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

• S. Uryasev
Chapter
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 49)

## Abstract

Probabilistic and quantile (percentile) functions are commonly used for the analysis of models with uncertainties or variabilities in parameters. In financial 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 specified confidence level? Percentiles are also used for defining 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 definitions 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.

## Keywords

Probability Function Credit Risk Constraint Function Quantile Function Stochastic Programming Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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