Frequency, Severity, and Pure Premium Decomposition

Why Decompose Loss Cost?

A total loss amount does not, by itself, explain how that amount arose. The same aggregate loss can result from many small claims or a few large ones, and it can arise from very different amounts of insured exposure. Separating claim frequency from average severity makes these components visible and provides a precise way to relate claim counts and loss amounts to exposure.

The key is to keep the measurement bases aligned. The claim count, loss amount, and exposure must refer to the same population, time period, and claim definition. For example, if exposure is measured in vehicle-years, frequency is claims per vehicle-year, severity is dollars per claim, and pure premium is dollars per vehicle-year.

Definition: For a specified experience population and period, let $e$ be the amount of exposure, $N$ the number of claims, and $L$ the aggregate loss amount on a stated basis. When $e>0$ and $N>0$, claim frequency is $f=N/e$, average severity is $s=L/N$, and pure premium (or loss cost) is $p=L/e$. Frequency is claims per exposure unit, severity is dollars per claim, and pure premium is dollars per exposure unit.

The word “loss” in this definition must be made specific in an application. It may mean, for example, paid loss or loss on an incurred basis. The amount $L$ and the resulting severity and pure premium must all use that same basis. Unless stated otherwise, expenses and premium are not included in $L$: pure premium here means loss per exposure, not a full rate indication.

The Product Relationship

The relationship among the three measures follows directly from their definitions. It is an exact accounting identity for a data set with positive exposure and at least one claim; it does not require a probability model or an assumption that claim counts and severities are independent.

f s = \left(\frac{N}{e}\right)\left(\frac{L}{N}\right) = \frac{L}{e} = p

The claim count cancels. Equivalently, multiplying frequency by severity first gives the aggregate loss per exposure unit. Multiplying that result by the amount of exposure recovers aggregate losses: $e p=L$. The units provide a useful check: claims per exposure unit multiplied by dollars per claim leaves dollars per exposure unit.

Fix the measurement basis.
Specify the experience population, period, exposure unit, claim definition, and loss basis before calculating any ratio.

Calculate frequency and severity.
Divide the claim count by exposure for frequency, and divide aggregate losses by the claim count for average severity.

Reconcile to pure premium.
Multiply frequency by severity and check that the result equals aggregate losses divided by exposure.

There is also a population-level version. For a fixed amount $e>0$ of exposure, let $N$ and $L$ be random claim count and aggregate loss. If $\operatorname{E}[N]>0$, define expected frequency as $\operatorname{E}[N]/e$, expected severity on a claim-weighted basis as $\operatorname{E}[L]/\operatorname{E}[N]$, and expected loss cost as $\operatorname{E}[L]/e$. Their product is again exact:

\frac{\operatorname{E}[N]}{e} \cdot \frac{\operatorname{E}[L]}{\operatorname{E}[N]} = \frac{\operatorname{E}[L]}{e}

This version does not assert that a generic average of claim severities can always be multiplied by expected frequency. The severity ratio shown is weighted by claim counts through the ratio of expected losses to expected claims. If different risks or types of claims have different severities, an unweighted average of subgroup severities need not be the appropriate severity for the whole population.

Worked Example: Decomposing One Portfolio’s Loss Cost

A hypothetical fleet program has 2,400 vehicle-years of exposure. During the experience period, it records 36 claims and $198,000 of incurred losses. Calculate frequency, average severity, and pure premium, then check the product relationship.

Method. Use the same vehicle-year exposure, claim count, and incurred-loss total for all three measures.

Frequency. Divide claims by vehicle-years:

f=\frac{36}{2{,}400}=0.015 \quad\text{claims per vehicle-year.}

That is 1.5 claims per 100 vehicle-years. The decimal form, $0.015$, is used in the product calculation.

