University of Cambridge > Talks.cam > Cambridge Finance Workshop Series > Portfolio Choice with Model Misspecification: A Foundation for Alpha and Beta Portfolios

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

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Hedge funds such as Bridgewater Associates o↵er two kinds of portfolios: “alpha”portfolios (a strategy with both long and short positions with overall zero market risk)and “beta” portfolios (a long-only strategy with exposure to market risk); similarly,sovereign wealth funds such as Norges Bank separate the management of their alpha and beta funds. Moreover, hedge funds and sovereign funds hold a large number of assets in their portfolios, ranging from several hundred to thousands (the portfolio of Norges Bank has over 9,000 assets). In this paper, we provide a rigorous foundation for “alpha” and “beta” portfolio strategies and characterize their properties when the number of assets is asymptotically large and returns are given by the Arbitrage Pricing Theory (APT). The APT is ideal for this analysis because it allows for alphas, while still imposing no arbitrage. Our first contribution is to extend the interpretation of the APT to show that it can capture not just small pricing errors that are independent of factors but also large pricing errors that arise from mismeasured or missing factors. Our second contribution is to show that under the APT , the optimal mean-variance portfolio in the presence of a risk-free asset can be decomposed into two components: an “alpha” portfolio that depends only on pricing errors and a “beta” portfolio that depends only on factor risk premia and their loadings. We then demonstrate that the alpha portfolio is the minimum-variance portfolio that is orthogonal to the beta portfolio, and vice versa, `a l a Roll (1980). This optimality property implies that the alpha and beta portfolios satisfy properties similar to those of the optimal mean-variance portfolio in terms of the relation between portfolio mean and variance. Moreover, their optimality implies that the squares of their Sharpe ratios sum to the square of the Sharpe ratio of the optimal mean-variance portfolio. Our third contribution is to characterize alpha and beta portfolios when the number of assets is asymptotically large: in this setting, we show that the portfolio weights of the alpha portfolio typically dominate the weights of the beta portfolio. We obtain similar decompositions and asymptotic results for the tangency portfolio, the global-minimum-variance portfolio, and the portfolios that comprise the Markowitz efficient frontier. Our fourth contribution is to show how these results about the decomposition of various portfolio weights, together with the restriction arising from the extended APT , can and should be used to improve the estimation of portfolio weights in the presence of model misspecification.

This talk is part of the Cambridge Finance Workshop Series series.

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