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

Abstract

A popular risk measure, conditional value-at-risk (CVaR), is called expected shortfall (ES) in financial applications. The research presented involved developing algorithms for the implementation of linear regression for estimating CVaR as a function of some factors. Such regression is called CVaR (superquantile) regression. The main statement of this paper is: 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 the CVaR quadrangle 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 of parameters, the minimization of error from the CVaR quadrangle is equivalent to the minimization of the Rockafellar error from the mixed-quantile quadrangle. Alternatively, a two-stage procedure based on the 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 the CVaR quadrangle for 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 on 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.

Global Sensitivity Analysis of Quantiles: New Importance Measure Based on Superquantiles and Subquantiles

“…It can be noted that superquantiles are fundamental building blocks for estimates of risk in finance [64] and engineering [65]. In finance, the superquantile has various names, such as expected tail loss [66], conditional value-at-risk (CVaR) [67][68][69][70] or tail value-atrisk [71], average value at risk [72], expected shortfall [73,74]. Subquantile is not such a widespread concept.…”

A New Family of Expectiles and its Properties

“…VaR and CVaR are considered as statistics within the framework of the fundamental risk quadrangle concept [10,11]. There are regular VaR and CVaR quadrangles.…”

“…The points  where CVaR is not smooth correspond to the discontinuity of VaR. (see for example [11,20] This result is intuitively obvious, because the expectile (optimal value * C in (1)) divides the whole interval of a random variable into two subintervals with different weights. The same division should produce the optimal value   in the formula (10) or the optimal t  in the formula (9).…”

“…Our next goal is to consider a new family of expectile functions using a power parameter 0  for the coefficients in formula (11) for different  as functions of .  Each example contains 5 atoms with the probability 0.2.…”

CoCDaR and mCoCDaR: New Approach for Measurement of Systemic Risk Contributions

“…This approach is motivated by the idea of identifying a risk contribution of each institution that is independent of contributions of other institutions. The estimation of CoCDaR and mCoCDaR was performed with CVaR regression developed in Rockafellar et al (2014) and Golodnikov et al (2019). The CVaR regression in CoCDaR uses drawdowns, while CoCVaR uses returns.…”