Expected Returns and the Arbitrage Pricing Theory Introduction We have now completed our treatment of fundamentals. The next three chapters cover expected returns and valuation. The arbitrage pricing theory (APT) is an interesting and powerful alternative to the CAPM for forecasting expected returns. This chapter describes the APT and emphasizes its implications for active managers. The conclusions are: • The APT is a model of expected returns. • Application of the APT is an art, not a science. • The APT points the quantitative manager toward the relationship between factors and expected returns. • APT factors can be defined in a multitude of ways. These may be fundamental, technical, or macro factors. • The flexibility of the APT makes it inappropriate as a model for consensus expected returns, but an appropriate model for a manager's expected returns. • The APT is a source of information to the active manager. It should be flexible. If all active managers shared the same information, it would be worthless. The APT requires less stringent assumptions than the CAPM and produces similar results. This makes it sound as if the APT is a dominant theory. The difficulty is that the APT says that it ispossible to forecast expected stock returns but it doesn't tell you
Page 174 how to do so. It has been called the ''arbitrary" pricing theory for just this reason. The CAPM, in contrast, comes with a user's manual. The APT states that each stock's expected excess return is determined by the stock's factor exposures. The link between the expected excess return and the stock factor exposures is described in Eq. (7.2). For each factor, there is a weight (called a factor forecast) such that the stock's expected excess return is the sum over all the factors of the stock's factor exposures times the factor forecasts. The theory doesn't say what the factors are, how to calculate a stock's exposure to the factors, or what the weights should be in the linear combination. This is where science steps out and art steps in. In discussing the APT, one should be careful to distinguish among • Stories that motivate the APT. These usually involve basic economic forces that alter the relative valuation of stocks. The motivating stories may mislead some people into thinking that it is necessary for the APT to be based on exogenous macroeconomic factors. The applications described in this chapter indicate that this is not the case. • Attempts to implement the APT. The APT is by nature arbitrary. Different individuals' attempts to implement it will take different forms. One should not confuse a particular implementation with the theory. • The theory. The technical theory has evolved since its origins in the mid- to late 1970s. This chapter will provide an idiosyncratic view of the theory. Other ways to look at the APT are cited in the chapter notes. This chapter will first detail some weaknesses of the CAPM that the APT was designed to correct. It will then describe the APT and its evolution as a theory. The final sections of this chapter will deal with the problem of implementation and give some examples of ways in which people either have tried or could try to implement APT models. Trouble with the CAPM The CAPM is based on the notion that the market portfolio is not only mean/variance-efficient but, in fact, the fully invested
Page 175 portfolio with the highest ratio of expected excess return to volatility. In practice, the theory has been applied to say that common, broad-based stock indices are efficient: the S&P 500 in the United States, the FTA in the United Kingdom, and the TSE1 in Japan. If we consider a broader notion of the market, including all bonds, international investments, and other assets such as precious metals, real property, etc., then we can see that the market consists of more than the local stock index. Even if the CAPM is true in some broader context of a worldwide portfolio, it cannot be valid in the restricted single-market world in which it is ordinarily applied. All of the other assumptions underlying the CAPM (mean/ variance preferences, identical expectations of mean and variance, no taxes or transactions costs, no restrictions on stock positions, etc.) can be challenged, adding additional wounds. The most grievous of these is the CAPM requirement that all participants know every stock's expected excess return. This assumption should be viewed in the context of our quest to get a handle on the expected excess returns in the first place! Thus we would suspect that the CAPM can be only approximately true. It provides a guideline that should be neither ignored nor taken as gospel. One dramatic episode that points out the weakness of the CAPM occurred for U.S. equities in 1983 and 1984, during a period characterized by a considerable drop in interest rates. The equities most adversely affected had high betas, and the equities most beneficially affected had low betas. We can illustrate this episode by a simple experiment. In December 1982, take the stocks in the S&P 500, order them according to their predicted beta, and form ten portfolios, each with an equal amount of capitalization. Portfolio 1 has the lowest-beta stocks, portfolio 2 the next lowest group, and so on, with portfolio 10 holding the highest-beta stocks. Then follow the capitalization-weighted returns on these portfolios for the next 24 months. It turns out that over the out-of-sample period, the predicted betas were excellent forecasts of the realized betas. No problem here. In Fig. 7.1, we see the scatter diagram of predicted versus realized beta. The regression line in Fig. 7.1 has a slope of 0.93, and the R2 of this regression was 0.89. The prediction of beta was quite accurate. The CAPM would say that the alpha of each portfolio should be zero. It didn't turn out that way. Not only were several of
Page 176 Figure 7.1 the alphas significantly different from zero, they were perversely related to the portfolio betas. The results are shown in the scatter diagram in Fig. 7.2. Remember that we already checked our predictions of beta and found that they were quite accurate. The explanation for this event must lie elsewhere. Something, most likely changes in interest rates and changes in inflationary expectations, was making higher-beta stocks have negative residual returns and lower-beta stocks have higher residual returns.1 This pattern was common throughout the early 1980s. There are periods in which CAPM predictions of expected excess returns appear to have systematic defects. This episodic evidence is meant to be suggestive, not to dash the CAPM once and for all. In fact, financial statisticians have 1This points out the hazards of benchmark timing by tilting the portfolio toward higher-beta stocks. The market was up considerably over this period; however, higher-beta stocks had relatively bad results. Benchmark timing with futures would have been much more effective, since it did not entail a residual bet on the high-beta stocks.
