An Empiricist s Guide to The Arbitrage Pricing Theory

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1 An Empiricist s Guide to The Arbitrage Pricing Theory Ravi Shukla Finance Department School of Management Syracuse University Syracuse, NY First Draft: June 1987 This version: October The Market Model and the CAPM The Capital Asset Pricing Model (CAPM) is based on the sufficiency of the meanvariance framework for investment decision making The asset pricing relationship is given by: µ i = r f + β i (µ m r f ), (1) where µ i and µ m are the expected returns on security i and the market portfolio, respectively, and r f is the return on the risk-free security The market portfolio is a value weighted portfolio of all the securities in the universe β i measures the systematic risk of the security and is estimated as the sensitivity of the security s returns to those of the market portfolio using the market model regression: r it = a i + β i r mt + e it, (2) where r it and r mt are period t returns on security i and the market portfolio, respectively The market model, viewed as the return generating model consistent with the CAPM, is used in event studies If, over some period T, the average returns on security i, the market portfolio, and the risk-free security are r i, r m and r f, respectively, then in the absence of any superior or inferior performance, an ex-post version of the CAPM must hold, ie, r i = r f + β i ( r m r f ) (3) Therefore, the difference α i defined below may be used as a measure of performance: α i = r i [r f + β i ( r m r f )] (4) The performance measure α i, popularly known as Jensen s alpha, is usually estimated using an excess return market model: r it r f = α i + β i (r mt r f )+ɛ it (5) 1

2 Defining the market return deviation δ mt = r mt µ m, and the market risk premium γ m = µ m r f, equation 5 may be written as: r it r f = α i + β i (δ mt + γ m )+ɛ it (6) For empirical applications, CRSP equal and value weighted portfolios are the two popular proxies for the market portfolios used in the literature Some studies have also used the S&P500 index 2 The Factor Model and the APT The no-arbitrage APT of Ross (1976) is based on the popular intuition that the security returns are affected by several, presumably a small number k, factors This intuition is translated into a factor model expressed as: r it µ i = b i1 δ 1t + b i2 δ 2t + + b ik δ kt + e it, (7) where δ s are the standardized factor scores in that they have zero means and unit standarddeviationsandbs are the sensitivities of the security to the factors The factors are assumed to be pervasive ie, they affect all the securities The factors are designed so that they are orthogonal to each other and to the residual element e In matrix notation, we can write for a sample of n securities: R t M = B t + E t, (7 ) where R t and M are the n 1 vector of realized and expected returns, B is the n k matrix of sensitivities, t is the k 1 vector of factor scores, and E t is the n 1 vector of residuals Ross assumes that the market, which consists of infinitely many securities, 1 is efficient in the sense that the payoff on an arbitrage portfolio a zero cost portfolio with zero sensitivities is zero 2 The factor model and no-arbitrage assumptions combined with a few other minor assumptions lead to the pricing equation of the APT: Equation (8) can be written in matrix form as: µ i γ 0 + b i1 γ 1 + b i2 γ b ik γ k (8) M lγ 0 + BΓ, (8 ) where l is an n 1 vector of ones and Γ is a k 1 vector of γs The approximation ( ) in the APT pricing equation arises in economies with finite number of securities because the total risk (variance) of the arbitrage portfolio is not completely diversifiable in a finite economy Ross (1976), Dybvig (1983) and Grinblatt 1 This assumption is needed to make sure that the total risk of a portfolio is diversifiable 2 Strictly speaking, the APT shows that if the payoffs on a sequence of arbitrage portfolios are bounded while the variance is not, then the sum of squared deviations from the pricing equation will be bounded See Huberman (1982) 2

