Portfolio Optimization with Expectile and Omega Functions

TL;DR

This paper explores the mathematical relationships between portfolio optimization problems involving negative expectile and omega ratio, providing theoretical insights and practical solutions for such risk measures.

Contribution

It establishes equivalences between optimization problems with negative expectile and omega functions, and derives subgradients and gradients for negative expectile.

Findings

Derived subgradients for negative expectile from dual representations

Provided an elementary derivation of the gradient of negative expectile

Conducted a case study solving portfolio optimization with negative expectile constraints

Abstract

This paper proves equivalences of portfolio optimization problems with negative expectile and omega ratio. We derive subgradients for the negative expectile as a function of the portfolio from a known dual representation of expectile and general theory about subgradients of risk measures. We also give an elementary derivation of the gradient of negative expectile under some assumptions and provide an example where negative expectile is demonstrably not differentiable. We conducted a case study and solved portfolio optimization problems with negative expectile objective and constraint.