Excess-of-Loss Reinsurance

For attachment $a$ and limit $l$, the ceded amount is

and retained amount is

Equivalently,

The last line matters: after the layer is exhausted, additional loss returns to the insurer.

1. One limited layer

For £20m xs £10m:

For an unlimited layer, take $l=\infty$, so ceded loss is $(X-a)_+$ and retained loss is $\min(X,a)$.

2. Expected layer loss

For a non-negative loss $X$,

This identity shows why layer pricing depends on the survival function around attachment and exhaustion—not merely on $E[X]$.

Exponential example

If $X$ is Exponential with mean $\theta$,

With $\theta=£5{,}000$, $a=£10{,}000$, and $l=£20{,}000$:

This is expected ceded amount per ground-up loss, before premium, expenses, and dependence.

3. The aggregation unit changes the answer

Suppose one storm causes three covered risk losses: £8m, £9m, and £12m. Consider an unlimited £10m attachment.

Per-risk layer

Apply the treaty separately:

Per-occurrence layer

Aggregate the event first:

The difference is not a modelling nuance; it is the contract's subject loss.

4. Aggregate stop loss

Let annual aggregate covered loss be $S$. An annual aggregate layer with attachment $A$ and limit $L$ pays

It responds to a bad year caused by many claims, large claims, or both. It therefore depends on the complete aggregate distribution and any annual aggregate deductible, corridor, or limit.

5. Reinstatement and exhaustion

An occurrence layer's limit may be consumed by one event. A reinstatement restores all or part of capacity, often for additional premium. To model annual recovery, specify:

• number and size of reinstatements;
• free or paid reinstatement;
• pro rata as to amount and/or time;
• event ordering and hours clause;
• annual aggregate limit if any.

Ignoring reinstatements can understate available cover; assuming unlimited reinstatements can overstate it.

6. Basis and counterparty risk

Even a perfectly calculated layer may not protect the intended outcome because:

• policy loss and treaty loss definitions differ;
• multiple events fall inside or outside the hours clause;
• exclusions create uncovered loss;
• claims settle before recovery is received;
• reinsurer default or dispute delays payment.

These are part of retained risk, not footnotes to a severity formula.

Practice

For £15m xs £5m, calculate ceded and retained amounts for gross loss of:

1. £3m;
2. £12m;
3. £28m.

Next, see how inflation moves losses through fixed nominal layers.