Why Aggregation Level Matters
A ratemaking data set can describe a policy period, a coverage period, or an individual claim. These are different observation units: one policy period can contain multiple coverages, and one policy or coverage period can have zero, one, or several claims. Calculations made at one level therefore do not automatically have the same interpretation as calculations made at another.
For an aggregate portfolio result, frequency must relate the total number of claims to the total applicable exposure. Severity must relate total loss to total claims. Pure premium must relate total loss to total exposure. A simple average of policy or coverage results usually gives each row equal weight, even when rows represent very different amounts of exposure or different numbers of claims.
The exposure unit must match the risk being summarized. A policy-year exposure and a coverage-year exposure are not interchangeable merely because both are called “exposure.” In a coverage-level analysis, the count, loss, and exposure should all refer to the coverage being analyzed. Combining coverage results into one portfolio figure is meaningful only when the combined exposure basis and the intended target are clear.
Three Observation Levels
| Level | Typical row | What a row can summarize | Important limitation |
|---|---|---|---|
| Policy-level | One policy-period | Policy exposure, its total claim count, and its total loss | Policies may have unequal exposure and unequal claim counts |
| Coverage-level | One coverage-period | Exposure, claims, and losses for a particular coverage | Coverage exposure bases and claim definitions must be appropriate for the comparison |
| Claim-level | One claim | Loss for an individual claim, enabling claim-level severity summaries | Claim-only rows omit the exposure of claim-free risks, so they do not by themselves provide aggregate frequency |
A policy-level or coverage-level row may include a zero claim count. Such rows are essential when the target includes frequency or pure premium: their exposure belongs in the denominator even though they contribute no claims or loss. At the claim level, the natural denominator for average severity is the number of claim rows, not the number of policies.
How the Weights Arise
Aggregate frequency is an exposure-weighted average of row frequencies. For every row with positive exposure, its row frequency is \(n_j/e_j\). Multiplying that rate by its exposure returns its claim count:
Summing those identities and dividing by total exposure gives the aggregate rate:
Thus, rows receive weights proportional to exposure, not equal weights. A zero-exposure row does not have a defined row frequency and should not be assigned one for this calculation; the aggregate definition uses the total count and total exposure directly.
Likewise, aggregate severity is a claim-count-weighted average of row severities for rows with at least one claim. If \(n_j>0\), the row severity is \(L_j/n_j\), and \(n_j(L_j/n_j)=L_j\). Therefore:
A row with more claims contributes more to aggregate severity because it represents more claim observations. A zero-claim row has no defined row severity and contributes neither loss nor claim count; it must not be given a severity of zero and included as though it were a claim.
provided total exposure and total claim count are positive. The claim-count total cancels. This identity does not mean that a simple average of row frequencies multiplied by a simple average of row severities equals aggregate pure premium. Those averages generally use different, inappropriate weights for the aggregate target.
Worked Examples
Worked Example: Aggregating Policy-Period Results
A hypothetical book contains three policy-periods. Their exposure, total claim count, and total loss are shown below. Find aggregate frequency, severity, and pure premium, then compare aggregate frequency with the simple average of the policy frequencies.
| Policy-period | Exposure \(e_j\) | Claims \(n_j\) | Total loss \(L_j\) |
|---|---|---|---|
| A | 1 | 2 | $4,000 |
| B | 3 | 1 | $1,000 |
| C | 2 | 0 | $0 |
Method and assumptions. Sum counts, losses, and policy-year exposures before forming the aggregate ratios. These are deterministic hypothetical records; no interest rate, mortality assumption, or probability distribution is used or needed.
Step 1: Totals. Total exposure is \(1+3+2=6\) policy-years. Total claims are \(2+1+0=3\). Total loss is \(\$4{,}000+\$1{,}000+\$0=\$5{,}000\).
Step 2: Aggregate frequency. The result is \(3/6=0.5\) claims per policy-year.
Step 3: Aggregate severity. The result is \(\$5{,}000/3=\$1{,}666.666\ldots\), or $1,666.67 per claim rounded to cents.
