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A popular risk measure, Conditional Value-at-Risk (CVaR), is called Expected
Shortfall (ES) in financial applications. The paper developed algorithms for implementation of
linear regression for estimating CVaR as a function of some factors. Such regression is called
CVaR (Superquantile) regression. The main statement of the paper: CVaR linear regression can
be reduced to minimizing the Rockafellar Error function with linear programming. The theoretical
basis for the analysis is established with the Quadrangle Theory of risk functions. We derived
relationships between elements of CVaR Quadrangle and Mixed-Quantile Quadrangle for discrete
distributions with equally probable atoms. The Deviation in CVaR Quadrangle is an integral. We
presented two equivalent variants of discretization of this integral, which resulted in two sets of
parameters for the Mixed-Quantile Quadrangle. For the first set of parameters, the minimization
of Error from CVaR Quadrangle is equivalent to the minimization of Rockafellar Error from the
Mixed-Quantile Quadrangle. Alternatively, a two-stage procedure based on Decomposition
Theorem can be used for CVaR linear regression with both sets of parameters. This procedure is
valid because the Deviation in the Mixed-Quantile Quadrangle (called Mixed CVaR Deviation)
coincides with the Deviation in CVaR Quadrangle for the both sets of parameters. We illustrated
theoretical results with a case study demonstrating the numerical efficiency of the suggested
approach. The case study codes, data and results are posted at the website. The case study was
done with the Portfolio Safeguard (PSG) optimization package which has precoded Risk,
Deviation, and Error functions for the considered Quadrangles.
International Data Science & Engineering Symposium
IDSES
Stan Uryasev