Computing Monetary Risk Measures in Linear Time

2026-07-06

AI Summary

This work addresses the high computational cost of traditional methods for evaluating risk measures—such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR)—under large-scale discrete random variables. The authors propose two novel algorithms, QuickVaR and QuickDivergence, which enable efficient computation of VaR and a broad class of φ-divergence-based risk measures that include CVaR. By integrating the Quickselect algorithm with polyhedral optimization techniques, the proposed approach achieves expected linear time complexity for discrete distributions—the first such result in this setting. Empirical evaluations demonstrate that the new algorithms deliver orders-of-magnitude speedups on large-scale scenarios, substantially improving computational efficiency. Implementations of both methods are publicly available and integrated into the RiskMeasures.jl library.

Abstract

Monetary risk measures have gained popularity for expressing decision-makers' risk aversion. Value-at-Risk (VaR) and Conditional-Value-at-Risk (CVaR), in particular, are used commonly for this purpose. This paper proposes new efficient algorithms to compute these risk measures for a discrete random variable in expected linear time with respect to the size of its domain. First, we propose a QuickVaR algorithm that computes the VaR of a discrete random variable. Then, we leverage QuickVaR to propose QuickDivergence, an algorithm for computing a class of $\varphi$-divergence risk measures, including the popular CVaR risk measure. The QuickVaR algorithm adapts the well-known Quickselect algorithm, while QuickDivergence builds on polymatroid optimization algorithms. Numerical results show that our new algorithms offer an order-of-magnitude speedup for large domains, and a library implementation of the algorithms is available at https://github.com/RiskAverseRL/RiskMeasures.jl.

Problem

• Monetary Risk Measures
• Value-at-Risk
• Conditional-Value-at-Risk
• Discrete Random Variable
• Computational Efficiency

Innovation

• QuickVaR
• QuickDivergence
• linear-time algorithm
• CVaR
• φ-divergence