Surplus, Profit and Conditional Expectation

This paper presents an approach to the allocation of surplus and profit loads to different product lines or policies within a single insurance company. The method presented calculates the allocation for each line based on the expected value conditional on the full surplus being exhausted. The use of conditional expectations produces a familiar variance-based allocation for a number of specific cases, and variance is the least squares approximation in the general case. The formula is easily extended to include correlations between policies. It is the intention of this paper merely to highlight the mathematical connection between conditional expectations and variances. Setting profit loads based on variance is already a popular and practical technique, and this additional theoretical support should encourage actuaries to keep it as one method in their “toolbox”. Surplus, Profit and Conditional Expectation Background and Statement of the Problem One of the most important goals of a stock insurance company is to provide its stockholders with an attractive return on their investment. In order to operate, the company must have sufficient surplus to cover the possibility that actual losses are worse than the original projections, and therefore have the ability to make good on the monetary promise of the insurance policies sold. The total amount of surplus required may be determined by regulatory formulas such as risk-based capital or by “probability of ruin” models. Roughly speaking, this surplus is the investment made by the stockholders. In theory, the stockholders will demand a return on this investment based on their evaluation of the variability of results and the correlation with their other investments. The stockholder’s primary concern is the size and variability of the overall company return; the relative performance of individual lines of business or policies within the company’s book of business is largely a matter of indifference. The challenge to insurance company management is how to set profit targets for individual lines of business and policies such that the overall company target can be met. For purposes of this paper, we will assume that the overall dollars of profit load and surplus are given, and we will focus on the allocation problem only. For simplification, we will also ignore such real-world complications as differing investment income streams and federal income taxes. Outline of the Conditional Expectation Approach We begin by assuming that the company has only two lines of business, for which the losses will be denoted “x” and “y”. The pure premium for each of these lines is the mathematical expected value: E[x] or E[y]. The total amount of funds required to be available to cover these losses is given as “T”. This may be considered the total assets available to pay claims, and is made up of the pure premiums, profit loads and overall surplus. T = E[x] + P, + E[y] + P, + Surplus where P, = profit on line x P, = profit on line y For internal company analysis, the surplus may also be distributed to individual lines such that S = S,+S,. As a simplifying assumption, we will let the ratio of profit load to allocated surplus be a constant for all lines of business; that is, P,/S, = P,/S,. The allocation problem is to determine how to split the excess of “T” over E[x]+E[y] in a reasonable manner. This can be done based on the conditional expectation of each line of business, which answers the question: “what is the expected value of each line given that total losses for all lines equal T”? The notation is: E[x]x+y=T] = the expected value of x, given that x+y=T From this expression, it is obvious that T = E[x]x+y=T] + E[y]x+y=Tj = E[x]+P,+S, + E[y]+P,+S, . The profit and allocated surplus for line “x” is then P,+ S, = E[x]x+y=T] E[x]. The general form of the conditional expectation is given by: T I x fx Wfy CT XF E[xIx+y=T] = OT I f, Wf, v xw 0 The Normal Distribution As a first case, we will assume that each of the two lines of business are normally distributed with means and standard deviations uX , py , ox and o,,, and that the two lines are independent. The expression for the conditional expectation is: Based on a rearrangement of terms, the allocated profit and surplus for line x is therefore given in proportion to its variance: E[xIx+y=T]-E[x] = (T-E[x]-E[y]) For the Normal distribution, this expression is exact and can be easily generalized to include additional lines of business and correlations between lines: E[xIx+y+z=T]-E[x] = (T-E[x]-E[y]-E[z])Z, From this expression, the two extreme cases of perfect independence and perfect correlation result in an allocation by variance and standard deviation, respectively: Perfect Independence Perfect Correlation Pxy = P, = Pyr = 0 Pxy = P, = Pxz =l z,= (Tx ox +(3, +cr, Other Distributions: Gamma and Exponential The results above are derived from the Normal distribution, which does have some problems when applied to typical insurance operations. First, the Normal distribution allows for negative values, which cannot take place with insurance claims. Second, the distribution is symmetrical about the average value whereas many insurance distributions are highly skewed. Another example of conditional expectation is the use of two gamma distributions with a common scale parameter.