Why Loss Adjustment Expense Belongs in the Pure Premium
A pure premium measures the expected claim cost per exposure. If the cost of handling claims is part of the insurer’s expected cost of providing coverage, a pure premium intended to represent that cost should include loss adjustment expense (LAE) as well as claim losses. The key is consistency: first establish what the experience loss data contain, then include only the LAE that is missing from those data.
This is not a separate expense provision for all insurer expenses. LAE is tied to investigating, adjusting, and settling claims. Acquisition costs and general administrative expenses are not LAE merely because they are expenses of operating an insurance business. Including them in this calculation would change the measure being calculated.
LAE is often classified as allocated loss adjustment expense (ALAE) or unallocated loss adjustment expense (ULAE). ALAE can be tied to particular claims, such as a claim-specific legal fee or an investigation cost. ULAE is not assigned to one particular claim in the same way, such as the cost of general claim-handling staff. An insurer’s data definitions and accounting treatment determine which amounts appear in a particular loss extract; the labels alone do not establish whether an expense has already been included.
A pure premium can use ultimate, incurred, or paid amounts, provided the selected amounts are appropriate for the analysis and the loss and expense components are placed on compatible bases. In the calculations below, each problem states the basis being used. Exposure means the chosen exposure unit for that analysis, not premium.
Building a Consistent Loss-and-Expense Numerator
Before dividing by exposures, identify the components in the selected experience data. A useful working reconciliation is to separate claim losses, ALAE, and ULAE, and mark whether each component is already present in the selected loss amount. This avoids two opposite errors: omitting an expense that belongs in the intended cost measure, and adding an expense a second time because it was already included.
Specify whether the desired numerator is losses alone, losses plus ALAE, or losses plus both ALAE and ULAE.
Determine which amounts are already included in the selected loss or claim-cost figures. Do not infer inclusion from a column heading alone.
Use an expense amount or an explicitly defined allocation method for any missing LAE. Keep its denominator clear if the allocation is a percentage.
Use exposures corresponding to the same coverage and experience scope as the assembled numerator.
If the problem specifies a trend factor, apply it to the relevant experience cost before interpreting the result as a projected pure premium.
Worked Calculations
Worked Example: Trending Losses and Both LAE Components
A hypothetical book has 12,000 earned exposures. The selected ultimate loss data are $840,000, excluding LAE. The matching ultimate ALAE is $96,000 and ULAE is $54,000; neither expense amount is included in the loss figure. Assume the target is a pure premium including both LAE components, all amounts are on the same coverage and experience basis, and no discounting is applied. The indicated projection uses a supplied combined trend factor of 1.06. Calculate the projected pure premium. Round only the final result to the nearest cent.
Method. Add the missing expenses to losses, divide by earned exposures to obtain the experience pure premium, and apply the stated projection factor.
First assemble the historical numerator:
The untrended cost per exposure is:
As a component check, losses contribute \(840{,}000/12{,}000=70.00\) per exposure, ALAE contributes \(96{,}000/12{,}000=8.00\), and ULAE contributes \(54{,}000/12{,}000=4.50\). Their sum is \(70.00+8.00+4.50=82.50\).
Apply the supplied factor to the assembled cost:
The projected pure premium including both LAE components is $87.45 per exposure. Equivalently, the projected components are \(70.00(1.06)=74.20\), \(8.00(1.06)=8.48\), and \(4.50(1.06)=4.77\); they sum to $87.45.
Check and trap avoided. The projected result is above the untrended $82.50 because the factor exceeds 1. A common mistake is to calculate \(840{,}000/12{,}000=70.00\) and stop, omitting both expenses. Another is to add an expense amount that is already contained in the loss figure; here the problem explicitly says neither component is included.
Worked Example: Adding ULAE on a Stated Allocation Basis
A hypothetical liability program has 10,000 earned exposures. Its selected ultimate losses exclude LAE and total $800,000. Matching ALAE totals $120,000. ULAE is not separately observed in the extract, so the analyst’s stated assumption is to estimate it as 10% of losses plus ALAE. The percentage is applied to that combined subtotal, not to losses alone. Assume all amounts share the same basis, there is no trend or discounting, and calculate the pure premium including both LAE components. Round only the final result to the nearest cent.
Method. Estimate the missing ULAE using the specified base, add it to losses and ALAE, and divide by exposures.
The subtotal to which the ULAE percentage applies is:
The ULAE estimate is:
Thus the total numerator is:
Dividing by 10,000 exposures gives:
The pure premium including LAE is $101.20 per exposure. A component check gives $80.00 of losses, $12.00 of ALAE, and $9.20 of estimated ULAE per exposure, totaling $101.20.
Check and trap avoided. The estimated ULAE is 10% of $920,000, as specified. Applying 10% only to the $800,000 of losses would give $80,000 and a different answer; that calculation changes the stated allocation base.
