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.

The paper compares portfolio optimization with the Second-Order Stochastic Dominance (SSD) constraints with mean-variance and minimum variance portfolio optimization. As a distribution-free decision rule, stochastic dominance takes into account the entire distribution of return rather than some specific characteristic, such as variance. The paper is focused on practical applications of the portfolio optimization and uses the Portfolio Safeguard (PSG) package, which has precoded modules for optimization with SSD constraints, mean-variance and minimum variance portfolio optimization. We have done in-sample and out-of-sample simulations for portfolios of stocks from the Dow Jones, S&P 100 and DAX indices. The considered portfolios’ SSD dominate the Dow Jones, S&P 100 and DAX indices. Simulation demonstrated a superior performance of portfolios with SD constraints, versus mean-variance and minimum variance portfolios.

We consider the problem of finding checkerboard copulas for modeling multivariate distributions. A checkerboard copula is a distribution with a corresponding density defined almost everywhere by a step function on an m-uniform subdivision of the unit hyper-cube. We develop optimization procedures for finding copulas defined by multiply-stochastic matrices matching available information. Two types of information are used for building copulas: 1) Spearman Rho rank correlation coefficients; 2) Empirical distributions of sums of random variables combined with empirical marginal probability distributions. To construct checkerboard copulas we solved optimization problems. The first problem maximizes entropy with constraints on Spearman Rho coefficients. The second problem minimizes some error function to match available data. We conducted a case study illustrating the application of the developed methodology using property and casualty insurance data. The optimization problems were numerically solved with the AORDA Portfolio Safeguard (PSG) package, which has precoded entropy and error functions. Case study data, codes, and results are posted at the web.