Matching Frequency and Severity Data

Why Matching Definitions Matters

A frequency estimate and a severity estimate can each be calculated correctly and still produce a misleading combined result. The reason is that the two estimates may describe different claim populations, coverage, time periods, or loss measures. A frequency of claims per vehicle-year cannot be combined indiscriminately with severity per payment transaction; the resulting product does not describe a consistent cost per vehicle-year.

The essential check is whether both measures refer to the same exposure and the same set of claims, with the same coverage and loss valuation. This tutorial develops that check as a practical reconciliation of frequency-severity estimates with aggregate losses. It is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: For a specified population, coverage, experience period, and valuation basis, let $E$ be the exposure measured in a stated unit, $N$ the number of claims under a stated claim-count definition, and $L$ the aggregate loss for those same claims under a stated loss definition. If $E>0$ and $N>0$, observed frequency is $N/E$, and observed severity is $L/N$. The estimates match only when their definitions refer to the same exposure basis and claim cohort, and the loss total includes exactly the losses represented by the count.

The word claim needs particular care. It might mean an accident or other covered occurrence, an opened claim file, a claim with a positive insurer payment, or an individual payment transaction. These units are not interchangeable. Similarly, “loss” might mean ground-up loss, insurer-paid loss after a deductible, incurred loss, or a payment made during a calendar period. A valid comparison names the chosen meaning rather than relying on the word alone.

A Reconciliation Identity

For observed data with $N>0$, the frequency-severity product is a bookkeeping identity when the definitions match. It is not a claim that different data sources match automatically.

\frac{N}{E}\times\frac{L}{N}=\frac{L}{E}

The cancellation of $N$ explains why the product equals aggregate loss per exposure: the count in the frequency denominator is exactly the same count used to calculate severity. If the numerator of severity instead represents a different collection of claims or a different loss basis, the cancellation is only algebraic. The result may no longer equal the aggregate cost being analyzed.

Important condition: The identity applies to a common, consistently defined set of claims and losses. It does not make a paid-loss severity equal to an ultimate-incurred severity, or a calendar-year payment total equal to an accident-year loss total. If $N=0$, $L/N$ is undefined; do not assign an arbitrary severity and multiply it by zero.

Before combining results, check several dimensions. The table summarizes what must be specified; in an actual analysis, a difference on any of these dimensions calls for explanation or reconciliation.

Dimension	Question to resolve	Example of a mismatch
Coverage	Which insured losses and coverage terms are included?	Frequency counts claims with a deductible, while severity measures ground-up losses.
Experience period	Which accidents, policies, or other events belong to the experience?	Accident-year counts are combined with payments made in a calendar year.
Exposure	What is the denominator unit, for what period, and for which coverage?	Frequency per vehicle-year is treated as frequency per policy-year.
Claim-count unit	Is the count by occurrence, claim file, claimant, or payment?	One occurrence with several payments is counted several times in one source.
Loss measure and valuation	Are losses paid, incurred, or ultimate, and at what valuation date?	Ultimate counts are paired with paid amounts that are not developed to ultimate.

Worked Examples

Worked Example: Vehicle-Year Frequency and Policy-Year Frequency

Method: Reconcile the exposure unit first, then verify the product against aggregate losses. A hypothetical fleet program records 72 covered claim occurrences and $216,000 of ultimate incurred loss for a period. The claims and losses relate to 900 earned vehicle-years. The same policies represent 600 policy-years. Calculate the cost per vehicle-year and explain what happens if policy-years are substituted as the frequency denominator.

Step 1: Use the exposure unit attached to the loss experience. The loss total corresponds to 900 vehicle-years. Frequency on that basis is $72/900=0.08$ occurrences per vehicle-year.

Step 2: Calculate severity for the same 72 occurrences and loss measure. Severity is $216,000 divided by 72, or $3,000 per occurrence.