Page 177 Figure 7.2 thrown considerable empirical sophistication into attempts to prove or disprove the validity of the CAPM, without coming to any hard-and-fast conclusions.2 Those efforts should be enough to convince the investor to value the notion that the market plays a central—indeed, the most important—role in the formation of expected returns. However, the example should be enough to convince you that it is worthwhile to look for further explanations. The APT The APT postulates a multiple-factor model of excess returns. It assumes that there are K factors such that the excess returns can be expressed as 2Most recently, Fama and French (1992) have attempted to discredit the CAPM, with considerable publicity. However, as described in Black (1993) and Grinold (1993), these results require careful scrutiny.
Page 178 where Xn,k = the exposure of stock n to factor k. The exposures are frequently called factor loadings. For practical purposes, we will assume that the exposures are known before the returns are observed. bk = the factor return for factor k. These factor returns are either attributed to the factors at the end of the period or observed during the period. un = stock n's specific return, the return that cannot be explained by the factors. It is sometimes called the idiosyncratic return to the stock. We impose very little structure3 on the model. The astute reader will note that the APT model, Eq. (7.1), is identical in structure to the structural risk model, Eq. (3.16). However, the focus is now on expected returns, not risk. The APT is about expected excess returns. The main result is that we can express expected excess returns in terms of the model's factor exposures. The APT formula for expected excess return is where mk is the factor forecast for factor k. The theory says that a correct factor forecast will exist. It doesn't say how to find it. The APT maintains that the expected excess return on any stock is determined by that stock's factor exposures and the factor forecasts associated with those factors. Examples We can breathe some life into this formula by considering a few examples. The CAPM is the first example. For the CAPM, we have one factor, K = 1. The stock's exposure to that factor is the stock's beta; i.e., Xn,1 = βn. The expected return associated with the factor is m1 = E{rM} = fM, the expected excess return on the market. 3We assume that the specific returns u(n) are uncorrelated with the factor returns b(k); i.e., Cov{b(k),u(n)} = 0. In most practical applications, we assume that the specific returns are uncorrelated with each other; i.e., Cov{u (n),u(m)} = 0 if m n; however, we do not need this second assumption for the theory to hold.
Page 179 The second example is more substantial. We first classify stocks by their industry membership, then look at four other attributes of the companies. These attributes are chosen for the purpose of example only. (Why these attributes, and why defined in just this way? That question reinforces the notion that the APT model will be in the eye of the model builder. It will be as much a matter of taste as anything else.) The four attributes selected are • A forecast of earnings growth based on the IBES consensus forecast and past realized earnings growth • Bond beta: the response of the stock to returns on an index of government bonds • Size: the natural logarithm of equity capitalization • Return on equity (ROE): earnings divided by book The four attributes include a forecast (earnings growth), a macroeconomic characteristic (bond beta), a firm characteristic (size), and a fundamental data item (return on equity). It is easier to make comparisons across factors if they are all expressed in the same fashion. One way to do this is to standardize the exposure by subtracting the market average from each attribute and dividing by the standard deviation. In that way, the average exposure will equal zero, roughly 66 percent of the stocks will have exposures running from –1 up to +1, and only 5 percent of the stocks will have exposures above +2 or below –2. Table 7.1 displays the primary industry, standardized exposures to these four factors, and predicted beta for the 20 stocks in the Major Market Index as of December 1992. Note that these exposures are standardized across a broad universe of over 1000 stocks, so, for example, the size factor exposures for the Major Market Index stocks are all positive. The APT forecast is based on the factor forecasts for the four factors and a forecast for the chemical industry. The factor forecasts are 2.0 percent for growth, 2.5 percent for bond beta, –1.5 percent for size, and 0.0 percent for return on equity. (These forecasts are for illustration only.) We think that growth firms will do well, interest-rate-sensitive stocks (which are usually highly leveraged firms) will do well, smaller stocks will do well, and return on equity will be irrelevant. In addition, we forecast 8 percent for the chemical industry, and 6 percent for all other industries.
