Conditional Value-at-Risk for Elliptical Distributions

Conditional Value-at-Risk (CVaR) represents a significant improvement over the Value-at-Risk (VaR) in the area of risk measurement, as it catches the risk beyond the VaR threshold. CVaR is also theoretically more solid, being a coherent risk measure, which enables building more robust risk assessment and management systems. However, in its general form CVaR calculations require sophisticated mathematical software, whereas VaR can be easily calculated using spreadsheets. In this paper closed-form CVaR formulas are derived for the most important elliptical distributions, such as normal, Student's t-distribution, Laplace and logistic distributions. These closed-form formulas not only enable practitioners to estimate CVaR in spreadsheets, but also allow developing portfolio optimization models by providing closed-form objective functions for them. The results of variance-covariance CVaR estimation were compared with the historical CVaR, and the relative estimation error is between 6% and 10%, which is comparable with VaR estimation error