Surplus Share Capacity for Large Risks

Why Capacity Matters for a Large Risk

A surplus share treaty allows an insurer to cede amounts above a stated retention, subject to the treaty’s line limit. For a risk within that limit, the insurer can cede the amount above its retention to the treaty. For a larger risk, the treaty may not have enough capacity to accept the entire excess. The remaining amount must be placed separately, if possible, or it stays with the insurer.

The key is to distinguish the retention, the treaty’s maximum cession, and the total risk amount that those two components can support. This tutorial considers a fixed retention and a treaty with a specified number of surplus lines. It is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Let $S$ be a risk’s sum insured, $R$ the insurer’s per-risk retention, and $N$ the number of surplus lines available under the treaty. Assume each surplus line has capacity $R$. The treaty’s maximum cession is $NR$, and the maximum amount supported by the insurer’s retention plus the treaty is $(N+1)R$. Any amount above $(N+1)R$ is beyond the treaty’s capacity and requires separate placement if the insurer is to avoid retaining it.

Calculating the Three Amounts

The allocation is made in order: the insurer’s retention applies first, the surplus share treaty accepts the next amount up to its limit, and any remaining excess is outside the treaty. For a fixed retention and equal-sized lines, the amounts are:

\begin{aligned} \text{Insurer's contractual retention} \quad I &= \min(S,R),\\ \text{Treaty cession} \quad C &= \min\bigl(\max(S-R,0),NR\bigr),\\ \text{Amount beyond treaty capacity} \quad F &= \max\bigl(S-(N+1)R,0\bigr). \end{aligned}

Here, $F$ is the amount that needs separate placement, for example through facultative reinsurance. It is not automatically ceded: it is outside this surplus treaty. If separate coverage is not obtained, the insurer retains this amount in addition to its contractual retention.

The equations also cover risks that do not exceed the retention. When $S \leq R$, $I=S$, $C=0$, and $F=0$. For larger risks, the allocation reconciles to the original sum insured:

$$ I+C+F=S. $$

Why the Capacity Formula Works

The treaty covers only the layer above the insurer’s retention, and it can cover no more than $N$ lines. The amount available for cession before applying the treaty limit is $S-R$ when $S>R$. Ceding that amount without checking the limit could exceed what the treaty agreed to accept.

Find the amount above retention.
For $S>R$, the excess is $S-R$. For $S\leq R$, there is no excess to cede.

Apply the treaty limit.
The treaty can accept at most $N$ lines, or $NR$, so the cession is the smaller of the excess and $NR$.

Identify any uncovered excess.
The insurer and treaty together support at most $R+NR=(N+1)R$. Subtract that amount from $S$ if $S$ is larger.

To verify the allocation when $S>R$, first suppose the treaty is not full: $S-R \leq NR$. Then $C=S-R$, $F=0$, and $I=R$, giving $I+C+F=R+(S-R)=S$. If the treaty is full, then $S-R>NR$. In that case $C=NR$, $F=S-(N+1)R$, and $I=R$, giving $I+C+F=R+NR+S-(N+1)R=S$.

Worked Allocations

Worked Example: A Risk Within the Treaty’s Capacity

A hypothetical insurer has a retention of $80,000 and a surplus share treaty with six lines. A policy has a sum insured of $530,000. Calculate the insurer’s retention, the treaty cession, and any amount requiring separate placement.

Method. Calculate the treaty limit first, then compare the policy excess with that limit. This avoids assuming that every amount above retention must be ceded.

The treaty’s maximum cession is $6 \times \$80{,}000=\$480{,}000$. The total amount supported by the retention and treaty is $\$80{,}000+\$480{,}000=\$560{,}000$.

The amount above retention is $\$530{,}000-\$80{,}000=\$450{,}000$. Since $\$450{,}000$ is less than the treaty limit of $\$480{,}000$, the treaty can accept all of it.

The insurer retains $80,000, the treaty cedes $450,000, and the amount requiring separate placement is $0. The allocation checks: $\$80{,}000+\$450{,}000+\$0=\$530{,}000$.

Worked Example: A Large Risk That Fills the Treaty

A hypothetical policy has a sum insured of $790,000. The insurer’s retention is $120,000, and its surplus share treaty has four lines. Assume separate proportional coverage is obtained for any amount above treaty capacity. Find the retention, treaty cession, separate placement, and the allocation of a $395,000 covered claim.

Method. Split the sum insured into the insurer’s retention, the treaty’s capped cession, and the residual amount. Then apply the same proportions to the claim, as assumed under the proportional coverage terms.

The treaty limit is $4 \times \$120{,}000=\$480{,}000$, so the total amount supported by retention and treaty is $\$120{,}000+\$480{,}000=\$600{,}000$. The amount above retention is $\$790{,}000-\$120{,}000=\$670{,}000$, which exceeds the treaty limit.