TABLE 7.1 Stock Industry Growth Bond β Size ROE American Express Financial services 0.17 –0.05 0.19 –0.28 AT&T Telephones –0.16 0.74 1.47 –0.59 Chevron Energy reserves and production –0.53 –0.24 0.83 –0.72 Coca-Cola Food and beverage –0.02 0.30 1.41 1.48 Disney Entertainment 0.13 –0.86 0.71 0.42 Dow Chemical Chemical –0.64 –0.92 0.48 0.22 DuPont Chemical –0.10 –0.74 1.05 –0.41 Eastman Kodak Leisure –0.19 –0.30 0.39 –0.55 Exxon Energy reserves and production –0.67 0.03 1.67 –0.27 General Electric Heavy electrical –0.24 0.13 1.56 0.15 General Motors Motor vehicles 2.74 –1.80 0.73 –1.24 IBM Computer hardware 0.51 –0.62 1.16 –0.62 International Paper Forest products and paper –0.23 –1.08 0.01 –0.49 Johnson & Johnson Medical products –0.12 0.68 1.06 0.78 McDonalds Restaurant –0.16 0.28 0.55 0.24 Merck Drugs –0.04 0.46 1.37 2.28 3M Chemical –0.22 –0.69 0.78 0.20 Philip Morris Tobacco –0.01 0.30 1.60 1.22 Procter & Gamble Home products –0.32 0.80 1.12 0.41 Sears Department stores –0.34 –1.29 0.45 –0.69
Page 181 TABLE 7.2 Stock Industry APT CAPM APT- CAPM American Express Finance services 5.93% 6.96% –1.03% AT&T Telephones 5.33% 5.04% 0.29% Chevron Energy reserves and production 3.10% 4.20% –1.11% Coca-Cola Food and beverage 4.60% 6.36% –1.77% Disney Entertainment 3.05% 6.78% –3.74% Dow Chemical Chemical 3.70% 6.78% –3.08% DuPont Chemical 4.38% 5.58% –1.21% Eastman Kodak Leisure 4.29% 5.64% –1.36% Exxon Energy reserves and production 2.23% 4.26% –2.03% General Electric Heavy electrical 3.51% 6.60% –3.10% General Motors Motor vehicles 5.89% 7.50% –1.62% IBM Computer hardware 3.73% 6.66% –2.93% International Paper Forest products and paper 2.83% 6.48% –3.66% Johnson & Johnson Medical products 5.87% 6.42% –0.55% McDonalds Restaurant 5.56% 6.36% –0.81% Merck Drugs 5.02% 6.60% –1.59% 3M Chemical 4.67% 5.46% –0.80% Philip Morris Tobacco 4.33% 6.12% –1.79% Procter & Gamble Home products 5.68% 6.30% –0.62% Sears Department stores 1.42% 6.60% –5.18% The CAPM forecast is for 6 percent expected market excess return. We display the final stock forecasts in Table 7.2. Here we display the APT forecast, the CAPM forecast, and the residual APT forecast (the APT forecast less the CAPM forecast). We can see from this example that whatever the use of an APT model in forecasting expected returns, it can make a great talking point. By putting in different expectations for the various factors, we could automatically generate mountains of research. The APT Rationale The APT result follows from an arbitrage argument. What would happen if we had a returns model like Eq. (7.1), and the APT
Page 182 relationship [Eq. (7.2)] failed to hold? We could find an active position with zero exposure to all of the factors4 and expected excess return of 1 percent. Since the active position has zero factor exposure, it has almost no risk, and so we can achieve 1 percent expected excess return at low risk levels. We can therefore also add this active position to any portfolio P and improve our performance by increasing expected excess return with little additional risk. Arbitrage means a certain gain at zero expense. In the case described above, we have a nearly certain gain, since the expected return is positive and the risk is very, very small. There is no expense, since it is an active portfolio (zero net investment). We call this second situation a quasiarbitrage opportunity. If we rule out these quasi-arbitrage opportunities, then the APT must hold. How do we find the factor model that will do this wonderful trick? The APT is silent on that question. However, a little old-fashioned mean/standard deviation analysis can provide some clues. Portfolio Q and the APT The APT is marvelously flexible. We can concentrate on any desired group of stocks.5 Among any group of N stocks, there will be an efficient frontier for portfolios made up of the N risky stocks. Figure 7.3 shows a typical frontier. With each portfolio we can associate a risk, as measured by the standard deviation of its returns, and a reward, as measured by the expected excess return. Portfolio Q in Fig. 7.3 has the highest rewardto-risk ratio (Sharpe ratio). Knowledge of portfolio Q is sufficient to calculate all expected returns.6 Portfolio Q plays the same role as the market portfolio does in the CAPM; in fact, the CAPM is another way of saying that portfolio Q is the market portfolio.7 4This will typically imply net zero investment, unless the factor model does not have an intercept. Even then, we have built a portfolio with almost no risk and 1 percent expected excess return. We can combine this with the risk-free asset for a net zero investment with an expected excess return and almost no risk. 5Contrast this with the CAPM, where coverage should, in theory, be universal. 6And vice versa. See Chap. 2 for more details. 7See the technical appendix to Chap. 2.
