Parallelogram Adjustments for Nonuniform Writings

Why Writing Patterns Matter

The parallelogram method estimates the average rate level earned during an experience period by tracking how policies written at different times earn into that period. Its standard geometric form assumes that policy writings are distributed uniformly over time. If writings are strongly seasonal, or otherwise concentrated in particular months, that assumption can misstate how much exposure at each rate level was earned.

The issue is not simply that the number of policies written varies. A policy written near the beginning of an experience period may earn most or all of its annual exposure in that period, while one written near the end earns only a small portion. The rate level attached to each policy cohort therefore needs to be combined with the cohort’s actual earned exposure, not just its number of policies.

The adjustment in this tutorial retains the annual-policy and pro rata earning assumptions used for the standard parallelogram, but replaces the uniform writing pattern with actual cohort data. Assume that a policy receives the rate level in effect on its inception date and keeps that level for its one-year term. Any hypothetical insurance product and data below are illustrative.

Definition: A writing cohort is a group of policies written at the same time and assigned the same rate level. Its earned exposure in an experience period is the cohort’s number of exposure units multiplied by the fraction of each policy term that falls within that period. The earned rate level is the average of cohort rate levels weighted by their earned exposures.

Calculating Earned Exposure From Actual Cohorts

For annual policies, assume each policy earns exposure evenly over its one-year term. Let the experience period be $[A,B)$, measured in years. A policy in a cohort written at time $s_c$ is in force over $[s_c,s_c+1)$. The part of its term earned in the experience period is the length of the overlap between those intervals:

p_c=\max\left(0,\ \min(s_c+1,B)-\max(s_c,A)\right)

Here $p_c$ is an earned fraction between 0 and 1. If cohort $c$ contains $N_c$ annual exposure units, its earned exposure is $E_c=N_cp_c$. If its rate level is $L_c$, the average earned rate level is

\bar L_{\text{earned}}=\frac{\sum_c E_cL_c}{\sum_c E_c} =\frac{\sum_c N_cp_cL_c}{\sum_c N_cp_c}.

This calculation uses the actual number of exposures written in each cohort. If cohort sizes are supplied as policy counts, treat each policy as one equally sized exposure unit, as in the examples below. When policies represent different amounts of exposure, the weights must instead reflect consistently measured exposure units.

Why the Weighted Average Works

Suppose the base-rate premium for one annual exposure unit is $B_0$, and all rate changes scale that premium by the rate-level index $L_c$. The earned premium from cohort $c$ is then $B_0E_cL_c$. Adding the cohort premiums gives total earned premium $B_0\sum_c E_cL_c$. The total earned base-rate exposure premium is $B_0\sum_c E_c$. Dividing the first amount by the second cancels $B_0$, leaving the earned rate level:

\frac{B_0\sum_c E_cL_c}{B_0\sum_c E_c} =\frac{\sum_c E_cL_c}{\sum_c E_c} =\bar L_{\text{earned}}.

To adjust historical earned premium to a target rate level $L_T$, multiply it by

F_{\text{on-level}}=\frac{L_T}{\bar L_{\text{earned}}}.

This factor follows because the historical premium reflects the average level $\bar L_{\text{earned}}$, while the desired premium reflects $L_T$. The ratio converts one level to the other. The formula assumes the same exposure basis and rate-level scale are used in both calculations; it does not correct for a change in the underlying risk mix.

A Practical Calculation Sequence

Fix the experience period.
Write down its beginning and end dates. Use the same date boundaries for every cohort.

Assign each cohort a rate level.
Use the rate level applicable when that cohort was written under the stated policy rules.

Find each earned fraction.
Calculate the overlap of the cohort’s annual policy term with the experience period.

Weight and sum.
Multiply cohort size by earned fraction, then multiply that earned exposure by its rate level.

Calculate the indication.
Divide the weighted rate-level total by total earned exposure. Divide the target level by this average to obtain the on-level factor.

Worked Calculations

Worked Example: Seasonal Writings and One Rate Change

A hypothetical insurer has annual policies and an experience period of January 1 through December 31, 2025. The rate level is 1.00 through June 30 and 1.20 for policies written on or after July 1. The following cohorts are written on the first day of each listed month: 40 policies on October 1, 2024; 20 on January 1, 2025; 20 on April 1, 2025; 60 on July 1, 2025; and 100 on October 1, 2025. Find the earned rate level and the factor to level 1.20.

