Exposure Curves & ILFs

An exposure curve \$G(x)\$ expresses the expected proportion of a risk's total loss that falls below a retention expressed as a fraction \$x\$ of the sum insured (or MPL). It is the normalized limited expected value:

Exposure curve

$$G(x)=\frac{E[\min(Y,\,x)]}{E[Y]}=\frac{\int_0^x (1-F_Y(t))\,dt}{\int_0^1 (1-F_Y(t))\,dt}$$

Y is loss as a fraction of the exposure base; G(x) gives the share of loss retained below the fraction x.

The share of loss ceded to a layer between fractions \$a\$ and \$b\$ is simply \$G(b)-G(a)\$. Families such as the Swiss Re exposure curves and the MBBEFD parameterization are widely used for property.

Increased limits factors

For casualty, ILFs play the same role, the ratio of expected losses limited at a higher limit to those at a basic limit:

Increased limits factor

$$\text{ILF}(L)=\frac{E[\min(X,L)]}{E[\min(X,B)]}$$

Ratio of limited expected value at limit L to that at basic limit B; the layer L2 xs L1 costs proportionally to ILF(L2)-ILF(L1).

Why exposure rating

Exposure rating is independent of the cedant's own losses, so it stabilizes pricing for high or rarely-penetrated layers where experience is not credible, and provides the benchmark against which experience is credibility-weighted.

Exposure Curves & ILFs

Increased limits factors

Why exposure rating

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