Page 183 Figure 7.3 The expected excess return on any stock will be proportional to that stock's beta with respect to portfolio Q. We might assume that we can stop here. We cannot, since we don't know portfolio Q. However, all is not lost. We can learn something from portfolio Q. We need to separate two issues at this point. The first is defining a qualified model, and the second is finding the correct set of factor forecasts. A multiple-factor model [as in Eq. (7.1)] is qualified if it is possible to find factor forecasts m(k) such that Eq. (7.2) holds.
Page 184 Once we have a qualified model, we still must come up with the correct factor forecasts m(k). We argue in the next section that it should not be hard to find a qualified model. However, in the section following that, we argue that the ability to make correct factor forecasts requires both considerable skill and a model linked to investment intuition. The Easy Part: Finding a Qualified Model This section describes the technical properties required for a multiple-factor model to qualify for successful APT implementation. The details are in the technical appendix. More significantly, we argue by example that we can fairly easily find qualified models. Let's start with the technical result. A factor model of the type described in Eq. (7.1) is qualified, i.e., Eq. (7.2), will hold for some factor forecasts mk, if and only if portfolio Q is diversified with respect to that factor model. Diversified with respect to the factor model means that among all portfolios with the same factor exposures as portfolio Q, portfolio Q has minimum risk. No other portfolio with the same factor exposures is less risky than portfolio Q. How do we translate this technical rule into practice? A frontier portfolio like Q should be highly diversified in the conventional sense of the word. Portfolio Q will contain all the stocks, with no exceptionally large holdings. We want portfolio Q to be diversified with respect to the multiple-factor model. Common sense would say that any multiple-factor model that captures the important differential movements in the stocks will qualify. The arbitrary aspect of the APT can be a benefit as well as a drawback! It's a drawback in that we don't know exactly what to do; it's a benefit in that it suggests that there are reasonable steps that will possibly succeed. We can check this idea by devising some surrogates for portfolioQ and seeing if they are diversified with respect to the BARRA U.S. Equity model. The BARRA model was constructed to help portfolio managers control risk, not to explain expected returns. However, it does attempt to capture those aspects of the market that cause some groups of stocks to behave differently from others.
Page 185 TABLE 7.3 Portfolio Number of Assets Unexplained Variance (%) MMI 20 2.51 OEX 100 0.99 S&P 500 500 0.30 FR1000 1000 0.21 FR3000 3000 0.18 ALLUS ~6000 0.17 We find that well over 99 percent of the variance of highly diversified portfolios is captured by the factor component. Table 7.3 shows the percentage of total risk that is not explained by the model for several major U.S. equity indices as of December 1992. The ALLUS portfolio includes approximately the 6000 largest U.S. companies, which are followed by BARRA. Table 7.3 indicates that portfolios are highly diversified with respect to the BARRA model. It may be possible to find a portfolio that has the same factor exposures as the Frank Russell 3000 index and less risk. However, the two portfolios would have the same factor risk (recall that they have the same factor exposures), and so they can differ only in their specific risk. Since 99.82 percent of the risk is explained by the factor component of return, there is very little margin for improvement. We cannot find portfolios that have the same factor exposures and substantially less risk, since the amount of nonfactor risk is already negligible. Any factor model that is good at explaining the risk of a diversified portfolio should be (nearly) qualified as an APT model. The exact specification of the factor model may not be important in qualifying a model. What is important is that the model contain sufficient factors to capture movement in the important dimensions. The Hard Part: Factor Forecasts We have settled one part of the implementation question. Any reasonably robust risk model in the form of Eq. (7.1) should qualify.
Page 186 The next step is to find the amount of expected excess return, mk in Eq. (7.2), to associate with each factor. Forecasting the mk is not easy just because we may have 1000 stocks and only 10 factors. If it were true that ''fewer is better," then we could just concentrate on forecasting the returns to the bond market and the stock market. In fact, as the fundamental law of active management states, it is better to make more forecasts than fewer for a given level of skill. If we can forecast expected returns on 1000 stocks or 10 factors, then to retain the same value added, we need ten times more skill in forecasting the factor returns than in forecasting the stock returns. The simplest approach to forecasting mk is to calculate a history of factor returns and take their average. This is the forecast that would have worked best in the past—i.e., a backcast rather than a forecast. If we hope that the past average helps in the future, we are implicitly assuming an element of stationarity in the market. The APT does not provide any guarantees here. However, there is hope. One of the non-APT reasons to focus on factors is the knowledge that the factor relationship is more stable than the stock relationship. For example, it is probably more valuable to know how growth stocks have done in the past than to know the past returns of a particular stock that is currently classified as a growth stock. The problem is that the stock may not have been classified as a growth stock over the earlier period; the stock may have changed its stripes. However, the factor returns will give us some information on how it may perform in its current set of stripes! Haugen and Baker (1996) have proposed an APT model whose factor forecasts rely simply on historical factor returns. Their factors roughly resemble the BARRA model, except that they replace more than 50 industries with 10 sectors, and they replace 13 risk indices with 40 descriptors.8 Each factor forecast is then simply the trailing 12-month average of the factor returns. So they forecast the finance sector return, the IBES estimated earnings-to-price factor return, etc., as their past 12-month averages. Model structure can be very helpful in developing good forecasts. As we'll see in the next section, APT models can be either purely statistical or structural. The factors have some meaning in 8This leads to collinearity, which the BARRA model attempts to avoid.
