Choosing the Model for the Prediction You Need
A pure premium model predicts expected loss per unit of exposure. The practical choice is not simply between a model with more detail and one with less. It is between models that use the available data appropriately and produce the prediction the pricing task requires. A model for expected aggregate loss per policy-year may be unsuitable if the work also requires separate claim frequency and severity indications.
Three common choices are an aggregate pure premium model, a frequency-severity model, and a Tweedie model. The first models loss per exposure unit directly. The second models claim counts and claim amounts separately. A Tweedie model with variance power between one and two can represent an aggregate loss with a probability mass at zero and a continuous positive-loss component. None is automatically best: the choice depends on data structure, exposure, zeros, claim detail, and the target.
A Decision Process
Begin by identifying the observational unit and the prediction target. A record might represent a policy-term, a policy-month, a rating cell, or an individual claim. Confirm that the loss amount and exposure refer to the same period and coverage basis. Then ask what detail is available: do records include only aggregate loss, or also claim counts and individual claim amounts? Finally, specify what must be predicted: aggregate pure premium, claim frequency and severity separately, or a policy-level loss distribution.
State the loss measure, exposure unit, and prediction level. A policy-level expected loss and an expected loss per vehicle-year are different targets.
Aggregate loss and exposure support an aggregate model. Claim counts and claim amounts make a frequency-severity approach possible, provided the claim records align with the policy-period data.
Use exposure as the denominator or appropriate model weight/offset. Retain genuine zero-loss periods when they belong to the target population; do not treat a zero as missing.
Assess whether the chosen model’s distributional and exposure assumptions are plausible, then examine calibration and predictive performance on suitable validation data.
What Each Model Uses
| Model choice | Data structure | Useful when | Key caution |
|---|---|---|---|
| Aggregate pure premium | Aggregate loss and exposure for each observation | The target is expected loss per exposure and claim-level data are unavailable or unnecessary | Exposure differences must be handled; noisy individual loss rates can obscure the underlying pattern |
| Frequency-severity | Claim counts, claim amounts, and corresponding exposure/predictors | Separate frequency and severity predictions are needed, or their patterns are to be analyzed separately | Combining component predictions requires compatible definitions and attention to dependence or selection effects |
| Tweedie aggregate | Aggregate losses, including zeros, and exposure/predictors | A compound-Poisson-type aggregate model is suitable and claim-level detail is not required | The assumed aggregate distribution and exposure scaling must be reasonable for the data |
A log-link model often expresses a positive expected value multiplicatively. In an aggregate model, exposure can be incorporated so that expected aggregate loss is proportional to exposure, while the modeled rate varies with rating characteristics. In a frequency model, a log exposure offset serves the corresponding purpose for expected claim count. These are not interchangeable data preparations: the response and exposure relationship must match the modeled quantity.
A zero aggregate loss is informative. In aggregate data it contributes to the observed loss distribution and can inform a Tweedie fit. In a frequency-severity data set, a policy-period with zero claims informs frequency, while there is no claim severity observation for that period. Removing zero-claim periods from frequency fitting changes the population being modeled. Conversely, inventing a severity of zero for a period with no claim would mix the frequency and severity responses.
Why Frequency Times Severity Gives Pure Premium
Consider a frequency-severity model with exposure \(e\), claim count \(N\), and individual claim amounts \(X_1,\ldots,X_N\). Let aggregate loss be \(S=\sum_{j=1}^{N}X_j\). Under the standard conditional assumptions that claim amounts have mean \(\mu\) and are independent of the count, and that the expected count is \(E[N]=e\lambda\), the expected aggregate loss follows directly:
Dividing by exposure gives \(E[S]/e=\lambda\mu\): expected claims per exposure unit times expected amount per claim. This is the justification for multiplying modeled frequency by modeled severity. If severity and frequency are dependent after conditioning on the modeled predictors, or if the severity sample does not represent the claims generated by the frequency model, the simple product may not deliver the required expected loss. The model specification or prediction procedure must then address that mismatch.
For a Tweedie aggregate model with variance power between one and two, the distribution has a mass at zero and positive continuous outcomes under its compound Poisson-gamma interpretation. It can therefore model aggregate losses without separately recording counts and amounts. That convenience does not make it a substitute for claim-level outputs when those outputs are part of the required prediction.
Worked Example: Aggregate Data With Unequal Exposure
A hypothetical portfolio extract contains four policy-period records. Their exposures are \(0.50, 1.00, 0.75,\) and \(0.25\) policy-years, and their aggregate losses are \(\$600,\ \$0,\ \$1{,}800,\) and \(\$0\), respectively. Only aggregate loss and exposure are available. Calculate the portfolio pure premium and select a suitable model family for a larger data set with the same structure.
Method. Because the records contain aggregate losses but no claim counts or individual claim amounts, use an aggregate approach. A Tweedie model is a candidate if its distributional assumptions are appropriate; the four records themselves are not enough to validate a fitted model.
First add exposure:
Then add losses:
The portfolio pure premium is total loss divided by total exposure:
The check is \(\$960.00\times2.50=\$2{,}400\), the observed total loss. A common mistake is to divide by the two policies with positive losses, yielding \(\$1{,}200\) per policy. That is not the portfolio pure premium per policy-year: it excludes two policy-periods that had exposure and belong in the portfolio denominator. The model choice is based on available aggregate data, not on dropping zero-loss observations.
