Why the Response Choice Matters
A pure premium is an expected loss amount per unit of exposure. There is more than one way to organize the data used to estimate it. A model can use the number of claims, the amounts of individual claims, or the total loss for each policy period. These are not interchangeable responses: each has a different observational unit, includes different records, and estimates a different quantity directly.
The choice affects which rows belong in the model and how zero-claim policies are represented. It also determines whether the response is a count, a positive dollar amount, or a policy-period loss amount. A model may ultimately support the same pricing objective as another model, but the response definition must match the intended target and the data available.
Here, exposure might be one earned vehicle-year, one building-year, or another defined unit. A policy period with exposure \(e_i=0\) cannot have a defined loss-per-exposure response and should not be treated as a positive-exposure observation. Monetary responses must also use a consistent basis: for example, losses may need to be evaluated at the same development age, coverage definition, deductible, and limit.
Three Responses, Three Data Structures
| Response choice | One model row represents | Response | Direct target |
|---|---|---|---|
| Claim count | A policy period or other exposure unit | \(N_i\), including zero | Expected number of claims for that exposure |
| Claim severity | One claim | \(X_{ij}\), usually positive | Expected amount per modeled claim under the selected claim basis |
| Aggregate loss per exposure | A policy period or other exposure unit | \(L_i/e_i\) | Expected aggregate loss per exposure unit |
For claim counts, the usual response is an integer count for every eligible policy period, including periods with no claims. Under a Poisson count model with mean proportional to exposure, the model can use \(\log(e_i)\) as an offset. The offset adjusts the expected count for the amount of exposure; it is not a predictor whose coefficient is estimated.
In this specification, \(\lambda_i\) is the modeled expected claim count per exposure unit and \(\eta_i\) is the linear predictor for that rate. The proportionality assumption is that doubling exposure, all modeled characteristics held fixed, doubles expected claim count. If that assumption is not appropriate, the offset specification needs reconsideration.
For a severity model, each row represents a claim, and the response is an amount defined for that claim. A severity dataset generally contains no row for a policy period with no claims, because there is no claim amount to record for that period. The fitted mean is conditional on a claim being represented in the severity data and on the claim definition and selection process used. Deductibles, limits, censoring, or claim selection can change the severity target; the amount must be defined consistently with the intended application.
For an aggregate-loss response, each row represents a policy period and its response is that period's total covered loss divided by its exposure. A zero-claim policy period has \(L_i=0\) and therefore a zero response when \(e_i>0\). This approach preserves the policy-period unit and directly models loss per exposure. An alternative parameterization may model \(L_i\) as the dollar response with exposure handled in the model, but it must reflect the assumed relationship between expected loss and exposure. A rate response should not automatically be given the same offset as an unscaled dollar total.
Worked Examples
Worked Example: Claim Counts With Unequal Exposure
A hypothetical insurer has three policy periods with earned exposures of 0.5, 1.0, and 1.5 vehicle-years and reported claim counts of 0, 1, and 3. Calculate the observed portfolio frequency per vehicle-year. Then state the expected claim count for a policy period with 0.75 vehicle-years if its modeled frequency is 1.2 claims per vehicle-year.
Method. Use policy-period counts as the responses and divide total claims by total exposure for the observed portfolio rate. Under the stated proportional-exposure assumption, expected count equals exposure times frequency.
Total claims are \(0+1+3=4\). Total exposure is \(0.5+1.0+1.5=3.0\) vehicle-years. Thus the observed portfolio frequency is \(4/3=1.3333\) claims per vehicle-year.
For the specified policy period, expected count is \(0.75 \times 1.2=0.90\) claims. A mean count need not be an integer; realized claim counts are integers, but their expectation can be fractional.
Check. The observed rate times total exposure gives \((4/3)\times3.0=4\) claims, matching the data. The calculation avoids the common mistake of averaging the three policy-period count values, which gives \(4/3\) claims per policy period without accounting for unequal exposures.
Worked Example: Claim-Level Severity Responses
A hypothetical book has two policy periods. Policy A has exposure 1.0 vehicle-year and two covered claims of $900 and $1,500. Policy B has exposure 0.5 vehicle-year and one covered claim of $1,800. Using these three claims as the severity observations, calculate the claim-level arithmetic mean. Also calculate the aggregate loss per vehicle-year for the two policy periods together, and explain why it is a different response summary.
Method. A claim-level severity response has one row per claim, so each of the three claims is counted once in the arithmetic mean. Aggregate loss per exposure uses policy-period totals and total exposure.
Total claim dollars are \($900+$1,500+$1,800=$4,200\). There are three claim rows, so the mean claim amount is \($4,200/3=$1,400.00\).
Policy A's aggregate loss is \($900+$1,500=$2,400\), and Policy B's aggregate loss is \($1,800\). Total exposure is \(1.0+0.5=1.5\) vehicle-years. The combined aggregate loss per vehicle-year is \($4,200/1.5=$2,800.00\).
Check. The severity answer is dollars per claim; the aggregate answer is dollars per vehicle-year. They differ in both unit and observational basis. The common mistake of treating the three claim amounts as policy-period losses would omit that Policy A has two claims and would not produce the policy-period aggregate response.