Method. Calculate each cohort’s annual-policy overlap with 2025, then use earned exposure as the weight. This avoids treating a policy written late in the year as though it earned a full year in the experience period.

The October 2024 cohort earns January through September 2025, or $9/12=0.75$ of a year. Its earned exposure is $40(0.75)=30$, at level 1.00. The January 2025 cohort earns a full year: $20(1)=20$, at level 1.00. The April 2025 cohort earns April through December, or $9/12=0.75$: $20(0.75)=15$, also at level 1.00.

The July 2025 cohort earns six months, so its earned exposure is $60(6/12)=30$, at level 1.20. The October 2025 cohort earns three months: $100(3/12)=25$, also at level 1.20.

Total earned exposure at level 1.00 is $30+20+15=65$. Total at level 1.20 is $30+25=55$, making total earned exposure $65+55=120$. The weighted level total is $65(1.00)+55(1.20)=65+66=131$. Thus the average earned rate level is $131/120=1.0917$, rounded to four decimal places.

The factor to level 1.20 is $1.20/1.0916667=1.0992$, rounded to four decimal places. The check is $1.0916667(1.0992366)\approx1.2000$ before rounding. The main mistake avoided is assigning every written policy a full earned exposure.

Worked Example: Multiple Changes With a Seasonal Writing Pattern

A hypothetical annual-policy book has an experience period of calendar year 2026. Rate levels are 1.00 for policies written before April 1, 2026, 1.10 for policies written from April 1 through September 30, and 1.21 for policies written on or after October 1. Cohort data are: 20 policies written October 1, 2025; 60 written January 1, 2026; 30 written April 1; 20 written July 1; and 70 written October 1, 2026. Calculate the earned rate level and factor to 1.21.

The October 2025 cohort earns three quarters of its annual term in 2026: $20(0.75)=15$ exposure units at 1.00. The January cohort earns a full year: $60(1)=60$ at 1.00. Therefore, level 1.00 contributes $15+60=75$ earned exposures.

The April cohort earns nine months: $30(0.75)=22.5$ at 1.10. The July cohort earns six months: $20(0.50)=10$ at 1.10. Together, level 1.10 contributes $22.5+10=32.5$ earned exposures.

The October 2026 cohort earns three months: $70(0.25)=17.5$ at 1.21. Total earned exposure is $75+32.5+17.5=125$. The weighted level total is $75(1.00)+32.5(1.10)+17.5(1.21)=75+35.75+21.175=131.925$.

The earned rate level is $131.925/125=1.0554$. The factor to level 1.21 is $1.21/1.0554=1.1465$, rounded to four decimal places. As a check, the exact ratio is $151.25/131.925$, approximately 1.1464847; multiplying the unrounded average by that ratio gives 1.21.

A tempting but incorrect calculation weights by policy counts alone: $[80(1.00)+50(1.10)+70(1.21)]/200=219.7/200=1.0985$. It would give a factor of about $1.1015$, not $1.1465$. Policy counts do not show how much of each annual term is earned in the experience period. Here, many policies at the newest level were written late and contribute only one quarter of a year each.

Worked Example: A Full-Year Experience Window With Uneven Cohorts

A hypothetical insurer measures experience from April 1, 2025, through March 31, 2026. Annual policies written January 1, 2025, have level 1.00; policies written April 1 have level 1.00; those written July 1 have level 1.10; those written October 1 have level 1.10; and those written January 1, 2026, have level 1.20. The respective cohort sizes are 40, 20, 100, 40, and 80 policies. If the base-rate premium is $1,000 per annual exposure unit, find the earned rate level and the on-level premium at 1.20.

The January 2025 cohort is in force for nine months of the experience period, so its earned exposure is $40(0.75)=30$ at 1.00. The April cohort contributes $20(1)=20$ at 1.00. The July cohort is in force for nine months, contributing $100(0.75)=75$ at 1.10. The October cohort contributes $40(0.50)=20$ at 1.10. The January 2026 cohort contributes three months, or $80(0.25)=20$, at 1.20.

