Why Pure Premium Calculations Go Wrong
A pure premium is an expected loss amount per unit of exposure. Its calculation can look simple, yet errors arise when a calculation combines quantities that describe different units or different loss amounts. A mean claim severity is not itself a per-policy pure premium; a policy with no claims still belongs in a policy-level average; and exposure should scale a prediction once, not twice. Severity must also represent the same payment basis as the target loss.
These distinctions matter whether a calculation uses frequency and severity components or a direct aggregate-loss target. Before calculating, specify the unit being averaged, the exposure period, and what amount of each claim is included. The earlier material on exposure scaling and zero-loss policies provides useful context, but the checks below state the relevant definitions directly.
For a homogeneous portfolio, if the expected claim count per exposure unit is \(\lambda\), and each claim has expected payment \(\mu\) on the target basis, the frequency-severity pure premium is \(\lambda\mu\). This product applies when the frequency and severity quantities refer to a consistent population and loss basis. For risk classes with different frequencies or severities, portfolio totals must be combined with the appropriate exposure and claim weights; an unweighted average of class-level figures need not represent the portfolio.
Check the Sampling Unit
A policy-level dataset has one observation per policy or policy period, including policies with no claims. A claim-level dataset has one observation per claim. Claim-level data naturally estimate a mean payment per claim. Policy-level data naturally estimate aggregate loss per policy period or per exposure unit. Confusing these sampling units can produce a plausible number with the wrong interpretation.
The connection between the two units is clearest in aggregate totals. Across a portfolio, total loss equals claim count multiplied by the average payment per claim, provided the claim mean is calculated over those same claims. Dividing by total exposure gives a pure premium. This identity does not mean the claim mean alone is a pure premium.
Worked Example: Reconciling Claim and Policy Summaries
A hypothetical book has 100 one-year policies in Class A and 20 one-year policies in Class B. Class A has 10 claims with average payment $1,000; Class B has 10 claims with average payment $3,000. Calculate the overall claim severity, claim frequency per policy-year, and pure premium. Identify the error in using the claim severity as the pure premium.
Method and assumptions. Treat each policy as one policy-year of exposure. Claim counts and payments are on the same loss basis, and the stated class-average payments summarize all claims in each class.
Step 1: Calculate aggregate losses. Class A loss is \(10\times \$1{,}000=\$10{,}000\). Class B loss is \(10\times \$3{,}000=\$30{,}000\). Total loss is \(\$10{,}000+\$30{,}000=\$40{,}000\).
Step 2: Calculate the claim-weighted severity. There are \(10+10=20\) claims. The overall mean payment per claim is \(\$40{,}000/20=\$2{,}000\). This is claim-weighted: each claim contributes one observation.
Step 3: Calculate frequency per policy-year. Total exposure is \(100+20=120\) policy-years, so frequency is \(20/120=0.1667\) claims per policy-year, to four decimals.
Step 4: Calculate the pure premium. The portfolio pure premium is \(\$40{,}000/120=\$333.33\) per policy-year. Equivalently, \(0.1667\times \$2{,}000\approx \$333.33\), using the unrounded frequency \(1/6\) for the product.
Common mistake avoided. Reporting $2,000 as the pure premium confuses dollars per claim with dollars per policy-year. The check is units: \((\text{claims}/\text{policy-year})\times(\text{dollars}/\text{claim})=\text{dollars}/\text{policy-year}\). The answer also checks against aggregate loss divided by total exposure: \(\$40{,}000/120=\$333.33\).
Notice that a claim-level average gives more weight to classes that produce more claims, not necessarily to classes with more policies. In the example, each class contributes 10 claims even though Class A has five times as many policies as Class B. That weighting is appropriate for the overall mean payment per claim. It is not a substitute for specifying the policy-level target.
Include Zero-Loss Policies
When the target is expected loss per policy or per exposure unit, policies with no losses contribute zero to the numerator but remain in the denominator. Restricting the denominator to policies with claims instead estimates an average for a selected group: policies that had positive losses during the observation period. That conditional average generally exceeds the all-policy average.
Worked Example: Restoring the Zero-Loss Policies
A hypothetical insurer observes four one-year policies with aggregate losses of $0, $0, $600, and $1,200. The two positive-loss policies each have one claim. Calculate the average aggregate loss per policy-year, the frequency per policy-year, the mean severity, and verify the pure premium by both methods.