Worked Example: Reconciling Two Data Extracts Without Double Counting
For a hypothetical product, an analyst is given 5,000 earned exposures and two extracts describing the same experience. The claim-system extract reports $300,000 of losses plus $30,000 of ALAE, but excludes ULAE. A reconciled accounting extract reports $369,600 of losses and claim adjustment expense combined, including ULAE. The accounting extract is consistent with the claim-system amounts plus ULAE estimated at 12% of losses and ALAE. Assume both extracts cover the same experience and no trend or discounting is required. Find the pure premium including all three components. Round only the final result to the nearest cent.
Method. Reconcile the components once, then use the complete combined amount. The extracts are alternative views of the same experience, not separate amounts to add together.
From the claim-system extract, losses and ALAE total:
The stated ULAE allocation is:
The reconciled total is therefore:
which agrees with the accounting extract. Divide that total by the matching exposures:
The pure premium including all LAE is $73.92 per exposure. The component check is $60.00 of losses, $6.00 of ALAE, and $7.92 of ULAE per exposure, again totaling $73.92.
Check and trap avoided. Adding the claim-system total of $330,000 to the accounting total of $369,600 would count the same losses and ALAE twice. The accounting total already includes the ULAE reconciliation and is not an additional experience period.
Why the Formula Works
The calculation follows from the meaning of average cost per exposure. Suppose the target claim-related cost consists of a loss amount \(L\) and a missing LAE amount \(A\). Their combined total is \(L+A\). If that total is attributed to \(N\) exposures on the selected basis, average cost per exposure is the total divided by the number of exposures. This is exactly the definition of the pure premium for that experience.
The last equality follows from division over addition. It shows that, when the same exposure denominator and compatible basis apply, the combined pure premium can also be checked by adding the loss cost per exposure and the LAE cost per exposure. No independence assumption between losses and LAE is needed for this arithmetic identity. The substantive requirements are that the amounts are correctly measured, not duplicated, and matched to the exposures.
If different components use different bases, periods, coverage definitions, or exposure populations, simply adding their ratios may not represent the intended cost per exposure. First place the components on the basis required for the comparison. Similarly, a percentage-based expense estimate is meaningful only when its percentage and its application base are explicitly understood.
Interpretation and Common Pitfalls
Including LAE changes the target from loss cost alone to claim-related cost including the selected adjustment expenses. It does not, by itself, determine whether an indicated premium is adequate or what other expense provisions belong in a rate. Those questions require their own assumptions and calculations. For this topic, the central discipline is to make the numerator agree with the intended cost measure.
- Check the data definition. “Losses,” “incurred losses,” and “claim costs” may include different expense components in different extracts. Confirm the actual definition.
- Do not add an included component again. If the chosen amount already includes ALAE, adding the same ALAE from a separate report overstates the numerator.
- Do not treat missing expenses as zero without support. If the target includes ULAE but the experience data omit it, the calculation needs an appropriate estimate or an explicit decision to exclude it.
- Keep percentage bases straight. A stated percentage of losses plus ALAE is not the same calculation as that percentage of losses alone.
- Match exposures to the costs. The denominator must represent the same experience scope and exposure unit as the numerator; premium is not substituted for exposure in this pure premium calculation.
These checks are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus: identify the cost components, assemble a consistent numerator, and calculate cost per matching exposure. In any practical application, document both the selected expense definition and the source of the expense estimate so another reviewer can reproduce the calculation.
Check Your Understanding
For each problem, identify what is already included before calculating. Amounts are hypothetical, and round money per exposure to the nearest cent where needed.
- A program has 6,000 earned exposures, $390,000 of ultimate losses excluding LAE, and $54,000 of matching LAE, all on the same basis. Calculate the pure premium including LAE. (Answer: $74.00 per exposure)
- A hypothetical book has 9,000 exposures, $630,000 of losses, $72,000 of ALAE, and $45,000 of ULAE. All three amounts are separate and ultimate. Calculate the pure premium including both LAE components. (Answer: $83.00 per exposure)
- A data extract for 12,000 exposures contains $960,000 of losses and $144,000 of ALAE, but omits ULAE. Estimate ULAE as 8% of losses plus ALAE. Find the pure premium including all three components. (Answer: $99.36 per exposure)
- A report for 5,000 exposures contains $300,000 of losses and $30,000 of ALAE, but excludes ULAE. Estimate ULAE as 12% of losses plus ALAE. What is the pure premium including all LAE? (Answer: $73.92 per exposure)
- A hypothetical portfolio has 16,000 exposures and $1,120,000 of losses plus $160,000 of LAE, with both components already included as separate totals on the same ultimate basis. Apply a one-year supplied trend factor of 1.05 to the combined cost. What is the projected pure premium? (Answer: $84.00 per exposure)