Portfolio Choice with Model Misspecification: A Foundation for Alpha and Beta Portfolios

Abstract

Our objective is to investigate the effect of model misspecification on mean-variance portfolios and to show how asset-pricing theory and asymptotic analysis (for large number of assets) can be used to provide powerful solutions to mitigate misspecification. The starting point of our analysis is the Arbitrage Pricing Theory (APT). We extend the APT to show that it can capture not just small pricing errors that are independent of factors, but also large pricing errors from mismeasured or missing factors. Our key insight is that, instead of treating misspecification directly in the mean-variance portfolio, it is better to first decompose the portfolio into components that correspond to the two components of returns in the APT: a ``beta'' portfolio that depends on factor risk premia and an ``alpha'' portfolio that depends only on pricing errors. For the beta portfolio, we treat misspecification using asymptotic analysis: as the number of assets increases, we show that the weights of the alpha portfolio dominate those of the beta portfolio, leading to an expression for mean-variance portfolio weights that is immune to beta misspecification. For the alpha portfolio, we treat misspecification by imposing the APT restriction on alphas, which serves both as an identification condition and a shrinkage constraint. Finally, we demonstrate that our approach achieves an out-of-sample Sharpe ratio that is more than double that of the equally weighted portfolio.