How a Surplus-Share Treaty Allocates a Policy Loss
A surplus-share treaty shares a policy’s insured amount between an insurer and a reinsurer. The insurer first retains a specified amount per policy, called its retention. The reinsurer then accepts a stated number of additional lines, with each line equal to that retention. Because the policy’s insured amount determines the shares, the ceded proportion can differ from one policy to another.
Once those shares are determined, a covered loss on the policy is allocated in the same proportions, provided the policy’s loss is proportional to its insured amount. A policy that exceeds the treaty’s total capacity is not automatically fully protected: if there is no additional facultative reinsurance, the insurer retains the portion beyond capacity as well as its own retention line.
The line count in this definition is explicit: \(n\) is the number of reinsurer lines, in addition to the insurer’s one retention line. Treaty wording should always be checked, because a stated line count is useful only when it is clear whether it refers to the reinsurer’s capacity or the combined total.
Deriving the Proportional Loss Allocation
The ceded amount of insured value cannot be negative and cannot exceed the treaty’s capacity. For a policy with sum insured \(S\), the amount above the insurer’s retention is \(\max(S-r,0)\), and the reinsurer’s maximum is \(nr\). The ceded insured amount is therefore the smaller of those two quantities. Under proportional sharing, the ceded fraction of the policy is that ceded insured amount divided by \(S\).
If the covered loss \(X\) is shared in the same proportion as the insured amount, multiplying the ceded fraction by \(X\) gives the ceded loss. The retained loss is the balance. This also explains why the same percentage applies to a partial loss and to a total loss on the same policy: the treaty’s share is determined by the policy’s insured amount, not by the size of the particular loss.
Record \(r\) and \(n\). Calculate the reinsurer’s maximum ceded insured amount \(nr\), and the combined supported amount \((n+1)r\).
Subtract the retention from \(S\), but not below zero; then cap the result at \(nr\).
Multiply \(X\) by the ceded insured amount divided by \(S\).
Calculate \(X^R=X-X^C\) and verify that \(X^C+X^R=X\).
Worked Applications
Worked Example: A Policy Within Treaty Capacity
A hypothetical insurer retains \(r=\$80{,}000\) per policy under a surplus-share treaty with \(n=3\) reinsurer lines. A policy has sum insured \(S=\$200{,}000\), and a covered loss of \(X=\$30{,}000\) occurs. Find the ceded and retained losses.
Method: calculate the ceded insured amount, then apply its proportion to the loss. The reinsurer’s capacity is
The policy amount above the insurer’s retention is
Since \(\$120{,}000\) is less than the \(\$240{,}000\) reinsurer capacity, the ceded insured amount is \(\$120{,}000\). Thus the ceded fraction is
The ceded and retained losses are
Check: \(\$18{,}000+\$12{,}000=\$30{,}000\), the original loss. The insurer’s retained proportion is \(\$80{,}000/\$200{,}000=0.4\), which also gives \(0.4(\$30{,}000)=\$12{,}000\). The common mistake avoided here is treating the three reinsurer lines as the policy’s total supported capacity; the insurer’s retention is a separate line.
Worked Example: A Policy Above Treaty Capacity
A hypothetical insurer retains \(\$100{,}000\) per policy under a two-line surplus-share treaty. A policy has sum insured \(S=\$450{,}000\) and a covered loss \(X=\$90{,}000\). Find the ceded and retained losses, assuming there is no facultative cover.
Method: cap the ceded insured amount at the reinsurer’s two-line capacity. Here \(r=\$100{,}000\) and \(n=2\), so the reinsurer can accept at most
The policy amount above the retention is
The treaty accepts the smaller of \(\$350{,}000\) and \(\$200{,}000\), so the ceded insured amount is \(\$200{,}000\). The ceded fraction is
Therefore,
Check: the loss shares total \(\$90{,}000\). The insurer retains \(\$50{,}000\) of the loss, not merely the share corresponding to its \(\$100{,}000\) retention line. The combined supported amount is \((2+1)(\$100{,}000)=\$300{,}000\), leaving \(\$150{,}000\) of insured amount above capacity with the insurer. That excess represents \((\$150{,}000/\$450{,}000)(\$90{,}000)=\$30{,}000\) of the retained loss; the retention line represents another \((\$100{,}000/\$450{,}000)(\$90{,}000)=\$20{,}000\). Together they give \(\$50{,}000\). A frequent error is to calculate \(90{,}000(450{,}000-100{,}000-200{,}000)/450{,}000\) as the cession. That expression uses the amount above capacity, not the ceded amount. The cession is based on the treaty’s \(\$200{,}000\) capacity, giving \(\$40{,}000\).
Worked Example: A Policy Exactly at Total Capacity
A hypothetical insurer retains \(r=\$75{,}000\) per policy, and its surplus-share treaty provides \(n=2\) reinsurer lines. A policy has sum insured \(S=\$225{,}000\) and incurs a covered loss of \(X=\$60{,}000\). Calculate both shares and explain the boundary case.
The reinsurer’s capacity is
The policy amount above the retention is
The amount above retention exactly equals the treaty capacity. Therefore the ceded insured amount is \(\$150{,}000\), and the ceded fraction is
The ceded loss is \((2/3)(\$60{,}000)=\$40{,}000\). The retained loss is \(\$60{,}000-\$40{,}000=\$20{,}000\). Check: the insurer’s retention line is one-third of the insured amount, so its loss share is \((1/3)(\$60{,}000)=\$20{,}000\), and the two shares sum to \(\$60{,}000\). At exactly total capacity, the treaty is fully used but there is no amount above capacity for the insurer to retain.
