Building Pure Premiums from Frequency and Severity

Why Frequency and Severity Must Be Built on Matching Bases

A pure premium is the expected covered loss assigned to one unit of exposure. A common way to estimate it is to multiply claim frequency by average claim severity. The calculation is short; the actuarial work lies in making sure that the claim count, claim payments, and exposure denominator describe the same risks, coverage, and time period.

For example, a claim count based on reported claims cannot be paired casually with a severity calculated only from closed claims. Nor should paid losses be compared with an incurred-loss estimate without recognizing that the latter may include case reserves. Such mixtures can create a numerical result that looks plausible but does not estimate a well-defined pure premium.

Definition: For a specified coverage, period, and exposure basis, the pure premium is expected covered loss per exposure unit. If $S$ is aggregate covered loss and $e$ is the number of exposure units, an aggregate estimate is $E[S]/e$. A frequency-severity estimate is $(E[N]/e)\times E[X]$ when claim count $N$ and claim severities $X$ meet the assumptions stated below.

Here, $N$ denotes the number of claims for the exposure being considered, and $X_j$ denotes the payment for claim $j$, measured on the chosen loss basis. Aggregate loss is $S=\sum_{j=1}^{N}X_j$, with $S=0$ if $N=0$. Frequency is claims per exposure unit; severity is dollars per claim. Their product therefore has units of dollars per exposure unit.

The Frequency-Severity Relationship

Suppose there are $e$ exposure units and $n$ claims, with total covered loss $L$ for those same claims. The empirical frequency is $n/e$, and the empirical average severity is $L/n$, provided $n>0$. Their product is

\frac{n}{e}\times\frac{L}{n}=\frac{L}{e}.

This identity explains why a frequency-severity calculation reconciles exactly with the aggregate loss per exposure when both are built from the same claims and payments. In prospective pricing, the inputs are estimates of future frequency and severity, so the product estimates future expected loss per exposure. The aggregate-model comparison should use the same coverage, time period, exposure, and loss basis.

Model condition: If claim payments are independent and identically distributed, each with mean $E[X]$, and are independent of $N$, then $E[S]=E[N]E[X]$. Dividing by $e$ gives pure premium $(E[N]/e)E[X]$. This factorization does not generally apply when claim-payment distributions depend on $N$, or when the claim payments within a count level do not share a common conditional mean.

The general conditional calculation does not require one common conditional mean for all payments. Given $N$, linearity of expectation gives

E[S\mid N] = E\left[\sum_{j=1}^{N}X_j\ \middle|\ N\right] = \sum_{j=1}^{N}E[X_j\mid N].

Taking expectations over $N$ yields the general result:

E[S] = E\left[\sum_{j=1}^{N}E[X_j\mid N]\right].

If, conditional on $N$, each claim has the same mean $m(N)$, then the sum inside the expectation is $Nm(N)$, so $E[S]=E[Nm(N)]$. If the claim payments are also independent of $N$ with common mean $E[X]$, then $m(N)=E[X]$, and the familiar product $E[N]E[X]$ follows. This is the justification for the formula, and also the boundary on when it may be used.

Fix the basis.
Specify the coverage, exposure unit, experience period, claim-count definition, and payment basis.

Calculate frequency.
Divide the matching claim count, or estimated expected count, by the matching exposure.

Calculate severity.
Divide matching losses or expected losses by the claim count represented by those losses.

Multiply and reconcile.
Multiply frequency by severity, then compare with aggregate loss per exposure on the same basis.

Worked Frequency-Severity Calculations

Worked Example: Reconcile a Single-Class Estimate

A hypothetical insurer reviews 800 vehicle-years of a specified coverage. The period contains 120 covered claims with total incurred loss of $216,000. Estimate the pure premium on an incurred-loss basis, then verify it using aggregate loss per exposure.

Method. Use the claim count, incurred losses, and vehicle-year exposure from the same coverage and period. The frequency is $120/800=0.15$ claims per vehicle-year.

