Understanding the law of large numbers

Healthcare Financial Management, Oct, 1995 by J. Bruce Ryan, Scott B. Clay

The law of large numbers is the foundation of all insurance and must be understood by organizations assuming financial risk associated with capitated contracts. The law of large numbers is too complex to fully explain and demonstrate in this column. In simple terms, however, the law of large numbers states that the actual outcome of a statistical process converges toward the expected value as the number of observations increases. In other words, the probability distribution around the expected value narrows as the number of observations increases.

The following example, which is based on data presented in the Managed Care column entitled "How to Determine Financial Reserves for Capitated Contracts," (HFM, March 1995, p. 18), illustrates the key concepts of the law of large numbers.

Assume that under a specific capitated contract, the mean cost of hospitalization (C) is $6,597 and the standard deviation of that cost (SD) is $5,492. If the hospitalization use rate (U) of the members covered by a capitation contract is 0.04781 (de, 47.81 admissions per 1,000 covered lives), then the expected cost of hospitalization per member per month (CM) is:

[Mathematical Expression Omitted]

And the standard deviation of the cost per member per month (SDM) is:

[Mathematical Expression Omitted]

Exhibit 1 presents the means and standard deviations calculated for varying levels of covered lives on a per-member-per-month basis. In each case, the average cost per member per month is $26.28. However, the standard deviation around the mean decreases in proportion to the square root of the number of covered lives (M) [ie, SDM,N=SDM [divided by] (M)1/2]. For example, the standard deviation for 100 covered lives is $346.66 [divided by] (100)1/2 or $34.67 per member per month.

Exhibit 1: Expected costs and standard deviations of
expected costs of hospitalization
              Number of covered lives
Per member per month     100    1,000   10,000   50,000

Expected cost (CM)     $26.28   $26.28  $26.28  $26.28
Standard deviation of
expected cost (SDM)    $34.67   $10.96   $3.47   $1.55