Expected Cessions Under Proportional Treaties

Why Expected Cessions Depend on the Treaty Allocation

A proportional reinsurance treaty assigns the reinsurer a specified share of covered losses. The expected ceded loss is therefore determined by two ingredients: the claim-level allocation and the expected covered loss to which that allocation applies. Quota share uses a fixed percentage; surplus share determines a percentage from the policy’s sum insured and the treaty’s retention and capacity.

The distinction matters in a portfolio. A quota-share percentage can be applied directly to expected gross loss. A surplus-share percentage can vary from policy to policy, so expected gross loss alone is generally not enough: the policy sums insured and corresponding expected losses are needed.

Definition: For a covered claim with gross loss $X$, let $Y^L$ denote the reinsurer’s payment per loss. Under a proportional allocation with cession fraction $q$, $Y^L=qX$, and the insurer’s retained loss is $(1-q)X$. The fraction $q$ is fixed for a quota-share treaty and is determined by policy-level treaty terms for a surplus-share treaty.

Quota-Share Cessions

Under a quota-share treaty, the reinsurer cedes the same fraction $q$ of every covered loss. If the treaty covers all losses in a portfolio and annual gross aggregate loss is $S$, then annual ceded loss is $qS$. Consequently,

E[\text{annual ceded loss}]=qE[S].

This result does not require a particular frequency or severity distribution. If annual claim count $N$ is independent of identically distributed claim severities $X_1,X_2,\ldots$, and $E[N]$ and $E[X]$ are finite, then $E[S]=E[N]E[X]$, where $S=\sum_{k=1}^{N}X_k$. Thus the expected cession can also be calculated as $qE[N]E[X]$.

Worked Example: Annual Cession Under Quota Share

A hypothetical equipment insurer buys a 30% quota-share treaty covering all claims in a portfolio. The annual claim count is $N\sim\operatorname{Poisson}(4)$. Claim severities are independent and identically distributed exponential random variables with mean $\$18{,}000$, independent of $N$. Find the expected annual ceded loss. No discounting or mortality model applies; losses are measured in dollars during one year.

Step 1: Identify the cession fraction. The treaty cedes 30% of every covered claim, so $q=0.30$.

Step 2: Find expected gross annual loss. For a compound frequency-severity model with independent claim count and severities, $E[S]=E[N]E[X]$. Here $E[N]=4$ and $E[X]=\$18{,}000$, giving $E[S]=4(\$18{,}000)=\$72{,}000$.

Step 3: Apply the treaty share. Expected annual ceded loss is $0.30(\$72{,}000)=\$21{,}600$.

Check. Expected annual retained loss is $0.70(\$72{,}000)=\$50{,}400$, and $\$21{,}600+\$50{,}400=\$72{,}000$, the expected gross loss. The method avoids treating the 30% treaty share as a reduction in claim frequency; it is a share of loss dollars.

Surplus-Share Cessions

A surplus-share treaty determines the reinsurer’s share from the policy’s sum insured. Let $R$ be the insurer’s retention per policy line, and let $m$ be the number of retention lines of reinsurance capacity. The treaty can cede at most $mR$ of a policy’s sum insured. For a policy with sum insured $S$, the ceded sum insured and cession fraction are

C_{\mathrm{SI}}=\min\bigl((S-R)^+,\,mR\bigr), \qquad q(S)=\frac{C_{\mathrm{SI}}}{S},

where $(z)^+=\max(z,0)$, and $S>0$. If $S\le R$, there is no cession. If $R<S\le (m+1)R$, the treaty cedes the amount above the retention. If $S>(m+1)R$, the treaty is at capacity; any amount beyond the ceded capacity is not ceded under this treaty.

For a proportional surplus allocation, a covered claim on that policy is divided using $q(S)$: the reinsurer pays $q(S)X$. If the policy has expected annual claim count $\lambda$ and claim severity $X$ with finite mean, its expected annual cession is $\lambda q(S)E[X]$, provided frequency and severity satisfy the usual independence assumptions for that calculation. For several policy classes, calculate each class’s expected cession using its own $q(S)$, then add the results.

