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Lambda Expected Shortfall

Published: December 29, 2025 | arXiv ID: 2512.23139v1

By: Fabio Bellini , Muqiao Huang , Qiuqi Wang and more

Potential Business Impact:

Helps banks better guess money risks.

Business Areas:
Venture Capital Financial Services, Lending and Investments

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.

Country of Origin
🇨🇦 🇮🇹 🇺🇸 Canada, United States, Italy

Page Count
31 pages

Category
Quantitative Finance:
Mathematical Finance