Conditional Value-at-Risk for Log-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. This paper addresses the derivation of the closed-form CVaR formulas for log-normal, log-logistic, log-Laplace and log-hyperbolic secant distributions, which are relevant for modeling the returns of financial assets. In many cases financial risk managers assume that not the returns themselves but their logarithms adhere to a particular distribution, and the samples of log-returns are considered. In such cases, the appropriate way to assess risk would be to use a log-distribution for CVaR and VaR estimation. We show how to use the log-normal and the log-logistic distributions for assessing the risk for different asset classes in 2003-2007 and 2007-2009. The log-Laplace and the log-hyperbolic secant distributions have fatter tails, and they may provide a higher accuracy for the tail risk assessment during the crisis periods