Introduction to the Theory of Probabilistic Functions and Percentiles (Value-at-Risk)

Abstract

Probabilistic and quantile (percentile) functions are commonly used for the analysis of models with uncertainties or variabilities in parameters. In nan-cial 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 speciied conndence level? Percentiles are also used for deening 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 deenitions and reviews several topics: sensitivities of probabilistic functionss sensitivities of percentiles (VaR)) optimization approaches for CVaR.