Page 187 the structural model; they don't in a purely statistical model. In statistical models, we have very little latitude in forecasting the factor returns. In a structural model, with factors that are linked to specified characteristics of the stocks, factor forecasts can be interpreted as forecasts for stocks that share a similar characteristic. We can apply both common sense and alternative statistical procedures to check those forecasts. Factor forecasts are easier if there is some explicit link between the factors and our intuition. Consider, for example, a "bond market beta" factor. This factor will show how the stock will react as bond prices (i.e., interest rates) change. A forecast for this factor is in fact a forecast for the bond market. This doesn't mean that forecasting future interest rates is easy. It means that the knowledge that you are, in fact, forecasting interest rates should make the task clearer. A similar result would hold for a factor defined in terms of a stock's fundamentals. Consider a "growth" factor, with consensus growth expectations as the factor exposures. Now our forecast is an expression of our outlook for growth stocks. Once again, we haven't guaranteed that we can forecast the outlook for growth stocks correctly. We have simply made the task more explicit, and thus given ourselves a greater chance for success. This suggests an opportunistic approach to building an APT model. We should take advantage of our conclusion that we can easily build qualified APT models. We should use factors that we have some ability to forecast. Suppose we are reasonably good at forecasting returns to gold, oil, bonds, yen, etc., and we have some skill at predicting economic fluctuations. We should work from our strengths and build an APT model based on those factors, then round out the model with some others (industry variables) to capture the bulk of the risk. There is no use building the world's greatest (most qualified) APT model if we cannot come up with the factor forecasts. Factor forecasts are difficult. Structure should help. The next section describes some approaches to building an APT model. Applications We have tried to emphasize the flexible nature of the APT. There are many ways to build APT models. The arbitrary nature of the
Page 188 APT leaves enormous room for creativity in implementation. Two equally well-informed scholars working independently will not come up with similar implementations. We've given six illustrations here. They fall into two catagories, structural and statistical. Structural models postulate some relationships between specific variables. The variables can be macroeconomic (unanticipated inflation, change in interest rates, etc.), fundamental (growth in earnings, return on equity, market share), or market-related (beta, industry membership). All types of variables can be used in one model. Practitioners tend to prefer the structural models, since these models allow them to connect the factors with specific variables and therefore link their investment experience and intuition to the model. Statistical models line up the returns data and turn a crank. Academics build APT models to test various hypotheses about market efficiency, the efficacy of the CAPM, etc. Academics tend to prefer the purely statistical models, since they can avoid putting their prejudgments into the model in that way. There are a great many ways to build an APT model. Structural Model 1: Given Exposures, Estimate Factor Returns The BARRA model, described in detail in Chap. 3, "Risk," can function as an APT model. As we saw above, it qualifies with ease, and it is just as easy (i.e., not very) to forecast returns to the BARRA factors as to any others. The BARRA model takes the factor exposures as given based on current characteristics of the stocks, such as their earnings yield and relative size. The factor returns are estimates. Structural Model 2: Given Factor Returns, Estimate Exposures In this model, the factor returns are given. For example, take the factor returns as the return on the value-weighted NYSE, gold, a government bond index, a basket of foreign currencies, and a basket of traded commodities. Set the exposure of each stock to the NYSE equal to 1. For the other factors, determine the past exposure of
Page 189 the stock to the factor returns by regressing the difference between the stock return and the NYSE return on the returns of the other factors. The factor forecasts are forecasts of the future values of the factor returns. Note that we hope that the estimated factor exposures are stable over time. Structural Model 3: Combine Structural Models 1 and 2 This model is the inevitable hybrid of structural models 1 and 2: Start with some primitive factor definitions, estimate the stock's factor exposure as in structural model 2, then attribute returns to the factors as in structural model 1. Statistical Model 1: Principal Components Analysis Look at the returns of a collection of stocks, or portfolios of stocks, over many months—say 50 stocks over 200 months. Calculate the 50 by 50 matrix of realized covariance between these stocks over the 200 months. Do a principal components analysis of the 50 by 50 covariance matrix. Typically, one will find that the first 20 components will explain 90 percent or more of the risk. Call these 20 principal component returns the factors. These factors are purely statistical constructs. We might as well call them Dick, Jane, Spot, . . . or red, green, blue. . . . The analysis will tell us the exposures of the 50 stocks to the factors. It will also give us the returns on those factors over the 200 months. The factor returns will be uncorrelated. We can determine the exposures to the factors of stocks that were not included in the original group by regressing the returns of the new stocks on the returns to the factors. The regression coefficients will measure the exposures to the factors; the fact that the factor returns are uncorrelated is useful at this stage. To implement this model, we need a forecast of the mk. The obvious forecast is the historical average of the factor returns. It may be the only possible forecast: Since the factors are by construction abstract, it would be hard to justify a forecast that differed from the historical average.
