Portfolio Optimization with Noisy Covariance Matrices

Mean-variance optimization provides a framework for constructing portfolios that have minimum risk for a given level of expected return. The required inputs are the expected asset returns, the asset covariance matrix, and a set of investment constraints. While portfolio optimization always leads to an increase in ex ante risk-adjusted performance, there is no guarantee that this performance improvement carries over ex post. The culprit is that both the expected return forecasts and the asset covariance matrix contain estimation error. In this paper, we explore the impact of sampling error in the covariance matrix when using mean-variance optimization for portfolio construction. In particular, we show that sampling error leads to several adverse effects, such as: (a) under-forecasting of risk, (b) increased out-of-sample volatility, (c) increased leverage and turnover, and (d) inefficient allocation of the risk budget.Moreover, we introduce a new framework to explain and understand the origin of these adverse effects. We decompose the optimal portfolio into an alpha portfolio which explains expected returns, and a hedge portfolio which has zero expected return but serves to reduce portfolio risk. We show that sampling error in the asset covariance matrix leads to systematic biases in the volatility and correlation forecasts of these portfolios.We also provide a geometric interpretation showing how these biases lead to the adverse effects described above