Comparing Proportional and Excess-of-Loss Recoveries

Why Compare Reinsurance Structures?

A reinsurance structure changes which part of a loss the insurer retains and which part is recoverable from reinsurers. A proportional treaty generally cedes a specified share of covered losses; an excess-of-loss treaty generally responds only when a loss exceeds a stated retention. Those different mechanics affect both the average retained loss and its variability.

A fair comparison therefore needs to hold the hypothetical gross-loss experience constant and calculate each treaty’s recovery and net loss from that same experience. Comparing only recovery amounts can be misleading: a larger expected recovery may simply reflect a larger expected transfer of loss, and treaty price, commissions, expenses, and collectibility are not included in a loss-only comparison.

Definition: For a gross covered loss $X$, the ceded loss is the amount recoverable under the treaty, and the net loss is the amount retained by the insurer. Ignoring other adjustments, $X=\text{ceded loss}+\text{net loss}$. A proportional treaty cedes a contractually specified proportion of loss; an excess-of-loss treaty cedes the portion above an attachment point, subject to any stated limit.

Proportional and Excess-of-Loss Mechanics

Under a quota share with constant ceding percentage $q$, where $0\le q\le1$, the ceded amount is $qX$ and the net amount is $(1-q)X$. The percentage applies to each covered loss, not just to losses above a threshold. A ceding commission can change the cash economics of the arrangement, but it does not change this basic allocation of loss.

A surplus share treaty is also proportional, but the proportion generally varies by policy. Let $T$ be the insurer’s retention per risk, measured in the same exposure units as the policy’s insured amount $S$. If the treaty can accept up to $C$ of ceded exposure, the ceded exposure is $\min((S-T)^+,C)$, where $(z)^+=\max(z,0)$. For a policy within treaty capacity, the cession percentage is the ceded exposure divided by $S$. Assuming losses are shared in that proportion, the same percentage applies to each covered loss on that policy.

For per-risk excess of loss with retention $R$ and limit $L$, the recovery on a loss $X$ is $\min((X-R)^+,L)$. The insurer’s net loss is $X-\min((X-R)^+,L)$. With an unlimited layer, the net loss simplifies to $\min(X,R)$. The retention and any limit apply separately to each covered risk; they do not automatically apply once to the portfolio total.

\begin{aligned} \text{Quota share:}\quad &\text{ceded}=qX,\qquad \text{net}=(1-q)X,\\ \text{Surplus share:}\quad &c(S)=\frac{\min((S-T)^+,C)}{S},\qquad \text{ceded}=c(S)X,\\ \text{Per-risk excess of loss:}\quad &\text{ceded}=\min((X-R)^+,L). \end{aligned}

The surplus share percentage formula assumes $S>0$ and that the treaty covers the policy as described. If the policy exceeds the treaty’s capacity, the portion beyond capacity is not included in the treaty recovery; it may require another arrangement or remain with the insurer. The formula does not determine the treatment of exclusions, coinsurance, or other contract terms.

How Proportional Cessions Change Variance

A constant quota share percentage scales each realization of the loss by the same amount. Under a finite-variance loss model, if $X$ is the gross loss and $N=(1-q)X$ is net loss, linearity of expectation and the scaling rule for variance give $E[N]=(1-q)E[X]$ and $\operatorname{Var}(N)=(1-q)^2\operatorname{Var}(X)$. This result describes loss variability before reinsurance cost and does not claim that all measures of risk or the insurer’s total financial variability scale in the same way.

E[(1-q)X]=(1-q)E[X],\qquad \operatorname{Var}((1-q)X)=(1-q)^2\operatorname{Var}(X).

The variance result follows directly from the definition. For a constant $a$, the deviation of $aX$ from its mean is $a(X-E[X])$. Squaring that deviation and taking its expectation yields $\operatorname{Var}(aX)=a^2E[(X-E[X])^2]=a^2\operatorname{Var}(X)$. Setting $a=1-q$ gives the quota share formula. The formula requires a constant share and a finite variance; it does not directly apply to a surplus share portfolio with differing policy-level percentages.

Worked Example: Comparing Quota Share and Per-Risk Excess of Loss

A hypothetical insurer has one covered risk. Its gross loss $X$, in thousands of dollars, has the following distribution: $P(X=0)=0.5$, $P(X=40)=0.3$, and $P(X=100)=0.2$. Compare a 40% quota share with an unlimited per-risk excess-of-loss cover attaching at $R=25$. Ignore premiums, commissions, expenses, and recoverable disputes.

