Expected Loss Versus Variability Transferred

Why Expected Cession and Variability Reduction Differ

A reinsurance treaty reallocates claim payments between an insurer and a reinsurer. The expected amount ceded is a measure of average cost transferred. It does not, by itself, measure how much the treaty reduces the insurer’s aggregate risk. A treaty can cede a smaller expected amount yet reduce aggregate variance more, because the variance effect depends on how ceded and retained payments move together and on the claim-count model.

The calculations below use hypothetical portfolios. The main model is a compound Poisson model: the number of claims is Poisson, claim amounts are independent and identically distributed, and claim amounts are independent of the claim count. These assumptions matter; with a fixed number of claims, the aggregate variance formula is different.

Definition: Let $X$ be the amount of one claim before reinsurance. Write $c(X)$ for the payment ceded to the reinsurer and $r(X)$ for the insurer’s retained payment. A valid allocation satisfies $X=c(X)+r(X)$ for every claim. If $N$ is the claim count, define aggregate loss $S=\sum_{j=1}^{N}X_j$, aggregate cession $C=\sum_{j=1}^{N}c(X_j)$, and aggregate retained loss $R=\sum_{j=1}^{N}r(X_j)$. Then $S=C+R$.

Compound Poisson Results

Suppose $N\sim\operatorname{Poisson}(\lambda)$, the claim amounts $X_j$ are independent copies of $X$, and $N$ is independent of those amounts. Assume the relevant first and second moments are finite. For any nonnegative per-claim payment function $g(X)$, the aggregate payment $\sum_{j=1}^{N}g(X_j)$ has mean $\lambda E[g(X)]$ and variance $\lambda E[g(X)^2]$. The second moment, not the variance, of the per-claim payment appears in the aggregate variance.

E[C]=\lambda E[c(X)],\qquad \operatorname{Var}(C)=\lambda E[c(X)^2]

E[R]=\lambda E[r(X)],\qquad \operatorname{Var}(R)=\lambda E[r(X)^2],\qquad \operatorname{Var}(S)=\lambda E[X^2]

The expected aggregate loss ceded is therefore $\lambda E[c(X)]$. The reduction in the insurer’s aggregate variance is instead $\operatorname{Var}(S)-\operatorname{Var}(R)$. It is not generally equal to either the expected cession or the variance of the aggregate cession.

Theorem: Under the compound Poisson assumptions above, the variance reduction from reinsurance is $\lambda E[X^2-r(X)^2]=\lambda E[2Xc(X)-c(X)^2]$. Also, $\operatorname{Var}(S)=\operatorname{Var}(C)+\operatorname{Var}(R)+2\operatorname{Cov}(C,R)$, with $\operatorname{Cov}(C,R)=\lambda E[c(X)r(X)]$.

To justify the aggregate variance formula, let $Z=\sum_{j=1}^{N}g(X_j)$. Conditional on $N=n$, $Z$ is a sum of $n$ independent copies of $g(X)$, so $E[Z\mid N]=N E[g(X)]$ and $\operatorname{Var}(Z\mid N)=N\operatorname{Var}(g(X))$. By the law of total variance,

\begin{aligned} \operatorname{Var}(Z) &=E[\operatorname{Var}(Z\mid N)] +\operatorname{Var}(E[Z\mid N])\\ &=\lambda\operatorname{Var}(g(X)) +\lambda(E[g(X)])^2\\ &=\lambda E[g(X)^2]. \end{aligned}

The first equality uses conditional variance; the second uses $E[N]=\operatorname{Var}(N)=\lambda$ for a Poisson count. Applying the result to $g=c$, $r$, or $X$ gives the formulas above. For the covariance, the same conditioning argument gives $\operatorname{Cov}(C,R)=\lambda\operatorname{Cov}(c(X),r(X))+\lambda E[c(X)]E[r(X)]=\lambda E[c(X)r(X)]$. Finally, substitute $X=c(X)+r(X)$ into $X^2-r(X)^2$ to obtain $2Xc(X)-c(X)^2$.

Worked Comparison: Quota Share and Excess-of-Loss

The following example uses the same portfolio under two treaties. The comparison makes clear that expected cession and variance reduction answer different questions.

Worked Example: More Expected Cession, but Less Variance Reduction

A hypothetical insurer has $N\sim\operatorname{Poisson}(4)$ claims. Each claim is independently either $\$100$ or $\$300$, with probability $0.5$ each. Compare (A) a 27% quota-share treaty, which cedes 27% of each claim, with (B) unlimited per-claim excess-of-loss attaching at $d=\$200$. Find the expected aggregate cession and the aggregate variance reduction under each treaty.

Step 1: Calculate the unreinsured moments. The claim mean and second moment are

E[X]=0.5(100)+0.5(300)=200,\qquad E[X^2]=0.5(100^2)+0.5(300^2)=50{,}000.

Thus $E[S]=4(200)=\$800$ and $\operatorname{Var}(S)=4(50{,}000)=\$200{,}000$.

