Finite-Sample Distortion Measures: Unified Risk and Gain via Scenario Weights

Abstract:

We study monetary and coherent measures of portfolio risk and gain when returns are represented by a finite set of scenarios, as in historical or Monte Carlo simulation. In this setting, many quantile-based functionals—including Value at Risk and Conditional Value at Risk—become linear forms in the order statistics and are therefore fully characterized by scenario-weight vectors. This leads to an implementation-friendly class of distribution measures that treats downside risk and upside potential within a unified framework. We derive explicit finite-sample weights for VaR/CVaR and their gain counterparts (GaR/CGaR), obtained by applying the risk measure to the short position. 

We further introduce equivalent weight vectors that preserve reported values on a given scenario set while enabling smoother or more robust weight profiles, and we present a matrix formulation for fully vectorized evaluation over large panels of measures. Numerical results for major digital assets illustrate the approach and quantify tail asymmetries between losses and gains.

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