Why Retention and Lines Determine Capacity
A surplus share treaty is a proportional reinsurance arrangement in which the insurer retains a specified amount for each eligible risk and may automatically cede additional amounts to reinsurers, subject to the treaty’s line capacity. The key capacity question is not just how many reinsurer lines are available: the dollar value of each line is tied to the insurer’s retention.
This tutorial focuses on the treaty’s per-risk capacity. It distinguishes the insurer’s retained line from the reinsurer lines, calculates the greatest amount the treaty can accept automatically, and determines the resulting total capacity for one eligible risk. It does not calculate how a particular risk’s amount is apportioned; the central task here is to understand the treaty limits that such an allocation must respect.
The word “line” needs careful attention. Unless a problem states otherwise, “four reinsurer lines” means four lines in addition to the insurer’s retained line. By contrast, “four total lines” includes the insurer’s line, leaving three reinsurer lines. This distinction changes both the maximum cession and the total capacity.
Capacity Formulas and Their Justification
Let \(R\) be the insurer’s per-risk retention and \(n\) the number of reinsurer lines. Assume that each line equals \(R\), that the risk is eligible for the treaty, and that the treaty’s stated line limit is the only capacity restriction being considered. Then the maximum automatic cession and total automatic capacity follow directly from counting the lines.
The justification is a sum of the amounts represented by each line. There are \(n\) reinsurer lines, and each is worth \(R\), so together they provide \(nR\) of reinsurance capacity. The insurer contributes one additional line worth \(R\). Adding the insurer’s line to the reinsurer lines gives \((n+1)R\). Thus, at the treaty’s maximum total capacity, the insurer’s own line remains \(R\); the additional capacity consists of the reinsurer lines.
These formulas describe the treaty’s maximum automatic capacity, not a promise that every risk can be placed under it. Eligibility rules, exclusions, special limits, or other treaty terms can reduce the capacity available for a particular risk. If the risk exceeds the automatic capacity, the excess is not made automatic by these formulas; it may require a separate arrangement or remain outside the treaty.
Record the insurer’s per-risk retention \(R\), which is also the dollar amount of one line.
Determine whether the stated count is reinsurer lines only or total lines including the insurer. Convert it to \(n\), the number of reinsurer lines.
Multiply \(nR\) for maximum automatic cession, then add the insurer’s \(R\) to find total automatic capacity.
The total must equal the insurer’s line plus all reinsurer lines. Confirm that the units are dollars per risk.
Worked Capacity Calculations
Worked Example: Four Reinsurer Lines
A hypothetical insurer has a surplus share treaty with a per-risk retention of $250,000 and four reinsurer lines. Assume the risk is eligible and no other treaty limit reduces capacity. Find the maximum automatic cession, the insurer’s retained amount at maximum capacity, and the total automatic capacity.
Method. Treat the stated four lines as reinsurer lines, in addition to the insurer’s line. The amount of each line is the $250,000 retention.
Maximum automatic cession. There are four reinsurer lines:
Insurer-retained amount. The insurer’s one line is $250,000. At the treaty’s maximum total capacity, the insurer retains this line.
Total automatic capacity. Add the insurer’s line to the four reinsurer lines:
Check. The treaty has five lines in total: one insurer line and four reinsurer lines. Five times $250,000 is $1,250,000, agreeing with the total. The common mistake avoided is treating the four reinsurer lines as the total number of lines.
Worked Example: A Stated Count Includes the Insurer
A hypothetical treaty has a $160,000 retention and provides six total lines, including the insurer’s line. Assume there are no additional restrictions. Calculate the number of reinsurer lines, the maximum automatic cession, and the total automatic capacity.
Method. Convert the total line count to the reinsurer line count before multiplying. One of the six lines belongs to the insurer, so the reinsurers have \(6-1=5\) lines.
Maximum automatic cession.
Total automatic capacity. The six total lines are each $160,000:
Equivalently, the total is the insurer’s $160,000 line plus the $800,000 maximum cession, or $960,000. Check. Both methods give the same capacity. Counting all six lines as reinsurer lines would incorrectly produce a $960,000 maximum cession and a $1,120,000 total capacity.
