Choosing Weights for Experience Periods

Why the Choice of Weights Matters

When experience from several years is combined, a year’s contribution depends not only on its result but also on how much weight it receives. An unweighted average gives every year the same influence. Exposure-based weights give more influence to years with more exposure; premium-based weights give more influence to years with more premium. Those choices can produce different combined results, so the weights must match the quantity being combined and the purpose of the analysis.

The experience periods being combined should already be aligned on the relevant exposure, premium, and loss bases. For example, premium weights should use the selected premium measure, such as on-level premium when the comparison requires a common rate level. This tutorial focuses on the weighting step, not on how those underlying measures are selected or adjusted. The approach is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: For experience period $j$, let $w_j$ be its weight and $x_j$ the measure being combined. The weighted average is $\sum_j w_jx_j/\sum_j w_j$. An exposure-weighted average uses exposures as $w_j$; a premium-weighted average uses premiums as $w_j$. Weights may be normalized to shares $w_j/\sum_k w_k$, which sum to one.

Two common combinations have useful interpretations. If $E_j$ is exposure and $L_j$ is loss for period $j$, its loss cost is $c_j=L_j/E_j$. Weighting these loss costs by exposure gives the combined loss cost. If $P_j$ is the selected premium and $r_j=L_j/P_j$ is the corresponding loss ratio, weighting loss ratios by premium gives the combined loss ratio. The loss and premium measures must be on bases appropriate to the analysis.

\bar{c}_E = \frac{\sum_j E_jc_j}{\sum_j E_j} = \frac{\sum_j L_j}{\sum_j E_j}, \qquad \bar{r}_P = \frac{\sum_j P_jr_j}{\sum_j P_j} = \frac{\sum_j L_j}{\sum_j P_j}.

These equalities follow directly from the definitions: $E_jc_j=L_j$, and $P_jr_j=L_j$. Thus, the weighted average is not an arbitrary convention. It reproduces the ratio of the corresponding aggregate totals. For the examples below, all scenarios and data are hypothetical. The calculations are deterministic; no interest rate, mortality assumption, probability distribution, or distribution parameters are assumed.

Exposure Weights and Loss Costs

An exposure-weighted loss cost answers: what is the total loss per unit of the selected exposure across the combined periods? A period with twice as much exposure as another receives twice as much influence on the combined loss cost. This is appropriate when loss cost per exposure unit is the quantity of interest and the exposure base is comparable across periods.

Worked Example: Combining Loss Costs by Exposure

A hypothetical equipment insurer is reviewing three aligned experience years. Loss amounts are in dollars, and exposure means one covered equipment-unit year. The selected premium amounts are also provided so the difference between exposure and premium shares can be checked. Calculate the exposure-weighted loss cost and compare it with the simple average of the annual loss costs.

Method and data. Use the exposure-weighted average of annual loss costs. Year 1 has 1,000 exposures, $600,000 of losses, and $900,000 of selected premium. Year 2 has 600 exposures, $420,000 of losses, and $720,000 of selected premium. Year 3 has 400 exposures, $360,000 of losses, and $600,000 of selected premium.

Step 1: Calculate each annual loss cost. Year 1: $600{,}000/1{,}000=\$600$ per exposure. Year 2: $420{,}000/600=\$700$ per exposure. Year 3: $360{,}000/400=\$900$ per exposure.

Step 2: Calculate exposure shares. Total exposure is $1{,}000+600+400=2{,}000$. The shares are $1{,}000/2{,}000=0.50$, $600/2{,}000=0.30$, and $400/2{,}000=0.20$.

Step 3: Weight the annual loss costs. The result is $0.50(\$600)+0.30(\$700)+0.20(\$900)=\$300+\$210+\$180=\$690$ per exposure. Equivalently, total losses are $600{,}000+420{,}000+360{,}000=\$1{,}380{,}000$, and $\$1{,}380{,}000/2{,}000=\$690$ per exposure.

Step 4: Compare with the unweighted average. The simple average is $(\$600+\$700+\$900)/3=\$733.33$ per exposure, rounded to the nearest cent. It differs because it gives each year one-third of the weight, rather than using the actual exposure shares.

