Expected Reinsurance Loss in Frequency-Severity Models

Why Condition on Claim Count?

A reinsurance layer specifies how much is ceded from each covered claim. To find the expected total recovery over a period, we must account for both the payment on an individual claim and the number of claims. Conditioning on the claim count separates these two parts: first calculate the expected payment for a fixed number of claims, then average over the frequency distribution.

This approach applies to a frequency-severity model, in which $N$ is the number of claims and $X_1,\ldots,X_N$ are their severities. It works for many frequency distributions; a Poisson assumption is not needed for the expected-loss calculation. Independence between frequency and severities, however, is important for the simplest formula.

Definition: Let $N$ be a nonnegative integer-valued claim count. Suppose the claim severities $X_1,X_2,\ldots$ are identically distributed as $X$, and are independent of $N$. For a per-claim treaty, let $Y_j^C$ and $Y_j^R$ be the ceded and retained payments on claim $j$. The aggregate ceded loss and aggregate retained loss are $C=\sum_{j=1}^{N}Y_j^C$ and $R=\sum_{j=1}^{N}Y_j^R$. A sum over no claims is zero.

Claim-Level Excess-of-Loss Payments

Consider a hypothetical per-claim excess-of-loss layer with attachment $d$ and payment limit $u$. The reinsurer pays nothing up to the attachment, then pays the amount above the attachment, subject to a maximum payment of $u$ per claim. Here $u$ denotes the layer’s maximum payment, not the maximum gross claim severity.

Y^C=\min\bigl((X-d)^+,u\bigr),\qquad Y^R=X-Y^C, \qquad (z)^+=\max(z,0).

The retained payment includes the part of the loss below attachment and, if the claim exhausts the layer, the part above the layer. Claim by claim, gross loss equals ceded payment plus retained payment. Therefore aggregate ceded and retained losses also add to aggregate gross loss.

Key result: If $E[N]$ is finite, severities have the same distribution, and the severities are independent of $N$, then $E[C]=E[N]E[Y^C]$ and $E[R]=E[N]E[Y^R]$. In particular, since $Y^R=X-Y^C$, $E[C]+E[R]=E[N]E[X]$.

Why the Expected-Loss Formula Holds

Condition on a fixed claim count $N=n$. Under the stated assumptions, each claim has expected ceded payment $E[Y^C]$, regardless of the value of $n$. By linearity of expectation, the expected sum of the $n$ ceded payments is $nE[Y^C]$. This does not require the claim payments to be independent of one another; linearity alone is enough for this step.

E[C\mid N=n] =E\left[\sum_{j=1}^{n}Y_j^C\mid N=n\right] =\sum_{j=1}^{n}E[Y_j^C\mid N=n] =nE[Y^C].

Taking expectations over the possible values of $N$ gives the result:

E[C]=E\bigl[E[C\mid N]\bigr] =E\bigl[NE[Y^C]\bigr] =E[N]E[Y^C].

The same reasoning gives $E[R]=E[N]E[Y^R]$. The step $E[Y_j^C\mid N=n]=E[Y^C]$ is justified by the assumption that claim severity is independent of claim count and has the same distribution for each claim. If severity or coverage depends on the claim count, calculate $E[C\mid N=n]$ using the relevant conditional severity distributions instead; multiplying $E[N]$ by an unconditional per-claim expectation may then be wrong.

Worked Examples

Worked Example: A Discrete Severity Model and a Capped Layer

A hypothetical portfolio has $N\sim\operatorname{Poisson}(2.5)$. Claim severities are independent of $N$, and each claim is $1,000 with probability 0.7 or $5,000 with probability 0.3. A per-claim layer attaches at $d=\$2{,}000$ and has payment limit $u=\$2{,}000$. Find the expected aggregate ceded and retained losses, and check their sum.

Method. Calculate the ceded and retained amounts for each possible claim severity, take their per-claim expectations, and multiply by $E[N]=2.5$. The frequency distribution is stated, but only its mean is needed for these expected losses.

