MULTIFACTOR MODELS AND RISK ESTIMATION
When it comes to putting theory into practice, one advantage of the CAPM framework is that the identity of the single risk factor (i.e., the excess return to the market portfolio) is well specified.
Thus, as noted earlier, the empirical challenge in implementing the CAPM successfully is to accurately estimate the market portfolio, a process that first requires identifying the relevant investment universe. As we saw in the last chapter, however, this is not a trivial problem as an improperly chosen proxy for the market portfolio (e.g., using the S&P 500 index to represent the market when evaluating a fixed-income portfolio) can lead to erroneous judgments. However, we also saw that once the returns to an acceptable surrogate for the market portfolio are identified (i.e., Rm), the process for estimating the parameters of the CAPM is straightforward and can be accomplished by either of the following regression equations:
1. A security or portfolio’s characteristic line can be estimated via regression techniques using the single-index market model:
Rit=ai+biRmt+et
2. Alternatively, this equation can also be estimated in excess return form by netting the risk- free rate from the period t returns to security i and the market portfolio:
(Rit– RFRt) = αi+bi(Rmt– RFRt) +eit
In contrast to the CAPM, we have seen that the primary practical problem associated with implementing the APT is that neither the identity nor the exact number of the underlying risk factors are developed by theory and therefore must be specified in an ad hoc manner. Said dif- ferently, before the APT can be used to value securities or measure investment performance, the investor must fill in a considerable amount of missing information about the fundamental rela- tionship between risk and expected return.
As discussed earlier, the first attempts to implement a usable form of the APT relied on mul- tivariate statistical techniques, wherein many periods of realized returns for a large number of securities are analyzed simultaneously in order to detect recognizable patterns of behavior.22A consistent finding of these studies is that there appear to be as many as three or four “priced”
(i.e., statistically significant) factors, although researchers were not able to establish that the same set of factors was generated by different subsets of their sample. Indeed, we also saw that other researchers noted that the inability to identify the risk factors is a major limitation to the usefulness of the APT.23
A different approach to developing an empirical model that captures the essence of the APT relies on the direct specification of the form of the relationship to be estimated. That is, in a mul- tifactor model, the investor chooses the exact number and identity of risk factors in the follow- ing equation:
➤9.2 Rit=ai+[bi1F1t+bi2F2t+ . . . +biKFKt] +eit
22See, for instance, Richard Roll and Stephen A. Ross, “An Empirical Investigation of the Arbitrage Pricing Theory,”
Journal of Finance 35, no. 5 (December 1980): 1073–1103; and Nai-fu Chen, “Some Empirical Tests of the Arbitrage Pricing Theory,” Journal of Finance 38, no. 5 (December 1983): 1393–1414.
23See Jay Shanken, “The Arbitrage Pricing Theory: Is It Testable,” Journal of Finance 37, no. 5 (December 1982):
1129–1140.
where Fjtis the period t return to the jth designated risk factor and Ritcan be measured as either a nominal or excess return to security i. The advantage of this approach, of course, is that the investor knows precisely how many and what things need to be estimated to fit the regression equation. On the other hand, the major disadvantage of a multifactor model is that it is developed with little theoretical guidance as to the true nature of the risk-return relationship. In this sense, developing a useful factor model is as much an art form as it is a theoretical exercise.
A wide variety of empirical factor specifications have been employed in practice. A hallmark of each alternative model that has been developed is that it attempts to identify a set of economic influences that is simultaneously broad enough to capture the major nuances of investment risk but small enough to provide a workable solution to the analyst or investor. Two general approaches have been employed in this factor identification process. First, risk factors can be macroeconomic in nature; that is, they can attempt to capture variations in the underlying reasons an asset’s cash flows and investment returns might change over time (e.g., changes in inflation or real GDP growth in the example discussed earlier). On the other hand, risk factors can also be identified at a microeconomic level by focusing on relevant characteristics of the securities them- selves, such as the size of the firm in question or some of its financial ratios. A few examples rep- resentative of both of these approaches to the problem are discussed in the following sections.
