Matching Loss, Exposure, and Premium Bases

Why Matching the Bases Comes First

A rate indication compares the losses expected under a proposed coverage and period with the premium available to pay those losses. That comparison is meaningful only if the quantities share compatible bases. Losses from one coverage, premium from another, or exposure measured over a different period can produce a precise-looking but misleading rate change.

In experience rating, earned premium corresponds to the portion of coverage provided during the experience period. It is generally the appropriate premium base for losses arising from that period. Written premium records policies issued or renewed; it need not correspond to the same coverage period as the losses. When rates changed during the experience period, earned premium may also need to be adjusted to a common current rate level. Such premium is called on-level premium.

Definition: A rate indication is based on losses and premium made comparable with respect to (1) the experience or projection period, (2) the coverage terms, and (3) the exposure definition. If $L$ denotes losses adjusted to the intended basis, $P$ denotes earned premium on the corresponding current rate level, and $t$ is the target loss ratio, the indicated rate change is $$ \frac{L}{tP}-1. $$ This form assumes that $t$ is the appropriate target loss ratio for the premium basis being used and that the indicated rate change scales premium without changing the adjusted losses.

The exposure base must also be understood. It might be policy-years, vehicle-years, insured units, or another defined measure. If losses and premium cover the same exposures, dividing both by the same exposure count gives loss per exposure and premium per exposure; the exposure count then cancels in the indication. It does not cancel if the counts refer to different populations or periods.

Building a Comparable Indication

The basic calculation can be derived from the target loss ratio. Suppose the current on-level premium is $P$, adjusted losses are $L$, and the proposed rate change is $r$. If the rate change multiplies premium by $1+r$, the premium after the change is $P(1+r)$. To achieve target loss ratio $t$, require

\frac{L}{P(1+r)}=t.

Solving gives $1+r=L/(tP)$, and hence $r=L/(tP)-1$. The formula therefore compares losses with the amount of premium that would support them at the target loss ratio. Its validity depends on using losses and premium that have first been aligned.

Define the target basis.
Specify the loss period, intended coverage terms, exposure unit, and target loss ratio.

Make losses comparable.
Use the loss measure required by the analysis, such as ultimate losses when that estimate is supplied. Apply any stated period projection or coverage adjustment to the losses, not just to premium.

Make premium comparable.
Use earned premium for the matching experience and adjust it to a common rate level when necessary. Apply any specified coverage conversion consistently.

Calculate and check.
Compare adjusted losses with $t$ times on-level premium. A result above 1 indicates an increase; a result below 1 indicates a decrease.

A period projection and a rate-level adjustment are different operations. A loss trend factor, if supplied, projects losses between periods; an on-level factor adjusts premium for rate changes. Do not apply one factor to the other quantity merely because both appear in the same problem. Likewise, a coverage change requires its own stated loss or premium adjustment; it is not automatically represented by a trend factor.

Worked Example: Converting Both Losses and Premium to Proposed Coverage

A hypothetical portfolio has 600 policy-years of experience. On its existing deductible basis, losses are $270,000 and earned premium at the current rate level is $450,000. For the same exposures, a proposed deductible is estimated to increase losses by $18,000 and current-rate-level premium by $30,000. The target loss ratio on the proposed basis is 0.60. Calculate the indicated rate change on the proposed coverage basis.

Method. Convert both losses and premium to the proposed deductible basis, then compare the resulting losses with the target loss dollars. The stated adjustments apply to the same 600 policy-years.

Proposed-basis losses are

$$ L=270{,}000+18{,}000=288{,}000. $$

Proposed-basis current-rate-level premium is

$$ P=450{,}000+30{,}000=480{,}000. $$

At a target loss ratio of 0.60, the premium supports target loss dollars of

$$ tP=0.60(480{,}000)=288{,}000. $$

Therefore the indicated rate factor is

\frac{L}{tP}=\frac{288{,}000}{288{,}000}=1, \qquad r=1-1=0.

The indicated change is 0.00%. As a check, the proposed-basis loss ratio is $288{,}000/480{,}000=0.60$, exactly the target. The common mistake this method avoids is adjusting losses to the proposed deductible but leaving premium on the existing deductible basis.

Aligning the Experience and Projection Periods

When the experience losses and premium belong to an earlier period but the indication concerns a later period, first determine which quantities the problem says to project. For example, if losses are projected forward by a stated trend factor while on-level premium is already stated at the current rate level, apply the trend factor to losses and do not apply it again to premium. If the exposure count changes, distinguish a change in the number of exposures from a change in the cost or premium per exposure.

The following example assumes the experience is otherwise comparable: no coverage adjustment is needed, all 1,200 policy-years remain representative of the prospective exposure basis, and the supplied trend applies to losses only.

Worked Example: Projecting Losses Without Re-Trending Premium

A hypothetical program has 1,200 policy-years of experience. Estimated ultimate losses for the experience period are $468,000. The earned premium has already been adjusted to the current rate level and is $720,000. Losses are to be projected forward two years using a 5% annual trend. The target loss ratio is 0.65. Find the indicated rate change, assuming no change in exposure count or coverage.

Method. Project the losses for two years, retain the already on-level premium as the premium base, and calculate the target loss dollars.

