Algorithms for Optimization of Value-at-Risk

Abstract

This paper suggests two new heuristic algorithms for optimization of Value-at- Risk (VaR). By definition, VaR is an estimate of the maximum portfolio loss during a standardized period with some confidencelevel. The optimization algo- rithms are based on the minimization of the closely related risk measure Condi-tional Value-at-Risk (CVaR). For continuous distributions, CVaR is the expected loss exceedingVaR, and is also known as Mean Excess Loss or Expected Short- fall. For discrete distributions, CVaR is the weighted average of VaR and losses exceeding VaR. CVaR is an upper bound for VaR, therefore, minimization of CVaR also reducesVaR. The algorithms are tested by minimizing the credit risk of a portfolio of emerging market bonds. Numerical experiments showed that the algorithms are efficient and can handle a large number of instruments and scenarios. However, calculations identified a deficiency of VaR risk measure, compared to CVaR. Minimization of VaR leads to an undesirable stretch of the tail of the distribution exceedingVaR. For portfolios with skewed distributions, such as credit risk, minimization of VaR may result in a significant increase of high losses exceedingVaR. For the credit risk problem studied in this paper,VaR minimization leads to about 16% increase of the average loss for the worst 1% scenarios (compared to the worst1% scenarios in CVaR minimum solution). 1% includes 200 of 20000 scenarios, which were used for estimating credit risk in this case study.