Comparing Pure Premiums Across Experience Windows

Why Experience Windows Can Produce Different Pure Premiums

A pure premium is a measure of loss cost per exposure unit. Comparing pure premiums from two experience windows can help explain whether observed loss cost changed, but the comparison is meaningful only when the losses and exposures are measured on compatible bases. The windows may overlap, include different years, or contain different amounts of exposure. A larger total loss in one window does not, by itself, mean its pure premium is higher.

For this tutorial, an experience window is a stated collection of periods. Its aggregate losses and exposures are summed across those periods, then divided. Assume the selected losses are on the same basis in both windows—for example, both are ultimate losses, with the same treatment of loss adjustment expense and the same coverage definition. If the comparison requires losses to be adjusted to a common cost level, that adjustment must be applied consistently before comparing the results. Exposures must use the same exposure unit and coverage basis in both windows.

Definition: For a stated experience window $W$, let $L_W$ be the aggregate losses included in the window and $E_W$ the corresponding aggregate exposure units. The window pure premium is $P_W=L_W/E_W$, provided $E_W>0$. It is the selected loss amount per exposure unit, not loss divided by premium.

The unit matters. If $E_W$ is 1,000 vehicle-years, then $P_W$ is dollars per vehicle-year. If the exposure is stated in thousands of units, the result must be interpreted using that scale. A comparison can be numerically correct and still be misleading if the exposure units, coverage, or loss basis differ.

Compute Each Window Before Explaining the Difference

For each window, first identify which periods are included. Sum the losses and exposures over exactly those periods, then calculate the ratio of the sums. If the windows overlap, the shared periods appear in both window totals; they are not counted twice within either individual calculation. They do, however, cancel when describing what changes from one window to the other.

P_A=\frac{\sum_{t\in A} L_t}{\sum_{t\in A} E_t}, \qquad P_B=\frac{\sum_{t\in B} L_t}{\sum_{t\in B} E_t}, \qquad \Delta P=P_B-P_A

Here $L_t$ and $E_t$ are the selected loss and exposure for period $t$. The sign of $\Delta P$ indicates whether the pure premium for window B is greater or less than that for window A. It does not establish why the difference occurred; that requires examining what losses and exposures changed.

Worked Example: Two Overlapping Windows With the Same Pure Premium

A hypothetical fleet program compares two-year windows. Window A contains years 1 and 2; window B contains years 2 and 3. Losses are on the same selected basis and exposures are vehicle-years. Year 1 has losses of $300,000 and 5,000 vehicle-years; year 2 has $420,000 and 7,000 vehicle-years; year 3 has $390,000 and 6,500 vehicle-years. Calculate both pure premiums and explain the result.

Method. Sum the loss and exposure amounts for each stated window, then divide aggregate losses by aggregate exposures.

Window A. Its losses are $300,000 + $420,000 = $720,000. Its exposures are 5,000 + 7,000 = 12,000 vehicle-years. Therefore, $P_A=\$720{,}000/12{,}000=\$60.00$ per vehicle-year.

Window B. Its losses are $420,000 + $390,000 = $810,000. Its exposures are 7,000 + 6,500 = 13,500 vehicle-years. Therefore, $P_B=\$810{,}000/13{,}500=\$60.00$ per vehicle-year.

The total losses rose by $90,000, and exposures rose by 1,500 vehicle-years. Those changes are proportional: $90,000/1,500 = $60.00 per added net exposure unit. Thus, the pure premiums are equal even though the aggregate totals differ. As a check, each year in these windows also has a pure premium of $60.00: $300,000/5,000, $420,000/7,000, and $390,000/6,500.

Common mistake avoided: Comparing $810,000 with $720,000 and concluding that window B has higher loss cost. The relevant comparison is loss per exposure: both windows equal $60.00 per vehicle-year.

