Claim-Level Versus Policy-Level Severity Sampling

Two Sampling Units, Two Different Targets

A severity model can be fitted to individual claims, while a pricing indication is often needed for a policy or unit of exposure. These are related quantities, but they answer different questions. Claim-level severity asks, among the claims that occur, how large is a claim on average at these rating characteristics? Policy-level loss cost asks, how much loss is expected per policy or per unit of exposure, including the chance of no claim?

The distinction matters because policies do not all produce the same number of claims. A policy with four claims contributes four observations to an ordinary claim-level severity sample; a policy with no claims contributes none. Yet the zero-claim policy still belongs in the portfolio when estimating loss per policy or per exposure. Treating those sampling units as interchangeable can distort both severity estimates and pricing calculations.

Definition: For policy $i$, let $N_i$ be its claim count during the observation period, and let $X_{ij}$ be the severity of its $j$th claim, for $j=1,\ldots,N_i$. Its aggregate loss is $L_i=\sum_{j=1}^{N_i}X_{ij}$, with $L_i=0$ when $N_i=0$. Claim-level severity describes the distribution or mean of $X_{ij}$ among claims. Policy-level loss cost is expected aggregate loss per policy or per unit of exposure; empirical loss cost is total observed loss divided by total exposure.

A claim-level GLM uses claim records as its observations and models a severity response conditional on a claim having occurred. The covariates may describe the policy or risk to which the claim belongs. A policy with no claims supplies no severity response: there is no claim amount to enter as an observation. This does not mean that the policy is irrelevant. Its exposure and zero claim count are relevant to frequency and to the portfolio’s total loss cost.

If policies have different exposures, dividing total losses by the number of policies is not generally the desired rate. Let $e_i$ denote policy $i$'s exposure in a specified unit. The empirical loss cost per exposure is $\sum_i L_i/\sum_i e_i$. This ratio includes zero-loss policies through the exposure denominator. It differs from an unweighted average of policy-specific loss costs when exposures differ.

How Severity and Frequency Relate to Loss Cost

Within a rating group, define expected claim count per policy as $m=E[N]$. Define the claim-weighted mean severity as the expected total claim amount divided by the expected number of claims, provided $E[N]>0$. This definition weights policies in proportion to the claims they produce; it is the population counterpart of total claim dollars divided by total claim count.

\mu_{\text{claim}}=\frac{E\!\left[\sum_{j=1}^{N}X_j\right]}{E[N]}, \qquad E[L]=E[N]\mu_{\text{claim}}.

The second identity follows directly from the first definition: multiplying both sides of the definition of $\mu_{\text{claim}}$ by $E[N]$ gives $E[\sum_{j=1}^{N}X_j]=E[N]\mu_{\text{claim}}$, and the sum is aggregate loss $L$. The identity does not require claim count and severity to be independent, because $\mu_{\text{claim}}$ is defined using expected total claim dollars per expected claim. If instead one uses a severity mean defined for a different sampling population, or conditions severity on policy characteristics in a way that does not match the claims being counted, the product need not represent the intended loss cost.

A familiar special case assumes that, within the rating group, each claim has the same conditional mean severity $\mu$ and that this mean applies across policies and claim positions. Then the expected aggregate loss is $E[N]\mu$. If exposure is $e$ and the expected count is $e\lambda$, where $\lambda$ is expected frequency per exposure, expected loss per exposure is $\lambda\mu$. Frequency accounts for whether and how often claims occur; severity accounts for the amount conditional on a claim.

This decomposition is useful for interpreting a pricing indication, but it does not make claim-level severity and policy-level loss cost the same response. In particular, a zero-claim policy contributes zero to observed aggregate loss and positive exposure to the loss-cost denominator, while it contributes no claim amount to a claim-level severity fit.

Identify the unit of analysis.
Claim records estimate a claim-conditional severity target. Policy records with exposure and aggregate losses can inform loss cost, which includes claim occurrence.

Match the denominator to the target.
For average claim severity, divide total claim dollars by total claims. For loss cost per exposure, divide total loss dollars by total exposure, including exposure on zero-claim policies.

Combine compatible frequency and severity measures.
Use a frequency per the same exposure basis and a claim-weighted severity for the corresponding risks. Their product gives expected loss per exposure under the stated definitions.

Worked Calculations

Worked Example: Zero-Claim Policies and the Severity Denominator

A hypothetical insurer observes four one-year policies, each with one policy-year of exposure. Policy A has two claims of $1,000 and $2,000. Policy B has no claims. Policy C has one claim of $3,000. Policy D has no claims. Calculate the mean observed claim severity and the average policy loss per policy-year. Explain why the answers differ.