3 and Titman (1983) provide theoretical arguments to show that the average pricing error would empirically be negligibly small Shanken (1982) argues that even if the average pricing error is small, individual pricing errors may be large Dybvig and Ross (1985) show that Shanken s arguments hold under very special conditions which are not likely to be encountered in real situations Robin and Shukla (1991) show that the pricing errors for some securities are large A version of the APT based on equilibrium [Chen and Ingersoll (1983), Connor (1984), Wei (1988)], rather than no-arbitrage, shows that the strict pricing equality will hold if one of the factors is the market or a residual market factor Most APT applications assume that the pricing errors are negligible and use the pricing equation as if it were a strict equality In the pricing equation, γ 0 is interpreted as the zero-beta or the risk-free rate and γ j, j =1,,k, is interpreted as the risk premium corresponding to the risk from the j th factor γ j may be calculated as the excess expected return on the basis portfolios A basis portfolio has unit sensitivity to a factor and zero sensitivity to the others For example, consider the j th basis portfolio w j with unit sensitivity to the j th factor and zero sensitivity to the others Pre-multiplying equation (8 ) by wj T and assuming strict equality of the pricing equation, we get: wj T M = wt j lγ 0 + wj T BΓ, µ j = γ 0 + γ j, γ j = µ j γ 0, since wj Tl =1because the portfolio weights add up to one, and wt j B = ut j by design where u j is a vector with zeros in all locations except the j th which has a one Substituting µ i from equation (8) into equation (7), and assuming strict equality of the pricing relationship, we get r it γ 0 = b i1 (δ 1t + γ 1 )+b i2 (δ 2t + γ 2 )+ + b ik (δ kt + γ k )+e it (9) If, over a period, a security s average returns conform to the APT, then the α i defined below would be zero: α i = r it [γ 0 + b i1 (δ 1t + γ 1 )+b i2 (δ 2t + γ 2 )+ + b ik (δ kt + γ k )] (10) Therefore, the following equation may be used to estimate a security s superior performance: r it γ 0 = α i + b i1 (δ 1t + γ 1 )+b i2 (δ 2t + γ 2 )+ + b ik (δ kt + γ k )+e it (11) Equation (11) is the counterpart to equation (6) in the CAPM framework Connor and Korajczyk (1986) show that the intercept in equation (11) has the same interpretation as the intercept in equation (6) In matrix form, equation (11) may be written as: 3 Two Applications R t lγ 0 = A + B( t + Γ)+E t (11 ) Two common applications of the asset pricing frameworks are event studies and performance evaluations Figure 1 shows a typical event study time line The total time 3

4 Figure 1: A typical event study time line Event Date Estimation Period (T e) Testing Period (T t) period under study, T, is divided into two parts: an estimation period T e during which the model coefficients are estimated, and a testing period T t during which the effect of the event is studied The two periods are separated by the event date The estimation period is considered to be free of the influences of the event so the coefficients estimated during this period are unbiased The abnormal returns estimated using these coefficients during the testing period are used to measure the impact of the event Specifically, the event study using the market model proceeds in two steps: 1 Estimate the market model regression, equation (2), for the security and estimate the coefficients â i and ˆβ i for the security: r it =â i + ˆβ i r mt +ê it t T e 2 Estimate the abnormal returns (deviations from the market model) ar it as: ar it = r it (â i + ˆβ i r mt ) t T t These abnormal returns or cumulated abnormal returns are then used in statistical hypothesis testing For performance measurement, the excess return market model, equation (5), is estimated over the period of interest and the intercept provides a measure of performance The event study and performance evaluation methodologies associated with the market model can be replicated in the factor model and APT framework using equation (11) as a multiple regression equation This, however, necessitates the knowledge of the values of (δ + γ)s, the factor scores plus the risk premia As Figure 2 shows, these factor scores and risk premia may be estimated using a set of factor providing securities Once the factor scores and risk-premia are estimated, they may be used in the application of interest The rest of the paper, therefore, focuses on techniques for estimating the factor scores and the risk-premia 4 Macroeconomic and Statistical Factors Chen, Roll, and Ross (1986), Burmeister and Wall (1986), and Chen, Grundy, and Stambaugh (1990) use innovations in a set of intuitively appealing macroeconomic variables such as industrial production, unanticipated inflation, change in expected inflation, risk premium, term premium, etc, as factors Alternatively, Roll and Ross 4