Page 190 Statistical Model 2: Maximum Likelihood Factor Analysis Here we perform a massive maximum likelihood estimation, looking at Eq. (7.1) over 60 months. To make this possible, we assume that the stock's exposures Xn,k are constant over the 5-year period. If we applied this to 500 stocks over 60 months and looked for 10 factors, we would be using 500 · 60 = 30,000 returns to estimate 500 · 10 = 5000 exposures and 60 · 10 = 600 factor returns. Statistical Model 3: The Dual of Statistical Model 2 This is quite imaginative. A detailed description is difficult, but see Connor and Korajczyk (1988). When N stocks are observed over T time periods, N is usually much greater than T. Instead of analyzing the principal components of the N by N historical covariance matrix, we look at the T by T matrix of covariances. This analysis reverses the role of factor exposure and factor return! Summary We have described the APT and talked about its relevance for active management. The APT is a powerful theory, but difficult to apply. The APT does not in any way relieve the active manager of the need for insight and skill. It is a framework that can help skilled and insightful active managers harness their abilities. Problems 1. According to the APT, what are the expected values of the un in Eq. (7.1)? What is the corresponding relationship for the CAPM? 2. Work by Fama and French, and others, over the past decade has identified size and book-to-price ratios as two critical factors determining expected returns. How would you build an APT model based on those two factors? Would the model require additional factors?
Page 191 3. In the example shown in Table 7.2, most of the CAPM forecasts exceed the APT forecasts. Why? Are APT forecasts required to match CAPM forecasts on average? 4. In an earnings-to-price tilt fund, the portfolio holdings consist (approximately) of the benchmark plus a multiplec times the earnings-to-price factor portfolio (which has unit exposure to earnings-toprice and zero exposure to all other factors). Thus, the tilt fund manager has an active exposure c to earnings-to-price. If the manager uses a constant multiple c over time, what does that imply about the manager's factor forecasts for earnings-to-price? 5. You have built an APT model based on industry, growth, bond beta, size, and return on equity (ROE). This month your factor forecasts are Heavy electrical industry 6.0% Growth 2.0% Bond beta –1.0% Size –0.5% ROE 1.0% These forecasts lead to a benchmark expected excess return of 6.0 percent. Given the following data for GE, Industry Heavy electrical Growth –0.24 Bond beta 0.13 Size 1.56 ROE 0.15 Beta 1.10 what is its alpha according to your model? Notes In the early 1970s, Sharpe (1977) (the paper was written in 1973), Merton (1973), and Rosenberg (1974) advocated multiple-factor approaches to the CAPM. Their arguments were based on reasoning similar to that underlying the CAPM. The results were a multiple-
Page 192 beta form of the CAPM, identical in form to Eq. (7.2). In fact, an active money management product called a "yield tilt fund" was launched based on the notion that higher-yielding stocks had higher expected returns. In the mid-1970s, Ross (1976) proposed a different way of looking at expected stock returns. Ross used the notion of arbitrage, which was the foundation of Black and Scholes's work on the valuation of options. In certain cases the mere ruling out of arbitrage opportunities is sufficient to produce an explicit formula for a stock's value. Later other authors, notably Connor (1984) and Huberman (1982), added additional assumptions and structure to produce the exact form of Eq. (7.2). Modern theoretical treatments of the APT reserve a role for the market portfolio. See Connor (1986) for a discussion. Bower, Bower, and Logue (1984); Roll and Ross (1984); and Sharpe (1984) have expositions of the APT aimed at professionals. See also Rosenberg (1981) and Rudd and Rosenberg (1980) for a discussion of the CAPM and APT. The text by Sharpe and Alexander (1990) has an excellent discussion. Applications of the APT are described in Roll and Ross (1979), Chen, Roll, and Ross (1986), Lehmann and Modest (1988), and Connor and Korajczyk (1986). For a discussion of some of the econometric and statistical issues surrounding the APT, see Shanken (1982), Shanken (1985), and the articles cited in those papers. Actual implementations of APT models are described by Roll and Ross; Chen, Roll, and Ross; Lehmann and Modest; Connor and Korajczyk, etc. References Black, Fischer. "Estimating Expected Returns." Financial Analysts Journal, vol. 49, no. 5, 1993, pp. 36–38. Bower, D. H., R. S. Bower, and D. E. Logue. "A Primer on Arbitrage Pricing Theory." The Midland Journal of Corporate Finance, vol. 2, no. 3, 1984, pp. 31-40. Chamberlain, G., and M. Rothschild. "Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Asset Markets." Econometrica vol. 51, no. 5, 1983, pp. 1281–1304. Chen, N., R. Roll, and S. Ross. "Economic Forces and the Stock Market." Journal of Business, vol. 59, no. 3, 1986, pp. 383–404.
