Bias-variance trade-off in portfolio optimization under Expected Shortfall with $\ell_2$ regularization

TL;DR

This paper analytically studies how $ ext{l}_2$ regularization affects the bias-variance trade-off in portfolio optimization under Expected Shortfall, showing it reduces estimation error and phase transitions, especially in small sample regimes.

Contribution

It provides an analytical framework demonstrating how $ ext{l}_2$ regularization stabilizes portfolio optimization and reduces estimation error under Expected Shortfall, especially when data is limited.

Findings

Regularizer reduces divergence of estimation error.

In data-rich regimes, regularizer has minimal impact.

Strong regularization significantly improves estimation accuracy.

Abstract

The optimization of a large random portfolio under the Expected Shortfall risk measure with an $\ell_2$ regularizer is carried out by analytical calculation. The regularizer reins in the large sample fluctuations and the concomitant divergent estimation error, and eliminates the phase transition where this error would otherwise blow up. In the data-dominated region, where the number $N$ of different assets in the portfolio is much less than the length $T$ of the available time series, the regularizer plays a negligible role even if its strength $\eta$ is large, while in the opposite limit, where the size of samples is comparable to, or even smaller than the number of assets, the optimum is almost entirely determined by the regularizer. We construct the contour map of estimation error on the $N/T$ vs. $\eta$ plane and find that for a given value of the estimation error the gain in $N/T$ due to the regularizer can reach a factor of about 4 for a sufficiently strong regularizer.