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Conditional value at risk (CVaR) is both a coherent risk measure and a natural risk statistic. It is often used to measure the risk associated with large losses. In this paper, we study how to estimate the sensitivities of CVaR using Monte Carlo simulation. We first prove that the CVaR sensitivity can be written as a conditional expectation for general loss distributions. We then propose an estimator of the CVaR sensitivity and analyze its asymptotic properties. The numerical results show that the estimator works well. Furthermore, we demonstrate how to use the estimator to solve optimization problems with CVaR objective and/or constraints, and compare it to a popular linear programming-based algorithm.simulation, statistical analysis, applications, portfolio

Value-at-risk (VaR) and conditional value-at-risk (CVaR) are two widely used risk measures of large losses and are employed in the financial industry for risk management purposes. In practice, loss distributions typically do not have closed-form expressions, but they can often be simulated (i.e., random observations of the loss distribution may be obtained by running a computer program). Therefore, Monte Carlo methods that design simulation experiments and utilize simulated observations are often employed in estimation, sensitivity analysis, and optimization of VaRs and CVaRs. In this article, we review some of the recent developments in these methods, provide a unified framework to understand them, and discuss their applications in financial risk management.