End-to-End Modeling of Covered Aggregate Loss

From Individual Claims to an Aggregate Model

An aggregate-loss calculation is a chain of random quantities. A claim count determines how many losses occur; each loss has a ground-up severity; the policy changes each loss into a covered payment; and reinsurance may then reduce the insurer’s payment. The aggregate result depends on keeping these stages distinct. In particular, the claim count is not the number of positive policy payments, and a reinsurance recovery may apply either to each claim or to the total for a period.

A complete model therefore states the distribution of the claim count, the severity distribution, the coverage payment rule, the reinsurance rule, and the assumptions connecting claims. The model also identifies the aggregate quantity of interest: for example, total covered payments before reinsurance, total reinsurance recovery, or the insurer’s net aggregate payment.

Definition: Let $N$ be the number of claims in a specified period. Let $X_i$ be the ground-up loss on claim $i$. If the policy payment rule is $g$, define the payment per loss by $Y_i^L=g(X_i)$, including a value of zero when the claim produces no policy payment. For per-claim reinsurance, let $R_i$ be the recovery on claim $i$, and define the insurer’s net payment on that claim as $Z_i=Y_i^L-R_i$. The aggregate covered payment is $L=\sum_{i=1}^{N}Y_i^L$; the aggregate recovery is $R=\sum_{i=1}^{N}R_i$; and the aggregate net payment is $T=\sum_{i=1}^{N}Z_i=L-R$.

The superscript matters. $Y^L$ describes payment per loss and includes zero payments. If only positive payments are being counted, let $M=\sum_{i=1}^N \mathbf{1}(Y_i^L>0)$ be the number of positive payments. Then $Y^P$ denotes the payment amount conditional on a positive payment. A sum over all losses using $Y^L$ and a sum over positive payments using $Y^P$ can represent the same aggregate, but their count and severity variables are not interchangeable.

For a standard frequency-severity model, assume that $N$ is independent of the claim severities, and that the $X_i$ are identically distributed and independent across claims. A specified policy rule then gives the distribution of each $Y_i^L$. If reinsurance is applied per claim, a specified recovery rule gives the distribution of $Z_i$. Independence is a modeling assumption, not a consequence of the definitions. If severity depends on the number of claims, or claims share a common driver, the independent and identically distributed model may not apply.

Why the Expected Aggregate Has a Simple Form

Under the standard model, the expected aggregate payment follows by conditioning on the claim count. If the per-loss payment has finite mean and is independent of $N$, then, conditional on $N=n$, the aggregate is a sum of $n$ payments, each with mean $E[Y^L]$. Averaging that conditional mean over the possible values of $N$ gives:

E[L\mid N=n]=nE[Y^L], \qquad E[L]=E[N]E[Y^L].

The same argument applies to per-claim net payments: when the $Z_i$ are identically distributed and independent of $N$, $E[T]=E[N]E[Z]$. Linearity of expectation also gives $E[T]=E[L]-E[R]$, whether or not $L$ and $R$ are independent. This does not mean their distributions can be modeled separately and combined as if independent: a recovery defined from a covered payment is generally dependent on that payment.

Specify frequency.
State the claim-count random variable $N$, its distribution, its parameter values, and the period to which it refers.

Specify ground-up severity.
State the distribution of $X_i$ and whether claim severities are independent of one another and of $N$.

Apply the coverage rule.
Use the stated contract to define $Y_i^L$ from $X_i$. Retain zero payments when counting losses rather than positive payments.

Apply reinsurance and choose the aggregate.
Define each per-claim recovery, or define a recovery from the aggregate if the contract operates at that level. State whether the target is gross covered, recovered, or net payment.

Worked Chains

Worked Example: A Policy Layer Followed by Per-Claim Reinsurance

A hypothetical insurer covers a portfolio for one year. The number of claims is $N\sim\operatorname{Poisson}(2)$. Claim severities are independent of $N$ and of one another, with ground-up loss $X$ equal to $100, $300, or $600 with probabilities 0.20, 0.50, and 0.30, respectively. The policy pays the amount above a $100 deductible, subject to a $300 maximum payment per claim. Per-claim reinsurance then reimburses the amount of that policy payment above $150. Find the expected aggregate covered payment, expected reinsurance recovery, and expected net payment. Also find the probability of zero aggregate net payment.

