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CIO Springer TAP Compal BES FCT
FC3 Value-at-Risk

Chair: Vaclav Kozmik, Charles University in Prague, Faculty of Mathematics and Physics, Czech Republic
Room 3.1.11


Stochastic dominance, CVaR and enhanced indexation

Diana Roman (, Gautam Mitra (, Victor Zverovich (
A theoretically sound way of comparing profit and loss distributions is stochastic dominance, or stochastic ordering of random variables. Second-order Stochastic Dominance (SSD) is of particular importance to portfolio theory since it describes the preference of rational and risk-averse investors. However, until recently, it was considered computationally intractable and was used (only) as a powerful tool for analysis, rather than for making decisions. In the last years, tractable models that apply the concept of stochastic dominance have been proposed. We consider a model whose portfolio solution has a return distribution that is (a) efficient with respect to SSD and (b) tracks or improves on a target, “benchmark” distribution. The return distribution of the solution is thus shaped to the extent that is possible. The model presents interesting connections to modern risk measures like Conditional Value at Risk, sometimes called Expected Shortfall. It can also be used in the context of enhanced indexation / outperforming financial indices. Backtesting results show clear and consistent outperformance over financial indices, obtained with a relatively low number of stocks and little rebalance over time.

Keywords: Portfolio optimisation, index tracking, conditional value-at-risk


Superquantile Regression with Applications to Buffered Reliability, Uncertainty Quantification, and Conditional Value-at-Risk

R. Tyrrell Rockafellar (, Johannes Royset (, Sofia Miranda (
We present a generalized regression technique centered on a superquantile (also called conditional value-at-risk) that is consistent with that coherent measure of risk and yields more conservatively fitted curves than classical least-squares and quantile regressions. In contrast to other generalized regression techniques that approximate conditional superquantiles by various combinations of conditional quantiles, we directly and in perfect analog to classical regression obtain superquantile regression functions as optimal solutions of certain error minimization problems. We show the existence and possible uniqueness of regression functions, discuss the stability of regression functions under perturbations and approximation of the underlying data, and propose an extension of the coefficient of determination R-squared for assessing the goodness of fit. We present two numerical methods for solving the error minimization problems and illustrate the methodology in several numerical examples in the areas of uncertainty quantification, reliability engineering, and financial risk management.

Keywords: Superquantile regression, Uncertainty quantification, Conditional value-at-risk


On Variance Reduction of Mean-CVaR Monte Carlo Estimators

Vaclav Kozmik (
We formulate an objective based on the convex combination of expectation and risk, measured by the CVaR risk measure. The poor performance of standard Monte Carlo estimators applied on functions of this form is discussed and a variance reduction scheme based on importance sampling is proposed. We provide analytical solution for random variables based on normal distribution and outline the way for the other distributions, either by analytical computation or by sampling. Our results are applied in the framework of stochastic dual dynamic programming algorithm. Computational results which validate the previous analysis are given.

Keywords: Importance sampling, Risk-averse optimization, Monte Carlo sampling