Proportional Reinsurance in Aggregate Loss Models

Why Proportional Reinsurance Changes Aggregate Loss

Proportional reinsurance divides each covered claim between the insurer and a reinsurer according to an agreed share. The division may be a fixed percentage for every risk, as in quota-share reinsurance, or may depend on the amount of insurance on each risk, as in surplus-share reinsurance. These arrangements affect not only the payment on an individual claim, but also the mean and variance of the insurer’s aggregate losses.

The key calculation is to identify the retained fraction for each claim before calculating aggregate amounts. A fixed retained percentage scales every claim in the same way. A surplus-share treaty can assign different retained percentages to different risks, so a single percentage applied to the whole portfolio may give the wrong result.

Definition: Under quota-share reinsurance, the insurer retains a fixed proportion $r$ of each covered claim and cedes the remaining proportion $c=1-r$ to the reinsurer. Under surplus-share reinsurance, a retention line $M$ is specified for each risk. If the amount insured on a risk is $S$, the insurer generally retains the fraction $\min(1,M/S)$ of a covered claim and cedes the balance, subject to the treaty’s terms.

A retention line is an amount of insurance retained on a risk, not a deductible applied to each claim. If $S\leq M$, the risk is fully retained. If $S>M$, the insurer retains the fraction $M/S$. This fraction is applied to the covered claim amount under a proportional arrangement; it is not a fixed dollar reduction from the claim.

Premiums are also divided according to the treaty. Under quota share, the ceded premium is commonly the same percentage of the relevant gross premium as the ceded claim share, subject to treaty provisions such as ceding commissions. Under surplus share, premium is typically allocated in proportion to the ceded exposure at the applicable rates. The premium split need not equal the fraction of a particular realized claim that is ceded. The examples below assume no commission or other adjustment and use a uniform premium rate where stated.

From Individual Claims to Aggregate Loss

Let $X$ denote a gross claim amount and let $Z$ denote the insurer’s retained payment on that claim. Under a quota share with retained fraction $r$, $Z=rX$. The reinsurer’s payment is $W=(1-r)X$. Thus, for an individual claim, the retained and ceded amounts sum to the gross claim.

Z+W=X,\qquad E[Z]=rE[X],\qquad \operatorname{Var}(Z)=r^2\operatorname{Var}(X).

The variance uses the square of the retained percentage. This follows directly from the variance rule $\operatorname{Var}(aX)=a^2\operatorname{Var}(X)$ for a constant $a$. If the same quota share applies to every claim, it also applies to the aggregate: the retained aggregate is $r$ times the gross aggregate. Therefore its mean is multiplied by $r$, while its variance is multiplied by $r^2$.

For a surplus-share treaty, the corresponding retained fraction can vary by risk. A claim on risk $j$ with amount insured $S_j$ has retained fraction $r_j=\min(1,M/S_j)$, so its retained claim is $r_jX_j$. To calculate aggregate moments, use the distribution of the actual retained claim amounts; do not substitute one portfolio-wide retention percentage unless the risk mix and the required calculation justify doing so.

Aggregate-moment result: Suppose the claim count $N$ is independent of the claim amounts $Z_1,Z_2,\ldots$, and the retained amounts are independent and identically distributed with mean $\mu_Z$ and variance $\sigma_Z^2$. For retained aggregate loss $T=\sum_{i=1}^{N}Z_i$, $$ E[T]=E[N]\mu_Z,\qquad \operatorname{Var}(T)=E[N]\sigma_Z^2+\operatorname{Var}(N)\mu_Z^2. $$ If $N$ is Poisson with mean $\lambda$, this reduces to $\operatorname{Var}(T)=\lambda E[Z^2]$.

For the general result, condition on $N=n$. The conditional mean and variance are $n\mu_Z$ and $n\sigma_Z^2$. Taking the expectation of the conditional mean gives $E[T]=E[N]\mu_Z$. The law of total variance gives $\operatorname{Var}(T)=E[N\sigma_Z^2]+\operatorname{Var}(N\mu_Z)$, which is the displayed expression. For a Poisson count, $E[N]=\operatorname{Var}(N)=\lambda$, so the expression becomes $\lambda(\sigma_Z^2+\mu_Z^2)=\lambda E[Z^2]$. This result requires independence between the count and claim amounts and the stated identical-distribution assumption. If claim amounts depend on the count or the risk mix changes with the count, those assumptions must be reconsidered.

Worked Example: Quota Share and Aggregate Moments

A hypothetical insurer has a quota-share treaty under which it retains 40% of every claim and cedes 60%. The annual number of claims $N$ is Poisson with mean 10. Independent claim amounts $X$ are each \$500 or \$1,500 with probability 0.5 each. The annual gross premium is \$24,000, with no ceding commission. Find the mean and variance of the retained and ceded aggregate claims, and allocate the premium.

Method. First calculate the gross claim moments, then apply the fixed retention shares. Use dollars for claim amounts and dollars squared for variances.

