Aggregate Moments Under Quota Share

Why Aggregate Moments Change Under Quota Share

A quota-share treaty divides every covered claim in the same fixed proportions. If the insurer retains fraction $r$ and cedes fraction $q$, where $r+q=1$, each claim is split in those proportions. This makes aggregate calculations especially direct: retained and ceded aggregate losses are fixed multiples of gross aggregate loss. The mean scales linearly, but the variance scales with the square of the share.

The frequency model still matters. Two portfolios can have the same expected claim count and the same severity distribution but different aggregate variances if their claim counts have different variances. The formulas below keep frequency variability explicit, so they apply beyond the compound Poisson model.

Definition: Let $N$ be the number of claims, and let $X_1,X_2,\ldots$ be identically distributed claim severities, independent of $N$ and independent of one another. Write $m=E[X]$ and $s^2=\operatorname{Var}(X)$, where $X$ has the common severity distribution. The gross aggregate loss is $S=\sum_{j=1}^{N}X_j$, with $S=0$ when $N=0$. Under a fixed quota share, let $q$ be the ceded fraction and $r=1-q$ the retained fraction. Then $S_R=rS$ is the retained aggregate loss and $S_C=qS$ is the ceded aggregate loss.

These formulas assume that the quota-share fractions apply to every covered loss without a per-policy or aggregate cap, and that all losses in $S$ are in scope for the treaty. Premium allocation, expenses, commissions, and timing or discounting are not included in these loss-moment calculations. If shares vary by claim, or a treaty limit changes the payment as losses accumulate, the fixed-multiple relationships need not hold.

Compound Aggregate Moments

Let $E[N]=\mu_N$ and $\operatorname{Var}(N)=\sigma_N^2$. Conditional on $N=n$, the aggregate is a sum of $n$ independent severities. Its conditional mean is $nm$, and its conditional variance is $ns^2$. Applying the laws of total expectation and total variance gives the compound aggregate moments.

E[S]=\mu_N m,\qquad \operatorname{Var}(S)=\mu_N s^2+\sigma_N^2m^2

The variance has two components. The term $\mu_Ns^2$ reflects severity variation across the claims that occur. The term $\sigma_N^2m^2$ reflects variation in the number of claims. Omitting the second term is generally wrong when the claim count is random.

Deriving the Quota-Share Moment Formulas

Because the same fractions apply to every claim, summing retained claim payments gives $S_R=rS$, and summing ceded payments gives $S_C=qS$. For any constant $c$, $E[cS]=cE[S]$, $\operatorname{Var}(cS)=c^2\operatorname{Var}(S)$, and $\operatorname{Cov}(cS,S)=c\operatorname{Var}(S)$. Therefore:

\begin{aligned} E[S_R]&=r\mu_Nm, & \operatorname{Var}(S_R)&=r^2(\mu_Ns^2+\sigma_N^2m^2),\\ E[S_C]&=q\mu_Nm, & \operatorname{Var}(S_C)&=q^2(\mu_Ns^2+\sigma_N^2m^2),\\ \operatorname{Cov}(S_R,S_C)&=rq(\mu_Ns^2+\sigma_N^2m^2). \end{aligned}

The covariance is positive whenever both shares are positive and gross aggregate variance is positive. Large gross-loss outcomes are simultaneously large retained-loss and ceded-loss outcomes. In fact, the retained and ceded aggregates are perfectly linearly related: $S_C=(q/r)S_R$ when $r>0$.

A useful check is that the two shares add back to gross loss. Their means add to $E[S]$, and their variances add only after including twice their covariance:

\begin{aligned} E[S_R+S_C]&=E[S],\\ \operatorname{Var}(S_R+S_C) &=\operatorname{Var}(S_R)+\operatorname{Var}(S_C)+2\operatorname{Cov}(S_R,S_C)\\ &=(r^2+q^2+2rq)\operatorname{Var}(S) =(r+q)^2\operatorname{Var}(S)=\operatorname{Var}(S). \end{aligned}

Find the gross moments.
Use the frequency mean and variance and the severity mean and variance to calculate $E[S]$ and $\operatorname{Var}(S)$.

