Distortion Measures and Homogeneous Financial Derivatives
20 Pages Posted: 24 May 2017 Last revised: 24 Jan 2018
Date Written: June 1, 2017
Abstract
This paper extends the evaluation and allocation of distortion risk measures to apply to arbitrary homogeneous components of a portfolio (“financial derivatives,” e.g. reinsurance recovery, of primitive portfolio components, e.g. lines of business). It is argued that the allocation of the portfolio measure to the financial derivative takes the usual form of (distortion-) weighted “co-measure” expectation. Due to homogeneity, the allocation of the derivative’s value to further subcomponents (ultimately, the primitive elements of the portfolio), following Aumann-Shapley, is the exposure gradient. However, the gradient in this case consists of two terms. The first is the familiar distorted expectation of the gradient of the component with respect to the subcomponent. The second term involves the conditional covariance of the component with the subcomponent. Sufficient conditions for this second term to vanish are provided. A method for estimating the second component in a simulation framework is proposed.
Keywords: distortion measures, spectral measures, capital allocation, financial derivatives, Aumann-Shapley, risk, insurance, reinsurance
JEL Classification: C71, D81, G22
Suggested Citation: Suggested Citation