The contour map of estimation error of Expected Shortfall (ES) is constructed. It allows one to quantitatively determine the sample size (the length of the time series) required by the optimization under ES of large institutional portfolios for a given size of the portfolio, at a given...

The optimized portfolio that is calculated by a covariance matrix has large sensitivities to small eigen values of the covariance matrix. Estimation of sampling errors for small eigen values is quite important for fund managers who construct their portfolios from estimated covariance matrixes....