Facultative Reinsurance for Individual Risks

Introduction

An insurer may decide that one particular policy exposes it to more risk than it wants to retain. It can approach a reinsurer with that individual risk and ask whether the reinsurer will accept some of the exposure. Because the decision is made risk by risk, the arrangement is called facultative reinsurance.

The central calculation is an allocation: after the reinsurer accepts specified terms, determine the premium ceded to the reinsurer and the amount of a covered claim the reinsurer must reimburse. The insurer’s retained premium and loss are the corresponding amounts left with the insurer. The offer alone does not establish an allocation; use the terms that were actually accepted.

Definition: Facultative reinsurance is reinsurance negotiated for an individually identified risk. The insurer presents that risk to a reinsurer, and the reinsurer may accept, decline, or accept it on modified terms. The accepted agreement specifies the exposure covered, the reinsurer’s payment obligation, and the reinsurance premium.

Terms of an Individual Placement

The insurer seeking reinsurance is the ceding insurer, and the reinsurer is the party accepting the specified reinsurance obligation. A facultative submission describes the policy or risk being offered. The final agreement may cover only part of the policy’s exposure, and its terms need not copy every term of the original insurance policy.

For a calculation, distinguish three stages:

Identify the original policy amount.
Use the gross premium charged by the insurer and the covered claim amount payable under the original policy. The claim amount should already reflect the original policy’s applicable coverage terms.

Read the accepted reinsurance terms.
Identify the accepted share, layer, limit, premium formula, or other stated terms. Do not substitute the insurer’s requested terms if the reinsurer accepted something different.

Allocate premium and loss separately.
Calculate the reinsurance premium from its stated basis, then calculate the reinsurance recovery from the accepted claim terms. Subtract each ceded amount from its corresponding gross amount to find the insurer’s retention.

Let $G$ denote the original policy’s gross premium, $C$ the reinsurance premium, $L$ the covered claim payable by the insurer before reinsurance, and $R(L)$ the reinsurer’s payment for that claim. In the simple setting used here, with no ceding commission, expenses, or other adjustments:

\text{retained premium}=G-C,\qquad \text{retained claim}=L-R(L).

These balances are useful checks, but they do not determine $C$ or $R(L)$. Those amounts must come from the accepted contract. In particular, a stated percentage of claim payments does not automatically imply the same percentage of the original premium is ceded.

Key distinction: The ceded premium is the amount the ceding insurer pays for reinsurance. The ceded loss, or recovery, is the amount the reinsurer pays toward a covered claim. Calculate them from their respective contractual terms; do not assume they are equal percentages unless the agreement says so.

Calculating an Accepted Share

One possible individual-risk agreement assigns the reinsurer a stated fraction of covered claim payments. If the accepted fraction is $c$, where $0\leq c\leq 1$, and the contract applies that fraction to the entire covered claim, then $R(L)=cL$. If the premium agreement also says that the reinsurer receives fraction $c$ of the original premium, then $C=cG$. Both conditions matter: the premium allocation follows the premium term, not automatically the claim formula.

Worked Example: An Accepted Share of One Policy

A hypothetical insurer writes an individual equipment policy with a gross annual premium of $6,000. A reinsurer accepts 35% of the covered claim payments and, separately, 35% of the original premium. A covered claim payable under the policy is $80,000. There are no commissions or other adjustments. Calculate ceded and retained premiums and claim amounts.

Method and assumption. Apply the accepted 35% share to both premium and loss because the agreement explicitly specifies both. A common mistake this avoids is using a claim share to infer a premium share when no premium term has been stated.

Ceded premium. The reinsurer receives 35% of $6,000:

$$ C=0.35(6{,}000)=2{,}100. $$

The insurer retains $6{,}000-2{,}100=\$3{,}900$ of premium.

Ceded claim. The reinsurer pays 35% of the $80,000 covered claim:

$$ R(L)=0.35(80{,}000)=28{,}000. $$

The insurer retains $80{,}000-28{,}000=\$52{,}000$ of the claim.

Check. The premium amounts reconcile: $2{,}100+3{,}900=6{,}000$. The claim amounts also reconcile: $28{,}000+52{,}000=80{,}000$.

Calculating an Individual-Risk Layer

A facultative agreement can instead specify a payment layer for the individual risk. Suppose the reinsurer pays losses above an attachment amount $A$, up to a maximum payment $U$. For a covered claim $L$, the recovery is

R(L)=\min\bigl(\max(L-A,0),U\bigr).

If $L\leq A$, there is no recovery. If $A<L<A+U$, the recovery is $L-A$. If $L\geq A+U$, the recovery is capped at $U$. This formula follows from the contract’s payment boundary: only the amount above $A$ is eligible, so the eligible amount is $\max(L-A,0)$; the reinsurer cannot pay more than $U$, so the final amount is the smaller of that eligible amount and $U$.

The insurer’s retained claim is $L-R(L)$. It consists of the amount below the attachment, plus any amount above the layer’s upper boundary. The reinsurance premium remains a separate calculation: it may be stated as a fixed amount or calculated by another explicit premium term.

Worked Example: A Facultative Layer With a Stated Premium

A hypothetical insurer has a policy with a gross premium of $42,000. For this individual risk, a reinsurer accepts a layer attaching at $400,000 and paying at most $250,000 per covered claim. The agreed reinsurance premium is a fixed $9,000. The insurer’s covered claim is $640,000. Calculate the recovery and both parties’ retained amounts.