5 Figure 2: Steps in application of the APT Step 1 Factor Providing Securities Step 2 Factor Loadings Step 3 Factor Scores & Risk Premia Step 4 Experimental Securities Step 5 Actual Application (1980), Chen (1983), and others use statistical procedure to estimate the factor model While statistically estimated factors lack the intuitive appeal of macroeconomic factors, they do not suffer the problems associated with ad hoc choice of factors Using the factor model equation, one can write: Σ = BB T + Φ, (12) where Σ is the covariance matrix of security returns and Φ is the residual covariance matrix This decomposition of total covariance matrix into factor loadings and residual components provides estimates of B, the factor loadings, and Φ, the residual covariance matrix The appropriate statistical procedure to be used for this purpose is maximum likelihood factor analysis (MLFA) Chamberlain and Rothschild (1983) show that, for large samples, the principal component analysis (PCA) can be used to estimate the decomposition; the eigenvectors being the factor loadings The relative ease of computing the principal components makes them quite attractive Shukla and Trzcinka (1990) show that the MLFA estimates are not necessarily superior to the PCA estimates An important issue in estimating the factor model is that of the number of factors The APT does not specify the number of factors Dhrymes, Friend, and Gultekin (1984) show that the number of factors extracted using a statistical procedure increases with the number of securities in the sample Therefore, they argue, the number of pervasive factors may not be small Trzcinka (1986) shows that while the number of statistically estimated factors increases with the sample size, the first factor remains dominant This suggests that we may have a one factor model, the one factor being the market Roll and Ross (1984) argue that what matters is the number of priced factors and not the number of statistical factors extracted from the covariance matrix While there is no clear answer about the number of factors, most of the literature uses 1, 5 or 10 factors Lehmann and Modest (1987) show that of all the decision choices, the number of factors has the least affect on the model estimates 5

6 5 Estimation of Factor Scores and Risk Premia To estimate the factor scores, δs,andriskpremiaγs, or their sum (δ + γ)s, one has to rely on the statistical methods For this process, one has to choose the securities to be used for estimating the factor scores and the risk-premia These factor providing securities may either be the same as the experimental securities or may be an independent set of securities To avoid the possibility of any confounding affects, one should use a set of securities other than the experimental securities This, however, necessitates the assumption that the factors estimated using a separate set of securities are identical to the factors that affect the behavior of the experimental securities In other words, one has to assume that the factors are pervasive To ensure that the factors are indeed pervasive, one should use as many securities as possible to estimate the factor scores Lehmann and Modest (1987) show that the choice of the number of securities does indeed make a difference on the characteristics of the estimated model Some papers have used size or industry based portfolios instead of individual stocks as factor providing securities to eliminate the idiosyncratic noise associated with individual stock returns For example, Chan, Hendershott, and Sanders (1988) use size based groups and Chen and Jordan (1993) and Kim, Shukla, and Tomas (1996) used industry based portfolios If a macroeconomic factor model is being used, the factor scores, δs, are directly available and need not be estimated However, one still needs to estimate the riskpremiaγs, or the sum of the factor scores and the risk-premia, (δ + γ)s For statistical factor model, both δs and(δ + γ)s need to be estimated Most papers have used the basis portfolio approach to estimate these quantities Connor and Korajczyk (1986) and Connor and Korajczyk (1988) have proposed the asymptotic principal components as an alternative 51 The Basis Portfolio Approach The basis portfolio based estimation procedure is implemented in two steps In the first step, the factor loadings or sensitivities and the residual covariance matrix are estimated for the factor providing securities In the second step, these sensitivities and the residual covariance matrix are used to estimate the factor scores If the factors are specified macroeconomic shocks, one uses a simple regression of the factor providing security returns on the factor scores to estimate the factor loadings If the factors are to be estimated statistically, then MLFA or PCA is used for this purpose 3 For the macroeconomic factors case, the residual covariance matrix Φ is estimated as Σ BΩB T where Ω is the k k covariance matrix of the factors For the statistical factors, the residual matrix, is calculated as Σ BB T 4 In the next step, the values of δs or(δ + γ)s are estimated by regarding the factor providing securities as a cross-section whose returns have the relationship described by equation (7) or (9) with the unknown δs or(δ + γ)s through the sensitivities β ij s To 3 FORTRAN programmers may use the IMSL routine FACTR while SAS users may use PROC FACTOR 4 The procedures used for factor analysis calculate this matrix and return it as an output 6

