Zero-Claim Policies in Frequency-Severity Fitting

Why Zero-Claim Policies Matter

A policy with no claims has no claim amount to contribute to a claim-level severity fit. It does, however, provide information about how often claims occur: the observed count is zero during a known period of coverage. That distinction is essential when fitting a frequency-severity model. Removing zero-count policies from the frequency data changes the population being analyzed, while adding artificial zero severities to a claim-level severity data set changes the meaning of the severity response.

This tutorial considers policy-level claim counts and claim-level severities. Assume policies are observed over known exposure, claims are completely recorded for that exposure, and the severity sample consists of reported claim amounts. When using the simple frequency calculations below, assume a constant claim rate per exposure unit. More complex models can allow the rate and mean severity to vary with policy characteristics, but the distinction between the two data sets remains.

Definition: For policy $i$, let $e_i$ be its exposure, $N_i$ its number of claims, and $X_{ij}$ the amount of its $j$th claim. Frequency is the expected number of claims per exposure unit. Severity is the expected amount per claim. Under a frequency-severity model with the relevant conditional expectations, the expected loss per exposure unit is the frequency multiplied by the mean severity.

A frequency record is one policy-period record: it includes the policy’s exposure and its count $N_i$, including $N_i=0$. A severity record is one claim record: it includes the amount $X_{ij}$ for an observed claim. A zero-claim policy contributes a count and exposure to frequency estimation, but contributes no claim amount to severity estimation. The severity data set is therefore not a list of all policies with zeroes inserted for policies without claims.

Frequency and Severity Use Different Observational Units

For a simple constant-rate model, suppose $N_i$ has a Poisson distribution with mean $e_i\lambda$, where $\lambda$ is the claim rate per exposure unit. Assume the policy counts are independent conditional on their exposures, and that exposure scales the expected count proportionally. The maximum likelihood estimate of $\lambda$ is total claims divided by total exposure:

\widehat{\lambda} = \frac{\sum_i N_i}{\sum_i e_i}.

A zero count contributes no claims to the numerator, but its positive exposure remains in the denominator. Under a Poisson frequency GLM, the count likelihood includes the probability of observing zero claims, and the log of exposure is used as an offset when the model has a log link. Omitting zero-count policies discards both their observed count outcomes and their exposure.

For severity, suppose the observed claim amounts are modeled as independent observations from a severity distribution with mean $\mu$, conditional on being claims in the modeled claim population. The severity fit uses the claim amounts, not one row per policy. A zero-claim policy has no $X_{ij}$ value, so it cannot add a zero to the claim-level severity sample. The exact severity model and any weighting depend on the intended claim population and data structure; the basic rule here is that policy exposure is not itself a claim severity.

Worked Example: Fitting Both Components From Policy and Claim Records

A hypothetical insurer observes eight policies, each with one exposure unit. Their claim counts are $0,1,0,2,0,1,0,0$. The four claim amounts are $600, $1,000, $800, and $800. Under a constant Poisson frequency model and a common-mean severity model, calculate the frequency estimate, mean severity, and pure premium per exposure unit.

Method and frequency calculation. Retain all eight policy exposures, including the five policies with zero claims. Total claims are $0+1+0+2+0+1+0+0=4$, and total exposure is $8$. Thus $\widehat{\lambda}=4/8=0.5$ claims per exposure unit.

Severity calculation. Use the four claim amounts only. Their total is $$600+$1,000+$800+$800=$3,200$, so the mean severity estimate is $\widehat{\mu}=$3,200/4=$800$ per claim. The policies with zero claims do not supply additional severity observations.

Pure premium calculation. The estimated expected loss per exposure unit is $0.5\times $800=$400.00$. The result is $400.00 per exposure unit.

Check and common mistake avoided. The four claims are consistent with a mean frequency of $4/8=0.5$, and the claim amounts average $800. Treating each of the five zero-claim policies as a zero-severity claim would incorrectly change the severity sample and its target.

Why Excluding Zero Counts Biases Frequency Upward

The effect of excluding zero-count policies can be shown directly under the Poisson model. For a policy with exposure $e$, let its expected claim count be $m=e\lambda$. The probability of a positive count is $1-\exp(-m)$. Because a zero count contributes nothing to the count total, conditioning on a positive count raises the average observed count:

E[N\mid N>0] = \frac{E[N]}{P(N>0)} = \frac{m}{1-\exp(-m)}.

The first equality follows because $N=0$ whenever the policy is outside the selected group, so $E[N\mathbf{1}_{\{N>0\}}]=E[N]=m$. Dividing by $P(N>0)$ gives the conditional expectation. Since $0<1-\exp(-m)<1$ for $m>0$, this conditional mean exceeds $m$. Dividing that mean by exposure estimates a count rate among claim-positive policies, not the rate among all insured exposure.

Worked Example: Quantifying the Zero-Truncation Effect

A hypothetical portfolio has 100 policies, each with one exposure unit. Suppose claim counts follow a Poisson distribution with mean $m=0.5$ per policy. An analyst keeps only policies with at least one claim. Calculate the positive-policy proportion, the expected count per retained policy, and the naïve count-per-retained-exposure rate. Compare it with the true rate.

Method. For a Poisson count with mean $0.5$, the probability of no claims is $P(N=0)=\exp(-0.5)=0.6065$ to four decimals. Thus $P(N>0)=1-0.6065=0.3935$. The expected number of positive policies among 100 is approximately $100(0.3935)=39.35$.