Worked Example: Applying a Surplus Treaty at Its Capacity

A hypothetical insurer covers a policy with sum insured $S=\$300{,}000$. Its surplus treaty has retention $R=\$100{,}000$ and capacity $m=1$ retention line. The policy generates $N\sim\operatorname{Poisson}(2)$ claims per year. Each claim has deterministic covered severity $X=\$120{,}000$. Find expected annual ceded loss. No discounting or mortality model applies.

Step 1: Calculate the amount above the retention. $(S-R)^+=\$300{,}000-\$100{,}000=\$200{,}000$.

Step 2: Apply the treaty capacity. The maximum ceded sum insured is $mR=1(\$100{,}000)=\$100{,}000$. Therefore, $C_{\mathrm{SI}}=\min(\$200{,}000,\$100{,}000)=\$100{,}000$.

Step 3: Find the claim-level cession fraction. $q(S)=\$100{,}000/\$300{,}000=1/3$.

Step 4: Calculate expected annual cession. The expected gross annual loss is $E[N]E[X]=2(\$120{,}000)=\$240{,}000$. The expected ceded loss is $(1/3)(\$240{,}000)=\$80{,}000$.

Check. Expected retained loss is $(2/3)(\$240{,}000)=\$160{,}000$; the two expected shares sum to $\$240{,}000$. The method avoids applying the $\$200{,}000$ excess of sum insured as though all of it were ceded: the treaty’s capacity limits the ceded sum insured to $\$100{,}000$.

Why the Expected-Loss Formula Works

The expected cession formula follows from the claim-level allocation and linearity of expectation. For a policy with a fixed cession fraction $q$, annual ceded loss is $C=\sum_{k=1}^{N}qX_k$. Conditional on $N=n$, the expected cession is $E[C\mid N=n]=qnE[X]$, because each of the $n$ claims has mean severity $E[X]$. Taking expectations over $N$ gives

E[C]=E\bigl[E[C\mid N]\bigr] =qE[N]E[X].

This derivation assumes that claim severity has the same mean for each claim and that claim count does not change that mean; independence of $N$ and the severities is a standard sufficient condition. More generally, for any covered aggregate loss $S$ with a fixed proportional share $q$, $C=qS$ directly implies $E[C]=qE[S]$. No distributional form such as Poisson or exponential is required for that last identity.

For surplus share, the same reasoning applies within each policy or class only after its cession fraction has been determined. If policy $j$ has fraction $q_j$, expected annual claim count $\lambda_j$, and mean severity $E[X_j]$, its expected cession is $\lambda_jq_jE[X_j]$. Adding expected cessions across classes is valid by linearity of expectation; the classes do not have to be independent.

Worked Example: Comparing Two Proportional Treaties Across Policy Classes

A hypothetical portfolio has three policy classes. A surplus treaty has retention $R=\$100{,}000$ and capacity $m=2$ lines. Class A has sum insured $\$80{,}000$, annual claim count $N_A\sim\operatorname{Poisson}(2)$, and exponential claim severity with mean $\$20{,}000$. Class B has sum insured $\$150{,}000$, $N_B\sim\operatorname{Poisson}(3)$, and exponential severity with mean $\$30{,}000$. Class C has sum insured $\$350{,}000$, $N_C\sim\operatorname{Poisson}(1)$, and exponential severity with mean $\$70{,}000$. Counts and severities are independent within each class. Compare expected annual cessions under the surplus treaty and a 25% quota-share treaty covering the same claims. No discounting or mortality model applies.

Step 1: Calculate the surplus cession fraction for each class. The capacity is $mR=2(\$100{,}000)=\$200{,}000$.

For Class A, $S_A=\$80{,}000\le R$, so $C_{\mathrm{SI},A}=0$ and $q_A=0$.

For Class B, $C_{\mathrm{SI},B}=\min(\$150{,}000-\$100{,}000,\$200{,}000)=\$50{,}000$. Thus $q_B=\$50{,}000/\$150{,}000=1/3$.

For Class C, $C_{\mathrm{SI},C}=\min(\$350{,}000-\$100{,}000,\$200{,}000)=\$200{,}000$. Thus $q_C=\$200{,}000/\$350{,}000=4/7$.

Step 2: Calculate expected gross loss by class. Class A has $2(\$20{,}000)=\$40{,}000$; Class B has $3(\$30{,}000)=\$90{,}000$; Class C has $1(\$70{,}000)=\$70{,}000$. Total expected gross annual loss is $\$40{,}000+\$90{,}000+\$70{,}000=\$200{,}000$.

