Claim Severity Sampling and Policy-Level Targets

Why the Sampling Unit Matters

A claim-level data set and a policy-level data set answer different questions. A claim-level record describes one reported claim, so averaging those records estimates the average amount per claim. A policy-level record describes one policy over a specified period, including policies with no claims, so averaging policy losses estimates the average amount per policy or exposure. Confusing these targets can produce a pure premium that is too high, too low, or expressed on the wrong exposure basis.

For frequency-severity ratemaking, the basic components must refer to the same portfolio, time period, loss definition, and exposure basis. The frequency is the number of claims per unit of exposure; severity is the average amount per claim. A portfolio's pure premium is its total loss divided by its exposure. When the components are calculated consistently, their product equals the policy-level pure premium. That identity is a useful check, not permission to combine averages with incompatible weights.

Definition: Let $N_j$ be the claim count for policy $j$, $L_j$ its aggregate loss during the observation period, and $e_j$ its exposure. For $n$ policies, let $N=\sum_{j=1}^n N_j$, $L=\sum_{j=1}^n L_j$, and $E=\sum_{j=1}^n e_j$. The empirical claim frequency is $N/E$, the claim-weighted average severity is $L/N$ when $N>0$, and the exposure-based pure premium is $L/E$. A policy-level average loss per policy, when each policy is counted once, is $L/n$; this is not necessarily the same target as $L/E$ when exposures differ.

What Each Average Measures

The claim-weighted average severity gives each claim one unit of weight, regardless of which policy generated it. If one policy has four claims and another has one, the first policy contributes four severity observations and the second contributes one. This is appropriate when the target is the average amount of a claim in the portfolio's observed claim population.

A policy-level loss average instead gives each policy one record. A zero-claim policy has $L_j=0$ and contributes zero to the numerator, but it remains in the denominator. Consequently, policy-level data include both claim occurrence and claim amount. Claim-only data do not contain the zero-claim policies and cannot, by themselves, estimate the average loss per policy.

When exposures vary, the denominator also matters. Dividing total losses by the number of policies gives dollars per policy for the observed period. Dividing by total exposure gives dollars per exposure unit. For example, if exposure is measured in policy-years, $L/E$ is dollars per policy-year. A half-year policy should not receive the same denominator weight as a two-year policy when the target is loss per policy-year.

Quantity	Numerator	Denominator	Interpretation
Claim-weighted severity	Total claim loss	Number of claims	Average dollars per claim
Frequency	Number of claims	Total exposure	Claims per exposure unit
Policy-level pure premium	Total policy loss	Total exposure	Dollars of loss per exposure unit
Average loss per policy	Total policy loss	Number of policies	Dollars of loss per policy for the period

Worked Example: Including Zero-Claim Policies

A hypothetical insurer observes four one-year policies. Policy A has two claims of $600 and $900, Policy B has no claims, Policy C has one claim of $1,500, and Policy D has one claim of $1,000. Treat all amounts as losses on the same valuation and coverage basis. Find the claim-weighted severity, frequency per policy-year, policy-level average loss, and pure premium. The data are observed amounts; no interest, mortality, or probability distribution assumption is needed for these empirical averages.

Method. Calculate each target using its own denominator, then verify that frequency times claim-weighted severity equals total loss per policy-year. This avoids dropping the zero-claim policy from the policy-level denominator.

There are $N=2+0+1+1=4$ claims. Total loss is $L=600+900+1{,}500+1{,}000=\$4{,}000$, and total exposure is $E=4$ policy-years.

The claim-weighted severity is $L/N=\$4{,}000/4=\$1{,}000$ per claim. Frequency is $N/E=4/4=1$ claim per policy-year. Therefore, the frequency-severity product is $1\times\$1{,}000=\$1{,}000$ per policy-year.

The policy losses are $1,500, $0, $1,500, and $1,000. Their average is $(1{,}500+0+1{,}500+1{,}000)/4=\$1{,}000$ per policy. Since each policy has one year of exposure, this is also $1,000 per policy-year. The product and policy-level calculation agree. The zero-claim policy contributes no loss, but it must remain in the policy denominator.

The Empirical Frequency-Severity Identity

The relationship between the two components follows directly from their definitions. It does not require a particular distribution for claim counts or claim amounts. The essential conditions are that the counts, losses, and exposures come from the same set of observations and use the same claim and loss basis.

