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THE JOURNAL OF FINANCE VOL xXXV, NO 5. DECEMBER 1980 The fournal of FINANce VOL. XXXV DECEMBER 1980 No 5 An Empirical Investigation of the Arbitrage Pricing Theory RICHARD ROLL and STEPhEN A. ROSS ABSTRACT Empirical tests are reported for Ross'[48] arbitrage theory of asset pricing Usin lata for individual equities during the 1962-72 period, at least three and probably four riced"factors are found in the generating process of returns. The theory is supported in that estimated expected returns depend on estimated factor loadings, and variable expected returns, do not add any further explanatory power to that of the factor THE ARBITRAGE PRICING THEORY(APT) formulated by Ross[48 ]offers a testable alternative to the well-known capital asset pricing model(CAPM)introduced by Sharpe [51], Lintner [30] and Mossin [38]. Although the CAPM has been predominant in empirical work over the past fifteen years and is the basis of modern portfolio theory, accumulating. research has increasingly cast doubt on its ability to explain the empirical constellation of asset returns More than a modest level of disenchantment with the CAPM is evidenced by the number of related but different theories, e.g., Hakansson [18], Mayers [34] Merton [35 Kraus and Litzenberger [23]; by anomalous empirical evidence, e.g Ball [2], Basu [4], Reinganum [40]; and by questioning of the CAPM's viability as a scientific theory, e. g, Roll [41]. Nonetheless, the CAPM retains a central portfolio managers, investment advisors, and security analysd titioners such as There is good reason for its durability: it is compatible with the single most widely-acknowledged empirical regularity in asset returns, their common varia bility. Apparently, intuition readily ascribes such common variation to a single factor which, with a random disturbance generates returns for each individual asset via some (linear) functional relationship. Oddly, though, this intui wholly divorced from the formal CaPm theory. To the contrary, elegant deriva Graduate School of Management, University of California, Los Angeles, and School of orgar and Management, Yale University, respectively 1073

1074 The Journal of finance tions of the CAPM equation have been concocted beginning from the first principles of utility theory; but the models popularity is not due to such analyses, for they make all too obvious the assumptions required for the CAPm's validity and make no use of the common variability of returns. a review of recent finance exts(e. g, Van Horne, [54, pp. 57-63])reveals that rationalizations of the CAPM are based instead on the dichotomy between diversifiable and non-diversifiable risk, a distinction which refers to a linear generating process, not to the CAPm derived from utility theory The aPt is a particularly appropriate alternative because it agrees perfectly with what appears to be the intuition behind the CAPM. Indeed, the APt is based on a linear return generating process as a first principle, and requires no utility assumptions beyond monotonicity and concavity. Nor is it restricted to a single period; it will hold in both the multiperiod and single period cases. Though consistent with every conceivable prescription for portfolio diversification,no particular portfolio plays a role in the aPt. Unlike the CAPM, there is no requirement that the market portfolio be mean variance efficient. There are two major differences between the aPt and the original Sharpe[50] diagonal"model, a single factor generating model which we believe is the intuitive grey eminence behind the CAPM. First, and most simply, the APT allows more than just one generating factor. Second, the aPt demonstrates that since any market equilibrium must be consistent with no arbitrage profits, every equilibrium will be characterized by a linear relationship between each asset,s expected return and its returns response amplitudes, or loadings, on the common factors. With minor caveats, given the factor generating model, the absence of riskless arbitrage profits-an easy enough condition to accept a priori-leads immediately to the Its modest assumptions and its pleasing implications surely render the APT worthy of being the object of empirical testing To our knowledge, though, there has so far been just one published empirical study of the aPt, by gehr [17]. He began with a procedure similar to the one reported here. We can claim to have extended Gehr's analysis with a more r omprehensive set of data(he used 24 industry indices and 41 individual stocks) nd to have carried the analysis farther--to a stage actually required if the tests are to be definitive. Nonetheless, Gehr's paper is well worth reading and it must be given precedence as the first empirical work directly on this subject. Another empirical study related to the aPt is an early paper by Brennan [6], which is unfortunately still unpublished. Brennans approach was to decompose the residuals from a market model regression. He found two factors present in the residuals and concluded that "the true return generating process must be represented by at least a two factor model rather than by the single factor diagonal model"(p. 30). Writing before the aPt, Brennan saw clearly that "it is ot possible to devise cross-sectional tests of the Capital Asset Pricing Model, since only in the case of a single factor model is it possible to relate ex ante and ex post returns"(p. 34). Of course, the APT's empirical usefulness rests precisely in its ability to permit such cross-sectional tests whether there is one factor or The possibility of multiple generating factors was recognized long ago. Farrar