Worked Example: Claim Detail Supports Frequency-Severity
A hypothetical book has two rating groups. Group A has 100 policy-years, 80 claims, and total claim amount \(\$240{,}000\). Group B has 50 policy-years, 30 claims, and total claim amount \(\$150{,}000\). Assume claim amounts and counts are measured on the same coverage and period basis. Calculate each group’s empirical frequency, average severity, and pure premium, then predict expected loss for 0.40 policy-year in Group B.
Method. The claim counts and claim amounts are both available, and the task calls for component rates. Use frequency-severity calculations. These group averages are empirical indications, not a claim that a more detailed fitted model would have the same estimates.
For Group A, frequency is \(80/100=0.80\) claims per policy-year. Average severity is \(\$240{,}000/80=\$3{,}000\) per claim. Thus its pure premium is \(0.80\times\$3{,}000=\$2{,}400\) per policy-year.
For Group B, frequency is \(30/50=0.60\) claims per policy-year. Average severity is \(\$150{,}000/30=\$5{,}000\) per claim. Its pure premium is \(0.60\times\$5{,}000=\$3{,}000\) per policy-year.
For the 0.40 policy-year Group B risk, expected claim count is \(0.40\times0.60=0.24\). Expected loss is \(0.24\times\$5{,}000=\$1{,}200.00\), also \(0.40\times\$3{,}000=\$1{,}200.00\) using the group pure premium. The agreement checks the exposure scaling. A common error is to use 30 claims as the frequency numerator and call it a rate; frequency must be divided by its 50 policy-years.
Worked Example: Choose a Model for a Component Prediction
A hypothetical insurer has 200 policy-years of records for a new coverage. It has verified 160 claims and \(\$640{,}000\) of total claim amounts. A pricing team needs both expected claim count and expected aggregate loss for a proposed cohort of 75 policy-years with the same modeled risk characteristics. Assume the historical rates are suitable for this illustrative prediction. Which approach matches the requested outputs, and what are the indicated predictions?
Method. Choose frequency-severity because the requested claim count is a separate output and both count and severity data are available. The historical frequency is \(160/200=0.80\) claims per policy-year. Historical average severity is \(\$640{,}000/160=\$4{,}000\) per claim.
For 75 policy-years, the expected claim count is \(75\times0.80=60\) claims. The indicated expected aggregate loss is \(60\times\$4{,}000=\$240{,}000.00\). Equivalently, the pure premium is \(0.80\times\$4{,}000=\$3{,}200\) per policy-year, and \(75\times\$3{,}200=\$240{,}000.00\).
The two calculations agree, providing an arithmetic check. An aggregate Tweedie model could be considered for aggregate loss prediction, but aggregate loss alone does not supply a separately modeled expected claim count. Selecting by the required output avoids using a model merely because it can produce one of the requested quantities.
Validation and Common Selection Traps
Model choice is a specification decision, not a substitute for validation. For any candidate, check that the response, exposure, and predictors are aligned; that zero-loss records are treated consistently with the target; and that the fitted predictions are calibrated for relevant groups. Compare predictive performance on data not used to choose or fit the model when feasible. A more elaborate model is not preferable merely because it has more components, and a simple aggregate model is not preferable merely because its response is convenient.
- Ignoring exposure variation. A half-year record and a full-year record do not represent equal amounts of risk time. Use an exposure-consistent target and denominator or model structure.
- Dropping zero-loss periods. This can distort an aggregate model and can remove essential frequency information. Keep eligible observations even when their loss is zero.
- Using claim severity as if it were pure premium. Average amount per claim is not expected loss per exposure; frequency is also required.
- Choosing a model without checking the prediction target. If the task requires claim counts as well as aggregate loss, an aggregate-only prediction does not meet the full requirement.
- Assuming component predictions always multiply correctly. The frequency-severity product relies on compatible definitions and assumptions; assess dependence and population alignment.
Check Your Understanding
For each scenario, identify a suitable model approach and calculate any requested indication.
- A hypothetical portfolio has aggregate losses of \(\$3{,}600\) over 4.5 policy-years. Claim counts and amounts are unavailable. What is the observed pure premium, and what kind of data-based model is directly supported? (Answer: \(\$800.00\) per policy-year; an aggregate pure premium model, with Tweedie as a candidate if its assumptions are suitable.)
- A hypothetical group has 60 claims over 80 policy-years and total claim amounts of \(\$180{,}000\). Calculate frequency, average severity, and pure premium. (Answer: \(0.75\) claims per policy-year; \(\$3{,}000\) per claim; \(\$2{,}250\) per policy-year.)
- A portfolio has \(\$4{,}500\) aggregate loss over 6.0 policy-years, including periods with zero loss. Should the zero-loss periods be removed before calculating the portfolio pure premium? Give the rate. (Answer: No; retain eligible exposure, and the pure premium is \(\$750.00\) per policy-year.)
- A pricing request requires both expected claim counts and expected aggregate losses. The data contain policy-period exposure, claim counts, and claim amounts. Which model structure most directly matches the requested outputs? (Answer: Frequency-severity, subject to compatible definitions and appropriate assumptions.)
- A Tweedie aggregate model predicts expected loss per policy-year for a risk with 0.30 policy-year of exposure. Its indicated rate is \(\$2{,}400\) per policy-year. What aggregate expected loss does this imply, and does the prediction by itself give expected claim count? (Answer: \(\$720.00\); no, the aggregate prediction alone does not give expected claim count.)