Deriving the Frequency-Severity Pure Premium

Why Frequency Times Severity Gives Pure Premium

A pure premium is the expected loss cost for one unit of exposure, before provisions for expenses, profit, or contingencies. The frequency-severity method expresses that cost as two components: how many claims are expected per exposure unit, and how much is expected to be paid for each claim. This decomposition is useful only when the count, claim amount, exposure, and time period all refer to the same underlying risk.

Definition: Let $e$ be a fixed amount of exposure, $N$ the number of claims arising from it, $X_j$ the loss amount for claim $j$, and $S=\sum_{j=1}^{N}X_j$ the aggregate loss. Expected claim frequency per exposure is $E[N]/e$. Expected claim severity is the expected loss amount per claim, $E[X_j]$, when claims share a common severity distribution. The expected pure premium per exposure is $E[S]/e$.

Here, a claim amount must be defined consistently. For example, it could mean insurer-paid indemnity, or insurer-paid indemnity plus a specified measure of claim adjustment expense. The definition must be the same in the aggregate loss and severity. Similarly, the exposure unit must be stated, such as one vehicle-year or one policy-year, and the count and losses must cover the same experience period and claim population.

Deriving the Identity

Assume the exposure amount $e$ is fixed, $N$ has finite expectation, and each claim amount has the same finite expected value $\mu$. A sufficient standard model is that claim severities are identically distributed with mean $\mu$ and independent of $N$. More generally, the result holds if the expected severity of each claim, conditional on the claim count, is $\mu$. No interest discounting or mortality model is used here: the loss amounts are measured on a common, undiscounted basis.

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

Taking expectations of both sides and using the law of total expectation gives the aggregate-loss expectation. Dividing by the fixed exposure amount then yields the pure premium identity.

E[S]=E\big[E[S\mid N]\big]=E[N\mu]=E[N]\mu

\frac{E[S]}{e} = \frac{E[N]}{e}\,E[X_j]

Thus, expected loss per exposure equals expected claim count per exposure multiplied by expected loss per claim. If $E[N]>0$, expected severity can also be expressed as $E[S]/E[N]$; substituting that expression gives the same result. This is a ratio of expectations, not generally the expectation of a random ratio such as $S/N$. When there are no claims, $S/N$ is undefined, even though aggregate loss and pure premium remain well defined.

Condition: The product is valid when frequency and severity use the same claim definition, exposure base, experience period, and loss-cost basis. If severity varies with the claim count, the simple form using one unconditional mean severity requires care: the relevant average must reflect the severity distribution of the claims generated by that frequency model.

Worked Calculations

Worked Example: Pure Premium Under a Simple Count-and-Loss Model

A hypothetical insurer models one-year losses for a risk class using one vehicle-year as the exposure unit. The claim count $N$ is Poisson with parameter $\lambda=0.12$ per vehicle-year. Claim amounts are independent of $N$ and are independently drawn from a discrete distribution: $X_j=\$400$ with probability $0.25$, and $X_j=\$1{,}200$ with probability $0.75$. Find the expected pure premium and check it using 10,000 vehicle-years. No interest rate or mortality assumption applies; losses are undiscounted.

Method and severity. The severity mean is $E[X_j]=0.25(\$400)+0.75(\$1{,}200)=\$100+\$900=\$1{,}000$. The Poisson parameter is the expected count per exposure unit, so $E[N]/e=0.12$ per vehicle-year.

Pure premium. Multiplying frequency by severity gives $0.12\times\$1{,}000=\$120$ per vehicle-year.

Check at 10,000 vehicle-years. The expected count is $10{,}000(0.12)=1{,}200$ claims. Expected aggregate loss is $1{,}200(\$1{,}000)=\$1{,}200{,}000$. Dividing by 10,000 vehicle-years gives $\$120$ per vehicle-year, matching the product.

The common mistake this avoids is treating the Poisson parameter as a dollar amount. It is a claim count per exposure unit; it must be multiplied by severity to obtain dollars per exposure unit.

Worked Example: Verifying the Identity for a Larger Exposure Base

A hypothetical book has 2,400 exposure units for one coverage period. Its claim count is modeled as Poisson with mean $0.075$ claims per exposure unit. Each claim amount is independent of the count and equals $\$800$ with probability $0.60$, or $\$2{,}300$ with probability $0.40$. Calculate the expected pure premium and verify it from expected total claims and losses. Interest and mortality are not applicable; amounts are undiscounted.

Step 1: Calculate severity. $E[X_j]=0.60(\$800)+0.40(\$2{,}300)=\$480+\$920=\$1{,}400$.

Step 2: Calculate expected frequency and pure premium. The expected frequency is $0.075$ claims per exposure unit. Therefore, $0.075(\$1{,}400)=\$105$ per exposure unit.

Step 3: Check using the whole book. Expected claims are $2{,}400(0.075)=180$. Expected aggregate loss is $180(\$1{,}400)=\$252{,}000$. Per exposure unit, this is $\$252{,}000/2{,}400=\$105$.

The method is the expected-count times expected-severity decomposition. It avoids dividing the expected loss by the exposure count while accidentally using a frequency expressed on a different base, such as per 100 exposure units.

Worked Example: Combining Two Risk Groups Correctly

A hypothetical portfolio contains 1,000 exposure units in Group A and 500 in Group B. Group A has a Poisson claim count with mean $0.08$ per exposure unit; its claim amount is $\$1{,}000$ or $\$3{,}000$, each with probability $0.50$. Group B has a Poisson claim count with mean $0.24$ per exposure unit; its claim amount is $\$1{,}000$ with probability $0.25$ or $\$5{,}000$ with probability $0.75$. Within each group, claim amounts are independent of counts. Find the portfolio pure premium and verify it both as aggregate expected loss per exposure and as portfolio frequency times portfolio severity. No interest or mortality assumption applies; amounts are undiscounted.