Macroeconomic-Based Risk Factor Models One particularly influential model was developed by Chen, Roll, and Ross, who hypothesized that security returns are governed by a set of broad economic influences in the following fashion:24
➤9.3 Rit=ai+[bi1Rmt+bi2MPt+bi3DEIt+bi4UIt+bi5UPRt+bi6UTSt] +eit
where:
Rm= the return on a value-weighted index of NYSE-listed stocks MP = the monthly growth rate in U.S. industrial production
DEI = the change in inflation, measured by the U.S. consumer price index UI = the difference between actual and expected levels of inflation
UPR = the unanticipated change in the bond credit spread (Baa yield – RFR) UTS = the unanticipated term structure shift (long-term less short-term RFR)
In estimating this model, the authors used a series of monthly returns for a large collection of securities from the Center for Research in Security Prices (CRSP) database over the period 1958–1984. Exhibit 9.3 shows the factor sensitivities (along with the associated t-statistics in parentheses) that they established. Notice two things about these findings. First, the economic significance of the designated risk factors changed dramatically over time. For instance, the inflation factors (DEI and UI ) appear to only be relevant during the 1968–1977 period. Second, the parameter on the stock market proxy is never significant, suggesting that it contributes little to the explanation beyond the information contained in the other macroeconomic risk factors.
Burmeister, Roll, and Ross analyzed the predictive ability of a model based on a different set of macroeconomic factors.25Specifically, they define the following five risk exposures: (1) con- Multifactor Models
in Practice
24Nai-fu Chen, Richard Roll, and Stephen A. Ross, “Economic Forces and the Stock Market,” Journal of Business 59, no. 3 (April 1986): 383–404.
25Edwin Burmeister, Richard Roll, and Stephen A. Ross, “A Practitioner’s Guide to Arbitrage Pricing Theory,” in John Peavy, ed., A Practitioner’s Guide to Factor Models (Charlottesville, Va.: Research Foundation of the Institute of Char- tered Financial Analysts, 1994.
fidence risk, based on unanticipated changes in the willingness of investors to take on investment risk; (2) time horizon risk, which is the unanticipated changes in investors’ desired time to receive payouts; (3) inflation risk, based on a combination of the unexpected components of short-term and long-term inflation rates; (4) business cycle risk, which represents unanticipated changes in the level of overall business activity; and (5) market-timing risk, defined as the part of the Standard & Poor’s 500 total return that is not explained by the other four macroeconomic factors. Using monthly data through the first quarter of 1992, the authors estimated risk premia (i.e. the market “price” of risk) for these factors:
They also compared the factor sensitivities for several different individual stocks and stock port- folios. Panel A and Panel B of Exhibit 9.4 show these factor beta estimates for a particular stock (Reebok International Ltd.) versus the S&P 500 index and for a portfolio of small-cap firms ver- sus a portfolio of large-cap firms. Also included in these graphs is the security’s or portfolio’s exposure to the BIRR composite risk index, which is designed to indicate which position has the most overall systematic risk. These comparisons highlight how a multifactor model can help investors distinguish the nature of the risk they are assuming when they hold with a particular position. For instance, notice that Reebok has greater exposures to all sources of risk than the S&P 500, with the incremental difference in the business cycle exposure being particularly dra- matic. Additionally, smaller firms are more exposed to business cycle and confidence risk than larger firms but less exposed to horizon risk.
Microeconomic-Based Risk Factor Models In contrast to macroeconomic-based explanations of the connection between risk and expected return, it is also possible to specify
RISKFACTOR RISKPREMIUM
Confidence 2.59%
Time horizon –0.66
Inflation –4.32
Business cycle 1.49
Market timing 3.61
MULTIFACTORMODELS ANDRISKESTIMATION 293 ESTIMATING A MULTIFACTOR MODEL WITH MACROECONOMIC RISK FACTORS
PERIOD CONSTANT RM MP DEI UI UPR UTS
1958–84 10.71 –2.40 11.76 –0.12 –0.80 8.27 –5.91
(2.76) (–0.63) (3.05) (–1.60) (–2.38) (2.97) (–1.88)
1958–67 9.53 1.36 12.39 0.01 –0.21 5.20 –0.09
(1.98) (0.28) (1.79) (0.06) (–0.42) (1.82) (–0.04)
1968–77 8.58 –5.27 13.47 –0.26 –1.42 12.90 –11.71
(1.17) (–0.72) (2.04) (–3.24) (–3.11) (2.96) (–2.30)
1978–84 15.45 –3.68 8.40 –0.12 –0.74 6.06 –5.93
(1.87) (–0.49) (1.43) (–0.46) (–0.87) (0.78) (–0.64)
EXHIBIT 9.3
Source: Nai-fu Chen, Richard Roll, and Stephen A. Ross, “Economic Forces and the Stock Market,” Journal of Business 59, no. 3 (April 1986).
