Worked Example: Pure Premium Under Two Exposure Bases

Why the Exposure Unit Matters

A pure premium is a loss amount per unit of exposure. Its numerical value depends on the unit used: a loss cost per policy-year will generally differ from a loss cost per vehicle-year, even when both describe exactly the same experience. A meaningful comparison therefore requires more than dividing losses by an exposure count. The analyst must also confirm that the exposure count and the rate being compared use compatible units.

This tutorial focuses on reconciling pure premium indications for the same hypothetical experience measured on two exposure bases. Assume that the loss amount is already on the intended basis for the analysis—for example, the selected losses are already adjusted to the required coverage and period. Loss development, trend selection, and expense provisions are not recalculated here.

Definition: For a selected loss amount $L$ and exposure $E$, the pure premium per exposure unit is $L/E$. If the same losses are divided by two different exposure counts, the resulting pure premiums have different units. A conversion relationship between exposure bases is needed to compare them. An indication factor is the indicated pure premium divided by a current pure premium or rate expressed on the same exposure basis.

For example, if 1 policy-year corresponds to 2.5 vehicle-years in a particular portfolio, then 400 policy-years correspond to 1,000 vehicle-years. The pure premium per policy-year will be 2.5 times the pure premium per vehicle-year. That difference is a unit conversion, not a disagreement about the losses.

Conversion and Reconciliation

Let $E_1$ and $E_2$ be two exposure counts for the same experience, and suppose their stated conversion relationship is $E_2=cE_1$, where $c$ is positive. Let $L$ denote the same selected loss amount under both calculations. The pure premiums are $P_1=L/E_1$ and $P_2=L/E_2$.

P_2 = \frac{L}{E_2} = \frac{L}{cE_1} = \frac{P_1}{c}

This gives the basic reconciliation rule: when the second exposure count is $c$ times the first, the pure premium per second-basis unit is the first-basis pure premium divided by $c$. The losses must be held fixed. If the loss amount, coverage, or experience period changes between calculations, this relationship alone cannot reconcile the results.

The same unit logic applies to indication factors. Suppose current pure premium benchmarks $R_1$ and $R_2$ are also expressed on the two bases and are compatible with the conversion: $R_2=R_1/c$. Then

\frac{P_2}{R_2} = \frac{P_1/c}{R_1/c} = \frac{P_1}{R_1}

Thus the indicated pure premium changes when the exposure unit changes, but the indication factor does not, provided that both the indicated and current amounts are converted consistently. This is the key check for a two-basis calculation.

Important condition: A single conversion factor is valid only when the stated relationship applies to the experience being analyzed. If the number of units per policy varies by class, the aggregate conversion must be calculated from the total exposures. Do not assume that an average from another portfolio, period, or mix applies.

Worked Calculations

Worked Example: Policy-Years and Vehicle-Years

A hypothetical fleet insurer selects $360,000 of losses for a period. The experience contains 480 policy-years and, under the stated exposure definition, 1,200 vehicle-years. The current pure premium benchmarks are $600 per policy-year and $240 per vehicle-year. Calculate the pure premium on both bases and the indication factor on each basis.

Method and units. Divide the same selected losses by each exposure count. The stated conversion is $1{,}200/480=2.5$ vehicle-years per policy-year.

Policy-year pure premium.

\frac{\$360{,}000}{480\text{ policy-years}} = \$750\text{ per policy-year}

Vehicle-year pure premium.

\frac{\$360{,}000}{1{,}200\text{ vehicle-years}} = \$300\text{ per vehicle-year}

The conversion check is $\$750/2.5=\$300$ per vehicle-year. On the policy-year basis, the indication factor is $\$750/\$600=1.25$. On the vehicle-year basis, it is $\$300/\$240=1.25$. Both bases therefore indicate 1.25 times the respective current benchmark, or a 25% increase.

Check. The current benchmarks are also consistent: $\$600/2.5=\$240$. The indicated and current amounts therefore use the same conversion. The common mistake is to compare $750 per policy-year directly with $240 per vehicle-year; that ratio mixes units and is not an indication factor.

Worked Example: Reconcile a Changed Exposure Count

A hypothetical delivery program has $195,000 of selected losses, 260 policy-years, and 650 vehicle-years. The insurer’s current pure premium benchmarks are $700 per policy-year and $280 per vehicle-year. Calculate both indicated pure premiums, verify the exposure conversion, and determine whether the two indication factors agree.

Method and conversion. The exposure relationship is $650/260=2.5$ vehicle-years per policy-year. Use the same $195,000 loss amount in both calculations.

The policy-year pure premium is

\frac{\$195{,}000}{260} = \$750\text{ per policy-year}.

The vehicle-year pure premium is

\frac{\$195{,}000}{650} = \$300\text{ per vehicle-year}.

Converting the first result gives $\$750/2.5=\$300$ per vehicle-year, so the two indicated amounts reconcile. The policy-year indication factor is

\frac{\$750}{\$700}=1.0714.

The vehicle-year indication factor is

\frac{\$300}{\$280}=1.0714.

To four decimal places, both bases indicate a factor of 1.0714, or approximately a 7.14% increase. The current benchmarks reconcile as well: $\$700/2.5=\$280$ per vehicle-year.

Check and common mistake. If the analyst mistakenly divides the $195,000 by 260 and labels the result “per vehicle-year,” the resulting $750 is overstated by the conversion factor of 2.5 for that unit. The correction is to divide by 650 for a vehicle-year pure premium, or explicitly convert the policy-year result by dividing by 2.5.

