Why the Prediction Target Matters
A pricing analysis can start with the same policies and claim records but ask different questions. How many claims should the insurer expect? How large is a claim, given that one occurs? What is the expected claim cost per unit of exposure? These are related questions, but their targets are not interchangeable. Choosing one without checking the decision and the data can produce a model that is statistically competent but operationally irrelevant.
For example, a hypothetical insurer estimating expected claim dollars needs more than a model of claim counts unless claim amounts are also supplied or assumed. Conversely, a team planning the number of claims to review may need a frequency prediction, not a dollar-cost prediction. The choice should be made before fitting a model: define the outcome, its unit, the population to which it applies, and the decision the prediction will support. This target-selection problem is aligned with the SOA Exam PA (Predictive Analytics) syllabus.
A zero-claim policy has a frequency of zero and a pure premium of zero over its observed period. It has no observed claim severity: \(L_i/N_i\) is undefined when \(N_i=0\). This distinction matters when constructing a modeling table. A severity model is ordinarily fit to claim-level records or to policy periods with claims, whereas a frequency or pure-premium target can include zero-claim periods.
Three Targets, Three Interpretations
| Target | Typical response | Unit and useful decision |
|---|---|---|
| Claim frequency | Claim count \(N_i\), with exposure accounted for, or a rate \(N_i/e_i\) | Claims per exposure unit; useful for expected claim counts or claim workload |
| Claim severity | Cost per claim, for observed claims | Dollars per claim; useful for estimating claim amounts conditional on a claim |
| Pure premium | Aggregate cost \(L_i\) relative to exposure \(e_i\) | Dollars per exposure unit; directly targets expected claim cost per exposure |
The ratio identity \(L_i/e_i=(N_i/e_i)(L_i/N_i)\) holds for a policy period with \(N_i>0\). It explains why frequency and severity can be combined to form a pure-premium estimate. However, the identity alone does not determine how a model should be fitted. The targets have different response distributions, use different records, and answer different questions. A model of severity among claimants does not directly predict which policies will have claims; a frequency model does not say how costly those claims will be.
Worked Example: Calculate Rates Without Averaging Them Incorrectly
Worked Example: Calculate Rates Without Averaging Them Incorrectly
A hypothetical insurer summarizes two groups of policy-years. Group A has 80 exposure units, 8 claims and $24,000 of total claim cost. Group B has 20 exposure units, 6 claims and $30,000 of total claim cost. Calculate each group’s frequency, severity and pure premium, then calculate the overall rates. Identify the appropriate overall pure premium for a portfolio-level cost estimate.
Method. Divide claims by exposure for frequency, claim cost by claims for severity, and claim cost by exposure for pure premium. For the combined portfolio, add the numerators and denominators before taking ratios.
For Group A: \(8/80=0.10\) claims per exposure unit; \(\$24{,}000/8=\$3{,}000\) per claim; and \(\$24{,}000/80=\$300\) per exposure unit.
For Group B: \(6/20=0.30\) claims per exposure unit; \(\$30{,}000/6=\$5{,}000\) per claim; and \(\$30{,}000/20=\$1{,}500\) per exposure unit.
The portfolio has \(80+20=100\) exposure units, \(8+6=14\) claims and \(\$24{,}000+\$30{,}000=\$54{,}000\) of claim cost. Its frequency is \(14/100=0.14\), its severity is \(\$54{,}000/14=\$3{,}857.142857\ldots\), and its pure premium is \(\$54{,}000/100=\$540\). Rounding only the final dollar results to cents gives a severity of \(\$3{,}857.14\) per claim and a pure premium of \(\$540.00\) per exposure unit.
As a check, the unrounded overall frequency multiplied by the unrounded overall severity is \(0.14\times \$3{,}857.142857\ldots=\$540\). The overall pure premium is also the total cost divided by total exposure, as required.
Common mistake avoided. The unweighted mean of the two group pure premiums is \((\$300+\$1{,}500)/2=\$900\). That is not the portfolio rate because the groups have unequal exposure. Dividing total cost by total exposure, rather than averaging group rates equally, gives the correct \(\$540\).
How Frequency and Severity Combine
The relationship between frequency and severity can be expressed as an expectation, not just a ratio of observed totals. Suppose a policy with characteristics \(x\) has \(N\) claims over one exposure unit, and the cost of claim \(j\) is \(Z_j\). Total cost is \(L=\sum_{j=1}^{N} Z_j\). If expected claim count is \(\lambda(x)=E[N\mid x]\), and each claim has conditional expected cost \(m(x)\) for this policy class, then the expected total cost per exposure unit is the expected claim count times expected cost per claim.
