Why Separate Frequency From Severity?
A portfolio’s total loss depends on two different features: how many claims occur and how large each claim is. Separating these features helps an actuary describe observed losses clearly. It also prevents a common unit error: confusing an average amount per claim with an average amount per policy, location, or year.
The distinction is useful only when the underlying records are defined consistently. A claim count needs a rule for what qualifies as one claim, while a severity amount needs a rule for what financial amount is assigned to that claim. The exposure basis and time period must also be stated. For example, “0.08 claims” is incomplete unless it says 0.08 claims per what, over which period.
Claim-Level and Aggregate Notation
Let \(N\) be the number of claims in a defined portfolio and observation period. Let \(X_j\) be the loss amount assigned to the \(j\)th claim, for \(j=1,\ldots,N\). The aggregate loss, denoted \(S\), is the sum of those claim amounts. When there are no claims, the sum is defined to be zero.
For a set of observed claims, the average severity is the aggregate loss divided by the number of claims, provided at least one claim occurred. If \(E\) denotes the amount of exposure measured over the same period, the observed frequency rate is the claim count divided by exposure. The units follow directly from the denominators.
For example, if \(E\) is measured in policy-years, then \(N/E\) is measured in claims per policy-year. If exposure is measured in vehicle-months instead, the rate is claims per vehicle-month. These rates are not automatically probabilities: a rate can exceed 1, and its interpretation depends on the exposure definition and time period.
What Must Be Defined Before Counting?
Before calculating frequency or severity, specify the claim and loss definitions. A claim could be counted by reported claim file, by occurrence, or by another clearly defined unit. A claim’s amount could be the insurer’s payment, the covered loss before a deductible, or another specified amount. Those choices produce different counts or severities, so they must not be mixed within one calculation.
Also choose a consistent observation basis. If the claim count covers six months but the exposure covers a full year, dividing one by the other does not give a rate for either clearly defined period. Similarly, a count for one portfolio should not be divided by exposure from a different portfolio unless the calculation is deliberately defined that way.
- Frequency numerator: the number of claims meeting the stated counting rule.
- Exposure denominator: the matching units at risk during the stated period, such as policy-years or machine-months.
- Severity amount: the amount assigned to each counted claim under one consistent financial definition.
- Aggregate loss: the sum of those same claim amounts over the same scope and period.
Worked Example: Claim Rate and Severity for a Policy Cohort
A hypothetical insurer observes 60 policy-years during one calendar year. Three covered claims are recorded, with assigned amounts of $1,200, $2,500, and $4,300. Calculate the aggregate loss, average severity, and observed frequency rate.
Assumptions and definitions. Each listed amount is the insurer’s loss amount for one distinct claim, all amounts are in dollars, and the three claims and 60 policy-years refer to the same cohort and calendar year. No interest discounting or mortality assumption is used. The data are treated as observed values; no probability distribution is assumed.
Step 1: Count the claims. There are \(N=3\) claims.
Step 2: Add the claim amounts.
The aggregate loss is $8,000.
Step 3: Divide by claim count for average severity.
Rounded only at the final step to the nearest cent, average severity is $2,666.67 per claim.
Step 4: Divide claim count by exposure.
The result is 0.05 claims per policy-year, or 5 claims per 100 policy-years. A common mistake is to divide $8,000 by 60 and call the result severity. That calculation is $133.33 per policy-year, an aggregate-loss measure per exposure unit, not an amount per claim. As a check, the three claim amounts sum to $8,000, and \(3/60\) has the stated frequency units.
Why Aggregate Loss Is a Sum
The aggregate formula is an accounting identity, not a claim about how claims are generated. In the chosen period, list each qualifying claim once and assign it its defined amount \(X_j\). The total amount attributable to those claims is obtained by adding every listed amount. If there are no qualifying claims, there are no amounts to add, so the total is zero.
Choose the portfolio, claim definition, loss definition, and time period.
The resulting count is \(N\).
Use \(X_j\) for the amount assigned to claim \(j\), applying the same financial definition throughout.
The aggregate is \(S=\sum_{j=1}^{N}X_j\); for \(N=0\), it is zero.
This reasoning requires no assumption that claim counts and claim amounts are independent, or that either follows a particular distribution. It only requires a defined set of claims and an amount assigned to each. Any later probabilistic model must add its own assumptions; the sum itself does not supply them.
Worked Example: Matching a Partial-Year Count to Exposure
A hypothetical equipment portfolio has 600 machine-months of exposure during a six-month observation period. Eight claims are recorded in that same period, with total assigned loss of $36,000. Calculate the average severity and the frequency rate per machine-year.
