1. Exposure curves
An exposure curve \(G(x)\) gives the expected share of a risk’s total loss that falls below a retention equal to the fraction x of its maximum probable loss (MPL). The share of loss ceded to a layer between fractions a and b is simply the difference of the curve.
The industry-standard family for property is the MBBEFD (Bernegger) curve, controlled by a single parameter that sets how heavy the tail is: the larger the parameter, the more the large risks contribute to the high layers.
2. An illustrative curve
| Fraction x of MPL | Exposure curve G(x) |
|---|
| 0.1 | 0.45 |
| 0.2 | 0.63 |
| 0.3 | 0.74 |
| 0.5 | 0.87 |
| 0.8 | 0.96 |
| 1.0 | 1.00 |
Illustrative exposure curve. Property pricing uses fitted MBBEFD (Bernegger) or Swiss Re curve families.
3. Allocating loss to a layer
Worked example
A property risk has an MPL of 10000 and an expected ground-up loss cost of 200. A layer covers from 3000 to 8000, that is fractions \(a=0.3\) to \(b=0.8\). Using the curve above, the share to the layer is \(G(0.8)-G(0.3)=0.96-0.74=0.22\); the layer’s expected loss is \(0.22\times 200=44\).
4. Per-risk vs catastrophe
Exposure curves price the per-risk layers. Natural-catastrophe accumulation is priced separately with catastrophe models, which produce a return-period loss curve (the exceedance-probability curve) for the whole portfolio. See exposure curves and capital & solvency.
