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Lambda Expected Shortfall

Author

Listed:
  • Fabio Bellini
  • Muqiao Huang
  • Qiuqi Wang
  • Ruodu Wang

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.

Suggested Citation

  • Fabio Bellini & Muqiao Huang & Qiuqi Wang & Ruodu Wang, 2025. "Lambda Expected Shortfall," Papers 2512.23139, arXiv.org, revised Jan 2026.
    • Handle: RePEc:arx:papers:2512.23139
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    File URL: http://arxiv.org/pdf/2512.23139
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