Facultative and Treaty Cessions in One Portfolio

Why the Allocation Order Matters

A portfolio may contain risks covered individually by facultative reinsurance and other risks protected under a treaty. To calculate the total ceded premium or loss, the insurer must first identify which contract applies to each risk. Applying the treaty to a facultatively placed risk as well can count the same exposure twice, unless the contracts explicitly provide for that result.

This tutorial uses hypothetical portfolios with fixed gross premiums and losses for a common accounting period. There is no interest, mortality, or probability model in the calculations: each premium, loss, and reinsurance term is treated as given. Unless an example says otherwise, premiums are compared on the same basis, and expenses, commissions, reinstatement premiums, and other adjustments are excluded.

Definition: Facultative reinsurance is arranged for an individually considered risk; the cession and terms apply to that risk as specified by its contract. A treaty applies automatically to risks meeting the treaty’s stated eligibility conditions. For a portfolio calculation, classify each risk by the contract that applies before adding ceded amounts.

Classify Risks Before Calculating

For each policy, record its gross premium, its gross loss for the period, whether facultative coverage applies, and—if it does not—whether it is eligible for the treaty. A risk that is outside both arrangements has no cession in this calculation. The relevant categories are therefore: facultative risks, treaty-eligible risks not assigned facultatively, and risks covered by neither arrangement.

For a proportional contract, the ceding percentage applies to both premium and loss when the contract specifies that same share for each. For example, if 40% of a risk is ceded under a proportional facultative contract, 40% of its stated premium and covered loss are ceded. A treaty quota share works the same way for each eligible policy, using the treaty’s quota-share percentage.

Proportional cession rule: If a contract cedes proportion $c$ of a policy’s premium $P_i$ and covered loss $L_i$, then ceded premium and loss are $cP_i$ and $cL_i$, respectively. The retained portions are $(1-c)P_i$ and $(1-c)L_i$. Apply this rule only to amounts and coverages included in that contract.

A useful portfolio method is to calculate each risk’s cession under its assigned arrangement, then add the cessions. In the common situation where facultative risks are excluded from the treaty, the two groups are disjoint. That separation is the key to avoiding double counting.

Classify each risk.
Mark whether facultative coverage applies. If not, test the policy against the treaty’s eligibility conditions.

Apply the correct contract terms.
Use the facultative terms for facultative risks and the treaty terms for treaty-eligible risks. Do not apply a treaty to an excluded risk.

Add cessions and determine retained amounts.
Sum the ceded premiums and losses across the portfolio. For each amount, retained equals gross less the cession.

Why the Portfolio Calculation Works

Suppose every risk is assigned to exactly one of three nonoverlapping groups: facultative group $F$, treaty group $T$, or uncovered group $O$. If facultative cessions are proportional, with share $c_i$ for risk $i$, and the treaty is quota share $q$, then the total ceded premium is the sum of the facultative cessions for $F$ and the treaty cessions for $T$. The same partition applies to losses.

This follows directly by summing the policy-level amounts. For premiums, each facultative risk contributes $c_iP_i$, each treaty risk contributes $qP_i$, and each uncovered risk contributes zero. Since the groups do not overlap, no policy’s premium is included in both sums.

P_{\text{ceded}} = \sum_{i\in F} c_iP_i + \sum_{i\in T} qP_i, \qquad L_{\text{ceded}} = \sum_{i\in F} c_iL_i + \sum_{i\in T} qL_i.

For a nonproportional facultative contract, the loss cession is determined by its attachment and limit, rather than by multiplying the loss by a share. Its ceded premium is also determined by the contract; it cannot generally be inferred by applying the loss recovery percentage to the gross premium.

Worked Example: Proportional Facultative Cessions and a Quota-Share Treaty

A hypothetical insurer has four policies for one accounting period. Policy A has gross premium $1,200 and gross loss $800; it has 50% facultative coverage. Policy B has gross premium $2,000 and gross loss $2,500; it has 40% facultative coverage. Policies C and D are not facultatively covered and are eligible for a 25% quota-share treaty. C has premium $1,600 and loss $400; D has premium $1,200 and loss $2,000. Calculate total ceded and retained premiums and losses. All listed losses are covered losses for the stated arrangements.

Method. Apply each facultative share to its own policy, and apply the 25% treaty share only to C and D. The contract groups are disjoint, so no policy receives two cessions.

