Quota Share Cession and Retention Percentages

Introduction

A quota share divides a specified portion of covered business between an insurer (the cedent) and a reinsurer. The central calculation is direct: apply the agreed cession percentage to the amount being shared, then assign the remainder to the cedent. The main complication is not the multiplication; it is identifying exactly which amount the percentage applies to and what a stated maximum limits.

This tutorial focuses on the cession and retention percentages and their application to individual covered premiums and losses. It assumes the amount being considered is within the treaty’s defined scope. It does not include adjustments such as ceding commissions or other treaty accounting items. The calculations and terminology are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: In a quota share arrangement, the cession percentage $q$ is the proportion of an eligible amount assigned to the reinsurer. The retention percentage is $1-q$, the proportion assigned to the cedent. For an eligible amount $z$, the ceded amount is $qz$, and the retained amount is $(1-q)z$, provided no limit changes the amount to which the percentage applies.

The Basic Quota Share Calculation

The cession and retention percentages are complementary: together they equal 100%. If a treaty cedes 35%, for example, the cedent retains 65%. Apply these percentages to each covered amount being considered. A covered premium and a covered loss are separate amounts, so calculate each separately; do not apply a percentage to the premium to find the loss share.

q+(1-q)=1,\qquad \text{ceded amount}=qz,\qquad \text{retained amount}=(1-q)z

Here, $z$ is the eligible premium or loss amount. For a percentage stated as $p\%$, use $q=p/100$ in the calculation. For example, 28% means $q=0.28$, not $q=28$. The same quota share percentage ordinarily applies to covered premiums and covered losses under the stated treaty terms, although the amount eligible for sharing must still be determined from the problem wording.

Worked Example: Dividing a Covered Premium and Loss

A hypothetical treaty cedes 35% of each covered premium and loss, with no stated maximum. For one risk, the covered premium is $2,400 and the covered loss is $6,800. Calculate the ceded and retained portions of both amounts.

Method. Convert the cession percentage to a decimal and use its complement for the retention percentage: $q=0.35$ and $1-q=0.65$.

Premium ceded. $0.35(2{,}400)=840$, so the reinsurer’s share of the premium is $840.

Premium retained. $0.65(2{,}400)=1{,}560$, so the cedent retains $1,560 of the premium.

Loss ceded. $0.35(6{,}800)=2{,}380$, so the reinsurer’s share of the loss is $2,380.

Loss retained. $0.65(6{,}800)=4{,}420$, so the cedent retains $4,420 of the loss.

Check. The premium portions add to $840+1{,}560=2{,}400$; the loss portions add to $2{,}380+4{,}420=6{,}800$. The common mistake this check catches is using the cession percentage for both parties rather than using complementary percentages.

Why the Two Shares Reconcile

The retained amount is not an independent estimate; it is the part of the original eligible amount that was not ceded. This gives a useful derivation and a reliable check on every calculation.

Start with the full eligible amount.
The ceded and retained portions must together account for $z$.

Assign the cession share.
The reinsurer receives $qz$.

Find the remainder.
The cedent retains $z-qz=(1-q)z$, by factoring out $z$.

Verify the total.
$qz+(1-q)z=z$, so the two portions exactly reconcile to the original amount.

This reasoning also explains why the retention percentage is the complement of the cession percentage, not a second percentage chosen independently. If the two percentages do not add to 100%, the stated division leaves an amount unassigned or allocates more than the full amount.

When a Maximum Amount Is Stated

A maximum can modify a quota share calculation in more than one way. Read the wording carefully: a maximum eligible amount limits the amount to which the percentage applies, while a maximum ceded amount limits the result after the percentage is applied. These are different treaty terms and generally produce different ceded amounts.

Key distinction: If only the first $M$ of an amount $z$ is eligible for sharing, the ceded amount is $q\min(z,M)$. If instead the ceded amount itself cannot exceed $C$, the ceded amount is $\min(qz,C)$. In either case, when the entire original amount remains the cedent’s responsibility except for the actual cession, retained amount equals $z-\text{ceded amount}$.

For an eligibility limit, any amount above $M$ is outside the quota share and remains with the cedent. For a cession cap, the quota share is first calculated on the amount specified by the treaty, but any amount above $C$ is not ceded and therefore remains with the cedent. A problem may state a maximum in other terms, such as a maximum insured amount or treaty capacity. Do not assume which interpretation applies: translate the stated condition into the amount it limits before calculating.

\begin{aligned} \text{Eligibility limit } M:\quad &\text{ceded}=q\min(z,M),& &\text{retained}=z-q\min(z,M),\\ \text{Ceded-amount cap } C:\quad &\text{ceded}=\min(qz,C),& &\text{retained}=z-\min(qz,C). \end{aligned}

Worked Example: A Maximum Eligible Amount

A hypothetical treaty cedes 40% of each covered loss, but only the first $12,000 of each loss is eligible for the quota share. A covered loss is $17,000. Find the ceded and retained amounts.

