Conditional Value-at-Risk for General Loss Distributions

TLDR

Conditional value‑at‑risk (CVaR) has fundamental properties that make it a superior risk measure to VaR, especially for loss distributions that may be discrete, which are common in scenario‑based and finite‑sample models. The authors use linear‑programming techniques to provide optimization shortcuts that enable practical, large‑scale CVaR calculations that would otherwise be infeasible. CVaR quantifies risks beyond VaR and is coherent, and its numerical efficiency and stability are demonstrated in case studies, including an index‑tracking example.

Abstract

Fundamental properties of conditional value-at-risk, as a measure of risk with significant advantages over value-at-risk, are derived for loss distributions in finance that can involve discreetness. Such distributions are of particular importance in applications because of the prevalence of models based on scenarios and finite sampling. Conditional value-at-risk is able to quantify dangers beyond value-at-risk, and moreover it is coherent. It provides optimization shortcuts which, through linear programming techniques, make practical many large-scale calculations that could otherwise be out of reach. The numerical efficiency and stability of such calculations, shown in several case studies, are illustrated further with an example of index tracking.