Allocating Surplus Share Premiums and Claims

Why Proportional Allocation Matters

A surplus share treaty divides each covered risk between the insurer and the reinsurer according to the amounts each retains and cedes. Once those shares are known, the same proportions generally determine how the risk’s gross premium and covered claims are divided. Applying the percentages consistently makes the allocation auditable: the insurer’s amount plus the reinsurer’s amount must equal the original premium or covered claim.

The key is to use the risk’s total sum insured as the denominator when calculating the proportions. The retention is the amount kept by the insurer; the amount above it is ceded, subject to the treaty’s terms and available capacity. The examples below assume the treaty can accept the stated cession and that the premium and claim are subject to the same proportional shares.

Definition: For a risk with sum insured $S$ and insurer retention $R$, where $0 \leq R \leq S$, the retained amount is $R$ and the ceded amount is $S-R$. If $S>0$, the retained and ceded proportions are $$ r=\frac{R}{S}, \qquad c=\frac{S-R}{S}. $$ Thus $r+c=1$. If $S \leq R$, the risk is fully retained and $c=0$. For gross premium $P$ and treaty-covered claim payment $L$, the proportional allocation is $rP$ and $cP$ for premium, and $rL$ and $cL$ for the claim.

Calculating the Shares and Applying Them

The retained proportion is not the retention divided by the ceded amount. Both proportions are fractions of the same total sum insured. For example, if the sum insured is $500,000 and the retention is $100,000, the insurer retains $100{,}000/500{,}000=20\%$ of the risk. The remaining $400,000 is ceded, so the reinsurer’s share is $400{,}000/500{,}000=80\%$.

These percentages allocate the gross premium before any separate ceding commission or other adjustment. They also allocate the amount of a claim that the policy actually pays and that is covered by the treaty. If a deductible or policy limit determines the covered payment, first identify that payment; then apply the treaty shares. Do not allocate an amount the policy does not pay.

P_{\text{retained}}=rP,\qquad P_{\text{ceded}}=cP

L_{\text{retained}}=rL,\qquad L_{\text{ceded}}=cL

These relationships assume a proportional surplus share arrangement for the risk in question: the reinsurer’s share of premium follows its share of the insured amount, and its share of a covered claim follows that same share. A separate commission, exclusion, limit, or other treaty provision could affect the final settlement; it should be applied only when the problem specifies it.

Why the Same Proportion Applies

The allocation follows from dividing the insured amount into two parts. If premium is allocated in proportion to insured amount, the premium per dollar of sum insured is $P/S$. The insurer’s share is that rate multiplied by its retained amount $R$; the reinsurer’s share is the same rate multiplied by the ceded amount $S-R$.

\frac{P}{S}R=P\frac{R}{S}=rP, \qquad \frac{P}{S}(S-R)=P\frac{S-R}{S}=cP

For a covered claim, the treaty assigns each dollar of the covered payment according to the same retained and ceded proportions. Therefore, a payment of $L$ gives the insurer $rL$ and the reinsurer $cL$. In both allocations, the parts add back to the original amount because $r+c=1$:

rP+cP=(r+c)P=P,\qquad rL+cL=(r+c)L=L.

This derivation also gives a useful check. If the allocated premium does not add to the original gross premium, or the allocated claim does not add to the covered payment, there is an arithmetic or setup error.

Worked Example: Allocating One Risk’s Premium and Claim

A hypothetical commercial property has a sum insured of $500,000. The surplus share treaty retains $100,000 on this risk, and the treaty can accept the full excess. The gross premium is $2,400. A covered claim payment is $75,000. Allocate the premium and claim.

Method. Find the retained and ceded proportions from the sum insured, then apply both proportions to the gross premium and covered claim.

Step 1: Find the ceded amount. The insurer retains $100,000, so the ceded amount is $500{,}000-100{,}000=\$400{,}000$.

Step 2: Find the proportions. The retained proportion is $100{,}000/500{,}000=0.20$, or 20%. The ceded proportion is $400{,}000/500{,}000=0.80$, or 80%.

Step 3: Allocate the premium. The insurer’s premium share is $0.20(2{,}400)=\$480$. The reinsurer’s share is $0.80(2{,}400)=\$1{,}920$.

Step 4: Allocate the covered claim. The insurer’s claim share is $0.20(75{,}000)=\$15{,}000$. The reinsurer’s share is $0.80(75{,}000)=\$60{,}000$.

Check. Premium shares total $480+1{,}920=\$2{,}400$; claim shares total $15{,}000+60{,}000=\$75{,}000$. The calculation avoids treating the $400,000 cession as the denominator for the percentages.

Allocating Several Risks

In a portfolio, calculate the proportions risk by risk when the sum insured or retention differs across risks. Do not apply one portfolio-wide percentage unless the treaty terms and the portfolio data establish that a single percentage is appropriate. After calculating each risk’s allocations, add the retained amounts and ceded amounts separately.

Worked Example: Two Risks With Different Shares

A hypothetical insurer has two risks under a surplus share treaty with a $150,000 retention on each. The treaty can accept the full cession for both. Risk A has a $300,000 sum insured, a gross premium of $1,800, and a covered claim of $48,000. Risk B has a $600,000 sum insured, a gross premium of $3,600, and a covered claim of $90,000. Find the separate allocations and portfolio totals.

Method. Calculate each risk’s proportions independently, allocate its premium and covered claim, and then sum the results.

Risk A proportions. Its ceded amount is $300{,}000-150{,}000=\$150{,}000$. The retained proportion is $150{,}000/300{,}000=0.50$, and the ceded proportion is 0.50.

Risk A allocations. Retained premium: $0.50(1{,}800)=\$900$. Ceded premium: $0.50(1{,}800)=\$900$. Retained claim: $0.50(48{,}000)=\$24{,}000$. Ceded claim: $0.50(48{,}000)=\$24{,}000$.

Risk B proportions. Its ceded amount is $600{,}000-150{,}000=\$450{,}000$. The retained proportion is $150{,}000/600{,}000=0.25$, and the ceded proportion is 0.75.

Risk B allocations. Retained premium: $0.25(3{,}600)=\$900$. Ceded premium: $0.75(3{,}600)=\$2{,}700$. Retained claim: $0.25(90{,}000)=\$22{,}500$. Ceded claim: $0.75(90{,}000)=\$67{,}500$.

Portfolio totals. Retained premium is $900+900=\$1{,}800$, and ceded premium is $900+2{,}700=\$3{,}600$. Retained claims total $24{,}000+22{,}500=\$46{,}500$, and ceded claims total $24{,}000+67{,}500=\$91{,}500$.

Check. Total gross premium is $1{,}800+3{,}600=\$5{,}400$, matching the two premium allocations $1{,}800+3{,}600=\$5{,}400$. Total covered claims are $48{,}000+90{,}000=\$138{,}000$, matching $46{,}500+91{,}500=\$138{,}000$. Using Risk A’s 50% share for Risk B would be incorrect because the two risks have different sums insured.

Allocate the Covered Payment, Not an Unpaid Loss

The amount allocated as a claim is the amount payable under the policy and covered by the reinsurance arrangement, not automatically the ground-up loss. For example, if a policy has a deductible, the covered payment may be less than the loss amount. The policy calculation comes first; the surplus share allocation comes second. This separates two distinct questions: what does the policy pay, and how is that payment divided between insurer and reinsurer?

Worked Example: Applying the Treaty After a Deductible

A hypothetical property has a $400,000 sum insured, a $160,000 surplus share retention, and a $6,400 gross premium. The policy has a $12,500 deductible. A covered ground-up loss is $112,500, and assume no policy limit reduces the payment. The treaty can accept the full cession. Allocate the premium and claim.

Method. First determine the policy payment after the deductible. Then apply the treaty’s proportions to that payment and to the gross premium.

Step 1: Determine the covered payment. The policy payment is $112{,}500-12{,}500=\$100{,}000$.