Worked Example: Class-Specific Conversion in an Aggregate Portfolio

A hypothetical equipment portfolio has three classes. Class A has 100 policy-years, 200 equipment-years, and $90,000 of selected losses. Class B has 80 policy-years, 240 equipment-years, and $144,000 of selected losses. Class C has 40 policy-years, 40 equipment-years, and $36,000 of selected losses. The current aggregate benchmark is $1,100 per policy-year. Find the aggregate indicated pure premium on both bases and the indication factor.

Method and exposure totals. Add the exposures and losses across classes before calculating the aggregate pure premium.

E_{\text{policy}}=100+80+40=220\text{ policy-years}

E_{\text{equipment}}=200+240+40=480\text{ equipment-years}

L=\$90{,}000+\$144{,}000+\$36{,}000=\$270{,}000

The aggregate pure premium per policy-year is

\frac{\$270{,}000}{220} = \$1{,}227.27\text{ per policy-year}

The aggregate pure premium per equipment-year is

\frac{\$270{,}000}{480} = \$562.50\text{ per equipment-year}.

The aggregate conversion is $480/220=2.1818$ equipment-years per policy-year (shown to four decimal places). Using the unrounded ratio, $\$1{,}227.2727\ldots/ (480/220)=\$562.50$ per equipment-year. The current benchmark’s equivalent equipment-year amount is

\$1{,}100 \times \frac{220}{480} = \$504.1667\text{ per equipment-year}.

The policy-year indication factor is $\$1{,}227.2727\ldots/\$1{,}100=1.1157$. The equipment-year indication factor is $\$562.50/\$504.1667\ldots=1.1157$. Both indicate approximately an 11.57% increase.

Check and common mistake. The class conversion factors differ: Class A has 2 equipment-years per policy-year, Class B has 3, and Class C has 1. Using an assumed factor of 2.5 for the whole portfolio would be wrong. The correct aggregate factor comes from total exposures, $480/220$, not an unweighted average of the class factors.

Practical Checks and Exam Traps

A two-basis calculation is most reliable when treated as a units problem as well as an arithmetic problem. Before comparing results, verify that both calculations use the same selected loss amount and that each exposure count has the intended definition. Then make the conversion explicit and check the current benchmark on the second basis.

Fix the loss amount.
Confirm that the same selected losses are used for both calculations. A changed loss selection is not an exposure-unit conversion.

Label each denominator.
Write the exposure unit alongside each count, such as policy-years or equipment-years.

Calculate both pure premiums.
Divide the fixed loss amount by each exposure count, retaining the resulting units.

Convert and compare.
Apply the stated exposure relationship and verify that the current benchmark uses the equivalent conversion before comparing indication factors.

Several errors can make equivalent calculations appear to disagree:

Mixing numerator or denominator units. A loss amount per policy-year cannot be compared directly with a current amount per vehicle-year.
Reversing the conversion. If there are $c$ second-basis units per first-basis unit, divide the first-basis pure premium by $c$ to obtain the second-basis amount.
Using an unsuitable average conversion. When class-specific exposure relationships vary, calculate the conversion from aggregate exposure totals for the portfolio being analyzed.
Changing the losses between calculations. If the loss selections differ, any difference in pure premiums reflects more than a change of exposure unit.

In practice, the two bases are useful when different parts of a rating or reporting process naturally use different exposure units. The reconciliation is a diagnostic: if equivalent calculations do not produce the same converted pure premium or indication factor, check the exposure definitions, the conversion, and the units of the benchmark before drawing a conclusion about the indicated change.

Key takeaway: For the same losses, pure premiums under two exposure bases differ by the reciprocal of the exposure conversion factor. Indication factors agree only when the current benchmark is converted consistently and the exposure relationship is valid for the experience being measured.

Check Your Understanding

For each hypothetical portfolio, calculate the requested pure premiums or indication factor, keeping the exposure units explicit.

A small commercial program has $150,000 of selected losses, 200 policy-years, and 500 vehicle-years. What are the pure premiums per policy-year and per vehicle-year? (Answer: $750 per policy-year and $300 per vehicle-year.)
A hypothetical service fleet has $216,000 of selected losses, 300 policy-years, and 900 vehicle-years. Current benchmarks are $640 per policy-year and $213.3333 per vehicle-year. Find both indication factors and check the conversion. (Answer: $720 per policy-year, $240 per vehicle-year, and a factor of 1.1250 on each basis.)
A portfolio has 50 policy-years corresponding to 100 machine-years with $40,000 of losses, and 80 policy-years corresponding to 80 machine-years with $64,000 of losses. Calculate the aggregate pure premium on both bases. (Answer: $800 per policy-year and $577.78 per machine-year; aggregate conversion is 180/130 machine-years per policy-year.)
A hypothetical insurer selects $330,000 of losses from 250 policy-years and 600 vehicle-years. Current pure premium benchmarks are $1,200 per policy-year and $500 per vehicle-year. Calculate both indicated pure premiums and indication factors. (Answer: $1,320 per policy-year, $550 per vehicle-year, and a factor of 1.1000 on each basis.)
A hypothetical book has 120 policy-years, 300 vehicle-years, and $270,000 of selected losses. An analyst reports $2,250 per policy-year and $900 per vehicle-year, then claims the amounts do not reconcile because the conversion is 2.5 vehicles per policy. Identify the error and give the corrected vehicle-year pure premium. (Answer: The policy-year amount should be divided by 2.5; the corrected amount is $900 per vehicle-year, so the reported amounts do reconcile.)

What You'll Learn