### Abstract

We consider the variance portfolio optimization problem with a ban on short selling. We provide an analytical solution by means of the replica method for the case of a portfolio of independent, but not identically distributed, assets. We study the behavior of the solution as a function of the ratio r between the number N of assets and the length T of the time series of returns used to estimate risk. The no-short-selling constraint acts as an asymmetric regularizer, setting some of the portfolio weights to zero and keeping the out-of-sample estimator for the variance bounded, avoiding the divergence present in the non-regularized case. However, the ban on short positions does not prevent the phase transition in the optimization problem, only shifts the critical point from its non-regularized value of to 2, and changes its character: at the out-of-sample estimator for the portfolio variance stays finite and the estimated in-sample variance vanishes, while another critical parameter, related to the estimated portfolio weights and the condensate density, diverges at the critical value . We also calculate the distribution of the optimal weights over the random samples and show that the regularizer preferentially removes the assets with large variances, in accord with one's natural expectation.

Original language | English |
---|---|

Article number | 123402 |

Journal | Journal of Statistical Mechanics: Theory and Experiment |

Volume | 2017 |

Issue number | 12 |

DOIs | |

Publication status | Published - Dec 12 2017 |

### 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

*Journal of Statistical Mechanics: Theory and Experiment*,

*2017*(12), [123402]. https://doi.org/10.1088/1742-5468/aa9684

**Analytic solution to variance optimization with no short positions.** / Kondor, I.; Papp, G.; Caccioli, Fabio.

Research output: Contribution to journal › Article

*Journal of Statistical Mechanics: Theory and Experiment*, vol. 2017, no. 12, 123402. https://doi.org/10.1088/1742-5468/aa9684

}

TY - JOUR

T1 - Analytic solution to variance optimization with no short positions

AU - Kondor, I.

AU - Papp, G.

AU - Caccioli, Fabio

PY - 2017/12/12

Y1 - 2017/12/12

N2 - We consider the variance portfolio optimization problem with a ban on short selling. We provide an analytical solution by means of the replica method for the case of a portfolio of independent, but not identically distributed, assets. We study the behavior of the solution as a function of the ratio r between the number N of assets and the length T of the time series of returns used to estimate risk. The no-short-selling constraint acts as an asymmetric regularizer, setting some of the portfolio weights to zero and keeping the out-of-sample estimator for the variance bounded, avoiding the divergence present in the non-regularized case. However, the ban on short positions does not prevent the phase transition in the optimization problem, only shifts the critical point from its non-regularized value of to 2, and changes its character: at the out-of-sample estimator for the portfolio variance stays finite and the estimated in-sample variance vanishes, while another critical parameter, related to the estimated portfolio weights and the condensate density, diverges at the critical value . We also calculate the distribution of the optimal weights over the random samples and show that the regularizer preferentially removes the assets with large variances, in accord with one's natural expectation.

AB - We consider the variance portfolio optimization problem with a ban on short selling. We provide an analytical solution by means of the replica method for the case of a portfolio of independent, but not identically distributed, assets. We study the behavior of the solution as a function of the ratio r between the number N of assets and the length T of the time series of returns used to estimate risk. The no-short-selling constraint acts as an asymmetric regularizer, setting some of the portfolio weights to zero and keeping the out-of-sample estimator for the variance bounded, avoiding the divergence present in the non-regularized case. However, the ban on short positions does not prevent the phase transition in the optimization problem, only shifts the critical point from its non-regularized value of to 2, and changes its character: at the out-of-sample estimator for the portfolio variance stays finite and the estimated in-sample variance vanishes, while another critical parameter, related to the estimated portfolio weights and the condensate density, diverges at the critical value . We also calculate the distribution of the optimal weights over the random samples and show that the regularizer preferentially removes the assets with large variances, in accord with one's natural expectation.

KW - cavity and replica method

KW - quantitative finance

KW - risk measure and management

UR - http://www.scopus.com/inward/record.url?scp=85039781243&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85039781243&partnerID=8YFLogxK

U2 - 10.1088/1742-5468/aa9684

DO - 10.1088/1742-5468/aa9684

M3 - Article

AN - SCOPUS:85039781243

VL - 2017

JO - Journal of Statistical Mechanics: Theory and Experiment

JF - Journal of Statistical Mechanics: Theory and Experiment

SN - 1742-5468

IS - 12

M1 - 123402

ER -