Signature
E_S = n * mu; SD_S = sigma * sqrt(n); CV_S = sigma / (mu * sqrt(n))
| Inputs | Definition | Unit |
|---|---|---|
n | Number of covered members whose claims are pooled | count |
mu | Expected annual eligible claims per member net of specific recoveries | dollars per member |
sigma | Standard deviation of one member's annual net claims | dollars per member |
E_S | Expected annual net claims of the whole group | dollars |
|---|---|---|
SD_S | Standard deviation of the group's annual net claims | dollars |
CV_S | Standard deviation of total net claims divided by expected total net claims | ratio |
Function
Aggregate stop-loss payment and expected payment function
Maps a self-funded employer's eligible claims for the contract year, net of specific stop-loss recoveries, to the payment due from the aggregate stop-loss carrier and the claims the employer retains, given the aggregate attachment point and any maximum aggregate payment. Applied to the distribution of the year's claims instead of one realised year, the same rule gives the carrier's expected payment, the net stop-loss premium before any loading. The records follow the notation of the Aggregate Stop-Loss article, where S is net claims for the year, A is the aggregate attachment point and L is the carrier's payment.
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Implementations
Excel
Pooled claims mean, spread and coefficient of variation in three cells
With named cells Members, MeanMember and SDMember, the three formulas return expected total claims, their standard deviation and the coefficient of variation.
=Members*MeanMember; =SDMember*SQRT(Members); =SDMember/(MeanMember*SQRT(Members))
Assumptions
Independent members' claims in the pooled group
Members' net claims are independent, so their variances add. Shared shocks such as an epidemic make claims correlated, and the true standard deviation of total claims is then larger than sigma times the square root of n.
Common mean and standard deviation for every member
Every member has the same expected net claims and standard deviation. With different values the expected total is the sum of the members' means and the variance the sum of their variances.
Worked examples
Total claims distribution for a 200-member group
With mean net claims of $7,500 and a standard deviation of $15,000 per member, 200 members give expected claims of $1,500,000, a standard deviation of about $212,132 and a coefficient of variation of about 0.1414.
n = 200; mu = 7500; sigma = 15000; E_S = 1500000; SD_S = 212132.03; CV_S = 0.1414
Total claims distribution for a 20-member group
The same per-member figures for 20 members give expected claims of $150,000, a standard deviation of about $67,082 and a coefficient of variation of about 0.4472, more than three times that of the larger group.
n = 20; mu = 7500; sigma = 15000; E_S = 150000; SD_S = 67082.04; CV_S = 0.4472
Common errors
Scaling the standard deviation of total claims with group size
Multiplying the per-member standard deviation by the number of members gives 15,000 × 200 = 3,000,000 dollars for Employer 1 instead of about $212,132. The attachment point would then lie only 0.125 standard deviations above expected claims, and a normal approximation would put the chance of reaching it near 0.45 rather than 0.0385.
Assuming independence when claims share common shocks
Real pools are finite and members' claims can move together, for example in an epidemic or a high-utilisation year. The square-root rule then overstates how much pooling narrows the distribution, so a large group's protection from aggregate stop-loss looks less valuable than it is.
Sources
Arrow on pooling independent medical risks
Arrow KJ. Uncertainty and the welfare economics of medical care. American Economic Review. 1963;53(5):941-973. Section IV.B: under the assumption that medical risks on different individuals are independent, pooling them reduces the risk to the insurer to relatively small proportions, with the caveat that pools are finite and risks may be interdependent through epidemics.
Canonical Identity
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