Calculating Surplus Share Cessions

Why Surplus Share Cessions Depend on Each Risk

A surplus share treaty cedes a different proportion of each covered risk, depending on its insured amount and the treaty’s retention and capacity. A risk at or below the retention is not ceded under the treaty. A larger risk may be shared proportionally between the insurer and reinsurer, while an amount above treaty capacity remains with the insurer unless another arrangement applies.

The calculation therefore has two stages: first determine how much of the insured amount is ceded; then use that amount to find the ceded proportion of the risk. When the covered claim is shared proportionally on the same basis, this proportion determines the recovery. The examples below assume hypothetical treaties and risks.

Definition: Let $R$ be the insurer’s retention per risk, $n$ the number of surplus lines available, $S$ the risk’s sum insured, and $L$ its covered loss. The treaty’s maximum ceded capacity per risk is $nR$, and the maximum sum insured that can be supported by the retention plus treaty is $R+nR$. The ceded amount of sum insured is

$$ C=\min\bigl(\max(S-R,0),\,nR\bigr). $$ The ceded proportion is \(C/S\) when \(S>0\), and the insurer’s retained proportion is \(1-C/S\). Under proportional claims sharing, the recovery is \(L(C/S)\), and the insurer’s net claim cost is \(L-L(C/S)\).

From Treaty Terms to Ceded Proportion

The retention is the amount of each risk that the insurer keeps before the surplus share treaty begins to cede. Each line adds capacity equal to the retention. Thus, a treaty with $n$ lines can cede at most $nR$ of sum insured per risk. The total amount supported by the insurer’s retention and treaty is $R+nR$; this is not the same as the treaty’s ceded capacity.

The formula for $C$ handles three cases. If $S\le R$, then $\max(S-R,0)=0$, so there is no cession. If $R<S\le R+nR$, the treaty cedes $S-R$, leaving the insurer with the retention. If $S>R+nR$, the treaty cedes its full capacity $nR$, and the insurer retains both the initial retention and the portion above the treaty’s capacity.

Under a proportional surplus share arrangement, the cession is a share of the risk, not a fixed dollar recovery against every claim. If $C$ is the ceded amount of sum insured, then the ceded fraction of that risk is $C/S$. The same fraction is applied to a covered loss only when the contract shares covered losses proportionally on that basis. A claim subject to a different coverage basis, exclusion, or contractual limit would require those terms to be considered separately.

Why the Recovery Formula Holds

Suppose the treaty cedes a fraction $C/S$ of a risk and the covered loss is shared in that same proportion. The reinsurance payment is then that fraction of the loss. The insurer’s net claim is the remaining fraction. The two amounts must add back to the original covered loss.

Find the risk share ceded.
The ceded proportion is the ceded insured amount divided by the total insured amount, $C/S$.

Apply the share to the loss.
With proportional claims sharing, the recovery is $L(C/S)$.

Calculate the retained claim.
The insurer pays the rest, $L-L(C/S)=L(1-C/S)$. Therefore, recovery plus net claim equals $L$.

This also explains why the ceded amount $C$ cannot be used directly as the recovery: $C$ is an insured amount, whereas $L$ is the covered loss. The ceded fraction links the two.

Worked Example: Risks Below, Within, and Above Capacity

A hypothetical treaty has a retention of $R=\$200{,}000$ and three surplus lines. It applies to three risks with sums insured of $\$100{,}000$, $\$500{,}000$, and $\$800{,}000$, and covered claims of $\$30{,}000$, $\$120{,}000$, and $\$200{,}000$, respectively. Assume each claim is shared proportionally with the ceded sum insured. Find the cession, recovery, and net claim for each risk, then the totals.

Method. Calculate treaty capacity, find $C$ separately for each risk, and apply each risk’s own ceded proportion to its claim. This avoids applying one portfolio-wide share to risks with different sums insured.

The treaty’s ceded capacity is $nR=3(\$200{,}000)=\$600{,}000$. The total amount supported by retention plus treaty is $\$200{,}000+\$600{,}000=\$800{,}000$.

Risk 1: $S=\$100{,}000$, $L=\$30{,}000$. Since $S<R$, the cession is

C=\min(\max(\$100{,}000-\$200{,}000,0),\$600{,}000)=\$0.

The ceded proportion is $0/\$100{,}000=0$. Recovery is $\$30{,}000(0)=\$0$, and net claim is $\$30{,}000-\$0=\$30{,}000$.