Here the assumptions are that the claim costs being averaged correspond to the same policy class and exposure basis as the frequency prediction, and that \(m(x)\) is the appropriate expected cost for each claim in that class. To justify the result, condition first on the claim count. Given \(N=n\), if the \(n\) claims have mean cost \(m(x)\), the expected sum is \(E[L\mid N=n,x]=nm(x)\), by additivity of expectation. Averaging over the possible values of \(N\) gives \(E[L\mid x]=E[E[L\mid N,x]\mid x]=E[Nm(x)\mid x]=m(x)E[N\mid x]\).
The result does not say that every frequency and severity model can be multiplied without thought. The severity mean must apply to the claims generated by the risks being predicted. If claim size differs systematically across risks, relevant predictors must be reflected in the severity estimate or the groups being combined. If the unit of exposure differs between the models, it must be reconciled before combining predictions.
Select a Modeling Route From the Decision and Data
Decide whether the output must be expected claim counts, cost per claim or claim dollars per exposure unit. Specify the time period and exposure basis; an annual rate and a half-year expected amount are not the same quantity.
Counts and exposure support a frequency analysis. Claim amounts support severity analysis among claims. Aggregate cost and exposure support a direct pure-premium target. Do not treat an unavailable outcome as if it were present in the data.
A count model can use counts with exposure accounted for, for example through an exposure offset in a log-mean model. A severity model uses claim amounts on claim records. A direct pure-premium model uses cost per exposure or models aggregate cost with exposure reflected in its expected value. The response distribution, link and fitting approach must suit the response and the data structure.
A frequency prediction alone is not a dollar estimate. A severity prediction alone is conditional on a claim. For expected cost, use a pure-premium prediction or combine compatible frequency and severity estimates, then express the result in the decision’s units.
For a count model with a log link, one common specification is \(\log E[N_i\mid x_i,e_i]=\log(e_i)+x_i^\mathsf{T}\beta\). The exposure term is an offset: it accounts for the amount of time or risk observed, while the modeled rate varies with predictors \(x_i\). This route is useful when counts and exposure are available and expected counts are the target. A Poisson model additionally assumes that, conditional on its predictors and exposure, the count has a Poisson distribution; that distributional assumption should not be asserted merely because the response is a count.
A direct pure-premium route is attractive when the pricing decision is about expected dollars per exposure unit and claim cost is available. A frequency-severity route is useful when both counts and claim amounts are available and separate estimates are valuable. These routes need not give identical fitted predictions: they can use different response structures and estimation procedures. Their predictions should be interpreted on a common exposure basis, and the route should be chosen for the decision rather than because one response column is easiest to construct.
Worked Example: Choose Frequency for a Claim-Workload Decision
Worked Example: Choose Frequency for a Claim-Workload Decision
A hypothetical portfolio has recorded 7 claims over 100 exposure units for a particular risk group. A fitted frequency model predicts 0.09 claims per exposure unit for a new portfolio. The operations team must estimate claim workload for 40 policies, each expected to contribute 0.5 exposure units. A separate pricing exercise has supplied an assumed average severity of $4,200. Select the target for the workload decision and calculate the implied expected claim count and cost, while distinguishing what the frequency data establish from what the severity assumption supplies.
Method. Use frequency for workload, convert the policies to total exposure, and multiply the predicted rate by that exposure. Use severity only for the additional dollar-cost calculation.
The historical group’s observed frequency is \(7/100=0.07\) claims per exposure unit. The model’s prediction for the new portfolio is \(0.09\), which is the prediction to use here. Total new exposure is \(40\times0.5=20\) units. Thus the expected claim count is \(0.09\times20=1.8\) claims. For one half-exposure policy, the corresponding expected count is \(0.09\times0.5=0.045\) claims.
The expected cost per exposure unit implied by the supplied severity assumption is \(0.09\times \$4{,}200=\$378\). For 20 exposure units, the implied expected cost is \(20\times \$378=\$7{,}560\); for one half-exposure policy, it is \(0.5\times \$378=\$189\).
The check is \(1.8\) expected claims multiplied by \(\$4{,}200\) per claim, giving \(\$7{,}560\). The workload estimate comes from the frequency model. The dollar estimate additionally depends on the supplied severity assumption; the count data alone do not establish that average claim cost.
