Detecting Inconsistent Premium Earning Assumptions

Why Earning Assumptions Matter

An on-level premium calculation adjusts historical premium to a common rate level so that premium can be compared on a consistent basis. The adjustment depends not only on the dates of rate changes, but also on which rate level applies to the policies and how much of each policy’s premium is earned in the experience period. If the calculation assumes a different policy effective-date pattern from the actual portfolio, its estimated rate level can be biased.

This is a question of matching weights. A rate level that applies to a large share of earned exposure should have more influence on earned premium than a rate level applying to a small share. Written policy counts, calendar-time averages, and midyear rate levels are not substitutes for earned exposure weights unless the circumstances make them equivalent.

Definition: An earning assumption specifies how written premium becomes earned over time and how much earned exposure is associated with each policy rate level in the experience period. Inconsistent premium earning assumptions occur when an on-level calculation uses weights or timing that do not represent the actual premium earning pattern of the portfolio being adjusted.

Earned Exposure Weights and the On-Level Factor

Let $r_j$ be the relative rate level applicable to exposure earned under policy group $j$, and let $B_j$ be that group’s base premium exposure: the amount of earned exposure expressed at a common base rate level. For example, if the base annual premium is $1,000 and a group contributes 40 earned policy-years, then $B_j=\$40{,}000$. Let $r_c$ be the target, or current, rate level.

Assume that premium is proportional to the rate-level index; the same base premium measure is used for each group; and the rate applicable to a policy is the rate level under which its premium is written. Under these assumptions, the historical earned premium and its equivalent at the target rate level are:

P_{\mathrm{hist}}=\sum_j B_j r_j, \qquad P_{\mathrm{target}}=\sum_j B_j r_c =r_c\sum_j B_j.

The on-level factor for historical earned premium is therefore the target-rate-level premium divided by the historical earned premium:

F_{\mathrm{OL}} =\frac{P_{\mathrm{target}}}{P_{\mathrm{hist}}} =\frac{r_c\sum_j B_j}{\sum_j B_jr_j} =\frac{r_c}{\bar r_E}, \qquad \bar r_E=\frac{\sum_j B_jr_j}{\sum_j B_j}.

Here, $\bar r_E$ is the average rate level weighted by the base premium exposure actually earned. It is not automatically the average rate level by calendar date, by written policy count, or by written premium.

Justification: Under proportional rating, each unit of base premium exposure earned at rate level $r_j$ contributes $r_j$ units of premium per base unit. Repricing that same earned exposure at $r_c$ replaces every $r_j$ with $r_c$. The ratio of repriced to historical premium is therefore $r_c\sum_j B_j/\sum_j B_jr_j$, which gives the on-level factor above.

The factor applies when the premium being adjusted has the same exposure basis and earning pattern represented by the $B_j$ weights. If a calculation instead uses written premium, a calendar-time average, or an assumed uniform distribution of effective dates, it must be checked against the actual pattern. A simplified method can be appropriate when its assumptions are a reasonable representation of that pattern; it is not appropriate merely because the dates look evenly spaced.

Worked Examples

Worked Example: Policies Earning Across a Midyear Rate Change

A hypothetical insurer has one-year policies with a base annual premium of $1,200. The rate level is 1.00 before July 1 and 1.20 from July 1 onward. Assume policies receive the rate level in effect on their effective date, with no midterm repricing. In the experience year, 100 policies take effect on January 1 and 100 take effect on July 1. Premium is earned uniformly over each policy’s one-year term. Calculate the factor to bring calendar-year earned premium to the 1.20 rate level.

Method. Divide earned exposure by the rate level attached to the policy, then reprice those same earned exposures at 1.20. This respects both the effective dates and the one-year earning period.

January policies. Each earns 6 months at the old rate level and 6 months at the new calendar period, but its premium remains at the 1.00 level because it was written before the change. Each policy earns $6/12=0.5$ policy-years in the first half and $6/12=0.5$ in the second half. Thus the 100 January policies contribute 100 earned policy-years at rate level 1.00.

July policies. Each contributes $6/12=0.5$ earned policy-years during the experience year, all at rate level 1.20. The 100 July policies contribute $100(0.5)=50$ earned policy-years at 1.20.