Assumptions and definitions. All eight claims belong to the stated six-month period, and the $36,000 is the sum of their assigned loss amounts. The exposure includes the same machines over that period. No interest discounting or mortality assumption is used, and no probability distribution is assumed.
Step 1: Convert exposure to machine-years. Twelve machine-months equal one machine-year, so
Step 2: Calculate average severity. The claim count is \(N=8\), so
Step 3: Calculate the frequency rate.
The frequency rate is 0.16 claims per machine-year, or 16 claims per 100 machine-years. The common time-unit trap is to divide eight by 600 and report 0.0133 as claims per machine-year. That number is claims per machine-month; converting the exposure first gives the requested annual unit. The check is that 50 machine-years corresponds to 600 machine-months, preserving the six-month observation basis.
Combining Groups Without Losing the Weights
When a portfolio contains subgroups, aggregate losses and claim counts can be added across groups if their definitions and periods match. Overall average severity is then the combined aggregate divided by the combined claim count. It is generally not the simple average of subgroup average severities: subgroups with more claims contribute more individual severities to the overall average.
Worked Example: Combining Two Claim Categories
A hypothetical property portfolio is observed for one year. Category A has two claims of $10,000 and $20,000. Category B has three claims of $2,000, $4,000, and $6,000. The categories together correspond to 200 policy-years. Find the overall aggregate loss, average severity, and frequency rate.
Assumptions and definitions. The categories are mutually exclusive, each listed amount belongs to one distinct claim, and all claim amounts use the same loss definition and calendar year. The 200 policy-years cover the combined portfolio for that year. No interest discounting or mortality assumption is used; the observations are not assigned a probability distribution.
Step 1: Find each category’s count and aggregate.
Step 2: Add counts and aggregate losses.
Step 3: Calculate overall average severity.
Step 4: Calculate frequency per exposure.
The overall average severity is $8,400 per claim, and the frequency rate is 0.025 claims per policy-year. A tempting but incorrect calculation averages the subgroup severities without weights: Category A’s mean is $15,000 and Category B’s is $4,000, giving an unweighted average of $9,500. The correction weights by claim counts: \((2\times15{,}000+3\times4{,}000)/5=\$8{,}400\). The result checks against the combined aggregate of $42,000 across five claims.
Zero Claims and Other Interpretation Traps
If a period has no claims, its aggregate loss is zero under the defined claim and loss basis. However, average severity for that period is not zero; it is undefined, because there is no claim amount over which to average. Treating severity as zero would confuse the absence of claims with the presence of claims that each have zero amount.
Keep the following distinctions in view:
| Measure | Numerator and denominator | Units |
|---|---|---|
| Claim count | Number of qualifying claims | Claims |
| Average severity | Aggregate loss divided by claim count | Currency per claim |
| Frequency rate | Claim count divided by exposure | Claims per exposure unit |
| Aggregate loss | Sum of claim amounts | Currency |
Another trap is to treat frequency and severity as interchangeable simply because both help describe loss experience. A portfolio can have many small claims or a few large claims, and the aggregate can be similar even though the claim pattern differs. Reporting all three components—count, claim amounts or severity, and aggregate—preserves that distinction.
Check Your Understanding
For each hypothetical situation, use the stated claim definition and period. Round monetary results to cents only when needed.
- A portfolio has 80 policy-years and three claims of $900, $2,100, and $3,000 in the same year. Find aggregate loss, average severity, and frequency per policy-year. (Answer: $6,000 aggregate; $2,000 per claim; 0.0375 claims per policy-year.)
- A repair program records 12 claims over 240 machine-months, with aggregate loss of $18,000. Convert exposure to machine-years, then find average severity and frequency per machine-year. (Answer: 20 machine-years; $1,500 per claim; 0.6 claims per machine-year.)
- In one year, Unit A has four claims averaging $7,000 each, and Unit B has six claims averaging $3,000 each. Find total claim count, aggregate loss, and the combined average severity. (Answer: 10 claims; $46,000 aggregate; $4,600 per claim.)
- A group has 25 policy-years and no claims during the observation period. State its aggregate loss, its observed frequency rate, and whether average severity is defined. (Answer: $0 aggregate; 0 claims per policy-year; average severity is undefined.)
- Two categories have aggregate losses of $24,000 and $16,000, from 3 and 8 claims respectively. Find the combined average severity. Explain why the unweighted mean of category averages is not the correct calculation. (Answer: $40,000 divided by 11, or $3,636.36 per claim; the category averages must be weighted by their claim counts.)