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Random Sampler

Allocating Surplus—Not!

by Gary G. Venter

Recently a fair amount of actuarial attention has been focused on the problem of how to allocate surplus to business units. Fortunately, this is usually just an academic exercise and is not used by carriers in their business planning process. I say fortunately because actually trying to use allocated surplus to make business decisions is a risky undertaking and can easily lead to wrong conclusions.

An Intermediate Calculation
Allocating surplus is not an end in itself. Surplus is allocated in order to make some other computation, most often to calculate the return (or risk-adjusted return1) on surplus for each business unit, perhaps for incentive compensation or development of growth strategies.

I will argue that there are other, better ways to accomplish these goals than allocating surplus. Further, there are so many difficult issues in allocation methods that it is not likely that an appropriate distribution will be produced.

Difficult Issues
To start, some of statutory surplus is taken up by statutory reserving requirements, including the difference between the fair value of liabilities and the undiscounted expected value that must be carried. It seems logical to allocate this portion of surplus to the lines with the offending reserves. However, these reserves do not necessarily increase the economic surplus that the company wants to carry—they just hide part of it. The fact that this portion of surplus is hidden in a reserve account does not mean that the line generating the reserve is actually using up that surplus. It is still there and may be protecting all of the policyholders against true insolvency. Making the wrong call on this step of the allocation could end up penalizing truly profitable business.

Another difficult issue is how to handle long-tailed payouts. You could treat the existing reserves as part of the line and allocate surplus to them. Or you could forecast the time that reserves will be needed on new business and allocate surplus to future years. But in the latter case you have to deal with the question of how to charge the future surplus to today's results.

Although there are a number of different allocation methods in the literature, a large class of them can be formulated as a two-step process. In the first step, pick a risk measure such as variance, VAR, tail-VAR2, and the like. Then pick an allocating principle that will allocate surplus as a function of the selected risk measure. Candidates include allocating in proportion to one of the following: the risk measure applied to the line losses, the marginal risk measure the line adds to the rest of the company, the marginal risk measure the last peso of premium in the line adds to the rest of the company, or the average (taken over all possible coalitions the line or a policy from that line can enter into) of the marginal risk measure the line or policy adds to the coalition. This average is called the game theory or Shapely approach, after an early developer of game theory.

The two-step method has some appeal but also a degree of caprice. (I guess it is not unique in this.) One problem is that there is no strong financial theory to tie the return definitively on a line to the allocated surplus. Making compensation and growth decisions on such a basis may not be optimal.

Just as an example of the differences the choices of the two steps can make, in 2001 there was a call for papers to analyze a hypothetical insurer, recommend a reinsurance program, allocate capital, and a few other things.
  PP BW PP/BW BW/PP
Workers Compensation 41% 11% 3.73 0.27
Auto Liability 26% 29% 0.90 1.12
Home/CMP(Property) 11% 51% 0.22 4.64
Auto Physical Damage, etc. 1% 1% 1.00 1.00
GL/CMP(Liability) 21% 8% 2.63 0.38
PP- Philbrick and Painter; BW- Bohra and Weist
The papers are published in the 2001 Spring Forum. Two of the papers responding were from actuaries working at U.S. subsidiaries of Swiss Re and Munich Re: "DFA Insurance Company Case Study, Part II: Capital Adequacy and Capital Allocation" by Stephen W. Philbrick and Robert A. Painter (Swiss Re), and "Preliminary Due Diligence of DFA Insurance Company" by Raju Bohra and Thomas E. Weist (Munich Re).

With apologies to the authors for some fudging, to get the results in a common format and to add up to 100 percent, the capital was allocated to line approximately in the chart above.

The allocation methods were not all that different. Both papers used the game theory approach. Neither separately evaluated the surplus from statutory vs. fair value reserves. The Philbrick and Painter paper's risk measure was tail VAR, while the Bohra and Weist paper's risk measure appears to be variance, but the main difference in allocation seems to arise from a different treatment of the time the capital is needed. Thus one approach hits long-tailed lines hard, while the other hits catastrophe-prone business.

Both methods seem to be based on reasonable although somewhat arbitrary assumptions. But since one method assigns four times as much capital to a line than does the other method, the same profit will generate very different returns. A given line of business could look extremely profitable or a waste of effort, depending on the method chosen.

Alternatives To Allocating
There are other methods of allocating capital besides the two-step method, but there is no room here to go into those. However, reasons for not allocating capital go beyond the fact that it is difficult to do so. For example, allocating could lead to violations of the economic principle of marginal pricing.

Suppose that writing a new policy in a line of business requires $x in capital over and above what is required for the existing book, and it costs $y to get this capital. If the expected profits from the policy exceed $y, then the firm adds value by writing this policy. This is marginal pricing—policy profits should cover the cost of writing that policy. The result could be different from allocating all the capital of the firm to policies in proportion to the marginal capital needed. That could end up allocating more than the marginal capital to the new policy. If the policy did not generate enough profits to cover the extra allocated capital, it would look like a losing proposition, when in fact it adds value to the firm.

This is just like fixed and marginal production costs in manufacturing. If unit pricing is more than the marginal unit costs, you should sell more. This does not necessarily cover the fixed costs, but the more you sell at a marginal profit the better chance you have of covering the fixed costs. Allocating all the capital is like trying to cover existing average fixed costs in every policy, and thus could lead to wrong decisions when evaluating growth opportunities.

Marginal capital can be evaluated by the increase the policy produces in some risk measure for the firm. However, the cost of the marginal capital is the key element in this analysis, and there are ways to estimate this cost without directly calculating how much marginal capital is needed. Recent approaches try to evaluate the marginal capital cost as the change in the cost of an option—for instance, the value of the default option inherent in the limited liability of the corporate form—that is produced by writing the policy. For more information on this topic, read Stewart C. Myers and James Read's AIB Working Paper, "Capital Allocation for Insurance Companies," which was published in August 2001 by the Automobile Insurers Bureau of Massachusetts. This paper is available at www.aib.org/RPP/Myers-Read.pdf.

Another way to evaluate growth opportunities is to look at policy pricing in comparison to a good theory of risk-based market pricing. If the actual price exceeds the market price, then selling it covers the risk associated. The CAS Risk Premium Project and recent papers by Shaun Wang develop such pricing theories. This method provides a theoretically sound direct measure to tell if a book of business is generating adequate profits.

Of course, if you really want to allocate surplus, you could allocate enough surplus to a policy so that the return from the market risk pricing equals your target return.

1A typical adjustment is to replace actual catastrophe experience with its expected value. It seems unusual to call this a risk adjustment. Does this serve to equalize the target return across lines of business?

2 A consistent risk measure has been defined as a function H of the aggregate loss distribution F(x) that meets certain consistency axioms. It has been shown that all such can be represented by a probability distortion function g(y) on the unit interval that satisfies the formula

The tail VAR at the 1 percent level is the special case where g(x) = (x - .99)/.01 for x>0.99, and g(x) = 0 otherwise. Thus it is the expected aggregate loss for the largest 1 percent of aggregate losses, i.e., E(X|X>99th percentile). Numerous other consistent risk measures can be defined using other g functions, such as g(u)=ua, or