How a Quota-Share Treaty Divides Risk
A quota-share treaty transfers a fixed proportion of specified business to a reinsurer. Because the proportion is fixed, each covered loss and the corresponding gross premium are divided using the same ceded percentage. The insurer retains the balance. This proportional allocation is useful to calculate carefully: a correct answer must identify the treaty share, apply it to the right amount, and reconcile the two parties’ shares to the original total.
The examples use hypothetical insurance arrangements. Unless an example states otherwise, assume the treaty applies to every loss and premium amount shown, there are no exclusions or limits, and there is no ceding commission. A quota share does not by itself specify which business is covered; the treaty terms determine that. Always check that the amount in the problem is subject to the treaty before allocating it.
The ceded percentage and retained percentage are complementary: \(c+r=1\). For example, a 30% quota share means that the reinsurer takes 30% and the insurer retains 70%. It does not mean that the insurer retains 30%. State explicitly which percentage is ceded before calculating.
Why the Allocation Formula Works
The treaty assigns each covered amount in the same fixed proportions. For a loss \(X\), the two assigned amounts are \(cX\) and \((1-c)X\). Their sum is the entire loss:
The same reasoning applies to a gross premium \(P\). Thus proportional allocation preserves the total amount: it divides a loss or premium but does not create or remove any part of that amount. This identity is also a practical check on arithmetic. If the ceded and retained amounts do not sum to the original amount, either a percentage or a calculation is wrong.
A quota share is applied to the amount specified by the treaty. If the treaty applies to each covered claim, multiplying each covered claim by \(c\) gives its ceded amount; adding those ceded amounts gives the total ceded losses. Since the percentage is fixed, the same result follows by first totaling the covered losses and then multiplying that total by \(c\).
This equality is an allocation identity, not a claim about the probability distribution of aggregate losses. It requires that the same share \(c\) apply to all the listed covered amounts. If different business segments have different shares, allocate each segment using its own treaty percentage before combining the results.
Worked Example: Allocating Claims and Gross Premium
A hypothetical insurer has a quota-share treaty under which 35% of each covered loss and 35% of the stated gross premium are ceded. Three covered claims are $8,000, $12,500, and $0. The gross premium subject to the treaty is $24,000. Find the ceded and retained losses and premiums.
Method. Use \(c=0.35\) and \(r=1-0.35=0.65\). Apply each percentage to the covered amount. No commission is specified, so report gross premium shares rather than a net remittance.
For the $8,000 claim, ceded loss is \(0.35(8{,}000)=\$2{,}800\), and retained loss is \(0.65(8{,}000)=\$5{,}200\). The check is \(\$2{,}800+\$5{,}200=\$8{,}000\).
For the $12,500 claim, ceded loss is \(0.35(12{,}500)=\$4{,}375\), and retained loss is \(0.65(12{,}500)=\$8{,}125\). The check is \(\$4{,}375+\$8{,}125=\$12{,}500\). The zero claim produces zero ceded and retained loss.
Total covered loss is \(\$8{,}000+\$12{,}500+\$0=\$20{,}500\). Total ceded loss is \(\$2{,}800+\$4{,}375+\$0=\$7{,}175\). Total retained loss is \(\$5{,}200+\$8{,}125+\$0=\$13{,}325\). The total check is \(\$7{,}175+\$13{,}325=\$20{,}500\).
Ceded premium is \(0.35(\$24{,}000)=\$8{,}400\), and retained premium is \(0.65(\$24{,}000)=\$15{,}600\). These sum to \(\$24{,}000\). The common mistake to avoid is using 35% as the retained share: that would reverse the parties’ allocations.
Loss Allocation and Premium Allocation Are Related, Not Identical Cash Flows
The same quota-share percentage applies to covered losses and to the gross premium specified by the treaty, but the two allocations answer different questions. Loss allocation identifies each party’s share of covered claim payments. Premium allocation identifies the share of the stated gross premium assigned to the reinsurer and the share retained by the insurer. A stated gross premium is not automatically the same as a net premium remittance when the treaty includes a ceding commission.