Confidence Horizon Inflation Business Cycle
Timing BIRR Index
–2.5 2.5
0.0 5.0
Size of Exposure
S&P 500
Reebok International Ltd.
EXHIBIT 9.4 MACROECONOMIC RISK EXPOSURE PROFILES A. Reebok International LTD. versus S&P 500 Index
B. Large-Cap versus Small-Cap Firms
Confidence Horizon Inflation Business Cycle
Timing BIRR Index
–2.5 5.0
2.5
0.0 7.5
Size of Exposure
High-Cap (Portfolio) Low-Cap (Portfolio)
Copyright 1994, The Research Foundation of the Institute of Chartered Financial Analysts. Reproduced and republished from A Practitioner’s Guide to Arbitrage Pricing Theory with permission from The Research Foundation of the Association for Investment Management and Research. All Rights Reserved.
risk in microeconomic terms using certain characteristics of the underlying sample of securities.
Typical of this characteristic-based approach to forming a multifactor model is the work of Fama and French, who use the following functional form:26
➤9.4 (Rit– RFRt) = αi+bi1(Rmt– RFRt) +bi2SMBt+bi3HMLt+eit
where, in addition to the excess return on a stock market portfolio, two other risk factors are defined:
SMB (i.e., small minus big) is the return to a portfolio of small capitalization stocks less the return to a portfolio of large capitalization stocks
HML (i.e., high minus low) is the return to a portfolio of stocks with high ratios of book-to- market values less the return to a portfolio of low book-to-market value stocks
In this specification, SMB is designed to capture elements of risk associated with firm size while HML is intended to distinguish risk differentials associated with “growth” (i.e., low book-to- market ratio) and “value” (i.e., high book-to-market) firms. As we saw earlier, these are two dimensions of a security—or portfolio of securities—that have consistently been shown to mat- ter when evaluating investment performance. Also, notice that without the SMB and HML fac- tors this model simply reduces to the excess returns form of the single-index market model.
As part of their analysis of the role that SMB and HML play in the return-generating process, Fama and French examined the behavior of a broad sample of stocks grouped into quintile port- folios by their price-earnings (P-E) ratios on a yearly basis over the period from July 1963 to December 1991. The results for both the single-index and multifactor versions of the model for the two extreme quintiles are shown in Exhibit 9.5 (t-statistics for the estimated coefficients are listed parenthetically). There are several important things to note about these findings. First, while the estimated beta from the single-factor model indicates that there are substantial differ- ences between low and high P-E stocks (i.e., 0.94 versus 1.10), this gap is dramatically reduced MULTIFACTORMODELS ANDRISKESTIMATION 295
26Eugene F. Fama and Kenneth R. French, “Common Risk Factors in the Returns on Stocks and Bonds,” Journal of Financial Economics 33, no. 1 (January 1993): 3–56.
ESTIMATING A MULTIFACTOR MODEL WITH CHARACTERISTIC-BASED RISK FACTORS
PORTFOLIO CONSTANT MARKET SMB HML R2
(1) SINGLE-INDEX MODEL
Lowest P-E 0.46 0.94 — — 0.78
(3.69) (34.73)
Highest P-E –0.20 1.10 — — 0.91
(–2.35) (57.42)
(2) MULTIFACTOR MODEL
Lowest P-E 0.08 1.03 0.24 0.67 0.91
(1.01) (51.56) (8.34) (19.62)
Highest P-E 0.04 0.99 –0.01 –0.50 0.96
(0.70) (66.78) (–0.55) (–19.73)
EXHIBIT 9.5
Reprinted from Eugene F. Fama and Kenneth R. French, “Common Risk Factors in the Returns on Stocks and Bonds,” Journal of Financial Economics 33, no. 1 (January 1993), with permission from Elsevier Science.
in the multifactor specification (i.e., 1.03 versus 0.99). This suggests that the market portfolio in a one-factor model serves as a proxy for some, but not all, of the additional risk dimensions pro- vided by SMB and HML. Second, it is apparent that low P-E stocks tend to be positively corre- lated with the small-firm premium, but the reverse is not reliably true for high P-E stocks.