Method and assumptions. Each policy has one policy-year of exposure. The losses shown are the complete aggregate losses on the chosen basis, and each of the two positive-loss policies has exactly one claim.
Step 1: Calculate policy-level loss and exposure totals. Total loss is \(\$0+\$0+\$600+\$1{,}200=\$1{,}800\). Total exposure is 4 policy-years. Thus the all-policy pure premium is \(\$1{,}800/4=\$450\) per policy-year.
Step 2: Calculate frequency. There are two claims, so frequency is \(2/4=0.5\) claims per policy-year.
Step 3: Calculate severity. Total claim payments are $1,800 across two claims, so mean severity is \(\$1{,}800/2=\$900\) per claim.
Step 4: Verify the frequency-severity result. The product is \(0.5\times \$900=\$450\) per policy-year, agreeing with the direct policy-level calculation.
Common mistake avoided. Dividing $1,800 by only the two positive-loss policies gives $900 per positive-loss policy, not $450 per policy-year for the full portfolio. The all-policy denominator includes both zero-loss policies. The check is that total loss divided by total exposure must agree with the frequency-severity product on the same basis.
Apply Exposure Once
Exposure can enter a calculation in different places depending on how the response and prediction are defined. A rate per exposure unit is multiplied by exposure to obtain an expected aggregate amount. In contrast, a prediction already expressed as expected aggregate loss for the observed policy period should not be multiplied by exposure again. The key is not whether exposure appears in a model, but whether the quantity being used is a rate or an already-scaled total.
Worked Example: Avoiding a Second Exposure Adjustment
A hypothetical portfolio has three policies with exposures 0.5, 1.0, and 2.0 policy-years. A model supplies expected aggregate losses of $200, $400, and $800 for those respective policy periods. Calculate total expected loss and the portfolio pure premium per policy-year. Then show the result of the common double-exposure error.
Method and assumptions. The supplied model predictions are already expected aggregate losses for each policy's stated exposure. The total portfolio exposure is the sum of the three exposures.
Step 1: Sum the predictions. Total expected loss is \(\$200+\$400+\$800=\$1{,}400\).
Step 2: Sum the exposures. Total exposure is \(0.5+1.0+2.0=3.5\) policy-years.
Step 3: Calculate the portfolio pure premium. The pure premium is \(\$1{,}400/3.5=\$400\) per policy-year.
Step 4: Diagnose double counting. Multiplying the predictions by exposure again would give \((0.5\times \$200)+(1.0\times \$400)+(2.0\times \$800)=\$100+\$400+\$1{,}600=\$2{,}100\). Dividing this incorrect total by 3.5 gives $600 per policy-year, an overstatement caused by scaling totals a second time.
Common mistake avoided. Treating every model output as a rate and multiplying by exposure without checking its scale. Here, the predictions were already totals. As a check, the correct predictions equal exposure times a $400 rate in each case: \(0.5\times \$400=\$200\), \(1.0\times \$400=\$400\), and \(2.0\times \$400=\$800\).
Keep Severity on the Target Loss Basis
A severity estimate is meaningful only after specifying what counts as a claim payment. Ground-up loss, payment after a deductible, and payment subject to a limit are different amounts. If the pure premium target is insurer payment after a limit, a ground-up severity estimate cannot be substituted without adjustment. The claim count, severity, and target pure premium must all describe the same covered payment basis.
For a fixed per-claim limit \(u\) with no deductible, the insurer payment for ground-up loss \(X\) is \(\min(X,u)\). If the target is this limited payment and all included losses generate a positive payment, frequency can be combined with the mean limited payment. Using the ground-up mean instead answers a different question: expected ground-up loss per exposure.
Worked Example: Matching Severity to a Per-Claim Limit
A hypothetical portfolio has 10 policy-years and two claims with ground-up losses of $10,000 and $30,000. The insurer pays each claim subject to a $20,000 per-claim limit, with no deductible. Calculate the limited-loss pure premium and compare it with the incorrect result from using ground-up severity.
Method and assumptions. Each stated claim is one claim, each has a positive insurer payment, and the target is insurer payment after applying the per-claim limit. Total exposure is 10 policy-years.
Step 1: Apply the limit to each claim. The first payment is \(\min(\$10{,}000,\$20{,}000)=\$10{,}000\). The second is \(\min(\$30{,}000,\$20{,}000)=\$20{,}000\).
