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Empirical Evidence of the Arbitrage Pricing Model

Jamal Munshi, Sonoma State Univesity, 1992
All rights reserved

The simplest and most commonly used asset pricing model in finance is a one factor model called the CAPM. It is 'one factor' in the sense that there is only one explanatory variable and that variable is the risk premium of the market as a whole. Its simplicity was attacked by Ross (1976) who felt that there must be more than one dimension to asset pricing and by Roll (1977) who claimed that the CAPM is not 'theory' since it cannot be refuted or tested. In its place Ross (1976) and Ross and Roll (1980) proposed a multi-factor model which they called the arbitrage pricing theory or the APT. Several macro-economic variables are used to explain asset pricing.

Wherease the CAPM relates stock returns to only the 'market' in the linear equation

Ri = Ro + (Rm-Ro)*B1

where Rm is the market rate of return, the APT model states the asset returns as a risk free return plus a linear combination of factors as :

Ri = Go + (G1-Go)*B1 + (G2-Go)*B2 + ....

where Go can be interpreted as the risk free rate of return and the (Gi-Go) terms are risk premia demanded for each class of risk defined by the factors.

Empirical tests of the APT have been inconclusive because no two researchers could agree on the value of the coefficients of any of the exogenous variables (Chen 1983, Chen, Roll and Ross 1983, Roll and Ross 1980, Kryzanowski et al 1994). Kryzanowski et al (1994) show that the explanatory variables are correlated. Efforts to generate orthogonal factors results in one dominant factor and APT models that retain multiple explanatory variables are unstable. No empirical investigation of the APT has produced results that were considered by the researchers to be superior to the CAPM. The entire APT epoch in financial research turned out to be a multicollinearity dead end.

Empirical tests of the APT are characterized by the emotional zeal of the authors and their universal dislike for the CAPM rather than objective scientific inquiry. A close look at Chen (1983) reveals these aspects of APT research. Chen, a great fan of the APT, reports that he was unable to find any evidence that the APT is not valid. In each case, his null hypothesis was that the APT is valid; and in each case, he was unable to reject this hypothesis.

It is clear that he had also set out to establish that the APT is better than the CAPM (as a predictor); but could only come up with "the APT performs VERY WELL against the CAPM". (i.e., it's pretty good).

The APT model was built using five factors (a very common number to use). To his great credit Chen fixed the predictor variables and number of factors a priori to avoid 'data dredging', that is, to keep adding predictors and factors until you prove what it is you are out to prove.

An important difference between CAPM and APT in the regression portion of the empirical test is that while the CAPM does not require a statistically significant relationship to exist between Ri and Rm (it only seeks to extract whatever covariance that might exist), the APT depends on it. The APT model cannot be built if the regression null hypothesis cannot be rejected. The validity of the linear model is tested with the hypothesis

Ho: b1=b2=b3=b4=b5=0
Ha: At least one of the regression weights is non-zero

This is, of course, statistical voodoo. In the APT the entire regression model has to be correct and valid. Thus the hypothesis should have been

Ho: at least one of the bi=0 against
Ha: none of the bi=0

Only a rejection of this hypothesis will lead to the conclusion that the model is correctly specified - a necessary condition for APT validation. Each of the regression weights should be tested with a t-test with the appropriate Bonferonni type adjustment. If any of the weights are not significantly different from zero, the model is incorrectly specified and the experiment is over. Conclusion; reject APT

But even with the slanted hypothesis, the data do not suggest that the null hypothesis can be rejected. Chen presents the regression weights along with the F-values but no p-values. I computed the p-values and I find that most of the data do not support the regression hypothesis at the 5% level. At this point the author retreats to the 10% level and pushes on. The t-values computed for each of the 20 regression weights (5 predictor variables times 4 periods of study) show that only eight of these are statistically significant. This does not provide much support for the model.

Two of the periods studied (1971-1974 and 1975-1978) actually support a one parameter hypothesis (ala CAPM). One period, 1967-1970 supports a 2-parameter hypothesis. None of the periods support the 5-parameter hypothesis. The only conclusion that can be reached is that the APT linear model is mis-specified.

The author shows that the residuals of the of the single parameter CAPM equation can be explained by throwing in more predictive variables (as one would expect). However, he does not apply the same test to his own 5 parameter APT model. Would addition of a 6th and 7th variable, for instance, reduce his error sum of squares in the APT?

As in other APT papers the author seems more interested in promoting the APT model that in carrying out objective scientific inquiry.

An asset's total variance of returns can be partitioned into two parts the systematic and the diversifiable as long as the linear model is valid (regardless of number of predictor variables). The regression itself is the process that makes this partition. There is no reason to believe that the predicted returns of two portfolios will be different purely on the basis of the difference between the total variance of the assets unless the linear model is incorrect. And there are better and more easily interpretable tests for the correctness of the linear model.

In Dybvig and Ross (1985) we find further evidence that research into asset pricing had deteriorated into open warfare between the CAPM camp and the APT camp. This paper responds to Shanken's charge that the APT suffers from the same testability problems that the APT camp uses to attack the CAPM. Although Ross goes to great lengths to refute Shanken's charge, it is clear that Shanken's attack has softened Ross's vitriol toward the CAPM. In this paper he takes a rather generous view of CAPM claiming now, that the CAPM and APT are really compatible and 'imply' each other. It's just a matter of how many factors we want to use (CAPM=1 factor, APT=k factors).

But he finally reverts to the Roll critique of the CAPM viz, that it is not testable since the 'market portfolio' is not identifiable. The APT, on the other hand, does not force the empiricist to define the market but allows him to use any subset of the market portfolio to validate the model. Ross and Roll thus prevail: the CAPM is not testable but the APT is. But this article does not really respond to the Shanken charge that there is no reason to believe that the eigenvalues of all subset portfolios will be the same. The APT camp says, in effect, that if the portfolios are 'sufficiently large' then the assumption 'is not a bad one'. But as Shanken shows, the definition of 'sufficiently large' suffers from the same empirical difficulty of defining the 'market portfolio' in the CAPM.


Chen, N.F, and Ingersoll, E., Exact pricing in linear factor models with finitely many assets: A note, Journal of Finance June 1983 page 985

Chen, Naifu, Richard Roll, and Stephen Ross, Economic forces and the stock market: testing the APT and alternate asset pricing theories, Working paper, December 1983

Chen, Naifu, Some empirical tests of the theory of arbitrage pricing, Journal of Finance, Dec 1983 pp 1393, p1414

Dybvig, Phillip, and Ross, Stephen, Yes, the APT is Testable, Journal of Finance, Sep, 1985

Fama, Eugene, and James MacBeth, Risk, return, and equilibrium, Journal of Political Economy, 1973, 81, p607

Kryzanowski, Lawrence, Simon Lalancette, and Minh Chau To, Some tests of APT mispricing using mimicking portfolios, Financial Review, v29: 2, p153, May 1994

Roll, Richard, A critique of the asset pricing theory's tests, Journal of Financial Economics, March 1977, p129

Roll, Richard and Stephen Ross, An empirical investigation of the arbitrage pricing theory, Journal of Finance, Dec 1980, p1073

Ross, Stephen, The arbitrage theory of capital pricing, Journal of Economic Theory, v13, p341, 1976

Sharpe, William, A simplified model for porftolio returns, Management Science, 1962, p277

Sharpe, William, Capital asset prices: a theory of market equilibrium under conditions of risk, Journal of Finance, v19, p425, 1964

Index of working papers

Original file name: apt

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