Reconcile What Was Added and Removed

For overlapping windows, a useful explanation starts with the periods that differ. A period present only in window B adds its losses and exposures to the comparison; a period present only in window A is removed. Shared periods do not affect the difference in aggregate amounts. This reconciliation makes the source of the change visible, but the resulting pure-premium change still depends on both the numerator and the denominator.

A short derivation shows why those two effects must be distinguished. Since $P_A=L_A/E_A$ and $P_B=L_B/E_B$, add and subtract $L_B/E_A$:

P_B-P_A = \left(\frac{L_B}{E_A}-\frac{L_A}{E_A}\right) + \left(\frac{L_B}{E_B}-\frac{L_B}{E_A}\right) = \frac{L_B-L_A}{E_A} + L_B\left(\frac{1}{E_B}-\frac{1}{E_A}\right)

This is an exact decomposition when both exposures are positive. The first term measures the effect of the change in losses using window A's exposure as the reference denominator. The second measures the effect of changing the exposure denominator, evaluated using window B's losses. A larger denominator, holding losses fixed, reduces the pure premium; a smaller denominator raises it. This order of decomposition is a choice: using a different reference order can assign the interaction between loss and exposure changes differently, even though the total change $P_B-P_A$ is unchanged.

Worked Example: A Higher Loss Total Partly Offset by More Exposure

A hypothetical property portfolio compares windows A (years 1–3) and B (years 2–4). All amounts use the same loss basis and exposure unit. Year 1 has losses of $100,000 and 2,000 exposure units; year 2 has $160,000 and 2,500; year 3 has $220,000 and 3,500; and year 4 has $250,000 and 3,000. Calculate the window pure premiums, reconcile the periods that change, and explain the difference.

Window A. Losses are $100,000 + $160,000 + $220,000 = $480,000. Exposures are 2,000 + 2,500 + 3,500 = 8,000. Thus, $P_A=\$480{,}000/8{,}000=\$60.00$ per exposure unit.

Window B. Losses are $160,000 + $220,000 + $250,000 = $630,000. Exposures are 2,500 + 3,500 + 3,000 = 9,000. Thus, $P_B=\$630{,}000/9{,}000=\$70.00$ per exposure unit. The increase is $10.00 per exposure unit.

Years 2 and 3 are shared. Year 1 is removed and year 4 is added, so losses increase by $250,000 − $100,000 = $150,000, while exposures increase by 3,000 − 2,000 = 1,000.

Decomposition. Using the formula with window A as the reference exposure, the loss effect is ($630,000 − $480,000)/8,000 = $18.75 per exposure unit. The exposure effect is $630,000 × (1/9,000 − 1/8,000) = $70.00 − $78.75 = −$8.75 per exposure unit. The net change is $18.75 − $8.75 = $10.00, matching $P_B-P_A$.

The added period has more exposure than the removed period, which offsets part of the loss-driven increase. This arithmetic explains the change in the aggregate pure premium; it does not prove that the changing periods caused an underlying change in risk.

Common mistake avoided: Treating the increase of $150,000 in losses as the entire explanation and reporting $18.75 as the pure-premium change. That calculation holds exposure at 8,000. The actual change also reflects the increase to 9,000 exposures, so the net increase is $10.00.

Pooling Periods Versus Averaging Their Pure Premiums

The pure premium for a combined window is an exposure-weighted average of its period pure premiums, not generally their unweighted arithmetic average. To see this, write each period's loss as $L_t=E_tP_t$. Then:

\frac{\sum_t L_t}{\sum_t E_t} = \frac{\sum_t E_tP_t}{\sum_t E_t}

Thus, periods with more exposure receive more weight in the combined result. The arithmetic average gives every period equal weight, regardless of its exposure. It agrees with the pooled pure premium when period exposures are equal, or in special cases where the different weights do not change the average. When exposures differ, the two calculations answer different questions.

The same principle applies when comparing windows: calculate each window's own ratio of aggregate losses to aggregate exposures. Do not average the two window pure premiums unless an explicitly defined weighting makes that operation appropriate for the question.

