Kernel-based estimation of spectral risk measures

Suparna Biswas and Rituparna Sen

Abstract

Spectral risk measures (SRMs) belong to the family of coherent risk measures. A natural estimator for the class of SRMs takes the form of L-statistics. Various authors have studied and derived the asymptotic properties of the empirical estimator of SRMs; we propose a kernel-based estimator. We investigate the large-sample properties of general L-statistics based on independent and identically distributed observations and dependent observations and apply them to our estimator. We prove that it is strongly consistent and asymptotically normal. Using Monte Carlo simulation, we compare the finite-sample performance of our proposed kernel estimator with that of several existing estimators for different SRMs and observe that our proposed kernel estimator outperforms all the other estimators. Based on our simulation study, we estimate the exponential SRM for heavily traded futures (that is, the Nikkei 225, Deutscher Aktienindex, Financial Times Stock Exchange 100 and Hang Seng futures). We also discuss the use of SRMs in setting the initial-margin requirements of clearinghouses. Finally, we perform an SRM backtesting exercise.