December 2nd, 2009

A Survey of Capital Allocation Metrics: Standard Deviation

Posted at 1:00 AM ET

Susan Witcraft, Managing Director, Financial Intelligence Team, Instrat
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Standard deviation has long been a common allocation basis in actuarial literature. There have been many papers published discussing the use of standard deviation as a basis for allocating capital for pricing either explicitly or implicitly — e.g., in calculating increased limits factors. With this approach, the standard deviation of each segment is calculated independently. Capital is allocated to each segment in the same proportion as its standard deviation bears to the sum of the standard deviations.

The standard deviations by line for the combinations of reserves and current accident year results are shown in Figure 2. Assuming the totality of capital is to be allocated among lines,1 the percentage and dollar amounts of capital allocated are also shown.

The primary advantages of standard deviation are ease of calculation and familiarity to most senior management. Drawbacks include its consideration of both favorable and adverse deviations from expectations and its lack of recognition of correlations among risks. Further, the allocations derived from standard deviation are not coherent: the sum of the allocations for subsets of a segment will equal the allocation for the segment as a whole only by chance.

To illustrate this last drawback, we can compare the surplus allocated when only casualty and property are considered, rather than allocation among the three casualty segments shown in Figure 2. The casualty standard deviation is USD43 million, i.e., USD21 million less than the sum of the standard deviations for the three segments. As such, if capital were allocated using only property and casualty USD133 million would be allocated to casualty, with USD217 million going to property. The balance in allocated capital between property and casualty changes significantly in this illustration because of the assumption that the correlations among the three casualty segments are less than perfect (i.e., less than 100 percent).

Footnote:

1. There are two aspects to this choice, both of which are outside the scope of this article. The first is whether to allocate the totality of an insurer’s capital or just a portion of it if the company perceives that it is carrying excess capital for future opportunities. The second is whether to allocate all capital to underwriting and therefore include all of the related investment risks with each line of business or to allocate capital separately to underwriting, investment and any other significant sources of risk within the organization.

Previous articles in this series:

Part I: Introduction >>

Part II: Illustration >>