Arbitrage Pricing Theory (APT)

Let's digress a bit on the underlying theory of factor models. Under some fairly general assumptions and using a no-arbitrage argument, we can show that the factor returns (betas) in factor models must be related across assets and that the asset's return is linear in the factors. The seminal work is Ross (1976).

Other references on APT and multifactor models can be found at www.kellogg.northwestern.edu/faculty/korajczy/htm/aptlist.htm.

Not only is the no-arbitrage argument less restrictive than the equilibrium requirement underlying CAPM, arbitrage pricing theory (APT) expands the number of factors beyond just the market return. This is a powerful result, as we see further on where we try to estimate parameters for large portfolios of assets; APT requires estimating several factor returns that can then be used to model the covariance matrix of returns across assets. This greatly reduces the data requirements necessary for estimation purposes. Recall that in the N asset case, we need to estimate covariances and N variances. For large portfolios, we will see that there are insufficient observations available to do so as N begins to overwhelm the number of time periods T. The S&P 500, for example, requires estimates for 500 variances and 124,750 covariances. This would require about 10,000 years of monthly returns, which, obviously, are not available. On the other hand, a k-factor model basically reduces the dimensionality of the problem to the k factors. So, in a sense, APT provides the basis for multifactor models that extend the concept of the CAPM as a pricing model but with fewer restrictions. We develop these ideas more fully further on. I am going to borrow liberally from Luenberger (1998). To begin, let's focus on a single linear factor model, letting f stand for some factor (for example, the return to the market portfolio) and write the return to asset i as r_i = α + β_if. Do the same for asset j: r_j = α + β_jf. Now form a portfolio of these two assets with weights w and (1 – w) having return r equal to:

Here is where the no-arbitrage argument comes in. The factor f is the source of systematic risk. Suppose we solve for the weight that makes this risk zero. Doing so will then imply that the return on the portfolio is risk free. A little algebra easily shows that the weight that makes [wβ_i + (1 – w)β_j]f = 0 is:

Substituting this into the previous equation and solving for the portfolio return confirms that it is risk free. For example, it is no longer a function of the systematic risk embodied in the factor.

This return must be equal to the risk-free return, r_f. Otherwise, an arbitrage opportunity presents itself, since we could borrow at the risk-free rate and fund this portfolio earning the premium (r – r_f). Therefore it must be true that:

Rearranging terms indicates that the relationship between the parameters across assets in the portfolio is not arbitrary:

Indeed, the no-arbitrage argument establishes that the risk-adjusted premium (the parameter α is the asset's alpha—its return in excess of its risk-adjusted return) over the risk-free rate of return must be constant across assets. The numerator is therefore a premium over the riskless rate, which is standardized by the return to the systematic source of risk from the factor) over the riskless rate of return must be constant across assets:

This is a powerful and subtle result. Solving for α on asset j:

Substituting this in for α in the factor model and taking expectations produces the expected return to the asset, given the factor f:

Thus, the expected return is the risk-free rate plus a risk premium (scaled by the constant C) where β measures the return to being exposed to the source of risk embodied in the factor. Statistically, β is the covariance between the asset's return and the risk factor f scaled by the factor variance:

We can generalize APT to many factors. The multifactor model takes the general form:

whereupon substituting the no-arbitrage constraint implications for α yields:

In practice, analysts are interested in estimating the parameters of this model to help them form an expectation of the asset's return. For estimation purposes, the APT is written as the linear multiple regression model, with error term ε, which captures the asset's idiosyncratic (or specific) risk, that is, the variance in return not explained by the systematic sources of risk represented in the factors.

Specific risk is unsystematic and can therefore be diversified away. To see why, let's construct a portfolio of N assets with weights indicating that the portfolio is fully invested, that is,

and write the return to the portfolio as follows:

In general, we diversify by adding assets to the portfolio. If we assume that the weights are all approximately the same (note that we are not assuming naïve investing as discussed in the chapter on anomalies), that is, w_i is approximately equal to 1/N, then if the idiosyncratic risks are bounded (they aren't infinite), then clearly, as N increases, must go to zero in the limit. In practice, factor selection can be designed to maximize the explanatory power of the model so that is minimized as well.

I made the claim earlier that CAPM is a single-factor model with the market rate of return as the factor. Let's establish now the link between APT and CAPM. Suppose we have an asset whose returns are determined by two factors:

In the CAPM, we would be interested in the covariance between r_i and r_m, that is, in estimating the asset's beta.

If we divide both sides by , we get the asset's beta as follows:

The connection to APT is that the CAPM beta is a weighted average of the underlying factor returns. The weights are a function of the covariances between the factors and the market return. We can generalize this to show that the CAPM beta is a weighted average of factors in any factor model. Therefore, CAPM is a special case of APT.

The parameters of the Fama-French three-factor model would be estimated using the following regression specification: