Why Pure Premium Equals Frequency Times Severity

Why the Identity Matters

Ratemaking often separates the expected cost of claims into two questions: how often claims occur and how costly they are on average. That separation is useful for organizing data and analyzing rating variables. But the familiar relationship between pure premium, frequency, and severity is exact only when all three quantities refer to the same claims, loss amounts, exposure period, and exposure unit.

The identity begins with arithmetic on observed aggregate data. It does not, by itself, establish that frequency and severity are statistically independent, or that a particular model is appropriate. First define the quantities consistently; then decide whether their observed relationship is a suitable basis for a rate indication.

Definition: For a specified group of risks and a fixed observation period, let $E$ be total earned exposure in a stated unit, $N$ the number of claims under a stated claim definition, and $C_j$ the loss amount for claim $j$ on a stated loss basis, for $j=1,\ldots,N$. The aggregate loss is $L=\sum_{j=1}^{N} C_j$. When $E>0$, define aggregate pure premium as $L/E$. When $N>0$, define claim frequency as $N/E$ and average claim severity as $L/N$.

For example, if exposure is measured in earned vehicle-years, frequency is claims per earned vehicle-year and pure premium is dollars of loss per earned vehicle-year. Changing the exposure unit changes the numerical frequency and pure premium, so the unit must be stated. The claim definition also matters: the count must represent the same claim units whose amounts make up the aggregate loss.

The Identity and Its Conditions

When $E>0$ and $N>0$, substitute the definitions of frequency and severity:

\text{Frequency}\times\text{Severity} = \frac{N}{E}\times\frac{L}{N} = \frac{L}{E} = \text{Aggregate pure premium}.

The cancellation of $N$ explains why the relationship holds. It is an exact identity for a given set of observed claims, provided all quantities use matching definitions and bases. It does not require a probability distribution, an interest assumption, or a mortality assumption. Those assumptions are not applicable to this arithmetic identity.

The loss basis must match as well. If $L$ is gross loss, severity must be calculated from gross claim amounts. If $L$ is capped or otherwise adjusted loss, severity must use those same adjusted amounts. A change in the loss basis does not necessarily change the claim count, but it does change the loss total and average severity.

A short expected-value version is sometimes useful. Suppose, conditional on observing $N$ claims, each claim amount has the same mean $\mu$, so that $E[L\mid N]=N\mu$. Then the law of iterated expectation gives $E[L]=E[N]\mu$. Dividing by fixed exposure $E$ gives:

\frac{E[L]}{E} = \frac{E[N]}{E}\,\mu.

Here $\mu$ is the mean amount per claim under the stated assumptions. The conditional-mean assumption matters: if claim severity changes with claim count or with the mix of claims, the expected loss can still be expressed as expected claim count times an appropriately claim-weighted mean severity, but that mean need not be a fixed $\mu$ independent of the count.

Worked Applications

Worked Example: Calculating All Three Quantities on One Basis

A hypothetical small commercial auto portfolio has 240 earned vehicle-years of exposure during the observation period. It has four claims, with gross loss amounts of $600, $900, $1,500, and $2,400. Calculate claim frequency, average gross severity, and aggregate gross pure premium, and verify the identity.

Method and basis. Use all four claims in the count and use their gross loss amounts in both aggregate loss and severity. The denominator is 240 earned vehicle-years. The data are fixed observed amounts; no probability distribution, interest rate, or mortality assumption is needed.

Step 1: Aggregate gross loss. Add the four claim amounts:

$$ L=600+900+1{,}500+2{,}400=5{,}400. $$

Step 2: Claim frequency. Divide the claim count by exposure:

\frac{N}{E}=\frac{4}{240}=0.0166667 \quad\text{claims per earned vehicle-year}.

Step 3: Average gross severity. Divide the gross loss total by the same four claims:

\frac{L}{N}=\frac{5{,}400}{4}=\$1{,}350.00 \quad\text{per claim}.

Step 4: Aggregate gross pure premium. Divide the same loss total by the same exposure:

\frac{L}{E}=\frac{5{,}400}{240}=\$22.50 \quad\text{per earned vehicle-year}.