Method and assumption. Apply the accepted layer to the covered claim and use the fixed premium as stated. This avoids the common error of calculating the premium as a percentage of the claim layer without contractual support.

Step 1: Amount above attachment.

$$ 640{,}000-400{,}000=240{,}000. $$

Step 2: Apply the layer limit. The amount above attachment, $240,000, is less than the maximum payment of $250,000. Therefore:

R(640{,}000)=\min(240{,}000,250{,}000)=240{,}000.

Step 3: Calculate retained claim.

$$ 640{,}000-240{,}000=400{,}000. $$

Step 4: Calculate premium amounts. The reinsurance premium is $9,000, so the insurer retains:

$$ 42{,}000-9{,}000=33{,}000. $$

Check. The reinsurer’s $240,000 recovery is within the $250,000 limit. The claim allocation reconciles to $640,000, and the premium allocation reconciles to $42,000.

Use the Accepted Terms, Not the Requested Terms

Facultative placements are negotiated for the individual risk. A reinsurer may accept a smaller share, a different layer, or a different premium from the terms the insurer first proposed. Once terms are changed, all calculations must use the accepted agreement. A useful exam habit is to mark the final accepted terms before beginning any arithmetic.

Also distinguish the basis used for the premium from the basis used for claim recovery. A contract might specify a percentage of the original premium but a different percentage of each covered claim. Alternatively, it might charge a fixed reinsurance premium while covering a claim layer. Neither arrangement is contradictory: the premium and payment provisions are separate contractual terms.

Worked Example: The Reinsurer Accepts a Different Share

An insurer offers a reinsurer 50% of an individual property risk. The reinsurer accepts only 40% of covered claim payments and charges a reinsurance premium equal to 30% of the original policy premium. The policy’s gross premium is $24,000, and a covered claim payable by the insurer is $175,000. Assume there are no commissions or other adjustments. Calculate the ceded and retained amounts.

Method and assumption. Use the accepted 40% claim share and the separately agreed 30% premium rate. This avoids treating the insurer’s original 50% offer as though it had been accepted, or incorrectly applying the claim share to the premium.

Ceded premium. The premium term is 30% of $24,000:

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

The insurer retains $24{,}000-7{,}200=\$16{,}800$ of premium.

Ceded claim. The accepted claim share is 40%, not the 50% initially offered:

$$ R(L)=0.40(175{,}000)=70{,}000. $$

The insurer retains $175{,}000-70{,}000=\$105{,}000$ of the claim.

Check. The reinsurer’s recovery is 40% of the claim, while its premium is 30% of the original premium, as specified. The totals reconcile: $7{,}200+16{,}800=24{,}000$ and $70{,}000+105{,}000=175{,}000$.

Common Mistakes and a Reliable Check

The most consequential errors usually come from confusing the proposed placement with the accepted one, or from mixing premium terms with claim terms. Use the following checks before finalizing a calculation:

Confirm acceptance. A submission or offer alone does not establish the reinsurer’s obligation. Use the final accepted terms.
Keep the units and bases clear. A premium percentage applies to its stated premium base; a claim percentage applies to its stated loss base.
Apply a layer to the covered claim. For a layer, account for the attachment and limit. Do not pay the reinsurer on the portion below attachment or above the covered layer.
Reconcile each allocation separately. Ceded premium plus retained premium should equal gross premium. Ceded claim plus retained claim should equal the covered claim before reinsurance.
Do not infer unprovided terms. If the reinsurance premium is not specified, a claim share alone is insufficient to calculate it.

These calculations are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. The key skill is not merely identifying that a risk has been reinsured; it is translating the accepted individual agreement into a precise premium allocation and claim recovery.

Check Your Understanding

For each hypothetical placement, use only the stated accepted terms. Premiums and claims have no commissions or other adjustments unless specified.

A reinsurer accepts 20% of both the original premium and covered claim payments for one policy. The gross premium is $9,500 and the covered claim is $30,000. Find the ceded and retained premium and claim amounts. (Answer: ceded premium $1,900; retained premium $7,600; ceded claim $6,000; retained claim $24,000.)
For an individual risk, the accepted claim share is 30%, while the reinsurance premium is 25% of the original premium. The gross premium is $16,000 and the covered claim is $90,000. Find all four ceded and retained amounts. (Answer: ceded premium $4,000; retained premium $12,000; ceded claim $27,000; retained claim $63,000.)
A facultative layer for one risk attaches at $150,000 and has a maximum payment of $100,000. Its agreed premium is $3,500. The gross policy premium is $22,000 and the covered claim is $310,000. Find the recovery, retained claim, and retained premium. (Answer: recovery $100,000; retained claim $210,000; retained premium $18,500.)
An insurer initially offers 60% of a risk, but the reinsurer accepts 45% of covered claims and charges 40% of the original premium. The gross premium is $50,000 and the covered claim is $240,000. Find the ceded and retained amounts. (Answer: ceded premium $20,000; retained premium $30,000; ceded claim $108,000; retained claim $132,000.)
A reinsurer accepts a layer attaching at $500,000 with a maximum payment of $300,000. The reinsurance premium is 12% of the original premium, which is $75,000. A covered claim is $910,000. Find the recovery, retained claim, ceded premium, and retained premium. (Answer: recovery $300,000; retained claim $610,000; ceded premium $9,000; retained premium $66,000.)

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