Inferring Existing Factors from Earned Premiums

Why Infer Factors From Earned Premium?

A rating plan may specify a base rate and classification factors, but a portfolio’s earned premiums show what rates were actually earned across its exposures. Dividing earned premium by earned exposure gives an observed average premium rate for each class. Comparing these averages can provide evidence about the relative factors embedded in the current plan.

The comparison is useful only when its interpretation is clear. A class’s average rate can reflect more than its own factor: other rating variables, the mix of risks within the class, and the timing of rate changes can all matter. The basic calculation is straightforward; deciding what the result does and does not reveal is the actuarial work.

Definition: For class $j$, let $EP_j$ be its earned premium over a stated period and $E_j$ its earned exposure on the rating plan’s exposure basis. The observed average premium rate per exposure is $$ \hat{p}_j=\frac{EP_j}{E_j}. $$ For a reference class $b$, the observed premium-rate ratio for class $j$ is $$ \hat{f}_j=\frac{\hat{p}_j}{\hat{p}_b}. $$ This ratio estimates the relative class factor only when the comparison is made on a consistent rate level and the classes are comparable with respect to other factors and exposure mix.

An exposure must use the basis on which the premium is rated. Depending on the product, that could be a vehicle-year, building-year, payroll unit, or another defined unit. A policy count is not automatically an exposure count: policies can have different terms or insured amounts. Premium and exposure also need to refer to the same population and period.

From Earned Premium to an Observed Factor

Suppose a simple plan charges a base rate $b$ per exposure and multiplies it by a class factor $f_j$. If class $j$ has no other rating differences and its premiums are earned at the same rate level, then $$ EP_j=E_j b f_j \quad\text{and therefore}\quad \hat{p}_j=b f_j. $$ If the reference class has factor $f_b$, the ratio of the class rates is $f_j/f_b$. When the reference factor is 1.00, that ratio is the class factor itself. If the reference factor is not 1.00, the ratio is relative to that reference; it is not the absolute factor.

Worked Example: Inferring Factors Relative to a Reference Class

A hypothetical insurer reviews earned premiums for three classes of a small commercial product. Each class is rated on the same annual exposure basis, and all premiums are at the same rate level. Assume no other rating differences affect the comparison. Class Oak is the reference class with factor 1.00. Calculate each observed average rate and the factors implied relative to Oak. The current plan lists Birch at 1.15.

Method: divide earned premium by earned exposure, then divide by the reference-class rate.

For Oak, earned premium is $720,000 and exposure is 1,200: $$ \hat{p}_{Oak}=\frac{\$720{,}000}{1{,}200}=\$600 \quad\text{per exposure}. $$

For Birch, earned premium is $864,000 and exposure is 1,200: $$ \hat{p}_{Birch}=\frac{\$864{,}000}{1{,}200}=\$720. $$ Its observed ratio to Oak is $$ \hat{f}_{Birch}=\frac{\$720}{\$600}=1.20. $$

For Cedar, earned premium is $1,008,000 and exposure is 1,400: $$ \hat{p}_{Cedar}=\frac{\$1{,}008{,}000}{1{,}400}=\$720, \qquad \hat{f}_{Cedar}=\frac{\$720}{\$600}=1.20. $$

The observed ratios imply factors of 1.20 for both Birch and Cedar relative to Oak. Birch’s observed ratio is 1.20, compared with the plan’s listed factor of 1.15. This difference is a signal to investigate, not by itself proof that the filed factor is wrong. Check: applying the implied rates gives Oak $1{,}200\times\$600=\$720{,}000$, Birch $1{,}200\times\$720=\$864{,}000$, and Cedar $1{,}400\times\$720=\$1{,}008{,}000$, matching the stated premiums. The common mistake avoided is comparing earned premiums directly: Birch and Oak have equal exposures here, but that need not be true in other comparisons.

Why Other Rating Factors Can Distort the Ratio

A class-level average rate can include the effects of other factors in the rating plan. This is especially important when the distribution of those other factors differs by class. A useful way to see the issue is to express the premium for an exposure in class $j$ and category $k$ of another rating variable as $b f_j g_k$, where $b$ is the base rate, $f_j$ the class factor, and $g_k$ the other factor.

