Why Surplus Share Capacity Matters
A surplus share treaty is a proportional reinsurance arrangement in which the insurer retains a specified amount on each risk and cedes the amount above that retention, subject to a maximum number of reinsurer lines. Because risks can have different sums insured, the treaty’s cession proportion generally varies by risk. A correct calculation therefore starts with the risk’s insured amount, not its premium or loss.
The key distinction is between the reinsurer’s capacity and the total risk amount the treaty can accommodate. If the insurer retains one amount and the reinsurer provides a stated number of lines of that amount, the treaty can cover the retention plus the available cession capacity. A risk exceeding that total may still have a portion ceded to the treaty, but the excess above capacity is not covered by this treaty.
A line is the amount of the insurer’s retention. Thus, a treaty with a $50,000 retention and four reinsurer lines has $200,000 of reinsurance capacity per risk and $250,000 of total capacity per risk. The reinsurer does not automatically take four lines on every risk: the cession depends on how much of that risk exceeds the retention.
From Capacity to a Cession Proportion
For \(S\le R\), the risk is within the insurer’s retention, so \(C=0\) and \(q=0\). For \(R<S\le (N+1)R\), the cession is the amount above the retention, \(C=S-R\). For \(S>(N+1)R\), the treaty’s cession is capped at \(NR\); the insurer retains both its initial retention and the amount above the treaty’s total capacity.
| Risk sum insured | Treaty cession \(C\) | Insurer-retained amount |
|---|---|---|
| \(S\le R\) | 0 | \(S\) |
| \(R<S\le (N+1)R\) | \(S-R\) | \(R\) |
| \(S>(N+1)R\) | \(NR\) | \(S-NR\) |
When the treaty cedes a proportion of a risk, the same proportion is applied to that risk’s premium and covered loss, assuming the treaty terms define proportional cession on that basis. If \(P\) is the gross premium and \(L\) is the gross covered loss for that risk, then the ceded amounts are \(qP\) and \(qL\). The retained amounts are \((1-q)P\) and \((1-q)L\). This allocation is per risk: do not calculate a portfolio-wide average cession percentage and apply it indiscriminately to every policy.
Why Premium and Loss Use the Same Proportion
The treaty divides a risk by insured amount. For a risk within treaty capacity, the insurer retains \(R\) of \(S\) and the reinsurer takes \(S-R\). The two parties’ shares of the risk therefore add to the whole:
Under proportional terms, premium and covered loss follow these same shares. The reinsurer’s proportion is \(q=(S-R)/S\) when the risk is above retention and within capacity. If the risk exceeds capacity, replace \(S-R\) in the numerator with the capped cession \(NR\). Applying the cession proportion to premium and loss preserves the treaty’s proportional allocation; it does not mean the ceded loss is the amount of sum insured ceded.
Record the retention \(R\) and reinsurer line count \(N\). Calculate cession capacity \(NR\) and total capacity \((N+1)R\).
Compare its sum insured \(S\) with the retention and capacity. Apply \(C=\min\{\max(S-R,0),NR\}\).
For \(S>0\), divide the cession amount by the risk’s sum insured: \(q=C/S\).
Multiply each gross amount for that risk by \(q\) to find its ceded amount. Subtract the ceded amount from gross to find the retained amount.
Check, risk by risk and in total, that ceded plus retained premium equals gross premium, and ceded plus retained loss equals gross loss.
Worked Applications
Worked Example: Capacity and the Cession on a Large Risk
A hypothetical property insurer has a surplus treaty with a $50,000 retention and four reinsurer lines. A policy has a $290,000 sum insured, $14,500 gross premium, and $58,000 covered loss. Calculate treaty capacity, the cession proportion, and the ceded and retained premium and loss. The values are deterministic; no interest rate, mortality assumption, or probability distribution is used.
Step 1: Calculate capacity. The reinsurer’s cession capacity is \(4\times \$50,000=\$200,000\). Total capacity, including the insurer’s retention, is \(\$50,000+\$200,000=\$250,000\).
