Lambda R{\'e}nyi entropic value-at-risk

TL;DR

This paper introduces the Lambda R{\'e}nyi entropic value-at-risk ($\Lambda$-EVaR), a new risk measure combining flexible confidence levels with higher-moment sensitivity, supported by theoretical properties and robustness analysis.

Contribution

It defines $\Lambda$-EVaR, establishes its key properties, provides dual representations, and analyzes its robustness, bridging adaptive risk tolerance with moment-sensitive risk assessment.

Findings

$\Lambda$-EVaR unifies confidence level flexibility with higher-moment sensitivity.

Derived dual representation and Rockafellar-Uryasev-type formula for efficient computation.

Closed-form expressions for worst-case behavior under Wasserstein and mean-variance uncertainty.

Abstract

This paper introduces the Lambda extension of the R\'{e}nyi entropic value-at-risk ($\Lambda$-EVaR), a novel family of risk measures that unifies the flexible confidence level structure of the $\Lambda$-framework with the higher-moment sensitivity of EVaR. We define $\Lambda$-EVaR, establish its foundational properties including monotonicity, cash subadditivity, and quasi-convexity, and provide a complete axiomatic characterization showing that convexity, concavity in mixtures and cash additivity hold only when $\Lambda$ is constant. A dual representation and an extended Rockafellar-Uryasev-type formula are derived, enabling efficient computation. We further analyze the worst-case behavior of $\Lambda$-EVaR under Wasserstein and mean-variance uncertainty, obtaining closed-form expressions that reveal its robustness properties. The proposed measure bridges the gap between adaptive risk tolerance and moment-sensitive risk assessment, offering a versatile tool for modern risk management.