Worked Example: Why an Unweighted Average Can Misstate the Combined Pure Premium

A hypothetical equipment policy has two periods on an identical coverage and loss basis. Period 1 has losses of $60,000 and 1,000 exposure units. Period 2 has losses of $200,000 and 5,000 exposure units. Find the pure premium for each period, the pooled two-period pure premium, and the unweighted average of the period pure premiums.

Period pure premiums. Period 1 is $60,000/1,000 = $60.00 per exposure unit. Period 2 is $200,000/5,000 = $40.00 per exposure unit.

Pooled result. Aggregate losses are $60,000 + $200,000 = $260,000. Aggregate exposures are 1,000 + 5,000 = 6,000. The two-period pure premium is $260,000/6,000 = $43.33 per exposure unit, rounded to the nearest cent.

Unweighted average. ($60.00 + $40.00)/2 = $50.00 per exposure unit. This is $6.67 greater than the pooled result. Period 2 has five times the exposure of period 1, so it should carry five times the weight in the pooled calculation. Indeed, $(1{,}000\times \$60+5{,}000\times \$40)/6{,}000=\$43.33$, confirming the aggregate calculation.

Common mistake avoided: Calling $50.00 the combined pure premium. It is the simple average of two period ratios; the combined pure premium is $43.33 because it uses total losses divided by total exposures.

Interpretation and Exam Traps

A pure-premium comparison is a descriptive calculation based on selected data. It does not, by itself, establish that an indicated rate should rise or fall. Before interpreting the result, verify that both windows use comparable coverage, exposure units, loss development treatment, and loss adjustment expense treatment. If losses have been adjusted for trend or another stated basis, ensure that both windows have been brought to the intended basis. A mismatch in selection or measurement can look like a change in loss cost.

Comparing totals instead of ratios. A larger portfolio or longer window can have more total losses and also more exposure. Compare losses per exposure unit.
Using a premium denominator. The pure-premium denominator is the corresponding exposure, not premium. Premium is not a substitute for exposure in this calculation.
Mixing exposure units. A value per policy-year cannot be directly compared with a value per vehicle-year unless the units are converted consistently.
Overlooking overlap. For overlapping windows, identify the removed and added periods rather than treating the entire difference in totals as new experience.
Averaging ratios without weights. An unweighted average of period pure premiums need not equal aggregate losses divided by aggregate exposures.

Key takeaway: Calculate each window as aggregate selected losses divided by its matching aggregate exposures. Then reconcile which periods—and therefore which losses and exposures—differ. Interpret the resulting pure-premium change only after confirming that the underlying bases are comparable.

Check Your Understanding

For each problem, use the stated loss and exposure bases and calculate aggregate pure premiums from the included periods.

A hypothetical crop policy window has selected losses of $525,000 and 7,500 exposure units. What is its pure premium per exposure unit? (Answer: $70.00 per exposure unit)
Window A has losses of $1,080,000 and 18,000 exposure units. Window B has losses of $1,260,000 and 21,000 exposure units. Calculate both pure premiums and the change from A to B. (Answer: $60.00 for each window; change $0.00 per exposure unit)
A two-period window has losses of $240,000 and 4,000 exposures. A later window adds a period with losses of $180,000 and 2,000 exposures, with all earlier experience retained. Calculate the later window's pure premium and its change from the original. (Answer: $70.00 per exposure unit; increase of $10.00)
Window A has losses of $500,000 and 10,000 exposures. Window B removes a period with $120,000 of losses and 2,000 exposures, then adds a period with $210,000 of losses and 3,000 exposures. Calculate window B's pure premium and the change from A. (Answer: $53.64 per exposure unit; increase of $3.64)
Two periods have losses and exposures of $60,000 and 1,000, and $200,000 and 5,000, respectively. What is the pooled pure premium, and how does it compare with the unweighted average of the two period pure premiums? (Answer: pooled $43.33 per exposure unit; unweighted average $50.00, which is $6.67 higher)

What You'll Learn