The base premium exposure at each rate level is therefore $B_{\mathrm{old}}=100(\$1{,}200)=\$120{,}000$ and $B_{\mathrm{new}}=50(\$1{,}200)=\$60{,}000$. Historical earned premium is:

P_{\mathrm{hist}} =(\$120{,}000)(1.00)+(\$60{,}000)(1.20) =\$120{,}000+\$72{,}000 =\$192{,}000.

The total earned policy-years are $100+50=150$. At the target rate level, premium is $150(\$1{,}200)(1.20)=\$216{,}000$. Thus:

F_{\mathrm{OL}} =\frac{\$216{,}000}{\$192{,}000} =1.125.

Applying the factor gives $\$192{,}000(1.125)=\$216{,}000$, as required. The factor is checked by reproducing the target-level premium for the same 150 earned policy-years.

Common mistake avoided. The experience year is split evenly at the July 1 rate change, but that does not mean half of earned exposure is at each rate level. In this example, 100 earned policy-years are at 1.00 and 50 are at 1.20. Using equal weights would give an average rate level of 1.10 and a factor of $1.20/1.10=1.0909$, which would understate the correct factor of 1.125.

Worked Example: A Portfolio With More Old-Rate Exposure

A hypothetical book has a base annual premium of $1,000 per policy-year. The rate level increases from 1.00 to 1.15 on July 1, and rates apply at policy inception with no midterm repricing. During the calendar year, 20 policies effective the previous October 1 earn 9 months each; 80 policies effective January 1 earn a full year each; and 100 policies effective July 1 earn 6 months each. Find the correct factor to the 1.15 level, then compare it with a calculation that assumes equal earned exposure at the old and new rate levels.

Method. Convert each cohort’s time in the experience year to earned policy-years and classify that exposure by the policy’s effective-date rate level.

The October cohort earns $20(9/12)=15$ policy-years at 1.00. The January cohort earns $80(12/12)=80$ policy-years at 1.00. The July cohort earns $100(6/12)=50$ policy-years at 1.15. Therefore, the total is 95 old-rate policy-years and 50 new-rate policy-years, or $95+50=145$ policy-years.

At $1,000 of base annual premium per policy-year, historical earned premium is:

P_{\mathrm{hist}} =95(\$1{,}000)(1.00)+50(\$1{,}000)(1.15) =\$95{,}000+\$57{,}500 =\$152{,}500.

The same 145 policy-years at the target rate level would produce $145(\$1{,}000)(1.15)=\$166{,}750$. Thus the correct factor is $\$166{,}750/\$152{,}500=1.0934426$, or 1.0934 to four decimal places. Multiplication checks: $\$152{,}500(1.0934426)$ is approximately $\$166{,}750$.

Under the equal-exposure assumption, the average rate level would instead be $(1.00+1.15)/2=1.075$, producing an assumed factor $1.15/1.075=1.0697674$. Applying that factor gives approximately $\$163{,}139.53$, which is about $\$3{,}610.47$ below the correct target-level premium. The assumed weighting under-corrects because actual earned exposure is concentrated at the old rate level.

Common mistake avoided. A rate change halfway through the year does not imply that half of the earned exposure is at each rate. Effective dates and policy terms determine how much exposure earns under each policy rate.

Worked Example: Why Written Policy Counts Can Give the Wrong Weight

A hypothetical portfolio has 80 policies written at rate level 1.00 and 20 at rate level 1.12. Although the written counts are 80 and 20, the experience period contains 40 base-premium exposure units earned at 1.00 and 60 at 1.12. Each exposure unit has a base premium of $900. Find the factor to the 1.12 level and quantify the error from weighting the rate levels by written policy counts.

Method. Use earned base-premium exposure as the weight. The written counts are provided as a trap: they do not describe the earning pattern in the experience period.

Historical earned premium is $40(\$900)(1.00)+60(\$900)(1.12)$. The first component is $\$36{,}000$, and the second is $\$60{,}480$, so total historical earned premium is $\$96{,}480$. Total exposure is $40+60=100$ units. At rate level 1.12, target premium is $100(\$900)(1.12)=\$100{,}800$.

The correct factor is $\$100{,}800/\$96{,}480=1.0447761$, or 1.0448 to four decimal places. The adjusted premium is approximately $\$100{,}800$.

