Reinsurance Pricing

Study guides: Exam 9, Exam GI-301

Reinsurance Pricing estimates the expected loss cost to a reinsurance contract — from the cedant's own adjusted history (experience rating) or from its current exposure profile run through severity or exposure curves (exposure rating) — and then loads that loss cost for the reinsurer's expenses, risk and the timing of cash flows. The price is usually quoted as a rate on the cedant's subject premium.

$\text{Loss cost rate} = \frac{\sum \text{trended, developed layer losses}}{\sum \text{on-level subject premium}}$

$\text{EF} = \frac{E[X \wedge u] - E[X \wedge d]}{E[X \wedge PL]}$

$\text{Premium} = \frac{\text{Loss cost}\,(1 + \text{ULAE}) + \text{Fixed}}{1 - \text{Variable \%}}$

Clark's pricing paradox: if a contract's experience is stable enough to price precisely, the cedant will not want to buy it. Reinsurance covers what is hard to price, so the basic tools are a starting point for judgment.
Experience rating (a burning cost when unadjusted): on-level the subject premium; trend each large loss individually and keep every loss that would pierce the layer after trend; apply the layer terms (ALAE included if it is covered pro rata); develop with excess-layer development factors; divide. The yearly loss costs should scatter randomly about their average — a drift signals wrong trend or development, or a changed mix. Proportional treaties follow the same logic on loss ratios, with catastrophe and shock losses removed and a modelled catastrophe load added back.
Exposure rating prices the current profile. In the exposure factor $\text{EF}$ , $PL$ is the policy limit, $u = \min(PL, AP + Lim)$ and $d = \min(PL, AP)$ for attachment point $AP$ and layer limit $Lim$ ; $\text{EF}$ times subject premium times the expected loss ratio is the layer loss (Limited Expected Value). Casualty uses increased limits factors from a severity distribution (excess loss factors for workers compensation); property uses exposure curves with retention and limit expressed as a share of insured value.
Reconciling the two: free cover — no trended loss reaching the top of a layer — is priced from exposure-rating relativities; credibility rests on the claim count expected in the experience period (not the count observed) and on how stable the yearly loss costs are.
Catastrophe covers were once priced by payback — premium set so the limit is repaid over a chosen number of years — and are now priced from catastrophe-model output (Catastrophe Expected Loss Cost). Reinstatements are usually paid pro rata as to amount.
Loss-sensitive features — Sliding Scale Commissions, profit commissions, Loss Corridors, annual aggregate deductibles — respond non-linearly to results, so they are priced over an aggregate loss distribution, not at the expected loss ratio (Reinsurance Contract Provisions).
The final price adds variable expenses (ceding commission, brokerage), fixed overhead and reinsurer ULAE; unlike the cedant, the reinsurer pays no premium tax. The traditional "100/80" load treats all expenses as a $20\%$ variable charge. Risk load and investment income sit beyond this formula (Risk Loads).

Experience Rating a Per-Risk Excess Layer A cedant buys $500K xs $500K per risk. Three years are shown for brevity ($000s; losses already trended to the treaty period, premium at the treaty period's rate level):

Year On-level subject premium Trended large losses Excess LDF

2021 $10{,}000$ $700$ , $1{,}200$ $1.10$

2022 $11{,}000$ $450$ , $900$ $1.35$

2023 $12{,}000$ $1{,}300$ , $650$ $1.90$

Projected subject premium is $13{,}000$ . Find the loss cost and the premium with ULAE of $3\%$ of losses, fixed expense of $50$ and $10\%$ brokerage.
Answer Each loss contributes $\min(\max(X - 500, 0), 500)$ : $200 + 500 = 700$ in 2021, $0 + 400 = 400$ in 2022 and $500 + 150 = 650$ in 2023.
 $\begin{align*} \text{Developed} &= 700(1.10) + 400(1.35) + 650(1.90) \\ &= 770 + 540 + 1{,}235 \\ &= 2{,}545 \\ \text{Loss cost rate} &= \frac{2{,}545}{33{,}000} \\ &= 7.71\% \\ \text{Expected loss} &= 0.07712(13{,}000) \\ &= 1{,}003 \\ \text{Premium} &= \frac{1{,}003(1.03) + 50}{1 - 0.10} \\ &= 1{,}203 \end{align*}$ 
The indicated rate is $1{,}203/13{,}000 = 9.25\%$ of subject premium. The yearly loss costs — $7.7\%$ , $4.9\%$ , $10.3\%$ — scatter without a clear trend, but two or three losses a year is thin experience: the result should be credibility-weighted with an exposure rate.

Year	On-level subject premium	Trended large losses	Excess LDF
2021	$10{,}000$	$700$ , $1{,}200$	$1.10$
2022	$11{,}000$	$450$ , $900$	$1.35$
2023	$12{,}000$	$1{,}300$ , $650$	$1.90$

Exposure Rating a Casualty Layer With ILFs A treaty covers $1.5M xs $500K per occurrence, loss and ALAE pro rata. The cedant's projected subject premium is $4.0M on policies with $1M limits and $2.0M on policies with $2M limits; its expected loss and ALAE ratio is $65\%$ . ILFs are $1.55$ at $500K, $1.80$ at $1M and $2.00$ at $2M. Find the treaty loss cost.
Answer The layer's top is $500 + 1{,}500 = 2{,}000$ ($000s), capped at each policy limit:
 $\begin{align*} \text{EF}_{\$1\text{M}} &= \frac{1.80 - 1.55}{1.80} \\ &= 0.1389 \\ \text{EF}_{\$2\text{M}} &= \frac{2.00 - 1.55}{2.00} \\ &= 0.2250 \\ \text{Layer loss} &= 4{,}000(0.65)(0.1389) + 2{,}000(0.65)(0.2250) \\ &= 361.1 + 292.5 \\ &= 653.6 \end{align*}$ 
The loss cost is $653.6/6{,}000 = 10.9\%$ of subject premium. The $2M policies supply almost half the layer loss from one third of the premium, because more of each of their limits sits inside the layer — which is why the limits profile, not last year's claims, drives an exposure rate.