The expected gross claim is $E[X]=0.5(500)+0.5(1{,}500)=1{,}000$. Its second moment is $E[X^2]=0.5(500^2)+0.5(1{,}500^2)=125{,}000+1{,}125{,}000=1{,}250{,}000$. Thus $\operatorname{Var}(X)=1{,}250{,}000-1{,}000^2=250{,}000$.

Because the claim count is Poisson, the gross aggregate mean is $10(1{,}000)=\$10{,}000$, and its variance is $10(1{,}250{,}000)=\$12{,}500{,}000$. Applying the retained share $r=0.40$, the retained aggregate mean is $0.40(\$10{,}000)=\$4{,}000$, and its variance is $0.40^2(\$12{,}500{,}000)=\$2{,}000{,}000$.

The ceded share is $0.60$. The ceded aggregate mean is $0.60(\$10{,}000)=\$6{,}000$, and its variance is $0.60^2(\$12{,}500{,}000)=\$4{,}500{,}000$. The retained and ceded means add to the gross mean. Their variances do not add to the gross variance by themselves because the two portions of each claim move together: their covariance is $0.40(0.60)(\$12{,}500{,}000)=\$3{,}000{,}000$. Therefore the gross variance is \$2,000,000 + \$4,500,000 + $2(\$3,000,000)=\$12,500,000$, as required.

With no commission, the insurer retains $0.40(\$24{,}000)=\$9{,}600$ of premium and cedes $0.60(\$24{,}000)=\$14{,}400$. These amounts sum to \$24,000. The premium allocation describes the treaty terms; it does not imply that realized claims will occur in those same proportions.

Check. Retained and ceded expected claims sum to \$10,000, and the premium shares sum to \$24,000. The variance check including covariance recovers the gross variance. The method avoids the common error of multiplying variance by the retention percentage rather than its square.

Applying a Surplus-Share Retention Line

For each risk, compare the amount insured $S$ with the retention line $M$. The retained fraction is 1 when $S\leq M$, and $M/S$ when $S>M$. If a common rate applies to all insured amounts, the premium can be split using the same retained and ceded exposure fractions. Actual treaty wording and rating terms determine the precise premium calculation.

Worked Example: Surplus Share Across Three Risks

A hypothetical insurer has a surplus-share retention line of \$100,000 per risk. Three risks have amounts insured of \$80,000, \$200,000, and \$500,000. Their respective covered claims are \$20,000, \$50,000, and \$125,000. Assume the claims are each 25% of the amount insured. A uniform gross premium rate of 2% applies to the insured amounts, with no commission. Calculate retained and ceded claims and premiums.

Method. Calculate the retained fraction separately for each risk, then apply it to that risk’s claim and insured amount.

For the \$80,000 risk, $S\leq M$, so the retained fraction is $1$. The insurer retains the full \$20,000 claim and cedes \$0. Its gross premium is $0.02(\$80{,}000)=\$1{,}600$, all retained.

For the \$200,000 risk, the retained fraction is $100{,}000/200{,}000=0.50$. The insurer retains $0.50(\$50{,}000)=\$25{,}000$ and cedes \$25,000. Retained insured exposure is $0.50(\$200{,}000)=\$100{,}000$. The gross premium is $0.02(\$200{,}000)=\$4{,}000$, of which \$2,000 is retained and \$2,000 ceded.

For the \$500,000 risk, the retained fraction is $100{,}000/500{,}000=0.20$. The insurer retains $0.20(\$125{,}000)=\$25{,}000$ and cedes \$100,000. Retained insured exposure is \$100,000. The gross premium is $0.02(\$500{,}000)=\$10{,}000$, split into \$2,000 retained and \$8,000 ceded.

Across the three risks, gross claims total \$195,000. Retained claims total $\$20{,}000+\$25{,}000+\$25{,}000=\$70{,}000$, and ceded claims total $\$0+\$25{,}000+\$100{,}000=\$125{,}000$. Gross premiums total \$15,600; retained premiums total \$5,600 and ceded premiums total \$10,000.

Check. Retained and ceded claims sum to \$195,000, and retained and ceded premiums sum to \$15,600. The method avoids using the \$100,000 line as a dollar deductible or applying one retention percentage to all three risks.

Aggregate Moments When Surplus Retention Varies

When risks have different insured amounts, their retained claim distribution can differ from the gross claim distribution in more than just scale. If claims arise from a mix of risk types, first determine the retained payment for each type and then calculate the mean and second moment of that retained payment. These moments can then be used with the specified claim-count model.

Worked Example: Surplus Share With a Poisson Claim Count

A hypothetical portfolio has a surplus-share retention line of \$20,000 per risk. Each claim is equally likely to be from type A or type B. A type A risk has \$50,000 insured and a fixed covered claim of \$10,000; a type B risk has \$100,000 insured and a fixed covered claim of \$30,000. The annual number of claims is Poisson with mean 4 and is independent of claim type. Find the retained and ceded aggregate means and variances.

