Why Calculate Pure Premiums by Risk Group?
A portfolio may contain groups with different expected claim frequencies and different average claim severities. A single portfolio average can conceal those differences. The frequency-severity approach first estimates expected loss cost for each group, then combines group results using the amount of exposure in each group.
A pure premium is an expected loss amount per unit of exposure. It represents expected losses only; it is not, by itself, a final charged premium that includes expenses, profit provisions, or other adjustments. The exposure basis must be stated: for example, a vehicle-year, a policy-year, or a specified amount of insured property exposure.
Frequency and severity must refer to compatible periods and claim definitions. If frequency is annual claims per vehicle-year, severity must be dollars per claim for the same type of covered claim. If severity is measured per payment rather than per claim, the frequency and severity definitions must be aligned before multiplying them.
Why Frequency Times Severity Gives Expected Loss
The multiplication follows from the expected value of aggregate loss. For one exposure unit in a particular risk group, let \(N\) be the number of claims and let \(X_j\) be the loss for claim \(j\). Total loss is \(S=\sum_{j=1}^{N}X_j\). Conditional on \(N=n\), if each claim has mean severity \(\mu_g\), the expected total loss is \(n\mu_g\). Taking expectation over the claim count gives:
Dividing by one exposure unit yields \(E[N]\mu_g\) per exposure unit. More generally, if the expected claim count for \(e_g\) exposure units is \(e_g\lambda_g\), then expected aggregate loss is \(e_g\lambda_g\mu_g\). This derivation uses a common conditional mean severity and an expected count that scales with exposure. If severity depends on claim count or on the event that generated the count, the simple product of separate marginal estimates need not equal expected loss; the dependence structure must then be reflected in the estimate.
Worked Example: Estimating Group Pure Premiums From Experience
A hypothetical insurer reviews one year of experience for two groups of small commercial vehicles. Group A has 300 vehicle-years, 18 claims, and total claim payments of $45,000. Group B has 200 vehicle-years, 24 claims, and total claim payments of $60,000. Calculate each group's frequency, severity, pure premium, and the portfolio pure premium. Treat the observed experience ratios as the supplied estimates, with claim amounts measured in dollars per claim.
Method and assumptions. Estimate frequency as claim count divided by vehicle-years and severity as total claim payments divided by claim count. This calculation treats each group's average claim payment as its severity estimate and uses the same one-year period for both groups.
Group A frequency. There are 18 claims over 300 vehicle-years:
Group A severity. The average payment per claim is:
Therefore, Group A's pure premium is \(0.06\times \$2{,}500=\$150\) per vehicle-year. Its expected loss at this estimated rate is \(300\times \$150=\$45{,}000\).
Group B frequency. There are 24 claims over 200 vehicle-years:
Group B severity. The average payment per claim is:
Group B's pure premium is \(0.12\times \$2{,}500=\$300\) per vehicle-year. Its expected loss is \(200\times \$300=\$60{,}000\).
The portfolio expected loss is \(\$45{,}000+\$60{,}000=\$105{,}000\), and total exposure is \(300+200=500\) vehicle-years. Thus the portfolio pure premium is \(\$105{,}000/500=\$210\) per vehicle-year. As a check, the exposure-weighted rate is \((300\times \$150+200\times \$300)/500=\$210\).
Common mistake avoided: averaging the two group pure premiums equally would give \((\$150+\$300)/2=\$225\), which is not the portfolio rate. The groups have different exposure amounts, so the correct weights are 300 and 200 vehicle-years.
Combining Group Estimates Correctly
Once each group has a pure premium, multiply that rate by the group's exposure to estimate its expected losses. Add those expected losses to obtain the portfolio expected loss, then divide by total exposure. This order makes the weighting explicit and helps prevent a group with little exposure from receiving the same influence as a much larger group.
The weight for group \(g\) is its share of total exposure, \(e_g/\sum_h e_h\). These weights sum to one, so the portfolio rate is a weighted average of the group rates. This averaging is valid only when the exposures use a common basis. For example, vehicle-years should not be added directly to property-years unless the problem defines a conversion to a common exposure measure.
Worked Example: Weighting Model Estimates Across Three Groups
A hypothetical rating analysis supplies annual frequency and severity estimates for three classes of equipment coverage. The estimates are: Class 1, frequency 0.075 claims per insured unit-year and severity $3,200 per claim; Class 2, frequency 0.11 and severity $2,200; Class 3, frequency 0.16 and severity $4,100. The portfolio has 400, 250, and 150 insured unit-years in the respective classes. Calculate each class pure premium, expected portfolio losses, and the portfolio pure premium.
Method and assumptions. The frequency estimates and exposures are annual and use the same insured unit-year basis. Each supplied severity is the expected payment per claim for its class. Calculate \(P_g=\lambda_g\mu_g\) first, then weight by exposure.
Class 1. Its pure premium is \(0.075\times \$3{,}200=\$240\) per unit-year. Expected losses are \(400\times \$240=\$96{,}000\).
