An elementary proof of the dual representation of Expected Shortfall

TL;DR

This paper offers a simple, accessible proof of the dual representation of Expected Shortfall, making it suitable for educational purposes and providing a new proof of its subadditivity property.

Contribution

It presents an elementary proof of the dual representation of Expected Shortfall using basic properties of quantile functions, applicable in general probability spaces.

Findings

Elementary proof of dual representation of Expected Shortfall

New proof of subadditivity of Expected Shortfall

Accessible approach for graduate education

Abstract

We provide an elementary proof of the dual representation of Expected Shortfall on the space of integrable random variables over a general probability space. Unlike the results in the extant literature, our proof only exploits basic properties of quantile functions and can thus be easily implemented in any graduate course on risk measures. As a byproduct, we obtain a new proof of the subadditivity of Expected Shortfall.