Check. The weighted result of $690 per exposure equals total losses divided by total exposures. The common mistake this method avoids is averaging the three annual loss costs equally when the periods have different exposure volumes.

Premium Weights and Loss Ratios

A premium-weighted loss ratio answers: what is the total selected loss divided by total selected premium across the combined periods? More premium means more influence on the combined ratio. If the intended comparison is at a common rate level, the selected premiums should reflect that basis; otherwise, differences in historical rates can affect the weights as well as the annual loss ratios.

Worked Example: Why the Premium Basis Changes the Weights

A hypothetical specialty insurer has equal exposure in two experience years. Year 1 has 1,000 covered units, $600,000 of losses, and $500,000 of historical earned premium. Year 2 has 1,000 covered units, $800,000 of losses, and $2,000,000 of historical earned premium. For a common-rate-level comparison, the selected on-level premium is $1,000,000 in each year. Calculate the historical-premium-weighted loss ratio and the on-level-premium-weighted loss ratio.

Method and data. In each calculation, weight each period’s loss ratio by the premium basis named for that calculation. The data are hypothetical; no interest, mortality, or probability distribution is assumed.

Step 1: Calculate period loss ratios on historical earned premium. Year 1 is $600{,}000/500{,}000=1.20=120\%$. Year 2 is $800{,}000/2{,}000{,}000=0.40=40\%$.

Step 2: Weight those ratios by historical earned premium. The combined ratio is $\{500{,}000(1.20)+2{,}000{,}000(0.40)\}/(500{,}000+2{,}000{,}000)$. The numerator is $600{,}000+800{,}000=\$1{,}400{,}000$; the denominator is $\$2{,}500{,}000$. Thus, the result is $1{,}400{,}000/2{,}500{,}000=56\%$.

Step 3: Calculate period loss ratios on the selected on-level premium. Year 1 is $600{,}000/1{,}000{,}000=60\%$. Year 2 is $800{,}000/1{,}000{,}000=80\%$.

Step 4: Weight those ratios by on-level premium. The result is $\{1{,}000{,}000(0.60)+1{,}000{,}000(0.80)\}/2{,}000{,}000$. The numerator is $600{,}000+800{,}000=\$1{,}400{,}000$, and the denominator is $\$2{,}000{,}000$, giving $70\%$.

Check and trap. Both calculations reproduce losses divided by the premium basis used: $1{,}400{,}000/2{,}500{,}000=56\%$ and $1{,}400{,}000/2{,}000{,}000=70\%$. The 56% result is correct for the historical-premium basis, but it does not represent the stated common-rate-level comparison. Using the historical premium weights without checking the intended basis gives the wrong answer for that objective.

Deriving the Aggregate Relationship

The weighted-average formulas can be justified in a few lines. Suppose each period’s loss ratio is $r_j=L_j/P_j$, where the loss and premium measures are consistently defined. Multiplying a period ratio by its premium weight cancels the period premium in the denominator:

P_jr_j=P_j\left(\frac{L_j}{P_j}\right)=L_j.

Adding this equality over all periods gives $\sum_jP_jr_j=\sum_jL_j$. Dividing by the total premium gives $\sum_jP_jr_j/\sum_jP_j=\sum_jL_j/\sum_jP_j$, which is the combined loss ratio. The same argument applies to loss costs: since $c_j=L_j/E_j$, multiplying by $E_j$, summing, and dividing by total exposure produces $\sum_jL_j/\sum_jE_j$.

This also explains a key limitation: the aggregate relationship depends on using the same denominator basis in each period’s metric and in its weight. If period loss ratios use one premium measure but the weights use another, the cancellation above does not establish that the weighted average equals aggregate losses divided by the intended premium total.

Different Weights, Different Combined Measures

It is possible to apply either kind of weight to a series of annual results, but the interpretation changes. An exposure-weighted average of loss ratios describes an exposure-weighted mean of the annual ratios. A premium-weighted average describes the combined loss ratio on the selected premium basis. They need not agree when premium per exposure differs across years.

Worked Example: Comparing Two Weighting Choices

A hypothetical property insurer has the following three years of aligned experience. Year 1 has 400 exposures, a 50% loss ratio, and $400,000 of selected premium. Year 2 has 1,000 exposures, a 70% loss ratio, and $1,500,000 of selected premium. Year 3 has 600 exposures, a 90% loss ratio, and $1,200,000 of selected premium. Calculate the exposure-weighted and premium-weighted averages of the annual loss ratios, then interpret the difference.