Arbitrage Pricing [15] and King[22], for example, employed factor analytic methods. Their work focused on industry influences and was pure(and very worthwhile)empiricism Since the aPT was not available to predict the cross-sectional effects of industry actors on expected returns, no tests were conducted for the presence of such More recently, Rosenberg and Marathe [44] have analyzed what they term extra-market"components of return. They find unequivocal empirical support for the presence of such components. Rosenberg and Marathe's work employs extraneous"descriptor variables to predict intertemporal changes in the CAPM's parameters. They state that"the appropriateness of the multiple-factor model of security returns, with loadings eq ual to predetermined descriptors, as opposed to a single-factor or market model, is conclusively demonstrated"(p 100).But,they do not ascertain the separate influences of these multiple factors on individual expected returns, and focus instead on a combined influence working through the market portfolio. In other words, they assume the CAPM and decompose the single market beta into its constitutent parts. Regarding the market portfolio as a construct which captures the influences of many factors follows the theoretical ideas in Rosenberg [45] and Sharpe [52] Thus, Rosenberg and Marathe 's work does not provide a definitive test of the APT There are a number of other recent papers which are more or less related to In contain evidence of more than just a single market factor influencing returns. In contrast, Kryzanowski and To [24] give a formal test for the presence of additional factors but find"that only the first factor is non-trivial"(p. 23) Nevertheless, there seems to be enough evidence in past empirical work to conclude that there may exist multiple factors in the returns generating processes of assets. The aPt provides a solid theoretical framework for ascertaining hether those factors, if they exist, are "priced, "i. e, are associated with risk premia. The purpose of our paper is to use the APT framework to investigate both the existence and the pricing questions In the following section, (1), a more complete discussion of the unique testable features of the aPT is provided. Then section II gives our basic tests. It concludes that three factors are definitely present in the"prices"(actually in the expected returns)of equities traded on the New York and American Exchanges. A fourth factor may be present also but the evidence there is less conclusive Sections III and IV present two additional tests of the APT. The most mportant and powerful is in section Ill, where the APT is compared against a specific alternative hypothesis that"own"variance influences expected returns If the apt is true, theown"variance should not be important, even though its sample value is known to be highly correlated cross-sectionally with sample mean returns. We find that the"own"variance's sample influence arises spuriously from skewness in the returns distribution. In section IV, we present a test of the consistency of the APT across groups of assets. Although the power of this test is probably weak, it gives no indication whatsoever of differences among groups

The Journal of finance Our conclusion is that the aPT performs well under empirical scrutiny and that it should be considered a reasonable model for explaining the cross-se variation in average asset returns I. The APt and its Testability A. The APT This section outlines the APT in a fashion that makes it suitable for empirical work. a detailed development of theory is presented in Ross [47, 48] and the intent here is to highlight those conclusions of the theory which are tested in subsequent sections. The theory begins with the traditional neoclassical assumptions of perfectly competitive and frictionless asset markets. Just as the CaPm is derived from the assumption that random asset returns follow a multivariate normal distribution the APt also begins with an assumption on the return generating process Individuals are assumed to believe(homogeneously) that the random returns he set of assets being considered are governed by a k-factor generating model of the form: F=E1+b161+…+b18k+∈, The first term in (1), EL, is the expected return on the i asset. The next k terms are of the form b,s, where 5, denotes the