Worked Example: Comparing Two Treaty Proposals
A hypothetical insurer compares two eligible surplus share proposals. Proposal A has a $120,000 retention and seven reinsurer lines. Proposal B has a $210,000 retention and four reinsurer lines. For each proposal, calculate maximum automatic cession and total automatic capacity, then compare the results.
Proposal A. Its maximum automatic cession is seven lines of $120,000:
The insurer contributes one $120,000 line, so Proposal A’s total capacity is:
Proposal B. Its maximum automatic cession is four lines of $210,000:
The insurer contributes one $210,000 line, so Proposal B’s total capacity is:
Comparison and check. Both proposals have the same maximum automatic cession, $840,000. Their total automatic capacities differ: Proposal A has $960,000, while Proposal B has $1,050,000. Proposal B’s higher retention accounts for the $90,000 difference in total capacity. The check is that each total equals its maximum cession plus its own insurer-retained line. Equal cession capacity does not imply equal total capacity.
Reading the Results Correctly
“Maximum automatic cession” and “total automatic capacity” answer different questions. The first is the greatest amount assigned to the reinsurers under the available lines. The second includes that amount plus the insurer’s own retention. For an eligible risk at the treaty’s capacity ceiling, the insurer’s line is still part of the total; it is not another reinsurer line.
| Quantity | Meaning | Formula when there are \(n\) reinsurer lines |
|---|---|---|
| One line | The insurer’s per-risk retention | \(R\) |
| Maximum automatic cession | Capacity supplied by the reinsurer lines | \(nR\) |
| Total automatic capacity | Insurer line plus reinsurer lines | \((n+1)R\) |
A treaty with a larger number of reinsurer lines does not necessarily have a larger total capacity than another treaty. Retention also matters. Compare the products \((n+1)R\) for total capacity and \(nR\) for maximum cession; do not compare line counts alone. Similarly, treaties can have equal maximum automatic cessions but different total capacities if their retentions differ.
Common Exam Traps
- Counting the insurer’s line twice. If the problem gives reinsurer lines, add one insurer line when finding total capacity. Do not add an extra line when the stated count already includes the insurer.
- Calling total capacity the maximum cession. The insurer’s retention is part of total capacity but not part of the reinsurers’ maximum cession.
- Forgetting that line size equals retention. A line count alone is not a dollar amount. Multiply by the per-risk retention.
- Comparing only the number of lines. A treaty with fewer lines can have a greater total capacity if its retention is sufficiently larger.
- Assuming a treaty automatically covers any larger risk. The formulas give the stated automatic limit. They do not remove eligibility conditions or create capacity above that limit.
These definitions and calculations are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. On an exam-style problem, state what the line count means, show the multiplication, and label cession and total capacity separately.
Check Your Understanding
For each hypothetical treaty, assume the risk is eligible and no additional term reduces the stated capacity. Give amounts in dollars.
- A treaty has a $90,000 retention and three reinsurer lines. Find the maximum automatic cession and total automatic capacity. (Answer: maximum cession $270,000; total capacity $360,000.)
- A treaty has a $225,000 retention and five total lines, including the insurer’s line. Find the number of reinsurer lines, maximum automatic cession, and total capacity. (Answer: 4 reinsurer lines; maximum cession $900,000; total capacity $1,125,000.)
- A treaty has a $140,000 retention and six reinsurer lines. Find the maximum automatic cession and the insurer-retained amount at maximum total capacity. (Answer: maximum cession $840,000; insurer-retained amount $140,000.)
- Proposal A has a $180,000 retention and four reinsurer lines. Proposal B has a $240,000 retention and three reinsurer lines. Compare their maximum automatic cessions and total capacities. (Answer: both maximum cessions are $720,000; A’s total capacity is $900,000 and B’s is $960,000.)
- A treaty has a $325,000 retention and two reinsurer lines. A hypothetical eligible risk has an amount of $1,100,000. Find the treaty’s maximum automatic cession and total automatic capacity, and state how much of the risk exceeds that total capacity. (Answer: maximum cession $650,000; total capacity $975,000; $125,000 exceeds the automatic total capacity.)