For a $1,000 claim, which is below attachment, the ceded payment is $\min((1{,}000-2{,}000)^+,2{,}000)=\$0$; the retained payment is $1,000. For a $5,000 claim, the amount above attachment is $3,000, but the layer payment is capped at $2,000. Thus the ceded payment is $2,000 and the retained payment is $3,000.

The expected ceded payment per claim is $0.7(0)+0.3(2{,}000)=\$600$. The expected retained payment per claim is $0.7(1{,}000)+0.3(3{,}000)=\$1{,}600$. The expected gross severity is $0.7(1{,}000)+0.3(5{,}000)=\$2{,}200$.

Therefore, expected aggregate ceded loss is $2.5(600)=\$1{,}500$, and expected aggregate retained loss is $2.5(1{,}600)=\$4{,}000$.

Check. Their sum is $\$1{,}500+\$4{,}000=\$5{,}500$. Expected aggregate gross loss is $2.5(2{,}200)=\$5{,}500$, so the amounts reconcile. A common mistake is to treat the $5,000 claim’s ceded payment as $3,000, forgetting the $2,000 layer limit. Another is to multiply the per-claim expected payments by something other than $E[N]$.

Worked Example: Conditioning on a Binomial Claim Count

A hypothetical program has $N\sim\operatorname{Binomial}(4,0.5)$, so $E[N]=4(0.5)=2$. Independent claim severities are $500 with probability 0.2, $1,500 with probability 0.5, and $4,000 with probability 0.3. A per-claim layer attaches at $d=\$1{,}000$ and has limit $u=\$2{,}000$. Find the expected aggregate ceded and retained losses.

Method. First calculate the expected treaty payments for one claim. Then condition on $N=n$, where the expected ceded amount is $n$ times the one-claim expectation. Averaging over the binomial count is equivalent to multiplying by its mean of 2.

For a $500 claim, the ceded payment is $0 and the retained payment is $500. For a $1,500 claim, the amount above attachment is $500, so the ceded payment is $500 and the retained payment is $1,000. For a $4,000 claim, the amount above attachment is $3,000, so the ceded payment is limited to $2,000 and the retained payment is $2,000.

Thus the expected ceded payment per claim is $0.2(0)+0.5(500)+0.3(2{,}000)=250+600=\$850$. The expected gross severity is $0.2(500)+0.5(1{,}500)+0.3(4{,}000)=100+750+1{,}200=\$2{,}050$. The expected retained payment per claim is $\$2{,}050-\$850=\$1{,}200$. Directly, this is also $0.2(500)+0.5(1{,}000)+0.3(2{,}000)=100+500+600=\$1{,}200$.

The expected aggregate ceded loss is $2(850)=\$1{,}700$. The expected aggregate retained loss is $2(1{,}200)=\$2{,}400$.

Check. The expected aggregate gross loss is $2(2{,}050)=\$4{,}100$, and $\$1{,}700+\$2{,}400=\$4{,}100$. The cap matters for the $4,000 claim: using $3,000 rather than $2,000 as its ceded payment would overstate expected recovery by $0.3(\$1{,}000)=\$300$ per claim.

Worked Example: A Uniform Severity and Layer Payment

A hypothetical portfolio has $N\sim\operatorname{Poisson}(1.2)$. Each claim severity is independent of $N$ and uniformly distributed from $0 to $6,000. A per-claim layer attaches at $d=\$1{,}000$ and has payment limit $u=\$3{,}000$. Find the expected aggregate ceded and retained losses.

Method. Integrate the layer payment across the severity distribution. The payment is zero below $1,000, rises linearly from zero to $3,000 between $1,000 and $4,000, and remains at $3,000 above $4,000.

The uniform density is $1/6{,}000$. The expected ceded payment per claim is the area under the payment function, divided by 6,000. Between $1,000 and $4,000 the area is a triangle with base $3,000 and height $3,000, or $\tfrac12(3{,}000)(3{,}000)=4{,}500{,}000$. Between $4,000 and $6,000 the payment is $3,000, giving area $(2{,}000)(3{,}000)=6{,}000{,}000$. The area below $1,000 contributes zero.

