Arbitrage Pricing Theory and Large Pricing Errors

Arbitrage Pricing Theory and Large Pricing Errors

A recent paper by Raman Uppal of EDHEC Business School and Paolo Zaffaroni of Imperial College Business School drills down into the Arbitrage Pricing Theory, a theory that addresses the relationship of an asset’s price to contextual economic facts.

The title of the Uppal/Zaffaroni paper, “Portfolio Choice with Model Misspecification,” doesn’t really give away the most interesting portion of its contents. What is fascinating is the paper’s reliance on APT; a theory that’s been around for nearly forty years but that is still a good deal less known than is the Capital Asset Pricing Model. APT’s relationship with CAPM is an equivocal one. From one point of view APT is the more general theory, and CAPM fits nicely into it as a special case. From another point of view, though, APT addresses a different issue. It is not so concerned with how rational investors should act so as to stay on an optimal frontier. It concerns itself rather with how exogenous macro factors (such as shifts in the yield curve or in inflation numbers) can change the equilibrium price of a given asset.

Uppal and Zaffaroni use APT as a meta-model: a model that can be employed to check the errors in the specification of first-order models that in turn are used to value assets.

How They Got There

To see how they got there it helps to understand that they concern themselves with creating “a rigorous foundation for alpha and beta portfolio strategies where the number of assets is asymptotically large,” that is, as the number of assets approaches infinity.

One of their goals is modeling model risk, that is, the risk that certain portfolio assets (presumably those for which marking to market is not feasible) will have been mispriced due to the use of a flawed valuation method. It is in working on this point that they “extend the interpretation of the APT to show that it can capture not just small pricing errors that are independent of factors but large pricing errors arising from mismeasured or missing factors,” that is, misspecified models.

Uppal and Zaffaroni equate a qualitative difference with a quantitative difference here. The difference in the quality of errors (errors resulting from simply sloppy management versus errors resulting from a fundamentally wrong model) is expected to correlate with a difference in quantity (the former errors will reliably prove smaller than the latter).

Practical Consequences

As to the optimal mean-variance portfolio, the authors observe in particular that it decomposes into an alpha sub-portfolio (which depends upon pricing errors) and a beta sub-portfolio, which depends on factor risk premia. Following from that, they find that as the number of assets approaches infinity, the weights of the alpha side of the portfolio come to dominate the weights of the beta side.

Separately, they look at the global mean-variance portfolio, and they find that this decomposes into two portions much as does its optimizing counterpart. This time there is a sub-portfolio that depends only on the residual covariance matrix on the one hand, and a sub-portfolio that depends only on the factor covariance matrix on the other.

Their final task, then, is to illustrate “how these results can and should be used to improve the estimation of portfolio weights when either the alpha or beta component of returns are potentially misspecified.”

All of this makes of the APT, as Uppal and Zaffaroni say with a bit of understatement, “much more than just a statistical model of returns.”

Who Are These Guys

Raman Uppal, portrayed above, holds his bachelor’s degree from St. Stephen’s College, Delhi University, his Ph.D. from the University of Pennsylvania, Wharton School of Business. He is the co-author, with Piet Sercu, of a highly regarded work on exchange-rate volatility, trade and capital flows (2000).

Paolo Zaffaroni teaches financial econometrics at the Imperial College in London. He has also taught at the London School of Economics and at Cambridge. He and two other authors have an article on a topic closely related to this one in a forthcoming issue of The Journal of Empirical Finance, “Model Averaging in Risk Management with an Application to Futures Markets.”

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