Why Policy Term Length Changes the Result
The parallelogram method represents how premium moves from one rate level to another as policies renew. The length of the policy term determines how long that transition takes: when terms are shorter, more policies can renew after a rate change during a given period. Consequently, a greater share of earned premium can reflect the new rate level.
The comparison is meaningful only when the other assumptions are held fixed. In this tutorial, policies are written uniformly over time, premium is earned uniformly over each policy term, and a rate change applies to policies written or renewed on or after its effective date. There are no other changes in exposure, coverage, mix, or rate level. These are assumptions of the simplified parallelogram model, not facts that must hold for every portfolio.
For the main comparison, suppose the rate change takes effect at the beginning of an experience period of length \(E\). Set the old rate level equal to 1 and write the new rate level as \(1+r\), where \(r\) is the rate change expressed as a decimal. For example, a 12% increase gives \(r=0.12\) and a new rate level of 1.12.
Deriving the New-Rate Share
Measure time \(t\) from the effective date of the change. Under uniform writings and uniform earning, during the first \(T\) time units after the change, the share of premium being earned from post-change policies increases linearly from zero to one. At time \(t\), that share is \(t/T\). After \(T\), all premium being earned is at the new rate level. Thus, for \(0\leq t\leq E\), the new-rate share at time \(t\) is \(\min(t/T,1)\).
The new-rate share for the whole experience period is the average of this time-varying share. This is an area-under-the-parallelogram calculation: divide the area representing new-rate premium by the experience-period length.
When \(E\leq T\), the area is a triangle with base \(E\) and height \(E/T\), so its area is \(E^2/(2T)\); dividing by \(E\) gives \(E/(2T)\). When \(E\geq T\), the area consists of a triangle of area \(T/2\) and a rectangle of area \(E-T\). Dividing their sum by \(E\) gives \((T/2+E-T)/E=1-T/(2E)\). At \(E=T\), both expressions give a share of \(1/2\), so the result is continuous at the boundary.
The average earned rate level is the weighted average of the old level and the new level. If \(q\) is the new-rate share, then the old-rate share is \(1-q\), so
To bring the experience premium to the new rate level, divide the target level by the average earned level:
This factor is not the same as the average earned rate level. The factor is a multiplier applied to the experience premium; multiplying the average level by the factor must recover the target level.
Comparing Term Lengths
For a fixed experience period and a change effective at its beginning, the new-rate share decreases as the term length increases. In the \(E\leq T\) case, the share is \(E/(2T)\), which falls as \(T\) grows. In the \(E\geq T\) case, it is \(1-T/(2E)\), which also falls as \(T\) grows. Shorter terms therefore produce a faster transition toward the new rate level, all else equal.
The same result can be seen when the rate change occurs after the experience period has started. Let \(L\) be the time from the rate change to the end of the experience period. Premium before the change is all at the old level. The post-change new-rate area is triangular if \(L\leq T\), and otherwise is a triangle plus a rectangle. Therefore the new-rate share over the full experience period is
Here \(E\) is the length of the entire experience period, not just the time remaining after the change. The numerator measures the area at the new level after the change; the denominator measures all earned premium time in the experience period. This formula assumes the same uniform-writing and uniform-earning model as before.
Worked Example: One-Year Experience Period and Two Term Lengths
A hypothetical portfolio has a 12-month experience period. A 12% rate increase takes effect at the beginning of the period. Compare the average earned rate level and factor to the new rate level for annual policies and six-month policies. Assume uniform writings and earning, and that the change applies at policy inception or renewal.
Method. Calculate the new-rate share separately for each term length, then calculate the average level and the factor to 1.12.
For annual policies, \(E=12\) months and \(T=12\) months. Since \(E=T\),
The average earned rate level is
The factor to the new rate level is
For six-month policies, \(E=12\) months and \(T=6\) months. Since \(E\geq T\),
The average earned rate level is
The factor to the new rate level is
The six-month policies have a larger new-rate share (0.7500 rather than 0.5000), because more renewals can occur during the year. Their average earned level is therefore closer to 1.12, and the required factor is smaller. Check: \(1.0600(1.0566038\ldots)=1.12\), and \(1.0900(1.0275229\ldots)=1.12\). The results are rounded to four decimal places.
Common mistake to avoid: Do not mistake the average level 1.0900 for the factor, or use a factor that does not return the target level. For example, a proposed factor of 1.1009 would give \(1.0900(1.1009)=1.199981\), not 1.12. The correct factor is \(1.12/1.0900\approx1.0275\).