Page 193 Connor, Gregory. "A Unified Beta Pricing Theory." Journal of Economic Theory, vol. 34, no. 1, 1984, pp. 13–31. ———. "Notes on the Arbitrage Pricing Theory." In Frontiers of Financial Theory, edited by G. Constantinides and S. Bhattacharya (Boston: Rowman and Littlefield, 1986). Connor, Gregory, and Robert A. Korajczyk. "Performance Measurement with the Arbitrage Pricing Theory." Northwestern University working paper, 1986. ———. "Risk and Return in an Equilibrium APT: Application of a New Test Methodology." Journal of Financial Economics vol. 21, no. 2, 1988, pp. 255–290. Fama, Eugene F., and Kenneth R. French. "The Cross-Section of Expected Stock Returns." Journal of Finance, vol 67, no. 2, 1992, pp. 427–465. Grinold, R. "Is Beta Dead Again?" Financial Analysts Journal, vol 49, no. 4, 1993, pp. 28–34. Haugen, Robert A., and Nardin L. Baker. "Commonality in the Determinants of Expected Stock Returns." Journal of Financial Economics, vol. 41, no. 3, 1996, pp. 401–439. Huberman, G. "A Simple Approach to Arbitrage Pricing Theory." Journal of Economic Theory, vol. 28, 1982, pp. 183–191. Lehmann, Bruce N., and David Modest. "The Empirical Foundations of the Arbitrage Pricing Theory." Journal of Financial Economics, vol. 21, no. 2, 1988, pp. 213–254. Mayers, D., and E. M. Rice. "Measuring Portfolio Performance and the Empirical Content of Asset Pricing Models." Journal of Financial Economics, vol. 7, no. 2, 1979, pp. 3–28. Merton, R. C. "An Intertemporal Capital Asset Pricing Model." Econometrica, vol. 41, no. 1, 1973, pp. 867–887. Pfleiderer, P. "A Short Note on the Similarities and the Differences between the Capital Asset Pricing Model and the Arbitrage Pricing Theory." Stanford University Graduate School of Business working paper, 1983. Roll, Richard. "A Critique of the Asset Pricing Theory's Tests." Journal of Financial Economics, vol. 4, no. 2, 1977, pp. 129–176. Roll, Richard, and Stephen A. Ross. "An Empirical Investigation of the Arbitrage Pricing Theory." Journal of Finance, vol. 35, no. 5, 1979, pp. 1073–1103. ———. "The Arbitrage Pricing Theory Approach to Strategic Portfolio Planning."Financial Analysts Journal, vol. 40, no. 3, 1984, pp. 14–26. Rosenberg, Barr. "Extra-Market Components of Covariance in Security Returns."Journal of Financial and Quantitative Analysis, vol. 9, no. 2, 1974, pp. 263–274. ———. "The Capital Asset Pricing Model and the Market Model." Journal of Portfolio Management, vol. 7, no. 2, 1981, pp. 5–16. Ross, Stephen A. "The Arbitrage Theory of Capital Asset Pricing." Journal of Economic Theory, vol. 13, 1976, pp. 341–360. Rudd, Andrew, and Henry K. Clasing, Jr. Modern Portfolio Theory, 2d ed. (Orinda, Calif.: Andrew
Rudd, 1988). Rudd, Andrew, and Barr Rosenberg. "The 'Market Model' in Investment Management."Journal of Finance, vol. 35, no. 2, 1980, pp. 597–607.
Page 194 Shanken, J. "The Arbitrage Pricing Theory: Is It Testable?" Journal of Finance, vol. 37, no. 5, 1982, pp. 1129–1140. ———. "Multi-Beta CAPM or Equilibrium APT? A Reply to Dybvig and Ross."Journal of Finance, vol. 40, no. 4, 1985, pp. 1189–1196. ———. "The Current State of the Arbitrage Pricing Theory." Journal of Finance, vol. 47, no. 4, 1992, pp. 1569–1574. Sharpe, William F. "Factor Models, CAPMs, and the APT." Journal of Portfolio Management, vol. 11, no. 1, 1984, pp. 21–25. Sharpe, William F. "The Capital Asset Pricing Model: A 'Multi-Beta' Interpretation." In Financial Decision Making under Uncertainty, edited by Haim Levy and Marshall Sarant (New York: Academic Press, 1977). Sharpe, William F., and Gordon J. Alexander. Investments (Englewood Cliffs, N.J.: Prentice-Hall, 1990). Technical Appendix This appendix contains • A description of factor models of stock return • A derivation of the APT in terms of factor models Factor Models A factor model represents excess returns as where X is the N by K matrix of stock exposures to the factors, b is the vector of K factor returns, and u is the specific return vector. For any portfolio P with holdings hP of the risky assets, the portfolio's factor exposures are Recall that portfolio C is the fully invested portfolio with minimum variance and that portfolio Q is the fully invested portfolio that has the highest ratio of expected excess return to risk. In the technical appendix to Chap. 2, we established that the expected excess return on each asset is proportional to that asset's beta with respect to portfolio Q. We assume that • fc > 0, and thus portfolio Q exists and fQ > 0.