Worked Example: Applying the Treaty Separately to Multiple Policies
A hypothetical insurer has a \(\$25{,}000\) retention per policy and a three-line surplus-share treaty. In one event, two separate policies each have a covered loss: Policy A has \(S_A=\$20{,}000\), \(X_A=\$8{,}000\); Policy B has \(S_B=\$175{,}000\), \(X_B=\$35{,}000\). Find the total ceded and retained losses across the two policies.
Method: apply the treaty to each policy before adding the results. The reinsurer capacity per policy is
Policy A is below the \(\$25{,}000\) retention, so its ceded insured amount is zero. Hence \(X_A^C=\$0\) and \(X_A^R=\$8{,}000\).
For Policy B, the amount above the retention is \(\$175{,}000-\$25{,}000=\$150{,}000\). The treaty capacity is only \(\$75{,}000\), so the ceded insured amount is \(\$75{,}000\). Its share of Policy B is
Thus \(X_B^C=(3/7)(\$35{,}000)=\$15{,}000\), and \(X_B^R=\$35{,}000-\$15{,}000=\$20{,}000\). Adding the policy allocations gives total ceded loss \(\$0+\$15{,}000=\$15{,}000\) and total retained loss \(\$8{,}000+\$20{,}000=\$28{,}000\). Check: \(\$15{,}000+\$28{,}000=\$43{,}000\), equal to the two original losses, \(\$8{,}000+\$35{,}000\). The treaty capacity is applied separately to each policy; it is not a single \(\$75{,}000\) cap on the event’s combined losses.
Capacity Cases and Exam Traps
The formula can be read as three cases. If \(S\le r\), the policy does not exceed the insurer’s retention, so nothing is ceded. If \(r<S\le(n+1)r\), the whole amount above retention is ceded, and the insurer retains only its retention proportion. If \(S>(n+1)r\), the treaty is exhausted; the ceded insured amount is \(nr\), and the insurer also retains the amount above total capacity.
| Policy sum insured | Ceded insured amount | Insured amount retained by insurer |
|---|---|---|
| \(S\le r\) | \(0\) | \(S\) |
| \(r<S\le(n+1)r\) | \(S-r\) | \(r\) |
| \(S>(n+1)r\) | \(nr\) | \(S-nr\) |
The table describes insured amounts. To obtain loss amounts, multiply each share by \(X/S\), as long as the loss is shared proportionally. These are the main checks to make in a calculation:
- Count lines correctly. With \(n\) reinsurer lines, the reinsurer capacity is \(nr\), while total supported capacity is \((n+1)r\).
- Cap the cession, not the insurer’s retention. Above total capacity, the cession stops increasing at \(nr\). The extra insured amount remains with the insurer unless other reinsurance applies.
- Use the policy’s own sum insured. Determine the cession fraction policy by policy. Do not apply one portfolio-wide fraction when policy sums insured differ.
- Allocate the loss, not just the insured amount. After finding the ceded fraction, multiply it by the policy loss \(X\). The ceded insured amount itself is not the ceded loss.
- Check conservation. Under a full allocation with no other payment source, \(X^C+X^R=X\).
Surplus-share calculations are useful because they distinguish proportional sharing that varies by policy from a fixed percentage applied to every policy. A policy below the retention receives no cession; one within capacity cedes its amount above retention; and one above capacity reaches the treaty maximum. These policy-level distinctions are essential when interpreting a treaty’s line count and when combining results for a portfolio.
Check Your Understanding
For each problem, assume proportional allocation of the covered loss and no facultative reinsurance unless stated otherwise. Apply the treaty separately to each policy.
- A hypothetical insurer retains \(\$60{,}000\) and has two reinsurer lines. A policy has \(S=\$150{,}000\) and \(X=\$30{,}000\). Find the ceded and retained losses. (Answer: ceded \(\$18{,}000\); retained \(\$12{,}000\))
- A hypothetical insurer retains \(\$40{,}000\) under a two-line treaty. A policy has \(S=\$320{,}000\) and \(X=\$64{,}000\). Find the ceded and retained losses, including the effect of amount above capacity. (Answer: ceded \(\$16{,}000\); retained \(\$48{,}000\))
- A hypothetical insurer retains \(\$90{,}000\) and has two reinsurer lines. A policy at exactly total capacity has \(S=\$270{,}000\) and \(X=\$54{,}000\). Find the ceded and retained losses. (Answer: ceded \(\$36{,}000\); retained \(\$18{,}000\))
- A hypothetical insurer retains \(\$25{,}000\) under a three-line treaty. Policy A has \(S_A=\$20{,}000\), \(X_A=\$8{,}000\); Policy B has \(S_B=\$175{,}000\), \(X_B=\$35{,}000\). Find the total ceded and retained loss. (Answer: total ceded \(\$15{,}000\); total retained \(\$28{,}000\))
- A hypothetical insurer retains \(\$100{,}000\) under a four-line treaty. A policy has \(S=\$650{,}000\) and \(X=\$130{,}000\). Find the ceded and retained losses, and identify the retained amount attributable to insured value above total treaty capacity. (Answer: ceded \(\$80{,}000\); retained \(\$50{,}000\), of which \(\$30{,}000\) is attributable to insured value above capacity)