Combining Frequency and Severity Predictions

From Two Model Predictions to a Pure Premium

A frequency model predicts how often claims are expected, while a severity model predicts the average cost of a claim. For ratemaking, the practical task is to combine those predictions on the same policy and coverage basis. The result can be a policy-level pure premium per unit of exposure, an expected loss for a particular policy term, or an exposure-weighted portfolio indication.

The key distinction is between a frequency rate and an expected claim count. A frequency rate is expressed per unit of exposure; an expected count applies to the policy’s actual exposure. Multiplying a per-exposure frequency rate by mean severity gives pure premium per exposure. Multiplying the expected claim count by mean severity gives expected loss for the policy term.

Definition: For policy $i$, let $e_i$ be its exposure, $\lambda_i$ its expected claim frequency per exposure unit, and $m_i$ its expected severity per claim. If severity is measured on the same coverage and claim basis as frequency, the policy’s pure premium per exposure unit is $\lambda_i m_i$. Its expected aggregate claim cost over the stated exposure is $e_i\lambda_i m_i$.

For example, exposure might be earned vehicle-years, insured building-years, or another clearly defined unit. Severity is measured in dollars per claim. Therefore, frequency in claims per vehicle-year multiplied by severity in dollars per claim gives dollars per vehicle-year. This unit check is a quick way to detect many calculation errors.

Combining Fitted Values

The frequency and severity predictions may come from separate fitted models. For a frequency model with a log link and exposure offset, a typical count specification is:

\log E[N_i]=\log(e_i)+\boldsymbol{x}_i^\mathsf{T}\boldsymbol{\beta}, \qquad \hat{\lambda}_i=\exp(\boldsymbol{x}_i^\mathsf{T}\hat{\boldsymbol{\beta}}).

Here $N_i$ is the claim count for policy $i$, and $\boldsymbol{x}_i$ contains its rating variables. The offset accounts for exposure in the count mean. The fitted rate $\hat{\lambda}_i$ is the predicted claim count per exposure unit; the fitted expected count for the policy is $e_i\hat{\lambda}_i$. A severity model supplies a predicted conditional mean severity, denoted $\hat m_i$, for claims on the corresponding policy and coverage basis.

Once the fitted values are available, the calculation is direct:

\widehat{PP}_i=\hat{\lambda}_i\hat m_i, \qquad \widehat{L}_i=e_i\hat{\lambda}_i\hat m_i.

The first quantity, $\widehat{PP}_i$, is the predicted pure premium per exposure unit. The second, $\widehat{L}_i$, is expected loss over the policy’s stated exposure. If the frequency model output is already an expected claim count for the policy term rather than a rate, multiply that count by severity to obtain expected loss; do not apply exposure a second time.

Condition: The severity prediction must represent expected cost per claim for the same coverage, claim definition, valuation basis, and relevant policy characteristics as the frequency prediction. The product gives expected cost when the conditional mean severity is appropriate for the claims counted by the frequency model. If the claim populations or bases differ, reconcile them before multiplying.

That alignment includes practical details such as policy limits, deductibles, and whether the severity target is paid or incurred. Earlier material on severity targets and censoring explains why these distinctions matter: a mean for capped payments is not automatically interchangeable with a mean for uncapped losses. The product is only meaningful after the two fitted predictions refer to compatible outcomes.

Why the Product Gives Expected Cost

The justification follows from conditional expectation. Suppose a policy has $N_i$ claims and each claim has severity $X_{ij}$. For the stated policy characteristics, assume each counted claim has conditional mean severity $m_i$, so that $E[X_{ij}\mid N_i]=m_i$. Then the expected total cost conditional on the claim count is:

E\left[\sum_{j=1}^{N_i}X_{ij}\,\middle|\,N_i\right] =\sum_{j=1}^{N_i}E[X_{ij}\mid N_i] =N_i m_i.

Taking expectations again gives $E[\sum_{j=1}^{N_i}X_{ij}]=E[N_i]m_i$. If expected claim count is $e_i\lambda_i$, expected total cost is $e_i\lambda_i m_i$, and dividing by exposure gives pure premium $\lambda_i m_i$. This argument does not require claim count to be independent of severity if the conditional mean for claims at each count remains $m_i$. If severity changes systematically with claim count or with an unmodeled feature associated with frequency, the simple product of the two marginal predictions may not represent the appropriate expected cost.

