Lambda R{é}nyi entropic value-at-risk

Abstract

This paper introduces the Lambda extension of the Rényi entropic value-at-risk ($\hat{I}›$-EVaR), a novel family of risk measures that unifies the flexible confidence level structure of the $\hat{I}›$-framework with the higher-moment sensitivity of EVaR. We define $\hat{I}›$-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 $\hat{I}›$ 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 $\hat{I}›$-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.