How a Quota Share Allocates Premium and Loss
A quota share treaty transfers a fixed proportion of specified business to a reinsurer. Because the same percentage applies to both the treaty’s subject premium and covered losses, the ceding insurer can calculate its retained amounts directly from the gross amounts. The central accounting check is simple: ceded plus retained must equal gross, separately for premium and loss.
The calculations are straightforward, but the contract terms and accounting basis matter. A percentage applies only to the business and amounts included in the treaty. A premium amount might be written or earned, for example; the calculation must use the basis specified in the problem. The examples below state their basis explicitly.
Here, a gross loss is the covered claim amount before reinsurance, after any applicable policy terms. “Net” means the amount retained by the ceding insurer after the quota share recovery; it does not mean the amount after expenses, taxes, or other reinsurance. A total loss means the full covered loss amount for the claim in the stated scenario, not necessarily the insured value of a policy.
For every subject premium and every covered loss, the allocation should reconcile:
The same calculation applies claim by claim. If multiple claims are subject to the treaty and share the same percentage, their ceded amounts can be added to obtain the aggregate recovery. This tutorial assumes a standard proportional quota share with no cession cap, per-risk cap, commission adjustment, exclusion, or other contract modification. Those terms can change the calculation and must not be assumed absent in an actual contract unless the problem says so.
Worked Claim and Premium Allocations
Worked Example: A Partial Claim and a Total Claim
A hypothetical insurer has a quota share with a 30% cession on covered business. For one accounting period, subject gross written premium is $200,000. Two covered claims are reported: a partial loss of $40,000 and a total covered loss of $160,000. Calculate ceded and retained premium and loss. Treat all amounts as undiscounted dollars for this period; no interest rate, mortality assumption, or probability distribution is used because the inputs are given amounts.
Method. Apply \(\alpha=0.30\) to the stated subject premium and to each covered claim. The remaining 70% is retained.
Premium. Ceded premium is \(0.30\times \$200{,}000=\$60{,}000\). Retained premium is \(0.70\times \$200{,}000=\$140{,}000\). The check is \(\$60{,}000+\$140{,}000=\$200{,}000\), the subject gross written premium.
Partial claim. Ceded loss is \(0.30\times \$40{,}000=\$12{,}000\). Retained loss is \(0.70\times \$40{,}000=\$28{,}000\). These add to \(\$40{,}000\).
Total claim. Ceded loss is \(0.30\times \$160{,}000=\$48{,}000\). Retained loss is \(0.70\times \$160{,}000=\$112{,}000\). These add to \(\$160{,}000\).
Aggregate loss check. Gross loss is \(\$40{,}000+\$160{,}000=\$200{,}000\). Total ceded loss is \(\$12{,}000+\$48{,}000=\$60{,}000\), and total retained loss is \(\$28{,}000+\$112{,}000=\$140{,}000\). Thus, \(\$60{,}000+\$140{,}000=\$200{,}000\), as required. The common mistake this method avoids is applying the share to only one claim, or treating a total claim as automatically ceded in full.
Worked Example: Checking the Cession Percentage
A hypothetical treaty cedes 25% of subject premium and covered loss. Subject gross premium is $6,000, and the covered gross loss is $15,000. Find the ceded and retained amounts for each. Assume the figures are undiscounted dollars for the same accounting basis; no interest rate, mortality assumption, or probability distribution is required.
Method. The stated 25% is the ceded share, so \(\alpha=0.25\) and the retained share is \(1-0.25=0.75\).
Premium. Ceded premium is \(0.25\times \$6{,}000=\$1{,}500\). Retained premium is \(0.75\times \$6{,}000=\$4{,}500\). Check: \(\$1{,}500+\$4{,}500=\$6{,}000\).
Loss. Ceded loss is \(0.25\times \$15{,}000=\$3{,}750\). Retained loss is \(0.75\times \$15{,}000=\$11{,}250\). Check: \(\$3{,}750+\$11{,}250=\$15{,}000\).
Common mistake avoided. If 25% is incorrectly treated as the retained share, the loss would be reported as $3,750 retained and $11,250 ceded. That reverses the treaty allocation. Label the percentage as ceded before calculating, then calculate the retained percentage by subtraction.
Aggregating Losses and Expected Amounts
A quota share with a fixed percentage preserves the same proportion at both the individual-claim and aggregate levels. If \(X_1,\ldots,X_n\) are covered gross claims, the total ceded loss is the sum of each claim’s ceded share:
This equality follows because the same constant \(\alpha\) multiplies every claim, so it can be factored outside the sum. The retained aggregate is similarly \((1-\alpha)\sum_{j=1}^{n}X_j\). Consequently, claim-by-claim calculations and an aggregate calculation must agree when all claims are subject to the same share.
The same proportionality applies to expected loss when \(X\) has a finite expectation. Since \(\alpha\) is a constant, \(E[\alpha X]=\alpha E[X]\). This is an accounting relationship for the ceded and retained loss amounts; it does not by itself specify the treaty premium, which is an input determined on the stated premium basis.
Worked Example: Expected Loss Under a Discrete Distribution
A hypothetical insurer’s annual covered gross loss \(X\) has the following discrete distribution: \(X=\$0\) with probability 0.50, \(X=\$100{,}000\) with probability 0.30, and \(X=\$300{,}000\) with probability 0.20. A quota share cedes 40%. Subject gross earned premium for the year is $50,000. Calculate ceded and retained premium, the ceded and retained losses in each outcome, and the expected ceded and retained losses. All amounts are undiscounted dollars for the year. No interest rate or mortality assumption applies; the stated discrete distribution and its probabilities are the complete loss model.
