The Promise and the Peril of Parametric Value-at-Risk (VaR) Analysis

Abstract

Leptokurtosis, or the risk lurking in “fat tails,” poses the deepest epistemic threat to economic forecasting. Parametric value-at-risk (VaR) models are extremely vulnerable to kurtosis in excess of the levels associated with a normal, Gaussian distribution. This article provides step-by-step guidance on the use of Student’s t-distribution to enhance the statistical robustness of VaR forecasts. For degrees of freedom greater than 4, Student’s t-distribution can emulate any level of kurtosis exceeding that of a Gaussian distribution. Because VaR is elicitable from historical data, observed levels of excess kurtosis can inform the proper use of Student’s t-distribution to measure value-at-risk. In addition, the calculation of parametric VaR according to the number of degrees of freedom implied by historical levels of excess kurtosis leads directly to the corresponding value of expected shortfall. Conducted in this fashion, parametric VaR not only exploits the elicitability of that quantile-based measure, but also informs the computation of expected shortfall as a theoretically coherent risk measure.