Frequency, Severity, and Pure Premium

Why Frequency and Severity Belong Together

A loss cost answers a practical question: how much claim loss is expected for one unit of exposure? For a hypothetical insurer, that unit might be one vehicle-year, one policy-year, or another consistently defined exposure unit. A loss cost helps describe the expected claim burden attached to that unit, before expenses, profit, or other premium components are included.

Two quantities explain the loss cost. Claim frequency measures the number of claims relative to exposure. Average severity measures the amount of loss per claim. Their product is the pure premium, also called the loss cost per exposure. The calculation is meaningful only when the claim counts, loss amounts, and exposures refer to compatible populations, time periods, and claim definitions.

Definition: For a portfolio with $E$ exposure units, $N$ claims, and total loss amount $L$, empirical claim frequency is $N/E$, empirical average severity is $L/N$ when $N>0$, and empirical pure premium is $L/E$. Thus, when $N>0$, empirical pure premium equals claim frequency multiplied by average severity.

Frequency is measured in claims per exposure unit, severity in dollars per claim, and pure premium in dollars per exposure unit. Keeping those units visible is a useful check: claims cancel when frequency is multiplied by severity.

\frac{\text{claims}}{\text{exposure}} \times \frac{\text{dollars}}{\text{claim}} = \frac{\text{dollars}}{\text{exposure}}

From Aggregate Loss to the Frequency-Severity Product

Let $N$ be the number of claims in a portfolio, and let $X_j$ be the loss amount associated with claim $j$, for $j=1,\ldots,N$. The aggregate loss is $S_N=\sum_{j=1}^{N}X_j$. For observed data, the total loss $L$ is the sum of the observed claim amounts. Dividing that total by exposure gives the observed pure premium.

\frac{L}{E} = \frac{N}{E}\times\frac{L}{N}, \qquad N>0

This identity follows because the factors $N$ cancel: $(N/E)(L/N)=L/E$. It is an exact rearrangement of the observed totals, not an approximation. The factor $N/E$ is the observed frequency, while $L/N$ is the observed average severity.

For a prospective calculation, the corresponding quantities are expected values under the assumptions selected for the future period. Suppose the expected number of claims per exposure unit is $\lambda$, and each claim has expected loss $m$. If claim severities have the same mean $m$ and the number of claims is independent of the severities, then expected aggregate loss per exposure unit is $\lambda m$. The independence and common-mean assumptions justify multiplying expected claim count by the expected amount per claim. If claim severity differs systematically with claim count or risk characteristics, a single frequency and a single severity may not be adequate summaries.

E[S_N\mid \text{one exposure}] = E[N]\,m = \lambda m

The prospective pure premium is therefore expected loss per exposure unit under the specified assumptions. It is not automatically a complete premium: expenses, risk margins, profit provisions, and other components are outside this calculation. Nor does the calculation itself adjust historical data for changes in time or coverage. The loss amounts and exposures used must already be on an appropriate, consistent basis for the calculation being made.

Key distinction: Observed pure premium is total observed loss divided by exposure. Prospective pure premium is expected loss per exposure under selected prospective frequency and severity assumptions. The arithmetic form is parallel, but the first describes experience and the second describes an expectation.

Worked Calculations

Worked Example: Calculating a Loss Cost From Portfolio Totals

A hypothetical book of small-equipment coverage has 240 equipment-years of exposure, 36 reported claims, and total losses of $108,000. Calculate observed frequency, average severity, and pure premium. Then estimate the losses represented by 800 exposure units if the observed pure premium is used as the estimate.

Method. Divide claims by exposure for frequency, losses by claims for severity, and losses by exposure for pure premium. The units are equipment-years, claims, and dollars.

Frequency. There are $36/240=0.15$ claims per equipment-year.

Severity. Average loss per claim is $\$108{,}000/36=\$3{,}000$.

Pure premium. Multiplication gives $0.15\times \$3{,}000=\$450$ per equipment-year. Direct division checks the result: $\$108{,}000/240=\$450$ per equipment-year.

Losses for 800 exposure units. Applying the same loss cost gives $800\times \$450=\$360{,}000$. Equivalently, the estimated claim count is $800\times0.15=120$, and $120\times\$3{,}000=\$360{,}000$.

The direct and frequency-severity calculations agree. This projection assumes the selected frequency and severity continue to apply to those 800 exposure units; it is not a guarantee of the realized losses. The common mistake avoided here is reporting $3,000, the average loss per claim, as though it were loss per equipment-year.

Worked Example: Combining Two Segments Correctly

A hypothetical warranty portfolio has two segments. Segment A has 400 exposure units, 40 claims, and $96,000 of losses. Segment B has 150 exposure units, 30 claims, and $90,000 of losses. Calculate frequency, severity, and pure premium for each segment and for the combined portfolio.

Segment A. Frequency is $40/400=0.10$ claims per exposure. Severity is $\$96{,}000/40=\$2{,}400$ per claim. Its pure premium is $0.10\times\$2{,}400=\$240$ per exposure; checking directly, $\$96{,}000/400=\$240$.

Segment B. Frequency is $30/150=0.20$ claims per exposure. Severity is $\$90{,}000/30=\$3{,}000$ per claim. Its pure premium is $0.20\times\$3{,}000=\$600$ per exposure; direct division gives $\$90{,}000/150=\$600$.