Average severity. Divide incurred losses by claims:

s=\frac{\$198{,}000}{36}=\$5{,}500\text{ per claim.}

Pure premium. Divide incurred losses by vehicle-years:

p=\frac{\$198{,}000}{2{,}400}=\$82.50\text{ per vehicle-year.}

Check. The component product is $0.015\times \$5{,}500=\$82.50$ per vehicle-year. Multiplying $82.50 by 2,400 vehicle-years returns $198,000. The direct and decomposed calculations agree.

Common mistake avoided. Treating 1.5 claims per 100 vehicle-years as the decimal $1.5$ in the product would give $1.5\times \$5{,}500=\$8{,}250$, which is 100 times too large. Convert the percentage-style statement to a per-unit decimal before multiplying.

Combining Groups Correctly

For a portfolio split into groups, calculate the aggregate frequency and pure premium using total claims and total exposure. Calculate aggregate severity using total losses and total claims. The same identity then holds for the combined portfolio. Group results can help explain the total, but simple unweighted averages of group measures generally do not reproduce it.

Measure	Portfolio calculation	Weight implicit in combining groups
Frequency	Total claims divided by total exposure	Exposure
Average severity	Total losses divided by total claims	Claim count
Pure premium	Total losses divided by total exposure	Exposure

For example, overall frequency is $(N_1+N_2)/(e_1+e_2)$, not generally the simple average of $N_1/e_1$ and $N_2/e_2$. Overall severity is $(L_1+L_2)/(N_1+N_2)$, so groups with more claims receive more weight. These weights follow from the definitions rather than from a separate adjustment rule.

Worked Example: Combining Two Risk Groups

A hypothetical insurer reviews two groups on the same incurred-loss basis. Group A has 800 exposure units, 20 claims, and $160,000 of losses. Group B has 1,200 exposure units, 18 claims, and $216,000 of losses. Calculate each group’s measures and the combined portfolio measures. Explain why the unweighted group averages do not give the portfolio results.

Group A. Frequency is $20/800=0.025$ claims per exposure unit. Severity is $\$160{,}000/20=\$8{,}000$ per claim. Pure premium is $\$160{,}000/800=\$200$ per exposure unit. The check is $0.025\times \$8{,}000=\$200$.

Group B. Frequency is $18/1{,}200=0.015$ claims per exposure unit. Severity is $\$216{,}000/18=\$12{,}000$ per claim. Pure premium is $\$216{,}000/1{,}200=\$180$ per exposure unit. The check is $0.015\times \$12{,}000=\$180$.

Combine the data. Total exposure is $800+1{,}200=2{,}000$ units, total claims are $20+18=38$, and total losses are $\$160{,}000+\$216{,}000=\$376{,}000$.

The combined frequency is $38/2{,}000=0.019$ claims per exposure unit. The combined severity is $\$376{,}000/38=\$9{,}894.7368\ldots$, or $9,894.74 per claim to the nearest cent. The combined pure premium is $\$376{,}000/2{,}000=\$188.00$ per exposure unit.

Check and weighting. Using the unrounded combined severity, $0.019\times \$9{,}894.7368\ldots=\$188.00$. The simple average of group frequencies would be $(0.025+0.015)/2=0.020$, which is not the combined frequency; the groups have different exposure amounts. The simple average of group severities, $10,000, is also not the combined severity; the groups have different claim counts. The totals, rather than an unweighted average of group ratios, produce the correct combined measures.

When the Identity Needs Care

The formulas have clear limits. If $N=0$, frequency and pure premium are still defined when exposure is positive: frequency is zero, and if there are no claims and no losses, pure premium is zero. Average severity $L/N$, however, is undefined because division by zero is not permitted. Do not report a zero average severity for a claim-free group; report that it is not defined for that group.

The product identity also depends on consistent definitions. If the numerator of severity includes one set of claims while the frequency denominator uses a different claim count, the cancellation in the derivation does not apply. Similarly, paid severity combined with a frequency intended to represent a different claim population will not necessarily reconcile to a paid-loss pure premium. State the basis and use it consistently.