A ceding commission is a separate treaty payment, commonly made by the reinsurer to the insurer under the treaty terms. It does not change the gross ceded premium share. If a commission is a specified fraction \(k\) of ceded premium, its amount is \(k(cP)\). The reinsurer’s net premium remittance after that commission is \(cP-k(cP)\), assuming that is the cash-flow convention specified. Do not subtract a commission unless the problem provides one.
Worked Example: Gross Ceded Premium and a Commission
A hypothetical insurer has a 40% quota-share treaty. The covered claims are $4,600, $8,250, and $12,150, and the gross premium subject to the treaty is $186,000. The reinsurer pays a ceding commission equal to 5% of ceded premium. Calculate the ceded and retained losses, the gross premium shares, the commission, and the net premium remittance to the reinsurer.
Method. First allocate covered losses and gross premium using \(c=0.40\) and \(r=0.60\). Then calculate the commission from the gross ceded premium. Keeping these steps separate avoids treating the commission as a change to the quota-share percentage.
Total covered losses are \(\$4{,}600+\$8{,}250+\$12{,}150=\$25{,}000\). Ceded losses are \(0.40(\$25{,}000)=\$10{,}000\), and retained losses are \(0.60(\$25{,}000)=\$15{,}000\). The check is \(\$10{,}000+\$15{,}000=\$25{,}000\).
Gross ceded premium is \(0.40(\$186{,}000)=\$74{,}400\). Gross retained premium is \(0.60(\$186{,}000)=\$111{,}600\). These shares reconcile to \(\$74{,}400+\$111{,}600=\$186{,}000\).
The commission is \(0.05(\$74{,}400)=\$3{,}720\). Under the stated convention, net premium remittance to the reinsurer is \(\$74{,}400-\$3{,}720=\$70{,}680\). The gross ceded premium remains \(\$74{,}400\); the commission is a separate cash flow. The answer is checked by confirming that the commission is 5% of the ceded premium, not 5% of the full \(\$186{,}000\) gross premium.
Separate Treaty Shares Must Be Allocated Separately
A portfolio can contain distinct groups of business subject to different quota-share percentages. In that case, there is no single treaty percentage for the combined portfolio unless the problem explicitly establishes one. Calculate ceded and retained amounts for each group using its own share, then add the group totals. An unweighted average of the percentages generally gives an incorrect allocation because the groups may have different premiums or losses.
Worked Example: Combining Two Groups With Different Shares
A hypothetical insurer has two business groups, each covered by a different quota-share treaty. Group A has a 25% ceded share, gross premium of $18,400, and covered claims of $5,600 and $12,000. Group B has a 40% ceded share, gross premium of $31,500, and covered claims of $9,250 and $3,750. Find the ceded and retained totals for losses and gross premium.
Method. Allocate each group separately, then combine the dollar amounts. There is no commission or other adjustment.
For Group A, total covered loss is \(\$5{,}600+\$12{,}000=\$17{,}600\). Ceded loss is \(0.25(\$17{,}600)=\$4{,}400\), and retained loss is \(0.75(\$17{,}600)=\$13{,}200\). Group A gross ceded premium is \(0.25(\$18{,}400)=\$4{,}600\); retained premium is \(0.75(\$18{,}400)=\$13{,}800\). Each pair sums to its group total.
For Group B, total covered loss is \(\$9{,}250+\$3{,}750=\$13{,}000\). Ceded loss is \(0.40(\$13{,}000)=\$5{,}200\), and retained loss is \(0.60(\$13{,}000)=\$7{,}800\). Group B gross ceded premium is \(0.40(\$31{,}500)=\$12{,}600\); retained premium is \(0.60(\$31{,}500)=\$18{,}900\). These amounts also reconcile to their group totals.
Combined covered losses are \(\$17{,}600+\$13{,}000=\$30{,}600\). Total ceded losses are \(\$4{,}400+\$5{,}200=\$9{,}600\), and total retained losses are \(\$13{,}200+\$7{,}800=\$21{,}000\). The check is \(\$9{,}600+\$21{,}000=\$30{,}600\).
