### 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 language | English |
---|---|

Article number | 013402 |

Journal | Journal of Statistical Mechanics: Theory and Experiment |

Volume | 2019 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 4 2019 |

### Fingerprint

### 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

**
Bias-variance trade-off in portfolio optimization under expected shortfall with ℓ
_{2}
regularization
.** / Papp, G.; Caccioli, Fabio; Kondor, I.

Research output: Contribution to journal › Article

_{2}regularization ',

*Journal of Statistical Mechanics: Theory and Experiment*, vol. 2019, no. 1, 013402. https://doi.org/10.1088/1742-5468/aaf108

}

TY - JOUR

T1 - Bias-variance trade-off in portfolio optimization under expected shortfall with ℓ 2 regularization

AU - Papp, G.

AU - Caccioli, Fabio

AU - Kondor, I.

PY - 2019/1/4

Y1 - 2019/1/4

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.

AB - 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.

KW - cavity and replica method

KW - quantitative finance

KW - risk measure and management

UR - http://www.scopus.com/inward/record.url?scp=85062524716&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85062524716&partnerID=8YFLogxK

U2 - 10.1088/1742-5468/aaf108

DO - 10.1088/1742-5468/aaf108

M3 - Article

VL - 2019

JO - Journal of Statistical Mechanics: Theory and Experiment

JF - Journal of Statistical Mechanics: Theory and Experiment

SN - 1742-5468

IS - 1

M1 - 013402

ER -