A family of coherent risk measures: an infinite weighted average of Value-at-Risk

Abstract

Value-at-Risk (VaR) remains one of the most widely used risk measures in finance due to its simplicity, interpretability, and regulatory prominence, particularly within the Basel framework. However, VaR is not a coherent risk measure, as it generally fails to satisfy the subadditivity property. This paper demonstrates that, although finite weighted combinations of VaR-type functionals do not necessarily yield coherent risk measures, appropriately constructed infinite weighted averages of quantiles can generate coherent risk measures. Motivated by this observation, we introduce a two-parameter weight function that jointly depends on the quantile level and a position parameter, and propose a novel coherent risk measurement framework termed generalized quantile regression (GQR). In particular, we establish explicit and verifiable conditions on the weight function under which the resulting risk functional satisfies coherence, reversibility, monotonicity, and comparability across quantile levels. The proposed framework is flexible, interpretable, and unifying: it encompasses a broad class of existing coherent risk measures as special cases while also generating several new and economically meaningful risk measures. These results provide a characterization that is not explicitly available in conventional distortion or spectral risk measure frameworks. We further develop corresponding nonparametric estimators, including in multi-dimensional settings, and investigate their asymptotic properties. Empirical studies demonstrate the effectiveness of the proposed GQR framework in both risk assessment and portfolio optimization.