on-leveling premium (parallelogram method)
Suppose you want to judge whether today's prices are adequate by looking at last year's results. There is a trap: the premium you actually collected last year was charged at the rates in force back then, and you may have changed your rates once or twice since. Comparing old premium to old losses tells you about old rates, not today's. On-leveling fixes this by restating historical premium as if every policy had been written at the current rate level.
The parallelogram method is the classic way to do it. Picture a square whose horizontal axis is calendar time and whose vertical axis is the fraction of a one-year policy's term that has elapsed. A rate change on a given date slices this square along a diagonal line, because policies written before the change keep their old rate until they renew, so the change phases in gradually over the following year — forming parallelogram-shaped regions, each earned under one rate level. By computing the area of each region, you find the portion of a year's earned premium that was at each historical rate, and you can then bring it all up to current rates with an on-level factor. For example, if a calendar year earned 60 percent of its premium at an old rate and 40 percent after a +10 percent change, the average rate level was below current, and the on-level factor scales the whole year up accordingly.
On-leveling is essential to the loss ratio method of rate indication and to any honest comparison of premium across years. Its standard form assumes rate changes affect all policies uniformly and that writing is spread evenly through the year (a uniform distribution of policy issuance) — assumptions that can fail for seasonal lines or large mid-year changes. When they fail, actuaries use the more data-intensive extension-of-exposures method, which re-rates each policy individually instead of relying on the geometric shortcut.
A +10% rate change took effect on 1 July. For annual policies issued evenly through the year, the parallelogram method shows the calendar year earned about 87.5% of its premium at the old rate and 12.5% at the new — so its average rate level is below current and needs an on-level factor above 1.
The diagonal of a rate change carves a calendar year into rate-level regions.
The parallelogram method assumes policies are written uniformly through the year and one-year terms. For seasonal business or non-annual terms it can mislead; the extension-of-exposures method (re-rating each policy) is more accurate but heavier.