Method and data. First form normalized shares under each weighting scheme; then multiply the annual ratios by those shares. The scenario is hypothetical and deterministic, with no assumed interest, mortality, or probability distribution.

Step 1: Calculate exposure shares. Total exposure is $400+1{,}000+600=2{,}000$. The shares are $0.20$, $0.50$, and $0.30$.

Step 2: Calculate the exposure-weighted average ratio. The result is $0.20(50\%)+0.50(70\%)+0.30(90\%)=10\%+35\%+27\%=72\%$.

Step 3: Calculate premium shares. Total selected premium is $400{,}000+1{,}500{,}000+1{,}200{,}000=\$3{,}100{,}000$. The shares are $400/3{,}100=0.129032$, $1{,}500/3{,}100=0.483871$, and $1{,}200/3{,}100=0.387097$, where the premium amounts are in thousands of dollars.

Step 4: Calculate the premium-weighted average ratio. In dollars, the weighted numerator is $400{,}000(0.50)+1{,}500{,}000(0.70)+1{,}200{,}000(0.90)=200{,}000+1{,}050{,}000+1{,}080{,}000=\$2{,}330{,}000$. Dividing by $\$3{,}100{,}000$ gives $0.7516129$, or $75.16\%$ to two decimal places.

Check and interpretation. The period losses implied by the ratios are $200,000, $1,050,000, and $1,080,000. Their total divided by total premium is $\$2{,}330{,}000/\$3{,}100{,}000=75.16\%$, confirming the premium-weighted result. The two answers differ because premium weights place more emphasis on periods with more premium per exposure. Neither average is inherently the right choice without a defined objective; the premium-weighted result is the aggregate loss ratio on the selected premium basis.

Practical Selection and Common Traps

Before combining years, identify what the result is intended to measure. If the quantity is loss dollars per exposure unit, use exposure weights and verify the exposure definition is consistent. If it is losses relative to premium, use the corresponding premium weights and verify the premium basis matches the loss ratio. A weight should reflect the quantity being pooled, not merely whichever column is easiest to retrieve.

Do not confuse a weighted average with a simple average. A simple average assigns equal weight to each period, regardless of its exposure or premium volume.
Do not use mismatched bases. Period loss ratios and their premium weights should refer to the same premium measure. A historical premium-weighted ratio and a common-rate-level premium-weighted ratio may differ.
Do not treat exposure and premium as interchangeable. Premium per exposure can differ by rate level, mix, or other features. Consequently, the two sets of weights can assign different importance to the same year.
Do not round weights too early. Calculate with full precision where practical, then round the final result to the requested precision.

Key takeaway: Exposure weights combine results in proportion to exposure; premium weights combine results in proportion to the selected premium. State the measure being combined, check that its denominators match the weights, and confirm the result against the corresponding aggregate totals.

Check Your Understanding

For each problem, identify the requested weighting basis and check the result using aggregate totals where possible.

A hypothetical program has 250 exposures at an annual loss cost of $480 and 750 exposures at an annual loss cost of $720. Calculate the exposure-weighted loss cost. (Answer: $660 per exposure.)
Two periods have selected premiums of $600,000 and $400,000, with loss ratios of 50% and 80%, respectively. Calculate the premium-weighted loss ratio. (Answer: 62.00%.)
Three experience years have exposures of 500, 500, and 1,000 and loss costs of $400, $600, and $800 per exposure. Calculate the combined exposure-weighted loss cost. (Answer: $650 per exposure.)
Two years have common-rate-level premiums of $800,000 and $1,200,000 and corresponding loss ratios of 55% and 75%. Find the premium-weighted loss ratio and verify it using aggregate losses. (Answer: 67.00%; aggregate losses are $1,340,000.)
Two periods have exposures of 500 and 1,000, loss ratios of 60% and 80%, and selected premiums of $400,000 and $1,600,000. Calculate both the exposure-weighted and premium-weighted averages of the loss ratios. (Answer: exposure-weighted 73.33%; premium-weighted 76.00%.)

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