Worked Example: A Rate Change Partway Through the Period
A hypothetical 9-month experience period has a rate increase of 8% effective after its first 3 months. Compare six-month and 12-month policy terms. Assume uniform writings and earning, with the change applying at policy inception or renewal.
Method. There are \(E=9\) months in the full period and \(L=9-3=6\) months from the change to the end. Calculate the post-change new-rate area for each \(T\), then divide by the full 9-month period.
For six-month policies, \(L=T=6\). The new-rate area is triangular:
Thus \(q=3/9=1/3\). The average earned rate level is
The factor to 1.08 is
For 12-month policies, \(L=6<T=12\), so the new-rate area is
The new-rate share is \(q=1.5/9=1/6\). The average earned rate level and factor are
The longer-term policies have less new-rate premium in this experience period, so their average level is farther from the target and their factor is larger. Check: \(1.0266667\ldots(1.0519481\ldots)=1.08\), and \(1.0133333\ldots(1.0657895\ldots)=1.08\). Results are rounded to four decimal places.
Common mistake to avoid: The denominator is the full 9-month period, not the 6 months after the change. Using \(3/6\) as the six-month policies’ share would omit the premium earned before the change and would overstate the new-rate share for the full experience period.
Worked Example: Comparing Shorter Terms Over Nine Months
A hypothetical rate increase of 15% takes effect at the start of a 9-month experience period. Compare four-month and eight-month policy terms. Assume uniform writings and earning and no other changes in exposure or coverage.
Method. Since \(E=9\) months is greater than both term lengths, use \(q=1-T/(2E)\).
For four-month policies,
The average earned rate level is
The factor to the new level is
For eight-month policies,
The average earned rate level and factor are
The four-month term produces a larger new-rate share than the eight-month term: \(7/9>5/9\). Check: the factors return the target level, since \((67/60)(1.15/(67/60))=1.15\) and \((13/12)(1.15/(13/12))=1.15\). Results are rounded to four decimal places.
Common mistake to avoid: Do not apply the triangle-only formula \(E/(2T)\) when \(E>T\). For both term lengths here, the share includes a rectangle at the new rate after the transition is complete; omitting it would understate the new-rate share.
Assumptions, Interpretation, and Checks
The term-length comparison isolates one feature of the earning pattern. It does not establish that two real portfolios have the same total earned premium or the same underlying risk. The result depends on the assumed writing pattern and on how premium earns during a policy term. If writings are not uniform, terms are mixed, or policies earn nonuniformly, the simple formulas may not represent the actual earned-premium weights. Use the parallelogram model only when its assumptions are a reasonable approximation to the data and rating process.
A reliable calculation keeps three quantities distinct:
- New-rate share \(q\): the proportion of earned premium in the experience period at the new level.
- Average earned level \(\overline{L}\): the weighted level represented by all earned premium in that period.
- On-level factor \(F_{\text{new}}\): the multiplier that converts the average level to the target level.
A useful final check is \(\overline{L}F_{\text{new}}=1+r\). If the equality fails, inspect whether the term and experience lengths use consistent units, whether the share formula matches the timing, and whether the factor was divided in the right direction. In particular, a factor to the new level is target level divided by average earned level, not the reverse.
Check Your Understanding
Use the stated uniform-writing and uniform-earning assumptions. Round rate levels and factors to four decimal places where needed.
- A rate increase of 10% takes effect at the start of a 12-month experience period. Policies have 3-month terms. Find the new-rate share, average earned rate level, and factor to the new level. (Answer: \(q=0.8750\), average level \(=1.0875\), factor \(=1.0115\).)
- A 9% increase takes effect after the first 2 months of a 10-month experience period. Policies have 12-month terms. Find the new-rate share and factor to the new level. (Answer: \(q=0.2667\), factor \(=1.0645\).)
- A 5% increase takes effect at the start of a 15-month experience period. Compare the new-rate shares and average earned rate levels for 6-month and 12-month terms. (Answer: 6-month term: \(q=0.8000\), average level \(=1.0400\); 12-month term: \(q=0.6000\), average level \(=1.0300\).)
- A 12% increase takes effect after the first 4 months of a 12-month experience period. Policies have 6-month terms. Find the new-rate share, average level, and factor to the new level. (Answer: \(q=0.4167\), average level \(=1.0500\), factor \(=1.0667\).)
- A 20% increase takes effect at the beginning of a 12-month experience period. Policies have 6-month terms. Find the average earned rate level and factor to the new level, and verify their product. (Answer: \(q=0.7500\), average level \(=1.1500\), factor \(=1.0435\); product \(=1.2000\).)