Method and gross-loss moments. First calculate the gross expected loss:

$$ E[X]=0(0.5)+40(0.3)+100(0.2)=12+20=32. $$

Next calculate the second moment and variance:

E[X^2]=0^2(0.5)+40^2(0.3)+100^2(0.2)=480+2000=2480, \qquad \operatorname{Var}(X)=2480-32^2=1456.

Thus gross expected loss is $32,000 and gross variance is $1456$ thousand-dollars squared.

Quota share. The insurer retains $1-0.40=0.60$ of each loss. The net outcomes are $0$, $24$, and $60$, each in thousands, with the original probabilities. Therefore:

E[N_{\mathrm{QS}}]=0(0.5)+24(0.3)+60(0.2)=7.2+12=19.2,

and

E[N_{\mathrm{QS}}^2]=24^2(0.3)+60^2(0.2)=172.8+720=892.8, \qquad \operatorname{Var}(N_{\mathrm{QS}})=892.8-19.2^2=524.16.

The formula check is $0.60^2(1456)=0.36(1456)=524.16$, matching the direct calculation. Expected ceded loss is $32-19.2=12.8$ thousand dollars.

Excess of loss. With an unlimited layer attaching at 25, the net loss is $\min(X,25)$. Net outcomes are therefore $0$, $25$, and $25$ thousand dollars. The expected net loss and its variance are:

E[N_{\mathrm{XOL}}]=0(0.5)+25(0.3)+25(0.2)=12.5, \qquad E[N_{\mathrm{XOL}}^2]=25^2(0.3)+25^2(0.2)=312.5,

$$ \operatorname{Var}(N_{\mathrm{XOL}})=312.5-12.5^2=156.25. $$

Expected ceded loss is $32-12.5=19.5$ thousand dollars. As a check, recoveries are $0$, $15$, and $75$ thousand dollars; their expectation is $0(0.5)+15(0.3)+75(0.2)=19.5$. The excess-of-loss structure gives lower expected net loss and variance for these terms and this distribution, but that is not a conclusion about which treaty is economically preferable once pricing and other contract features are considered. The comparison avoids the common mistake of treating the excess-of-loss recovery as a fixed percentage of each loss.

Surplus Share: The Proportion Depends on the Risk

Quota share uses one percentage across the covered portfolio. Surplus share instead relates the cession to the amount insured on each policy, subject to the treaty’s retention and capacity. Consequently, two policies can have different cession percentages even when the same treaty applies. For a particular policy, however, the policy’s surplus share percentage is applied proportionally to its covered losses.

Worked Example: Applying Surplus Share Across Three Policies

A hypothetical insurer writes three policies with insured amounts of $100,000, $250,000, and $400,000. A surplus share treaty retains the first $100,000 per risk and has capacity to accept three additional lines of $100,000 each, for a maximum ceded exposure of $300,000 per risk. The observed covered losses are respectively $40,000, $200,000, and $200,000. Calculate each cession percentage, recovery, and net loss.

Method. For each policy, calculate the amount above the $100,000 retention, cap it at $300,000, divide by the insured amount to get the cession percentage, then apply that percentage to the loss.

Policy 1: Insured amount is $100,000, so the exposure above retention is $100{,}000-100{,}000=0$. The ceded exposure is $0 and the cession percentage is $0/100{,}000=0\%$. Recovery is $0\%\times\$40{,}000=\$0$; net loss is $40,000.

Policy 2: Exposure above retention is $250{,}000-100{,}000=\$150{,}000$, below the $300,000 treaty capacity. The cession percentage is $150{,}000/250{,}000=60\%$. Recovery is $0.60\times\$200{,}000=\$120{,}000$; net loss is $\$200{,}000-\$120{,}000=\$80{,}000$.

Policy 3: Exposure above retention is $400{,}000-100{,}000=\$300{,}000$, exactly the treaty capacity. The cession percentage is $300{,}000/400{,}000=75\%$. Recovery is $0.75\times\$200{,}000=\$150{,}000$; net loss is $\$200{,}000-\$150{,}000=\$50{,}000$.

Portfolio check: Gross losses total $\$40{,}000+\$200{,}000+\$200{,}000=\$440{,}000$. Recoveries total $\$0+\$120{,}000+\$150{,}000=\$270{,}000$. Net losses total $\$40{,}000+\$80{,}000+\$50{,}000=\$170{,}000$, and $\$270{,}000+\$170{,}000=\$440{,}000$. The common mistake is to apply one portfolio-wide quota share percentage: the treaty gives policy-specific percentages of 0%, 60%, and 75%.

Comparing Portfolio Volatility

When separate risks are independent, the variance of their aggregate net loss is the sum of their individual net loss variances. If risks are dependent, covariance terms must also be included. The distinction matters because a treaty may reduce the severity of individual claims without eliminating the effect of losses occurring together.

