Some Empirical Tests of the Theory of Arbitrage Pricing TORIo Nai-Fu Chen The Journal of finance, Vol. 38, No. 5. (Dec, 1983), pp. 1393-1414 Stable url: http://inks.jstor.org/sici?sici=0022-1082%028198312%02938%3a5%3c1393%3asetott%3e2.0.c0%3b2-2 The Journal of finance is currently published by American Finance Association Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.htmlJstOr'sTermsandConditionsofUseprovidesinpartthatunlessyou have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://wwwjstor.org/journals/afina.html Each copy of any part of a jSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission jStOR is an independent not-for-profit organization dedicated to creating and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact support@jstor. org http://www」]stor.org Sat Apr823:41:102006
THE JOURNAL OF FINANCE VOL. XXXVIIl NO. 5. DECEMBER 1983 Some Empirical Tests of the Theory of Arbitrage Pricing NAI-FU CHEN* ABSTRACT We estimate the parameters of Ross' s Arbitrage Pricing Theory(APT). Using daily return data during the 1963-78 period, we compare the evidence on the aPt and the Capital Asset Pricing Model(CAPM)as implemented by market indices and find that the apt performs well. The theory is further supported in that estimated expected returns depend on estimated factor loadings, and variables such as own variance and firm size do not contribute additional explanatory power to that of the factor loadings THE ARBITRAGE PRICING THEORY(APT), originally formulated by Ross 135, 36 and extended by Huberman [23 and Connor [13], is an asset pricing model that explains the cross-sectional variation in asset returns. Like the Capital Asset Pricing Model(CAPM)of Sharpe [39, Lintner [26], and Black [2], the APT begins with an assumption on the return generating process: each asset return linearly related to several, say k, common"global "factors plus its own idiosyn cratic disturbance. Then in a well-diversified, frictionless, and perfectly compet 7e economy, the no arbitrage condition requires that the expected return vector must lie(asymptotically) in the k+ 1 dimensional vector space spanned by a vector of all one' s and the k vectors of asset response amplitudes (to the k common global factors The initial empirical evidence on the model has been rather encouraging(see Gehr [19]and Roll and Ross [34). In this paper, we shall compare the empirical performance of the APT with that of the CAPM. We shall also test whether the APT can explain some of the empirical anomalies"related to the CAPM in recent years. The paper has six sections. In Sections I and Il, some basic results related to the aPt are given so that testable implications of the model can be clearly identified and the parameters of the model can be estimated. Section III contains the cross-sectional results of the APT In Sections IV and V, we attempt to reject the apt by looking at variables that are known to be highly correlated with returns to see if they have any additional explanatory power after the aPt parameters are included. W marize our findings in Section VI. Some Graduate School of Business, University of Chicago. I thank my dissertation committee chairman Richard Roll, for his guidance and encouragement; Eugene Fama for his suggestions; and Glenn Graves for his assistance in the mathematical programmings. i have also benefited from the comments of Armen Alchian, George Constantinides, Tom Copeland, Robert Geske, Jack Hirshleifer, Robert Hamada, Herb Johnson, Edward Leamer, Robert Litzenberger, David Mayers, Merton Miller, Marc Reinganum, Fred Weston, Arnold Zellner, and especially Stephen Brown and Michael Gibbons.This esearch was partially supported by the Research Program in Competition and Business policy at the University of California, Los angeles, and a University of California, Santa Barbara Academic ate grant8581557074277 1393
The Journal of finance mathematical derivations and the estimation procedure for the factors are de- scribed in the appendix. I. The arbitrage Pricing Theory and Its Implications A. A Brief Review of the aPt Assume that asset markets are perfectly competitive and frictionless and that dividuals believe that returns on assets are generated by a k-factor model, so that the return on the ith asset can be written as: Ei+ bl51 +.. bik Sk where E, is the expected return; 5,, j=1,., k, are the mean zero factors common to all assets; bi is the sensitivity of the return on asset i to the fluctuations in factor j; and E is the"nonsystematic"risk component idiosyncratic to the ih asset with E(6,18,=0 for all j. In a well diversified economy with no arbitrage opportunity, the equilibrium expected return on the ith asset is given by Ei= o +a,bi i2b2+ If there exists a riskless(or a"zero beta")asset, its return will be Xo. The other parameters, A1 Ak, can be interpreted as risk premiums corresponding to risk factors 81,..., Sk. In other words, A, is the expected return per unit of long investment of a portfolio with zero net investment and bpi 1 and bp2 B. Factor Analysis and the estimation of the Factor Loadings The procedure to estimate factor loadings (i.e., the bi 's)for all assets corre- sponding to the same set of common factors is quite involved and expensive we first do a factor analysis on an initial subset of assets, and then we extend the factor structure of the subset to the entire sample. This is accomplished via a large scale mathematical programming exercise. Section II contains a brief It is clear that the development of the theory of arbitrage pricing is quite separate from the factor analysis. We use factor analysis here only as a statistical tool to uncover the pervasive forces(factors) in the economy by examining how asset returns covary together. As with any statistical method, its result is meaningful only when the method is applied to a representative sample. In the present context, the initial subset to which the factor analysis is applied should consist of a large random sample of securities of net positive supply in the economy; thus the sample would be closely representative of the risks borne by investors. In a recent article, Shanken [37 points out some of the potential pitfalls of testing the aPt when the factored covariance matrix is unrepresen tative of the covariation of assets in the economy. By forming portfolios from See Ross[35, 36], Huberman [23], and Connor[ 13] for the formal development. See also Ingersoll [24], Chen and Ingersoll [10], Grinblatt and Titman [22], and Dybvig [16]
