The method really does not need surplus in its considerations, either. In the same way that amount of written premiums could be used instead of surplus in the first method, amount of marginal profits can be used here.

However, the biggest problem with applying this method is stability. Profits of property casualty insurers are extremely volatile, even when positive!

A very different approach was taken by Robert Butsic. Butsic has developed a technique for considering the difference in riskiness between lines of insurance that uses "imputed equity values" to adjust for risk. In Branch Office Profit Measurement for Property-Liability Insurers (1985 CAS Call Paper Program), he wrote:

"A suggested method for subjectively balancing risk for various product lines is:

1. Select a product line, say Commercial Multiple Peril, with an average perceived risk. Assign to it an arbitrary premium/equity ratio in the neighborhood of the long-term industry average premium/equity ratio for all lines; e.g., 2.5-to-l.


2. Select another line, compare it to the standard line (CMP) and set a premium/equity ratio at which you would be indifferent to writing this line compared to the standard line. For example, Fire (having a fast loss payout and a relatively complete pricing data base) at a 4-to-1 premium/equity ratio might be considered equally risky as CMP at 2.5-to-l.


3. Repeat the process for all applicable product lines. Of course, the method can be extended to sublines or even new types of insurance.

This procedure or one which actually attempts to measure the relative systematic risk . . . will produce imputed equity values for each line based upon the respective premiums written. The aggregate all-lines imputed equity need not equal the "actual" equity reported externally, since our intent is to measure relative profitability between lines without having to be concerned about their different absolute levels of risk." [Emphasis in original.]

Butsic is using his technique to measure the relative success of profit centers. His calculations would assign as "imputed equity" amounts equal to l/4 of each center's Fire premiums and l/2.5 of its CMP premiums.

The calculation is really comparing profits to risk-adjusted premiums on the basis of these subjective weights. While risk-adjusted premiums are an excellent tool for gauging managers‘ performance, the concept of surplus is actually not needed here. Furthermore, like the other methods that used premiums in allocations, lines that are growing or shrinking may be misrepresented, changes in rate level can distort the allocations, and the method uses an arbitrary one-year period as its base.

Butsic suggests subjective risk adjustments. An objective quantified technique would prove to be difficult because the risk, as measured by the likely error in projecting losses, is not stable. This instability can be seen in recent actual results.

The IS0 and the NAII jointly collect quarterly incurred losses and earned premiums from insurers that have elected to participate in the Fast Track Monitoring System. A time series of incurred losses and earned premiums from each quarter between the first quarter of 1975 and the second quarter of 1986 is available for a consistent set of insurers, for several lines of insurance (9). The Fast Track System reports both losses from accidents in the current quarter, as well as changes in reserves for accidents in prior quarters. If surplus is to be allocated in proportion to the relative riskiness of the various lines, the unpredictability of these Fast Track losses can be used. To measure the error in projecting losses at various dates, the analysis divided the data into five-year periods ending in the fourth quarters of each year between 1979 and 1985, and in the second quarter of 1986. In each period, and for each line of insurance, a regression model developed using the SAS computer language calculated the total squared error in these models. The model for line #i in the period beginning in Quarter to :

where Ai, Bi and Di are fitted regression coefficients that vary by line. The Ci(to + t) are fitted seasonality constants that take on one value if to+ t is the first quarter of a year and a different value if it is a second quarter, etc.

After SAS developed the total squared error in equation (10) for the eight time periods (results are shown in Table IIIA), Chi-Square statistics are developed and shown in Table IIIB. Since Workers' Compensation experience is at industry total levels, and Fast Track is based on a smaller sample, these results are only meaningful for comparisons between time periods and not between lines. Each line must be brought to a comparable level for that comparison.

An alternative presentation is shown in Table IIIC. There, December 31, 1985 surplus for the industry Is allocated in proportion to the square root of the total squared error in equation (10) for each line of insurance, brought to 1985 industry loss volume levels as shown:

For the period beginning in quarter to, for line #i

If
		(11)
then
		(12)

Where L i, t refers to the actual incurred iosses in the t-th quarter and , L* i, t refers to the estimate. L i is the industry total losses incurred for 1985 from Best's Aggregates and Averages. S is December 31, 1985 industry surplus.

________________________________________________________________
(9) The author wishes to thank Mr. John Pergola of Insurance Services Office, Inc. for providing the Fast Track data used in this analysis. Workers' Compensation results are taken from the A.M. Best quarterly underwriting results.

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