Step 1: Find each group's severity. For Group A, $E[X_A]=0.50(\$1{,}000)+0.50(\$3{,}000)=\$2{,}000$. For Group B, $E[X_B]=0.25(\$1{,}000)+0.75(\$5{,}000)=\$4{,}000$.

Step 2: Find group expected counts and losses. Group A has $1{,}000(0.08)=80$ expected claims and expected loss $80(\$2{,}000)=\$160{,}000$. Group B has $500(0.24)=120$ expected claims and expected loss $120(\$4{,}000)=\$480{,}000$.

Step 3: Divide total expected loss by total exposure. The portfolio has $80+120=200$ expected claims and $\$160{,}000+\$480{,}000=\$640{,}000$ expected loss, across $1{,}000+500=1{,}500$ exposure units. Thus its pure premium is $\$640{,}000/1{,}500=\$426.666\ldots$, or $\$426.67$ per exposure unit to the nearest cent.

Step 4: Check using portfolio frequency and severity. Portfolio frequency is $200/1{,}500=0.133333\ldots$ claims per exposure unit. Portfolio severity is $\$640{,}000/200=\$3{,}200$ per claim. Their product is $0.133333\ldots(\$3{,}200)=\$426.666\ldots$, again $\$426.67$ per exposure unit to the nearest cent.

The method weights group severity by expected claim counts when finding portfolio severity. A simple average of the two group severities would be $(\$2{,}000+\$4{,}000)/2=\$3{,}000$, which ignores that Group B generates more claims. Using that unweighted average would give $0.133333\ldots(\$3{,}000)=\$400$, an incorrect portfolio pure premium.

Observed Experience and Weighting

For a fixed observed portfolio, let $n$ be the number of claims, $s$ the sum of their loss amounts, and $e$ the exposure. If $n>0$, the observed frequency is $n/e$ and the observed severity is $s/n$. Multiplying them cancels the claim count:

\frac{n}{e}\times\frac{s}{n}=\frac{s}{e}

This is an arithmetic identity for observed data, not by itself a guarantee that observed loss per exposure equals the future expected pure premium. Observed counts and amounts can differ from their expectations. Also, when $n=0$, observed severity $s/n$ is undefined, while observed aggregate loss per exposure is simply zero.

When combining multiple groups, calculate portfolio frequency as total expected claims divided by total exposure, and portfolio severity as total expected loss divided by total expected claims. Equivalently, calculate each group's expected loss, add those losses, and divide by total exposure. These approaches preserve the relationship between the claim mix and the resulting cost. Using an unweighted average of group frequencies or severities generally does not.

Common Mistakes and Practical Checks

Mixing exposure units. If frequency is stated per 100 exposure units, convert it to a per-unit basis before reporting pure premium per exposure unit, or retain the per-100 basis consistently.
Using inconsistent loss amounts. A severity based on paid losses cannot be multiplied by a frequency whose claim definition or valuation basis differs without an adjustment.
Using the wrong average across groups. Portfolio severity is claim-weighted, not generally the unweighted mean of class severities. Portfolio frequency is exposure-weighted, not generally the unweighted mean of class frequencies.
Confusing expectation with observed experience. The identity for expected values follows from the model assumptions. A particular period's loss per exposure is a realized result and need not equal the expected pure premium.
Ignoring a zero-claim case. Severity calculated as aggregate loss divided by claim count is undefined at zero claims. Define expected severity from the model or use aggregate expected loss directly rather than dividing by zero.

Key takeaway: On a consistent basis, $E[S]/e=(E[N]/e)E[X_j]$. For a portfolio of groups, add expected claims and expected losses first, then divide by total exposure; this automatically gives the correct portfolio frequency, severity, and pure premium.

Check Your Understanding

For each hypothetical model, check that the count, severity, and exposure refer to the same period and basis.

A one-year risk unit has $N\sim\operatorname{Poisson}(0.05)$ claims. Each claim is $\$1{,}000$ or $\$3{,}000$ with equal probability, independently of $N$. Find the pure premium per risk unit. (Answer: $\$100$ per risk unit.)
A hypothetical portfolio has 900 exposure units, expected frequency $0.20$ claims per unit, and expected severity $\$750$ per claim. Find expected claims, expected aggregate loss, and pure premium per exposure unit. (Answer: 180 expected claims; $\$135{,}000$ expected loss; $\$150$ per exposure unit.)
Group A has 600 exposure units, frequency $0.10$, and severity $\$1{,}200$. Group B has 400 exposure units, frequency $0.20$, and severity $\$1{,}800$. Find the portfolio pure premium using the combined expected claims and losses. (Answer: $\$216$ per exposure unit.)
A risk class has frequency $0.06$ claims per exposure unit. Claim amounts are $\$2{,}000$ with probability $0.80$ and $\$7{,}000$ with probability $0.20$. For 5,000 exposure units, find expected claims, expected loss, and pure premium. (Answer: 300 expected claims; $\$900{,}000$ expected loss; $\$180$ per exposure unit.)
A hypothetical book has 12 claims per 100 exposure-years and average claim severity of $\$2{,}500$. Convert frequency to a per-exposure-year basis and find pure premium per exposure-year. (Answer: frequency $0.12$ claims per exposure-year; pure premium $\$300$ per exposure-year.)

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