-diversified portfolio P as:2(10.5)where βP now denotes the beta against the well-diversified market index.We know that risk premiums must rise in proportion to beta. Therefore, if a portfolio has (let’s say) twice the beta against the macro factor as the market index, its beta with respect to the index will be 2, and it should have twice the risk premium. More generally, for any well-diversified portfolio P, the expected excess return must be:(10.6)In other words, the risk premium (i.e., the expected excess return) on portfolio P is the product of its beta and the risk premium of the market index. Equation 10.6 thus establishes that the SML of the CAPM must also apply to well-diversified portfolios simply by virtue of the “no-arbitrage” requirement of the APT.Individual Assets and the APT We have demonstrated that if arbitrage opportunities are to be ruled out, each well-diversified portfolio’s expected return must satisfy the SML predicted by the CAPM. The natural question is whether this relationship tells us anything about the expected returns on the component stocks. The answer is that if this relationship is to be satisfied by all well-diversified portfolios, it must be satisfied by almost all individual securities, although a rigorous proof of this proposition is somewhat difficult. We can illustrate the argument less formally.page 317 Suppose that the expected return–beta relationship is violated for all single assets. Now create a pair of well-diversified portfolios from these assets. What are the chances that in spite of the fact that for any pair of assets the relationship does not hold, the relationship will hold for both well-diversified portfolios? The chances are small, but it is perhaps possible that the relationships among the single securities are violated in offsetting ways so that somehow it holds for the pair of well-diversified portfolios.Now construct yet another well-diversified portfolio. What are the chances that the violations of the relationships for single securities are such that this third portfolio also will fulfill the no-arbitrage expected return–beta relationship? Obviously, the chances are smaller still. Continue with a fourth well-diversified portfolio, and so on. If the no-arbitrage expected return–beta relationship has to hold for each of these different, well-diversified portfolios, it must be virtually certain that the relationship holds for all but a small number of individual securities.We use the term virtually certain advisedly because we must distinguish this conclusion from the statement that all securities surely fulfill this relationship. The reason we cannot make the latter statement has to do with a property of well-diversified portfolios.Recall that to qualify as well diversified, a portfolio must have very small positions in all securities. If, for example, only one security violates the expected return–beta relationship, then the effect of this violation on a well-diversified portfolio will be too small to be of importance for any practical purpose, and meaningful arbitrage opportunities will not arise. But if many securities violate the expected return–beta relationship, the relationship will no longer hold even approximately for well-diversified portfolios, and arbitrage opportunities will be available. Consequently, we conclude that the no-arbitrage condition for a single-factor security market implies the expected return–beta relationship for all well-diversified portfolios and for all but possibly a small number of individual securities.Well-Diversified Portfolios in PracticeWhat is the effect of diversification on portfolio standard deviation in practice, where portfolio size is not unlimited? To illustrate, consider the residual standard deviation of a 1,000-stock portfolio with equal weights on each component stock. If the annualized residual standard deviation for each stock is σ(ei) = 40%, then the portfolio achieves a small but still not negligible standard deviation of .What is a “large” portfolio? Many widely held ETFs or mutual funds hold hundreds of different shares, but very few hold more than 1,000. Therefore, for plausible portfolios, even broad diversification is not likely to achieve the risk reduction of the APT’s “well-diversified” ideal. This is a shortcoming in the model. On the other hand, even the levels of residual risk attainable in practice should make the APT’s security market line at the very least a good approximation to the risk–return relation. We address the comparative strengths of the APT and the CAPM as models of risk and return in the next section.10.3 The APT and the CAPMThe APT serves many of the same functions as the CAPM. It gives us a benchmark for rates of return that can be used in capital budgeting, security valuation, or investment performance evaluation. Moreover, it highlights the crucial distinction between nondiversifiable risk (factor risk), which requires a reward in the form of a risk premium, and diversifiable risk, which does not.page 318 In many ways, the APT is an extremely appealing model. It does not require that almost all investors be mean-variance optimizers. Instead, it is built on the highly plausible assumption that a rational capital market will