Weighting Experience Years With Different Exposures

Why Different Experience Years Need Different Weights

A multi-year experience modification summarizes results from more than one experience year. If the years represent different volumes of business, they should not necessarily contribute equally. A year with twice the expected losses of another year generally represents about twice as much expected-loss volume in a calculation that weights by expected losses.

The key calculation is a weighted average of annual modifications. The weights may be computed from supplied year-specific expected losses or provided directly by the rating plan. A simple arithmetic average is appropriate only when the years have equal weights. The examples below use hypothetical data and assume that the annual modifications are on a consistent basis and that the specified expected-loss weights are the weights the plan directs the user to apply.

Definition: Let $M_t$ be the annual experience modification for year $t$, and let $E_t$ be that year’s expected losses on the basis specified by the plan. When expected losses are the prescribed weights, the multi-year modification is $$ M_{\mathrm{multi}}=\frac{\sum_{t=1}^{n}E_tM_t}{\sum_{t=1}^{n}E_t}. $$ Equivalently, define the normalized weight $w_t=E_t/\sum_{j=1}^{n}E_j$. Then $M_{\mathrm{multi}}=\sum_{t=1}^{n}w_tM_t$, where the weights sum to 1.

How to Calculate the Weighted Modification

First establish which weights the plan requires. If it specifies expected losses for each year, use those expected losses. If it supplies weights directly, use them and normalize them if they do not already sum to 1. Do not substitute the number of years, policy counts, payroll, or another exposure measure unless the plan says that measure determines the weights.

Expected losses can reflect both exposure volume and the expected loss cost per unit of exposure. Thus, when expected losses are not supplied but the plan calls for them, calculate each year’s expected losses using the applicable exposure and expected loss rate before finding the weights. A larger exposure count does not automatically mean a larger expected-loss weight if expected loss costs per exposure differ.

List the annual modifications and their corresponding weights.
Keep the years aligned: each modification must be paired with the expected losses or supplied weight for that same year.

Normalize weights if necessary.
For expected losses, divide each year’s expected losses by the total. For supplied weights, divide each raw weight by the sum of all raw weights.

Multiply and add.
Multiply each annual modification by its normalized weight, then sum the products.

Check the result.
The weights should sum to 1, and the weighted result should lie between the smallest and largest annual modifications when all weights are nonnegative.

Worked Example: Weighting by Annual Expected Losses

A hypothetical account has annual modifications of 0.90, 1.10, and 1.20. The corresponding expected losses are $80,000, $120,000, and $200,000. Calculate the multi-year modification using expected losses as weights.

Method. Use the expected-loss-weighted mean. The total expected losses are $80,000 + $120,000 + $200,000 = $400,000. The normalized weights are therefore $80{,}000/400{,}000=0.20$, $120{,}000/400{,}000=0.30$, and $200{,}000/400{,}000=0.50$. They sum to $0.20+0.30+0.50=1.00$.

Multiply each annual modification by its weight: $0.20(0.90)=0.18$, $0.30(1.10)=0.33$, and $0.50(1.20)=0.60$. Add the weighted contributions: $0.18+0.33+0.60=1.11$.

Check. The answer, 1.11, lies between the smallest annual modification, 0.90, and the largest, 1.20. The year with the largest expected losses has the greatest influence. The common mistake avoided here is averaging the three years equally; that would give $(0.90+1.10+1.20)/3=1.0667$, which does not use the stated expected-loss weights.

Why the Formula Works

The weighted formula follows from combining expected-loss volumes rather than treating each year as an equally sized unit. In the special case where each annual modification is the annual actual-to-expected ratio, $M_t=A_t/E_t$, with $A_t$ denoting actual losses on the same basis as expected losses, the pooled ratio is the ratio of total actual losses to total expected losses:

\frac{\sum_{t=1}^{n}A_t}{\sum_{t=1}^{n}E_t} = \frac{\sum_{t=1}^{n}E_t(A_t/E_t)}{\sum_{t=1}^{n}E_t} = \frac{\sum_{t=1}^{n}E_tM_t}{\sum_{t=1}^{n}E_t}.

The equality holds because $E_tM_t=E_t(A_t/E_t)=A_t$ for each year when $E_t>0$. Thus, the multi-year ratio gives each year influence in proportion to its expected losses. This justification applies directly to annual actual-to-expected ratios on a common basis. If annual modifications include other plan features, such as year-specific credibility or ballast, do not assume that pooling their underlying components is equivalent to this formula. Use the weights and combination rule specified for the calculation.

Worked Example: Deriving Weights From Exposure and Expected Loss Rates

A hypothetical service-contract portfolio has two experience years. In Year 1, there are 600 exposure units and expected losses are $150 per unit; its annual modification is 0.80. In Year 2, there are 1,000 exposure units and expected losses are $120 per unit; its annual modification is 1.10. The plan calls for expected-loss weighting. Calculate the multi-year modification.