Finally, low P-E stocks also tend to have high book-to-market ratios while high P-E stocks tend to have low book-to-market ratios (i.e., estimated HML parameters of 0.67 and –0.50, respec- tively). Not surprisingly, relative levels of P-E and book-to-market ratios are both commonly employed in practice to classify growth and value stocks.
Extensions of Characteristic-Based Risk Factor Models There have been other interesting characteristic-based approaches to estimating a multifactor model of risk and return.
Three of those approaches are described here. First, Carhart directly extends the Fama-French three-factor model by including a fourth common risk factor that accounts for the tendency for firms with positive (negative) past returns to produce positive (negative) future returns.27He calls this additional risk dimension a momentum factor and estimates it by taking the average return to a set of stocks with the best performance over the prior year minus the average return to stocks with the worst returns. In this fashion, Carhart defines the momentum factor—which he labels PR1YR—in a fashion similar to SMB and HML. Formally, the model he proposes is:
➤9.5 (Rit– RFRt) = αi+bi1(Rmt– RFRt) +bi2SMBt+bi3HMLt+bi4PR1YRt+eit
He demonstrates that the typical factor sensitivity (i.e., factor beta) for the momentum variable is positive and its inclusion into the Fama-French model increases explanatory power by as much as 15 percent.
A second type of security characteristic-based method for defining systematic risk exposures involves the use of index portfolios (e.g., S&P 500, Wilshire 5000) as common factors. The intu- ition behind this approach is that, if the indexes themselves are designed to emphasize certain investment characteristics, they can act as proxies for the underlying exposure that determines returns to that characteristic. Examples of this include the Russell 1000 Growth index, which emphasizes large-cap stocks with low book-to-market ratios, or the EAFE (Europe, Australia, and the Far East) index that selects a variety of companies that are domiciled outside the United States. Typical of these index-based factor models is the work of Elton, Gruber, and Blake, who rely on four indexes: the S&P 500, the Lehman Brothers aggregate bond index, the Prudential Bache index of the difference between large- and small-cap stocks, and the Prudential Bache index of the difference between value and growth stocks.28Ferson and Schadt have developed an interesting variation on this approach, which, in addition to using stock and bond indexes as risk factors, also includes other “public information” variables, such as the shape of the yield curve and dividend payouts.29
BARRA, a leading risk forecasting and investment consulting firm, provides a final example of the microeconomic approach to building a multifactor model. In its most expansive form, the BARRA model for analyzing U.S. equities includes as risk factors 13 characteristic-based variables and more than 50 industry indexes.30Exhibit 9.6 provides a brief description of the 13 characteristic-
27Mark M. Carhart, “On Persistence in Mutual Fund Performance,” Journal of Finance 52, no. 1 (March 1997): 57–82.
28Edwin J. Elton, Martin J. Gruber, and Christopher R. Blake, “The Persistence of Risk-Adjusted Mutual Fund Perfor- mance,” Journal of Business 69, no. 2 (April 1996): 133–157.
29Wayne R. Ferson and Rudi W. Schadt, “Measuring Fund Strategy and Performance in Changing Economic Conditions,”
Journal of Finance 51, no. 2 (June 1996): 425–462.
30A more complete description of the BARRA approach to analyzing investment risk can be found in Richard Grinold and Ronald N. Kahn, “Multiple-Factor Models for Portfolio Risk,” in John Peavy, ed., A Practitioner’s Guide to Factor Models (Charlottesville, Va.: Research Foundation of the Institute of Chartered Financial Analysts, 1994.
based factors that form the heart of the BARRA approach to decomposing investment risk. One use- ful application for this model is to understand where the investment “bets” in an actively managed portfolio are being placed relative to a performance benchmark. Exhibit 9.7 illustrates this sort of comparison for a small-cap-oriented mutual fund (POOL2) versus the S&P 500 index (SAP500). As you would expect, there are dramatic differences between the fund and the benchmark in terms of the firm-size risk factors (i.e., size, SIZ, and size nonlinearity, SNL). However, it also appears that POOL2 contains more highly leveraged companies (LEV) with more emphasis on earnings momen- tum (MOM).