Step 3: Combine and check. The product is $0.08\times\$3{,}000=\$240$ per vehicle-year. Directly, aggregate loss per vehicle-year is $\$216{,}000/900=\$240$. The two calculations agree.

If the analyst instead divides the claim count by 600 policy-years, the calculated frequency is $72/600=0.12$ claims per policy-year. Multiplying this by the $3,000 severity gives $360 per policy-year. That amount is not the cost per vehicle-year; it has a different exposure unit. It may be a meaningful policy-year result only if the numerator, loss basis, and policy-year exposure are all consistently defined for that target. Common mistake avoided: treating policy-years and vehicle-years as equivalent simply because both are annual exposure measures.

Worked Example: Matching a Deductible to the Loss Measure

Method: Match the severity basis to the coverage represented by the claim count and aggregate loss. A hypothetical insurer covers physical-damage claims with a $500 deductible and no limit affecting these claims. There are 2,000 earned exposure units and 40 covered occurrences, all with ground-up losses greater than $500. The total ground-up loss is $240,000. Calculate insurer-paid cost per exposure.

Step 1: Calculate frequency. The 40 occurrences generate positive insurer payments, so frequency is $40/2{,}000=0.02$ occurrences per exposure unit.

Step 2: Calculate the insurer-paid aggregate. Each claim's insurer payment equals its ground-up loss minus the $500 deductible. Since all 40 losses exceed the deductible and no limit applies, aggregate insurer payments are $\$240{,}000-(40\times\$500)=\$220{,}000$.

Step 3: Use severity on the insurer-paid basis. Paid severity is $\$220{,}000/40=\$5{,}500$ per occurrence. The frequency-severity product is $0.02\times\$5{,}500=\$110$ per exposure unit.

Step 4: Check against the target aggregate. Paid aggregate loss per exposure is $\$220{,}000/2{,}000=\$110$. The result reconciles. Using ground-up severity instead would give $\$240{,}000/40=\$6{,}000$ and $0.02\times\$6{,}000=\$120$, which describes ground-up loss per exposure, not insurer-paid loss per exposure. Common mistake avoided: combining claims counted under the insurer’s coverage with severity that has not had the deductible applied.

Worked Example: Accident-Year Ultimate Loss and Calendar-Year Payments

Method: Keep the experience cohort and valuation basis consistent. A hypothetical book has 3,000 earned vehicle-years for accident year 2024. Its ultimate claim count is 120 occurrences, and its ultimate incurred losses are $720,000. A calendar-year 2025 payment report shows $570,000 in payments: $480,000 on accident-year 2024 claims and $90,000 on accident-year 2023 claims. Compare the correct accident-year ultimate result with a tempting mismatched calculation.

Step 1: Calculate accident-year frequency. Using the 2024 ultimate count and 2024 exposure, frequency is $120/3{,}000=0.04$ occurrences per vehicle-year.

Step 2: Calculate matching ultimate severity. Ultimate severity for those same 2024 claims is $\$720{,}000/120=\$6{,}000$ per occurrence.

Step 3: Reconcile the 2024 ultimate result. The product is $0.04\times\$6{,}000=\$240$ per vehicle-year. Directly, $\$720{,}000/3{,}000=\$240$.

The calendar-year payment report has a different period definition and includes $90,000 for 2023 accidents. Dividing its full $570,000 by the 2024 ultimate count gives an apparent severity of $\$570{,}000/120=\$4{,}750$; multiplying by 0.04 gives $190 per vehicle-year. That is not the 2024 ultimate result. Even the $480,000 paid on 2024 claims alone is a paid-to-date amount: dividing by 120 gives $4,000 per ultimate claim and a product of $160 per vehicle-year, not the ultimate-incurred cost. Common mistake avoided: assuming that payments made during a calendar year are the same thing as ultimate losses for an accident year.

A Practical Matching Procedure

A disciplined sequence makes discrepancies easier to locate. Record the definitions before calculating a product, and reconcile the result to the aggregate measure that the analysis is intended to represent.