Method and assumptions. Treat the records as the complete observed experience for these four policy-years. Claim-level severity is total claim dollars divided by the number of claims. Policy loss per policy-year is total aggregate loss divided by four policy-years.

First total the claim amounts: Policy A has $ \$1{,}000+\$2{,}000=\$3{,}000$; Policy B has $\$0$; Policy C has $\$3{,}000$; and Policy D has $\$0$. Total loss is $\$3{,}000+\$0+\$3{,}000+\$0=\$6{,}000$. There are $2+0+1+0=3$ claims.

The mean observed claim severity is $\$6{,}000/3=\$2{,}000$ per claim. The average policy loss per policy-year is $\$6{,}000/4=\$1{,}500$. The latter is lower because the denominator includes all four policy-years, including the two with no claims.

Check. Policy losses are $\$3{,}000,\$0,\$3{,}000,\$0$, whose sum is $\$6{,}000$; their average is $\$1{,}500$. The claim amounts also sum to $\$6{,}000$, and dividing by three claims gives $\$2{,}000$. Common mistake avoided: dividing claim dollars by four policies and calling $\$1{,}500$ the mean claim severity. That calculation is a policy-level loss average, not a claim-level average.

Worked Example: Unequal Exposures and Two Empirical Averages

In a hypothetical book, Policy A has 0.5 exposure units and two claims of $\$600$ and $\$900$. Policy B has 1.0 exposure unit and no claims. Policy C has 2.0 exposure units and four claims of $\$1{,}200$, $\$1{,}500$, $\$1{,}800$, and $\$2{,}100$. Calculate the claim-level mean severity and the loss cost per exposure unit. Also calculate the unweighted average of the three policy-specific loss costs.

Method. Use total claims as the claim-severity denominator and total exposure as the loss-cost denominator. Policy-specific loss cost is that policy’s aggregate loss divided by its own exposure.

Policy A’s aggregate loss is $\$600+\$900=\$1{,}500$. Policy B’s is $\$0$. Policy C’s is $\$1{,}200+\$1{,}500+\$1{,}800+\$2{,}100=\$6{,}600$. Thus total loss is $\$1{,}500+\$0+\$6{,}600=\$8{,}100$, total claims are $2+0+4=6$, and total exposure is $0.5+1.0+2.0=3.5$ units.

Mean claim severity is $\$8{,}100/6=\$1{,}350$. Loss cost per exposure unit is $\$8{,}100/3.5=\$2{,}314.2857\ldots$, or $\$2{,}314.29$ to the nearest cent.

The policy-specific loss costs are Policy A: $\$1{,}500/0.5=\$3{,}000$ per unit; Policy B: $\$0/1.0=\$0$; and Policy C: $\$6{,}600/2.0=\$3{,}300$. Their unweighted average is $(\$3{,}000+\$0+\$3{,}300)/3=\$2{,}100$ per unit. It does not equal the portfolio loss cost, because it gives the 0.5-unit and 2.0-unit policies equal weight. The exposure-weighted calculation is $(0.5(\$3{,}000)+1.0(\$0)+2.0(\$3{,}300))/3.5=\$8{,}100/3.5=\$2{,}314.29$.

Check and trap. The total of the policy losses is $\$8{,}100$, and both the claim and exposure calculations use that same numerator. Averaging policy-specific rates without exposure weights gives the wrong portfolio loss cost; it is not the same as total loss divided by total exposure.

Worked Example: Combining Claim-Weighted Severity and Frequency

A hypothetical portfolio has 80 one-unit policies in class A and 20 one-unit policies in class B. In each class, claim count follows a Poisson distribution: $N\sim\operatorname{Poisson}(0.1)$ for class A and $N\sim\operatorname{Poisson}(1)$ for class B. Conditional on a claim, severity has a Gamma distribution with shape parameter 2 and scale parameter $\$500$ in class A, and shape parameter 2 and scale parameter $\$1{,}000$ in class B. For a Gamma distribution with shape $\alpha$ and scale $\theta$, the mean is $\alpha\theta$. Calculate expected loss per policy exposure unit for the combined portfolio, and show why simply averaging the two class severity means is inappropriate.

Method and assumptions. The stated Poisson means are expected claim counts per one-unit policy. Claim severities have the stated class-specific Gamma means; for expected aggregate loss, only those means and expected counts are needed. Calculate expected claim counts and losses by class, then divide total expected loss by total exposure.