Step 2: Calculate limited severity. Total limited payments are \(\$10{,}000+\$20{,}000=\$30{,}000\). Across two claims, mean limited severity is \(\$30{,}000/2=\$15{,}000\).
Step 3: Calculate frequency and pure premium. Frequency is \(2/10=0.2\) claims per policy-year. The limited-loss pure premium is \(0.2\times \$15{,}000=\$3{,}000\) per policy-year. Directly, \(\$30{,}000/10=\$3{,}000\).
Step 4: Compare the mismatched calculation. Ground-up severity is \((\$10{,}000+\$30{,}000)/2=\$20{,}000\). Multiplying it by the same frequency gives \(0.2\times \$20{,}000=\$4{,}000\) per policy-year. That is the ground-up pure premium, not the limited-payment pure premium.
Common mistake avoided. Combining a limited-payment target with ground-up severity. The direct limited-payment total divided by exposure confirms the correct answer of $3,000 per policy-year.
A Consistency Check Before Accepting a Result
A short audit catches many errors before a result is used. The derivation behind the frequency-severity identity also explains why these checks work. Suppose each exposure unit has expected claim count \(\lambda\), and claim payments on the target basis have common mean \(\mu\), independent of the count and identically distributed across claims. Conditional on \(N=n\), the expected total payment is \(n\mu\). Averaging over the claim count gives:
If exposure is \(e\), then \(E[N]=e\lambda\). Therefore \(E[L]/e=\lambda\mu\), the pure premium per exposure unit. This justification relies on a common claim-payment mean independent of claim count, or on conditions that make the same expected-sum identity valid. If claim severity depends on frequency or risk mix, a single unqualified product of separately averaged quantities may not represent the target. In that case, the calculation must preserve the relevant joint structure and weights.
Decide whether each observation is a claim, a policy period, or an exposure unit. Include zero-loss policies for an all-policy target.
Write down whether each number is claim frequency, severity per claim, pure premium per exposure, or aggregate loss for a period.
Confirm that the severity and target use the same treatment of deductibles, limits, and other payment adjustments.
Where possible, compare the result with aggregate loss divided by total exposure. Make sure exposure has not been applied to an already-scaled total.
The most reliable exam habit is to write units beside intermediate quantities. If severity is dollars per claim and frequency is claims per policy-year, their product has units of dollars per policy-year. If the result has the wrong units, or fails the aggregate-loss-over-exposure check, revisit the sampling unit, denominator, exposure scaling, and loss basis before proceeding.
Check Your Understanding
For each problem, state the relevant unit and use a direct total-loss check where possible.
- A hypothetical portfolio has 50 policy-years, 8 claims, and total payments of $24,000 on the stated target basis. Find the claim severity and pure premium per policy-year. (Answer: severity $3,000 per claim; pure premium $480 per policy-year.)
- Six one-year policies have aggregate losses of $0, $0, $300, $500, $700, and $900. Find the all-policy average and the average among positive-loss policies. Which is the pure premium for all six policy-years? (Answer: all-policy pure premium $400; positive-loss average $600; $400 is the all-policy pure premium.)
- A model's expected aggregate-loss predictions are $150 and $750 for policy periods with exposures 0.5 and 1.5 years. Find total expected loss and the pure premium per exposure unit. What error results from multiplying each prediction by exposure again? (Answer: total expected loss $900; pure premium $450 per exposure unit; double-scaling gives $1,200 total, or $600 per exposure unit.)
- A hypothetical book has 20 exposure units and three claims with ground-up amounts of $4,000, $12,000, and $24,000. Each payment is subject to a $10,000 limit, with no deductible. Find the limited-loss severity and pure premium. (Answer: limited payments $4,000, $10,000, and $10,000; severity $8,000 per claim; frequency 0.15 claims per exposure unit; pure premium $1,200 per exposure unit.)
- Two hypothetical classes each have 10 exposure units. Class A has 1 claim averaging $2,000; Class B has 4 claims averaging $5,000. Find the portfolio claim-weighted severity, frequency per exposure unit, and pure premium. Explain why the unweighted average of the class severities is not the portfolio claim severity. (Answer: total loss $22,000; claim-weighted severity $4,400; frequency 0.25 claims per exposure unit; pure premium $1,100 per exposure unit; the class severities must be weighted by claim counts, not equally.)