Excess and Deductible Rating

Study guide: Exam 8

Excess and Deductible Rating is the pricing of coverage that pays only part of each loss — above a deductible or retention, up to a limit, or between an attachment and an exhaustion point — by splitting the expected ground-up loss among the parties with a severity distribution and its limited expected values.

 $\begin{aligned} E[X] = {} & E[X \wedge d] \\ & + \left(E[X \wedge u] - E[X \wedge d]\right) \\ & + \left(E[X] - E[X \wedge u]\right) \end{aligned}$

$X$ is the ground-up loss per occurrence, $d$ the deductible or retention and $u$ the limit. The first term over $E[X]$ is the LER behind deductible credits; the second is the insured layer, priced by ILFs; the third is what excess coverage or excess-of-loss reinsurance pays. Multiply by the expected claim count $E[N]$ for aggregate cost.
Problems common to every line. Few claims reach the upper layers, so pricing leans on fitted curves and industry data, and the tail fit drives the answer. Data are truncated below deductibles and censored at policy limits. A fixed retention leverages inflation into the excess layer. Excess losses report and settle later. Parameter uncertainty is large relative to the expected loss, which calls for a risk load.
Expenses do not scale with the layer. Claims handling, fixed expense and the risk provision stay behind when the expected loss moves to the insured, so a high-deductible or high-layer premium looks large relative to its expected loss.
Selection. Insureds choose their limits and deductibles knowing their own risk. The GLM monograph therefore estimates coverage-option factors outside the model, by loss elimination and ILF methods, and enters them as an offset.
Line-specific problems. In property, loss is capped by the insured value and damage ratios fall as risk size grows, hence Exposure Curves; catastrophes accumulate across risks. In liability, nothing below the limit caps the loss: tails are heavy, reporting is slow, and whether ALAE sits inside or outside the limit matters. In workers compensation, statutory benefits have no policy limit, excess cost is driven by rare fatal and permanent-total claims, and large-deductible reimbursements carry credit risk.

Inflation Is Leveraged Into the Excess Layer Severity is Pareto with $\alpha = 2$ , so $E[X \wedge u] = \dfrac{\theta u}{\theta + u}$ , with $\theta = \$50{,}000$ . Severity inflation of $10\%$ raises $\theta$ to $\$55{,}000$ . Compare the change in expected loss per claim for the primary layer $\$100{,}000$ xs $0$ and for $\$500{,}000$ xs $\$500{,}000$ .
Answer Primary layer:
 $\begin{align*} E[X \wedge 100\text{K}]_{\text{before}} &= \frac{50{,}000 \times 100{,}000}{150{,}000} \\ &= 33{,}333.33 \\[4pt] E[X \wedge 100\text{K}]_{\text{after}} &= \frac{55{,}000 \times 100{,}000}{155{,}000} \\ &= 35{,}483.87 \end{align*}$ 
Excess layer, $E[X \wedge 1\text{M}] - E[X \wedge 500\text{K}]$ :
 $\begin{align*} \text{Before} &= 47{,}619.05 - 45{,}454.55 \\ &= 2{,}164.50 \\[4pt] \text{After} &= 52{,}132.70 - 49{,}549.55 \\ &= 2{,}583.15 \end{align*}$ 
The ground-up trend is $+10\%$ . The primary layer grows $+6.5\%$ and the excess layer $+19.3\%$ , because more claims now pierce $\$500{,}000$ . A rate for the excess layer that is trended at the ground-up $10\%$ is inadequate.

Premium for a Large Deductible Policy A workers compensation account has expected ground-up losses of $\$800{,}000$ . Losses limited to a $\$250{,}000$ per-occurrence deductible are expected to be $\$560{,}000$ , and $\$540{,}000$ after a $\$900{,}000$ aggregate deductible limit. The insurer adjusts every claim at $10\%$ of ground-up loss and has fixed expenses of $\$60{,}000$ . It targets a $\$25{,}000$ profit and risk provision, and premium tax is $3\%$ . Deductible reimbursements are not premium.

Compute the premium and the insured's expected total cost.
Answer The insurer's expected loss is the per-occurrence excess plus the aggregate excess:
 $\begin{align*} \text{Insured loss} &= (800{,}000 - 560{,}000) + (560{,}000 - 540{,}000) \\ &= \$260{,}000 \\[6pt] \text{Premium} &= \frac{260{,}000 + 80{,}000 + 60{,}000 + 25{,}000}{1 - 0.03} \\ &= \$438{,}144 \end{align*}$ 
The insured's expected cost is $\$438{,}144 + \$540{,}000 = \$978{,}144$ .

Only $59\%$ of the premium is expected loss. The claims handling on the deductible layer, the fixed expense and the risk provision all remain. Against a guaranteed-cost premium of $(800{,}000 + 80{,}000 + 60{,}000 + 40{,}000)/0.97 = \$1{,}010{,}309$ , the insured saves $\$32{,}165$ . That saving comes from no premium tax on the $\$540{,}000$ of reimbursements and from a smaller risk provision. In exchange, the insurer now carries credit risk on the reimbursements.