Risk 2: $S=\$500{,}000$, $L=\$120{,}000$. The amount above retention is $\$500{,}000-\$200{,}000=\$300{,}000$, which is below the $\$600{,}000$ capacity. Thus $C=\$300{,}000$, and the ceded proportion is $\$300{,}000/\$500{,}000=0.60$. Recovery is $\$120{,}000(0.60)=\$72{,}000$. Net claim is $\$120{,}000-\$72{,}000=\$48{,}000$.

Risk 3: $S=\$800{,}000$, $L=\$200{,}000$. The amount above retention is $\$600{,}000$, exactly the treaty capacity. Therefore $C=\$600{,}000$, the ceded proportion is $\$600{,}000/\$800{,}000=0.75$, recovery is $\$200{,}000(0.75)=\$150{,}000$, and net claim is $\$200{,}000-\$150{,}000=\$50{,}000$.

Total covered claims are $\$30{,}000+\$120{,}000+\$200{,}000=\$350{,}000$. Total recovery is $\$0+\$72{,}000+\$150{,}000=\$222{,}000$, and total net claim is $\$30{,}000+\$48{,}000+\$50{,}000=\$128{,}000$. The check is $\$222{,}000+\$128{,}000=\$350{,}000$, equal to total covered claims.

When a Risk Exceeds Treaty Capacity

A risk above $R+nR$ still has a ceded amount of only $nR$. The insurer’s retained sum insured is then $S-nR$, which exceeds the original retention. Because the ceded amount is capped while the total sum insured grows, the ceded proportion $nR/S$ decreases as $S$ increases beyond the treaty’s supported amount.

This has an important consequence: a treaty capacity limit applies to the amount ceded, not directly to the dollar amount of a claim. To calculate a recovery, first find the risk’s ceded fraction. Then apply it to the covered claim, subject to the proportional-sharing assumption and the treaty’s terms.

Worked Example: Two Risks and an Amount Above Capacity

A hypothetical treaty has a retention of $\$250{,}000$ and two surplus lines. Risk A has a sum insured of $\$600{,}000$ and a covered claim of $\$210{,}000$. Risk B has a sum insured of $\$1{,}000{,}000$ and a covered claim of $\$360{,}000$. Assume proportional sharing of covered claims. Calculate each recovery and net claim, as well as the totals.

Method. Find the $\$500{,}000$ ceded capacity first. Then calculate each cession separately. This distinguishes the treaty’s capacity from its total supported sum insured, which is $\$250{,}000+\$500{,}000=\$750{,}000$.

Risk A. The amount above retention is $\$600{,}000-\$250{,}000=\$350{,}000$, less than capacity, so $C_A=\$350{,}000$. Its ceded proportion is $\$350{,}000/\$600{,}000=7/12$. Recovery is $\$210{,}000(7/12)=\$17{,}500(7)=\$122{,}500$. Net claim is $\$210{,}000-\$122{,}500=\$87{,}500$.

Risk B. The amount above retention is $\$1{,}000{,}000-\$250{,}000=\$750{,}000$, which exceeds capacity. Thus $C_B=\min(\$750{,}000,\$500{,}000)=\$500{,}000$. Its ceded proportion is $\$500{,}000/\$1{,}000{,}000=0.50$. Recovery is $\$360{,}000(0.50)=\$180{,}000$. Net claim is $\$360{,}000-\$180{,}000=\$180{,}000$.

The total claim is $\$210{,}000+\$360{,}000=\$570{,}000$. Total recovery is $\$122{,}500+\$180{,}000=\$302{,}500$, and total net claim is $\$87{,}500+\$180{,}000=\$267{,}500$. Check: $\$302{,}500+\$267{,}500=\$570{,}000$.

Portfolio Calculation and a Common Exam Trap

For a portfolio, calculate each risk’s cession and claim share before summing. It is generally incorrect to combine all sums insured and claims first, then apply a single proportion: the treaty retention and capacity operate per risk, and each risk can fall in a different capacity range.

Common mistake—using capacity divided by supported sum insured as the cession percentage. For Risk B in the preceding example, an incorrect calculation might use $\$500{,}000/\$750{,}000=2/3$ as the ceded proportion, then report recovery of $\$360{,}000(2/3)=\$240{,}000$. That calculation uses the total supported amount, rather than the risk’s sum insured, as the denominator. The treaty cedes $\$500{,}000$ of Risk B’s $\$1{,}000{,}000$, so the correct proportion is $\$500{,}000/\$1{,}000{,}000=0.50$. Correct recovery is $\$360{,}000(0.50)=\$180{,}000$, not $\$240{,}000$.

