On-Leveling

Study guides: Exam FAM, Exam 5, Exam GI-101

On-Leveling is the procedure that restates historical earned premium to the current rate level, producing on-level premium. Werner & Modlin give two methods: the parallelogram method, which works from aggregate premium and the dates of past rate changes, and extension of exposures, which re-rates every historical policy under the current manual.

$\text{On-Level Factor} = \frac{\text{Current cumulative index}}{\text{Average index for the year}}$

The parallelogram method. Picture a unit square whose horizontal axis is calendar time and whose vertical axis is the fraction of the policy term elapsed. A rate change effective on date $d$ enters as a diagonal line; each region's area is the share of the year's earned premium written at that rate level.

For annual policies written uniformly, a rate change effective at time $p$ through year $Y$ earns over a triangle of area $\tfrac{1}{2}(1-p)^2$ in year $Y$ — so a $7/1$ change affects only $0.125$ of that year's earned premium, and a $1/1$ change affects $0.500$ .
The method's assumptions are its weaknesses: premium written uniformly through the year, a single policy term, and rate changes applied uniformly to all risks. Seasonal writing (Werner's example: boat owners policies, bought before the boating season) or a mid-year term change breaks it, and where the rate changes varied by class the on-level premium is unusable for classification ratemaking.
Extension of exposures has none of those assumptions — it simply re-rates each historical policy at today's rates — and it automatically captures distributional shifts. Its cost is data: policy-level detail for every historical year, and a rating engine that still knows the old policies' characteristics.
A special case: if the rate change applies to in-force policies rather than only new and renewal business — Werner's case is a change mandated by law — it takes effect immediately across the whole book and is drawn as a vertical line rather than a diagonal one. The areas are still found by geometry, and the on-level factor is still the current cumulative index over the year's average index.
On-levelling handles rate changes only (law-mandated ones included); changes in the average premium from distributional shifts in the business written are premium trend, and the effect of a coverage or benefit-level change on losses is a separate adjustment on the loss side.

Parallelogram Factor for a Mid-Year Change Rates increased $12\%$ effective $7/1/2023$ for policies written on or after that date. All policies are annual and premium is written uniformly. There have been no other changes.

Compute the on-level factor for calendar year $2023$ and for calendar year $2024$ .
Answer Set the pre-change index at $1.000$ , so the current index is $1.120$ .

CY 2023. The change takes effect at $p = 0.5$ , so the area earned at the new level is

$\tfrac{1}{2}(1 - 0.5)^2 = 0.125$
 $\begin{align*} \text{Avg index}_{2023} &= 0.875(1.000) + 0.125(1.120) \\ &= 1.0150 \\[4pt] \text{OLF}_{2023} &= \frac{1.120}{1.0150} = 1.1034 \end{align*}$ 
CY 2024. By symmetry, the portion of $2024$ still earning at the old level is the complementary triangle, $\tfrac{1}{2}(0.5)^2 = 0.125$ :
 $\begin{align*} \text{Avg index}_{2024} &= 0.125(1.000) + 0.875(1.120) \\ &= 1.1050 \\[4pt] \text{OLF}_{2024} &= \frac{1.120}{1.1050} = 1.0136 \end{align*}$ 
Calendar year $2025$ would be fully at the new level, with an on-level factor of $1.000$ . Notice how a single mid-year change takes two full years to work through earned premium — the reason rate changes feel slow to show up in results.

When the Parallelogram Method Breaks A crop insurer writes $80\%$ of its annual policies in March and the rest through the year. It took a $+15\%$ rate change effective $7/1/2024$ .

Should the parallelogram method be used?

Answer No. The method assumes premium is written uniformly, and it is not: by $7/1$ this book has already written most of its year at the old rates. The parallelogram formula would say $12.5\%$ of CY 2024 earned premium is at the new level; the true figure is close to zero for the March cohort's remaining term and only the small tail of later writings is affected.

Two valid alternatives:

Extension of exposures — re-rate the historical policies at current rates. Exact, and it also picks up the mix shift, at the cost of policy-level data.

Adjust the parallelogram weights empirically — replace the geometric areas with the actual earned-premium distribution by month from the insurer's own records, then weight the indices by those shares.

The general rule: the parallelogram method is a shortcut that substitutes geometry for data. When the writing pattern is known to violate uniformity — seasonal lines, a book in rapid growth, a mid-year change in policy term — use the data instead.