Calculating Multi-Year Experience Modifications

Why Pool Experience Across Years?

An experience modification reflects how an insured’s observed losses compare with the losses expected under a rating plan. The experience period may contain several years. A sound calculation combines those years on the loss basis specified by the plan, then applies the plan’s formula to the resulting totals. Calculating a separate modification for each year and averaging those factors is generally not the same calculation.

This tutorial focuses on pooling annual amounts when the plan specifies fixed component weights and a single ballast for the pooled experience. The expected losses and any primary and excess amounts are assumed to have already been calculated on the plan’s required basis. The calculation below produces a raw modification; it does not apply any separate floor, cap, or other adjustment.

Definition: For experience year $j$, let $A_{p,j}$ and $E_{p,j}$ be actual and expected primary losses, and let $A_{e,j}$ and $E_{e,j}$ be actual and expected excess losses, all measured on the plan’s specified basis. The pooled amounts are the sums across the $n$ included years. With fixed primary and excess weights $w_p$ and $w_e$, and pooled ballast $B$, the plan formula is $$ M=1+\frac{w_p(A_p-E_p)+w_e(A_e-E_e)}{E_p+E_e+B}. $$ Here $M$ is the raw experience modification. A single-component plan can be written $M=1+w(A-E)/(E+B)$.

The component weights determine how much each component’s actual-to-expected deviation affects the modification. A weight of zero removes that component’s deviation from the numerator; a positive weight gives it influence. Ballast is a stabilizing amount in the denominator, so it reduces the size of the adjustment for a given weighted deviation. Use the weights and ballast specified by the hypothetical plan rather than inferring them from the loss data.

The formula assumes the plan applies the same weights across the included years and applies $B$ once to the pooled expected amount. Annual loss amounts must use consistent units and definitions. If the plan instead specifies year-specific weights or another pooling rule, this formula cannot be used unchanged.

Pooling Before Applying the Formula

For each loss component, add actual amounts across all included years and separately add expected amounts. Then calculate the actual-minus-expected deviation for each pooled component. Apply the corresponding weight to each deviation, add the weighted deviations, and divide by total pooled expected losses plus ballast.

Confirm the experience basis.
Check which years and loss components are included, and confirm that actual and expected values are defined consistently.

Sum by component.
Pool actual primary, expected primary, actual excess, and expected excess amounts separately.

Apply the stated plan formula.
Weight each pooled deviation, use the pooled expected amount and ballast in the denominator, and calculate $M$.

Check the result.
Confirm that the sign of the modification’s departure from 1 agrees with the weighted deviations, and that ballast was included as specified.

Worked Example: Pooling Three Years Under a Single-Component Plan

A hypothetical plan uses one loss component and calculates $M=1+0.80(A-E)/(E+B)$ from pooled experience. Its ballast is $B=\$50{,}000$. Annual actual losses are $\$92{,}000$, $\$118{,}000$, and $\$75{,}000$; annual expected losses are $\$100{,}000$, $\$110{,}000$, and $\$90{,}000$. Calculate the pooled modification. Report the final factor to four decimal places.

Method. Pool the three years first, then apply the single-component formula. The common mistake to avoid is averaging three annual modifications instead.

Pooled actual losses are $A=\$92{,}000+\$118{,}000+\$75{,}000=\$285{,}000$. Pooled expected losses are $E=\$100{,}000+\$110{,}000+\$90{,}000=\$300{,}000$. The actual-minus-expected deviation is $A-E=\$285{,}000-\$300{,}000=-\$15{,}000$.

The denominator is $E+B=\$300{,}000+\$50{,}000=\$350{,}000$. The weighted deviation is $0.80(-\$15{,}000)=-\$12{,}000$. Therefore,

M=1+\frac{-\$12{,}000}{\$350{,}000} =1-0.0342857\ldots =0.9657.

The factor is below 1 because actual pooled losses are below expected losses. As a check on the aggregation method, the annual factors would be $1-6{,}400/150{,}000=0.9573$, $1+6{,}400/160{,}000=1.0400$, and $1-12{,}000/140{,}000=0.9143$. Their simple average is approximately $0.9705$, not $0.9657$. That average uses three separate denominators and is not the stated pooled plan formula.