Worked Example: Aggregate Recoveries Across Different Risks

A hypothetical surplus share treaty has a retention of $\$300{,}000$ and three lines. Three covered risks have the following data: Risk 1 has a sum insured of $\$900{,}000$ and claim of $\$180{,}000$; Risk 2 has a sum insured of $\$1{,}500{,}000$ and claim of $\$350{,}000$; Risk 3 has a sum insured of $\$250{,}000$ and claim of $\$70{,}000$. Assume proportional claims sharing. Find total recovery and total net claim.

Method. The ceded capacity per risk is $3(\$300{,}000)=\$900{,}000$. Compute each risk’s cession, proportion, and recovery, then aggregate. The trap to avoid is applying the same percentage to all three claims.

Risk 1. The amount above retention is $\$900{,}000-\$300{,}000=\$600{,}000$, within capacity. The ceded proportion is $\$600{,}000/\$900{,}000=2/3$. Recovery is $\$180{,}000(2/3)=\$120{,}000$, and net claim is $\$60{,}000$.

Risk 2. The amount above retention is $\$1{,}500{,}000-\$300{,}000=\$1{,}200{,}000$, greater than capacity. The cession is capped at $\$900{,}000$, so the ceded proportion is $\$900{,}000/\$1{,}500{,}000=0.60$. Recovery is $\$350{,}000(0.60)=\$210{,}000$, and net claim is $\$350{,}000-\$210{,}000=\$140{,}000$.

Risk 3. Its sum insured is below retention, so its cession and recovery are both zero. Net claim is $\$70{,}000$.

Total claims are $\$180{,}000+\$350{,}000+\$70{,}000=\$600{,}000$. Total recovery is $\$120{,}000+\$210{,}000+\$0=\$330{,}000$. Total net claim is $\$60{,}000+\$140{,}000+\$70{,}000=\$270{,}000$. The check is $\$330{,}000+\$270{,}000=\$600{,}000$.

Calculation Checklist

A reliable solution keeps insured amounts and claim amounts separate until the ceded proportion is known. For each risk:

Identify the per-risk retention $R$, number of lines $n$, sum insured $S$, and covered claim $L$.
Calculate ceded capacity $nR$.
Calculate $C=\min(\max(S-R,0),nR)$.
Find the ceded proportion $C/S$, if $S>0$.
Under proportional claims sharing, calculate recovery $L(C/S)$ and net claim $L-L(C/S)$.
For multiple risks, add the individual recoveries and net claims, then check that they sum to the total covered claims.

The calculations are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus. The central discipline is to respect the per-risk structure: retention and capacity determine each risk’s cession, and that risk-specific percentage—not the treaty capacity alone—determines its recovery.

Check Your Understanding

For each problem, assume covered claims are shared proportionally according to the ceded share of sum insured. Treat each risk separately unless the question asks for totals.

A treaty has a $\$100{,}000$ retention and two lines. A risk has a sum insured of $\$160{,}000$ and a covered claim of $\$40{,}000$. Find the ceded amount, recovery, and net claim. (Answer: cession $\$60{,}000$; recovery $\$15{,}000$; net claim $\$25{,}000$.)
A treaty has a $\$400{,}000$ retention and three lines. A risk has a sum insured of $\$1{,}000{,}000$ and a covered claim of $\$180{,}000$. Find the ceded proportion, recovery, and net claim. (Answer: ceded proportion $0.6000$; recovery $\$108{,}000$; net claim $\$72{,}000$.)
A treaty has a $\$150{,}000$ retention and two lines. A risk has a sum insured of $\$600{,}000$ and a covered claim of $\$240{,}000$. Find the cession and claim split, and explain whether the risk exceeds total supported sum insured. (Answer: cession $\$300{,}000$; recovery $\$120{,}000$; net claim $\$120{,}000$; total supported sum insured is $\$450{,}000$, so the risk exceeds it.)
A treaty has a $\$200{,}000$ retention and two lines. Risk A has sum insured $\$120{,}000$ and claim $\$24{,}000$; Risk B has sum insured $\$700{,}000$ and claim $\$140{,}000$. Find total recovery and total net claim. (Answer: total recovery $\$80{,}000$; total net claim $\$84{,}000$.)
A treaty has a $\$250{,}000$ retention and three lines. A risk has sum insured $\$1{,}400{,}000$ and claim $\$280{,}000$. Calculate the cession, ceded proportion, recovery, and net claim. (Answer: cession $\$750{,}000$; ceded proportion $15/28$; recovery $\$150{,}000$; net claim $\$130{,}000$.)

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