# Fama-French vs. Arbitrage Pricing Theory of Ross

Is there a specific reason for why Fama-French papers on CAPM extensions do not refer to APT of Ross? In textbooks, APT is always an extension of CAPM and the foundation of extending the set of risk factors beyond market portfolio. However, I do not see this explicitly stated in Fama-French papers.

• You might be in interested in this video from John Cochrane about What is the FF3 model?'' – Kevin Sep 24 '20 at 14:41
• I have heard that Fama French got their idea from the ICAPM model of R. C. Merton and not from the APT model of Ross. But who said this I don't know. – noob2 Sep 25 '20 at 18:46
• Yes, the origins of their approach are puzzling to me. For example, yes, small cap firms outperform large cap firms. But why does this result in (S-L) variable that they construct? Why don't they take, say, the fraction S/L as the factor. – Qwerty Sep 25 '20 at 19:00

First, Fama French is not a model. It is a falsification of the Capital Asset Pricing Model. The APT is not an extension of the CAPM. It is conceptually similar, but it is laid upon a different conceptual framework.

Let us begin with the Fama-French. It is not a model. It does not describe the behavior of humans. The Capital Asset Pricing Model does describe the behavior of humans if it is true.

Fama and French added variables, probably incorrectly called factors, to the CAPM as a test of the CAPM. If the CAPM were true, then the slopes of the added variables should be zero. They were not.

Now, this is where it all gets a bit strange. In 1940 Abraham Wald set the foundations for what is now called Decision Theory. There are two branches of decision theory, a Bayesian and a Frequentist branch.

Both the CAPM and the APT are Frequentist models. However, all admissible models are Bayesian models. Frequentist models are admissible if they map to a Bayesian solution either at the limit or in every sample. Admissible Frequentist estimators are Bayes estimators. That is the linkage between the two interpretations of probability.

There is no requirement that an estimator is admissible, of course. The sample median of the normal distribution is not admissible under squared loss, but that doesn't make it less valid of an estimator. It just isn't an admissible estimator.

In Frequentist decision theory, you have an acceptance region and a rejection region. If the result of an experiment puts the statistic of interest in the acceptance region, then you accept the null as if true. If it is in the rejection region, then you reject the null and accept the alternative as if true.

In rejecting the CAPM, economics is to behave as if the Fama French alternative is true, subject to further exploration. So, one should, in the next case, start with the Fama French as the null model and then explore alternatives. If the Fama French is not rejected often enough, then it should be accepted as if true.

The problem is that Fama-French does not work out-of-sample. So it rejects the CAPM, but it also rejects itself.

The APT is a different creature. It is built off of the idea of orthogonalizing variables to force compliance with the requirements of regression. As such, its error terms do not have the same interpretation as CAPM error terms. The APT is not an extension of the CAPM and so Fama-French does not falsify it. It is falsified by other research. It also has never passed a validation test.