Tail Risk Constraints and Maximum Entropy

TL;DR

This paper explores how imposing tail risk constraints like VaR and CVaR influences portfolio distribution shapes, leading to a natural emergence of barbell payoffs under maximum entropy principles.

Contribution

It introduces a maximum entropy framework constrained by tail risk measures, highlighting their dominance over other distribution properties in portfolio construction.

Findings

Tail constraints override other distribution considerations.

Barbell payoff structure naturally emerges from the model.

Portfolio components become less relevant under tail constraints.

Abstract

In the world of modern financial theory, portfolio construction has traditionally operated under at least one of two central assumptions: the constraints are derived from a utility function and/or the multivariate probability distribution of the underlying asset returns is fully known. In practice, both the performance criteria and the informational structure are markedly different: risk-taking agents are mandated to build portfolios by primarily constraining the tails of the portfolio return to satisfy VaR, stress testing, or expected shortfall (CVaR) conditions, and are largely ignorant about the remaining properties of the probability distributions. As an alternative, we derive the shape of portfolio distributions which have maximum entropy subject to real-world left-tail constraints and other expectations. Two consequences are (i) the left-tail constraints are sufficiently powerful to overide other considerations in the conventional theory, rendering individual portfolio components of limited relevance; and (ii) the "barbell" payoff (maximal certainty/low risk on one side, maximum uncertainty on the other) emerges naturally from this construction.