mean zero j th factor common to the returns of all assets under consideration. The coefficient b, quantifies the sensi tivity of asset is returns to the movements in the common factor 8. The common factors capture the systematic components of risk in the model. The final term, Ei, is a noise term, i.e, an unsystematic risk component, idiosyncratic to the i th asset. It is assumed to reflect the random influence of information that is unrelated to other assets. In keeping with this assumption, we also have that E{e|8}=0 and that Ei is(quite)independent of E, for all i andj. Too strong a dependence in he ais would be like saying that there are more than simply the k hypothesized common factors. Finally, we assume for the set of n assets under consideration, that n is much greater than the number of factors, k Before developing the theory, it is worth pausing to examine (1)in a bit more detail. The assumption of a k-factor generating model is very similar in spirit to a restriction on the Arrow-Debreu tableau that displays the returns on the assets n different states of nature. If the Et terms were omitted, then(1) would say that each asset i has returns r that are an exact linear combination of the returns a riskless asset (with identical return in each state) and the returns on k other factors or assets or column vectors, & Sk. In such a setting the riskl return and each of the k factors can be expressed as a linear combination of k+ 1 other returns, say ri through r+l. Any other asset s return, since it is a linear combination of the factors must also be a linear combination of the first k+1

Arbitrage Pricing 1077 assets'returns. And thus, portfolios of the first k+ 1 assets are perfect substitutes for all other assets in the market. Since perfect substitutes must be priced equally, there must be restrictions on the individual returns generated by the model. This is the core of the APt: there are only a few systematic components of risk existing in nature. As a consequence, many portfolios are close substitutes and as such, they must have the same value What are the common or systematic factors? This question is equivalent to asking what causes the particular values of covariance terms in the CAPM. If there are only a few systematic components of risk, one would expect these to be related to fundamental economic aggregates, such as GNP, or to interest rates or weather(although no causality is implied by such relations). The factor model formalism suggests that a whole theoretical and empirical structure must be explored to better understand what economic forces actually affect returns systematically. But in testing the APT, it is no more appropriate for us to examine this issue than it would be for tests of the CaPm to examine what, if anything, causes returns to be multivariate normal. In both instances, the return generating process is taken as one of the primitive assumptions of the theory. We do consider the basic underlying causes of the generating process of returns to be a potentially important area of research, but we think it is an area that can be investigated separately from testing asset pricing theories Now let us develop the APT itself from the return generating pr Consider an individual who is currently holding a portfolio and is contemplating n alteration of his portfolio. Any new portfolio will differ from the old portfolio by investment proportions x(i=1 ) which is the dollar amount purchased or sold of asset i as a fraction of total invested wealth. The sum of the x. proportions, since the new portfolio and the old portfolio put the same wealth into the n assets. In other words, additional purchases of assets must be financed by sales of others Portfolios that use no wealth such as x =(xy xn) are called In deciding whether or not to alter his current holdings, an individual will examine all the available arbitrage portfolios. The additional return obtainable from altering the current portfolio by n is given by x=∑x (∑xE)+(∑xb1)61 (∑xbA)+∑x Consider the arbitrage portfolio chosen in the following fashion. First, we will eep each element, x, of order 1/n in size; i.e. we will choose the arbitrage fied. Second that An underscored symbol indicates a vector or matrix

1080 The Journal of finance portfolio probably should not possess much idiosyncratic risk. Thus, it might serve as a substitute for one of the factors Furthermore individual asset 6's calculated against the market portfolio would enter the pricing relationship and the excess return on the market would be the weight on these b s. But, it is important to understand that any well-diversified portfolio could serve the same function and that, in general, k well-diversified portfolios could be found that approximate the k factors better than any single market index. In general, the market portfolio plays no special role whatsoever in the aPt, unlike its pivotal role in the CAPm,(Cf. Roll [41, 42]and Ross [49) The lack of a special role in the APT for the market portfolios