Step 4: Aggregate pure premium. The result is \(\$5{,}000/6=\$833.333\ldots\), or $833.33 per policy-year rounded to cents.
The policy frequencies are \(2/1=2\), \(1/3=0.3333\ldots\), and \(0/2=0\). Their simple average is \((2+1/3+0)/3=7/9=0.7777\ldots\), not the aggregate frequency of \(0.5\). Exposure weighting gives \([1(2)+3(1/3)+2(0)]/6=3/6=0.5\).
Check and mistake avoided. Using unrounded figures, \(0.5\times(5{,}000/3)=833.333\ldots\), matching aggregate pure premium. The common mistake is to average policy frequencies equally; that gives a policy-level average, not claims per total policy-year. Including policy C’s two exposure units in the denominator is also necessary despite its zero claims.
Worked Example: Combining Coverage-Level Summaries
A hypothetical portfolio has the following two coverage-level summaries. Calculate aggregate frequency, severity, and pure premium across these coverage exposures. Also compare the aggregate results with unweighted averages of the coverage rates.
| Coverage | Exposure | Claims | Total loss |
|---|---|---|---|
| Liability | 4 coverage-years | 2 | $6,000 |
| Collision | 2 coverage-years | 3 | $600 |
Method and assumptions. Treat the stated coverage-year units as a compatible basis for the requested combined coverage result, and use the corresponding coverage claim counts and losses. The figures are deterministic and hypothetical; no interest, mortality, or probability-distribution assumption is involved.
Step 1: Total each quantity. Exposure is \(4+2=6\) coverage-years; claims are \(2+3=5\); and loss is \(\$6{,}000+\$600=\$6{,}600\).
Step 2: Calculate portfolio measures. Aggregate frequency is \(5/6=0.8333\) claims per coverage-year. Aggregate severity is \(\$6{,}600/5=\$1{,}320.00\) per claim. Aggregate pure premium is \(\$6{,}600/6=\$1{,}100.00\) per coverage-year.
The coverage frequencies are \(2/4=0.5\) for liability and \(3/2=1.5\) for collision. Their unweighted average is \((0.5+1.5)/2=1.0\), whereas their exposure-weighted average is \([4(0.5)+2(1.5)]/6=5/6=0.8333\).
The coverage severities are \(\$6{,}000/2=\$3{,}000\) and \(\$600/3=\$200\). Their unweighted average is \((3{,}000+200)/2=\$1{,}600\), whereas their claim-count-weighted average is \([2(3{,}000)+3(200)]/5=\$6{,}600/5=\$1{,}320\). Coverage pure premiums are \(\$1{,}500\) and \(\$300\); their unweighted average is \(\$900\), not the aggregate \(\$1{,}100\).
Check and mistake avoided. \(0.8333\ldots\times\$1{,}320=\$1{,}100\) per coverage-year. The common mistake is to give each coverage equal weight when the target is loss per total coverage-year. If coverage exposures are not genuinely comparable, the combined ratio may not answer a meaningful portfolio question; the correct response is to define the intended exposure basis, not to force the average.
Worked Example: Why Claim Rows Alone Do Not Give Frequency
A hypothetical set of policy-period records has the following totals. Policy P2 has three separate claims, whose losses are listed individually.
| Policy-period | Exposure | Claim losses |
|---|---|---|
| P1 | 1.00 policy-year | $300 |
| P2 | 0.25 policy-year | $100, $200, $300 |
| P3 | 0.75 policy-year | None |
Method and assumptions. Use all policy-period exposures for frequency and pure premium, and all four claim records for severity. These are deterministic hypothetical data; no interest rate, mortality assumption, or probability distribution is assumed.
Step 1: Count claims and total loss. P1 has one claim and \(\$300\) loss. P2 has three claims and loss \(\$100+\$200+\$300=\$600\). P3 has zero claims and zero loss. Total claims are \(1+3+0=4\), and total loss is \(\$300+\$600=\$900\).