Conner has analyzed the ability of the BARRA model to explain the returns generated by a sample of U.S. stocks over the period from 1985 to 1993.31Interestingly, he found that the indus- try indexes, taken collectively, provided about four times the explanatory power as any single characteristic-based factor, followed in importance by volatility, growth, dividend yield, and momentum. Overall, the BARRA model was able to explain slightly more return variability than the other models to which it was compared, in part because of the large number of factors it employs.
MULTIFACTORMODELS ANDRISKESTIMATION 297 DESCRIPTION OF BARRA CHARACTERISTIC-BASED RISK FACTORS
• Volatility (VOL) Captures both long-term and short-term dimensions of relative return variability
• Momentum (MOM) Differentiates between stocks with positive and negative excess returns in the recent past
• Size (SIZ) Based on a firm’s relative market capitalization
• Size Nonlinearity (SNL) Captures deviations from linearity in the relationship between returns and firm size
• Trading Activity (TRA) Measures the relative trading in a stock, based on the premise that more actively traded stocks are more likely to be those with greater interest from institutional investors
• Growth (GRO) Uses historical growth and profitability measures to predict future earnings growth
• Earnings Yield (EYL) Combines current and historical earnings-to-price ratios with analyst forecasts under the assumption that stocks with similar earnings yields produce similar returns
• Value (VAL) Based on relative book-to-market ratios
• Earnings Variability (EVR) Measures the variability in earnings and cash flows using both historical values and analyst forecasts
• Leverage (LEV) Measures the relative financial leverage of a company
• Currency Sensitivity (CUR) Based on the relative sensitivity of a company’s stock return to movements in a basket of foreign currencies
• Dividend Yield (YLD) Computes a measure of the predicted dividend yield using a firm’s past dividend and stock price history
• Nonestimation Indicator (NEU) Uses returns to firms outside the equity universe to account for risk dimensions not captured by the other risk factors
EXHIBIT 9.6
Source: BARRA.
31Gregory Connor, “The Three Types of Factor Models: A Comparison of Their Explanatory Power,” Financial Analysts Journal 51, no. 3 (May/June 1995): 42–46.
Estimating Expected Returns for Individual Stocks One direct way in which to employ a multifactor risk model is to use it to estimate the expected return for an individual stock position. In order to accomplish this task, the following steps must be taken: (1) a specific set of K common risk factors must be identified, (2) the risk premia (Fj) for the factors must be esti- mated, (3) the sensitivities (bij) of the ith stock to each of those K factors must be estimated, and (4) the expected returns can be calculated by combining the results of the previous steps in the appropriate way.
As an example of this process, we will use the Fama-French model discussed earlier. This immediately solves the first step by designating the following three common risk factors: the excess return on the market portfolio (Rm), the return differential between small and large capi- talization stocks (SMB), and the return differential between high and low book-to-market stocks (HML). The second step is often addressed in practice by using historical return data to calculate the average values for each of the risk factors. However, it is important to recognize that these averages can vary tremendously depending on the time period the investor selects. For example, for the three-factor model, the top panel of Exhibit 9.8 lists the average annual risk premia over three different time frames: a five-year period ending in June 2000, a 20-year period ending in December 2000, and a 73-year period ending in December 2000.32Notice that, while data for the longest time frame confirm that small stocks earn higher returns than large stocks and value stocks outperform growth stocks (i.e., positive risk premia for the SMB and HML factors), this is not true over shorter periods. In particular, during the most recent five years, the opposite occurred in both cases.
To illustrate the final steps involved in estimating expected stock returns, risk factor sensitiv- ities were estimated by regression analysis for three different stocks using monthly return data Estimating Risk
in a Multifactor Setting: Examples
1.0
–2.5 –2.0 –1.5 –1.0 –0.5 0.0 0.5
Exposure
VOL EYL VAL EVR LEV CUR YLD NEU
Risk Index Exposures
GRO TRA SNL SIZ MOM
MGD - POOL 2 BMK - SAP500
EXHIBIT 9.7 BARRA RISK DECOMPOSITION FOR A SMALL-CAP FUND VERSUS S&P 500
32The data used in these calculations are available from Professor Kenneth French’s Web site at http://mba.tuck.
dartmouth.edu/pages/faculty/ken.french