Common mistake avoided. Reporting 0.09 as the expected claims for all 40 policies ignores that the portfolio has 20 exposure units. The rate must be multiplied by total exposure to produce the expected portfolio count.
Worked Example: Build an Expected Cost From Two Predictions
Worked Example: Build an Expected Cost From Two Predictions
For a hypothetical risk class, a frequency model predicts 0.12 claims per full exposure unit. A severity model, fitted to claim amounts for comparable risks, predicts an expected claim cost of $5,200. A policy is expected to contribute 0.75 exposure units. Calculate its expected claim count and cost using the frequency-severity route. A separate direct pure-premium model predicts $640 per full exposure unit; calculate its implied cost for the policy and explain why the two model results need not match.
Method. Put both component predictions on the same risk and exposure basis. First scale the expected claim count to the policy’s exposure, then multiply by expected severity.
For 0.75 exposure units, expected claim count is \(0.12\times0.75=0.09\). Expected cost per full exposure unit from the component models is \(0.12\times \$5{,}200=\$624\). The policy’s expected cost is therefore \(0.75\times \$624=\$468\). Equivalently, \(0.09\times \$5{,}200=\$468\).
The direct pure-premium prediction gives \(0.75\times \$640=\$480\). The two estimates differ by \(\$480-\$468=\$12\). This is not an arithmetic contradiction: one prediction comes from a direct cost model and the other from separately fitted frequency and severity models. Fitting and model assumptions can produce different estimates, even though the expectation identity holds when its assumptions apply. Both calculations use the same 0.75 exposure units.
Common mistake avoided. Multiplying the full-exposure pure-premium prediction of $640 by the full-exposure frequency-severity result of $624, or comparing one full-year figure with one 0.75-exposure amount, would mix units. The direct model’s $480 and the combined estimate’s $468 are both amounts for this 0.75-exposure policy.
Common Target-Selection Traps
- Choosing a frequency model for a dollar decision without severity. Frequency supplies expected claims per exposure, not dollars. State what severity information or assumption is needed before reporting expected cost.
- Fitting severity to all policy periods as though each had a claim. Severity is defined per claim. Zero-claim periods belong in frequency and pure-premium analyses, but they do not have observed claim severity.
- Using a simple average of unequal-exposure rates. A portfolio rate should reflect the exposure represented by each group. In general, combine totals and then divide by total exposure.
- Ignoring the prediction unit. A rate per exposure unit must be multiplied by the exposure in the decision period. Confirm the unit before translating a rate into expected counts or dollars.
- Assuming the direct and component routes must agree exactly. The expectation relationship is a useful consistency check under its assumptions, not a guarantee that separately fitted models produce identical estimates.
Check Your Understanding
For each hypothetical case, identify the relevant quantity and keep its units explicit.
- A group has 40 exposure units, 5 claims and $39,000 of aggregate claim cost. Calculate its observed frequency, severity and pure premium. (Answer: Frequency \(=5/40=0.125\) claims per exposure unit; severity \(=\$39{,}000/5=\$7{,}800\) per claim; pure premium \(=\$39{,}000/40=\$975\) per exposure unit.)
- A model predicts 0.06 claims per exposure unit for a hypothetical fleet. The fleet will contribute 1.5 exposure units. Calculate its expected claim count. (Answer: \(0.06\times1.5=0.09\) expected claims.)
- A hypothetical risk class has predicted frequency 0.15 claims per exposure unit and predicted severity of $3,600. For a policy with 0.4 exposure units, calculate expected claim cost. (Answer: Pure premium \(=0.15\times\$3{,}600=\$540\) per exposure unit; expected cost \(=0.4\times\$540=\$216\).)
- A portfolio has 60 exposure units and $18,000 in aggregate claim cost. Calculate its pure premium. Explain why an average claim amount cannot be calculated without a claim count. (Answer: Pure premium \(=\$18{,}000/60=\$300\) per exposure unit; severity cannot be calculated without the number of claims.)
- A hypothetical team has claim counts and exposure but no claim-cost records. Its decision is to estimate the number of claims expected over 25 exposure units, and a model predicts frequency of 0.08. It is also asked for expected dollars, with no severity estimate supplied. State the expected count and whether the dollar estimate is identified by the supplied information. (Answer: Expected count \(=0.08\times25=2\); expected dollars are not identified without a severity estimate or other claim-cost information.)