Why On-Level Historical Premiums
Historical premiums reflect the rates in force when the associated exposures were insured. If rates changed during the experience period, directly adding those premiums does not show what the same exposures would have produced at today’s rates. On-leveling adjusts each historical earned premium to a common, current rate level so the amounts can be compared on that basis.
This adjustment is about the premium rate applied to a given exposure, not about changes in the exposure itself. The method assumes that the relevant rate changes apply uniformly to the premiums being adjusted and that coverage, exposure characteristics, and other premium components are otherwise held constant. It does not, by itself, adjust for a change in exposure mix, loss costs, or the period in which losses occur.
An index can be set to 1.000 at a chosen base level. Its absolute value is only a reference: the ratio of levels determines the adjustment. If rates increase by 10%, the new level is 1.10 times the old level. A later 5% decrease produces a level of \(1.10(0.95)=1.045\) times the original level—not 1.05 and not 1.10 minus 0.05.
Assigning Rate Levels to Earned Premium
The key timing question is: which rates applied to the exposure as it earned premium? First divide the experience into periods with a constant rate level. Then adjust each period’s earned premium separately. A rate change effective on a particular date applies from that date onward; it does not change the level for earlier dates.
The earned premium for a segment is the amount associated with the exposures earned during that segment. It is not automatically the written premium for policies incepting then: a policy can earn premium across multiple rate-level periods. When a data source reports a single earned-premium amount spanning more than one level, the amount must be allocated or otherwise restated by level before applying the simple factor.
Here \(j\) indexes periods or segments, \(P_j\) is earned premium in segment \(j\), \(L_j\) is the rate level for that segment, and \(L_c\) is the level selected as current. If a segment is already at the current level, its factor is 1. The adjustment does not change the exposure count; it expresses the segment’s premium at the selected rate level.
Why the Factor Is a Ratio
Suppose an exposure would produce premium \(B\) at the base rate level 1. If the rate level is \(L_j\), its premium is \(B L_j\). To express that same exposure at current level \(L_c\), its premium would be \(B L_c\). Taking the ratio cancels the unknown base premium \(B\):
Thus multiplying the historical premium by \(L_c/L_j\) changes the rate basis while preserving the exposure represented by that premium. The argument requires premium to be proportional to the applicable rate level for the business being adjusted. If a change affects only some coverages or classifications, the adjustment should be applied only to the corresponding premium, using the applicable level for that component.
Worked Applications
Worked Example: Quarterly Premiums Around Two Rate Changes
A hypothetical insurer uses an index of 1.000 at the start of the year. A 10% increase takes effect April 1, and a 5% decrease from the then-current rates takes effect October 1. The current level is therefore the level in force from October 1 onward. The earned premiums are: Q1, $100,000; Q2, $132,000; Q3, $125,400; and Q4, $156,750. Find the total on-level premium at the current level. Treat the reported quarterly premiums as earned in their stated quarters, with no within-quarter rate change. Interest and mortality assumptions do not apply; these are deterministic premium amounts, not a probability model.
Method: determine each quarter’s level, then multiply by the current-to-historical level ratio. The April 1 increase makes the Q2 and Q3 level \(1.000(1.10)=1.100\). The October 1 decrease applies only to Q4, giving \(1.100(0.95)=1.045\). Therefore the current level is 1.045.
Q1: Its level is 1.000, so the factor is \(1.045/1.000=1.045\). The on-level premium is \(100{,}000(1.045)=\$104{,}500\).
Q2: Its level is 1.100, so the factor is \(1.045/1.100=0.95\). The on-level premium is \(132{,}000(0.95)=\$125{,}400\).
Q3: It remains at level 1.100 because the October 1 decrease has not yet taken effect. Its factor is also 0.95, and its on-level premium is \(125{,}400(0.95)=\$119{,}130\).
Q4: It is already at the current level 1.045. Its factor is \(1.045/1.045=1\), so its on-level premium is \(\$156{,}750\).
The total is \(\$104{,}500+\$125{,}400+\$119{,}130+\$156{,}750=\$505{,}780\). The calculation is in whole dollars, so no rounding is required. As a check on direction, Q1 is adjusted upward because its level is below current; Q2 and Q3 are adjusted downward because their level is above current; Q4 is unchanged.
Common mistake avoided: Applying the October decrease to Q3 would incorrectly assign it level 1.045 and leave its premium unchanged. The effective date is October 1, so Q3 is still at 1.100 and must be adjusted by 0.95.
Worked Example: Compounding Several Changes
A hypothetical specialty policy portfolio has a base rate level of 1.000. Rates increase 8% on July 1, increase another 5% on January 1, and decrease 4% on April 1. The current level is the level after April 1. Earned premiums by level are $240,000 at 1.000, $324,000 at the level after the July increase, $283,500 at the level after the January increase, and $217,728 at the current level. Calculate total premium on-level to the current level. There are no interest or mortality assumptions; no random loss distribution is specified or needed.
Method: compound each percentage change against the level then in force. After July 1, the index is \(1.000(1.08)=1.080\). After January 1, it is \(1.080(1.05)=1.134\). After the April decrease, the current index is \(1.134(0.96)=1.08864\).