If the class contains exposures $e_{jk}$ in the different categories, its total earned premium, under a constant rate level, is $$ EP_j=\sum_k e_{jk}b f_j g_k. $$ Dividing by total exposure $E_j=\sum_k e_{jk}$ gives $$ \hat{p}_j=b f_j\left(\frac{\sum_k e_{jk}g_k}{E_j}\right). $$ The expression in parentheses is the exposure-weighted average of the other factor within that class. Thus, the ratio of two observed rates includes both the class-factor ratio and any difference between the classes’ average other factors. If those average factors are the same, they cancel; if not, the observed rate ratio need not equal the class-factor ratio.

Worked Example: A Different Territory Mix Changes the Observed Ratio

A hypothetical insurer uses a base rate of $500 per exposure. Class A has class factor 1.00; Class B has factor 1.20. A separate territory factor is 0.80 in Territory Low and 1.20 in Territory High. Class A has 100 exposures in each territory. Class B has 200 exposures, all in Territory High. Calculate the average rates and compare their ratio with the underlying class-factor ratio.

Method: calculate premium by class and territory before taking the class averages. In Class A, the Territory Low rate is $ \$500\times1.00\times0.80=\$400$, giving $100\times\$400=\$40{,}000$. The Territory High rate is $ \$500\times1.00\times1.20=\$600$, giving $100\times\$600=\$60{,}000$. Class A therefore has total premium of $100,000 and 200 exposures: $$ \hat{p}_A=\frac{\$100{,}000}{200}=\$500. $$

Class B’s rate in Territory High is $ \$500\times1.20\times1.20=\$720$. Its premium is $200\times\$720=\$144{,}000$, so $$ \hat{p}_B=\frac{\$144{,}000}{200}=\$720. $$ The observed ratio is $ \$720/\$500=1.44$. The underlying class-factor ratio is only $1.20/1.00=1.20$. The remaining difference arises because Class B has only the higher territory factor, while Class A’s equal territory mix averages to a territory factor of 1.00. Check: the two class rates equal their respective base rate times class factor times average territory factor: A is $ \$500\times1.00\times1.00=\$500$, and B is $ \$500\times1.20\times1.20=\$720$. The common mistake avoided is treating an aggregate rate ratio as a pure class factor without checking other rating variables.

Rate Timing Must Be Comparable

Earned premiums may reflect more than one rate level. If one class earned a greater share of its exposures after a rate change than another class, their raw premium-per-exposure figures may not be on the same basis. The comparison should use premiums brought to a common rate level, or an appropriate exposure-period calculation that accounts for the rates in force.

For a straightforward rate change, each exposure segment can be repriced to the selected common level. The calculation must use the segment’s actual rate level: a premium earned before an increase is adjusted, while a premium earned after the increase is not adjusted again. This calculation assumes that the exposure information correctly identifies when the exposure earned premium and that the rate change applies uniformly to the indicated business.

Worked Example: Comparing Classes After a Rate Increase

A hypothetical insurer raises rates by 10% effective halfway through a year. The old annual rates are $500 for Class A and $600 for Class B; the new rates are therefore $550 and $660. Class A earns 600 exposures before the change and 300 after it. Class B earns 400 before and 500 after. Calculate the raw rate ratio and the ratio after bringing both classes to the new rate level.

Method: first calculate actual earned premium by period, then reprice the pre-change exposure to the new level. For Class A, the premium earned before the change is $600\times\$500=\$300{,}000$. After the change it is $300\times\$550=\$165{,}000$. Total earned premium is $465,000 on 900 exposures, so its raw average rate is $$ \hat{p}_A=\frac{\$465{,}000}{900}=\$516.67 $$ to the nearest cent.

For Class B, the pre-change premium is $400\times\$600=\$240{,}000$; the post-change premium is $500\times\$660=\$330{,}000$. Total earned premium is $570,000 on 900 exposures: $$ \hat{p}_B=\frac{\$570{,}000}{900}=\$633.33. $$ The raw observed ratio is $ \$633.33/\$516.67$, or approximately 1.226 when calculated from the unrounded rates.

