Paper: arXiv 2605.06220

Authors: Ilaria Peri, Linus Wunderlich

Abstract

Lambda quantiles, originally introduced as lambda value at risk, generalise the classical value at risk by allowing for a variable confidence level. This work presents efficient algorithms for computing lambda quantiles and demonstrates their application in portfolio optimisation. We first develop a robust algorithm, Λ-Newton-Bis, that combines Newton’s method with a bisection strategy to ensure global convergence. The algorithm handles potential discontinuities and achieves local quadratic convergence under standard regularity assumptions. To address cases with multiple roots, we also propose an interval analysis approach. We then demonstrate the algorithm’s computational efficiency and practical relevance within a portfolio optimization framework. To this end, we develop two alternative solution methods that incorporate the Λ-Newton-Bis procedure. Numerical experiments confirm the algorithm’s convergence properties and highlight its computational advantages in optimization tasks based on lambda quantiles.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 5.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper combines heavy theoretical grounding in functional analysis and root-finding numerical methods (Newton-Bis) with a clear, albeit limited, implementation in a portfolio optimization framework using specific numerical experiments.

Research Flowchart

  flowchart TD
    A[Research Goal: Robust $\Lambda$-Quantile Evaluation & Portfolio Optimization] --> B[Input: $\Lambda$-VaR Definitions & Portfolio Data]
    B --> C[Methodology: $\Lambda$-Newton-Bis & Interval Analysis]
    C --> D[Computational Process: Hybrid Newton-Bisection]
    D --> E[Outcome: Efficient Optimization Framework]
    E --> F[Findings: Guaranteed Convergence & Superior Performance]

Cited References (36)

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  7. Optimization Methods in Finance (2018)
  8. Numerical Optimization (2006)
and 28 more
  1. Risk contributions of lambda quantiles (2022)
  2. Newton Methods for Nonlinear Problems (2011)
  3. Minimization of functions having Lipschitz continuous first partial derivatives (1966)
  4. The method of steepest descent for non-linear minimization problems (1944)
  5. Nonlinear Programming (1951)
  6. An axiomatization of $\Lambda$ -quantiles (2022)
  7. Backtesting Lambda value at risk (2018)
  8. On the properties of the Lambda value at risk: robustness, elicitability and consistency (2017)
  9. Lambda value at risk and regulatory capital: a dynamic approach to tail risk (2018)
  10. Risk measures on $\mathcal P (\mathbb R )$ and value at risk with probability/loss function (2014)
  11. Risk sharing with Lambda value at risk (2024)
  12. Cash-subadditive risk measures without quasi-convexity (2025)
  13. Backtesting lambda value at risk (2018)
  14. Coherent measures of risk (1999)
  15. Cash subadditive risk measures and interest rate ambiguity (2009)
  16. Making and evaluating point forecasts (2011)
  17. Risk measures based on benchmark loss distributions (2020)
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  19. Fifty years of portfolio optimization - A European perspective (2023)
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  21. On elicitable risk measures (2015)
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  24. Optimal risk sharing for lambda value-at-risk (2024)
  25. Risk sharing with Lambda value at risk under heterogeneous beliefs — → 2408.03147
  26. Robust $\Lambda$ -quantiles and extremal distributions — → 2406.13539
  27. Optimal insurance design with Lambda-Value-at-Risk — → 2408.09799
  28. Max- and min-stability under first-order stochastic dominance (2025)

3 of 36 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.