For the base-level segment, the factor is \(1.08864/1.000=1.08864\), so the adjusted premium is \(240{,}000(1.08864)=\$261{,}273.60\).
For the segment at level 1.080, the factor is \(1.08864/1.080=1.008\), so the adjusted premium is \(324{,}000(1.008)=\$326{,}592.00\).
For the segment at level 1.134, the factor is \(1.08864/1.134=0.96\), so the adjusted premium is \(283{,}500(0.96)=\$272{,}160.00\).
The final segment is already at level 1.08864, so its factor is 1 and its adjusted premium remains \(\$217{,}728.00\). The total is \(\$261{,}273.60+\$326{,}592.00+\$272{,}160.00+\$217{,}728.00=\$1{,}077{,}753.60\). All results are reported to the cent. The factors have the expected directions: the base and 1.080 segments increase, the 1.134 segment decreases, and the current-level segment is unchanged.
Common mistake avoided: Adding the percentage changes gives \(8\%+5\%-4\%=9\%\), which is not the cumulative change. The changes apply successively to different prevailing levels; the correct current index is \(1.08(1.05)(0.96)=1.08864\).
Worked Example: A Rate Decrease Applied to the New Level
A hypothetical insurer begins a year at level 1.000, raises rates by 10% effective May 1, and decreases rates by 10% effective November 1. The earned premiums are $80,000 for January through April, $165,000 for May through October, and $59,400 for November through December. Find total premium at the current level. Assume each amount is earned entirely during its stated interval and that the changes apply uniformly. Interest, mortality, and a stochastic distribution are not relevant to this deterministic calculation.
Method: calculate the level at each effective date and adjust each interval separately. The May increase gives level \(1.000(1.10)=1.100\). The November decrease applies to that increased level, giving current level \(1.100(0.90)=0.990\).
January through April is at level 1.000. Its factor is \(0.990/1.000=0.99\), and its adjusted premium is \(80{,}000(0.99)=\$79{,}200\).
May through October is at level 1.100. Its factor is \(0.990/1.100=0.90\), and its adjusted premium is \(165{,}000(0.90)=\$148{,}500\).
November through December is already at level 0.990. Its factor is 1, and its adjusted premium remains \(\$59{,}400\). The total is \(\$79{,}200+\$148{,}500+\$59{,}400=\$287{,}100\), with no rounding required.
Common mistake avoided: Treating a 10% increase followed by a 10% decrease as returning to level 1.000 misses the base of the second percentage. The decrease is taken from 1.100, so the current level is 0.990. Consequently, the middle-period factor is 0.90, not \(0.990/1.000\).
Practical Checks and Common Traps
Before summing adjusted premiums, check both the timing and the arithmetic. A rate change’s effective date determines the first segment at the new level. Each later change compounds on the level then in force. The adjustment factor must be current level divided by historical level, not the reverse.
- Reversing the factor: \(L_j/L_c\) adjusts in the wrong direction. For example, if historical level is 1.10 and current level is 1.045, the correct factor is \(1.045/1.10=0.95\), not \(1.10/1.045\).
- Using the wrong effective period: A change effective October 1 applies to fourth-quarter earnings, not to third-quarter earnings.
- Adding percentage changes: Successive changes multiply. A 10% increase followed by a 5% decrease produces \(1.10(0.95)=1.045\), a net increase of 4.5%, not 5%.
- Adjusting an already-current amount: Its factor is 1; applying another increase would double-count the change.
- Mixing rate leveling with other adjustments: On-leveling changes the rate basis only. It does not independently correct for a different portfolio mix, coverage, or exposure base.
A useful final reasonableness check is to compare each segment’s historical and current levels. If current is higher, its premium should increase; if current is lower, it should decrease. A segment at the current level should stay exactly the same. These checks often reveal an incorrect level assignment before it affects the total.
Check Your Understanding
For each hypothetical portfolio, find the requested on-level premium. Assume each stated premium is earned entirely at its stated rate level, rate changes apply uniformly, and no other adjustments are required. Interest, mortality, and a probability distribution are not involved.
- A portfolio earned $75,000 at level 1.000. The current level is 1.060. What is the on-level premium? (Answer: $79,500.00)
- Rates start at 1.000, increase 5%, then increase a further 10%. Earned premiums are $200,000 at level 1.000 and $100,000 at level 1.050. Find the total at the final level. (Answer: $341,000.00)
- A 8% increase takes effect, followed later by a 5% decrease. Earned premiums are $50,000 at level 1.000 and $80,000 at level 1.080. Find the total at the current level. (Answer: $127,300.00)
- Rates increase 6%, then increase 4%, then decrease 5%. Premiums are $40,000 at level 1.060, $25,000 at level 1.1024, and $30,000 at the current level. Find the total on-level premium. (Answer: $93,270.00)
- A 10% increase takes effect April 1 and a 5% decrease from the new level takes effect October 1. Earned premiums are $60,000 in Q1, $88,000 in Q2, $99,000 in Q3, and $104,500 in Q4. Find the total on-level to Q4’s current level. (Answer: $344,850.00)