7 estimate δs using equation (7), the security return deviations, (r it µ i ), are regressed on their sensitivities, β ij s The average security return calculated from the time series of returns, r i, is used as an empirical estimate of µ i While estimating (δ + γ)s using equation (9), it is assumed that the factor providing securities are priced correctly Therefore, a regression of security returns on the factor loadings at each time point provides estimates of γ 0 and (δ + γ)s The cross-sectional regressions are estimated using the generalized least squares (GLS) method The GLS estimators of the δs at time t are: t =(B T Φ 1 B) 1 B T Φ 1 (R t µl) (13) This procedure is repeated for all time points in the sample to get the time series of δ s The same B and Φ are used for every t, assuming stationarity of factor loadings and residual covariance matrix This means that one can calculate (B T Φ 1 B) 1 B T Φ 1 once and multiply it with (R t µl) at each time point to get the factor scores at that point The rows of this fixed matrix can be viewed as the weights of a basis portfolios whose returns mimic individual factors The GLS estimates of the basis portfolios, therefore, are W T =(B T Φ 1 B) 1 B T Φ 1 Appendix A shows a derivation of the weights that justifies the use of the name basis portfolios and discusses an alternative, the minimum idiosyncratic risk portfolios To estimate (δ + γ)s, one estimates equation (9), rather than equation (7), and ( t + Γ) =(B T Φ 1 B) 1 B T Φ 1 R t (14a) γ 0t = R t [ β 1 (δ 1t + γ 1 )+ β 2 (δ 2t + γ 2 )+ + β k (δ kt + γ k )] (14b) where β j is the average sensitivity of the securities to the j th factor Note that this process gives a time series of γ 0t, rather than a single, fixed value γ 0, implying that the risk-free rate is not fixed over time This suggests that the proper interpretation of γ 0 may be a zero-beta rate 5 52 The Asymptotic Principal Components Approach To use the basis portfolios for estimating the factor scores, one has to create the n n covariance matrix Σ using T > n observations This creates a problem because to study a large number of securities one would have to use a large T AlargeT weakens the reliability of the stationarity assumption The Asymptotic Principal Components (APC) approach circumvents this problem Connor and Korajczyk (1986) show that the first k eigenvectors of the T T matrix Ξ are asymptotically (n ) a linear transformation of the k factor scores plus the risk premia, (δ+γ)s While estimating Ξ, the role of securities and time points get interchanged Therefore, the APC approach may be used on samples with a large number of securities without violating the stationarity assumption Another advantage of the asymptotic principal components approach 5 Also, γ 0 estimates include average pricing errors which have been ignored See Robin and Shukla (1991) for details 7

8 is that it directly gives the factor scores, rather than using two steps: first getting the sensitivities and then the factor scores Connor and Korajczyk (1988) suggest an iterative scheme that results in more efficient estimates of the factor scores The methodology may be implemented as follows: Estimate the eigenvectors of the matrix Ξ Suppose the k T matrix of eigenvectors is G Estimate the residual covariance matrix V = Ξ G T G Calculate Ξ = V 1/2 ΞV 1/2 Estimate the eigenvectors of Ξ These eigenvectors will be more efficient estimators of the true factor scores than the eigenvectors of Ξ This algorithm is based on the observation that a factor analysis decomposes the covariance matrix so that the residual covariance matrix is diagonal while no such effort is made in principal components analysis The iterative process brings the solution one step closer to the maximum likelihood factor analysis 6 Conclusion In conclusion, a step by step guide to implementing the factor model based procedure to event studies and performance evaluation is outlined below: 1 Select a large number (n f ) of securities that will be used as factor providing securities Make sure that this set does not include any experimental securities If possible, use portfolios rather than individual securities Collect returns for the full period of interest (estimation and testing period for event study, and evaluation period the performance study) for the factor providing securities Now follow one of the following approaches: To use the basis portfolio approach: Estimate the n f n f covariance matrix Use MLFA or PCA to get k factor loadings Estimate (δ + γ)s for the entire period using the method described in section 51 To use the asymptotic principal components approach: Calculate the T T covariance matrix Estimate the asymptotic principal components as described in section 52 The first k eigenvectors are the (δ + γ)s 2A For event study application: 8

