An Efficient Approach to Quantile Capital Allocation and Sensitivity Analysis

Abstract

In various fields of applications such as capital allocation, sensitivity analysis and systemic risk evaluation, one often needs to compute or estimate the expectation of a random variable given that another random variable is equal to its quantile at some pre-specified probability level. A primary example of such an application is the Euler capital allocation formula for the quantile (often called the Value-at-Risk), which is of crucial importance in financial risk management. It is well known that classic nonparametric estimation for the above quantile allocation problem has a slower rate of convergence than the standard rate. In this paper, we propose an alternative approach to the quantile allocation problem via adjusting the probability level in connection with an expected shortfall. The asymptotic distribution of the proposed nonparametric estimator of the new capital allocation is derived for dependent data under the setup of a mixing sequence. In order to assess the performance of the proposed nonparametric estimator, AR-GARCH models are proposed to fit each risk variable and further, a bootstrap method based on residuals is employed to quantify the estimation uncertainty. A simulation study is conducted to examine the finite sample performance of the proposed inference. Finally, the proposed methodology of quantile capital allocation is illustrated for a financial data set.

Keywords: Bootstrap, Capital Allocation, Expected Shortfall, Nonparametric Estimation, Sensitivity Analysis, Value-at-Risk