Method and assumptions. Map each possible ground-up loss through the policy and then through the per-claim reinsurance rule. The policy payment is $Y^L=\min((X-100)_+,300)$, where $a_+=\max(a,0)$. The recovery is $R=(Y^L-150)_+$, and the net payment is $Z=Y^L-R$.

Ground-up loss $X$	Probability	Policy payment $Y^L$	Recovery $R$	Net payment $Z$
$100	0.20	$0	$0	$0
$300	0.50	$200	$50	$150
$600	0.30	$300	$150	$150

The expected policy payment per loss is $E[Y^L]=0.20(0)+0.50(200)+0.30(300)=0+100+90=\$190$. Since $E[N]=2$, the expected aggregate covered payment is $E[L]=2(190)=\$380$.

The expected recovery per loss is $E[R]=0.20(0)+0.50(50)+0.30(150)=0+25+45=\$70$. Thus $E[R_{\text{aggregate}}]=2(70)=\$140$, and the expected aggregate net payment is $380-140=\$240$. As a check, the expected net payment per loss is $0.20(0)+0.50(150)+0.30(150)=\$120$, so $E[T]=2(120)=\$240$.

For zero aggregate net payment, every claim must have zero net payment. A claim has zero net payment with probability 0.20. Conditional on $N=n$, the probability that all $n$ claims have zero net payment is $0.20^n$, including probability 1 when $n=0$. Therefore $P(T=0)=\sum_{n=0}^{\infty}e^{-2}\frac{2^n}{n!}(0.20)^n =e^{-2}e^{0.4}=e^{-1.6}=0.2019$.

Check and common mistake. The expected net amount is less than the expected covered amount by exactly the expected recovery: $380-140=240$. A common error is to apply the reinsurance rule to the ground-up loss $X$; the stated recovery is based on the policy payment $Y^L$, not directly on $X$.

Worked Example: A Poisson Count and a Proportional Recovery

A hypothetical equipment policy has $N\sim\operatorname{Poisson}(1.5)$ claims in a year. Claim severities are independent of $N$ and have the following distribution: $X=$100$ with probability 0.50, $X=$200$ with probability 0.30, and $X=$400$ with probability 0.20. The policy pays the lesser of the ground-up loss and $250. A reinsurer then reimburses 25% of each policy payment. Calculate the expected aggregate policy payment, expected aggregate recovery, and expected aggregate net payment. Find the probability that the net aggregate is zero.

Method and assumptions. First apply the $250 per-claim policy limit, then apply the 25% recovery to that payment. Thus $Y^L=\min(X,250)$, $R=0.25Y^L$, and $Z=0.75Y^L$.

The policy payments corresponding to ground-up losses of $100, $200, and $400 are $100, $200, and $250. Hence $E[Y^L]=0.50(100)+0.30(200)+0.20(250)=50+60+50=\$160$. The expected aggregate policy payment is $E[L]=1.5(160)=\$240$.

The expected recovery per claim is $E[R]=0.25(160)=\$40$, so the expected aggregate recovery is $1.5(40)=\$60$. The expected net payment per claim is $160-40=\$120$, and the expected aggregate net payment is $1.5(120)=\$180$. The check $240-60=180$ agrees.

All three possible severities produce a positive net payment. Therefore the aggregate net payment is zero exactly when there are no claims. For a Poisson random variable with mean 1.5, $P(N=0)=e^{-1.5}=0.2231$.

Check and common mistake. The stated proportional recovery is 25% of the covered payment, not 25% of the ground-up loss. Applying it to the $400 ground-up loss would overstate the recovery because the policy payment on that loss is capped at $250.

Worked Example: Reinsurance Applied to the Annual Aggregate

A hypothetical portfolio has $N\sim\operatorname{Binomial}(3,0.5)$ claims. Each claim has a ground-up loss of $200, and the policy applies a $100 deductible with no other limit, so each covered payment is $100. An annual aggregate reinsurance contract reimburses covered aggregate payments above an attachment of $100. Calculate the expected covered aggregate, expected recovery, and expected net aggregate.

Method and assumptions. The individual covered payment is fixed at $100, so $L=100N$. The annual recovery is $R^A=(L-100)_+$, and the net aggregate is $T=L-R^A$. This recovery depends on the annual total, not separately on each claim.