9 Estimate the regression equation (9) for the experimental securities and estimate the coefficients ˆb ij, j = 1, 2,,k, for the securities during the estimation period: r it = γ 0 +b i1 (δ 1t +γ 1 )+b i2 (δ 2t +γ 2 )+ +b ik (δ kt +γ k )+e it t T e Discard the estimated intercept Calculate the abnormal returns (deviations from the factor model) ar it during the test period as: ar it = r it [γ 0t +ˆb i1 (δ 1t +γ 1 )+ˆb i2 (δ 2t +γ 2 )+ +ˆb ik (δ kt +γ k )] t T t Use the abnormal returns in statistical hypothesis testing 2B For performance evaluation: Estimate α i and β ij for the securities by estimating the regression equation (11): r it γ 0t = α i +b i1 (δ 1t +γ 1 )+b i2 (δ 2t +γ 2 )+ +b ik (δ kt +γ k )+e it Use α i for statistical hypothesis testing A The Basis Portfolio Weights A basis portfolio mimicking the j th factor is defined as one whose returns have unit sensitivity to the j th factor and zero sensitivity to the other factors Therefore, the basis portfolio, w j,forthej th factor is the solution to the following programming problem: Min w j wj T Φw j, st wj T B = ut j, (14) where u j is a k 1 vector with zeros in all the locations except the j th which has a 1 If we write the constraints in long form, we will see that the portfolio w j is being forced to have unit sensitivity to the j th factor and zero sensitivity to other factors Note that the portfolio weights are not scaled to sum to 1 The solution to the problem is obtained by forming the Lagrangian: Min w j, λ w T j Φw j 2(w T j B u T j )λ, (15) where λ is a k 1 vector The first order conditions give us: 2Φw j 2Bλ = z, (16) where z is a vector of zeros, and 9

10 w T j B = ut j (17) Equation (16) can be rewritten as w T j = λt B T Φ 1 (18) Post multiply (18) by B, substitute from (17) and simplify to get Substitute this λ T in(18toget λ T = u T j (BT Φ 1 B) 1 w T j = u T j (B T Φ 1 B) 1 B T Φ 1 (19) This is the composition of the j th basis portfolio The return on the j th basis portfolio at time t would be: δ jt = w T j R t = u T j (B T Φ 1 B) 1 B T Φ 1 R t (20) If we stack equations (19) for all js to get the full matrix mode, realizing that stacked u T j s will give us the identity matrix, we will get W T =(B T Φ 1 B) 1 B T Φ 1 (21) W is the n k matrix in which the j th column is the composition of the j th basis portfolio Similarly, writing the returns on the basis portfolios, equation (20, in the full matrix mode we get t =(B T Φ 1 B) 1 B T Φ 1 R t (22) To get the entire time series of all factor scores one just has to replace the vector R t by the n T matrix R in equation (22) The solution is identical to the GLS regression estimators Therefore, these basis portfolio are also known as GLS basis portfolios Lehmann and Modest (1988) argue that the requirement of unit sensitivity to the j th factor in the GLS basis portfolio method is unnecessary and that it causes the portfolio weights to be sensitive to the measurement errors Therefore, they suggest the minimum idiosyncratic risk portfolio (MIRP) technique in which the constraint of unit sensitivity to j th factor is replaced by the sum of portfolio weights being equal to one To get the MIRP basis portfolios the only change to be made in the GLS basis portfolio formulation is to replace matrix B by C which is identical to B except that the j th column of C is a vector of 1s Using MIRP technique, one can only obtain one basis portfolio at a time because a full matrix solution as written in equations (21) and (22) for the GLS method cannot be written for the MIRP method because C is different for each j References Burmeister, Edwin, and K Wall, 1986, The Arbitrage Pricing Theory and Macroeconomic Factor Measures, Financial Review, 21,