Facultative premium cessions. For A, $0.50(1{,}200)=600$. For B, $0.40(2{,}000)=800$. The facultative ceded premium is $600+800=\$1{,}400$.

Facultative loss cessions. For A, $0.50(800)=400$. For B, $0.40(2{,}500)=1{,}000$. The facultative ceded loss is $400+1{,}000=\$1{,}400$.

Treaty premium cessions. For C, $0.25(1{,}600)=400$. For D, $0.25(1{,}200)=300$. The treaty ceded premium is $400+300=\$700$.

Treaty loss cessions. For C, $0.25(400)=100$. For D, $0.25(2{,}000)=500$. The treaty ceded loss is $100+500=\$600$.

Portfolio totals. Total ceded premium is $1{,}400+700=\$2{,}100$. Gross premium is $1{,}200+2{,}000+1{,}600+1{,}200=\$6{,}000$, so retained premium is $6{,}000-2{,}100=\$3{,}900$. Total ceded loss is $1{,}400+600=\$2{,}000$. Gross loss is $800+2{,}500+400+2{,}000=\$5{,}700$, so retained loss is $5{,}700-2{,}000=\$3{,}700$.

Check. Ceded plus retained premium is $2{,}100+3{,}900=6{,}000$, and ceded plus retained loss is $2{,}000+3{,}700=5{,}700$. The totals match the gross amounts. A common error would be to include A and B in the treaty calculation as well as their facultative cessions; that would apply two contracts to those policies despite the stated treaty group.

Treaty Eligibility Is a Separate Test

A policy that is not facultatively covered is not automatically treaty-eligible. Treaty wording may restrict eligibility by policy type, location, attachment point, or other stated criteria. In a calculation, use the eligibility information provided rather than assuming that every remaining policy belongs to the treaty.

Worked Example: Excluding a Non-Facultative Risk From the Treaty

A hypothetical portfolio contains four policies. Policy M has premium $1,800 and loss $1,200, with 40% facultative coverage. Policies N and P are not facultative and are eligible for a 25% quota-share treaty: N has premium $2,400 and loss $900, while P has premium $1,600 and loss $2,000. Policy O has premium $1,000 and loss $1,600, is not facultative, and is explicitly ineligible for the treaty. Find total ceded and retained premiums and losses.

Method. Treat M under its facultative contract, N and P under the treaty, and O as uncovered by either arrangement. In particular, O has zero cession here because the problem states it is treaty-ineligible.

Facultative cessions for M. Ceded premium is $0.40(1{,}800)=\$720$, and ceded loss is $0.40(1{,}200)=\$480$.

Treaty cessions for N and P. Ceded premium is $0.25(2{,}400)+0.25(1{,}600)=600+400=\$1{,}000$. Ceded loss is $0.25(900)+0.25(2{,}000)=225+500=\$725$.

Portfolio totals. Total ceded premium is $720+1{,}000=\$1{,}720$. Gross premium is $1{,}800+2{,}400+1{,}000+1{,}600=\$6{,}800$, giving retained premium $6{,}800-1{,}720=\$5{,}080$. Total ceded loss is $480+725=\$1{,}205$. Gross loss is $1{,}200+900+1{,}600+2{,}000=\$5{,}700$, giving retained loss $5{,}700-1{,}205=\$4{,}495$.

Check. The ceded and retained amounts sum to the gross amounts: $1{,}720+5{,}080=6{,}800$ for premium and $1{,}205+4{,}495=5{,}700$ for loss. The common mistake is to place O in the treaty group merely because it has no facultative coverage. Eligibility must be established independently.

Facultative Excess-of-Loss With a Proportional Treaty

Facultative coverage need not be proportional. For an excess-of-loss contract attaching at $A$ with limit $U$, the recovery on a covered loss $L$ is the amount of loss above $A$, capped at $U$. Equivalently, it is $\min(\max(L-A,0),U)$. Use the contract’s stated definition of attachment and limit; here, $U$ is the maximum amount recoverable.

The premium for that facultative cover is a separate contract amount. Unless the problem supplies a rating rule and the information needed to use it, calculate the ceded premium from the stated facultative premium, not from the loss recovery formula.