Step 2: Apply the quota share. The ceded and retained payments are $c(X)=0.27X$ and $r(X)=0.73X$, so the allocation identity holds. The expected aggregate cession is $4(0.27)(200)=\$216$. The retained aggregate variance is

\operatorname{Var}(R) =4E[(0.73X)^2] =4(0.73^2)(50{,}000) =4(0.5329)(50{,}000) =\$106{,}580.

The variance reduction is $\$200{,}000-\$106{,}580=\$93{,}420$.

Step 3: Apply the excess-of-loss treaty. For each claim, $c(X)=(X-200)_+$, where $(z)_+=\max(z,0)$, and $r(X)=X-c(X)$. A $\$100$ claim has ceded payment $\$0$ and retained payment $\$100$. A $\$300$ claim has ceded payment $\$100$ and retained payment $\$200$. Each pair sums to the original claim. The expected aggregate cession is $4[0.5(0)+0.5(100)]=\$200$. The retained second moment per claim is

E[r(X)^2]=0.5(100^2)+0.5(200^2)=5{,}000+20{,}000=25{,}000.

Therefore $\operatorname{Var}(R)=4(25{,}000)=\$100{,}000$, and the variance reduction is $\$200{,}000-\$100{,}000=\$100{,}000$.

Check. The quota share cedes $\$216$ in expectation, compared with $\$200$ under excess-of-loss, but its variance reduction is $\$93{,}420$, compared with $\$100{,}000$ under excess-of-loss. The excess-of-loss treaty reduces variance by $\$6{,}580$ more, even though its expected cession is $\$16$ less. The expected totals also reconcile: $\$800=\$216+\$584$ for quota share and $\$800=\$200+\$600$ for excess-of-loss.

Method and mistake avoided. Calculate retained payments claim by claim, then use the compound Poisson second-moment formula. A $\$300$ claim under the attachment at $\$200$ retains $\$200$, not $\$100$: the treaty cedes only the $\$100$ above attachment.

Why Variance Reduction Is Not the Variance of Cession

Because $S=C+R$, the variance identity includes the covariance term. Under the compound Poisson model, $C$ and $R$ are generally positively associated: a claim that generates a large ceded payment may also leave a large retained payment. Consequently, subtracting $\operatorname{Var}(R)$ from $\operatorname{Var}(S)$ does not usually give $\operatorname{Var}(C)$.

For the excess-of-loss treaty in the preceding example, the ceded payment is $0$ or $100$, so $E[c(X)^2]=0.5(0^2)+0.5(100^2)=5{,}000$ and $\operatorname{Var}(C)=4(5{,}000)=\$20{,}000$. Also, $E[c(X)r(X)]=0.5(0)(100)+0.5(100)(200)=10{,}000$, giving $\operatorname{Cov}(C,R)=4(10{,}000)=\$40{,}000$. Thus the variance reduction is $\$20{,}000+2(\$40{,}000)=\$100{,}000$, not $\$20{,}000$.

Worked Example: A Limited Per-Claim Layer

A hypothetical portfolio has $N\sim\operatorname{Poisson}(3)$. Claim severity is $\$80$, $\$180$, or $\$400$, with probabilities $0.5$, $0.3$, and $0.2$, respectively. A per-claim layer attaches at $d=\$100$ and has limit $u=\$150$. Calculate the expected aggregate cession and variance reduction.

Step 1: Determine the claim-level allocation. The layer payment is $c(X)=\min((X-100)_+,150)$. For severities $80,180,400$, ceded payments are respectively $0,80,150$; retained payments are $80,100,250$. In each case, ceded plus retained equals the claim amount.

Step 2: Calculate expected cession.

$$ E[c(X)]=0.5(0)+0.3(80)+0.2(150)=24+30=54. $$

Hence $E[C]=3(54)=\$162$.

Step 3: Calculate the variance reduction. First,

\begin{aligned} E[X^2]&=0.5(80^2)+0.3(180^2)+0.2(400^2)\\ &=3{,}200+9{,}720+32{,}000=44{,}920. \end{aligned}

The unreinsured aggregate variance is $3(44{,}920)=\$134{,}760$. For retained payments,

E[r(X)^2]=0.5(80^2)+0.3(100^2)+0.2(250^2) =3{,}200+3{,}000+12{,}500=18{,}700.

Therefore the retained aggregate variance is $3(18{,}700)=\$56{,}100$, and the reduction is $\$134{,}760-\$56{,}100=\$78{,}660$.

Check. The mean claim amount is $0.5(80)+0.3(180)+0.2(400)=174$, so the portfolio’s expected aggregate loss is $3(174)=\$522$. The expected retained aggregate loss is $3[0.5(80)+0.3(100)+0.2(250)]=3(120)=\$360$. Thus $\$522=\$162+\$360$, consistent with the expected cession. The variance reduction is a separate calculation and is not $\$162$.

Method and mistake avoided. Apply the layer limit to the per-claim payment before calculating moments. Treating the layer as unlimited would incorrectly assign a $\$300$ cession to the $\$400$ claim, even though this layer’s maximum payment is $\$150$.

