Quota Share

Study guide: Exam 9

Quota Share is a proportional treaty in which the reinsurer takes a fixed percentage $a$ of the premium and of every loss (usually including ALAE) on a defined book. In return it pays the cedant a ceding commission for the expenses the cedant incurred to write the business.

$\begin{aligned} \text{Ceded premium} &= a \cdot P \\ \text{Ceded loss} &= a \cdot X \end{aligned}$

$\text{Reinsurer's ratio} = \text{ELR} + c + b + e$

$P$ is subject premium, $X$ subject loss, $\text{ELR}$ the expected loss ratio (including a catastrophe load), $c$ the ceding commission, $b$ brokerage and $e$ the reinsurer's own expenses, each as a ratio to ceded premium. The reinsurer judges this combined ratio, with investment income and risk, against its target return.
The treaty loss ratio is the cedant's loss ratio, so pricing is ratemaking on the cedant's data (Clark's steps): compile five or more years of subject experience; remove catastrophe and shock losses; develop, trend and bring premium to current rate level (the parallelogram method for a losses-occurring treaty); select the non-catastrophe ELR; add a catastrophe load, now usually from a catastrophe model; then add commission, brokerage and expenses.
What it does, and doesn't do. It provides capacity and surplus relief: the ceding commission recovers acquisition cost on the ceded unearned premium. It does not change the shape of the net result. Net loss is $(1-a)X$ , so its coefficient of variation equals gross. The cedant keeps $1-a$ of every catastrophe and every large loss.
When the two sides disagree on the ELR, adjustable features settle it: Sliding Scale Commissions, a Profit Commission, Loss Corridors, or a per-occurrence limit on ceded catastrophe loss. See Reinsurance Contract Provisions.
For reserving, ceded losses are a scalar multiple of gross and develop on the gross pattern, unlike Excess of Loss cessions. Compare Surplus Share, where the percentage varies by risk.

Pricing a Property Quota Share A cedant asks for a $40\%$ quota share on homeowners business with projected subject earned premium of $\$50$ million. Five years of developed, trended, on-level loss ratios excluding catastrophes are $52\%, 58\%, 55\%, 61\%, 54\%$ . A catastrophe model gives expected annual catastrophe loss of $\$5.6$ million on the subject book. Terms offered: ceding commission $28\%$ , brokerage $2\%$ ; the reinsurer's own expenses are $1.5\%$ . The reinsurer targets a $95\%$ combined ratio.
Answer
 $\begin{align*} \text{Non-cat ELR} &= \frac{52 + 58 + 55 + 61 + 54}{5} \\ &= 56.0\% \end{align*}$ 
 $\begin{align*} \text{Cat load} &= \frac{\$5.6\text{M}}{\$50\text{M}} \\ &= 11.2\% \end{align*}$ 
 $\begin{align*} \text{Combined} &= 67.2\% + 28\% + 2\% + 1.5\% \\ &= 98.7\% \end{align*}$ 
Ceded premium is $0.40 \times \$50\text{M} = \$20$ million and expected ceded loss is $0.672 \times \$20\text{M} = \$13.44$ million. At $98.7\%$ the terms miss the target. The commission that meets it is $95 - 67.2 - 2 - 1.5 = 24.3\%$ , so the actuary would counter at about $24\%$ , or propose an adjustable feature instead.

Effect of a Per-Occurrence Catastrophe Limit Same treaty. The cedant's annual catastrophe loss (one event per year at most) is $\$0$ with probability $0.70$ , $\$10$ M with $0.20$ , $\$30$ M with $0.08$ and $\$60$ M with $0.02$ . The reinsurer proposes limiting ceded catastrophe loss to $\$8$ million per occurrence. Find the new catastrophe load and combined ratio.
Answer Ceded catastrophe loss at $40\%$ is $0, 4, 12, 24$ (in $ millions) before the cap, and $0, 4, 8, 8$ after it:
 $\begin{align*} E[\text{uncapped}] &= 0.20(4) + 0.08(12) + 0.02(24) \\ &= 2.24 \end{align*}$ 
 $\begin{align*} E[\text{capped}] &= 0.20(4) + 0.08(8) + 0.02(8) \\ &= 1.60 \end{align*}$ 
On ceded premium of $\$20$ M the catastrophe load falls from $11.2\%$ to $1.60/20 = 8.0\%$ , so the ELR becomes $64.0\%$ and the combined ratio $64.0 + 28 + 2 + 1.5 = 95.5\%$ , close to target at the original commission.

The cost moves to the cedant. In the $\$60$ M event its net loss is $\$52$ M instead of $\$36$ M, so it needs a catastrophe excess cover on its net retention, which is exactly where such covers apply.