Brainstorms

Surplus Allocation Redux by Stephen W. Philbrick

Allocation of surplus, and its close cousin, risk load, are concepts that fascinate me. I've written several columns in the past, dealing with this subject. While it sounds like an esoteric subject, insurance as a product cannot be written without making some assumption.

Even those who argue for the indivisibility of surplus have to make an implicit assumption. Allocation of surplus is not a compartmentalization of surplus. It is not an end product—it is an intermediate calculation. For those who accept that actuarial pricing of risks includes determining an adequate return, allocated surplus is the "what" in the question, "Adequate return on what?"

I'm probably a bit ahead of myself. Some still argue that returns should be calculated on premium, not on surplus. The recently published monograph, Actuarial Considerations Regarding Risk and Return In Property-Casualty Insurance Pricing (www.casact.org/pubs/vfac/index.cfm?fa=toc) has its genesis in a debate over this distinction. This book has many fine articles on a variety of subjects, including this debate. I found myself in broad agreement with the proponents for both sides, however. I haven't yet determined whether my reading comprehension is deficient, or the authors were not really in dispute.

A number of the articles touched on the need to determine allocated surplus, as a start to determining the required return on a policy or block of policies. Frank Pierson's article, "Rate of Return," was particularly intriguing, both in terms of his argument that an insurance policy can be viewed as a "reserve" investment, as well as his exposition of an allocation algorithm.

Russ Bingham provided a useful service by showing the equivalence between two important cash-flow modeling approaches. What I would like to see is the extension of this concept to allocation techniques. Todd Bault started this process, with a discussion in the 1995 Proceedings (www.casact.org/pubs/proceed/proceed95/95078.pdf) attempting to show comparisons among competing risk load formulas.

Has the problem been solved? Are the apparently different approaches in the literature ultimately identical if we carefully identify assumptions and notation?

Dan Gogol wrote a paper in the 1996 Proceedings (www.casact.org/pubs/proceed/proceed96/96041.pdf) with a strong claim:

It will be shown below, by Theorem 1, that in a certain sense the above covariance of a category with surplus is proportional to the category's effect on surplus variability. It is shown by Theorem 2 that if surplus is allocated to each category of underwriting according to the above formula, and the appropriate risk-based loss discounting rate is used, the following is true. Each category will improve the risk-return relation of the insurer if, and only if, its rate of return on allocated surplus is greater than the rate of return on the total amount of surplus allocated to underwriting.

This isn't just an arbitrary approach. It appears to be an optimal algorithm, backed up by Theorems. Despite this ambitious claim, I haven't seen it either challenged or affirmed. The Actuarial Considerations monograph doesn't even mention his article, but it turns out that this is because the papers in that monograph were largely written prior to the publication of Gogol's paper, so the authors did not have a chance to weigh in on the subject.

I have written before about my fascination with an algorithm called the Shapley value. This algorithm can also be used to allocate surplus to line, and its developers make equally strong claims. I haven't yet determined whether the two approaches are identical, or talking about different problems.

I hope other actuaries interested in this issue will read some of the recent papers on the subject, and help me determine whether we have solved this important problem, or whether more research is still needed.