Class A’s severity mean is $2(\$500)=\$1{,}000$. Its expected claims are $80(0.1)=8$, so its expected loss is $8(\$1{,}000)=\$8{,}000$. Class B’s severity mean is $2(\$1{,}000)=\$2{,}000$. Its expected claims are $20(1)=20$, so its expected loss is $20(\$2{,}000)=\$40{,}000$.

Total expected loss is $\$8{,}000+\$40{,}000=\$48{,}000$, and total exposure is $80+20=100$ units. Therefore expected loss per exposure unit is $\$48{,}000/100=\$480$.

The combined expected claim count is $8+20=28$, so expected frequency per exposure unit is $28/100=0.28$. The claim-weighted severity across the two classes is $\$48{,}000/28=\$1{,}714.2857\ldots$. Multiplying gives $0.28(\$1{,}714.2857\ldots)=\$480$ per exposure unit, agreeing with the direct calculation. By contrast, the unweighted average of the two class severity means is $(\$1{,}000+\$2{,}000)/2=\$1{,}500$. Multiplying that by $0.28$ gives $\$420$, which is wrong: class B produces more claims per policy and therefore contributes more claims to the combined severity target.

Check and trap. The class expected losses sum to $\$48{,}000$, and dividing by 100 exposures gives $\$480$. Common mistake avoided: averaging class severity means equally when the number of claims, not the number of classes, determines their contribution to pooled claim-level severity.

Interpretation and Practical Checks

A claim-level severity model is appropriate when the response of interest is claim amount conditional on a claim and the claim records represent the intended claim population. Its fitted mean is not, by itself, a loss cost per policy. To obtain expected loss per exposure, it must be combined with a compatible expected frequency, or otherwise translated to an aggregate-loss target. If a policy’s covariates affect both frequency and severity, those effects belong in their respective components of the pricing calculation.

The claim sample is naturally claim-weighted: a policy generating more claims supplies more severity observations. That is not automatically a bias. It is the correct weighting for the mean severity of a randomly selected claim in the portfolio. It can, however, differ from an equally policy-weighted average of policy-specific severity means. When the objective is the severity distribution for a randomly selected policy rather than a randomly selected claim, the target and sampling scheme must be stated explicitly.

A useful diagnostic is to label every numerator and denominator before calculating. “Total claim dollars divided by total claims” is an average claim amount. “Total claim dollars divided by total exposure” is an empirical loss cost. “Average of policy-specific loss costs” is a third statistic and generally requires exposure weighting to recover the portfolio rate. Keep the exposure basis consistent with the frequency measure when using a frequency-severity decomposition.

Key takeaway: Zero-claim policies have no observed claim severity, so they are not severity records. They still contribute exposure and zero aggregate loss to policy-level loss cost. Claim severity, claim frequency, and exposure must be matched to their intended sampling units before they are combined.

Check Your Understanding

For each problem, distinguish the claim denominator from the policy or exposure denominator.

A hypothetical portfolio has four one-unit policies. Their claim records are: Policy A, $\$400$ and $\$600$; Policy B, no claims; Policy C, $\$1{,}000$; Policy D, $\$800$ and $\$1{,}200$. Find mean claim severity and loss per policy-year. (Answer: mean claim severity $\$800$; loss per policy-year $\$1{,}000$.)
Three hypothetical policies have exposures 0.5, 1.5, and 0.0 units. The first has one $\$900$ claim, the second has no claims, and the third has three claims of $\$1{,}200$, $\$1{,}500$, and $\$1{,}800$. Find the claim mean and loss cost per exposure unit. (Answer: claim mean $\$1{,}350$; loss cost $\$2{,}700$ per exposure unit.)
A hypothetical book has 60 one-unit policies with Poisson claim count mean 0.2 and mean claim severity $\$500$, and 40 one-unit policies with Poisson claim count mean 0.5 and mean claim severity $\$1{,}500$. Find total expected loss per exposure unit and the pooled claim-weighted mean severity. (Answer: $\$360$ per exposure unit; pooled claim-weighted severity $\$1{,}125$.)
Three hypothetical policies have exposure 0.25, 0.75, and 1.00 units. Their aggregate losses are $\$2{,}000$, $\$0$, and $\$3{,}000$, respectively; the positive losses consist of one claim and two claims, respectively. Find the claim mean and portfolio loss cost per exposure unit. (Answer: claim mean $\$1{,}666.67$; loss cost $\$2{,}500$ per exposure unit.)

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