preclude arbitrage opportunities. A violation of the APT’s pricing relationships will cause extremely strong pressure to restore them. Moreover, the APT provides an expected return–beta relationship using as a benchmark a well-diversified index portfolio rather than the elusive and impossible-to-observe market portfolio of all assets that underpins the CAPM. When we replace the unobserved market portfolio of the CAPM with a broad, but observable, index portfolio, we can no longer be sure that this portfolio will be an adequate benchmark for the CAPM’s security market line.In spite of these apparent advantages, the APT does not fully dominate the CAPM. The CAPM provides an unequivocal statement on the expected return–beta relationship for all securities, whereas the APT implies that this relationship holds for all but perhaps a small number of securities. Because the APT is built on the foundation of well-diversified portfolios, it cannot rule out a violation of the expected return–beta relationship for any particular asset. Moreover, we’ve seen that even large portfolios may have non-negligible residual risk.In the end, however, it is noteworthy and comforting that despite the very different paths they take to get there, both models arrive at the same security market line. Most important, they both highlight the distinction between firm-specific and systematic risk, which is at the heart of all modern models of risk and return.10.4 A Multifactor APTSo far, we’ve examined the APT in a one-factor world. In reality, as we pointed out above, there are several sources of systematic risk such as uncertainty in the business cycle, interest rates, energy prices, and so on. Presumably, exposure to any of these factors will affect a stock’s appropriate expected return. The APT can be generalized to accommodate these multiple sources of risk in a manner much like the multifactor CAPM.Suppose that we generalize the single-factor model expressed in Equation 10.1 to a two-factor model:(10.7)In Example 10.2, factor 1 was the departure of GDP growth from expectations and factor 2 was the unanticipated change in interest rates. Each factor has 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, ei, also has zero expected value. Extending such a two-factor model to any number of factors is straightforward.The benchmark portfolios in the APT are factor portfolios, which are well-diversified portfolios constructed to have a beta of 1 on one of the factors and a beta of zero on any other factor. We can think of each factor portfolio as a tracking portfolio. That is, the returns on such a portfolio track the evolution of one particular source of macroeconomic risk but are uncorrelated with other sources of risk. It is possible to form such factor portfolios because we have a large number of securities to choose from, and a relatively small number of factors. The multifactor SML predicts that the contribution of each source of page 319risk to the security’s total risk premium equals the factor beta times the risk premium of the factor portfolio tracking that source of risk. We illustrate with an example.Example 10.3 Multifactor SMLSuppose that the two factor portfolios, portfolios 1 and 2, have expected returns E(r1) = 10% and E(r2) = 12% and that the risk-free rate is 4%. The risk premium on the first factor portfolio is 10% − 4% = 6%, and that on the second factor portfolio is 12% − 4% = 8%.Now consider a well-diversified portfolio, portfolio A, with beta on the first factor portfolio, βA1 = .5, and beta on the second factor portfolio, βA2 = .75. The multifactor APT states that the overall risk premium on this portfolio should equal the sum of the risk premiums required as compensation for each source of systematic risk.The risk premium attributable to risk factor 1 is the portfolio’s exposure to factor 1, βA1, 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 βA1 [E(r1) − rf] = .5(10% − 4%) = 3%. Similarly, the risk premium attributable to risk factor 2 is proportional to the exposure to that factor. The total equilibrium expected return on P therefore would be the sum of the risk-free rate plus total compensation for all sources of risk:4% Risk-free rate rf = 4%+ 3% Risk premium for exposure to factor 1 βP1 × [E(r1) − rf] = .5 × 6%+ 6% Risk premium for exposure to factor 2 βP2 × [E(r2) − rf] = .75 × 8%= 13% Total expected return rf + βP1[E(r1) − rf] + βP2[E(r2) − rf] To generalize Example 10.3, note that the factor exposures of any portfolio, P, are given by its betas, βP1 and βP2. A competing portfolio, Q, can be formed by investing in factor portfolios with the following weights: βP1 in the first factor portfolio, βP2 in the second factor portfolio, and 1 − βP1 − βP2 in T-bills. By construction, portfolio Q will have betas equal to those of portfolio P and expected return of(10.8)This is a two-factor SML, and, as Example 10.4 shows, any well-diversified portfolio with the same betas must have the same expected return as long as capital markets do not allow for easy arbitrage opportunities.Example 10.4 Mispricing and ArbitrageUsing the numbers in Example 10.3:Suppose the expected return on portfolio A from Example 10.3 were 12% rather than 13%. This return would give rise to