Contingencies Online - September/October 2009

FiguRE 1: Sample Company Profile (amounts in millions)

net Earned

Premium

Expected net losses

CV of net
losses
Booked
Reserves
170 100 0.10 100
130 80 0.15 150
70 50 0.15 150

Policy type
Auto
General Liability
Workers’
Compensation
Property

CV of
Reserves
0.15
0.10
0.18

240

160

0.44

50

0.14

FiguRE 2: Capital Allocation Based on Standard Deviation
(in millions)

Policy type
Auto
General Liability
Workers’ Compensation
Property

Standard Deviation

18 19 27 70

Allocation
13.4%
14.2%
20.1%
52.2%

Allocated Capital 47 50 71 183

FiguRE 3: Capital Allocated According to Covariance (in millions)
Covariance
(trillions)
485
538
693
4,859

Policy type
Auto
General Liability
Workers’ Compensation
Property

Allocation
7.4%
8.2%
10.5%
73.9%

Allocated Capital
26
29
37
259

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 $43 million, i.e., $21 million less than the sum of the standard deviations for the three segments. As such, if capital were allocated using only property and casualty, $133 million would be allocated to casualty, with $217 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).

Investment and credit risk are ignored in this illustration.

Now, let’s take a look at how the different proportional capital allocation approaches affect this company.

Standard Deviation

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, the percentage and dollar amounts of capital allocated are also shown. (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 of 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.)

The primary advantages of standard

Covariance

Covariance is the expected value of the product of the deviations of two variables from their means. In each iteration, the difference between a segment’s value and its mean is multiplied by the difference between the total value and the total mean. The average of these products is the covariance. Capital is then allocated in proportion to the covariances. Figure 3 shows the covariances and allocated capital for the sample company.