\operatorname{Var}\left(\sum_{j=1}^{n}N_j\right) =\sum_{j=1}^{n}\operatorname{Var}(N_j) +2\sum_{j<k}\operatorname{Cov}(N_j,N_k).

For independent risks, each covariance is zero. For positively dependent losses, the covariance terms are positive, so summing individual variances alone understates aggregate variance. This is a loss-model comparison; it does not account for reinsurer default risk, treaty wording disputes, or premiums.

Worked Example: Two Independent Risks Under Both Structures

A hypothetical portfolio has two independent risks, each with the loss distribution used in the first example: $P(X=0)=0.5$, $P(X=40)=0.3$, and $P(X=100)=0.2$, in thousands of dollars. Compare the variance of aggregate net loss under a 40% quota share and under an unlimited per-risk excess-of-loss cover attaching at $25,000 on each risk.

Quota share. Each risk has net variance $524.16$ thousand-dollars squared, as calculated above. Independence gives:

\operatorname{Var}(N_{\mathrm{QS},1}+N_{\mathrm{QS},2}) =524.16+524.16=1048.32.

Each risk’s expected net loss is $19.2$ thousand dollars, so aggregate expected net loss is $19.2+19.2=38.4$ thousand dollars.

Per-risk excess of loss. Each risk has net variance $156.25$ thousand-dollars squared and expected net loss $12.5$ thousand dollars. Therefore:

\operatorname{Var}(N_{\mathrm{XOL},1}+N_{\mathrm{XOL},2}) =156.25+156.25=312.5, \qquad E[N_{\mathrm{XOL},1}+N_{\mathrm{XOL},2}]=12.5+12.5=25.

The aggregate variance difference is $1048.32-312.5=735.82$ thousand-dollars squared. The result depends on the independence assumption. If the two net losses had covariance $c$, their aggregate variance would be the sum of the individual variances plus $2c$.

The common trap is to apply the $25,000 retention once to the combined gross loss. This is per-risk excess of loss: each risk has its own attachment. For example, if both risks each produce a $40,000 loss, the per-risk net amounts are $25,000 and $25,000, or $50,000 total. Treating the $80,000 aggregate as one loss and applying a single $25,000 retention would produce $25,000 net, which describes a different structure, not the stated per-risk treaty.

Exam Traps and Practical Interpretation

Confusing per-risk and aggregate attachment. Apply a per-risk retention separately to each risk. Apply an aggregate retention to the defined aggregate only when the contract is an aggregate cover.
Using a single share for surplus treaties. Calculate each policy’s cession percentage from its insured amount, retention, and available capacity before applying it to that policy’s loss.
Comparing volatility without specifying the measure. Variance is in squared units; standard deviation is in the original units. A lower expected net loss is not itself a statement about the reinsurance’s cost or value.
Ignoring dependence. Variances add without covariance terms only for independent losses (or when covariance is otherwise known to be zero).
Forgetting limits. An excess-of-loss recovery with limit $L$ is capped at $L$; an unlimited layer cannot be substituted for a limited one.

Key takeaway: Proportional treaties transfer a stated share of covered losses, while excess-of-loss treaties respond to losses above an attachment. To compare them, calculate recoveries and net losses on the same gross-loss basis, state whether the cover is per risk or aggregate, and make the dependence assumptions explicit.

Check Your Understanding

For each problem, calculate the requested quantity before checking the answer.

A hypothetical loss $X$, in thousands of dollars, is 0 with probability 0.5, 40 with probability 0.3, and 80 with probability 0.2. A 25% quota share applies. What is expected net loss? (Answer: $21,000.)
A hypothetical claim is 10, 50, or 90 thousand dollars with probabilities 0.2, 0.5, and 0.3, respectively. An unlimited per-risk excess-of-loss cover attaches at $30,000. What is expected net loss? (Answer: $26,000.)
A hypothetical policy has an insured amount of $200,000. Surplus share retains $80,000 and has capacity for two additional $80,000 lines. A covered loss is $100,000. What are the cession percentage, recovery, and net loss? (Answer: 60%; $60,000 recovery; $40,000 net loss.)
Two independent risks each follow the loss distribution in the first worked example. What is aggregate net variance under a 40% quota share and under unlimited per-risk excess of loss attaching at $25,000 on each risk? (Answer: 1,048.32 and 312.5 thousand-dollars squared, respectively.)
For the loss distribution in the first worked example, compare expected ceded loss under a 40% quota share and unlimited per-risk excess of loss attaching at $25,000. (Answer: $12,800 under quota share and $19,500 under excess of loss.)

What You'll Learn