the following arbitrage opportunity.page 320 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. (The weight of −.25 on risk-free T-bills does not affect the sensitivity to either factor.)Now invest $1 in portfolio Q and sell (short) $1 in portfolio A. Your net investment is zero, but your expected dollar profit is positive and equal toMoreover, your net position is riskless. Your exposure to each risk factor cancels out because you are long $1 in portfolio Q and short $1 in portfolio A, and both of these well-diversified portfolios have exactly the same factor betas. Thus, if portfolio A’s expected return differs from that of portfolio Q’s, you can earn positive risk-free profits on a zero-net-investment position. This is an arbitrage opportunity.Because portfolio Q in Example 10.4 has precisely the same exposures as portfolio A to the two sources of risk, their expected returns also ought to be equal. So portfolio A also ought to have an expected return of 13%. If it does not, then there will be an arbitrage opportunity and great pressure on prices until the opportunity is eliminated.3 We conclude that any well-diversified portfolio with betas βP1 and βP2 must have the expected return given in Equation 10.8.Concept Check 10.3Using the factor portfolios of Example 10.3, find the equilibrium rate of return on a portfolio with β1 = .2 and β2 = 1.4.Finally, the extension of the multifactor SML of Equation 10.8 to individual assets is precisely the same as for the one-factor APT. Equation 10.8 cannot be satisfied by every well-diversified portfolio unless it is satisfied approximately by individual securities. Equation 10.8 thus represents the multifactor SML for an economy with multiple sources of risk.We pointed out earlier that one application of the CAPM is to provide “fair” rates of return for regulated utilities. The multifactor APT can be used to the same ends. The nearby box summarizes a study in which the APT was applied to find the cost of capital for regulated electric companies. Notice that empirical estimates for interest rate and inflation risk premiums in the box are negative, as we argued was reasonable in our discussion of Example 10.2.page 321 WORDS FROM THE STREETUsing the APT to Find Cost of CapitalElton, Gruber, and Mei* use the APT to derive the cost of capital for electric utilities. They consider six potential systematic risk factors: unanticipated developments in the term structure of interest rates, the level of interest rates, inflation rates, the business cycle (measured by GDP), foreign exchange rates, and a summary measure they devise to measure other macro factors.Their first step is to estimate the risk premium associated with exposure to each risk source. They accomplish this in a two-step strategy (which we will describe in considerable detail in Chapter 13):Estimate “factor loadings” (i.e., betas) of a large sample of firms. Regress returns of 100 randomly selected stocks against the five systematic factors. They use a time-series regression for each stock (e.g., 60 months of data), therefore estimating 100 regressions, one for each stock.Estimate the reward earned per unit of exposure to each risk factor. For each month, regress the return of each stock against the five betas estimated. The coefficient on each beta is the extra average return earned as beta increases (i.e., it is an estimate of the risk premium for that risk factor from that month’s data). These estimates are of course subject to sampling error. Therefore, average the risk premium estimates across the 12 months in each year. The average response of return to risk is less subject to sampling error.The risk premiums are in the middle column of the table in the next column.Notice that some risk premiums are negative. The interpretation of this result is that risk premium should be positive for risk factors you don’t want exposure to, but negative for factors you do want exposure to. For example, you should desire securities that have higher returns when inflation increases and be willing to accept lower expected returns on such securities; this shows up as a negative risk premium.Factor Factor RiskPremium Factor Betas forNiagara MohawkTerm structure  0.425  1.0615Interest rates −0.051 −2.4167Exchange rates −0.049  1.3235Business cycle  0.041  0.1292Inflation −0.069 −0.5220Other macro factors  0.530  0.3046The study finds that average returns are related to factor betas as follows:Finally, to obtain the cost of capital for a particular firm, the authors estimate the firm’s betas against each source of risk, multiply each factor beta by the “cost of factor risk” from the table above, sum over all risk sources to obtain the total risk premium, and add the risk-free rate.For example, the beta estimates for Niagara Mohawk appear in the last column of the table above. Therefore, its cost of capital isIn other words, the monthly cost of capital for Niagara Mohawk is .72% above the monthly risk-free rate. Its annualized risk premium is therefore .72% × 12 = 8.64%.* Edwin J. Elton, Martin J. Gruber, and Jianping 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. 