Combined gross premium is \(\$18{,}400+\$31{,}500=\$49{,}900\). Total ceded premium is \(\$4{,}600+\$12{,}600=\$17{,}200\), and total retained premium is \(\$13{,}800+\$18{,}900=\$32{,}700\). The check is \(\$17{,}200+\$32{,}700=\$49{,}900\).
A tempting but incorrect shortcut is to average 25% and 40% to obtain 32.5%, then apply that percentage to the combined losses or premium. The groups have different amounts and different treaty shares, so that unweighted average does not preserve their allocations. The correct method is to allocate by group and sum.
A Reliable Exam Calculation Sequence
For a quota-share calculation, use the following sequence. It keeps the share definition, covered amount, and any separate cash-flow adjustment distinct.
Determine which losses and which gross premium are subject to the treaty. Do not allocate amounts that the stated treaty excludes.
Set \(c\) equal to the ceded fraction and calculate \(r=1-c\). Label each percentage as ceded or retained.
Multiply covered losses and stated gross premium by the appropriate fractions. If groups have different shares, do this group by group.
Calculate any commission or other specified payment from its stated base; do not alter the gross quota-share allocation.
Check that ceded plus retained losses equal covered losses, and ceded plus retained gross premium equal gross premium.
Common Mistakes and Why the Topic Matters
Quota-share calculations are direct, but errors often arise from misreading which party’s percentage is stated or which amount forms the calculation base. A treaty described as “40% ceded” assigns 40% to the reinsurer and 60% to the insurer. A commission stated as a percentage of ceded premium uses \(cP\), not the full gross premium \(P\). If a problem gives several segments, each treaty percentage must stay attached to its segment.
- Reversing the shares: If 30% is ceded, the insurer retains 70%, not 30%.
- Allocating an inapplicable amount: Confirm that the loss or premium is within the treaty’s stated scope.
- Confusing gross ceded premium with net remittance: A commission affects the cash remittance only when specified; it does not rewrite the gross premium share.
- Averaging different treaty percentages: Allocate each group separately unless a valid weighted share has been explicitly established.
- Skipping the reconciliation: Check that both parties’ shares add back to the covered loss or gross premium.
The allocation identifies how covered losses and gross premium are divided between the insurer and reinsurer. This is a basic cash-flow calculation for proportional reinsurance and is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus. It should not be confused with calculating aggregate moments or estimating expected cessions, which require additional model information.
Check Your Understanding
For each hypothetical arrangement, calculate the requested allocation and check that the ceded and retained amounts sum to the stated total.
- A 20% quota share applies to a covered claim of $17,850 and gross premium of $42,500. Find ceded and retained amounts for the claim and premium. (Answer: Ceded loss $3,570; retained loss $14,280; ceded premium $8,500; retained premium $34,000.)
- A 35% quota share applies to covered claims of $3,200, $6,800, and $0, and gross premium of $76,000. Find total ceded and retained losses and premium. (Answer: Ceded loss $3,500; retained loss $6,500; ceded premium $26,600; retained premium $49,400.)
- Group A has covered loss of $20,000, gross premium of $50,000, and a 30% ceded share. Group B has covered loss of $10,000, gross premium of $20,000, and a 50% ceded share. Find combined ceded and retained losses and premium. (Answer: Ceded loss $11,000; retained loss $19,000; ceded premium $25,000; retained premium $45,000.)
- A 25% quota share applies to gross premium of $80,000. The ceding commission is 10% of ceded premium. Find gross ceded premium, gross retained premium, commission, and net premium remittance to the reinsurer. (Answer: Gross ceded premium $20,000; gross retained premium $60,000; commission $2,000; net remittance $18,000.)
- A 40% quota share applies to two covered claims of $7,500 and $2,500, while gross premium subject to the treaty is $55,000. Find the ceded and retained totals, then verify both reconciliations. (Answer: Ceded loss $4,000; retained loss $6,000; ceded premium $22,000; retained premium $33,000; totals reconcile to $10,000 in losses and $55,000 in premium.)