Identify both shares.
Convert the ceded percentage to $q$, then calculate $r=1-q$. Confirm that the two fractions sum to one.

Scale means and variances.
Multiply the gross mean by each share and the gross variance by the square of each share.

Check the allocation.
The means must add to the gross mean. If checking variances, include twice the retained-ceded covariance.

Worked Applications

Worked Example: Poisson Frequency and Exponential Severity

A hypothetical portfolio has a Poisson claim count with mean $\lambda=10$. Claim severities are independent exponential random variables with mean $\$1{,}000$, so their variance is $\$1{,}000{,}000$. A quota-share treaty cedes 30% of every loss. Find the gross, retained, and ceded means and variances, and the covariance of retained and ceded aggregate losses.

Method and gross moments. For a Poisson count, $E[N]=\operatorname{Var}(N)=10$. For the exponential severity, $m=1{,}000$ and $s^2=1{,}000{,}000$. Thus:

E[S]=10(1{,}000)=10{,}000, \qquad \operatorname{Var}(S)=10(1{,}000{,}000)+10(1{,}000)^2 =10{,}000{,}000+10{,}000{,}000=20{,}000{,}000.

The same variance can be written $\lambda E[X^2]$. Here $E[X^2]=s^2+m^2=2{,}000{,}000$, so $10E[X^2]=20{,}000{,}000$, which checks the calculation.

Apply the shares. The ceded fraction is $q=0.30$, and the retained fraction is $r=0.70$. Therefore:

\begin{aligned} E[S_R]&=0.70(10{,}000)=7{,}000, & \operatorname{Var}(S_R)&=0.70^2(20{,}000{,}000)=9{,}800{,}000,\\ E[S_C]&=0.30(10{,}000)=3{,}000, & \operatorname{Var}(S_C)&=0.30^2(20{,}000{,}000)=1{,}800{,}000,\\ \operatorname{Cov}(S_R,S_C)&=0.70(0.30)(20{,}000{,}000)=4{,}200{,}000. \end{aligned}

Check and trap avoided. The means add to $\$10{,}000$, and $9{,}800{,}000+1{,}800{,}000+2(4{,}200{,}000)=20{,}000{,}000$. A common error is to multiply the gross variance by 0.70 for the retained variance. The correct multiplier is $0.70^2$, because variance is quadratic in a scale factor.

Worked Example: Fixed Claim Count

A hypothetical equipment program has exactly five claims in the period. Each severity has mean $\$400$ and variance $\$3{,}600$, independently of the other claims. The reinsurer takes 25% of every loss. Calculate the gross and ceded aggregate moments, then the retained aggregate mean and variance.

Method. Here $N=5$ with certainty, so $E[N]=5$ and $\operatorname{Var}(N)=0$. The gross aggregate moments are:

E[S]=5(400)=2{,}000,\qquad \operatorname{Var}(S)=5(3{,}600)+0(400)^2=18{,}000.

The ceded fraction is $q=0.25$, so $r=0.75$. Thus:

\begin{aligned} E[S_C]&=0.25(2{,}000)=500, & \operatorname{Var}(S_C)&=0.25^2(18{,}000)=1{,}125,\\ E[S_R]&=0.75(2{,}000)=1{,}500, & \operatorname{Var}(S_R)&=0.75^2(18{,}000)=10{,}125,\\ \operatorname{Cov}(S_R,S_C)&=0.75(0.25)(18{,}000)=3{,}375. \end{aligned}

Check and trap avoided. The expected losses add to $\$2{,}000$, and $10{,}125+1{,}125+2(3{,}375)=18{,}000$. The frequency-variance term is zero because the number of claims is fixed. Do not insert a Poisson assumption merely because the problem concerns a claim count.

Worked Example: Overdispersed Claim Count

A hypothetical protection plan has a negative binomial claim count with mean 6 and variance 9. Claim severities are independent of the count and of one another, with mean $\$500$ and variance $\$250{,}000$. The treaty cedes 40% of each loss. Find the gross variance, retained and ceded means and variances, and their covariance.