Step 2: Find the cession amount. The policy exceeds the total treaty capacity by \(\$290,000-\$250,000=\$40,000\). Its amount above the insurer’s retention is \(\$290,000-\$50,000=\$240,000\), but the treaty can cede only $200,000. Thus \(C=\min(\$240,000,\$200,000)=\$200,000\).
Step 3: Find the cession proportion. \(q=\$200,000/\$290,000=20/29\), or approximately 0.689655. Keep the fraction for the monetary calculations.
Step 4: Allocate premium. Ceded premium is \(\$14,500\times20/29=\$10,000\). Retained premium is \(\$14,500-\$10,000=\$4,500\).
Step 5: Allocate loss. Ceded loss is \(\$58,000\times20/29=\$40,000\). Retained loss is \(\$58,000-\$40,000=\$18,000\).
Check. The cession is $200,000, exactly the treaty’s limit per risk. The insurer retains \(\$290,000-\$200,000=\$90,000\) of the insured amount. Premium reconciles: \(\$10,000+\$4,500=\$14,500\). Loss reconciles: \(\$40,000+\$18,000=\$58,000\). The common mistake avoided is ceding the full $240,000 above retention despite the $200,000 treaty capacity.
Worked Example: Different Sums Insured and a Portfolio Reconciliation
A hypothetical insurer has an $80,000 retention and two reinsurer lines. Three policies have the following sums insured, gross premiums, and covered losses: Policy A has $60,000, $4,000, and $20,000; Policy B has $180,000, $9,000, and $45,000; Policy C has $300,000, $15,000, and $90,000, respectively. Calculate total ceded premium and loss. These are deterministic contractual allocations; no interest rate, mortality assumption, or probability distribution is used.
Step 1: Calculate treaty capacity. Cession capacity is \(2\times \$80,000=\$160,000\) per risk. Total capacity is \(\$80,000+\$160,000=\$240,000\).
Policy A. Its $60,000 sum insured is below the $80,000 retention. Cession is $0 and the cession proportion is 0. Ceded premium is $0 and ceded loss is $0.
Policy B. Its sum insured exceeds retention by \(\$180,000-\$80,000=\$100,000\), which is within the $160,000 cession capacity. Its cession proportion is \(\$100,000/\$180,000=5/9\). Ceded premium is \(\$9,000\times5/9=\$5,000\). Ceded loss is \(\$45,000\times5/9=\$25,000\).
Policy C. Its amount above retention is \(\$300,000-\$80,000=\$220,000\), so the $160,000 cession limit applies. Its cession proportion is \(\$160,000/\$300,000=8/15\). Ceded premium is \(\$15,000\times8/15=\$8,000\). Ceded loss is \(\$90,000\times8/15=\$48,000\).
Step 2: Total the ceded amounts. Total ceded premium is \(\$0+\$5,000+\$8,000=\$13,000\). Total ceded loss is \(\$0+\$25,000+\$48,000=\$73,000\).
Check. Gross premium is \(\$4,000+\$9,000+\$15,000=\$28,000\), so retained premium is \(\$28,000-\$13,000=\$15,000\). Gross loss is \(\$20,000+\$45,000+\$90,000=\$155,000\), so retained loss is \(\$155,000-\$73,000=\$82,000\). The allocations reconcile. Policy A has no cession because it is below retention; Policy C is capped at $160,000 of cession.
Worked Example: Allocating a Premium and Loss When Capacity Binds
A hypothetical commercial insurer retains $60,000 per risk and buys three reinsurer lines. A policy has a $260,000 sum insured, $13,000 gross premium, and $65,000 covered loss. Calculate the ceded and retained amounts. The calculation is deterministic; no interest rate, mortality assumption, or probability distribution is used.
Step 1: Calculate capacity. Cession capacity is \(3\times\$60,000=\$180,000\), and total capacity is \(\$60,000+\$180,000=\$240,000\).