When the Claim Count Is Fixed

The compound Poisson result should not be used automatically when the claim count is fixed. If exactly $n$ independent claims occur, then for any per-claim payment $g(X)$, the aggregate variance is $n\operatorname{Var}(g(X))$, rather than $nE[g(X)^2]$. The random claim count in a Poisson model contributes additional variability.

Worked Example: Fixed Claim Count and Quota Share

A hypothetical portfolio has exactly $N=2$ claims. Each claim is independently $\$100$ with probability $0.75$ or $\$500$ with probability $0.25$. A quota-share treaty cedes 40% of every claim. Find the expected aggregate cession and the reduction in aggregate variance.

Step 1: Calculate the claim moments.

$$ E[X]=0.75(100)+0.25(500)=75+125=200, $$

and $E[X^2]=0.75(10{,}000)+0.25(250{,}000)=7{,}500+62{,}500=70{,}000$. Hence $\operatorname{Var}(X)=70{,}000-200^2=30{,}000$, and the aggregate variance before reinsurance is $2(30{,}000)=\$60{,}000$.

Step 2: Calculate expected cession. The per-claim cession is $c(X)=0.4X$, so the expected aggregate cession is $2(0.4)(200)=\$160$.

Step 3: Calculate retained variance and reduction. Since $r(X)=0.6X$, the retained aggregate variance is $2(0.6^2)(30{,}000)=2(0.36)(30{,}000)=\$21{,}600$. The variance reduction is $\$60{,}000-\$21{,}600=\$38{,}400$.

Check. The aggregate expected loss is $2(200)=\$400$. The retained expected loss is $2(0.6)(200)=\$240$, and $\$160+\$240=\$400$. The variance formula used the per-claim variance $30{,}000$, as required for a fixed count; using $E[X^2]$ as in the Poisson case would give the wrong aggregate variance.

Method and mistake avoided. Identify the claim-count model before applying a variance formula. The expected cession calculation is unchanged, but fixed-count and Poisson aggregate variances differ.

Practical Interpretation and Exam Traps

A useful comparison reports at least two quantities separately: expected aggregate cession and aggregate variance reduction. The former is in dollars of expected payment; the latter is in squared dollars, because it is a difference between variances. Do not describe the variance reduction as the reinsurer’s expected payment, and do not confuse it with the decrease in standard deviation.

Check the allocation on each claim. Confirm $X=c(X)+r(X)$ at values below attachment, inside a layer, and above its limit.
Use the correct moment. For a compound Poisson aggregate, variance is $\lambda E[g(X)^2]$; for a fixed number of independent claims, it is $n\operatorname{Var}(g(X))$.
Keep expectations and variances distinct. A treaty’s expected cession is $\lambda E[c(X)]$ under the compound Poisson model, while variance reduction is $\lambda E[X^2-r(X)^2]$.
Include covariance when decomposing total variance. In general, $\operatorname{Var}(S)-\operatorname{Var}(R)=\operatorname{Var}(C)+2\operatorname{Cov}(C,R)$, not just $\operatorname{Var}(C)$.

The comparison of expected cost and variability transferred is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus. A reliable solution states the frequency assumptions, defines ceded and retained payments, verifies their allocation, and then calculates the requested expectation and variance quantities separately.

Check Your Understanding

For each problem, identify the claim-count model and distinguish expected aggregate cession from variance reduction.

A hypothetical portfolio has $N\sim\operatorname{Poisson}(2)$, with severities $\$50$ and $\$150$ equally likely. A quota share cedes 20% of each claim. Find the expected aggregate cession and the variance reduction. (Answer: expected cession $\$40$; variance reduction $\$9{,}000$)
A hypothetical portfolio has $N\sim\operatorname{Poisson}(5)$, and every claim is $\$100$. A per-claim layer attaches at $d=\$60$ and has limit $u=\$20$. Find the expected aggregate cession and the variance reduction. (Answer: expected cession $\$100$; variance reduction $\$18{,}000$)
A hypothetical portfolio has $N\sim\operatorname{Poisson}(3)$, with severities $\$100$ and $\$300$ equally likely. A quota share cedes 25%. Find the expected aggregate cession and the retained aggregate variance. (Answer: expected cession $\$150$; retained aggregate variance $\$84{,}375$)
A hypothetical portfolio has exactly four independent claims, each $\$100$ or $\$300$ with probability $0.5$ each. Unlimited per-claim excess-of-loss attaches at $d=\$200$. Find the expected aggregate cession and variance reduction. (Answer: expected cession $\$200$; variance reduction $\$30{,}000$)
A hypothetical portfolio has $N\sim\operatorname{Poisson}(4)$, with severity $\$40$ with probability $0.75$ and $\$120$ with probability $0.25$. A per-claim layer attaches at $d=\$80$ and has limit $u=\$40$. Find the expected aggregate cession and variance reduction. (Answer: expected cession $\$40$; variance reduction $\$8{,}000$)

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