Wang’s capital allocation formula for elliptically contoured distributions

Abstract

There is a growing interest among insurance companies to be able not only to compute total company capital requirements but also to allocate this total capital across its various business units. Wang [A Set of New Methods and Tools for Enterprise Risk Capital Management and Portfolio Optimization, Working Paper, SCOR Reinsurance Company, 2002] recently recommended allocating the total cost of capital of an insurance company based on the idea of “exponential tilting”. Under the assumption that the risks or losses follow a multivariate normal distribution, the resulting allocation formula will be a function of the variance–covariance structure. We extend Wang’s idea into a larger class of multivariate risks called “elliptically contoured” multivariate distributions, of which the multivariate normal is a special case. In addition, this paper develops three criteria of what constitutes a “fair allocation” between lines of business of an insurance company: no undercut, symmetry, and consistency. We prove that the covariance-based allocation principle satisfies the requirements of a fair allocation. Because the resulting allocation reduces to the covariance-based principle, it follows that Wang’s allocation formula is also considered fair.