Efficient Density Estimation and Value at Risk Using Fejér-Type Kernel Functions

Abstract

This paper presents a nonparametric method for computing the Value at Risk (VaR) based on efficient density estimators with Fejér-type kernel functions and empirical bandwidths obtained from Fourier analysis techniques. The kernel-type estimator with a Fejér-type kernel was recently found to dominate all other known density estimators under the -risk, . This theoretical finding is supported via simulations by comparing the quality of the density estimator in question with other fixed kernel estimators using the common -risk. Two data-driven bandwidth selection methods, cross-validation and the one based on the Fourier analysis of a kernel density estimator, are used and compared to the theoretical bandwidth. The proposed nonparametric method for computing the VaR is applied to two fictitious portfolios. The performance of the new VaR computation method is compared to the commonly used Gaussian and historical simulation methods using a standard back-test procedure. The obtained results show that the proposed VaR model provides more reliable estimates than the standard VaR models.