To put the premiums at the new rate level, reprice only the pre-change exposures. Class A’s adjusted premium is $600\times\$550+300\times\$550=\$495{,}000$, giving an adjusted rate of $\$495{,}000/900=\$550$. Class B’s adjusted premium is $400\times\$660+500\times\$660=\$594{,}000$, giving an adjusted rate of $\$594{,}000/900=\$660$. The adjusted ratio is $ \$660/\$550=1.20$, the ratio of the new rates. Check: the adjusted rates agree with the stated current rates of $550 and $660. The common mistake avoided is comparing the raw averages as if both classes had the same share of exposure at the old and new levels.

Derivation and Interpretation

The factor-ratio formula follows directly from the assumed premium structure. With one factor per class, equal rate level, and no differences from other variables, $EP_j=E_jbf_j$, so $$ \hat{p}_j=\frac{EP_j}{E_j} =\frac{E_jbf_j}{E_j} =bf_j. $$ For reference class $b$, the same reasoning gives $\hat{p}_b=bf_b$. Dividing the two equations cancels the common base rate: $$ \frac{\hat{p}_j}{\hat{p}_b} =\frac{bf_j}{bf_b} =\frac{f_j}{f_b}. $$ This is why relative rates can reveal relative factors without knowing the base rate. The result depends on the assumptions: if the rate level or the average effects of other factors differ, the cancellation does not hold.

In practice, an observed ratio is a portfolio statistic, not automatically an isolated measure of a classification effect. Before interpreting it, verify that:

earned premium and earned exposures correspond to the same classes, period, and exposure basis;
the classes are compared at a common rate level, including any relevant rate changes;
other rating variables are controlled for or have comparable distributions; and
premium differences are not caused by different policy terms or other non-class factors included in the records.

These checks help distinguish a genuine difference in the current plan’s class factor from a difference in the business written or earned. If the evidence is not comparable, the ratio is still descriptive of the observed portfolio, but it should not be labeled an estimate of the isolated class factor without qualification.

Common Exam Trap

Common mistake: A class has $900,000 of earned premium and 1,500 exposures, while the reference class has $600,000 and 1,000 exposures. Using the premium ratio gives $900{,}000/600{,}000=1.50$, which is wrong as a rate comparison. The correct rates are $900{,}000/1{,}500=\$600$ and $600{,}000/1{,}000=\$600$, so the observed rate ratio is 1.00. Always divide each class’s premium by its own exposure before taking the ratio.

A second interpretive trap is to report an observed ratio to several decimal places as if it were necessarily the selected factor. The arithmetic may be precise while the comparison remains affected by rate timing, portfolio mix, or other variables. State what the calculation estimates and the conditions needed for that interpretation.

Key takeaway: Earned premium divided by earned exposure estimates a class’s observed average premium rate. The ratio of two such rates estimates the relative existing factor only when the comparison is on a common basis and differences in other rating effects do not distort the result.

Check Your Understanding

For each problem, calculate the requested rate or ratio and note any stated comparison limitation.

A hypothetical class has $528,000 earned premium and 880 earned exposures. Its reference class has an observed rate of $500 per exposure. What is the first class’s observed rate and ratio to the reference? (Answer: $600 per exposure; ratio 1.20.)
Class North has $1,035,000 earned premium on 1,500 exposures; its reference class rate is $600 per exposure. Assuming a common rate level and comparable mix, what factor is implied relative to the reference? (Answer: $690 per exposure; factor 1.15.)
A 10% increase changes two hypothetical class rates from $400 to $440 and from $500 to $550. Class P earns 300 exposures before and 700 after the change; Class Q earns 500 before and 500 after. What are their raw average rates and raw ratio, and what is the current-rate ratio? (Answer: P $428; Q $525; raw ratio approximately 1.2266; current-rate ratio 1.25.)
The base rate is $700. Class A has factor 1.00 and equal exposure in territories with factors 0.90 and 1.10. Class B has class factor 1.10 and all its exposure in the 1.10 territory. What are the observed class rates and their ratio? (Answer: Class A $700; Class B $847; ratio 1.21, not the underlying class-factor ratio of 1.10.)
A class earns $1,200,000 on 2,000 exposures, while a reference earns $960,000 on 1,600 exposures. What is the observed rate ratio? If both classes have the same exposure mix for other factors and share a rate level, what relative factor does the ratio imply? (Answer: both rates are $600 per exposure; ratio 1.00 and implied relative factor 1.00.)

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