Why Method Selection Matters
On-leveling adjusts historical earned premium to the rate level in effect at a chosen date. The two common approaches, extension and the parallelogram method, can target the same result but use different information. Extension uses exposure or policy records to calculate what the observed business would have paid at the target rates. The parallelogram method estimates the average rate level earned during a period from the timing of rate changes, writings, and policy earning.
The choice is therefore not simply a matter of which formula is shorter. It depends on whether the necessary data are available and whether the assumptions behind an aggregate estimate are credible. In particular, the analyst should consider policy terms, writing patterns, and whether the portfolio’s mix of risks has changed.
What Each Method Requires
The extension method recalculates premium for the observed exposure using the rates that were actually applicable and the rates at the target level. With sufficiently detailed records, this can account directly for the business written, including its classifications and policy terms. The method needs enough information to determine historical premium and to rate the same exposure at the target rates.
The parallelogram method estimates how much exposure earned under each rate level as rate changes flow into the portfolio. Its geometric model uses the dates of rate changes and assumptions about the timing and distribution of writings and the earning pattern of policies. It is especially useful when detailed exposure-level rerating data are unavailable but reliable aggregate information about writings, policy terms, or earned exposure by rate level is available.
| Consideration | Extension | Parallelogram |
|---|---|---|
| Primary data | Exposure or policy records, historical premiums, and rates needed to rerate the exposure | Rate-change dates, policy-term and earning information, and writing or in-force patterns |
| Basic approach | Rerate the observed exposure at the selected rate level | Estimate the distribution of earned exposure among rate levels |
| Key condition | Records allow a valid comparison of historical and target-rate premiums for the same exposure | Assumed or observed writing and earning patterns represent the business being analyzed |
| Policy mix | Can reflect the actual mix if records retain the relevant rating characteristics | May obscure mix changes unless the available weights and rate-level estimates account for them |
Neither method automatically solves a policy-mix problem. Extension can preserve the observed mix when the records include the details needed to rerate it. An aggregate parallelogram estimate generally relies on the mix represented by its weights; if the mix changes materially, a change in average premium may not be due to rate changes alone. The analyst should not treat a change in average premium as a pure rate-level change without checking what business is represented.
Extension: Rerating the Same Exposure
Let \(P_{\text{hist}}\) be the premium for a set of earned exposures at the historical rates, and let \(P_{\text{target}}\) be the premium obtained by rerating those same exposures at the selected rates. Then the extension factor is
The numerator and denominator must refer to the same exposure basis. If, for example, the historical premium is for earned exposure, the target premium should be calculated for that same earned exposure—not for a different set of written policies. The method does not require a uniform pattern of writings when the relevant exposure records are available; it uses the actual exposure being evaluated.
Worked Example: Extension With Two Rating Groups
A hypothetical insurer has earned exposures in two rating groups. Group A has 60 exposure units, a base premium of $700 per unit, and a historical rate-level multiplier of 0.95. Group B has 40 units, a base premium of $900 per unit, and a historical multiplier of 1.10. The selected rate level uses a multiplier of 1.08 for both groups. Calculate the extension factor and the on-level premium. Assume the listed exposures are the same under both calculations; no interest or mortality model is involved.
Method. Use extension: calculate historical premium and target-rate premium separately for each group, then divide the totals.
Historical premium for Group A: \(60 \times \$700 \times 0.95=\$39{,}900\).
Historical premium for Group B: \(40 \times \$900 \times 1.10=\$39{,}600\).
Thus, \(P_{\text{hist}}=\$39{,}900+\$39{,}600=\$79{,}500\).
Target premium for Group A: \(60 \times \$700 \times 1.08=\$45{,}360\).
Target premium for Group B: \(40 \times \$900 \times 1.08=\$38{,}880\).
Thus, \(P_{\text{target}}=\$45{,}360+\$38{,}880=\$84{,}240\), and
The on-level premium is \(\$79{,}500 \times 1.0596226\ldots=\$84{,}240\), which checks against the direct target-rate calculation. The factor is rounded to three decimals. A common mistake is to apply the target multiplier of 1.08 directly to the total historical premium: the two groups had different historical multipliers, so that shortcut does not rerate the observed exposure correctly.
Parallelogram: Estimating Earned Rate Level
The parallelogram method estimates the portion of earned exposure at each rate level. Once an average earned rate level is estimated, the on-level factor is the selected rate level divided by that average. For a portfolio with two levels, the average can be written using earned-exposure weights, provided those weights represent the relevant base-premium exposure and portfolio mix.
where \(R_j\) is a rate-level index and \(w_j\) is the proportion of the relevant earned exposure assigned to that level, with \(\sum_j w_j=1\). These weights should not be confused with shares of actual earned dollars: actual premium already reflects the rate level. If underlying premiums differ across groups, the weights must represent the exposure basis appropriate to the portfolio rate-level calculation.
Why the Geometric Assumption Produces an Average
Consider a hypothetical book of 12-month policies with renewal-effective rate changes, writings distributed uniformly through the year, and exposure earned evenly over each policy term. After a change takes effect, the share of the portfolio renewed at the new level grows approximately linearly from zero to one over the following 12 months. The average share over that transition is the area of a triangle divided by the width of the period:
Thus, over a full transition year, a rate increase from 1.00 to 1.12 gives an average rate level of \(1.00+0.5(0.12)=1.06\), under these assumptions. The geometry represents the gradual earning-in of the new rate, not a claim that all policies change rate at the same instant. If writings are not uniform, terms differ, or rate changes apply midterm rather than at renewal, the assumed shape changes; the corresponding parallelogram calculation must reflect those conditions.
