Excess of Loss Mathematics

An excess-of-loss layer \$\ell \text{ xs } d\$ transforms each loss \$X\$ into the ceded amount:

Layer loss function

$$L(X)=\min\big(\max(X-d,0),\;\ell\big)$$

The expected loss to the layer is the integral of the survival function \$S_X(x)=P(X>x)\$ across the layer, the layer-integration identity:

Expected layer severity

$$E[L]=\int_{d}^{d+\ell} S_X(x)\,dx=\int_{d}^{d+\ell}\big(1-F_X(x)\big)\,dx$$

The area under the survival curve between the attachment d and the exhaustion point d+l.

Reinstatements

Catastrophe XoL limits are often reinstated after a loss for an additional premium. With base premium \$P\$ and \$k\$ paid reinstatements at percentages \$c_j\$, the total expected premium and recoveries must balance (Mata, ASTIN). The reinstatement premium for the \$j\$-th reinstatement is proportional to the limit reinstated:

Reinstatement premium

$$RP_j=c_j\,P\,\frac{\min(\text{loss to layer in period }j,\ \ell)}{\ell}$$

Reinstatements cap the reinsurer aggregate exposure and are priced consistently with the layer expected losses.

Pricing the volatility

Because a layer's outcome is highly skewed, the technical premium exceeds the expected loss by a risk load. Under the standard-deviation principle:

Standard-deviation premium

$$\Pi = E[L] + \theta\,\sqrt{\operatorname{Var}(L)}$$

theta is the risk-load factor reflecting the reinsurer required return for bearing volatility.

Excess of Loss Mathematics

Reinstatements

Pricing the volatility

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