The two-year loss trend factor is

$$ (1.05)^2=1.1025. $$

Projected losses are

$$ L=468{,}000(1.1025)=515{,}970. $$

The target loss dollars supported by the on-level premium are

$$ tP=0.65(720{,}000)=468{,}000. $$

Thus the indicated rate factor and change are

\frac{L}{tP}=\frac{515{,}970}{468{,}000}=1.1025, \qquad r=1.1025-1=0.1025.

The indicated change is a 10.25% increase. To check using policy-years, current-rate-level premium per policy-year is $720{,}000/1{,}200=600$. Target loss dollars per policy-year are $0.65(600)=390$. Projected losses per policy-year are $515{,}970/1{,}200=429.975$, and $429.975/390-1=0.1025$. The exposure count cancels because both totals cover the same 1,200 policy-years. The common mistake avoided is trending the premium a second time even though it is already on the stated rate level.

Combining Segments Without Averaging Their Indications

When experience is divided into segments, the overall indication should be calculated from the aggregate adjusted losses and aggregate on-level premium, provided the segments share a common target loss ratio and compatible coverage and period bases. An unweighted average of segment rate changes gives each segment equal influence regardless of its premium volume. That is generally not the combined indication.

Worked Example: Aggregating Two Exposure Segments

A hypothetical insurer reviews two groups, each measured over the same experience period and on the same coverage basis. Group A has 1,000 policy-years, $300,000 of adjusted losses, and $500,000 of on-level earned premium. Group B has 800 policy-years, $280,000 of adjusted losses, and $400,000 of on-level earned premium. The common target loss ratio is 0.65. Calculate the combined indication and explain why averaging the group indications is not appropriate.

Method. Add losses and premium first, then calculate the indication using the combined totals.

Total exposure is $1{,}000+800=1{,}800$ policy-years. Total losses and premium are

L=300{,}000+280{,}000=580{,}000, \qquad P=500{,}000+400{,}000=900{,}000.

Target loss dollars are

$$ tP=0.65(900{,}000)=585{,}000. $$

The combined factor is $580{,}000/585{,}000=0.99145299\ldots$, so the indicated change is $-0.00854701\ldots$, or a 0.85% decrease to two decimal places.

For comparison, Group A's loss ratio is $300{,}000/500{,}000=0.60$, giving an indication of $0.60/0.65-1=-0.076923\ldots$, or a 7.69% decrease. Group B's loss ratio is $280{,}000/400{,}000=0.70$, giving $0.70/0.65-1=0.076923\ldots$, or a 7.69% increase. Their unweighted average is 0%, not the combined indication. The correct result weights the experience through aggregate premium: Group A contributes $500,000 of premium and Group B contributes $400,000. As a check, combined losses per policy-year are $580{,}000/1{,}800=322.222\ldots$, and premium per policy-year is $900{,}000/1{,}800=500$. The target loss amount per policy-year is $0.65(500)=325$, again giving $322.222\ldots/325-1=-0.008547\ldots$. This avoids the common mistake of averaging segment indications as if the segments had equal premium volume.

Common Basis Errors and Their Corrections

Using written premium automatically. Written premium may cover a different period from the losses. Use the premium measure specified for the experience analysis, usually earned premium, and put it on the appropriate rate level.
Changing only one side of the ratio. If a coverage change affects both expected losses and premium, apply the stated adjustments to both before calculating the indication.
Mixing historical and prospective periods. Project only the quantities and time interval specified. Confirm whether the premium is already on-level and whether a loss trend is intended.
Combining incompatible exposure counts. A loss total for 1,200 policy-years cannot be directly compared with premium for 1,000 policy-years unless the mismatch is resolved or the quantities are converted to a common basis.
Averaging indications instead of aggregating experience. With a common target, add the compatible losses and premiums, then calculate the combined result.
Misreading the direction of the ratio. If $L/(tP)>1$, adjusted losses exceed target loss dollars and the indication is an increase. If it is below 1, the indication is a decrease.

Key takeaway: Before calculating a rate change, label each loss, premium, and exposure quantity by period, coverage basis, and unit. Once those labels match, use $L/(tP)-1$; before they match, the formula cannot repair the inconsistency.

Check Your Understanding

For each problem, first check that the losses, premium, period, and coverage basis are comparable.

A hypothetical book has adjusted losses of $198,000 and on-level earned premium of $330,000 on the same coverage and period. The target loss ratio is 0.60. What is the indicated rate change? (Answer: 0.00%.)
A hypothetical portfolio has $240,000 of losses, which must be trended forward three years at 4% annually. On-level premium is $400,000, the target loss ratio is 0.65, and no exposure or coverage change is assumed. What is the indicated change, rounded to two decimal places? (Answer: 3.83% increase.)
Two compatible segments have adjusted losses and on-level premiums of $140,000 and $250,000, and $210,000 and $350,000, respectively. With a common target loss ratio of 0.60, calculate the combined indication. (Answer: 2.78% decrease.)
A hypothetical program has $96,000 of adjusted losses and $180,000 of on-level premium, both for 2,400 vehicle-months. The target loss ratio is 0.65. Verify the indication using both totals and per-vehicle-month amounts. (Answer: 17.95% decrease.)
On an existing coverage basis, losses are $420,000 and on-level premium is $700,000. For the same exposures, a proposed coverage increases losses by $35,000 and premium by $20,000. If the proposed-basis target loss ratio is 0.60, what is the indicated rate change? (Answer: 5.32% increase.)

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