Worked Example: Annual Policies With a Renewal-Effective Increase
A hypothetical book has 12-month policies, uniform writings through the year, and a 12% rate increase effective January 1, 2025, at renewal. Assume the portfolio mix is stable and policies earn evenly. The calendar-year 2025 earned premium at the actual rate levels is $1,060,000. Estimate the on-level premium at the new rate level. No interest or mortality assumption is involved; the writing distribution is uniform by date.
Method. Use the parallelogram method because the problem supplies aggregate earned premium and the timing assumptions needed to estimate the average rate level.
During 2025, the share of the portfolio at the new rate level rises linearly from zero to one as policies renew. Its average over the year is \(0.5\). The average earned rate level is therefore
The selected rate level is 1.12, so the on-level factor is
The estimated on-level premium is \(\$1{,}060{,}000 \times 1.0566038\ldots=\$1{,}120{,}000\). This equals the amount obtained by multiplying the underlying premium at level 1.00, \(\$1{,}060{,}000/1.06=\$1{,}000{,}000\), by 1.12. The result checks. The factor is rounded to three decimals if reported as a factor; the premium is shown to the nearest dollar. A common mistake is to use 1.12 as the factor: 1.12 is the target rate index, not the ratio of target level to average earned level.
Selecting a Method From the Data
A practical selection starts by asking what can be measured directly. If policy or exposure records permit the same earned exposure to be rerated at the target rates, extension is generally the more direct choice. If those records are not available but rate-change dates and credible aggregate earned-exposure information are, the parallelogram method may be suitable. Before relying on a parallelogram estimate, check whether its writing and earning assumptions match the portfolio.
For example, a method that assumes uniform writings may not represent a book with strongly seasonal production. Likewise, an annual-policy diagram may not represent a portfolio with many short-term policies unless policy-term effects are incorporated. Where the available records support a more specific calculation, use the observed patterns rather than applying a standard geometric shape mechanically.
Worked Example: Choosing an Aggregate Method With Uneven Writings
A hypothetical insurer lacks policy-level records needed to rerate its earned exposure, but it has reliable monthly in-force records. Those records account for nonuniform writings and policy terms and show that 60% of the relevant earned base-premium exposure was at rate level 1.00 and 40% was at 1.10. The portfolio mix is stable, and the target level is 1.10. Actual earned premium is $520,000. Select and apply a suitable method. The recorded shares are base-premium-exposure weights, not shares of actual earned dollars; no interest or mortality model is involved.
Method selection. Extension is not supported because policy-level rerating records are unavailable. Use the parallelogram approach based on the observed earned-exposure distribution. Because the monthly records reflect the uneven writings and policy terms, do not substitute a uniform-writing assumption.
The average earned rate level is
The factor to the target level is
The on-level premium is \(\$520{,}000 \times 1.0576923\ldots=\$550{,}000\). As a check, the underlying base-premium exposure is \(\$520{,}000/1.04=\$500{,}000\); at level 1.10 it produces \(\$500{,}000 \times 1.10=\$550{,}000\). The factor is rounded to three decimals. A common mistake is to treat the 60% and 40% as shares of actual premium and weight rate levels with those dollar shares. The problem specifies base-premium-exposure weights, which are the weights needed for the average rate-level calculation.
Common Traps and a Selection Checklist
The methods answer a related question, but their inputs and failure modes are different. Before calculating, confirm that the numerator and denominator refer to the same exposure basis and that the weights used in an aggregate average have the intended meaning.
If the same exposure can be rerated at historical and target rates, extension is feasible. If not, identify what aggregate rate-level and earning data are available.
For a parallelogram estimate, assess writing timing, policy terms, earning patterns, and whether the portfolio mix represented by the data is appropriate.
Divide target-level premium or rate level by historical premium or average earned rate level. A factor below 1 can be appropriate if the target level is lower.
Confirm that the on-level premium applies the factor to earned premium on the same exposure basis and that observed mix changes have not been mistaken for rate changes.
Check Your Understanding
For each problem, identify the relevant calculation or method and check that its assumptions match the data.
- A hypothetical earned exposure has historical premium of $48,000 and target-rate premium of $54,000 when rerated using the same records. Find the extension factor. (Answer: 1.125)
- A hypothetical book of annual policies has uniform writings, stable mix, and a renewal-effective 8% increase on January 1. For calendar-year experience during the transition year, find the average earned rate level and the factor to the new level. (Answer: average level 1.04; factor \(1.08/1.04=1.0385\))
- A hypothetical book of six-month policies has uniform writings and a 12% renewal-effective increase on July 1. Assume the new-rate share rises linearly from zero to one during July–December and is zero during January–June. Find the calendar-year average rate level and the factor to 1.12. (Answer: average level 1.03; factor \(1.12/1.03=1.0874\))
- A hypothetical insurer has no policy-level rerating records, but has monthly earned-exposure records that reflect seasonal writings and varying policy terms. Which approach is more directly supported, and what should be checked before using it? (Answer: a parallelogram-based estimate using the observed patterns; check that the records represent the relevant exposure and mix.)
- A hypothetical portfolio has 75% of earned base-premium exposure at level 1.00 and 25% at level 1.16. Find the average earned level and factor to level 1.16. (Answer: average level 1.04; factor \(1.16/1.04=1.1154\))