\frac{N}{E}\times\frac{L}{N}=\frac{L}{E}, \qquad N>0,\ E>0.

The numerator $N$ cancels because each claim counted in frequency is also one of the claims contributing to severity. The remaining ratio is total loss per exposure, exactly the empirical pure premium. If there are no claims, $N=0$, then the observed aggregate loss is zero and the empirical pure premium is zero; the observed average severity $L/N$ is undefined. A frequency-severity model cannot obtain an observed severity from a zero-claim sample alone.

A related policy-level form makes the role of zero losses explicit. If all policies have equal exposure $e$, then $E=ne$, and average loss per policy is $L/n$. Dividing that policy average by $e$ gives the loss per exposure unit:

\frac{L}{E}=\frac{L}{ne}=\frac{1}{e}\left(\frac{L}{n}\right).

If each policy has one full exposure unit, $e=1$, the average loss per policy and the pure premium per exposure unit have the same numerical value. With unequal exposures, however, a simple average across policies is generally not the average loss per exposure. Use total loss divided by total exposure for that target.

Worked Example: Unequal Policy Exposures

A hypothetical portfolio has four policies with exposures of 0.5, 1.0, 1.5, and 2.0 policy-years. Their claim records are: one claim of $400, two claims of $300 and $700, no claims, and one claim of $1,600, respectively. Find the pooled severity, frequency per policy-year, and pure premium per policy-year. Losses are on a common basis; no interest, mortality, or claim distribution assumption is required.

Method. Sum exposure rather than counting each policy as one full year. Then calculate the loss per claim and claims per exposure unit. This avoids using a policy-count denominator for a policy-year target.

Total claims are $N=1+2+0+1=4$. Total loss is $L=400+300+700+1{,}600=\$3{,}000$. Total exposure is $E=0.5+1.0+1.5+2.0=5.0$ policy-years.

The claim-weighted severity is $\$3{,}000/4=\$750$ per claim. Frequency is $4/5=$ 0.8 claims per policy-year. Their product is $0.8\times\$750=\$600$ per policy-year.

The policy losses are $400, $1,000, $0, and $1,600, which sum to $3,000. Dividing by 5.0 policy-years gives $\$3{,}000/5=\$600$ per policy-year, confirming the product. Dividing by four policies instead would give $750 per policy for the observed policy periods. That is a different denominator and, because exposures differ, is not the pure premium per policy-year.

Pooling, Weights, and the Target Portfolio

A pooled claim severity is weighted by claim counts: a cell with more claims contributes more observations to the pooled average. By contrast, a portfolio frequency is weighted by exposure when it is calculated as total claims divided by total exposure. These weights are not interchangeable with an unweighted average of cell-level severities or frequencies.

For a portfolio partitioned into cells $k$, the aggregate quantities are $N=\sum_k N_k$, $L=\sum_k L_k$, and $E=\sum_k E_k$. The pooled severity is $\sum_k L_k/\sum_k N_k$, not generally the simple average of the cell severities $L_k/N_k$. Similarly, pooled frequency is $\sum_k N_k/\sum_k E_k$, not generally the unweighted average of cell frequencies. The totals preserve the exact identity across the full portfolio.

This distinction matters when comparing a claim-level severity estimate with a policy-level target. The claim sample answers, “How large is an observed claim on average?” The policy-level target answers, “How much loss is generated per policy or exposure unit?” Claim severity alone cannot answer the second question. It must be combined with frequency using compatible portfolio weights, or the policy losses can be averaged directly using the intended exposure denominator.

Worked Example: Why Equal Cell Weights Can Misstate the Portfolio Target

Consider two hypothetical rating cells, each observed for a full year. Cell A has two policies, six claims, and $600 total loss. Cell B has eight policies, two claims, and $400 total loss. Find each cell's severity and frequency, the correct portfolio pure premium, and the result of incorrectly multiplying pooled frequency by the unweighted average of the two cell severities. No interest, mortality, or claim distribution assumption is needed; the amounts are observed losses on a common basis.

Method. Calculate each cell's quantities, then pool claims, losses, and exposure by summing their totals. Compare that result with an unweighted average of cell severities. This avoids giving a two-claim cell the same severity weight as a six-claim cell.

Cell A has $E_A=2$ policy-years, frequency $6/2=3$ claims per policy-year, and severity $\$600/6=\$100$ per claim. Its loss per exposure is $3\times\$100=\$300$.