Page 195 • The specific returns u are uncorrelated with the factor returns b. • The factor exposures X are known with certainty at the start of the period. With these assumptions, the N by N asset covariance matrix is where F is the K by K covariance of the factors and Δ is an N byN matrix that gives the covariance of the specific returns. We usually assume that Δ is a diagonal matrix, although that is not necessary. We refer to the factor model as (X, F, Δ). We say that a factor model explains expected excess returns f if we can express the vector of expected excess returns f as a linear combination of the factor exposures X. The model (X, F, Δ) explains expected excess returns if there is a K-element vector of factor forecasts m such that Equation (7A.4) gives us an expression for f. In the appendix to Chap. 2, we derived another expression for f involving portfolioQ. Let's look for the link between these two expressions. The N-element vector of stock covariances with respect to portfolio Q is From Eq. (2A.36) (Proposition 3 in the technical appendix to Chap. 2), we know that the expected excess returns are Compare Eqs. (7A.4) and (7A.6). We are getting perilously close to the APT result. As an initial stab, we could write m* = κQ · F · XT · hQ. Then One alternative is to ignore the second term and live with
Page 196 a little imperfection. To attain perfection, however, we need one additional assumption. We show below that this assumption works and that we need to make it; i.e., this assumption is both necessary and sufficient. First a definition: A portfolio P is diversified with respect to the factor model (X, F, Δ) if portfolio P has minimal risk among all portfolios that have the same factor exposures as portfolio P; i.e., of all portfolios h with XT · hP = xP, portfolio P has the least risk. Our assumption is that portfolio Q is diversified with respect to the factor model (X, F, Δ). Proposition 1 (APT) The factor model (X, F, Δ) explains expected excess returns if and only if portfolio Q is diversified with respect to (X, F, Δ). Proof Suppose first that portfolio Q is diversified with respect to (X, F, Δ). Now we can find the portfolio with exposures xQ that has minimal risk by solving The first-order conditions for this problem are satisfied by the optimal solution h* and a K-element vector of Lagrange multipliers π that satisfy Since portfolio Q is diversified with respect to (X, F, Δ), then hQ =h* is the optimal solution. Therefore Combining Eq. (7A.12) with Eqs. (7A.6), (7A.7), and (7A.4) leads to and the factor model (X, F, Δ) explains the expected excess returns. For the converse, suppose that the factor model (X, F, Δ) explains the expected excess returns and that portfolio Q is not diversified with respect to (X, F, Δ). Then there exists a portfolio P with
Page 197 the same exposures as portfolio Q, i.e., xP = XT · hP = xQ, and less risk than portfolio Q, i.e., . However, we have fP = fQ, since the factor exposures determine the expected returns, and portfolios P and Q have identical factor exposures. So . Portfolio P cannot be all cash, since the expected excess return on cash is zero. Therefore portfolio P is a mixture of cash and a nonzero fraction of some fully invested portfolio P*. It must be that fP/σP = fP*/σP*. Recall, however, that portfolio Q is the fully invested portfolio with the largest possible ratio of expected excess return to risk. This contradiction establishes the point: Portfolio Q must be diversified with respect to (X, F, Δ). Exercises 1. A factor model contains an intercept if some weighted combination of the columns of X is equal to a vector of 1s. This will, of course, be true if one of the columns of X is a column of 1s. It will also be true if X contains a classification of stocks by industry or economic sector. The technical requirement for a model to have an intercept is that there exists a K-element vector g such that e = X · g. Assume that the model contains an intercept, and demonstrate that we can then determine the fraction of the portfolio invested in risky assets by looking only at the portfolio's factor exposures. 2. Show that a model that does not contain an intercept is indeed strange. In particular, show there will be a fully invested portfolio with zero exposures to all the factors—a portfolio P with (fully invested) and xP = XT · hP = 0 (zero exposure to each factor). Applications Exercises 1. What is the percentage of unexplained variance in the CAPMMI portfolio? Does this portfolio qualify as highly diversified? How much would it be possible to lower the risk of the CAPMMI in a portfolio with identical factor exposures?
Page 198 2. Assume an excess return forecast of 5 percent per year for a value factor, excess return of –1 percent per year for a size factor, and excess return forecasts of zero for all other factors. Using the CAPMMI as a benchmark, what MMI asset has the highest alpha? What is its value?
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