Combined totals. Exposure is $400+150=550$, claims are $40+30=70$, and losses are $\$96{,}000+\$90{,}000=\$186{,}000$.

Combined frequency is $70/550=0.1272727$ claims per exposure. Combined severity is $\$186{,}000/70=\$2{,}657.1429$ per claim. Their product is $0.1272727\times\$2{,}657.1429=\$338.1818$ per exposure. Direct division confirms $\$186{,}000/550=\$338.1818$. To the nearest cent, combined pure premium is $338.18 per exposure.

The combined result can also be checked by weighting each segment’s pure premium by its share of exposure: $(400\times\$240+150\times\$600)/550=(\$96{,}000+\$90{,}000)/550=\$338.1818$. The common mistake avoided is taking the unweighted average of the segment pure premiums, $(\$240+\$600)/2=\$420$. That treats the segments as if they had equal exposure, although they do not.

Worked Example: Applying Prospective Assumptions

For a hypothetical future period, an insurer selects an expected claim frequency of 0.08 claims per policy-year and an expected severity of $4,250 per claim. A proposed portfolio contains 1,250 policy-years. Calculate the prospective pure premium and the expected aggregate losses under these assumptions.

Method. Multiply prospective frequency by prospective severity to obtain expected loss per policy-year, then multiply by the number of policy-years to obtain expected aggregate loss.

Prospective pure premium is $0.08\times\$4{,}250=\$340$ per policy-year. The units confirm the result: claims per policy-year multiplied by dollars per claim gives dollars per policy-year.

Expected claim count is $1{,}250\times0.08=100$ claims. Expected aggregate loss is $100\times\$4{,}250=\$425{,}000$. The exposure calculation gives the same answer: $1{,}250\times\$340=\$425{,}000$.

Thus the indicated prospective loss cost is $340 per policy-year, and the expected aggregate losses for the proposed exposure are $425,000. These are expected values, not predictions that exactly 100 claims or exactly $425,000 of loss will occur. The common mistake avoided is multiplying the 0.08 frequency by the number of policy-years and then treating the resulting 100 expected claims as a dollar amount without applying severity.

Weighting, Interpretation, and Exam Traps

When segments are combined, totals should be combined before calculating portfolio frequency and severity. Overall frequency is total claims divided by total exposure, not the simple average of segment frequencies. Overall severity is total losses divided by total claims, not the simple average of segment severities. These denominators differ because segments can have different exposures and different numbers of claims.

For segment $k$, with exposure $E_k$, claims $N_k$, and losses $L_k$, the combined pure premium is $\sum_k L_k/\sum_k E_k$. It is also the exposure-weighted average of segment pure premiums $L_k/E_k$. This follows by multiplying each segment pure premium by its exposure, summing to recover total losses, and dividing by total exposure.

\frac{\sum_k L_k}{\sum_k E_k} = \frac{\sum_k E_k(L_k/E_k)}{\sum_k E_k}

A frequent denominator error is to divide total losses by the number of claims and call the result pure premium. That calculation gives average severity, in dollars per claim. In the two-segment example, $\$186{,}000/70=\$2{,}657.14$ per claim; it is not the combined loss cost. Dividing by 550 exposure units gives $338.18 per exposure. The correction is to identify the requested denominator first: claims for severity, exposure for pure premium.

If there are no claims, empirical severity $L/N$ is undefined because $N=0$. If losses are also zero and exposure is positive, observed pure premium $L/E$ is still zero. A prospective estimate in that situation requires selected assumptions or other information; it cannot be obtained by calculating an average severity from zero observed claims.

A pure premium is useful because it expresses expected claim losses on a common exposure basis. The result is interpretable only if the exposure unit is stated and the underlying claim and loss definitions are consistent. Check both the arithmetic and the units, and distinguish expected loss cost from the full amount charged to a policyholder.

Exam check: Before multiplying, label frequency as claims per exposure and severity as dollars per claim. Before interpreting the answer, verify that the result is dollars per exposure and that the data or assumptions refer to the intended period and population.

Check Your Understanding

For each problem, calculate the requested quantities and check that the units and denominators are consistent.

A hypothetical bicycle-protection portfolio has 900 bicycle-years of exposure, 54 claims, and $145,800 in total losses. Find frequency, severity, and pure premium. (Answer: 0.06 claims per bicycle-year; $2,700 per claim; $162 per bicycle-year.)
A hypothetical appliance plan has 200 exposure units, 18 claims, and $45,000 of losses in one segment, and 300 exposure units, 42 claims, and $126,000 of losses in another. Find the combined pure premium. (Answer: $342 per exposure unit.)
A prospective portfolio has 1,600 policy-years. Selected frequency is 0.075 claims per policy-year and selected severity is $5,200 per claim. Find the pure premium, expected claim count, and expected aggregate loss. (Answer: $390 per policy-year; 120 claims; $624,000.)
A hypothetical device-coverage portfolio has 250 exposure units, 20 claims, and $80,000 of losses in one segment, and 450 exposure units, 90 claims, and $315,000 of losses in another. Find overall frequency, severity, and pure premium. (Answer: 0.1571 claims per exposure; $3,590.91 per claim; $564.29 per exposure.)
A segment has positive exposure but no observed claims and no observed losses. State its observed pure premium and whether its empirical average severity can be calculated. (Answer: Observed pure premium is $0 per exposure; empirical average severity is undefined because there are zero claims.)

What You'll Learn