Bias-Variance Trade-Off in Portfolio Optimization under Expected Shortfall with ℓ2 Regularization

16 Pages Posted: 17 Jan 2017

See all articles by Gabor Papp

Gabor Papp

Eötvös Loránd University

Fabio Caccioli

University College London

Imre Kondor

Parmenides Foundation; London Mathematical Laboratory

Date Written: January 14, 2017

Abstract

The optimization of a large random portfolio under the Expected Shortfall risk measure with an ℓ2 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 of different assets in the portfolio is much less than the length of the available time series, the regularizer plays a negligible role, while in the opposite limit (which occurs much more frequently in practice), 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. Our results show that the transition region between these two extremes is relatively narrow, and it is only here that one can meaningfully speak of a trade-off between fluctuations and bias. The cause of both the anomalously large fluctuations and the limited usefulness of regularization is the unboundedness of Expected Shortfall as a loss function, a property it shares with all the other coherent measures, but also with downside risk measures in general, including Value at Risk.

Keywords: Portfolio Optimization, Bias-Variance Trade-Off, Expected Shortfall

JEL Classification: C02, C13, C61, G11, G32

Suggested Citation

Papp, Gabor and Caccioli, Fabio and Kondor, Imre, Bias-Variance Trade-Off in Portfolio Optimization under Expected Shortfall with ℓ2 Regularization (January 14, 2017). Available at SSRN: https://ssrn.com/abstract=2899446 or http://dx.doi.org/10.2139/ssrn.2899446

Gabor Papp

Eötvös Loránd University ( email )

Pazmany Peter setany 1A
Budapest, -- H1117
Hungary

Fabio Caccioli

University College London ( email )

Gower Street
London, WC1E 6BT
United Kingdom

Imre Kondor (Contact Author)

Parmenides Foundation ( email )

Kirchplatz 1
Pullach
Munchen, 82049
Germany

London Mathematical Laboratory ( email )

14 Buckingham St
London, WC2N 6DF
United Kingdom

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