Estimating the VaR-Induced Euler Allocation Rule

The prominence of the Euler allocation rule (EAR) is rooted in the fact that it is the only return on risk-adjusted capital (RORAC) compatible capital allocation rule. When the total regulatory capital is set using the value-at-risk (VaR), the EAR becomes -- using a statistical term -- the quantile-regression (QR) function. Although the cumulative QR function has received considerable attention in the literature, a fully developed statistical inference theory for the QR function itself has been elusive. In the present paper we develop such a theory based on an empirical QR estimator, for which we establish consistency, asymptotic normality, and standard error estimation. This makes the herein developed results readily applicable in practice, thus facilitating decision making within the RORAC paradigm, conditional mean risk sharing, and current regulatory frameworks