Worked Example: Policy-Period Aggregate Loss per Exposure
A hypothetical small commercial portfolio records the following policy-period data: Period 1 has exposure 0.5 building-years and aggregate loss $600; Period 2 has exposure 1.0 building-year and aggregate loss $0; Period 3 has exposure 1.5 building-years and aggregate loss $3,600. Calculate each policy-period response \(L_i/e_i\) and the exposure-weighted portfolio aggregate loss per building-year. State which periods are rows in this response dataset.
Method. Divide each policy-period aggregate loss by that period's positive exposure. For the portfolio summary, divide total aggregate loss by total exposure; equivalently, weight each period's loss-per-exposure response by its exposure.
Period 1 response is \($600/0.5=$1,200\) per building-year. Period 2 response is \($0/1.0=$0\) per building-year. Period 3 response is \($3,600/1.5=$2,400\) per building-year. All three policy periods are rows, including the period with zero loss.
Total loss is \($600+$0+$3,600=$4,200\), and total exposure is \(0.5+1.0+1.5=3.0\) building-years. The portfolio rate is \($4,200/3.0=$1,400.00\) per building-year. As a check, the exposure-weighted average is \((0.5\times$1,200+1.0\times$0+1.5\times$2,400)/3.0=($600+$0+$3,600)/3.0=$1,400.00\).
Check. The unweighted average of the three policy-period responses is \(($1,200+$0+$2,400)/3=$1,200\), which is not the portfolio loss per exposure because the periods have unequal exposures. Use the exposure-weighted calculation, or equivalently total loss divided by total exposure.
Why Exposure Treatment Follows the Response
The response definition determines the scale to which exposure must be applied. For a count model, the response is an integer count, so exposure enters the expected count. For a policy-period rate response, the response has already been divided by exposure; its unit is loss per exposure. For a severity response, the row is a claim, not an exposure period, so policy exposure is not a substitute denominator for the claim amount.
The count offset follows directly from the assumed proportionality. Suppose policy periods with the same modeled characteristics have rate \(\lambda_i\) claims per exposure unit. A period with exposure \(e_i\) then has expected count \(e_i\lambda_i\). Taking logarithms gives \(\log(E[N_i])=\log(e_i)+\log(\lambda_i)\). If the rate has log-linear form \(\log(\lambda_i)=\eta_i\), the result is the log-link equation with \(\log(e_i)\) as a fixed offset. This explains why the offset coefficient is one: it is a consequence of the proportional-exposure assumption, not an additional fitted effect.
Aggregate-loss responses require particular care when fitting a GLM. One may define the response as a per-exposure rate, or define it as the total dollars for a policy period and specify exposure handling for that dollar-scale mean. These choices are related but are not automatically identical likelihood specifications. A modeler must use a response, variance structure, and exposure treatment that agree with the intended model. The response choice alone does not establish a distribution or guarantee that two fitted approaches give identical predictions.
Selection, Coverage, and Practical Checks
Before fitting any of the three responses, define the loss basis. If severity observations are limited to claims above a deductible, they represent a selected or conditional claim sample rather than an unrestricted claim-size sample. If a policy limit caps recorded payments, a paid severity amount may represent a limited payment rather than the underlying loss. Similarly, an aggregate policy-period response should include losses consistently with the coverage, deductible, limit, and valuation rules being modeled.
A useful data audit is to reconcile records across units. For each policy period, the claim counts should agree with the number of associated claim records under the chosen claim definition, and the aggregate loss should agree with the sum of the included claim-level amounts if both use the same basis. Any disagreement should be explained, such as differing development treatments or claim inclusion rules, before comparing model results.
- For counts, retain eligible zero-claim periods and confirm that exposure is positive and measured in the intended unit.
- For severity, confirm that each row is a claim and that the claim amount and claim selection mechanism match the target.
- For aggregate loss per exposure, retain eligible zero-loss periods, divide by the correct exposure, and account for unequal exposure when summarizing the portfolio.
Check Your Understanding
For each problem, identify the relevant response and keep its unit and denominator explicit.
- A hypothetical policy has 0.4 earned years and two claims. What is its observed count response, and what is its observed frequency per earned year? (Answer: Count response \(=2\) claims; frequency \(=2/0.4=5\) claims per earned year.)
- A severity dataset contains claim amounts of $700, $1,100, and $2,000 from three claims. What is the arithmetic mean severity, and how many response rows are there? (Answer: $1,266.67 per claim, rounded to cents; 3 rows.)
- Two policy periods have losses of $500 and $2,100, with exposures of 0.5 and 1.5 units. What are the policy-period loss-per-exposure responses and the combined portfolio rate? (Answer: $1,000 and $1,400 per exposure unit; combined rate \(=$2,600/2.0=$1,300\) per exposure unit.)
- A policy-period count model uses a log link and assumes expected count is proportional to exposure. If exposure is 2.5 units and the modeled frequency is 0.8 claims per unit, what is expected count? State the offset value. (Answer: Expected count \(=2.5(0.8)=2.0\) claims; offset \(=\log(2.5)\).)
- A policy-period dataset has exposures 0.25, 0.75, and 1.0, with losses $300, $0, and $2,400, respectively. Calculate the combined aggregate loss per exposure and explain why the zero-loss period remains a row. (Answer: \($2,700/2.0=$1,350\) per exposure unit; it is an eligible positive-exposure policy period and its aggregate-loss response is zero.)