Numerical methods for lambda quantiles: robust evaluation and portfolio optimisation

2026-05-07

AI Summary

This study addresses the limitations of traditional risk measures—such as Value-at-Risk (VaR)—which rely on fixed confidence levels and thus fail to accommodate dynamic risk preferences. While λ-quantiles offer variable confidence levels, their computation suffers from discontinuities, multiple roots, and convergence difficulties. To overcome these challenges, this work proposes the Λ-Newton-Bis hybrid algorithm, which integrates Newton’s method with bisection to ensure global convergence while achieving local quadratic convergence. By incorporating interval analysis, the algorithm effectively handles discontinuities and multiple roots. The paper further introduces two novel solution strategies that, for the first time, efficiently embed λ-quantiles into portfolio optimization frameworks. Numerical experiments demonstrate that the proposed approach significantly outperforms existing methods in terms of convergence, robustness, and computational efficiency.

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.

Problem

• lambda quantiles
• portfolio optimisation
• numerical methods
• value at risk
• computational efficiency

Innovation

• lambda quantiles
• robust algorithm
• Newton-bisection method
• portfolio optimisation
• interval analysis