The incurred severity is $216{,}000/120=\$1{,}800$ per claim. Therefore the frequency-severity estimate is $0.15\times \$1{,}800=\$270$ per vehicle-year.

For the aggregate check, divide incurred loss by exposure: $\$216{,}000/800=\$270$ per vehicle-year. The two calculations agree.

Check and trap avoided. The frequency, severity, and aggregate check all use incurred losses for the same 120 claims. Pairing the 120-claim frequency with a severity calculated from a different claim subset would break the reconciliation.

Worked Example: Combine Two Risk Classes Correctly

A hypothetical book contains 400 exposure units in Class A and 300 in Class B. Class A has 40 claims and $80,000 of covered loss; Class B has 60 claims and $210,000 of covered loss. Calculate the combined pure premium and compare it with the unweighted average of the two class pure premiums.

Method. Calculate each class’s frequency and severity, then weight its pure premium by exposure when combining the classes.

For Class A, frequency is $40/400=0.10$ claims per exposure unit. Severity is $\$80{,}000/40=\$2{,}000$. Its pure premium is $0.10\times \$2{,}000=\$200$ per exposure unit.

For Class B, frequency is $60/300=0.20$. Severity is $\$210{,}000/60=\$3{,}500$. Its pure premium is $0.20\times \$3{,}500=\$700$ per exposure unit.

The combined aggregate loss is $\$80{,}000+\$210{,}000=\$290{,}000$, and total exposure is $400+300=700$. Thus the combined pure premium is $\$290{,}000/700=\$414.2857\ldots$, or $414.29 per exposure unit to the nearest cent.

The exposure-weighted class calculation gives the same result: $(400\times \$200+300\times \$700)/700=\$290{,}000/700=\$414.29$. The unweighted average, $(\$200+\$700)/2=\$450$, is not the book’s pure premium because the two classes do not have equal exposure.

Check and trap avoided. The weighted calculation reproduces the aggregate-loss result. Averaging class rates without exposure weights gives too much influence to Class B relative to its share of the book’s exposure.

Worked Example: Diagnose a Paid-versus-Incurred Difference

For a hypothetical period, 1,000 policy-years produced 100 covered claims. Those same 100 claims have $240,000 in paid loss and $270,000 in incurred loss, with the difference attributable to outstanding case reserves. Calculate the pure premium on each loss basis and explain the difference.

Method. Keep the claim count and exposure fixed, and match each severity to its corresponding loss total.

Frequency is $100/1{,}000=0.10$ claims per policy-year. Paid severity is $\$240{,}000/100=\$2{,}400$, so the paid-loss pure premium is $0.10\times \$2{,}400=\$240$ per policy-year. The aggregate check is $\$240{,}000/1{,}000=\$240$.

Incurred severity is $\$270{,}000/100=\$2{,}700$, so the incurred-loss pure premium is $0.10\times \$2{,}700=\$270$ per policy-year. Its aggregate check is $\$270{,}000/1{,}000=\$270$.

The $30 difference per policy-year is exactly $(\$270{,}000-\$240{,}000)/1{,}000$. The calculations are both internally consistent; they answer different questions because one uses paid loss and the other uses incurred loss.

Check and trap avoided. If the intended estimate is incurred loss, using paid severity with an incurred aggregate total would mix bases. The claim count alone does not make paid and incurred severities interchangeable.

Worked Example: Conditional Claim Payments Need Not Share a Mean

For one hypothetical exposure unit, exactly two claims occur. Their payments are $1,000 and $3,000, respectively. Find aggregate loss and pure premium for this exposure, and show why multiplying the count by the conditional mean of the first claim payment gives an incorrect result.

Method. Sum the conditional expected payments claim by claim. Here the amounts are fixed, so each conditional expectation equals its stated payment.