Why the Pooled Calculation Works

Pooling is justified by the way the formula treats deviations. If the plan uses fixed component weights, the weighted deviation for a component across years is the sum of that component’s annual deviations multiplied by the same weight. This lets the calculation be organized either as sums first or as annual deviations first, provided the final denominator and ballast follow the pooled plan formula.

\begin{aligned} w_p(A_p-E_p)+w_e(A_e-E_e) &=w_p\left(\sum_{j=1}^{n}A_{p,j}-\sum_{j=1}^{n}E_{p,j}\right) +w_e\left(\sum_{j=1}^{n}A_{e,j}-\sum_{j=1}^{n}E_{e,j}\right)\\ &=\sum_{j=1}^{n}\left[w_p(A_{p,j}-E_{p,j})+ w_e(A_{e,j}-E_{e,j})\right]. \end{aligned}

The equality follows from the definition of the pooled totals and the distributive property: each fixed weight multiplies the sum of annual deviations, which equals the sum of the weighted annual deviations. The modification, however, is not generally the average of annual modifications. Those annual factors can have different denominators, and averaging ratios does not in general equal dividing pooled totals. In the stated formula, $B$ is also included once in the pooled denominator, not once per year.

Worked Example: Separate Primary and Excess Deviations

A hypothetical two-year plan uses $M=1+[0.80(A_p-E_p)+0.30(A_e-E_e)]/(E_p+E_e+B)$, with $B=\$30{,}000$. For year 1, primary actual and expected losses are $\$70{,}000$ and $\$80{,}000$, while excess actual and expected losses are $\$30{,}000$ and $\$40{,}000$. For year 2, the corresponding amounts are primary actual $\$105{,}000$, primary expected $\$100{,}000$, excess actual $\$55{,}000$, and excess expected $\$50{,}000$. Calculate $M$ to four decimal places.

Method. Pool each component independently, then apply its own weight. The common mistake is to combine primary and excess deviations before weighting them.

Pooled primary actual losses are $\$70{,}000+\$105{,}000=\$175{,}000$, and pooled primary expected losses are $\$80{,}000+\$100{,}000=\$180{,}000$. Thus the primary deviation is $-\$5{,}000$.

Pooled excess actual losses are $\$30{,}000+\$55{,}000=\$85{,}000$, and pooled excess expected losses are $\$40{,}000+\$50{,}000=\$90{,}000$. Thus the excess deviation is also $-\$5{,}000$. The weighted numerator is $0.80(-\$5{,}000)+0.30(-\$5{,}000)=-\$4{,}000-\$1{,}500=-\$5{,}500$.

Total pooled expected losses are $\$180{,}000+\$90{,}000=\$270{,}000$, so the denominator including ballast is $\$270{,}000+\$30{,}000=\$300{,}000$. Therefore,

M=1+\frac{-\$5{,}500}{\$300{,}000} =1-0.0183333\ldots =0.9817.

Both components have negative deviations, so the result is below 1, as expected. The calculation also confirms that the excess deviation is not given the primary weight: its contribution is $0.30(-\$5{,}000)$, not $0.80(-\$5{,}000)$.

Interpreting Signs and Performing Checks

After calculating a modification, compare it with 1 before rounding. A positive weighted numerator makes $M>1$, while a negative weighted numerator makes $M<1$, as long as the denominator is positive. If weighted deviations from different components have opposite signs, they can offset. It is therefore the weighted sum, not any single actual loss total, that determines the direction of the adjustment.

Worked Example: Opposing Component Deviations Across Three Years

A hypothetical plan uses $w_p=0.70$, $w_e=0.25$, and $B=\$40{,}000$ in the pooled formula. Across three years, primary actual losses are $\$140{,}000$, $\$80{,}000$, and $\$110{,}000$, while primary expected losses are $\$120{,}000$, $\$90{,}000$, and $\$100{,}000$. Excess actual losses are $\$20{,}000$, $\$75{,}000$, and $\$25{,}000$; excess expected losses are $\$30{,}000$, $\$60{,}000$, and $\$30{,}000$. Calculate $M$ to four decimal places.

Method. Find each pooled component deviation before applying its separate weight. The common mistake is to treat the positive primary deviation as if it alone determines the result.

