Improving Risk Forecasts for Optimized Portfolios

Sample covariance matrices tend to underestimate the risk of optimized portfolios. In this article, we identify special portfolios, termed “eigenportfolios,” that capture these systematic biases. Further, we present a methodology for estimating eigenportfolio biases and for adjusting the covariance matrix to remove these biases. We show that this procedure effectively removes the biases of optimized portfolios. We demonstrate that the adjusted covariance matrices are effective at reducing the out-of-sample volatilities of optimized portfolios.The Markowitz mean–variance framework provides the foundation for modern portfolio theory. Required inputs include a set of asset expected returns and a covariance matrix. The covariance matrix, in turn, is typically estimated from a finite sample of historical data. One problem with sample covariance matrices, however, is that they tend to systematically underestimate the risk of optimized portfolios. This problem presents an obvious challenge to practitioners of portfolio optimization because portfolio volatility cannot be reliably estimated with covariance matrices.In this article, we provide a practical solution to this investment problem. We investigate the sources of the biases of optimized portfolios. We identify and describe special portfolios, which we call eigenportfolios, that capture these systematic biases. The eigenportfolios are mutually uncorrelated and represent distinct combinations of the original assets. We demonstrate that the biases of eigenportfolios can be reliably estimated by numerical simulation. We found that the lowest-volatility eigenportfolios exhibit the largest underprediction biases and the sample covariance matrix slightly overpredicts the risk of the highest-volatility eigenportfolios. We show that the magnitude of these biases is extremely stable over time and depends on the degree of sampling error in the covariance matrix.We also describe how to adjust the covariance matrix to remove the biases of the eigenportfolios. Furthermore, we show that the adjustment procedure effectively removes the biases of optimized portfolios while still providing accurate forecasts for nonoptimized portfolios.Additionally, we examine the out-of-sample performance of optimized portfolios. We constructed portfolios optimized to have minimum volatility with a fixed alpha constraint. Relative to the conventional sample covariance matrices, we found that the adjusted covariance matrices produce portfolios with lower out-of-sample volatilities and, hence, better risk-adjusted performance.We also study how the performance gains in risk forecasting accuracy and reduced out-of-sample volatility depend on the degree of sampling error in the covariance matrix. A critical determinant of the degree of sampling error is the ratio of the number of stocks, N, to the number of periods, T, used to estimate the covariance matrix. For small values of N/T, the sampling error is relatively small, and we found modest performance gains from the adjustment methodology. As N/T increases, however, sampling error becomes more significant and the performance gains from the covariance adjustments become increasingly important.