is particularly important. As we have seen, the aPt pricing relationship was derived by considering any set of n assets which followed the generating process(1). In the CAPM, it is crucial to both the theory and the testing that all of the universe of available assets be included in the measured market portfolio. By contrast, the APT yields a statement of relative pricing on subsets of the universe of assets. As a consequence, the APT can, in principle, be tested by examining only subsets of the set of all returns. We think that in many discussions of the CAPm, scholars were actually thinking intuitively of the aPt and of process(1 )with just a single factor. Problems of identifying that factor and testing for others were not considered important To obtain a more precise understanding of the factor risk premia, E'-Eo, in 3), it is useful to specialize the aPt theory to an explicit stochastic environment within which individual equilibrium is achieved. Since Pt is valid intertemporal as well as static settings and in discrete as well as in continuous time, the choice of stochastic models is one of convenience alone. The only critical assumption is the returns be generated by (1)over the shortest trading period a particularly convenient specialization is to a rational anticipations intertem- poral diffusion model. (See Cox, Ingersoll and Ross [8] for a more elaborate version of such a model and for the relevant literature references. )Suppose there are k exogenous, independent(without loss of generality) factors, s which follow multivariate diffusion process and whose current values are sufficient statistics to determine the current state of the economy. As a consequence the current price, p,, of each asset i will be a function only of =(s sh)and the particular fixed contractual conditions which define that asset in the next differ ential time unit. Similarly the random return, dr, on asset i will depend on the random movements of the factors. By the diffusion assumption we can write d=E:dt+bnds2+…+bkds It follows immediately that the conditions of the APt are satisfied exactly-with E=0 and the APt pricing relationship (3)must hold exactly to prevent arbitrage. In this setting, however, we can go further and examine the premia If individuals in this economy are solving consumption withdrawal proble then the current utility of future consumption, e.g, the discounted expected value of the utility of future consumption, v, will be a function only of the individuals current wealth, w, and the current state of nature, s. The individual will optimize

Arbitrage Pricing by choosing a consumption withdrawal plan, c, and an optimal portfolio choice x,so as to maximize the expected increment in V; i.e max edv n optimum, consumption will be withdrawn to the point where its marginal equals the marginal utility of wealth The individual portfolio choice will result from the optimization of a locally quadratic form exactly as in the static CAPM theory with the additional feature that covariances of the change in wealth, du, with the changes in state variables, ds, will now be influenced by portfolio choice and will, in general, alter the optimal portfolio. By solving this optimization problem and using the marginal utility condition, u(c)=ve, the individual equilibrium sets factor risk premia E-Eo=(R/c)(ac/as)03; (wvu)/Vu, the individual coefficient of relative risk aver a, is the local variance of (independent) factor S,.(The interested reader referred to Cox, Ingersoll and Ross [8]for details. Notice that the premia E Eo can be negative if consumption moves counter to the state variable. In this case portfolios which bear positive factor s risk hedge against adverse movements in consumption, but too much can be made of this, since by simply redefining s to be-g the sign can be reversed. The sign, therefore, is somewhat arbitrary and we will assume it is normalized to be positive. Aggregating over individuals yields (3) One special case of particular interest occurs when state dependencies can be ignored. In the log case, R=l, for example, or any case with a relative wealth criteria(see Ross [48])the risk premia take the special form E-Eo=R(∑xb)02 where x is the individual optimal portfolio. This form emphasizes the general relationship between b, and o,. Normalizing 2x,by to unity by scaling s, we E-Eo= Roj The risk premium of factor j is proportional to its variance and the constant of proportionality is a measure of relative risk aversion For other utility functions, individual consumption vectors can be expressed in erms of portfolios of returns and similar expressions can be obtained. In effect, nce the weighted state consumption elasticities for all individuals satisfy the aPt pricing relationships, they must all be proportional 2 Breeden [5] has developed the observation that homogenous beliefs about Es and bs imply perfect correlation between individual random consumption changes. His results depend on th assumption, made also by aPt, that k<N