The instability of downside risk measures

TL;DR

This paper investigates the feasibility and noise sensitivity of portfolio optimization using downside risk measures, revealing a phase transition phenomenon and providing analytical insights into the critical behavior of estimation errors.

Contribution

It introduces a phase transition framework for understanding the feasibility of downside risk-based portfolio optimization and employs the replica method for analytical characterization.

Findings

Existence of an optimization phase transition depending on sample and parameters

Critical exponents for estimation error divergence at the phase transition

Analytical phase diagram corroborated by simulations

Abstract

We study the feasibility and noise sensitivity of portfolio optimization under some downside risk measures (Value-at-Risk, Expected Shortfall, and semivariance) when they are estimated by fitting a parametric distribution on a finite sample of asset returns. We find that the existence of the optimum is a probabilistic issue, depending on the particular random sample, in all three cases. At a critical combination of the parameters of these problems we find an algorithmic phase transition, separating the phase where the optimization is feasible from the one where it is not. This transition is similar to the one discovered earlier for Expected Shortfall based on historical time series. We employ the replica method to compute the phase diagram, as well as to obtain the critical exponent of the estimation error that diverges at the critical point. The analytical results are corroborated by Monte Carlo simulations.