Method. Calculate each year’s expected losses before forming the weights. Year 1 expected losses are $600\times \$150=\$90{,}000$. Year 2 expected losses are $1{,}000\times \$120=\$120{,}000$. Total expected losses are $\$90{,}000+\$120{,}000=\$210{,}000$.

The normalized weights are $90{,}000/210{,}000=3/7$ for Year 1 and $120{,}000/210{,}000=4/7$ for Year 2. Their sum is $3/7+4/7=1$. The weighted modification is $ (3/7)(0.80)+(4/7)(1.10)=2.40/7+4.40/7=6.80/7=0.971428\ldots $. Rounded only at the final step to four decimal places, the result is 0.9714.

Check. Year 2 has more exposure units, but its expected losses are only $120{,}000/90{,}000=4/3$ times Year 1’s, not $1{,}000/600=5/3$ times as large. The result is between 0.80 and 1.10 and is closer to 1.10 because Year 2 has the greater expected-loss weight. The mistake avoided is weighting by exposure counts alone when the expected loss per unit differs.

Using Supplied Weights

A plan or problem may provide raw weights instead of expected losses. Raw weights need not sum to 1: only their relative sizes matter. Normalize each by the total raw weight, then calculate the weighted mean. If the weights are already proportions that sum to 1, use them directly.

Worked Example: Normalizing Supplied Relative Weights

A hypothetical plan supplies relative weights of 2, 3, and 5 for three years. The annual modifications are 0.70, 1.20, and 1.10, respectively. Calculate the multi-year modification.

Method. The raw weights total $2+3+5=10$. Normalize them: $2/10=0.20$, $3/10=0.30$, and $5/10=0.50$. The weighted contributions are $0.20(0.70)=0.14$, $0.30(1.20)=0.36$, and $0.50(1.10)=0.55$.

Add the contributions: $0.14+0.36+0.55=1.05$. The multi-year modification is 1.0500 when reported to four decimal places.

Check. The normalized weights sum to 1, and 1.0500 lies between 0.70 and 1.20. The common mistake avoided is using the raw weights as though they were proportions: $2(0.70)+3(1.20)+5(1.10)$ is not a weighted average until divided by the raw-weight total of 10. In this example, the corrected weighted result is 1.0500.

Common Errors and Practical Checks

The central exam skill is not simply performing multiplication. It is identifying the weighting basis and preserving the correspondence between each year’s modifier and its weight. Before calculating, verify whether the information is stated as expected losses, normalized weights, or raw relative weights.

Taking an unweighted average. This gives every year equal influence, regardless of expected-loss volume. It is valid only when the prescribed weights are equal.
Weighting by the wrong exposure measure. Exposure counts or payroll may not be proportional to expected losses if expected loss rates vary. Use the plan’s specified basis.
Failing to normalize raw weights. Raw weights such as 2, 3, and 5 must be divided by their total before being used as proportions, or the weighted sum must be divided by that total.
Pairing values from different years. A year’s modification must be multiplied by that same year’s expected losses or supplied weight.
Rounding too early. Keep exact fractions or sufficient precision in normalized weights and intermediate products. Round only the final reported modification.

A useful independent check is to compare the answer with the annual modifications. With nonnegative weights summing to 1, a weighted average cannot be below the smallest annual modification or above the largest. Also check that a larger weight pulls the result more strongly toward its corresponding annual modification.

Key takeaway: For a multi-year calculation with unequal year volumes, apply the plan’s prescribed weights. When expected losses are the weights, divide the sum of expected loss times annual modification by total expected losses; do not give unequal years equal influence by taking a simple average.

Check Your Understanding

For each hypothetical scenario, identify the prescribed weights, calculate the multi-year modification, and check that it is consistent with the annual values.

Two years have modifications 0.95 and 1.05 and expected losses of $40,000 and $60,000. What is the expected-loss-weighted modification? (Answer: 1.0100)
Three years have expected losses of $75,000, $125,000, and $200,000 and annual modifications of 0.80, 1.00, and 1.15. Calculate the multi-year modification. (Answer: 1.0375)
A plan supplies raw weights of 1, 4, and 5. The corresponding annual modifications are 1.20, 0.90, and 1.10. Normalize the weights and calculate the result. (Answer: 1.0300)
A portfolio has 400 exposure units in Year 1 with expected losses of $200 per unit, and 600 units in Year 2 with expected losses of $150 per unit. The annual modifications are 1.10 and 0.90. Using expected-loss weights, calculate the multi-year modification to four decimal places. (Answer: 0.9941)
Three annual modifications are 0.90, 1.05, and 1.20, with corresponding expected losses of $50,000, $100,000, and $250,000. Calculate the weighted result and state whether an equal-weight average would give the same answer. (Answer: 1.1250; no, the equal-weight average is 1.0500)

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