Worked Problems on Premium On-Leveling

Why Premium Must Be On-Leveled

A loss ratio compares losses with the premium collected for the exposures that produced those losses. If rates changed during the experience period, the collected premium reflects more than one rate level. To interpret the experience at a common rate level, adjust the earned premium to that level before calculating the adjusted actual loss ratio.

Premium on-leveling changes the premium denominator; by itself, it does not trend or develop losses. The result depends on how much premium was earned at each rate level, not merely on the dates when rate changes were announced or policies were written. The calculations below assume hypothetical insurance portfolios.

Definition: Let a rate-level index equal 1.00 at a chosen base level. Let $R_c$ be the index for the selected current rate level, and let $\bar{R}$ be the average rate-level index applicable to earned premium during an experience period. The on-level factor is $R_c/\bar{R}$, and on-level earned premium is actual earned premium multiplied by that factor. The adjusted actual loss ratio is losses on the selected loss basis divided by on-level earned premium.

Build the Average Rate Level From Earned Premium

When rate levels change within an experience period, first divide the period into intervals over which the rate level is constant. If earned exposure is uniform over time, each interval's weight is its share of the period. More generally, use each interval's share of earned exposure or premium at the base level, if that distribution is supplied. The weighted average rate level is

\bar{R}=\sum_j w_j R_j,\qquad \sum_j w_j=1, \qquad \text{On-level earned premium} = \text{Actual earned premium}\times\frac{R_c}{\bar{R}}.

Here $R_j$ is the rate-level index for interval $j$, and $w_j$ is that interval's share of the relevant earned exposure. Uniform earning over calendar months gives equal weights for equal-length intervals. A rate change effective halfway through a year therefore does not mean that all annual earned premium is at the new level.

The standard calculation assumes the rate-level index applies proportionally to the premium being adjusted and that the exposure mix is suitable for this aggregate adjustment. If fixed charges, fees, or other amounts do not vary with rates, they should not automatically be multiplied by the same factor. If the exposure distribution is not uniform, use the actual earning pattern rather than equal calendar-time weights.

Why the On-Level Factor Works

The formula follows by comparing premium for the same exposures at two rate levels. Suppose the exposures would produce base-level premium $P_0$. Under the historical pattern, the premium is $P_0\bar{R}$: each portion of exposure contributes its base premium multiplied by its applicable rate index, and those contributions add to the weighted average. At the selected current level, those same exposures would produce $P_0R_c$. Thus,

\frac{\text{Premium at current level}}{\text{Historical earned premium}} = \frac{P_0R_c}{P_0\bar{R}} = \frac{R_c}{\bar{R}}.

Multiplying actual earned premium by this ratio puts the premium for the same exposures at the selected level. Keeping the chosen losses unchanged, the adjusted actual loss ratio is

\text{Adjusted actual loss ratio} = \frac{\text{Losses on the selected basis}}{\text{On-level earned premium}}.

This ratio is not a complete rate indication by itself. For example, it does not make a separate adjustment for loss trend or development. Those adjustments must be addressed on their own basis when required.

Set the target level.
Identify the selected current rate-level index and the period of earned premium being adjusted.

Map the earned pattern.
List rate levels by the intervals in which they apply to earned exposure. Use the supplied exposure weights, or uniform time weights only when uniform earning is assumed.

Calculate the weighted level.
Multiply each interval's rate index by its weight and add the products to obtain $\bar{R}$.

Adjust premium and calculate the ratio.
Multiply actual earned premium by $R_c/\bar{R}$. Divide the specified losses by the adjusted premium, and check that the result moves in the expected direction.

Worked Applications

Worked Example: A Midyear Rate Increase

A hypothetical insurer earned $600,000 of premium during a calendar year. A 10% rate increase became effective July 1. Assume exposure is earned uniformly by month, the rate level was 1.00 for January through June and 1.10 for July through December, and the selected current level is 1.10. Losses on the chosen basis are $420,000. Calculate on-level earned premium and the adjusted actual loss ratio.

Method and assumption. Use time-weighted average rate levels because earning is uniform by month. The first and second half-years each receive weight $6/12=0.50$.

Average historical level.

\bar{R}=(0.50)(1.00)+(0.50)(1.10)=0.50+0.55=1.05.

On-level factor. The target is 1.10, so the factor is $1.10/1.05=1.047619\ldots$. Thus

\$600{,}000\times\frac{1.10}{1.05} =\$600{,}000\times1.047619\ldots =\$628{,}571.43.

Adjusted actual loss ratio.

\frac{\$420{,}000}{\$628{,}571.43} =0.6681818\ldots =66.82\%.

The unadjusted ratio is $\$420{,}000/\$600{,}000=70.00\%$, so the adjusted ratio is lower because the on-level premium is higher. The common mistake avoided here is applying the full 10% increase to all annual premium; half the premium was earned before the increase. Money is rounded to the cent and the loss ratio to two decimal places.

Worked Example: Two Changes Within One Year

A hypothetical portfolio earned $1,000,000 during a calendar year. Rates were at level 1.00 in January through March, increased to 1.10 on April 1, and then decreased by 5% from that level on October 1. Assume uniform monthly earning. Losses on the selected basis are $650,000, and the target is the level after the October change. Find on-level earned premium and the adjusted actual loss ratio.