Worked Policy and Portfolio Calculations

Worked Example: Combining Predictions for Policies With Different Exposure

A hypothetical insurer has fitted frequency and severity models for a coverage measured in earned vehicle-years. The model predictions for three policies are shown below. Calculate each policy’s pure premium per vehicle-year, expected loss for its term, and the portfolio pure premium.

Policy	Exposure (vehicle-years)	Frequency rate (claims per vehicle-year)	Mean severity
A	1.00	0.08	$5,000
B	0.50	0.12	$4,000
C	2.00	0.05	$6,000

Method. Multiply frequency by severity for each policy’s rate, then multiply that rate by exposure for expected term loss.

Policy A has pure premium $0.08\times \$5{,}000=\$400.00$ per vehicle-year. Its expected claim count is $1.00\times0.08=0.08$, so its expected loss is $0.08\times \$5{,}000=\$400.00$.

Policy B has pure premium $0.12\times \$4{,}000=\$480.00$ per vehicle-year. Its expected claim count is $0.50\times0.12=0.06$, so its expected loss is $0.06\times \$4{,}000=\$240.00$.

Policy C has pure premium $0.05\times \$6{,}000=\$300.00$ per vehicle-year. Its expected claim count is $2.00\times0.05=0.10$, so its expected loss is $0.10\times \$6{,}000=\$600.00$.

Total exposure is $1.00+0.50+2.00=3.50$ vehicle-years. Total expected loss is $\$400.00+\$240.00+\$600.00=\$1{,}240.00$. Thus the portfolio pure premium is $\$1{,}240.00/3.50=\$354.2857\ldots$, or $354.29 per vehicle-year rounded to the nearest cent. The check is that $\$354.2857\ldots\times3.50=\$1{,}240.00$, before rounding.

Common mistake avoided: The unweighted mean of the three policy rates is $(\$400+\$480+\$300)/3=\$393.33$. That treats a half vehicle-year and two vehicle-years as equally representative. Portfolio pure premium must weight each policy rate by its exposure.

Worked Example: Applying Fitted Rating Relativities

For a hypothetical property coverage, a fitted frequency model has a base rate of $0.10$ claims per insured building-year, and a fitted severity model has a base mean of $\$4{,}500$ per claim. The models provide frequency and severity relativities for two policies. Policy D has frequency relativity $1.20$, severity relativity $1.10$, and exposure $0.75$ building-years. Policy E has relativities $0.80$ and $0.90$, respectively, and exposure $1.25$ building-years. Calculate the fitted rates and expected costs.

Method. Apply each model’s own relativity to its base prediction, multiply the resulting fitted frequency rate by fitted mean severity, and then apply exposure once.

For D, fitted frequency is $0.10\times1.20=0.12$ claims per building-year. Fitted severity is $\$4{,}500\times1.10=\$4{,}950.00$. Pure premium is $0.12\times\$4{,}950.00=\$594.00$ per building-year. Expected claim count is $0.75\times0.12=0.09$, and expected loss is $0.09\times\$4{,}950.00=\$445.50$.

For E, fitted frequency is $0.10\times0.80=0.08$ claims per building-year. Fitted severity is $\$4{,}500\times0.90=\$4{,}050.00$. Pure premium is $0.08\times\$4{,}050.00=\$324.00$ per building-year. Expected claim count is $1.25\times0.08=0.10$, and expected loss is $0.10\times\$4{,}050.00=\$405.00$.

Combined expected loss is $\$445.50+\$405.00=\$850.50$, and combined exposure is $0.75+1.25=2.00$ building-years. The portfolio pure premium is $\$850.50/2.00=\$425.25$ per building-year. As a check, $0.75\times\$594.00+1.25\times\$324.00=\$445.50+\$405.00=\$850.50$.

Common mistake avoided: Applying the frequency relativity to severity, or the severity relativity to frequency, would mix the two model components. Each relativity modifies only the fitted quantity it was developed for.

Worked Example: Converting a Frequency Rate With a Different Unit

A hypothetical equipment insurer’s frequency predictions are reported as claims per 100 exposure-years. For class R, the fitted frequency is 18 claims per 100 exposure-years, fitted mean severity is $\$3{,}200$, and exposure is 120 years. For class S, the corresponding values are 25 claims per 100 exposure-years, $\$2{,}800$, and 80 years. Calculate the class expected losses and combined pure premium per exposure-year.

Method. Convert both frequency rates to claims per exposure-year before multiplying by severity.