Method and gross moments. Use the given frequency moments directly; the variance exceeds the mean, so a Poisson variance assumption would not be appropriate. With $\mu_N=6$, $\sigma_N^2=9$, $m=500$, and $s^2=250{,}000$:

E[S]=6(500)=3{,}000,\qquad \operatorname{Var}(S)=6(250{,}000)+9(500)^2 =1{,}500{,}000+2{,}250{,}000=3{,}750{,}000.

Apply the shares. Here $q=0.40$ and $r=0.60$. The retained and ceded means are $0.60(3{,}000)=\$1{,}800$ and $0.40(3{,}000)=\$1{,}200$. Their variances and covariance are:

\begin{aligned} \operatorname{Var}(S_R)&=0.60^2(3{,}750{,}000)=1{,}350{,}000,\\ \operatorname{Var}(S_C)&=0.40^2(3{,}750{,}000)=600{,}000,\\ \operatorname{Cov}(S_R,S_C)&=0.60(0.40)(3{,}750{,}000)=900{,}000. \end{aligned}

Check and trap avoided. The means total $\$3{,}000$, and $1{,}350{,}000+600{,}000+2(900{,}000)=3{,}750{,}000$. Treating the count as Poisson would replace its variance 9 by 6 and understate gross variance by $(9-6)(500)^2=\$750{,}000$.

Common Exam Traps and Scope

Scaling a variance linearly. If $Y=cS$, then $\operatorname{Var}(Y)=c^2\operatorname{Var}(S)$, not $c\operatorname{Var}(S)$.
Ignoring count variability. The term $\operatorname{Var}(N)m^2$ disappears only for a fixed claim count. For Poisson frequency it equals $\lambda m^2$.
Adding the two component variances and stopping. Retained and ceded losses are not independent: both are multiples of the same gross loss. Their covariance is needed to recover gross variance.
Confusing a proportional treaty with a capped treaty. A fixed quota share gives $S_R=rS$ and $S_C=qS$. A per-claim limit, aggregate cap, or varying share breaks this simple proportional relationship.

These moment results are useful for comparing the variability carried by insurer and reinsurer, and for checking aggregate calculations. They do not by themselves determine tail probabilities or the distribution of the retained aggregate. Those questions require distributional information beyond the first two moments. The methods here are aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Check Your Understanding

Use the compound aggregate moment formulas first, then apply the quota-share fractions. Variances and covariances are in dollars squared.

A hypothetical portfolio has Poisson frequency with mean 4 and exponential severity with mean $\$300$. The treaty cedes 20% of every loss. Find the gross mean and variance, then the retained and ceded variances. (Answer: gross mean $\$1{,}200$; gross variance $\$720{,}000$; retained variance $\$460{,}800$; ceded variance $\$28{,}800$.)
There are exactly three independent claims, each with severity mean $\$800$ and variance $\$40{,}000$. A treaty cedes 35%. Find the retained and ceded means, variances, and covariance. (Answer: retained mean $\$1{,}560$, variance $\$50{,}700$; ceded mean $\$840$, variance $\$14{,}700$; covariance $\$27{,}300$.)
A hypothetical frequency model has $E[N]=5$ and $\operatorname{Var}(N)=7$. Severity has mean $\$200$ and variance $\$1{,}600$. If 25% is ceded, find gross aggregate variance, retained variance, ceded variance, and covariance. (Answer: gross variance $\$288{,}000$; retained variance $\$162{,}000$; ceded variance $\$18{,}000$; covariance $\$54{,}000$.)
A Poisson portfolio has mean claim count 12. Its independent severities have mean $\$500$ and variance $\$100{,}000$. The insurer retains 90%. Calculate the gross mean and variance, the ceded variance, and the retained-ceded covariance. (Answer: gross mean $\$6{,}000$, gross variance $\$4{,}200{,}000$, ceded variance $\$42{,}000$, covariance $\$378{,}000$.)
For any fixed quota share with $r+q=1$, show how the retained and ceded variances and covariance recombine to give gross variance. (Answer: $r^2\operatorname{Var}(S)+q^2\operatorname{Var}(S)+2rq\operatorname{Var}(S)=(r+q)^2\operatorname{Var}(S)=\operatorname{Var}(S)$.)

What You'll Learn