Estimation of Distortion Risk Measures

The concept of coherent risk measure was introduced in Artzner et al. (1999). They listed some properties, called axioms of ‘coherence’, that any good risk measure should possess, and studied the (non-)coherence of widely-used risk measure such as Value-atRisk (VaR) and expected shortfall (also known as tail conditional expectation or tail VaR). Kusuoka (2001) introduced two additional axioms called law invariance and comonotonic additivity, and proved that the class of coherent risk measures satisfying these two axioms coincides with the class of distortion risk measures with convex distortions. To be more specific, let X be a random variable representing a loss of some financial position, and let F (x) := P(X ≤ x) be the distribution function (df) of X. We denote its quantile function by F−1(u) := inf{x ∈ R : FX(x) ≥ u}, 0 < u < 1. A distortion risk measure is then of the following form