Cell B has $E_B=8$ policy-years, frequency $2/8=0.25$ claims per policy-year, and severity $\$400/2=\$200$ per claim. Its loss per exposure is $0.25\times\$200=\$50$.

For the pooled portfolio, $N=6+2=8$, $L=600+400=\$1{,}000$, and $E=2+8=10$ policy-years. Thus pooled frequency is $8/10=0.8$, pooled claim-weighted severity is $\$1{,}000/8=\$125$, and the pure premium is $0.8\times\$125=\$100$ per policy-year. Directly, $\$1{,}000/10=\$100$ per policy-year.

The unweighted average of the two cell severities is $(\$100+\$200)/2=\$150$. Multiplying this by the pooled frequency gives $0.8\times\$150=\$120$ per policy-year, which is incorrect for this portfolio. The error is weighting the two cell averages equally even though the cells have different claim counts. The claim-weighted pooled severity is $125, not $150.

Common Mistakes and Practical Checks

Dropping zero-claim policies. Claim-only records can estimate an average per claim, but do not provide the policy-level loss denominator. Include zero-loss policies when calculating a policy average or use total exposure for a pure premium.
Confusing per policy with per exposure. A policy count and a policy-year total are not interchangeable when exposure varies. State the unit alongside each result.
Using an unweighted average of cell severities. If the target is the portfolio's average claim amount, weight cell severity by claim counts or calculate total loss divided by total claims.
Combining incompatible components. Frequency for one period or exposure base cannot be multiplied by severity from a different claim definition, coverage basis, or population and expected to reproduce the same empirical pure premium.
Rounding too early. Retain exact totals and ratios through the product. For example, $N/E$ multiplied by $L/N$ equals $L/E$ exactly before rounding; rounded intermediate figures may differ slightly.

A useful audit is to calculate the target two ways: first as total loss divided by total exposure, and then as frequency times claim-weighted severity. If the calculations do not agree before rounding, check the claim counts, loss totals, exposure units, and weights. If there are no observed claims, the direct loss-per-exposure calculation remains defined, but the empirical severity does not.

Key takeaway: The sampling unit determines the average being estimated. Claims give a claim-weighted average amount per claim; policies or exposures give a loss average that includes zero-claim outcomes. For a consistent portfolio, $ (N/E)(L/N)=L/E $. Keep the population, period, loss basis, and weights aligned.

Check Your Understanding

For each problem, identify the relevant sampling unit and keep the denominator consistent with the requested target. The hypothetical loss amounts are empirical values; no interest, mortality, or probability distribution assumption is needed.

A four-policy portfolio has one-year exposures and claim counts of 0, 1, 2, and 2. Its five claims total $2,500. Find the claim-weighted severity, frequency per policy-year, and pure premium. (Answer: Severity $500 per claim; frequency 1.25 claims per policy-year; pure premium $625 per policy-year.)
Three policies have exposures of 0.5, 1.0, and 2.0 policy-years and aggregate losses of $0, $800, and $4,800. There are eight claims in total. Find pooled severity, frequency, and pure premium per policy-year. (Answer: Severity $700 per claim; frequency $8/3.5=2.2857$ claims per policy-year; pure premium $1,600 per policy-year.)
Four equal one-year policies have losses of $0, $400, $600, and $1,000. The positive losses each represent one claim. Find average loss per policy, claim-weighted severity, frequency, and the frequency-severity product. (Answer: Average loss $500 per policy; severity $666.67 per claim; frequency 0.75 claims per policy-year; product $500 per policy-year.)
Cell A has exposure of 5 policy-years, 10 claims, and $4,000 total loss. Cell B has exposure of 15 policy-years, 5 claims, and $3,000 total loss. Find the pooled severity, pooled frequency, and portfolio pure premium. (Answer: Severity $7{,}000/15=\$466.67$ per claim; frequency $15/20=0.75$ claims per policy-year; pure premium $350 per policy-year.)
A portfolio has 10 equal policy-years and two rating cells. Cell A has 4 claims with mean severity $250; Cell B has 6 claims with mean severity $500. A proposed calculation uses the simple average severity of $375 and pooled frequency. Is that portfolio pure premium correct? Calculate the correct amount. (Answer: No. Pooled frequency is 1 claim per policy-year, but pooled severity is $(4\times250+6\times500)/10=\$400$, so the correct pure premium is $400 per policy-year; the proposed result is $375.)

What You'll Learn