Specify the target.
Name the intended cost measure, such as insurer-paid or ultimate incurred loss per earned vehicle-year, and state the coverage and experience period.

Fix the exposure basis.
State the exposure unit and period. Do not silently substitute policy-years for vehicle-years, or one coverage's exposures for another's.

Define the count.
State whether $N$ counts occurrences, claim files, or another unit, and ensure the count covers the same population and period as the target losses.

Define the severity amount.
Specify the loss basis, including deductible and limit treatment when relevant, and its valuation date. Confirm that $L$ includes exactly the claims counted in $N$.

Reconcile.
Calculate $N/E$, $L/N$, and $L/E$ using matched data. Verify that $(N/E)(L/N)=L/E$; investigate any disagreement in definitions, totals, or arithmetic.

The final check is powerful but limited. Agreement confirms the arithmetic for the selected definitions; it does not establish that those definitions are appropriate for every purpose. For example, a paid-loss result can reconcile perfectly and still be unsuitable when the intended target is ultimate incurred loss. Likewise, a consistently calculated cost per policy-year answers a different question from cost per vehicle-year.

Common Traps

Using “claim” without defining the unit. One occurrence may produce several claim files or payment transactions. Use the same unit in the count and in the severity denominator.
Mixing ground-up and insured loss. A deductible changes the payment amount. A severity based on ground-up loss is not an insurer-paid severity.
Mixing accident and calendar periods. Calendar-year payments can include claims from multiple accident years, while accident-year counts group claims by when the events occurred.
Ignoring valuation maturity. Paid losses at a valuation date are not automatically ultimate losses. A count developed to ultimate paired with an undeveloped paid amount has inconsistent development bases.
Changing the exposure denominator silently. A product per policy-year cannot be compared directly with aggregate losses per vehicle-year without a justified conversion.

Key takeaway: Frequency and severity are comparable as a combined estimate only when the exposure, coverage, claim-count unit, experience period, and loss valuation are aligned. For matched observed data, their product must equal aggregate loss per the same exposure unit.

Check Your Understanding

For each problem, identify the relevant matching issue and calculate the requested result.

A hypothetical portfolio has 54 claims, $189,000 of matching incurred losses, and 1,800 earned vehicle-years. Calculate frequency, severity, and loss per vehicle-year, then verify the product. (Answer: 0.03 claims per vehicle-year; $3,500 per claim; $105 per vehicle-year.)
A hypothetical insurer has 25 claims, each with ground-up loss above a $1,000 deductible, and aggregate ground-up loss of $100,000. Exposure is 1,250 units; no limit applies. Calculate insurer-paid cost per exposure and state the mismatched result from using ground-up severity. (Answer: paid aggregate $75,000; frequency 0.02; paid severity $3,000; paid cost $60 per exposure; ground-up result $80 per exposure.)
For a hypothetical accident-year cohort, ultimate claims are 80, ultimate incurred loss is $640,000, and exposure is 2,000 units. A calendar-year payment report contains $300,000, of which $240,000 belongs to this cohort and $60,000 to an earlier cohort. Calculate the matched ultimate cost per exposure and the result from mistakenly using the full calendar-year payment amount as severity numerator. (Answer: matched ultimate cost $320 per exposure; mismatched calculation $150 per exposure.)
A hypothetical account has 90 occurrences and $405,000 of matching aggregate loss on 3,000 earned policy-years. Calculate the frequency-severity product and verify it against aggregate loss per policy-year. (Answer: 0.03 occurrences per policy-year; $4,500 per occurrence; $135 per policy-year.)
A hypothetical book has 48 occurrences, 60 payment transactions, $120,000 of loss, and 2,400 exposure units. If the loss total is consistent with both count views, calculate the occurrence-based and transaction-based products, then identify the result of crossing the transaction frequency with occurrence severity. (Answer: occurrence-based $50 per exposure; transaction-based $50 per exposure; crossed calculation $62.50 per exposure.)

What You'll Learn