Wang’s Capital Allocation Formula for Elliptically-Contoured Distributions

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 (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 is quite similar to the covariance-based principle, it follows that Wang’s allocation formula is also considered fair.