For R, the rate is $18/100=0.18$ claims per exposure-year. Its pure premium is $0.18\times\$3{,}200=\$576.00$ per exposure-year. Expected claim count is $120\times0.18=21.6$, and expected loss is $21.6\times\$3{,}200=\$69{,}120.00$.

For S, the rate is $25/100=0.25$ claims per exposure-year. Its pure premium is $0.25\times\$2{,}800=\$700.00$ per exposure-year. Expected claim count is $80\times0.25=20$, and expected loss is $20\times\$2{,}800=\$56{,}000.00$.

Combined exposure is $120+80=200$ years; combined expected loss is $\$69{,}120+\$56{,}000=\$125{,}120$. The combined pure premium is $\$125{,}120/200=\$625.60$ per exposure-year. Equivalently, the exposure-weighted calculation is $(120\times\$576+80\times\$700)/200=(\$69{,}120+\$56{,}000)/200=\$625.60$. The two calculations agree.

Common mistake avoided: Treating 18 claims per 100 years as 18 per year would overstate the frequency rate by a factor of 100. Convert the unit before multiplying.

Aggregation, Interpretation, and Exam Traps

For $k$ policies, the portfolio expected loss is the sum of their expected losses, and the portfolio pure premium per exposure unit is that total divided by total exposure:

\widehat{L}_{\mathrm{portfolio}} =\sum_{i=1}^{k}e_i\hat{\lambda}_i\hat m_i, \qquad \widehat{PP}_{\mathrm{portfolio}} =\frac{\sum_{i=1}^{k}e_i\hat{\lambda}_i\hat m_i} {\sum_{i=1}^{k}e_i}.

This is an exposure-weighted average of policy pure premiums, not generally their simple arithmetic average. If all policies have equal exposure, the two averages coincide; otherwise, each policy’s contribution depends on its exposure. For multiple coverage groups, perform the calculation on a consistent basis and aggregate the appropriate expected costs and exposures.

A frequent exam trap is applying exposure twice. If a model predicts an expected count of $0.09$ claims for a policy term and mean severity is $\$4{,}950$, expected loss is $0.09\times\$4{,}950=\$445.50$. Multiplying by the policy exposure again would be wrong because exposure is already reflected in the predicted count. By contrast, if $0.12$ is a rate per exposure unit and the policy has $0.75$ exposure units, first calculate the count $0.75\times0.12=0.09$.

A second trap is combining predictions with different bases. Frequency for one coverage limit multiplied by severity for another, or a paid-severity mean multiplied by counts defined on a different claim basis, does not automatically produce a valid pure premium. Verify units and definitions before doing arithmetic. These calculations produce expected loss; they do not, by themselves, include expenses, profit provisions, or other pricing adjustments.

Key takeaway: Identify whether frequency is a rate or a term-specific expected count. Multiply compatible fitted frequency and severity predictions, apply exposure exactly once, and aggregate expected losses before dividing by total exposure.

Check Your Understanding

For each problem, keep the frequency unit, severity basis, and exposure explicit.

A fitted rate is $0.07$ claims per machine-year, mean severity is $\$5{,}200$, and a policy has $1.4$ machine-years of exposure. Find the pure premium per machine-year and expected loss for the policy. (Answer: $364.00 per machine-year; expected loss $509.60.)
A policy has exposure $0.60$ years, a fitted frequency rate of $0.15$ claims per year, and fitted mean severity of $\$4{,}600$. Calculate its expected claim count and expected loss. (Answer: 0.09 claims; $414.00 expected loss.)
Class A has exposure 80, frequency $0.02$ claims per exposure unit, and mean severity $\$2{,}500$. Class B has exposure 120, frequency $0.03$, and mean severity $\$1{,}800$, on the same unit and coverage basis. Find the portfolio expected loss and pure premium. (Answer: $10,480.00 expected loss; $52.40 per exposure unit.)
A model reports 12 claims per 100 exposure-years. Mean severity is $\$2{,}750$, and the portfolio has 25 exposure-years. Convert the rate, then find expected count and expected loss. (Answer: 0.12 claims per exposure-year; 3 expected claims; $8,250.00 expected loss.)
For policy M, the fitted expected count for its term is 0.24 and mean severity is $\$7{,}000$; its exposure is 0.80 years. For policy N, the fitted expected count is 0.06 and mean severity is $\$5{,}000$; its exposure is 0.20 years. Calculate total expected loss and the combined pure premium per exposure-year. (Answer: $1,980.00 total expected loss; $1,980.00 per exposure-year.)

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