In practice, frequency can be expressed using different exposure units, such as policy-years, vehicle-years, or another appropriate measure. A change in the unit changes the numerical frequency and pure premium, but not the underlying aggregate losses if the conversion is performed consistently. For instance, expressing 0.015 claims per vehicle-year as 1.5 claims per 100 vehicle-years requires expressing pure premium per 100 vehicle-years as well; the corresponding severity remains dollars per claim.

Worked Example: Recovering Missing Loss Measures

A hypothetical property portfolio has 3,200 exposure units and 64 reported claims. Its observed pure premium on a specified loss basis is $147.50 per exposure unit. Find the aggregate losses, frequency, and average severity on that same basis, and verify the decomposition.

Aggregate losses. Multiply pure premium by exposure:

L=p e=\$147.50\times 3{,}200=\$472{,}000.

Frequency. Divide claims by exposure:

f=\frac{64}{3{,}200}=0.02\text{ claims per exposure unit.}

Average severity. Divide the recovered aggregate loss by claim count:

s=\frac{\$472{,}000}{64}=\$7{,}375\text{ per claim.}

Check. The product is $0.02\times \$7{,}375=\$147.50$ per exposure unit, as given. The aggregate check is $\$7{,}375\times64=\$472{,}000$.

Common mistake avoided. The $147.50 is loss per exposure unit, not average loss per claim. Dividing $147.50 by 64 would mix incompatible denominators. First use exposure to recover aggregate losses, then divide those losses by claims to obtain severity.

Common Exam Traps and Practical Checks

This decomposition is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus. A reliable calculation should include the units beside each ratio and a reconciliation to aggregate losses. Common errors include:

Mixing percentages and decimals. If frequency is 2 claims per 100 exposure units, use $0.02$ claims per exposure unit when multiplying by severity.
Using the wrong denominator for severity. Average severity is loss per claim, not loss per exposure unit and not loss per policyholder unless each policyholder has exactly one claim by definition.
Averaging groups without the right weights. Frequency and pure premium aggregate by exposure; severity aggregates by claim count.
Switching loss bases. Use a claim count and aggregate loss measure that refer to the same defined experience. Name whether the amount is paid, incurred, or another specified measure.
Reporting severity when no claims occurred. With $N=0$, $L/N$ is undefined, even if the group’s frequency and pure premium are both zero.

The final check is simple and powerful: calculate pure premium directly as $L/e$, then calculate it as $f s$. If the results differ, investigate the units, weights, claim count, or loss basis before interpreting the figures.

Check Your Understanding

For each hypothetical portfolio, use consistent units and check the frequency–severity product against pure premium.

A small equipment program has 900 exposure units, 27 claims, and $108,000 of losses. Find frequency, average severity, and pure premium. (Answer: Frequency $=0.03$ claims per exposure unit; severity $=\$4{,}000$ per claim; pure premium $=\$120$ per exposure unit.)
Group A has 400 exposure units, 8 claims, and $48,000 of losses. Group B has 600 exposure units, 18 claims, and $162,000 of losses. Find combined frequency, severity, and pure premium. (Answer: Frequency $=0.026$; severity $=\$8{,}076.92$ per claim; pure premium $=\$210$ per exposure unit.)
A hypothetical liability book has 1,500 exposure units and 42 claims. Its average severity is $3,750. Calculate aggregate losses and pure premium. (Answer: Aggregate losses $=\$157{,}500$; pure premium $=\$105$ per exposure unit.)
Portfolio A has 1,200 exposure units, 24 claims, and average severity of $4,500. Portfolio B has 800 exposure units, 20 claims, and aggregate losses of $140,000. Find combined frequency, severity, and pure premium. (Answer: Frequency $=0.022$; severity $=\$5{,}636.36$ per claim; pure premium $=\$124$ per exposure unit.)
A hypothetical portfolio has 54 claims, aggregate losses of $351,000, and pure premium of $130 per exposure unit. Recover exposure, frequency, and average severity. (Answer: Exposure $=2{,}700$ units; frequency $=0.02$; severity $=\$6{,}500$ per claim.)

What You'll Learn