Method. Work in thousands of dollars for individual claims. Find the two retained and ceded payment amounts, calculate their moments over the type mix, and apply the Poisson aggregate formula.

For type A, the retained fraction is $20/50=0.40$. The insurer retains \$4,000 and cedes \$6,000. For type B, the retained fraction is $20/100=0.20$. The insurer retains \$6,000 and cedes \$24,000.

The retained claim $Z$, measured in thousands of dollars, is 4 or 6 with probability 0.5 each. Therefore $E[Z]=(4+6)/2=5$, and $E[Z^2]=(4^2+6^2)/2=(16+36)/2=26$. With Poisson mean 4, the retained aggregate mean is $4(5)=20$ thousand dollars, or \$20,000. Its variance is $4(26)=104$ thousand-dollars squared, or \$104,000,000.

The ceded claim $W$, also in thousands of dollars, is 6 or 24 with probability 0.5 each. Thus $E[W]=(6+24)/2=15$ and $E[W^2]=(6^2+24^2)/2=(36+576)/2=306$. The ceded aggregate mean is $4(15)=60$ thousand dollars, or \$60,000, and its variance is $4(306)=1{,}224$ thousand-dollars squared, or \$1,224,000,000.

As a check, gross claims are \$10,000 or \$30,000 with equal probability. Their mean is \$20,000 per claim and their second moment is $(10^2+30^2)/2=500$ thousand-dollars squared. Gross aggregate mean is therefore \$80,000 and gross aggregate variance is $4(500)=2{,}000$ thousand-dollars squared, or \$2,000,000,000. The retained and ceded aggregate means sum to \$80,000.

The retained and ceded aggregates are not independent: each ceded amount is paired with a retained amount on the same claim. The expected product per claim is $E[ZW]=(4(6)+6(24))/2=(24+144)/2=84$ thousand-dollars squared. For a Poisson claim count, the aggregate covariance is $4(84)=336$ thousand-dollars squared. Hence $104+1{,}224+2(336)=2{,}000$, recovering the gross aggregate variance in thousand-dollars squared.

Check. Both aggregate means add to the gross aggregate mean, and the variance identity recovers the gross aggregate variance when covariance is included. The method avoids treating surplus-share retained claims as if one fixed percentage applied to every risk.

Common Mistakes and Exam Checks

Proportional calculations are compact, but several errors can produce plausible-looking answers. For example, if the gross aggregate variance is \$12,500,000 and the insurer retains 40%, the incorrect calculation $0.40(\$12{,}500{,}000)=\$5{,}000{,}000$ treats variance as if it scaled like a mean. The correction is $0.40^2(\$12{,}500{,}000)=\$2{,}000{,}000$, because the aggregate itself is multiplied by 0.40 and variance squares that factor.

For surplus share, another tempting error is to retain the line amount as a fixed portion of every claim. The line determines the retained fraction of the amount insured: for $S>M$, that fraction is $M/S$. A claim of \$50,000 on a \$200,000 risk with a \$100,000 line is therefore retained at 50%, or \$25,000—not reduced by a \$100,000 deductible.

Before finalizing a calculation, check that each retained and ceded claim sums to its gross claim, and that each retained and ceded premium sums to its gross premium under the assumed allocation rule. For aggregate variances, check whether retained and ceded amounts share the same underlying claims; if they do, their covariance matters when reconciling to the gross variance. These distinctions are central to proportional reinsurance calculations aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

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

A hypothetical quota-share treaty retains 75% of claims. Gross aggregate claims have mean \$6,000 and variance \$8,000,000. Find the retained and ceded aggregate means and variances. (Answer: retained mean \$4,500 and variance \$4,500,000; ceded mean \$1,500 and variance \$500,000.)
A surplus-share treaty has a \$40,000 line on a risk insured for \$100,000. A covered claim is \$18,000. At a uniform 3% premium rate, with no commission, find the retained and ceded claim and premium. (Answer: retained claim \$7,200, ceded claim \$10,800; retained premium \$1,200, ceded premium \$1,800.)
A Poisson claim count has mean 3. Each gross claim is \$2,000 or \$8,000 with probability 0.5 each. A quota-share treaty retains 70%. Find the retained aggregate mean and variance. (Answer: mean \$10,500; variance \$49,980,000.)
A surplus-share treaty has a \$50,000 line. With equal probability, a claim is either \$12,000 on a risk insured for \$50,000, or \$24,000 on a risk insured for \$200,000. The independent claim count is Poisson with mean 2. Find the retained aggregate mean and variance. (Answer: mean \$18,000; variance \$180,000,000.)
A fixed quota share retains 75% of an aggregate with mean \$2,500,000 and variance \$16,000,000. Find the ceded mean and variance, and the covariance between retained and ceded aggregate losses. (Answer: ceded mean \$625,000, ceded variance \$1,000,000, and covariance \$3,000,000.)

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