Step 2: Total exposure and aggregate frequency. Total exposure is \(1+0.25+0.75=2.00\) policy-years. Frequency is \(4/2=2\) claims per policy-year.
Step 3: Severity and pure premium. Severity is \(\$900/4=\$225.00\) per claim. Pure premium is \(\$900/2=\$450.00\) per policy-year.
The policy frequencies are \(1/1=1\), \(3/0.25=12\), and \(0/0.75=0\). Their unweighted average is \((1+12+0)/3=13/3=4.3333\ldots\), not the aggregate frequency of 2. At the claim level, the average loss is \((300+100+200+300)/4=\$225.00\), which correctly gives each claim equal weight.
Check and mistake avoided. \(2\times\$225=\$450\) per policy-year. A claim-only file contains the four claims but not P3’s \(0.75\) policy-year or the other policy exposures. It can calculate claim severity from these records, but it cannot calculate portfolio frequency or pure premium without the exposure and zero-claim information. Do not divide four claims by the number of claim rows; that would produce no frequency denominator.
Choosing and Checking a Summary
The appropriate aggregation level depends on the question. A policy-level table is useful when the target is defined per policy period. A coverage-level table is appropriate when the target is defined for a coverage and its exposure basis. Claim-level observations are suitable for summaries such as average loss per claim, but claims alone do not represent the risks that generated no claims.
Before accepting a reported rate, ask what one row represents, which exposure is in the denominator, and whether the count and loss definitions match that exposure. Then check the arithmetic using totals. When aggregate frequency, severity, and pure premium use consistent definitions and positive denominators, the identity \( \text{frequency}\times\text{severity}=\text{pure premium} \) is a useful audit. Small discrepancies may arise from rounding, but a material difference usually signals inconsistent populations, definitions, or weights.
- For frequency: divide total claims by total exposure; do not average row frequencies equally unless the intended measure is explicitly an average row rate.
- For severity: divide total loss by total claims; zero-claim rows have no row severity.
- For pure premium: divide total loss by total exposure and retain zero-claim exposure.
- For coverage comparisons: confirm that exposure units and claim definitions support the combined result.
Check Your Understanding
For each problem, use the stated exposure basis and keep the units attached to your results.
- A hypothetical portfolio has policy-period A with 0.5 policy-year, one claim, and $800 loss; policy-period B has 1.5 policy-years, no claims, and no loss. Find aggregate frequency, severity, and pure premium. (Answer: 0.5 claims per policy-year; $800.00 per claim; $400.00 per policy-year.)
- Two compatible coverage groups have these totals: group X has 3 coverage-years, 3 claims, and $2,400 loss; group Y has 1 coverage-year, 2 claims, and $600 loss. Find aggregate frequency, severity, and pure premium, and the unweighted average of the two group frequencies. (Answer: 1.25 claims per coverage-year; $600.00 per claim; $750.00 per coverage-year; unweighted frequency average 1.5.)
- A claim-level file lists four hypothetical losses: $150, $250, $400, and $800. The corresponding portfolio has 2.5 policy-years of exposure, including zero-claim policies. Find claim severity, frequency, and pure premium. (Answer: $400.00 per claim; 1.6 claims per policy-year; $640.00 per policy-year.)
- Coverage group A has 5 coverage-years, 2 claims, and $1,000 loss. Group B has 1 coverage-year, 2 claims, and $500 loss. Find the combined frequency, severity, and pure premium; compare with the unweighted average group frequency. (Answer: 0.6667 claims per coverage-year; $375.00 per claim; $250.00 per coverage-year; unweighted frequency average 1.2.)
- Two policy-periods each have one claim, with losses of $200 and $1,000. Their exposures are 0.5 and 1.5 policy-years, respectively. Find aggregate severity and explain why it differs from the unweighted average of the two policy-level severities. (Answer: aggregate severity is $600.00 per claim; the unweighted average is also $600.00 because each policy has one claim.)