How Quota Share Allocates Premiums and Claims
A quota share treaty divides specified premiums and claim costs between a ceding insurer and a reinsurer in fixed proportions. Once the ceded percentage is known, the same percentage applies to each covered premium amount and each covered claim amount, subject to the treaty’s terms. The calculation is straightforward, but a reversed percentage or an unclear definition of the amount subject to the treaty can change the result materially.
This tutorial assumes that the amounts supplied are covered by the treaty, that the treaty uses one fixed quota share percentage for them, and that no ceding commission or other adjustment is included. Premium and claim amounts are allocated separately; a premium share is not a claim recovery, and neither amount by itself describes the parties’ overall profit.
Here, “gross” means before this quota share allocation, not necessarily before every other adjustment that could apply to an insurance contract. If a problem supplies an amount already adjusted under the policy or another stated arrangement, use that amount as the treaty-subject premium or claim amount. Do not apply the quota share percentage to an amount outside the scope specified in the problem.
Allocation Formulas and a Reconciliation
The allocation can be written in a compact table. The ceded share is the part transferred to the reinsurer; the retained share remains with the ceding insurer.
| Amount being allocated | Reinsurer’s share | Ceding insurer’s share |
|---|---|---|
| Covered premium \(\Pi\) | \(q\Pi\) | \((1-q)\Pi\) |
| Covered claim \(C\) | \(qC\) | \((1-q)C\) |
The reason the same proportions apply to premium and claims is built into the quota share agreement: each party takes the stated fraction of the covered business’s premium and the same fraction of its covered claim amounts. Algebra also gives a useful check. Since \(r=1-q\), the two allocated amounts must add back to the original amount:
This conservation check is important in exam calculations. A correct allocation must satisfy both the treaty percentage and the original total. For example, if the ceded share is 35%, the retained share is 65%, not another 35%.
Worked Example: Allocating One Policy’s Premium and Claim
A hypothetical insurer has a quota share treaty that cedes 35% of each covered policy’s premium and claim amount. One covered policy has a treaty-subject annual premium of $2,400 and a covered claim of $1,600. Calculate the reinsurer’s and insurer’s shares of both amounts. Ignore commissions and other adjustments.
Method. Use \(q=0.35\) for the reinsurer and \(r=1-0.35=0.65\) for the ceding insurer. Apply each fraction separately to the premium and the claim.
Premium. The reinsurer receives \(0.35(2{,}400)=840\) dollars. The ceding insurer retains \(0.65(2{,}400)=1{,}560\) dollars. Check: \(840+1{,}560=2{,}400\).
Claim. The reinsurer bears \(0.35(1{,}600)=560\) dollars. The ceding insurer bears \(0.65(1{,}600)=1{,}040\) dollars. Check: \(560+1{,}040=1{,}600\).
Answer. The reinsurer receives $840 of premium and bears $560 of the claim. The ceding insurer retains $1,560 of premium and bears $1,040 of the claim. The common mistake avoided here is assigning the ceded percentage to the ceding insurer’s amounts.
Why the Same Share Applies to Each Amount
A quota share percentage describes a proportional division of the covered business. If the reinsurer accepts fraction \(q\) of the business, then, under the stated treaty terms, it receives fraction \(q\) of the premium and bears fraction \(q\) of each covered claim. The ceding insurer keeps the complementary fraction \(1-q\). The allocation follows directly by splitting any covered amount \(Z\) into two proportional parts:
Adding the parts gives \(qZ+(1-q)Z=[q+(1-q)]Z=Z\). Thus, there is no unexplained amount created or lost by the allocation. This is a calculation identity, not a statement that the two parties have identical expenses, cash-flow timing, or overall financial outcomes.
For example, the reinsurer’s allocated premium less its allocated claims is \(q\Pi-qC=q(\Pi-C)\), before any other items. The ceding insurer’s corresponding amount is \((1-q)\Pi-(1-q)C=(1-q)(\Pi-C)\). The treaty divides this simple premium-minus-claim result proportionally when both amounts are subject to the same share. Actual profitability may also depend on expenses, commissions, timing, and other contract terms that are not part of this calculation.
Worked Example: Allocating Several Covered Policies
A hypothetical insurer cedes 40% of the covered premium and claim amounts for a small portfolio. The portfolio has three covered policies with annual premiums of $1,500, $2,500, and $4,000. Their covered claims are $700, $1,200, and $2,100, respectively. Find the total ceded and retained premium and claim amounts.
Method. First total each category, then apply \(q=0.40\) to find the ceded amount and \(1-q=0.60\) to find the retained amount. Because every listed amount has the same treaty percentage, applying the percentage to the total gives the same result as allocating each policy separately.
Total premium. Add the premiums: \(1{,}500+2{,}500+4{,}000=8{,}000\) dollars. The reinsurer’s share is \(0.40(8{,}000)=3{,}200\) dollars. The ceding insurer’s share is \(0.60(8{,}000)=4{,}800\) dollars. Check: \(3{,}200+4{,}800=8{,}000\).
