Introduction to the Theory of Probabilistic Functions and Percentiles (value-at-risk)

Probabilistic and quantile (percentile) functions are commonly used for the analysis of models with uncertainties or variabilities in parameters. In nancial applications, the percentile of the losses is called Value-at-Risk (VaR). VaR, a widely used performance measure, answers the question: what is the maximum loss with a speci ed con dence level? Percentiles are also used for de ning other relevant performance measures, such as Conditional Value-at-Risk (CVaR). CVaR (also called Mean Excess Loss, Mean Shortfall, or Tail VaR) is the average loss for the worst x% scenarios (e.g., 5%). CVaR risk measure has more attractive properties compared to VaR. This introductory paper gives basic de nitions and reviews several topics: sensitivities of probabilistic functions; sensitivities of percentiles (VaR); optimization approaches for CVaR. The emphasis of this paper is on issues which have been relatively recently developed. University of Florida, Dept. of Industrial and Systems Engineering, PO Box 116595, 303 Weil Hall, Gainesville, FL 32611-6595, E-mail: [email protected] .edu, URL: http://www.ise.u .edu/uryasev