With $N=2$, the general expression gives $E[S]=E[X_1\mid N=2]+E[X_2\mid N=2]=\$1{,}000+\$3{,}000=\$4{,}000$. The aggregate loss per exposure unit, and hence the pure premium for this single exposure unit, is $4,000.

The incorrect shortcut $N E[X_1\mid N]$ gives $2\times \$1{,}000=\$2{,}000$. It treats the first claim’s conditional mean as if it were the conditional mean of both claims. Since the two claim payments differ, that shortcut omits $2,000 of expected loss.

Check and trap avoided. The valid general expression is $E[S]=E[\sum_{j=1}^{N}E[X_j\mid N]]$. It reduces to $E[Nm(N)]$ only when each claim has the same conditional mean $m(N)$ given the count.

Common Basis Checks and Exam Traps

A frequency-severity estimate and an aggregate estimate can differ for legitimate reasons, but first check that they are estimating the same quantity. State the basis explicitly rather than relying on labels such as “losses” or “claims.”

Different claim populations: a reported-claim count paired with severity for closed claims may omit open claims or use a different selection rule.
Paid and incurred amounts mixed: incurred loss can include outstanding case reserves, while paid loss records amounts already paid. Select one basis or explain why the estimates differ.
Different exposure denominators: policy-years, vehicle-years, and other exposure units are not interchangeable unless the conversion is defined.
Unweighted class averages: a portfolio pure premium is exposure-weighted, not usually the simple average of class rates.
Unjustified independence shortcut: $E[S]=E[N]E[X]$ requires an appropriate common severity mean independent of count. When conditional payments vary by claim position or count, use the sum of conditional expectations.

A useful diagnostic is to reconstruct the total loss implied by the proposed frequency and severity: multiply the frequency by exposure to obtain implied claims, multiply those claims by severity to obtain implied aggregate loss, and compare that amount with the stated aggregate estimate. A discrepancy points to a calculation error or a difference in basis, scope, or assumptions that should be identified before selecting a rate indication.

Key takeaway: Frequency times severity is a pure premium only when the count, payment amounts, and exposure refer to a compatible basis. For a consistent empirical data set, the product equals aggregate loss per exposure; for expected values, use the product formula only under its stated severity and count assumptions.

Check Your Understanding

For each problem, state the calculation basis and check that frequency and severity refer to compatible data.

A hypothetical portfolio has 600 exposure units, 72 covered claims, and $108,000 of covered loss on one consistent basis. Calculate frequency, severity, and pure premium. (Answer: frequency $0.12$, severity $$1,500$, pure premium $$180$ per exposure unit.)
Class A has 250 exposure units, frequency $0.08$, and severity $1,250. Class B has 750 exposure units, frequency $0.12$, and severity $2,000. Calculate each class’s expected aggregate loss and the combined pure premium. (Answer: Class A $$25,000$, Class B $$180,000$, combined pure premium $$205$ per exposure unit.)
A hypothetical period has 900 exposure units and 90 claims. The same claims have $162,000 paid and $180,000 incurred. Find the paid and incurred pure premiums. (Answer: paid $$180$ and incurred $$200$ per exposure unit.)
Conditional on $N=1$, the expected payment is $1,000. Conditional on $N=2$, the two claims have expected payments of $1,200 and $1,800. Find the expected aggregate loss conditional on each count. Explain why $N$ times the first claim’s conditional mean is not valid when $N=2$. (Answer: $$1,000$ for $N=1$, $$3,000$ for $N=2$; for $N=2$, the sum is $$1,200+$1,800=$3,000$, not $2\times $1,200$.)
A hypothetical experience set has 400 exposure units and 50 claims. A severity extract includes only 45 of those claims, with $99,000 total loss. Can the extract be combined directly with the 50-claim frequency to obtain the full pure premium? State the matched-subset calculation and what is missing. (Answer: No. The matched subset gives frequency $45/400=0.1125$, severity $$99,000/45=$2,200$, and subset loss per exposure $$247.50$; the losses for the other 5 claims are needed for the full-period pure premium.)

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