Step 2: Find the treaty proportions. The retained proportion is $160{,}000/400{,}000=0.40$. The ceded amount is $400{,}000-160{,}000=\$240{,}000$, so the ceded proportion is $240{,}000/400{,}000=0.60$.

Step 3: Allocate the premium. Retained premium is $0.40(6{,}400)=\$2{,}560$. Ceded premium is $0.60(6{,}400)=\$3{,}840$.

Step 4: Allocate the covered payment. Retained claim is $0.40(100{,}000)=\$40{,}000$. Ceded claim is $0.60(100{,}000)=\$60{,}000$.

Check. Premium shares sum to $2{,}560+3{,}840=\$6{,}400$, and claim shares sum to $40{,}000+60{,}000=\$100{,}000$, the policy payment. Applying 40% and 60% directly to the $112,500 ground-up loss would allocate $112,500, not the $100,000 payable under the policy.

Common Mistakes and Reconciliation

The arithmetic is usually short; the main exam challenge is choosing the right base and amount. In particular, keep the retention, cession, proportions, gross premium, ground-up loss, and covered claim payment distinct.

Using the wrong denominator. Divide each retained or ceded amount by the total sum insured, not by the other share.
Reversing the proportions. A larger ceded amount than retention means the reinsurer has the larger share; verify which party each fraction represents.
Applying different proportions to premium and claim without a stated reason. Under the proportional terms assumed here, both use the same risk-specific shares.
Allocating the wrong claim amount. Apply the treaty shares to the covered policy payment, after any applicable policy deductible or limit.
Ignoring a treaty adjustment that is specified. The allocations here are before a separate commission or other adjustment. Do not invent one when it is absent, and do not omit one when the problem explicitly supplies it.

A reliable calculation records four items for each risk: sum insured, retained amount, retained and ceded proportions, and the premium or covered payment to be allocated. Finish by reconciling both sides. The retained and ceded premium must sum to gross premium, and the retained and ceded claim must sum to the covered claim payment.

Key takeaway: Under a proportional surplus share treaty, calculate each risk’s retained and ceded proportions using its total sum insured. Apply those proportions to the gross premium and to the covered claim payment, and check that each pair of allocated amounts adds back to its original total.

Check Your Understanding

For each problem, assume the treaty can accept the full cession and that no separate commission or treaty adjustment applies. Allocate only the specified gross premium and covered claim.

A hypothetical risk has a $240,000 sum insured, a $60,000 retention, a $1,500 gross premium, and a $20,000 covered claim. Find the retained and ceded premium and claim. (Answer: Retained proportion 25%, ceded proportion 75%; retained/ceded premium $375/$1,125; retained/ceded claim $5,000/$15,000.)
A risk has a $750,000 sum insured, a $150,000 retention, a $9,000 gross premium, and a $75,000 covered claim. Find both allocations. (Answer: Retained proportion 20%, ceded proportion 80%; retained/ceded premium $1,800/$7,200; retained/ceded claim $15,000/$60,000.)
A treaty retains $100,000 per risk. Risk A has sum insured $250,000, premium $2,500, and covered claim $40,000. Risk B has sum insured $500,000, premium $4,000, and covered claim $125,000. Find the portfolio’s total retained and ceded premiums and claims. (Answer: Retained/ceded premiums $1,800/$4,700; retained/ceded claims $41,000/$124,000.)
A risk has a $600,000 sum insured, a $180,000 retention, and a $7,200 gross premium. Its covered ground-up loss is $95,000, with a $5,000 deductible and no limit reduction. Find the retained and ceded premium and claim. (Answer: Retained proportion 30%, ceded proportion 70%; retained/ceded premium $2,160/$5,040; covered payment $90,000; retained/ceded claim $27,000/$63,000.)
A risk has a $500,000 sum insured, a $200,000 retention, and a $12,000 gross premium. It has two covered claim payments, $30,000 and $80,000. Allocate the premium and each claim, then find total claims by party. (Answer: Retained/ceded premium $4,800/$7,200; retained/ceded claims are $12,000/$18,000 on the first payment and $32,000/$48,000 on the second; total retained/ceded claims $44,000/$66,000.)

What You'll Learn