Empirical Tests of Arbitrage Pricing any given set of assets, Shanken demonstrates that factor analysis can produce many different factor structures from the manipulated portfolios In the extreme case where the constructed portfolios are mutually uncorrelated factor analysis produces no common factor. Of course, forming uncorrelated portfolios by longing and shorting securities merely repackages the risk bearing and potential reward associated with the original securities and does not alter the fundamental forces and characteristics inherent in the economy. However, as a statistical tool, factor analysis can no longer detect those pervasive forces from such manipulated portfolios. This, of course, should not be construed as a criticism of the theory or of the testability of the aPt, but rather should serve as a reminder of the potential problems involved in doing statistical analysis on unrepresentative samples. In this paper, we select 180 securities for each of the initial factor analyses. If we miss an important factor because of unrepresentativeness, all the tests that follow will be biased against the apt C. Testable Hypothesis of the APT We regard Equation(2)as the main result of the apt that explains the cross- sectional differences in asset returns, and it is(2)that will be tested in the following sections A logical first step in testing(2)would be to look for priced factors. However, the task of finding priced factors turns out to be not particularly straightforward If we have determined that k factors exist (in the sense of Connor [13 ])in the generating process of asset returns in the economy, then the number of priced factors-as long as there is at least one-is not well defined. Intuitively this can be most easily seen by noting that a k factor pricing equation can always be collapsed into a single beta equation via mean-variance efficient set mathematics (and with an additional orthogonal transformation similar to that described below, the number of priced factors can be arbitrary) Mathematically, let u be the risk premium vector for a particular set of risk factors and let u be any vector of the same dimension with u'u=U'u. Then there exists an orthogonal trans formation that will transform the original set of factors to a new set of orthogonal factors whose associated risk premium vector is u. In other words, if it has been established that k factors are present, the number of priced factors can be any number between 1 and k. It should be emphasized here however, that those factors that are not priced are just as important as those that are priced in an individuals investment decision(see Breeden [4], Constantinides [12] and roll [32] for related issues). They are irrelevant only in predicting expected return This should be borne in mind when interpreting the cross-sectional results in Section Ill a question that naturally arises in this investigation is how the aPt fa against other asset pricing models. It is immediately apparent from Equations The term "pervasive forces"was made popular by Connor [13]. See Shanken [37 for his interpretation of Connor's result and its implications to the aPt, Contrary to some beliefs, Shanken's results were not driven by the idiosync erms in a finite sample or the approximate nature of Ross'original formulation. The no common factor result can be obtained in an economy with exactly k factors and no idiosyncratic risks
1396 The Journal of Finance (1)and(2)that any model that predicts a linear relation between"risk"and return is potentially consistent with the APT. For example, if we let &1 be the return on the market and &2 be changes in interest rate, we obtain Mertons [27] two factor pricing equation. Therefore, unless the"market "and/or the "factors can be properly identified, accepting the aPt(i.e. not being able to reject the aPT) does not necessarily reject the other pricing equations. however the results of APT can be compared with other models as popularly implemented. In Section Ill, we examine the cross-sectional results of the apt and the CaPm It has been suggested that the aPt is so general that it is not rejectable Fortunately, this is not true. The most important result of the APt is that only those risks that are reflected in the covariance matrix are priced, nothing else So if we are able to find a variable that is priced even after the factor loadings (FL)are accounted for, the APT would be rejected. For example, if the daily return to every small firm were increased by 1% per day, the returns to small firms would be statistically significantly higher than the returns to large firms no matter how many factors were extracted from the covariance matrix to account for risk ex ante, as long as the number of factors is small relative to the number of securitie In Sections IV and V, we attempt to reject the apt using variables such as own variance and size of firm equity. Those variables are chosen because of the well documented high correlation between them and the average returns Il. Estimation of the Factor Loadings A. The Data The data are described in Table I. All the parameters in Sections IIl, IV, and V are computed using only data within each subperiod so that we may have four dependent tests of each hypothesis The computation of the b,s, the factor loadings(FL), uses only data from odd days within each subperiod. Even day returns are reserved for testing purposes B. Methodology The details of the estimation procedure are described in the Appendix; the following is a brief outline (i)The first 180 stocks in the sample(alphabetically) are selected and their sample covariance matrix computed. The choice 180 is the upper limit It is often asserted that the CAPM is a special case of the aPt. This is true if ists a rotation of the factors such that one of the factors is the"market. "Ex ante, there is no ason to assume this is the case in our finite economy. Ex post, we discover in this study as well as ubsequent studies [ 7, 11] that the first factor is highly correlated with a measure of the "market ghted NYSE stock index. However, throughout this study, we maintain the ex ante position that the Capm and the aPt are nonnested for hypothesis testing the number of pervasive factors that influence the is no more than a few and certainly less than, say, 20. We insert"sm thological case where the number of factors is equal to the number of ittern of returns can be explained. In the present study, the maximum number 80, the size of the initial factor analysis, while the number of securities in each