Nonparametric estimation of systemic risk via conditional value-at-risk

Two forms of CoVaR have recently been introduced in the literature for measuring systemic risk, differing on whether or not the conditioning is on a set of measure zero. We focus on the former, and make allusions to the possibility of analogous results holding for the latter. After reviewing maximum likelihood estimation (MLE) and quantile regression methods, we introduce four new nonparametric estimators that are applicable given a bivariate random sample. Three of these employ results on concomitants of order statistics, while the fourth is novel in the way it uses saddlepoint approximations to invert the empirical (bivariate) moment generating function in order to recover the conditional distribution. All estimators are shown to be consistent under mild regularity conditions, and asymptotic normality is established for the saddlepoint-based estimator using M-estimation arguments. Simulations shed light on the quality of the finite-sample-based estimators, and the methodology is illustrated on a real data set. One surprising result to emerge is that, in spite of its asymptotic optimality, the MLE does not always dominate the remaining estimators in terms of basic accuracy measures such as absolute relative error. This finding may have important implications for practitioners seeking to make accurate CoVaR inferences.