Method and assumption. The second change is a 5% decrease applied to the then-current level, so the final index is $1.10(0.95)=1.045$, not 1.05. The interval weights are $3/12$, $6/12$, and $3/12$.

Average historical level.

\bar{R} =\frac{3}{12}(1.00)+\frac{6}{12}(1.10)+\frac{3}{12}(1.045) =0.25+0.55+0.26125 =1.06125.

On-level factor and premium.

\frac{R_c}{\bar{R}} =\frac{1.045}{1.06125} =0.984687868\ldots, \qquad \$1{,}000{,}000\times0.984687868\ldots =\$984{,}687.87.

Adjusted actual loss ratio.

\frac{\$650{,}000}{\$984{,}687.87} =0.6601076\ldots =66.01\%.

The result is below the unadjusted ratio of 65.00%? Check the comparison: $\$650{,}000/\$1{,}000{,}000=65.00\%$, which is lower than 66.01%. This is the expected direction: the selected level, 1.045, is below the experience-period average, 1.06125, so premium is adjusted downward and the ratio rises. The common mistake avoided is treating the two percentage changes as additive or applying the October decrease to the original level. Money is rounded to the cent and the loss ratio to two decimal places.

Worked Example: Combining Two Experience Periods

A hypothetical insurer has two nonoverlapping experience periods. In period A, actual earned premium is $400,000 and the average earned rate-level index is 0.90. In period B, actual earned premium is $700,000 and the average index is 1.02. The selected target index is 1.05 for both periods. Losses are $280,000 in period A and $460,000 in period B. Calculate the combined adjusted actual loss ratio.

Method. On-level each period's premium separately because the periods have different average rate levels. Then add the adjusted premiums and losses. This preserves the premium weighting across periods.

Period A. Its on-level factor is $1.05/0.90=1.1666667\ldots$, giving

\$400{,}000\times\frac{1.05}{0.90} =\$466{,}666.67.

Period B. Its on-level factor is $1.05/1.02=1.0294118\ldots$, giving

\$700{,}000\times\frac{1.05}{1.02} =\$720{,}588.24.

Combined ratio. Add premiums and losses on their corresponding bases:

\text{On-level premium} =\$466{,}666.67+\$720{,}588.24 =\$1{,}187{,}254.91, $$ $$ \text{Losses}=\$280{,}000+\$460{,}000=\$740{,}000, \qquad \frac{\$740{,}000}{\$1{,}187{,}254.91} =0.6232866\ldots =62.33\%.

The unadjusted aggregate ratio is $\$740{,}000/\$1{,}100{,}000=67.27\%$. The adjustment lowers it because both periods are brought to a target index higher than their respective averages. The common mistake avoided is taking a simple average of the two period loss ratios; the combined ratio must use aggregate losses divided by aggregate on-level premium. Money is rounded to the cent and the ratios to two decimal places.

Common Direction Checks and Exam Traps

The comparison between $R_c$ and $\bar{R}$ provides a fast arithmetic check. If $R_c>\bar{R}$, on-level premium increases and, with losses fixed, the adjusted actual loss ratio decreases. If $R_c<\bar{R}$, on-level premium decreases and the ratio increases. If the two levels are equal, premium is unchanged.

A frequent wrong calculation is to apply a rate increase directly to all premium earned during a period when the change applied only to part of that period. For the first example, multiplying $600,000 by 1.10 would give $660,000. That is not the premium at the current level for the same exposure: the original premium averages level 1.05, so the correct calculation is $\$600{,}000(1.10/1.05)=\$628{,}571.43$. The denominator of the adjustment matters.

Use effective dates and earning weights. A policy's written date alone does not determine the rate level on all premium earned from it.
Compound sequential changes. A decrease or increase stated as a percentage of the current level applies to that current level, not necessarily to the original index.
Keep the loss basis consistent. On-leveling premium does not by itself adjust losses for trend, development, or other changes.
Combine periods with totals. Divide summed losses by summed on-level premium rather than taking an unweighted average of period ratios.

Key takeaway: Determine the rate level applicable to each portion of earned exposure, calculate its weighted average, and multiply earned premium by the target-to-average rate-level ratio. Then calculate the adjusted actual loss ratio using losses on the stated basis.

Check Your Understanding

For each hypothetical problem, calculate the requested quantity using the stated earning assumptions.

A portfolio earned $800,000 uniformly during a year. The index was 1.00 for the first nine months and 1.08 for the final three months. The target index is 1.08. What is on-level earned premium? (Answer: $847,058.82)
A portfolio earned $540,000 uniformly over a year, half at index 1.20 and half at index 1.14. The target is 1.14. What is on-level earned premium? (Answer: $526,153.85)
A portfolio earned $300,000 uniformly by quarter. The quarterly indices were 1.00, 1.04, 1.04, and 1.092, reflecting a 4% increase effective April 1 and a further 5% increase effective October 1. With target index 1.092, what is on-level earned premium? (Answer: $314,093.96)
Two periods have earned premiums of $500,000 and $200,000, average rate indices of 1.10 and 0.95, respectively, and losses totaling $400,000. If both are adjusted to target index 1.045, what is the combined adjusted actual loss ratio? (Answer: 57.55%)
A hypothetical insurer earned $900,000 uniformly during a year; a 12% increase took effect halfway through the year. Losses are $630,000, and the target is the post-increase level. What are on-level earned premium and the adjusted actual loss ratio? (Answer: $950,943.40 and 66.25%)

What You'll Learn