Diversification quotient based on expectiles

TL;DR

This paper introduces an expectile-based diversification quotient (DQ) for portfolio risk assessment, offering computational simplicity, robustness to small samples, and advantageous mathematical properties over traditional measures.

Contribution

It develops a novel expectile-based DQ with explicit formulas, demonstrating its advantages and practical applicability in portfolio optimization.

Findings

Expectile-based DQ has simple formulas and links to the Omega ratio.

It is robust against small-sample tail data issues.

The optimization problem can be efficiently solved via linear programming.

Abstract

A diversification quotient (DQ) quantifies diversification in stochastic portfolio models based on a family of risk measures. We study DQ based on expectiles, offering a useful alternative to conventional risk measures such as Value-at-Risk (VaR) and Expected Shortfall (ES). The expectile-based DQ admits simple formulas and has a natural connection to the Omega ratio. Moreover, the expectile-based DQ is not affected by small-sample issues faced by VaR-based or ES-based DQ due to the scarcity of tail data. The expectile-based DQ exhibits pseudo-convexity in portfolio weights, allowing gradient descent algorithms for portfolio selection. We show that the corresponding optimization problem can be efficiently solved using linear programming techniques in real-data applications. Explicit formulas for DQ based on expectiles are also derived for elliptical and multivariate regularly varying distribution models. Our findings enhance the understanding of the DQ's role in financial risk management and highlight its potential to improve portfolio construction strategies.