Method. First identify the eligible amount. Since the loss exceeds the $12,000 eligibility maximum, the eligible amount is $\min(17{,}000,12{,}000)=12{,}000$.

Ceded amount. Apply the cession percentage to the eligible amount: $0.40(12{,}000)=4{,}800$.

Retained amount. The cedent retains both its 60% share of the eligible portion and the portion outside the treaty. Equivalently, subtract the actual cession from the full loss: $17{,}000-4{,}800=12{,}200$.

Check. The eligible portion has $4,800 ceded and $12{,}000-4{,}800=7{,}200$ retained. The remaining $17{,}000-12{,}000=5{,}000$ is outside the quota share and retained in full. Thus retained is $7{,}200+5{,}000=12{,}200$, and $4{,}800+12{,}200=17{,}000$. The common mistake is to calculate $40\%$ of the entire loss, ignoring the stated eligibility maximum.

Worked Example: A Maximum Ceded Amount

A hypothetical treaty cedes 30% of a covered loss, subject to a maximum cession of $6,000 for that loss. The covered loss is $24,000. Calculate the actual cession and the cedent’s retention.

Method. Calculate the uncapped share first, then compare it with the maximum ceded amount.

Uncapped cession. $0.30(24{,}000)=7{,}200$.

Apply the cap. The uncapped amount exceeds $6,000, so the actual cession is $\min(7{,}200,6{,}000)=6{,}000$.

Retention. The cedent retains $24{,}000-6{,}000=18{,}000$.

Check. The ceded and retained amounts total $6{,}000+18{,}000=24{,}000$. If the 30% share had been used without the cap, retention would have been $16,800, but that result would violate the stated maximum cession. The cap is binding because the uncapped cession is greater than $6,000.

Common Exam Traps

The most frequent errors arise from applying the right percentage to the wrong amount. The following incorrect calculation illustrates why a maximum needs to be classified before it is used.

Common mistake: For a $17,000 loss, a 40% quota share, and a maximum eligible amount of $12,000, a candidate calculates $0.40(17{,}000)=6{,}800$ ceded and $10{,}200$ retained. This ignores the eligibility maximum, so the cession is too high by $6{,}800-4{,}800=2{,}000$. The correction is to apply 40% to $\min(17{,}000,12{,}000)=12{,}000$, giving $4,800 ceded and $17{,}000-4{,}800=12{,}200$ retained.

Other useful checks are to write the cession and retention percentages explicitly, keep the premium calculation separate from the loss calculation, and confirm that the amounts assigned to the two parties sum to the original amount. When a cap applies, compare the uncapped cession with the cap; do not apply the cap automatically if it is not binding.

Percentage conversion: Convert a stated percentage such as 45% to 0.45 before multiplying.
Complement: Use $1-q$ for the retention percentage. Do not mistakenly retain $q$ as well as cede $q$.
Limit interpretation: Check whether the maximum limits eligible amount, ceded amount, or some other stated basis.
Reconciliation: Verify that actual ceded amount plus retained amount equals the full amount under consideration.

Key takeaway: First identify the amount eligible for the quota share and the meaning of any maximum. Then calculate the cession, apply any applicable cap, and find retention as the original amount minus the actual cession.

Check Your Understanding

For each problem, identify what the percentage applies to and check that the ceded and retained amounts reconcile.

A hypothetical treaty cedes 25% of each covered premium and loss, with no maximum. For a premium of $3,600 and a loss of $12,400, calculate ceded and retained amounts for each. (Answer: Premium: $900 ceded and $2,700 retained; loss: $3,100 ceded and $9,300 retained.)
A treaty cedes 45% of each covered loss, but only the first $12,000 of each loss is eligible. A covered loss is $20,000. Find the ceded and retained amounts. (Answer: $5,400 ceded and $14,600 retained.)
A 70% quota share applies to a $15,000 covered amount, but the maximum ceded amount is $8,000. Find the actual cession and retention. (Answer: $8,000 ceded and $7,000 retained.)
A treaty cedes 32% of each covered amount, with no maximum. For a covered premium of $2,500 and a covered loss of $8,750, calculate both parties’ shares. (Answer: Premium: $800 ceded and $1,700 retained; loss: $2,800 ceded and $5,950 retained.)
A hypothetical treaty cedes 55% of a covered loss of $18,000, subject to a maximum cession of $9,000. Determine whether the maximum binds and calculate retention. (Answer: The uncapped cession is $9,900, so the maximum binds; $9,000 is ceded and $9,000 is retained.)

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