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Brainstorms

Capital Allocation

by Stephen W. Philbrick

Efficient use of capital is rapidly becoming a catch phrase in insurance circles. Although the aggregate amount of capital in an insurance com-pany is arguably the most important issue, allocation of capital to lines of business is very important. Although there are legitimate criticisms of allocating capital to line, a notional allocation of capital to line is necessary to calculate rates of return on capital by line. While it is possible to calculate underwriting returns by line without allocating capital to line, any calculation of total return by line will implicitly or explicitly require a capital allocation.

In this column I briefly discuss three ways to calculate capital allocations. The first two are commonly used, easy to understand and easy to calculate, but inconsistent with one another. The third method is computationally more complex, but has some theoretical attributes that may make up for the complexity.

For ease of discussion, let us consider a company with only two lines of business (LOB), A and B. Assume that we set capital requirements equal to some number of standard deviations. Assume that this value for line A is 3,000, and for line B is 4,000. If we further assume independence of the two LOBs, the overall capital requirement for the entire company is 5,000 (square root of the sum of the squares). Clearly, the aggregate company requirement is less than the sum of the requirements of the LOBs. We will allocate the total company requirement to the LOBs, so there is a benefit to writing these two LOBs in a single company. The issue is how to determine how the benefit is allocated to each LOB.

In the first method, we calculate the capital requirements of each LOB as if it were a stand-alone company. We then rescale these amounts so that the sum adds up to the total requirement for the company. In this instance, the individual requirements are 3,000 and 4,000 respectively, for a total of 7,000. The total company requirement is 5,000, so we multiply each line's requirement by 5/7. The resulting capital requirements for the two LOBs are roughly 2,100 and 2,900 respectively.

In the second method, we will look at the marginal capital requirements for each LOB. For example, to determine the capital requirement for line B, we will imagine that our company has been writing line A and then adds line B. We calculate how much more capital is needed by the entire company than the company without the new line. The marginal capital requirement for line B is 2,000 (5,000 minus 3,000). The marginal capital requirement for line A is 1,000 (5,000 minus 4,000). In this case, the sum of the marginal capital requirements does not equal the total company requirement. Commonly, we scale up the marginal capital requirements so that the sum matches the total company requirement. In this case, we have to multiply each marginal capital requirement by 5/3. These revised capital requirements are roughly 1,700 and 3,400 respectively.

The second method is sometimes called the "last-in" method, because it determines the capital requirements as if the line of business were the last one added to a company. For similar reasons the first method is sometimes called the "first-in" method.

The third method is calculated as the average of the first-in and last-in methods. If there are more than two lines, the average is calculated over all "orders of entry." That is, calculate the marginal capital as if the line is the first one added, the second one added, the third one added, and so on. Then calculate the average of all of these values. While this may sound ad hoc, it has solid grounding in game theory. In fact, this method does not require any rescaling, so it is less ad hoc than the traditional first-in or last-in methods. The results of this method are referred to in game theory literature as the Shapley values.

I think there are good reasons to think this approach may be the best way to allocate capital to lines (and to individual risks). Readers interested in more details should read Donald Mango's excellent paper, "An Application of Game Theory…" in the 1998 Proceedings. The concept of Shapley values is also outlined in papers by Jean Lemaire in the 1984 and 1991 ASTIN Bulletins. I have created a spreadsheet outlining the calculations for this two-line example, as well as a four-line and an eight-line example.

Download the Excel spreadsheet by clicking here.

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Ruth Salzmann suggested a modification to the "New and Improved Bornhuetter-Ferguson" method outlined in the November column. Her suggestions can be found in "From the Readers".