Worked Example: Facultative Excess-of-Loss Alongside a Quota-Share Treaty

A hypothetical insurer has four policies. Policy R has gross premium $900 and gross loss $1,300. Its facultative contract covers $500 excess of $500, and its ceded premium is $210. Policies S and T are treaty-eligible but not facultative: S has premium $1,500 and loss $600; T has premium $1,100 and loss $1,900. The treaty cedes 30% of eligible premiums and losses. Policy V has premium $700 and loss $1,000, but is treaty-ineligible and has no facultative cover. Calculate total ceded and retained premiums and losses.

Method. Calculate R’s facultative recovery using its attachment and limit. Use the specified ceded premium for R. Apply the treaty’s 30% share to S and T only; V has no cession.

Facultative loss for R. The amount above the $500 attachment is $1{,}300-500=\$800$. The limit is $500, so the recovery is $\min(800,500)=\$500$. Its retained loss is $1{,}300-500=\$800$.

Facultative premium for R. The contract states a ceded premium of $210. Its retained premium is $900-210=\$690$. Do not apply the loss recovery percentage to R’s premium: the recovery is determined by the loss layer, while the premium is separately stated.

Treaty cessions. For S and T, ceded premium is $0.30(1{,}500)+0.30(1{,}100)=450+330=\$780$. Ceded loss is $0.30(600)+0.30(1{,}900)=180+570=\$750$.

Portfolio totals. Total ceded premium is $210+780=\$990$. Gross premium is $900+1{,}500+1{,}100+700=\$4{,}200$, so retained premium is $4{,}200-990=\$3{,}210$. Total ceded loss is $500+750=\$1{,}250$. Gross loss is $1{,}300+600+1{,}900+1{,}000=\$4{,}800$, so retained loss is $4{,}800-1{,}250=\$3{,}550$.

Check. Premium totals satisfy $990+3{,}210=4{,}200$, and loss totals satisfy $1{,}250+3{,}550=4{,}800$. The main trap is to treat the facultative recovery as a percentage of R’s loss and then use that percentage on its premium. This is an excess-of-loss contract, and its premium is given independently.

Common Allocation Errors

Using the wrong portfolio group. A risk is not treaty-eligible merely because it lacks facultative coverage. Check the stated treaty conditions.
Ceding a risk twice. If the problem assigns a risk to facultative coverage and excludes it from the treaty, do not also apply the treaty share to that risk.
Applying a proportional formula to an excess layer. A layer recovery depends on the loss amount, attachment, and limit; it is not generally a fixed percentage of the gross loss.
Inferring a facultative premium from a recovery. A nonproportional loss recovery does not determine the premium unless a premium rule is explicitly supplied.
Mixing accounting bases. Premiums must refer to a consistent period and basis before being added. A mismatch between written and earned premiums, for example, cannot be repaired by the cession arithmetic.

Key takeaway: Classify policies first, apply each contract only to its assigned risks and covered amounts, and then sum the cessions. For a proportional arrangement, use the specified share; for a facultative layer, calculate the recovery from its attachment and limit and use the separately stated premium terms.

Check Your Understanding

For each problem, identify the applicable contract before calculating the requested cession. All amounts are hypothetical and in dollars.

Policy A has premium $2,500 and loss $1,800, with 40% facultative coverage. Policy B has premium $3,000 and loss $5,000 and is treaty-eligible under a 20% quota-share treaty. What are the total ceded premium and loss? (Answer: ceded premium $1,600; ceded loss $1,720)
Policy C has premium $4,000 and loss $2,500 with 25% facultative coverage. Policy D has premium $2,000 and loss $1,000 and is treaty-eligible under a 35% quota-share treaty. Policy E has premium $1,000 and loss $300 but is ineligible for the treaty and has no facultative coverage. Find total ceded premium and loss. (Answer: ceded premium $1,700; ceded loss $975)
A facultative risk has premium $1,800, loss $1,200, and a 50% proportional cession. Two other risks are treaty-eligible under a 40% quota-share treaty: their respective premiums are $2,500 and $1,000, and their losses are $3,000 and $1,800. Find total ceded premium and loss. (Answer: ceded premium $2,300; ceded loss $2,520)
A facultative contract covers $800 excess of $400 on a risk with gross loss $1,900 and specifies ceded premium $175. Two separate treaty-eligible risks have combined premium $2,400 and combined loss $1,600 under a 25% quota-share treaty. What are total ceded premium and loss? (Answer: ceded premium $775; ceded loss $1,200)

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