CVaR Regression Based on the Relation between CVaR and Mixed-Quantile Quadrangles

摘要

A popular risk measure, conditional value-at-risk (CVaR), is called expected shortfall (ES)in financial applications. The research presented involved developing algorithms for theimplementation of linear regression for estimating CVaR as a function of some factors. Suchregression is called CVaR (superquantile) regression. The main statement of this paper is: CVaRlinear regression can be reduced to minimizing the Rockafellar error function with linearprogramming. The theoretical basis for the analysis is established with the quadrangle theory of riskfunctions. We derived relationships between elements of CVaR quadrangle and mixed-quantilequadrangle for discrete distributions with equally probable atoms. The deviation in the CVaRquadrangle is an integral. We present two equivalent variants of discretization of this integral,which resulted in two sets of parameters for the mixed-quantile quadrangle. For the first set ofparameters, the minimization of error from the CVaR quadrangle is equivalent to the minimizationof the Rockafellar error from the mixed-quantile quadrangle. Alternatively, a two-stage procedurebased on the decomposition theorem can be used for CVaR linear regression with both sets ofparameters. This procedure is valid because the deviation in the mixed-quantile quadrangle (calledmixed CVaR deviation) coincides with the deviation in the CVaR quadrangle for both sets ofparameters. We illustrated theoretical results with a case study demonstrating the numericalefficiency of the suggested approach. The case study codes, data, and results are posted on thewebsite. The case study was done with the Portfolio Safeguard (PSG) optimization package, whichhas precoded risk, deviation, and error functions for the considered quadrangles.