Lambda Expected Shortfall
Fabio Bellini, Muqiao Huang, Qiuqi Wang, Ruodu Wang
TL;DR
This work introduces $\mathrm{ES}_{\Lambda}$, a principled generalization of Expected Shortfall that serves as the natural counterpart to $\Lambda$-Value-at-Risk. By defining $\mathrm{ES}_{\Lambda}(X)=\sup_{x\in\mathbb{R}}\big(\mathrm{ES}_{\Lambda(x)}(X) \wedge x\big)$ for a decreasing $\Lambda$, the authors show that $\mathrm{ES}_{\Lambda}$ is the smallest risk measure that is quasi-convex and law-invariant and dominates $\mathrm{VaR}_{\Lambda}$; they also establish a dual representation and a Rockafellar–Uryasev (RU) style minimization form that facilitates optimization. The paper extends these results to $L^1$ spaces, discusses finiteness and dominance properties, and provides RU and dual formulations that connect to quasi-convex cash-subadditive risk measures. Collectively, these contributions offer a robust, flexible framework for risk quantification and optimization in settings where $\Lambda$-VaR is used, with practical implications for risk management and regulatory applications.
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.