Check. Multiplying frequency by severity gives $0.0166667\times \$1{,}350.00\approx \$22.50$ per earned vehicle-year. The identity is verified; the displayed frequency is rounded, while the calculation uses $4/240$ exactly. The common mistake this method avoids is dividing severity by exposure instead of by the claim count.

Worked Example: Combining Two Exposure Segments

A hypothetical property program has two segments. Segment A has 80 earned location-years and two gross claims of $500 and $1,500. Segment B has 120 earned location-years and three gross claims of $1,000, $2,000, and $3,000. Find the overall frequency, severity, and gross pure premium. Also explain why simply averaging the two segment severities would be incorrect.

Method and basis. Each claim is counted once in its segment, all losses are gross amounts, and exposure is measured in earned location-years. The observations are deterministic for this calculation; no distribution, interest, or mortality assumption applies.

Step 1: Calculate segment totals. Segment A has $N_A=2$ claims and $L_A=\$500+\$1{,}500=\$2{,}000$. Segment B has $N_B=3$ claims and $L_B=\$1{,}000+\$2{,}000+\$3{,}000=\$6{,}000$.

Step 2: Combine counts, losses, and exposures.

N=2+3=5,\qquad L=\$2{,}000+\$6{,}000=\$8{,}000,\qquad E=80+120=200.

Step 3: Calculate the overall quantities. Overall frequency is $5/200=0.025$ claims per earned location-year. Overall severity is $\$8{,}000/5=\$1{,}600.00$ per claim. Overall pure premium is $\$8{,}000/200=\$40.00$ per earned location-year.

Check. $0.025\times \$1{,}600.00=\$40.00$ per earned location-year.

Segment A severity is $\$2{,}000/2=\$1{,}000.00$, and Segment B severity is $\$6{,}000/3=\$2{,}000.00$. Their unweighted average is $(\$1{,}000+\$2{,}000)/2=\$1{,}500.00$, which is not the overall severity. The overall average weights each segment by its claim count: $(2\times \$1{,}000+3\times \$2{,}000)/5=\$1{,}600.00$. The common mistake avoided is giving a two-claim segment the same severity weight as a three-claim segment.

Worked Example: Keeping a Claim Limit Consistent

A hypothetical liability portfolio has 120 earned policy-years and three covered claims with gross amounts of $800, $2,400, and $4,800. For a separate analysis, each claim amount is capped at $2,000. Calculate the gross pure premium and the capped-loss pure premium, using the same count of covered claims, and verify each frequency-severity product.

Method and basis. The claim definition is three covered claims in both calculations. For the first calculation, use gross amounts; for the second, use the amount per claim after the $2,000 cap. Exposure remains 120 earned policy-years. These are fixed hypothetical amounts, so no probability distribution, interest rate, or mortality assumption is required.

Step 1: Calculate gross loss and gross severity.

L_{\text{gross}}=\$800+\$2{,}400+\$4{,}800=\$8{,}000, \qquad S_{\text{gross}}=\frac{\$8{,}000}{3}=\$2{,}666.6667.

Gross frequency is $3/120=0.025$ claims per earned policy-year. Gross pure premium is $\$8{,}000/120=\$66.6667$ per earned policy-year. The check is $0.025\times \$2{,}666.6667\approx \$66.6667$.

Step 2: Apply the cap to each amount. The capped amounts are $\min(\$800,\$2{,}000)=\$800$, $\min(\$2{,}400,\$2{,}000)=\$2{,}000$, and $\min(\$4{,}800,\$2{,}000)=\$2{,}000$. Thus capped aggregate loss is $\$800+\$2{,}000+\$2{,}000=\$4{,}800$, and capped severity is $\$4{,}800/3=\$1{,}600.00$ per claim.

The claim count and exposure are unchanged, so capped frequency is still $0.025$. Capped pure premium is $\$4{,}800/120=\$40.00$ per earned policy-year. The check is $0.025\times \$1{,}600.00=\$40.00$.