Pooled primary actual losses are $\$140{,}000+\$80{,}000+\$110{,}000=\$330{,}000$, and pooled primary expected losses are $\$120{,}000+\$90{,}000+\$100{,}000=\$310{,}000$. The primary deviation is $+\$20{,}000$.

Pooled excess actual losses are $\$20{,}000+\$75{,}000+\$25{,}000=\$120{,}000$, and pooled excess expected losses are $\$30{,}000+\$60{,}000+\$30{,}000=\$120{,}000$. The excess deviation is zero. The weighted numerator is $0.70(\$20{,}000)+0.25(\$0)=\$14{,}000$.

Total expected losses are $\$310{,}000+\$120{,}000=\$430{,}000$. Adding ballast gives a denominator of $\$430{,}000+\$40{,}000=\$470{,}000$. Thus,

M=1+\frac{\$14{,}000}{\$470{,}000} =1.0297872\ldots =1.0298.

The positive primary deviation is not offset by the excess component because excess actual and expected totals are equal. The result above 1 is consistent with the positive weighted numerator.

Common Aggregation Errors

Averaging annual modifications. This usually changes the calculation because the annual denominators differ. Use the pooled formula when the plan specifies pooled totals.
Weighting the wrong quantities. Apply component weights to their corresponding actual-minus-expected deviations, not to the expected amounts alone.
Applying ballast repeatedly. If the stated plan defines one ballast on pooled experience, include it once in the pooled denominator.
Mixing bases or units. Actual and expected amounts must refer to the same years, components, and monetary units before they are compared.
Reading the direction from one component. Check the sign of the full weighted numerator, especially when component deviations oppose one another.

A useful final audit is to verify the pooled actual and expected totals independently, recompute each weighted deviation, and check that the numerator’s sign agrees with the direction of $M-1$. Retain sufficient precision during intermediate calculations; round only the reported modification, to the precision requested by the problem.

Check Your Understanding

For each hypothetical plan, pool the stated years and apply its formula. Round each final modification to four decimal places.

A one-component plan uses $M=1+0.75(A-E)/(E+25{,}000)$. Two years have actual losses of $\$88{,}000$ and $\$102{,}000$, and expected losses of $\$100{,}000$ and $\$110{,}000$. Calculate the pooled modification. (Answer: $0.9362$)
A two-component plan uses $w_p=0.80$, $w_e=0.40$, and $B=\$20{,}000$. In year 1, $(A_p,E_p,A_e,E_e)=(\$60{,}000,\$70{,}000,\$20{,}000,\$25{,}000)$; in year 2, $(A_p,E_p,A_e,E_e)=(\$85{,}000,\$80{,}000,\$35{,}000,\$30{,}000)$. Calculate $M$. (Answer: $0.9822$)
A one-component plan uses $w=0.60$ and $B=\$10{,}000$. Across three years, actual losses are $\$44{,}000$, $\$68{,}000$, and $\$91{,}000$; expected losses are $\$50{,}000$, $\$60{,}000$, and $\$80{,}000$. Calculate the pooled modification. (Answer: $1.0390$)
A plan uses $w_p=0.70$, $w_e=0.20$, and $B=\$50{,}000$. Across two years, primary actual losses are $\$90{,}000$ and $\$120{,}000$, primary expected losses are $\$100{,}000$ and $\$110{,}000$, excess actual losses are $\$60{,}000$ and $\$25{,}000$, and excess expected losses are $\$40{,}000$ and $\$30{,}000$. Calculate $M$, accounting for the opposing component deviations. (Answer: $1.0091$)
A plan uses $w_p=0.60$, $w_e=0.40$, and $B=\$25{,}000$. For three years, primary actual losses are $\$75{,}000$, $\$95{,}000$, and $\$110{,}000$, and primary expected losses are $\$80{,}000$, $\$90{,}000$, and $\$100{,}000$. Excess actual losses are $\$15{,}000$, $\$40{,}000$, and $\$25{,}000$, and excess expected losses are $\$20{,}000$, $\$35{,}000$, and $\$30{,}000$. Calculate the pooled modification. (Answer: $1.0105$)

What You'll Learn