Class 2. Its pure premium is \(0.11\times \$2{,}200=\$242\) per unit-year. Expected losses are \(250\times \$242=\$60{,}500\).
Class 3. Its pure premium is \(0.16\times \$4{,}100=\$656\) per unit-year. Expected losses are \(150\times \$656=\$98{,}400\).
Total expected losses are \(\$96{,}000+\$60{,}500+\$98{,}400=\$254{,}900\). Total exposure is \(400+250+150=800\) unit-years. The portfolio pure premium is \(\$254{,}900/800=\$318.625\), or $318.63 per unit-year when rounded to the nearest cent.
Check. The exposure shares are \(400/800=0.5\), \(250/800=0.3125\), and \(150/800=0.1875\). Their weighted average is \(0.5(\$240)+0.3125(\$242)+0.1875(\$656)=\$120+\$75.625+\$123=\$318.625\), matching the aggregate-loss calculation.
Common mistake avoided: taking the unweighted mean of the three class rates would give \((\$240+\$242+\$656)/3=\$379.33\). That gives the smallest class the same weight as the other classes despite having fewer exposures; it does not represent portfolio expected losses per unit of exposure.
Keep the Calculation Unrounded Until the End
Displayed rates are often rounded for reporting, but the portfolio expected loss should be calculated from the original group estimates or from the unrounded portfolio rate. Multiplying a rounded portfolio rate back by exposure can produce a small discrepancy. The discrepancy is arithmetic rounding, not a change in the underlying expected loss.
Worked Example: Preserving the Exact Portfolio Total
A hypothetical portfolio has 355 exposure units and a total expected loss of $195,000 from its group calculations. Find the portfolio pure premium and show why the displayed rounded rate should not be used to reconstruct the exact expected loss.
Method. Divide total expected loss by total exposure, retaining the exact quotient for subsequent calculations:
The pure premium rounded to four decimal places is $549.2958. If this displayed value is multiplied by 355, the result is \(\$549.2958\times355=\$195{,}000.009\), which rounds to $195,000.01 to the nearest cent. It is close, but it is not exactly $195,000.
The exact expected loss remains $195,000, obtained from the underlying group amounts or from the unrounded quotient \(\$195{,}000/355\) multiplied by 355. Round the rate only when reporting it; do not substitute the rounded rate into a calculation that requires an exact total.
Common mistake avoided: claiming that the rounded rate times exposure equals the original total exactly. The corrected statement is either that the exact quotient reproduces $195,000, or that the four-decimal rate gives approximately $195,000.01.
Checks Before Reporting a Result
A reliable calculation can be reviewed with a short sequence of checks. These checks do not replace actuarial judgment about whether the input estimates are appropriate, but they catch frequent arithmetic and specification errors.
Confirm that frequency is claims per stated exposure unit for the same period and that severity is dollars per corresponding claim.
Multiply the group frequency by its group severity; do not substitute a portfolio-wide component unless it is explicitly the estimate to use.
Multiply each group pure premium by that group's exposure and sum the resulting dollar amounts.
Check that the portfolio rate equals total expected loss divided by total exposure and is an exposure-weighted average of group rates.
Retain unrounded values in intermediate calculations, and label the units and rounding precision of reported results.
The portfolio pure premium must lie between the smallest and largest group pure premiums when all exposures are positive and the exposure basis is common. If it does not, revisit the weights, units, or arithmetic. A result outside that range can arise from a miscalculation or from comparing rates defined on different exposure bases.
Check Your Understanding
For each problem, use the stated exposure basis and retain unrounded values until the final answer.
- A hypothetical property class has an annual frequency estimate of 0.04 claims per insured location-year and a severity estimate of $2,750 per claim. There are 600 location-years. Find the pure premium and expected losses. (Answer: $110 per location-year; $66,000 expected losses.)
- Two hypothetical groups have exposures of 200 and 300 policy-years. Their frequency and severity estimates are, respectively, 0.08 and $1,500 for Group A, and 0.05 and $2,400 for Group B. Find both group pure premiums and the portfolio rate. (Answer: $120 per policy-year for each group; $120 per policy-year for the portfolio.)
- A class has 450 exposure units, 36 claims, and total claim payments of $108,000 in the period. Estimate its frequency, severity, pure premium, and total expected losses using these experience ratios. (Answer: 0.08 claims per exposure unit; $3,000 per claim; $240 per exposure unit; $108,000.)
- A hypothetical portfolio has three groups: exposures 100, 250, and 150; annual frequencies 0.03, 0.12, and 0.08; and severities $4,000, $2,000, and $3,500, respectively. Calculate group rates and the portfolio rate. (Answer: group rates $120, $240, and $280; portfolio pure premium $228 per exposure unit.)
- A frequency estimate is 0.02 claims per policy-quarter, and the severity estimate is $5,000 per claim. There are 800 policy-years. Assuming the quarterly rate applies consistently for four quarters and severity is on the matching claim basis, find the annual pure premium and expected annual losses. (Answer: 0.08 claims per policy-year; $400 per policy-year; $320,000 expected losses.)