Abstract
This paper extends the evaluation and allocation of distortion risk measures to apply to arbitrary homogeneous operators (“financial derivatives,” e.g. reinsurance recovery) of primitive portfolio elements (e.g. line of business losses). Previous literature argues that the allocation of the portfolio measure to the financial derivative should take the usual special-case form of Aumann–Shapley, being a distortion-weighted “co-measure” expectation. This is taken here as the definition of the “distorted” measure of the derivative “with respect to” the underlying portfolio. Due to homogeneity, the subsequent allocation of the derivative’s value to the primitive elements of the portfolio again follows Aumann–Shapley, in the form of the exposure gradient of the distorted measure. However, the gradient in this case is seen to consist of two terms. The first is the familiar distorted expectation of the gradient of the financial derivative with respect to exposure to the element. The second term involves the conditional covariance of the financial derivative with the element. Sufficient conditions for this second term to vanish are provided. A method for estimating the second term in a simulation framework is proposed. Examples are provided.
1. Introduction
This paper discusses the allocation of capital or costs in the particular situation where (1) the capital or costs are computed by a distortion risk measure, (2) that measure is applied to a portfolio of liabilities whose total loss is the sum of the component loss random variables, (3) a nonlinear homogeneous risk transformation (e.g., reinsurance) is contemplated in order to decompose the total loss into a ceded portion that will be paid by another party and the complementary retained portion that will remain in the portfolio, (4) it is desired to evaluate the impact of that risk transformation on capital or costs, and (5) it is desired to allocate that impact back to the original component loss random variables.
7. Conclusion
This paper reviewed the basic mathematics of distortion risk measures – the only law-invariant comonotonic additive coherent risk measures – and the coherent allocation of such to portfolio components. It argued that since financial derivatives must have the portfolio risk measure allocated to them in the same way that portfolio components have – via ‘‘co-measure’’ distorted expectation – then that expectation is also a suitable definition for the ‘‘distorted’’ measure of a financial derivative ‘‘with respect to’’ (i.e., in the context of evaluating) an underlying portfolio. Examples of nonlinear homogeneous financial derivatives, motivated by reinsurance, were given.