Therefore, the insurer retains $120,000, the treaty cedes its full $480,000 capacity, and the remaining $\$790{,}000-\$600{,}000=\$190{,}000$ requires separate placement. Check: $\$120{,}000+\$480{,}000+\$190{,}000=\$790{,}000$.

The claim is exactly half the sum insured: $\$395{,}000/\$790{,}000=0.5$. Under the stated proportional terms, the insurer’s share is $0.5 \times \$120{,}000=\$60{,}000$; the treaty’s share is $0.5 \times \$480{,}000=\$240{,}000$; and the separate reinsurer’s share is $0.5 \times \$190{,}000=\$95{,}000$. The claim allocation checks: $\$60{,}000+\$240{,}000+\$95{,}000=\$395{,}000$.

Common mistake avoided: Ceding all $670,000 above retention would exceed the treaty’s $480,000 limit. Only the amount above that limit can be sought through separate placement.

Worked Example: An Unplaced Excess and the Insurer’s Actual Exposure

A hypothetical insurer writes a $510,000 policy. Its retention is $75,000, and its surplus treaty has five lines. No separate coverage is obtained for excess above the treaty limit. For a covered loss of $204,000, calculate the treaty cession, the amount that remains with the insurer, and the loss shares.

Method. First calculate the treaty allocation as if the excess were separately placeable. Then, because no separate coverage was obtained, add that excess to the insurer’s retention when determining the insurer’s actual exposure.

The treaty’s maximum cession is $5 \times \$75{,}000=\$375{,}000$. The retention plus treaty supports $\$75{,}000+\$375{,}000=\$450{,}000$. The treaty cedes its full $375,000, because the amount above retention is $\$510{,}000-\$75{,}000=\$435{,}000$. The excess beyond treaty capacity is $\$510{,}000-\$450{,}000=\$60{,}000$.

With no separate placement, the insurer retains both its $75,000 contractual retention and the $60,000 excess, for a total retained amount of $\$75{,}000+\$60{,}000=\$135{,}000$. Check: $\$135{,}000+\$375{,}000=\$510{,}000$.

The loss is 40% of the sum insured, since $\$204{,}000/\$510{,}000=0.4$. The insurer’s loss share is $0.4 \times \$135{,}000=\$54{,}000$, and the treaty’s share is $0.4 \times \$375{,}000=\$150{,}000$. These sum to the covered loss: $\$54{,}000+\$150{,}000=\$204{,}000$.

Common mistake avoided: The contractual retention remains $75,000, but the insurer’s actual net exposure is $135,000 when the $60,000 excess is unplaced. These are different amounts.

Common Capacity Traps

A line limit counts surplus lines, not the insurer’s retention line. If a treaty provides $N$ surplus lines of $R$ each, its cession capacity is $NR$, while the combined amount supported with the insurer’s retention is $(N+1)R$. Mixing up these two totals can overstate the treaty’s capacity by one line.

Also distinguish an amount requiring separate placement from an amount that has actually been reinsured. The first is calculated from the treaty limit; the second depends on whether a separate contract is obtained. When a policy has a loss, proportional claim allocation across the retention and placements is appropriate only if the contracts have the assumed proportional terms and cover the same loss. Different policy terms or separate reinsurance conditions can change the allocation.

Key takeaway: For a fixed retention $R$ and $N$ equal-sized surplus lines, cap the treaty cession at $NR$. The insurer’s retention plus that cession supports $(N+1)R$; any sum insured above that amount is outside the treaty and must be placed separately or retained.

Check Your Understanding

For each problem, assume a fixed retention and equal-sized surplus lines. Treat any amount above treaty capacity as requiring separate placement unless the question says otherwise.

A hypothetical insurer has a $50,000 retention and three surplus lines. A policy has a $180,000 sum insured. Find the retention, treaty cession, and amount requiring separate placement. (Answer: $50,000 retained; $130,000 ceded; $0 requiring separate placement.)
A $560,000 policy is written with a $90,000 retention and four surplus lines. Calculate the treaty limit, total amount supported by retention and treaty, and amount requiring separate placement. (Answer: $360,000 treaty limit; $450,000 supported; $110,000 requiring separate placement.)
A hypothetical risk has a $900,000 sum insured, a $125,000 retention, and six surplus lines. Calculate the three-part allocation and verify that it sums to the policy amount. (Answer: $125,000 retained; $750,000 ceded; $25,000 requiring separate placement; total $900,000.)
A $320,000 policy has an $80,000 retention and five surplus lines. Separate proportional coverage is obtained for any excess. If the covered claim is $96,000, calculate the claim shares for the insurer, treaty, and separate reinsurer. (Answer: insurer $24,000; treaty $72,000; separate reinsurer $0.)
A hypothetical $1,050,000 policy has a $200,000 retention and three surplus lines. No separate placement is obtained. Find the treaty cession and the insurer’s actual retained amount. If the covered loss is $420,000 and the treaty responds proportionally, calculate each party’s loss share. (Answer: treaty cession $600,000; insurer retains $450,000; insurer’s loss share $180,000; treaty’s loss share $240,000.)

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