Portfolio Optimization under Expected Shortfall: Contour Maps of Estimation Error

TL;DR

This paper uses contour maps to analyze estimation errors in portfolio optimization under Expected Shortfall, revealing that large sample sizes are needed for reliable estimates, which are often impractical.

Contribution

It introduces contour maps for quantifying estimation errors and sensitivities in ES-optimized portfolios, combining analytical methods from statistical physics with numerical simulations.

Findings

Estimation errors grow with portfolio size and confidence level.

Required sample sizes for accurate ES estimation are impractically large.

Sensitivity of portfolio weights to return changes is quantified.

Abstract

The contour maps of the error of historical resp. parametric estimates for large random portfolios optimized under the risk measure Expected Shortfall (ES) are constructed. Similar maps for the sensitivity of the portfolio weights to small changes in the returns as well as the VaR of the ES-optimized portfolio are also presented, along with results for the distribution of portfolio weights over the random samples and for the out-of-sample and in-the-sample estimates for ES. The contour maps allow one to quantitatively determine the sample size (the length of the time series) required by the optimization for a given number of different assets in the portfolio, at a given confidence level and a given level of relative estimation error. The necessary sample sizes invariably turn out to be unrealistically large for any reasonable choice of the number of assets and the confidence level. These results are obtained via analytical calculations based on methods borrowed from the statistical physics of random systems, supported by numerical simulations.