Total claims. Add the claims: \(700+1{,}200+2{,}100=4{,}000\) dollars. The reinsurer’s share is \(0.40(4{,}000)=1{,}600\) dollars. The ceding insurer’s share is \(0.60(4{,}000)=2{,}400\) dollars. Check: \(1{,}600+2{,}400=4{,}000\).
Answer. The reinsurer receives $3,200 of premium and bears $1,600 of claims. The ceding insurer retains $4,800 of premium and bears $2,400 of claims. The common mistake avoided is multiplying only one item in a multi-policy total by the quota share and leaving the other covered items unallocated.
When the Amounts Are Not the Same
The premium total and claim total will generally differ, so their dollar allocations will also differ even though the percentage is the same. A 30% quota share does not mean that the reinsurer receives 30% of premium dollars and 30% of the same dollar amount in claims; it means the reinsurer receives 30% of the covered premium total and bears 30% of the covered claim total.
If asked for a party’s premium-minus-claim amount, calculate its allocated premium and allocated claim separately, then subtract. Do not subtract the gross claim from the gross premium and assign the entire difference to one party. Under a single share applied to both amounts, the difference can also be checked by applying the share directly to the gross difference.
Worked Example: Comparing the Parties’ Premium-minus-Claim Amounts
A hypothetical book has $30,000 of covered premium and $18,000 of covered claims. Its quota share treaty cedes 25%. Calculate each party’s allocated premium and claims, then calculate each party’s premium minus claims before expenses and other adjustments.
Method. The reinsurer’s share is \(q=0.25\), and the ceding insurer’s share is \(1-0.25=0.75\). Allocate premium and claims first; then subtract within each party.
Reinsurer. Ceded premium is \(0.25(30{,}000)=7{,}500\) dollars. Ceded claims are \(0.25(18{,}000)=4{,}500\) dollars. The premium-minus-claim amount is \(7{,}500-4{,}500=3{,}000\) dollars.
Ceding insurer. Retained premium is \(0.75(30{,}000)=22{,}500\) dollars. Retained claims are \(0.75(18{,}000)=13{,}500\) dollars. The premium-minus-claim amount is \(22{,}500-13{,}500=9{,}000\) dollars.
Check. The gross premium-minus-claim amount is \(30{,}000-18{,}000=12{,}000\) dollars. The allocated amounts sum to \(3{,}000+9{,}000=12{,}000\). Equivalently, the reinsurer’s result is \(0.25(12{,}000)=3{,}000\), and the ceding insurer’s is \(0.75(12{,}000)=9{,}000\).
Answer. Before expenses and other adjustments, the reinsurer’s premium-minus-claim amount is $3,000 and the ceding insurer’s is $9,000. The common mistake avoided is treating either amount as profit without accounting for items excluded from the calculation.
Common Mistakes and Checks
- Reversing the percentages. If the treaty cedes 30%, the reinsurer gets 30% and the ceding insurer retains 70%. Always write \(r=1-q\) before calculating.
- Applying the percentage to the wrong amount. Confirm that the stated premium or claim is subject to the treaty before multiplying. An amount outside the treaty scope is not automatically allocated.
- Allocating premium but not claims. A quota share allocation includes both categories when both are covered by the stated treaty terms. Calculate them separately.
- Assuming a recovery equals a premium share. The reinsurer’s claim amount is its percentage of the covered claim, not its percentage of the premium.
- Failing to reconcile. Ceded plus retained premium must equal covered gross premium; ceded plus retained claims must equal covered gross claims.
- Calling a partial result profit. Premium less claims is only that difference before any excluded expenses, commissions, or other adjustments.
Check Your Understanding
For each problem, assume the stated amounts are covered by the treaty and that no commission or other adjustment applies.
- A treaty cedes 30% of a covered premium of $15,000 and a covered claim of $8,400. What are the ceded and retained amounts for each? (Answer: Premium: $4,500 ceded and $10,500 retained; claim: $2,520 ceded and $5,880 retained.)
- A 42% quota share applies to a $6,800 premium and a $9,100 claim. Calculate both parties’ shares of each amount. (Answer: Premium: $2,856 ceded and $3,944 retained; claim: $3,822 ceded and $5,278 retained.)
- A 25% quota share applies to two policies. Their premiums are $1,200 and $2,000, and their claims are $500 and $1,800. Find total ceded and retained premium and claims. (Answer: Premium: $800 ceded and $2,400 retained; claims: $575 ceded and $1,725 retained.)
- A treaty cedes 60% of a portfolio with $24,000 and $16,000 of covered premium, and $9,000 and $22,000 of covered claims. Find each party’s total premium, total claims, and premium-minus-claims amount. (Answer: Reinsurer: $24,000 premium, $18,600 claims, and $5,400 premium minus claims; ceding insurer: $16,000 premium, $12,400 claims, and $3,600 premium minus claims.)
- A reinsurer bears $13,500 of a covered claim of $30,000 under a quota share treaty. Find the ceded percentage, then allocate a covered premium of $48,000. (Answer: Ceded percentage: 45%; premium: $21,600 ceded and $26,400 retained.)