Premium. Ceded premium is \(0.40\times \$50{,}000=\$20{,}000\). Retained premium is \(0.60\times \$50{,}000=\$30{,}000\). Their sum is $50,000.
Outcome 1, probability 0.50. Gross loss is $0. Ceded loss is \(0.40\times \$0=\$0\), and retained loss is \(0.60\times \$0=\$0\).
Outcome 2, probability 0.30. Gross loss is $100,000. Ceded loss is \(0.40\times \$100{,}000=\$40{,}000\), and retained loss is \(0.60\times \$100{,}000=\$60{,}000\). They sum to $100,000.
Outcome 3, probability 0.20. Gross loss is $300,000. Ceded loss is \(0.40\times \$300{,}000=\$120{,}000\), and retained loss is \(0.60\times \$300{,}000=\$180{,}000\). They sum to $300,000.
Expected gross loss. Weight each outcome by its probability: \(0.50\times \$0+0.30\times \$100{,}000+0.20\times \$300{,}000=\$0+\$30{,}000+\$60{,}000=\$90{,}000\).
Expected ceded and retained losses. Expected ceded loss is \(0.50\times \$0+0.30\times \$40{,}000+0.20\times \$120{,}000=\$0+\$12{,}000+\$24{,}000=\$36{,}000\). Expected retained loss is \(0.50\times \$0+0.30\times \$60{,}000+0.20\times \$180{,}000=\$0+\$18{,}000+\$36{,}000=\$54{,}000\). The final check is \(\$36{,}000+\$54{,}000=\$90{,}000\), equal to expected gross loss. The method avoids confusing a loss outcome with its probability-weighted expected value.
What the Proportional Calculation Does—and Does Not—Say
The fixed-share equations determine the allocation once the subject amounts and treaty percentage are known. They do not establish whether a particular policy, claim, or premium is included in the treaty; that depends on the contract. Nor do they determine the appropriate premium to charge for the reinsurance. For this calculation, use the treaty’s stated subject premium amount and apply the agreed percentage.
Keep the accounting basis consistent. If the problem gives earned premium, calculate ceded and retained earned premium. If it gives written premium, calculate ceded and retained written premium. Do not combine one basis with the other, and do not infer a premium amount from a loss amount. Premium and loss are separate allocations, each of which must reconcile to its own gross amount.
For several claims, it is often clearest to calculate the ceded and retained parts of each claim and then total each column. If all claims have the same quota share, applying the percentage once to aggregate gross loss gives the same result. If different claims are not subject to the same percentage, that shortcut is not valid; calculate each treaty allocation on its applicable subject amount.
Determine whether the amount is gross premium or gross covered loss, and note the accounting period and basis stated in the problem.
Write \(\alpha\) as the share ceded to the reinsurer. The insurer’s retained share is \(1-\alpha\).
Multiply the gross subject amount by \(\alpha\) for the ceded amount and by \(1-\alpha\) for the retained amount.
Add ceded and retained amounts and verify that their sum equals the original gross amount, for premium and loss separately.
A frequent exam trap is to calculate only the reinsurer’s recovery and then report it as the insurer’s net loss. Under a 25% quota share, for instance, the reinsurer pays 25% of a covered loss; the insurer retains the other 75%. The recovery reduces the insurer’s gross loss, but the retained loss is not the recovery itself. Another error is to apply the cession percentage to premium but forget to apply it to covered losses, or vice versa. A reliable answer labels each amount as gross, ceded, or retained and checks the totals before proceeding.
Check Your Understanding
For each problem, identify the ceded share, calculate the retained share, and use the gross-to-ceded-to-retained reconciliation as a check.
- A hypothetical treaty cedes 35% of $120,000 gross written premium. A covered partial loss is $80,000. Calculate ceded and retained premium and loss. (Answer: premium ceded $42,000 and retained $78,000; loss ceded $28,000 and retained $52,000.)
- A 20% quota share applies to $15,000 of subject gross premium and a covered total loss of $15,000. Calculate all four ceded and retained amounts. (Answer: premium ceded $3,000 and retained $12,000; loss ceded $3,000 and retained $12,000.)
- A 40% treaty applies to two covered losses, $12,000 and $28,000, and to $90,000 of subject gross premium. Find the ceded and retained amounts, first by claim and then in aggregate. (Answer: premium ceded $36,000 and retained $54,000; aggregate loss ceded $16,000 and retained $24,000; the claim-level ceded losses are $4,800 and $11,200.)
- Annual gross covered loss \(X\) is $0 with probability 0.60, $40,000 with probability 0.30, and $200,000 with probability 0.10. A 25% quota share applies. Calculate expected gross, ceded, and retained losses. (Answer: expected gross loss $32,000; expected ceded loss $8,000; expected retained loss $24,000.)
- A hypothetical treaty cedes 30% of $250,000 in subject gross earned premium. During the period, covered losses are $18,000, $62,000, and $120,000. Calculate ceded and retained premium and aggregate loss, and verify the loss allocation by claim. (Answer: premium ceded $75,000 and retained $175,000; aggregate loss ceded $60,000 and retained $140,000; claim-level ceded losses are $5,400, $18,600, and $36,000.)