Lambda Expected Shortfall

2025/12/29 by Fabio Bellini, Muqiao Huang, Qiuqi Wang +1 · 1 voice
Economics, Econometrics and Finance · Mathematics · #q-fin.MF #math.PR #q-fin.RM

paper · pdf

arxiv published 2025/12/29 · arxiv updated 2026/01/07

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.

Citations

• Optimal insurance design with Lambda-Value-at-Risk
• ρ-arbitrage and ρ-consistent pricing for star-shaped risk measures
• Cash-subadditive risk measures without quasi-convexity
• Star-shaped Risk Measures
• Adjusted Expected Shortfall
• Robustness in the Optimization of Risk Measures
• On the properties of the Lambda value at risk: robustness, elicitability and consistency
• Backtesting Lambda Value at Risk
• Higher order elicitability and Osband's principle
• Extreme Negative Dependence and Risk Aggregation
• On the Measurement of Economic Tail Risk
• Coherence and elicitability
• Making and Evaluating Point Forecasts
• Cash Sub-additive Risk Measures and Interest Rate Ambiguity
• Conditional value-at-risk for general loss distributions

Discussions

• Fabio Bellini, Muqiao Huang, Qiuqi Wang, Ruodu Wang: Lambda Expected Shortfall https://arxiv.org/abs/2512.23139 https://arxiv.org/pdf/2512.23139 https://arxiv.org/html/2512.23139 [bsky, 0 points, 0 comments]

Related

• Expected Shortfall as a Tool for Financial Risk Management
• On the properties of the Lambda value at risk: robustness, elicitability and consistency
• Adjusted Expected Shortfall
• Value-at-Risk and expected shortfall for linear portfolios with elliptically distributed risk factors
• On the Concavity of Expected Shortfall