Arbitrage Pricing Theory

Overview

A natural extension of the single-index model is to use more than just one index. Before we start down that path, though, we should first consider why such an approach makes theoretical sense.

Suppose there are a large number of factors (e.g. thousands) and that each stock's return is determined simply by a weighted sum of these factors. Suppose, furthermore, that there are at least as many linearly independent stocks as there are factors. Finally, suppose that each factor is independent of all the others in the usual probabilistic sense.

From this it unavoidably follows that instead of choosing a set of stock weights to build your portfolio, you can choose a set of "factor weights" and reverse-engineer the required stocks with pretty basic linear algebra.

From this perspective, a stock's return is given by

where $\beta_{ij}$ is how much factor $j$ influences stock $i$ and $\beta_{jt}$ is the return of factor $j$ at time $t$.

The expected return of the asset is, in the same way, simply a weighted sum of the expected return of each factor.

In the single-factor model, we assumed stocks had a single risk-premium determined by their beta. In this model, we assume a stock has a number of risk premiums determined by a number of betas.

Like with CAPM if we assume the market is perfectly efficient, $\alpha_i = 0$ for every stock. The reason is that if a stock had a non-zero alpha, then there would exist a portfolio of short and long positions with a risk free return of $\alpha_i$ more than the risk-free interest rate. An investor could then leverage that portfolio to achieve an arbitrarily high risk-free return.

Why is this perspective called "arbitrage pricing theory?" The answer is that this is the perspective taken by arbitrage traders: they

1. Try to find a asset whose price is too low (high).
2. They construct a portfolio with the same common factors
3. They go long (short) on the asset and short (long) on the portfolio.
4. If they're right, they get an essentially risk-free return above the normal risk-free interest rate. Even if they're wrong about (1), their risk should be near zero.

Unlike the CAPM model, this model has not been falsified empirically. On the other hand, unlike the CAPM model, this model can not be falsified empirically: even if you prove a persistent non-zero alpha value exists after accounting for 100 factors, proponents can always just say your implementation is missing other important factors.

Fama-French Three-Factor Model

Any discussion of factors should probably start with Eugene Fama - the "father of modern finance" Eugene Fama and winner of the Nobel prize for his "empirical analysis of asset prices" The Prize in Economic Sciences 2013.

His most famous contribution is proposing the three levels of market efficiency, but this is closely followed by his extension of the single factor model based on the capital asset pricing model to include two additional observations: small firms and firms with high book-to-market ratios tend to outperform the market even after controlling for market risk ($\beta$) Fama–French three-factor model. . He actually found results much more impressive than just this, namely The Cross-Section of Expected Stock Returns:

• After controlling for the book-to-market ratio, leverage no longer correlates with higher return
• After controlling for market capitalization and the book-to-market ratio, the E/P ratio no longer correlates with higher returns.
• After controlling for market capitalization, $\beta$ no longer correlates with higher returns. In other words, they find the CAPM model to be completely wrong!

You may think this disproves the CAPM, and I think Fama would agree with you. You may think this disproves the efficient market hypothesis - and there Fama does not: he's pretty adamant that all a paper can do is prove that either the market is not efficient or that the pricing model is wrong, but that it's impossible to prove which is the case. For the skeptical, Fama cites an alternative reason small firms might yield better returns in an efficient market Chan

Is there any criticism of the study itself? I haven't seen any besides the obvious: just because the strategy worked in the past (1963 - 1990 in this case) doesn't mean it works now. If this is true, there are two possible reasons: (a) the market itself changed, presumably getting more efficient over time. and (b) the relationship is statistical luck.

The original study offers some indication of (a): $\beta$ did predict returns between 1941 and 1965 - but even this goes away after controlling for firm size, suggesting what changed was the relationship between $\beta$ and firm size, not either of them and returns. The authors reported no evidence that their favored model has weakened over time.

In my opinion (b) is only plausible for the market capitalization parameter since its p-values ranged from 0.05 to 0.00065. Once you consider all the plausible other parameters that might predict stock trends, this seems plausibly due to chance. The book-to-market ratio's p-values, however, ranged from roughly 1-in-100,000 to roughly 1-in-a-billion, which seems really implausible to be by chance alone.

Let's consider these two factors in more detail.

Other Factors