46–68.10.5 The Fama-French (FF) Three-Factor ModelThe APT shows us how multiple risk factors can result in a multifactor SML. But how can we identify the most likely sources of systematic risk? One approach comes from Merton’s multifactor CAPM, discussed in Chapter 9, in which the extra-market risk factors are due to hedging demands against a range of risks associated with either consumption or investment opportunities. Another approach, which is more pervasive today, uses firm characteristics that seem on empirical grounds to proxy for exposure to systematic risk. The factors chosen are variables that on past evidence have predicted average returns well and therefore may be capturing risk premiums. One example of this approach is the Fama and French three-factor model and its variants, which have come to dominate empirical research in security returns:4(10.9)page 322 where5SMB = Small Minus Big (i.e., the return of a portfolio of small stocks in excess of the return on a portfolio of large stocks).HML = High Minus Low (i.e., the return of a portfolio of stocks with a high book-to-market ratio in excess of the return on a portfolio of stocks with a low book-to-market ratio).Note that in this model the market index does play a role and is expected to capture systematic risk originating from macroeconomic factors.These two extra-market factors are chosen because of long-standing observations that firm size, measured by market capitalization (the market value of outstanding equity), and the book-to-market ratio (book value per share divided by stock price) predict deviations of average stock returns from levels consistent with the CAPM. Fama and French justify this model on empirical grounds: While SMB and HML are not themselves obvious candidates for relevant risk factors, the argument is that these variables may proxy for hard-to-measure more-fundamental variables. For example, Fama and French point out that firms with high book-to-market ratios are more likely to be in financial distress and that small stocks may be more sensitive to changes in business conditions. Thus, these variables may capture sensitivity to risk factors in the macroeconomy. More evidence on the Fama-French model appears in Chapter 13.The problem with empirical approaches such as the Fama-French model is that the extra-market factors in these models cannot be clearly identified with a source of risk that is of obvious concern to a significant group of investors. Black6 points out that when researchers scan and rescan the database of security returns in search of explanatory factors (an activity often called data-snooping), they may eventually uncover past “patterns” that are due purely to chance. However, Fama and French have shown that size and book-to-market ratios have predicted average returns in different time periods and in markets all over the world, thus mitigating potential effects of data-snooping.The risk premiums associated with Fama-French factors raise the question of whether they reflect a multi-index ICAPM based on extra-market hedging demands or just represent yet-unexplained anomalies, where firm characteristics are correlated with alpha values. This is an important distinction for the debate over the proper interpretation of the model because the validity of FF-style models may either signify a deviation from rational equilibrium (as there is no obvious reason to prefer one or another of these firm characteristics per se) or indicate that firm characteristics identified as empirically associated with average returns are correlated with other (harder to specify) risk factors.The issue is still unresolved and is revisited in Chapter 13.Estimating and Implementing a Three-Factor SMLIn Chapter 8, we estimated the single-index model for Amazon. Now we are ready to estimate a three-factor model and see what it implies about the equilibrium expected rate of return on Amazon stock. We begin by estimating Amazon’s beta on each of the page 323Fama-French factors. Therefore, we generalize regression Equation 10.1 of the single-factor model and fit the following multiple regression equation:(10.10)The three betas on the right-hand side of Equation 10.10 measure Amazon’s sensitivities to the three hypothesized sources of systematic risk: the market index (M), the value-versus-growth factor (HML), and the size factor (SMB).Table 10.1 shows estimates of both the single-factor and three-factor models. The intercept in each regression is the estimate of Amazon’s (monthly) alpha over the sample period. The coefficient on the market index excess return, rM − rf , is the estimate of Amazon’s market risk, while the coefficients on the returns of SMB and HML are estimates of the betas against the two extra-market risk factors.Amazon is a large firm and a growth firm (with a very low book-to-market ratio). Perhaps not surprisingly, then, its betas against both SMB and HML are negative and statistically significant. Given that significance, it is also not surprising that the R-square in the three-factor model is substantially greater than in the single-factor model, and the residual standard deviation is lower. While Amazon’s beta against the market index is higher in the three-factor model, that systematic risk is offset to some degree by its negative betas on the two extra-market risk factors.Using the