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...

Risk analysis involves gaining deeper insight into the sources of risk, and evaluating whether these risks accurately reflect the views of the portfolio manager. In this paper, we show how to extend standard volatility analytics to shortfall, a measure of extreme risk. Using two examples, we...