Common mistake avoided. Multiplying capped frequency by gross severity gives $0.025\times \$2{,}666.6667\approx \$66.6667$, not the capped pure premium. That calculation mixes capped frequency and gross severity terminology—the frequency is numerically the same here, but the severity is not on the capped loss basis. The correction is to use capped severity, $1,600.00, with the count and exposure for that same portfolio.

Where the Identity Can Mislead

The algebra is simple; assembling compatible inputs is the actuarial work. Before multiplying frequency by severity, check the following:

Claim unit: Decide what one counted claim means for the analysis. A count of covered claim records should not be paired with a loss total that combines or splits those records differently without reconciling the units.
Loss basis: Match gross, capped, or otherwise adjusted losses to the severity being used. A changed loss basis can change severity and pure premium even if frequency is unchanged.
Exposure denominator: Use the same exposure quantity and unit in frequency and pure premium. Claims per earned vehicle-year cannot be multiplied by a severity calculated for a different claim population and treated as a consistent indication.
Observation period: Make sure the claims and earned exposure correspond to the intended period and coverage. Otherwise the numerator and denominator describe different experience.
Aggregation weights: Overall severity is total loss divided by total claims, not generally the arithmetic average of segment severities. Overall frequency and pure premium likewise use combined counts and losses over combined exposure.

There is an important boundary case. If $N=0$, then the observed aggregate loss from the defined claims is zero, but $L/N$ is undefined. Therefore the identity written using observed average severity cannot be calculated for that claim-free group. The aggregate ratio $L/E$ is still defined when $E>0$. This distinction is another reason not to treat every policy-level ratio as an ordinary value that can be averaged.

Key takeaway: On a consistent aggregate basis, pure premium equals claim frequency times average claim severity because $(N/E)(L/N)=L/E$. The identity is exact when the claim definition, loss basis, period, and exposure denominator align; it is not a substitute for checking how the data were constructed.

Check Your Understanding

For each problem, use the stated claim definition, loss basis, and exposure unit. Give money to the nearest cent and frequencies to four decimal places where needed.

A hypothetical equipment portfolio has 150 earned machine-years and three covered claims of $400, $1,100, and $2,000. Find frequency, average severity, and pure premium, then verify the identity. (Answer: frequency $=0.0200$ claims per earned machine-year; severity $=\$1{,}166.67$; pure premium $=\$23.33$ per earned machine-year.)
A hypothetical program has 60 earned site-years, two claims totaling $3,000, and 90 earned site-years, three claims totaling $6,000. Find the combined frequency, severity, and pure premium. (Answer: frequency $=0.0333$ claims per earned site-year; severity $=\$1{,}800.00$; pure premium $=\$60.00$ per earned site-year.)
A hypothetical portfolio has 80 earned policy-years and two claims with gross amounts $1,000 and $5,000. Each amount is capped at $3,000, with no change to the covered-claim count. Find capped severity and capped pure premium. (Answer: capped severity $=\$2{,}000.00$ per claim; capped pure premium $=\$50.00$ per earned policy-year.)
Two hypothetical segments each have one claim. Segment A has 40 earned exposure units and a $1,200 loss; Segment B has 160 earned exposure units and a $3,600 loss. Find the overall severity and pure premium, and explain why the unweighted average segment severity is not the overall severity. (Answer: overall severity $=\$2{,}400.00$ per claim; pure premium $=\$24.00$ per exposure unit; the unweighted average severity is $2,400 here, which happens to agree because each segment has one claim.)
A hypothetical portfolio has 200 earned vehicle-years, five claims, and aggregate loss of $12,500 on a specified adjusted loss basis. An analyst reports frequency as $5/200$ and severity as $2,500 per claim, but compares the product with a pure premium calculated from a different loss basis of $10,000. What is the pure premium on the stated adjusted basis, and what consistency correction is required? (Answer: adjusted-basis pure premium $=\$62.50$ per earned vehicle-year; use adjusted-basis severity of $\$12{,}500/5=\$2{,}500.00$ with frequency $0.025$, rather than comparing that product with the different-basis value $\$10{,}000/200=\$50.00$.)

What You'll Learn