Therefore $E[Y^C]=(4{,}500{,}000+6{,}000{,}000)/6{,}000=\$1{,}750$. For a uniform distribution from $0 to $6,000, $E[X]=(0+6{,}000)/2=\$3{,}000$. Hence $E[Y^R]=E[X]-E[Y^C]=\$1{,}250$.

Since $E[N]=1.2$, expected aggregate ceded loss is $1.2(1{,}750)=\$2{,}100$, and expected aggregate retained loss is $1.2(1{,}250)=\$1{,}500$.

Check. Expected aggregate gross loss is $1.2(3{,}000)=\$3{,}600$, equal to $\$2{,}100+\$1{,}500$. The capped portion must include all severities from $4,000 through $6,000; omitting it would leave out $6,000,000 of area in the per-claim calculation.

When the Shortcut Does Not Apply

The useful shortcut $E[C]=E[N]E[Y^C]$ is not a statement that any frequency model can be paired with any unconditional severity average. It relies on the expected payment per claim being the same conditional on each possible claim count. If claim severities change with frequency, or the treaty terms or covered population change with $N$, condition explicitly:

E[C]=\sum_{n=0}^{\infty}P(N=n)E[C\mid N=n].

For example, if high claim counts are associated with higher severities, then $E[Y_j^C\mid N=n]$ may increase with $n$. In that case $E[C\mid N=n]$ must use the severity distribution conditional on $N=n$; it cannot automatically be replaced by $nE[Y^C]$ for an unconditional severity $X$.

A second exam trap is to calculate expected aggregate payments from an expected claim count while silently changing the treaty basis. The formulas here concern a payment determined separately for each claim. A treaty that applies a deductible or limit to the annual aggregate uses a payment based on total loss, not a sum of independent claim-level treaty payments. The claim-level calculation does not by itself give the expected payment under that different arrangement.

Key takeaway: For a claim-level treaty, calculate the expected ceded and retained payment per claim, then multiply each by expected claim count when severities are identically distributed and independent of frequency. Always check that expected ceded plus expected retained equals expected gross loss.

Check Your Understanding

For each problem, use claim-level payments and state the expected aggregate ceded and retained losses or the requested conditional calculation.

A hypothetical portfolio has $E[N]=3$. Claim severity is $1,000 with probability 0.4 and $3,000 with probability 0.6, independently of frequency. A per-claim layer attaches at $1,500 and has limit $1,000. Find expected aggregate ceded and retained losses. (Answer: ceded $1,800; retained $4,800.)
A hypothetical portfolio has $N\sim\operatorname{Binomial}(5,0.4)$. Independent claim severities are $2,000 with probability 0.25 and $6,000 with probability 0.75. A layer attaches at $1,000 and has limit $2,500. Find expected aggregate ceded and retained losses. (Answer: ceded $4,250; retained $5,750.)
A hypothetical portfolio has $E[N]=1.5$, independent of severity, where $X$ is uniform from $0 to $4,000. A per-claim layer attaches at $1,000 and has limit $2,000. Find expected aggregate ceded and retained losses. (Answer: ceded $1,500; retained $1,500.)
A hypothetical portfolio has $E[N]=1.9$, with claim severities independent of frequency: $1,000 with probability 0.5 and $4,000 with probability 0.5. A per-claim layer attaches at $2,000 and has limit $1,500. Find expected aggregate ceded and retained losses. (Answer: ceded $1,425; retained $3,325.)
In a hypothetical frequency-dependent model, $P(N=1)=P(N=2)=0.5$. Given $N=1$, the one claim is certain to be $2,000; given $N=2$, both claims are certain to be $4,000. A per-claim layer attaches at $1,000 and has limit $1,000. Find $E[C\mid N=1]$, $E[C\mid N=2]$, and $E[C]$. (Answer: $E[C\mid N=1]=\$1{,}000$, $E[C\mid N=2]=\$2{,}000$, and $E[C]=\$1{,}500$.)

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