Portfolio Optimization with Spectral Measures of Risk

TL;DR

This paper extends spectral risk measures to portfolio optimization, providing analytical properties that enable efficient minimization and revealing a unified view of risk and return optimization.

Contribution

It introduces a generalized methodology for spectral measures, showing their minimization is equivalent to a convex, piecewise linear optimization problem, unifying risk and return trade-offs.

Findings

Spectral measures can be minimized efficiently using convex optimization techniques.

Classical risk-reward problems are equivalent to optimizing a single spectral measure.

The approach simplifies portfolio optimization by unifying risk and return considerations.

Abstract

We study Spectral Measures of Risk from the perspective of portfolio optimization. We derive exact results which extend to general Spectral Measures M_phi the Pflug--Rockafellar--Uryasev methodology for the minimization of alpha--Expected Shortfall. The minimization problem of a spectral measure is shown to be equivalent to the minimization of a suitable function which contains additional parameters, but displays analytical properties (piecewise linearity and convexity in all arguments, absence of sorting subroutines) which allow for efficient minimization procedures. In doing so we also reveal a new picture where the classical risk--reward problem a la Markowitz (minimizing risks with constrained returns or maximizing returns with constrained risks) is shown to coincide to the unconstrained optimization of a single suitable spectral measure. In other words, minimizing a spectral measure turns out to be already an optimization process itself, where risk minimization and returns maximization cannot be disentangled from each other.