Step 3: Calculate surplus-share cessions by class. Class A cedes $0(\$40{,}000)=\$0$. Class B cedes $(1/3)(\$90{,}000)=\$30{,}000$. Class C cedes $(4/7)(\$70{,}000)=\$40{,}000$. Total expected surplus-share cession is $\$0+\$30{,}000+\$40{,}000=\$70{,}000$.

Step 4: Calculate quota-share cession. The quota-share treaty cedes 25% of total expected gross loss, so its expected cession is $0.25(\$200{,}000)=\$50{,}000$. The surplus-share cession is therefore $\$70{,}000-\$50{,}000=\$20{,}000$ higher.

Check. Under the surplus treaty, expected retained loss is $\$200{,}000-\$70{,}000=\$130{,}000$; under quota share it is $\$200{,}000-\$50{,}000=\$150{,}000$. In each case, retained plus ceded expected loss equals the same $\$200{,}000$ gross expectation. The calculation avoids assigning one portfolio-wide surplus fraction when the policy sums insured imply three different fractions.

Common Mistakes and Applicability

A surplus-share treaty’s capacity applies to the ceded sum insured, not directly to the expected claim severity or annual aggregate loss. First determine the ceded sum insured for each policy, divide by that policy’s sum insured to obtain its cession fraction, and only then apply the fraction to expected covered claim loss.

A common incorrect calculation in the single-policy example would be to use the amount above retention, $\$200{,}000$, as the ceded sum insured. That gives an incorrect fraction of $\$200{,}000/\$300{,}000=2/3$, and an incorrect expected cession of $(2/3)(\$240{,}000)=\$160{,}000$. The treaty has only one line of capacity, so the correct ceded sum insured is $\min(\$200{,}000,\$100{,}000)=\$100{,}000$. The correct fraction is $1/3$, and the correct expected annual cession is $\$80{,}000$.

The surplus formula also presumes that the treaty’s proportional share applies to the covered loss on the policy. If claim payments are subject to separate coverage terms, exclusions, or policy limits, use the covered loss amount to which the treaty actually applies. An expected aggregate loss for the entire portfolio is sufficient for a quota-share calculation, but generally not for surplus share unless the relevant policy-level cession fractions and expected losses are also known.

Key takeaway: For quota share, multiply expected covered aggregate loss by the fixed treaty percentage. For surplus share, calculate the capacity-limited ceded sum insured and cession fraction policy by policy, then apply those fractions to the corresponding expected losses.

Check Your Understanding

For each hypothetical arrangement, calculate the requested expected cession. Round dollar answers to the nearest cent.

A 20% quota-share treaty covers a portfolio with $N\sim\operatorname{Poisson}(5)$ claims per year and independent exponential severities with mean $\$10{,}000$. Find expected annual ceded loss. (Answer: $\$10{,}000.00$)
A policy has sum insured $\$200{,}000$. A surplus treaty has $R=\$80{,}000$ and $m=2$. Its annual claim count has mean $1.5$, and mean claim severity is $\$40{,}000$. Find expected annual ceded loss. (Answer: $\$36{,}000.00$)
A policy has sum insured $\$420{,}000$, with $R=\$100{,}000$ and $m=2$. Its annual claim count has mean 3, and mean severity is $\$42{,}000$. Find the surplus treaty’s expected annual cession. (Answer: $\$60{,}000.00$)
A hypothetical portfolio has three classes under a surplus treaty with $R=\$50{,}000$, $m=3$: Class A has $S=\$40{,}000$, expected annual gross loss $\$20{,}000$; Class B has $S=\$100{,}000$, expected annual gross loss $\$40{,}000$; Class C has $S=\$200{,}000$, expected annual gross loss $\$40{,}000$. Find expected surplus cession and compare it with a 40% quota share on the same portfolio. (Answer: surplus $\$50{,}000.00$; quota share $\$40{,}000.00$; surplus is $\$10{,}000.00$ higher.)
A surplus-share treaty applies to a portfolio, and only the portfolio’s expected annual gross loss is known. Is that information generally enough to calculate expected ceded loss? (Answer: No. Policy sums insured and corresponding expected losses, or equivalent information sufficient to determine the policy-level cession fractions, are generally also required.)

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