Claim count $N$	Probability	Covered aggregate $L$	Aggregate recovery $R^A$	Net aggregate $T$
0	0.125	$0	$0	$0
1	0.375	$100	$0	$100
2	0.375	$200	$100	$100
3	0.125	$300	$200	$100

The probabilities follow from the binomial model: $P(N=0)=0.5^3=0.125$, $P(N=1)=3(0.5)^3=0.375$, $P(N=2)=3(0.5)^3=0.375$, and $P(N=3)=0.5^3=0.125$. The expected covered aggregate is $E[L]=100E[N]=100(3)(0.5)=\$150$.

Using the table, the expected recovery is $E[R^A]=0.125(0)+0.375(0)+0.375(100)+0.125(200)=0+0+37.50+25=\$62.50$. The expected net aggregate is $150-62.50=\$87.50$. Directly from the table, it is $0.125(0)+0.375(100)+0.375(100)+0.125(100)=\$87.50$, which checks.

Check and common mistake. Treating the annual attachment as a $100 per-claim attachment would give zero recovery on every $100 payment. That is the wrong contract: the attachment applies to $L$, the annual aggregate. Once there are at least two claims, the aggregate exceeds $100 and a recovery is due.

What Determines an Aggregate Distribution?

The mean can often be calculated from $E[N]$ and a per-claim mean, but an aggregate distribution requires more information. Under the independent frequency-severity assumptions, the distribution of a per-claim amount and the distribution of $N$ determine the distribution of their random sum. In particular, if $Z_i$ are identically distributed per-claim net payments independent of $N$, then for any threshold $t$,

P(T\le t)=\sum_{n=0}^{\infty}P(N=n)\, P\left(\sum_{i=1}^{n}Z_i\le t\right),

where the sum inside the probability is defined to be zero when $n=0$. The conditional term depends on the full distribution of $Z_i$, not just its mean. That distribution is obtained by carrying ground-up severity through the coverage and per-claim reinsurance rules. Zero payments must be retained if the sum is over all claims.

If reinsurance is applied to the annual aggregate, the net amount is instead defined from $L$, for example $T=L-R^A(L)$. It generally cannot be represented as a sum of independently adjusted per-claim net payments: whether a claim is recovered may depend on the other claims in the year. The contract wording and the level at which recovery is calculated therefore determine which random variables are needed.

Common Modeling Traps

Skipping the policy-payment stage. A reinsurance rule may apply to the insurer’s covered payment, not to ground-up severity. State which amount is the recovery base.
Dropping zero payments without changing the count. $Y^L$ includes zeros over all losses. $Y^P$ describes positive payments and must be paired with the number of positive payments.
Using a per-claim rule for an aggregate contract. A recovery based on an annual total depends on the aggregate and may not be separable claim by claim.
Using only a mean to describe a distribution. Expected aggregate payment can require only the claim-count mean and per-claim mean under the stated assumptions. An aggregate probability, such as the probability of exceeding a threshold, needs the relevant distributions as well.
Assuming independence without checking the model. If the claim count and severities are dependent, or if claims share a common source of variation, the standard independent frequency-severity calculation may not hold.

Key takeaway: Track the model in order: claim count, ground-up severity, payment per loss, reinsurance recovery, and the aggregate quantity of interest. The distinction between per-claim and aggregate reinsurance determines how those random variables combine.

Check Your Understanding

For each problem, identify the relevant payment stages before calculating.

A hypothetical portfolio has $N\sim\operatorname{Poisson}(3)$. Each claim produces a net payment of $120 with probability 0.75 and zero otherwise, independently across claims and independently of $N$. Find the expected aggregate net payment and the probability it is zero. (Answer: $270; 0.1054)
A hypothetical insurer has $N\sim\operatorname{Binomial}(5,0.20)$. Each claim produces a fixed covered payment of $400, with no reinsurance. Find the expected aggregate covered payment. (Answer: $400)
Two independent claims occur with certainty. Each ground-up severity is $100 or $300 with equal probability. A policy has a $100 deductible and no limit; a reinsurer reimburses 20% of each covered payment. Find the expected aggregate net payment. (Answer: $160)
A hypothetical portfolio has $N\sim\operatorname{Poisson}(1)$. Each covered payment is $50 or $150 with equal probability. Per-claim reinsurance reimburses the amount above $100. Find the expected aggregate net payment and the probability it is zero. (Answer: $75; 0.3679)
A hypothetical portfolio has $N\sim\operatorname{Binomial}(2,0.50)$, and each claim produces a covered payment of $80. Reinsurance applies to the annual aggregate and reimburses the amount above an $80 attachment. Find the expected recovery and expected net aggregate. (Answer: $20; $60)

What You'll Learn