Under non-proportional reinsurance, the reinsurer indemnifies losses that exceed a stated retention \\(d\\) (priority or attachment), up to a limit \\(\ell\\). The premium prices the layer of risk assumed, not a share of the original premium.
A layer tower: the cedant keeps the retention; each layer sits excess of the one below.
The layer loss function
Layer loss
$$L(X)=\min\big(\max(X-d,\,0),\;\ell\big)$$
The payout is zero below the retention d, rises one-for-one through the layer, then is capped at the limit.
Worked example
Layer \(5\text{ xs }5\). A \(7\text{M}\) loss pays \(\min(\max(7-5,0),5)=2\text{M}\). A \(12\text{M}\) loss pays the full \(5\text{M}\) limit. A \(4\text{M}\) loss pays nothing.
Forms of excess of loss
Per-risk XoL, applies to each individual risk loss.
Per-occurrence / catastrophe XoL, applies to the accumulation from one event.
Aggregate XoL / stop-loss, applies to total losses over a period. See stop-loss.
A \(5\text{ xs }5\) layer priced at a \(10\%\) ROL costs \(0.10\times 5 = 0.5\text{M}\), implying a 10-year payback. With one paid reinstatement, exhausting the limit triggers a reinstatement premium proportional to the limit reinstated.
Capacity, EPI and uses
Capacity of the layer
$$\text{Capacity} = d + \ell$$
Retention d plus limit l is the reinsurer maximum exposure per loss on the layer.
The layer premium is often quoted as a rate on the cedant’s estimated premium income (EPI), the base the cover protects.
Homogenizes the cedant’s portfolio by removing the peaks.
Lower cost than proportional: the reinsurer does not pay small and medium losses.
High retention of original premium, supporting growth.
Protection against single large losses and against the accumulation of many small ones.
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