The expected total number of claims across all 100 policies is $100(0.5)=50$. Conditional on a policy being positive, its expected count is \[ E[N\mid N>0]=\frac{0.5}{1-\exp(-0.5)} =\frac{0.5}{0.393469}=1.2707. \] Because each policy has one exposure unit, the naïve rate calculated within the retained group is 1.2707 claims per retained exposure unit. The target rate across all policies is $0.5$ claims per exposure unit.

Check and common mistake avoided. The inflation factor is $1.2707/0.5=2.5414$ (using unrounded values). This matches $1/P(N>0)$. The count per claim-positive policy is not the count per insured policy; discarding the zero-count denominator changes the frequency target.

In a realized sample, the numerical estimate depends on the observed counts and exposures. The derivation explains the direction of the selection effect under the stated Poisson model; it does not mean every small sample estimate must be larger than the corresponding full-data estimate. If exposure varies across policies or underlying risk differs by class, conditioning on positive counts can also change the mix of policies retained. A single overall adjustment generally cannot repair that altered mix.

Putting the Components Together Across Classes

When frequency and severity differ by rating class, calculate the expected loss per exposure unit within each class before aggregating. Portfolio pure premium is weighted by exposure, not by the number of claim records and not by an unweighted average of class pure premiums. The exposure totals must include all policies in each class, including those with zero claims.

Worked Example: Correctly Weighting Class Pure Premiums

A hypothetical two-class portfolio has Class A exposure of 4 units and 2 claims, with mean claim severity of $700. Class B has exposure of 6 units and 3 claims, with mean severity of $500. Both exposure totals include zero-claim policies. Calculate each class frequency and pure premium, then the portfolio pure premium.

Step 1: Calculate class frequencies. For Class A, $\widehat{\lambda}_A=2/4=0.5$ claims per exposure unit. For Class B, $\widehat{\lambda}_B=3/6=0.5$ claims per exposure unit.

Step 2: Calculate class pure premiums. Class A has $0.5\times $700=$350.00$ per exposure unit. Class B has $0.5\times $500=$250.00$ per exposure unit.

Step 3: Weight by exposure. The portfolio has 10 exposure units. Expected loss is $4($350)+6($250)=$1,400+$1,500=$2,900$. Therefore the portfolio pure premium is $$2,900/10=$290.00$ per exposure unit.

Check and common mistake avoided. There are 5 claims in 10 exposure units, giving portfolio frequency $5/10=0.5$. The claim-weighted mean severity is $(2($700)+3($500))/5=$2,900/5=$580$. Their product is $0.5\times $580=$290.00$, confirming the exposure-weighted result. Averaging $350 and $250 without exposure weights would give $300.00, which is incorrect because the two classes have different exposures.

Practical Checks Before Fitting

Define the frequency record.
Specify the policy or policy-period unit, the claim count, and the matching exposure period. Confirm that policies with zero claims remain in the data.

Define the severity record.
Specify what counts as a claim and which claim amounts enter the severity sample. Do not create a claim-level zero amount for a policy that had no claim.

Check the selected population.
If records were selected because they had claims, recognize that the frequency sample is zero-truncated. Its conditional count distribution is not the original all-policy count distribution.

Combine and aggregate consistently.
Multiply frequency by the corresponding mean severity, then use the relevant exposure weights to calculate a portfolio result.

A frequent data-processing error is to construct the frequency file by starting from claim transactions and then treating each remaining policy as one policy with claims. That file has lost the zero-count policies and often their exposure as well. Conversely, a claim-level severity file should not be expanded with policy records that have no claim. These are different omissions: the first changes frequency estimation; the second incorrectly treats nonclaims as claims of zero severity.

Key takeaway: Frequency estimation needs the full set of policy exposures and their claim counts, including zeroes. Claim-level severity estimation uses observed claim amounts and has no severity observation for a zero-claim policy. Keeping those data structures separate avoids selection bias and preserves the intended pure-premium target. This distinction is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Check Your Understanding

For each problem, distinguish the policy-level frequency data from the claim-level severity data before calculating.

A hypothetical portfolio has 12 exposure units and 6 claims with total claim amounts of $9,000. The exposure total includes policies with no claims. Calculate the frequency, mean severity, and pure premium per exposure unit. (Answer: 0.5 claims per exposure unit; $1,500.00 mean severity; $750.00 pure premium per exposure unit.)
For a one-unit policy, $N\sim\operatorname{Poisson}(0.4)$. Calculate the expected count conditional on at least one claim, and the factor by which it exceeds the unconditional frequency. Give probabilities or rates to four decimals. (Answer: 1.2133 claims per positive policy; factor 3.0332.)
Class A has 3 exposure units, 3 claims, and mean severity $400; Class B has 7 exposure units, 7 claims, and mean severity $200. The exposure totals include zero-claim policies. Calculate both class pure premiums and the exposure-weighted portfolio pure premium. (Answer: Class A $400.00, Class B $200.00, portfolio $260.00 per exposure unit.)
Four policy records have exposure-count pairs $(0.5,0)$, $(1.5,1)$, $(2,0)$, and $(1,2)$. Calculate frequency using all policies, then calculate the misleading rate if both zero-count records are dropped. (Answer: 0.6 claims per exposure unit using all records; 1.2 claims per exposure unit after dropping zero-count records.)
A hypothetical book has 10 total exposure units and 3 claims with amounts $500, $700, and $900. One policy contributes 4 of the exposure units but has no claims. Calculate frequency, claim-level mean severity, and pure premium; state whether the zero-claim policy supplies a severity observation. (Answer: 0.3 claims per exposure unit; $700.00 mean severity; $210.00 pure premium per exposure unit; no severity observation from the zero-claim policy.)

What You'll Learn