Signature
z = (A - E_S) / SD_S
| Inputs | Definition | Unit |
|---|---|---|
A | Aggregate attachment point for the contract year | dollars |
E_S | Expected annual net claims of the group | dollars |
SD_S | Standard deviation of the group's annual net claims | dollars |
z | Number of standard deviations of total claims by which the attachment point lies above expected claims | standard deviations |
|---|
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
Standardised distance and tail probability for the aggregate attachment point
The first formula returns z from named cells AttachPoint, ExpClaims and SDClaims; the second returns the normal approximation to the probability of reaching the attachment point from the cell holding z, named ZScore.
=(AttachPoint-ExpClaims)/SDClaims; =1-NORM.S.DIST(ZScore,TRUE)
Assumptions
Normal approximation to a group's total net claims
Reading z as a tail probability assumes total net claims are approximately normal. Claims are skewed and can share common shocks, so the approximation understates the upper tail, especially for small groups, and the result shows the direction of an effect rather than a price.
Worked examples
Distance to the attachment point for a 200-member group
Employer 1's attachment point lies $375,000 above expected claims, about 1.7678 standard deviations. The normal approximation puts the chance of reaching it at about 0.0385, roughly one year in 26.
A = 1875000; E_S = 1500000; SD_S = 212132.03; z = 1.7678
Distance to the attachment point for a 20-member group
Employer 2's attachment point lies $37,500 above expected claims, about 0.5590 standard deviations. The normal approximation puts the chance of reaching it at about 0.2881, a little more than one year in four and more than seven times Employer 1's chance.
A = 187500; E_S = 150000; SD_S = 67082.04; z = 0.5590
Common errors
Reversing the sign of the standardised distance
Computing z as expected claims minus the attachment point gives minus 1.7678 for Employer 1, and 1-NORM.S.DIST then returns about 0.9615 instead of 0.0385, so the carrier appears to pay in almost every year.
Reading the normal tail probability as reliable for a small group
For 20 members the claims distribution is strongly skewed, and a few very costly members or a shared shock can push claims far above the attachment point more often than the normal curve implies. The 0.2881 figure for Employer 2 is a lower guide, not an estimate to price on.
Sources
Normal approximation to the distribution of total claims
Kaas R, Goovaerts M, Dhaene J, Denuit M. Modern Actuarial Risk Theory: Using R. 2nd ed. Berlin: Springer; 2008. Chapter 2, The individual risk model, which studies the distribution of the total claim amount of a portfolio, the probability that the amounts paid exceed a fixed threshold, and approximations to that distribution, especially in the far tail.
Canonical Identity
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