If the written policy counts are incorrectly used as weights, the assumed average rate level is $0.80(1.00)+0.20(1.12)=1.024$. That gives a factor of $1.12/1.024=1.09375$. Applying this factor to historical premium gives $\$96{,}480(1.09375)=\$105{,}525$, which overstates target premium by $\$105{,}525-\$100{,}800=\$4{,}725$.

Common mistake avoided. Written counts describe how many policies were written at each level, not how much of their premium was earned in the experience period. Use them only if they correctly imply the earned exposure weights for the calculation.

Diagnosing an Incompatible Assumption

A practical check is to trace each rate level through the policies that contribute earned premium. For every group, ask which rate applied at policy inception, what share of the policy term falls in the experience period, and what base premium exposure that share represents. Then compare those weights with the weights used by the on-level method.

Identify the premium being adjusted.
Confirm whether it is earned or written premium and specify the experience period and target rate level.

Map policies to rate levels.
Use effective dates and the applicable contract rules to determine which rate level produced each policy’s premium. Do not assume midterm repricing unless it applies.

Calculate earned exposure by rate level.
Account for policy terms and the portion of each term that contributes to the experience period. Convert to base-premium weights when base premiums differ.

Compare assumed and actual weights.
Check whether the method’s uniform-writing, midpoint, or other earning assumption represents the calculated exposure pattern.

Reprice and verify.
Calculate target premium for the same earned exposure, divide by historical earned premium, and check that applying the factor reproduces the target premium.

The direction of bias can often be diagnosed before doing the full calculation. If the actual portfolio has more earned exposure at an old, lower rate level than the assumed weights imply, actual historical earned premium is lower than the assumption suggests. The true factor to a higher target level will therefore be larger than the assumed factor, so the assumed method under-corrects. If actual exposure is more concentrated at the higher level, the reverse can occur.

This comparison must use consistent units. If base premium varies by territory or coverage, policy counts alone do not provide the $B_j$ weights. If a component is fixed rather than proportional to the rate level, it should not be scaled as if it varied proportionally. Likewise, the simple calculation above does not handle midterm endorsements, nonuniform earning patterns, or rate changes applied to existing policies unless those features are explicitly represented in the exposure weights.

Common Traps and Practical Checks

Using the calendar midpoint as the portfolio average. A rate change halfway through the year does not establish equal exposure at each rate level.
Confusing effective dates with earning dates. A policy may earn premium during a period after its effective date while retaining the rate level attached to its inception.
Weighting by written premium or policy count without checking. Those quantities need not match the base premium exposure actually earned.
Changing the exposure pattern during repricing. The target premium must represent the same earned exposure as the historical premium, changed only to the target rate level.
Applying a factor to incompatible premium data. The factor is derived for a particular premium measure and period; check that the premium being adjusted uses that same basis.

Key takeaway: An on-level factor for earned premium is determined by the rate levels attached to the earned base-premium exposure—not by a presumed average date or a count of policies. Calculate or validate those weights before applying the factor.

Check Your Understanding

For each problem, use the stated earned-exposure weights rather than inferring them from calendar time or written counts.

A portfolio has 30 earned exposure units at rate level 1.00 and 70 at 1.10. Find the factor to 1.10. (Answer: $1.10/[0.30(1.00)+0.70(1.10)]=1.0280$, to four decimal places.)
Earned exposure is 75 units at 1.00 and 25 at 1.20. Find the on-level factor to 1.20. (Answer: $1.20/[0.75(1.00)+0.25(1.20)]=1.1429$, to four decimal places.)
A book has 20 earned exposure units at 1.00 and 80 at 1.08. Base premium is $500 per unit. Calculate historical earned premium and the factor to 1.08. (Answer: historical earned premium is $53,200; target premium is $54,000; factor $=54{,}000/53{,}200=1.0150$, to four decimal places.)
Actual earned exposure is 20 units at 1.00 and 80 at 1.25. An analyst incorrectly assumes equal exposure at the two levels. Find the correct and assumed factors to 1.25, and state the direction of bias. (Answer: correct factor $=1.25/[0.20(1.00)+0.80(1.25)]=1.0417$; assumed factor $=1.25/1.125=1.1111$; the assumed method over-corrects.)

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