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Stop-Loss & Aggregate Cover

Stop-loss caps the aggregate loss of a portfolio, its price is the expected excess of aggregate losses over the retention.

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Stop-loss (aggregate excess of loss) responds to the total losses \\(S\\) of a portfolio over a period, above an aggregate retention \\(d\\). Its net premium is the net stop-loss premium:

Net stop-loss premium
$$\Pi(d)=E\big[\max(S-d,0)\big]=\int_{d}^{\infty}\big(1-F_S(s)\big)\,ds$$
The expected excess of aggregate losses over the retention d, the area under the aggregate survival curve.

With an upper limit \\(m\\) (a capped aggregate layer), the expected recovery is:

Capped aggregate layer
$$E\big[\min(\max(S-d,0),\,m)\big]=\int_{d}^{d+m}\big(1-F_S(s)\big)\,ds$$

Why the aggregate distribution matters

Pricing stop-loss requires the full distribution \\(F_S\\) of aggregate losses, obtained by convolution, Panjer recursion, the fast Fourier transform, or Monte-Carlo simulation.

Where Power Re uses it

Stop-loss and aggregate structures are central to Medical & Health and Agriculture, where frequency shocks and systemic weather risk drive the aggregate result.

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