Step 2: Calculate the cession. The amount above retention is \(\$260,000-\$60,000=\$200,000\), which exceeds the treaty cession capacity. Therefore \(C=\$180,000\), and the insurer retains \(\$260,000-\$180,000=\$80,000\) of the sum insured.
Step 3: Calculate the proportion. \(q=\$180,000/\$260,000=9/13\). The insurer’s retained proportion is \(1-9/13=4/13\).
Step 4: Allocate premium and loss. Ceded premium is \(\$13,000\times9/13=\$9,000\); retained premium is \(\$13,000-\$9,000=\$4,000\). Ceded loss is \(\$65,000\times9/13=\$45,000\); retained loss is \(\$65,000-\$45,000=\$20,000\).
Check. The ceded sum insured is exactly the $180,000 treaty capacity, and \(\$180,000+\$80,000=\$260,000\). The premium totals \(\$9,000+\$4,000=\$13,000\), and the loss totals \(\$45,000+\$20,000=\$65,000\). The common mistake avoided is applying the fraction \(200/260\), which would incorrectly cede the entire amount above retention, including $20,000 beyond treaty capacity.
Common Traps and Practical Checks
A surplus treaty’s line limit applies to the ceded amount of sum insured, not directly to a claim payment. Once the risk-specific cession proportion is determined, the covered loss is allocated proportionally. It is not correct to subtract the retention from the loss or to cap the ceded loss at the cession capacity measured in sum insured.
A second frequent error is using the same cession proportion for every risk in a portfolio. A risk below retention has no cession; another risk may be ceded only partly; a sufficiently large risk may be limited by capacity. Calculate \(C\) and \(q\) separately for each risk, then aggregate ceded amounts.
The formulas assume the treaty covers the stated risks and defines cessions proportionally on the stated sum-insured basis, with premium and covered loss allocated in the same proportion. Actual treaty wording can impose distinct definitions, exclusions, limits, or accounting adjustments. Those terms must be applied if specified; capacity calculations alone do not establish coverage. The distinctions among retention, attachment, and limit are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
Check Your Understanding
For each problem, calculate the requested amounts by risk, applying treaty capacity before allocating premium or loss.
- A treaty has a $75,000 retention and three reinsurer lines. A $270,000 risk has an $18,000 premium and a $72,000 covered loss. Find the cession capacity, cession proportion, ceded premium, and ceded loss. (Answer: Capacity $225,000; cession $195,000; proportion \(13/18\); ceded premium $13,000; ceded loss $52,000.)
- A treaty has a $100,000 retention and two lines. A $80,000 risk has a $6,000 premium and $24,000 loss; a $150,000 risk has a $9,000 premium and $30,000 loss; and a $360,000 risk has an $18,000 premium and $81,000 loss. Find total ceded premium and loss. (Answer: Ceded premium $13,000; ceded loss $55,000.)
- A treaty has a $60,000 retention and four lines. A $340,000 risk has a $17,000 premium and a $51,000 covered loss. Find its cession amount, cession proportion, ceded premium, and ceded loss. (Answer: Cession $240,000; proportion \(12/17\); ceded premium $12,000; ceded loss $36,000.)
- A treaty has a $50,000 retention and three lines. Three risks have sum insured, premium, and covered loss as follows: Risk A: $40,000, $2,000, and $8,000; Risk B: $125,000, $10,000, and $25,000; Risk C: $260,000, $13,000, and $52,000. Find total ceded premium and total ceded loss. (Answer: Ceded premium $13,500; ceded loss $45,000.)
- A treaty has a $40,000 retention and five reinsurer lines. A $310,000 risk has a $15,500 premium and a $62,000 covered loss. Find the treaty’s total capacity, the amount of insured value retained, and the ceded and retained premium and loss. (Answer: Total capacity $240,000; insured value retained $110,000; ceded premium $10,000 and retained premium $5,500; ceded loss $40,000 and retained loss $22,000.)