11 Chamberlain, Gary, and Michael Rothschild, 1983, Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Asset Markets, Econometrica, 51, Chan, K C, Patrick H Hendershott, and A B Sanders, 1988, Risk and Return on Real Estate: Evidence from Equity REITs, Journal of American Real Estate and Urban Economics Association, 18, Chen, Nai-Fu, 1983, Some Empirical Tests of the Theory of Arbitrage Pricing, Journal of Finance, 38, Chen, Nai-Fu, Bruce Grundy, and Robert F Stambaugh, 1990, Changing Risk, Changing Risk Premiums, and Dividend Yield Effects, Journal of Business, 63(1), S51 S70 Chen, Nai-Fu, and Jonathan Ingersoll, 1983, Exact Pricing in Linear Factor Models with Finitely Many Assets: A Note, Journal of Finance, 38, Chen, Nai-Fu, Richard R Roll, and Stephen A Ross, 1986, Economic Forces and the Stock Market, Journal of Business, 59(3), Chen, Su-Jane, and Bradford D Jordan, 1993, Some Empirical Tests in the Arbitrage Pricing Theory: Macrovariables vs Derived Factors, Journal of Banking and Finance, 17, Connor, Gregory, 1984, A Unified Beta Pricing Theory, Journal of Economic Theory, 34, Connor, Gregory, and Robert A Korajczyk, 1986, Performance Measurement with the Arbitrage Pricing Theory: A New Framework for Analysis, Journal of Financial Economics, 15(3), , 1988, Risk and Return in an Equilibrium APT: Application of a New Test Methodology, Journal of Financial Economics, 21(2), Dhrymes, Phoebe, Irwin Friend, and Mustafa Gultekin, 1984, A Critical Re- Examination of the Empirical Evidence on the Arbitrage Pricing Theory, Journal of Finance, 39(2), Dybvig, Philip, 1983, An Explicit Bound on Deviations from APT Pricing in a Finite Economy, Journal of Financial Economics, 12, Dybvig, Philip, and Stephen A Ross, 1985, Yes, the APT is Testable, Journal of Finance, 40(4), Grinblatt, Mark, and Sheridan Titman, 1983, Factor Pricing in a Finite Economy, Journal of Financial Economics, 12, Huberman, Gur, 1982, A Simple Approach to Arbitrage Pricing Theory, Journal of Economic Theory, 28,

12 Kim, Moon K, Ravi K Shukla, and Michael J Tomas, 1996, Mutual Fund Objective Misclassification, Unpublished manuscript Lehmann, Bruce, and David Modest, 1988, The Empirical Foundations of the Arbitrage Pricing Theory, Journal of Financial Economics, 21, Lehmann, Bruce N, and David M Modest, 1987, Mutual Fund Performance Evaluation: A Comparison of Benchmarks and Benchmark Comparisons, Journal of Finance, 42(2), Robin, Ashok J, and Ravi K Shukla, 1991, The Magnitude of Pricing Errors in the Arbitrage Pricing Theory, Journal of Financial Research, 14, Roll, Richard R, and Stephen A Ross, 1980, An Empirical Investigation of the Arbitrage Pricing Theory, Journal of Finance, 35, , 1984, A Critical Reexamination of the Arbitrage Pricing Theory: A Reply, Journal of Finance, 39, Ross, Stephen A, 1976, The Arbitrage Theory of Capital Asset Pricing, Journal of Economic Theory, 13, Shanken, Jay, 1982, The Arbitrage Pricing Theory? It is Testable, Journal of Finance, 37, Shukla, Ravi K, and Charles A Trzcinka, 1990, Sequential Tests of the Arbitrage Pricing Theory: A Comparison of Principal Components and Maximum Likelihood Factors, Journal of Finance, 45, Trzcinka, Charles A, 1986, On the Number of Factors in the Arbitrage Pricing Model, Journal of Finance, 41(2), Wei, K C John, 1988, An Asset-Pricing Theory Unifying CAPM and APT, Journal of Finance, 43,