Reinsurance layers
Excess of loss with the algebra made visible: the claim density split at the retention, the insurer keeping the purple, the reinsurer paying the gold, and the expected costs, shares and volatility relief recomputed as you slide M.
E[X] whole claim 6.17
E[Y] insurer 5.43
E[Z] reinsurer 0.75
Reinsurer share 12.1%
sd(X) gross 5.83
sd(Y) net 3.47
Volatility kept 59.4%
The trade the retention slider makes: lowering M hands the reinsurer a bigger expected share, but look at how much faster the NET standard deviation falls — excess of loss buys volatility relief cheaply because the tail is where the variance lives. Switch to Pareto claims and the same retention cedes a far bigger share: with a heavy tail the excess layer carries serious expected cost, which is the mean excess result from extreme value theory wearing pounds. Claim inflation against a FIXED M quietly grows the reinsurer’s share every year — the classic exam point.
What the exam wants from this picture
Under individual excess of loss the insurer pays Y = min(X, M) and the reinsurer Z = max(0, X − M), so E[Z] is the integral of (x − M) times the density above the retention — the gold area, weighted by how far each claim overshoots. Two standard results fall out of playing with the sliders. First, the variance asymmetry: the tail owns the variance, so ceding it cuts the insurer’s standard deviation much faster than its expected cost — that is the actuarial case for excess of loss over quota share when the worry is solvency rather than volume. Second, the inflation trap: claims inflate, M stays fixed, and a growing share of every claim spills over the retention — the reinsurer’s expected cost rises faster than inflation, which is why retentions are indexed in practice and why the examiners keep asking.
The tail’s own theory. Why Pareto layers cost so much more is the extreme value theory tool’s whole subject, and aggregate claims with reinsurance feed the ruin theory simulator. Memori is a flashcard app built by actuarial students, with a ready-made CS2 set in the shop. Join the beta.
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