Total earned exposure is $30+20+75+20+20=165$. The weighted level total is $30(1.00)+20(1.00)+75(1.10)+20(1.10)+20(1.20)=30+20+82.5+22+24=178.5$. Thus $\bar L_{\text{earned}}=178.5/165=1.0818$, rounded to four decimal places.

The factor to level 1.20 is $1.20/(178.5/165)=198/178.5=1.1092$, rounded to four decimal places. The actual earned premium is $178.5($1,000)=$178,500$. At level 1.20, the corresponding premium is $165(1.20)($1,000)=$198,000$. Their ratio is $198,000/178,500=1.1092437$, which verifies the factor before rounding.

The common mistake avoided is using the calendar dates alone to infer the weights. Each cohort’s policy term must be intersected with the actual experience period; the experience window here does not begin on January 1.

Assumptions, Interpretation, and Exam Traps

The standard parallelogram is a convenient representation when writings are uniform through time and policies earn pro rata. With nonuniform writings, the geometric shape based on a uniform flow no longer represents the actual cohort volumes. The weighted-cohort calculation makes the replacement explicit: actual cohort sizes determine how much exposure each writing date contributes, and overlap determines how much of that exposure is earned.

The calculation is appropriate under the stated annual-policy assumptions when policies receive the rate level applicable at inception and earn evenly during the term. It does not automatically apply when policy terms differ, rate changes apply differently to in-force policies, or exposures do not earn pro rata. Those conditions require a method that represents the actual policy and rating rules.

Do not assume uniform writings without checking. A concentrated seasonal peak can shift the earned exposure toward particular rate levels.
Do not weight only by policies written. Use $N_cp_c$, not $N_c$, because only part of a policy term may be earned in the experience period.
Keep rate levels and date intervals aligned. A cohort written on a rate-change date must be assigned the level specified for that date.
Apply the factor in the correct direction. To move from the average earned level to a higher target, use $L_T/\bar L_{\text{earned}}$; reversing the ratio moves premium the wrong way.
Separate rate level from other changes. This factor isolates the rate-level adjustment under a consistent exposure basis; it does not correct for changes in the underlying insured population.

Key takeaway: When policy writings are not uniform, calculate earned exposure from the actual writing cohorts and their overlap with the experience period. Weight each cohort’s rate level by that earned exposure before calculating the on-level factor.

Check Your Understanding

For each problem, use annual policies that earn pro rata and receive the rate level in force at inception. Round factors to four decimal places.

A calendar-year 2025 experience period has level 1.00 through June and level 1.10 from July 1. Cohorts are 40 policies written October 1, 2024; 30 written January 1, 2025; 20 written April 1; 50 written July 1; and 80 written October 1. Find the earned rate level and factor to 1.10. (Answer: earned level $1.0375$; factor $1.0602$.)
For calendar year 2026, levels are 1.00 before April 1, 1.20 from April 1 through September 30, and 1.30 from October 1 onward. Cohorts are 60 policies written October 1, 2025; 40 written January 1, 2026; 20 written April 1; 100 written July 1; and 40 written October 1. Find the earned rate level and factor to 1.30. (Answer: earned level $1.1000$; factor $1.1818$.)
An annual-policy cohort of 40 policies is written August 1. The experience period is January 1 through December 31 of that same year. What earned exposure does the cohort contribute? If its rate level is 1.20, what is its contribution to the weighted level total? (Answer: $40(5/12)=16.6667$ earned exposures; weighted level contribution $16.6667(1.20)=20.0000$, before rounding.)
For calendar year 2025, levels are 1.00 through June 30 and 1.15 from July 1. Cohorts are 80 policies written October 1, 2024; 20 written January 1, 2025; 60 written April 1; 40 written July 1; and 120 written October 1. Find the earned rate level and factor to 1.15. (Answer: earned level $1.0429$; factor $1.1027$.)
A candidate weights cohorts by policy counts, without considering earned fractions. Explain why this does not generally produce the earned rate level for a calendar-year experience period with annual policies. (Answer: policy counts measure writings, not earned exposure; the correct weight for cohort $c$ is $N_cp_c$, where $p_c$ is the fraction of its policy term overlapping the experience period.)

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