Lambda Expected Shortfall

TL;DR

This paper introduces Lambda Expected Shortfall, a new risk measure generalizing Expected Shortfall, with explicit formulas and properties, linking it to Lambda-VaR for improved risk management and optimization.

Contribution

It defines Lambda-ES as a novel, explicit, and law-invariant risk measure that extends ES and relates to Lambda-VaR, with theoretical properties and optimization insights.

Findings

Lambda-ES has an explicit formula.

Lambda-ES is the smallest quasi-convex, law-invariant risk measure dominating Lambda-VaR.

The paper explores dual representation and optimization problems for Lambda-ES.

Abstract

The Lambda Value-at-Risk (Lambda-VaR) is a generalization of the Value-at-Risk (VaR), which has been actively studied in quantitative finance. Over the past two decades, the Expected Shortfall (ES) has become one of the most important risk measures alongside VaR because of its various desirable properties in the practice of optimization, risk management, and financial regulation. Analogously to the intimate relation between ES and VaR, we introduce the Lambda Expected Shortfall (Lambda-ES), as a generalization of ES and a counterpart to Lambda-VaR. Our definition of Lambda-ES has an explicit formula and many convenient properties, and we show that it is the smallest quasi-convex and law-invariant risk measure dominating Lambda-VaR under mild assumptions. We examine further properties of Lambda-ES, its dual representation, and related optimization problems.