Casualty Actuarial Society


Capital Allocation Irrelevance

Two new papers on the relationship of capital and pricing are in draft form at university websites.

"Solvency, Capital Allocation and Fair Rate of Return in Insurance"
by Michael Sherris of the University of New South Wales

"Capital Allocation For Insurance Companies - What Good Is It?"
by Helmut Gründl and Hato Schmeiser of Humboldt-Universität ,Berlin

Both papers claim that allocation of capital to business unit or contract for pricing or performance comparisons is irrelevant in the following sense: there is an algorithm that will find the minimal acceptable return on allocated capital for each business unit for any amount of capital allocated to it. The resulting minimal acceptable price for the risk taken will be the same no matter what the capital allocation is. In fact the algorithm is just to compute the market value of the risk transfer, divide by the allocated capital, and make that the target return.

Here "minimal acceptable" for price and return is that which gives the market value of the risk transfer. This is consistent with the fact that in a competitive market everyone is a price taker, so the pricing problem is to find reservation prices.

Still, many managements prefer to phrase their decision making in terms of return on capital. It is no problem to translate reservation prices into return targets. The easiest thing for management to understand would be to allocate capital in proportion to reservation prices, so that every business unit ends up with the same target return. Shaun Wang had already proposed this several years ago, but these papers argue that this is the only allocation that makes the target return constant across business units.

That's not to say the calculation is an easy thing to do. These papers provide a simplified pricing model. A more complicated model was developed by the risk premium project (see the last 2 papers of the CAS 2004 Winter Forum). There are yet more refinements possible. These include higher co-moments, as in Kozik and Larson and their cited papers. Also Chung, Y. Peter, Johnson, Herb & Schill Michael J., (2001), "Asset Pricing When Returns Are Nonnormal: Fama-French Factors vs. Higher-order Systematic Co-Moments," to appear in the Journal of Business shows that higher co-moments provide an appealing alternative explanation to Fama-French factors. Also jump risk appears to need special attention in pricing. There is evidence to suggest that the same degree of risk from a jump is more expensive than from a continuous process, perhaps because it is harder to hedge. And the role of non-systematic risk appears larger all the time. Both the risk aversion of customers and differentials in the cost of capital between retained earnings and outside sources make non-systematic risk important.

Gary Venter

A more detailed discussion of the Sherris paper by Stephen Mildenhall follows.

Solvency, Capital Allocation and Fair Rate of Return in Insurance
Sherris' new paper is an interesting addition to the actuarial literature for those researching capital allocation questions.

The author begins by splitting equity into the value of the default option and the rest (E=D+S); nothing profound so far. But next he does two important things.

  1. He shows how to allocate D in an additive manner. This additive allocation of D avoids the homogeneity problem noted for Myers-Read.
  2. He shows that you only have to allocate D in order to price insurance policies and that there is no natural allocation for S. However, since you only have to allocate D to price, what does it matter if you cannot allocate the remaining capital? It has no economic meaning!

Sherris' allocation of D follows Phillips et al.'s ground-breaking 1998 JRI paper; in-fact the allocation method is already implicit in Phillip's paper. The idea is to allocate D according to the call on assets that each line has in the event of insolvency. It is easy to see this is an additive allocation, and it will allocate capital to arbitrary subdivisions of the book: by-line, by-business unit, or even by-policy. It is just a co-measure allocation, per Kreps. Sherris makes the very important point, missed by MR and others (including me), that to extend from an allocation of D to an allocation of E, while very tempting, is inappropriate and rests on assumptions which typically do not hold.

Thoughtlessly extending the allocation from D to E can lead to counter-intuitive results. Consider a large multi-line insurer with a concentration in Workers Compensation, but which also has a substantial property book. For such a carrier, a property catastrophe is the most likely immediate trigger for an insolvency, but even a large cat will generate total losses which are relatively small compared to the outstanding workers compensation reserves---because of the latter's very long payout tail. Using the equal priority rule for allocating assets under insolvency WC will get allocated most of the capital charge whereas the real culprit, cat lines, will get a relatively small capital allocation. Sherris solves this apparent paradox: only the default option D can be allocated like this, not all of the surplus.

Sherris' paper raises an another interesting issue. We can think about capital allocation in two parts. The first part is capital allocation conditional upon end of period solvency. The company loses some money but remains solvent; this is a kind of smoothing capital. The second part is capital allocation given end of period insolvency. Phillips and Sherris have categorically answered the second question: you allocate based on expected recovery by line. If you think about a scenario-style analysis where you have a grid of simulations with losses by BU in each row, then what I am suggesting is that a different allocation is in order for the cases where you lose money but stay solvent vs. the last few scenarios where you actually become insolvent. For example, when you remain solvent it is legitimate to use expected future profits to offset losses in some lines or units. When an total insolvency occurs you can no longer count on future income.

Stephen Mildenhall