Workers' Compensation Pool: Correlation and Stop-Loss Adequacy
A workers' compensation pool covers 800 member employers. Each member has at most one claim in the policy year. For a randomly selected member, the probability of no claim is 0.80. Given a claim, the ground-up loss is uniformly distributed from 0 to 10,000. The policy pays 80% of the loss above a 1,000 per-claim deductible, with no policy limit. The pool retains all claim costs subject to an aggregate stop-loss with a 300,000 attachment. The base model assumes independent claims. A consultant notes that 15% of members belong to one construction trade group whose injury experience is correlated. The board asks whether the current stop-loss premium, computed from the independent model, is adequate and what additional information to request.
1. Calculate the expected insurer payment per member and the expected aggregate retained claim cost before stop-loss. Show your steps.
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Let $L$ be ground-up loss given a claim. For $L \sim U(0,10000)$ and deductible $d=1000$, the expected excess loss is
$$E[(L-d)_+] = \frac{(10000-1000)^2}{2(10000-0)} = \frac{81{,}000{,}000}{20{,}000}=4050.$$
With 80% coinsurance, the conditional insurer payment is $0.8 \times 4050 = 3240$. The probability of a claim is $1-0.80=0.20$, so the unconditional expected payment per member is
$$0.20 \times 3240 = 648.$$
For 800 members, expected retained claim cost before stop-loss is
$$800 \times 648 = 518{,}400.$$
The 300,000 stop-loss attachment is well below this mean, so the stop-loss is expected to pay frequently, not only in remote tail events. That makes the shape of the aggregate distribution, not just the mean, central to pricing the cover.
2. Explain how the construction-group correlation changes the economics of the stop-loss cover even though each member's marginal claim distribution is unchanged.
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Correlation does not change the expected value of each member's claim, so the independent-model mean of 518,400 still holds if the marginal distributions are correct. It changes the variance and upper tail of the aggregate distribution. A common construction-group shock makes many members more likely to have claims in the same year, so large aggregate losses occur more often than under independence.
A stop-loss payment is $E[(S-300000)_+]$, where $S$ is aggregate retained cost. This function is convex in $S$: a fixed increase in the tail has more effect than an equal decrease near the mean. Positive dependence therefore raises the expected stop-loss payment and the required premium even when $E[S]$ is unchanged. The independent model will understate the stop-loss premium, required capital, and probability of a large assessment. The result is not just a pricing issue; it can also affect solvency and member assessments.
3. Recommend two data requests and one stress test before renewing the stop-loss.
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Request member-level construction-group identifiers and claim counts by group for at least five years, so the actuary can estimate within-group correlation rather than assume it. Also request payroll or exposure by group and loss amounts, not just claim counts, because correlation in frequency and correlation in severity have different effects on the aggregate tail.
Run a common-shock stress test: assume all members in the construction group face the same elevated claim probability in one year, calibrated so that the group's expected count rises materially, and recalculate the aggregate distribution and stop-loss expected payment. Compare that result with the independent model. If the stressed stop-loss cost exceeds the quoted premium by a wide margin, the cover is underpriced or the retention should be reconsidered.