three-factor model to estimate expected returns requires forecasts of the premiums on the two extra-market risk premiums as well as that on the market index. Suppose the risk-free rate is rf = 1%, the market risk premium is E(rM) − rf = 6%, and the risk premiums on SMB and HML are each 2%. Then Amazon’s expected return would be calculated using these benchmark risk premiums and the betas estimated in Table 10.1:If, instead, we used the beta estimate from the single-index model, we would obtain an expected return ofTable 10.1Estimates of single-index and three-factor Fama-French regressions for Amazon, monthly data, 5 years ending June 2018.Source: Authors’ calculations.page 324 Amazon’s expected return in the multifactor model is lower due to the considerable hedging value it offers against the size and value risk factors. This is ignored by the single-factor model.Notice that in neither case do we add the regression estimate of alpha to the forecast of expected return. Equilibrium expected returns depend only on risk; the expectation of alpha in market equilibrium must be zero. While a security may have outperformed its benchmark return in a particular sample period (as reflected in a positive alpha), we would not expect that performance to continue into the future. In fact, as an empirical matter, individual firm alphas show virtually no persistence over time.Smart Betas and Multifactor ModelsWe’ve seen that both the CAPM and the APT have multifactor generalizations. There are at least two important implications of these generalizations. First, investors should be aware that their portfolios are subject to more than one systematic source of risk, and that they need to think about how much exposure they wish to establish to each systematic factor. Second, when they evaluate investment performance, they should be aware that risk premiums come from exposure to several risk factors and that alpha needs to be computed controlling for each of them.A new product called smart-beta ETFs has important implications for both of these issues. They are analogous to index ETFs, but instead of tracking a broad market index using market capitalization weights, they are funds designed to provide exposure to specific characteristics such as value, growth, or volatility. Among the more prominent themes are the extra-market factors of the Fama-French three-factor model: size (SMB) and value (HML). Another common factor is momentum (WML, for Winners Minus Losers), which is the return on a portfolio that buys recent well-performing stocks and sells poorly performing ones. Other recently considered factors are based on volatility, as measured by the standard deviation of stock returns; quality, the difference in returns of stocks with high versus low return on assets or similar measures of profitability; investment, the difference between returns on firms with high versus low rates of asset growth; and dividend yield.Smart-beta ETFs allow investors to tailor portfolio exposures either toward or away from a range of extra-market risk factors using easy-to-trade index-like products. They are therefore well-suited to a multifactor environment. They also raise the question of appropriate performance evaluation. When investors can cheaply and effectively manage exposure to multidimensional sources of systematic risk, investment success is captured by a multifactor alpha, providing a clean measure of the success of security selection. This is the message of the multifactor SML presented in Equation 7.13.In this regard, interesting preliminary research by Cao, Hsu, Xiao, and Zhan7 indicates that since smart-beta ETFs have been introduced, investors have increasingly shifted toward multifactor SMLs in assessing mutual fund performance. Their evidence is that since the advent of active trading in these ETFs, the flow of money into or out of mutual funds has more closely tracked multifactor alphas rather than traditional one-factor CAPM alphas.page 325 SUMMARYMultifactor models seek to improve the explanatory power of single-factor models by explicitly accounting for the various components of systematic risk. These models use indicators intended to capture a wide range of macroeconomic risk factors.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.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.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.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.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 individual securities also must satisfy this relationship, at least approximately.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.A multifactor APT generalizes the single-factor model to accommodate several sources of systematic 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.The multifactor extension of the single-factor CAPM, the ICAPM, predicts the same multidimensional security market line as the multifactor APT. The ICAPM suggests that priced extra-market risk factors will be the ones that lead to significant hedging demand by a substantial fraction of investors. Other approaches to the multifactor APT are more empirically based, where the extra-market factors are selected based on past ability to predict risk premiums.