Bias-variance trade-off in portfolio optimization under expected shortfall with ℓ 2 regularization

G. Papp, Fabio Caccioli, I. Kondor

Research output: Contribution to journalArticle

Abstract

The optimization of a large random portfolio under the expected shortfall risk measure with an ℓ 2 regularizer is carried out by analytical calculation for the case of uncorrelated Gaussian returns. 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 di?erent 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 η 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 versus η 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 four for a su?ciently strong regularizer.

Original languageEnglish
Article number013402
JournalJournal of Statistical Mechanics: Theory and Experiment
Volume2019
Issue number1
DOIs
Publication statusPublished - Jan 4 2019

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Expected Shortfall
Portfolio Optimization
Estimation Error
Regularization
Trade-offs
optimization
Risk Measures
Blow-up
Phase Transition
Eliminate
Time series
Fluctuations
Optimization
Portfolio optimization
Estimation error
Expected shortfall
Assets

Keywords

  • cavity and replica method
  • quantitative finance
  • risk measure and management

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

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N2 - The optimization of a large random portfolio under the expected shortfall risk measure with an ℓ 2 regularizer is carried out by analytical calculation for the case of uncorrelated Gaussian returns. 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 di?erent 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 η 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 versus η 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 four for a su?ciently strong regularizer.

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