Why the Loss Measure Must Be Explicit
An indication is only interpretable when its loss numerator has a clear, consistent definition. A total labelled “losses” might mean indemnity alone, indemnity plus allocated loss adjustment expense, paid amounts, or paid amounts plus case reserves. These are different measures. Choosing among them starts with the purpose and scope of the data, not with which total happens to be largest or easiest to retrieve.
For a loss ratio calculation, the selected loss measure is compared with premium on the intended basis. The loss numerator should represent the same coverage, experience period, and claim-cost components throughout the calculation. A paid amount from one field should not be combined with an incurred amount from another as though they were interchangeable. This tutorial is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
Specify the Measure by Component
A useful way to describe a loss measure is to state both its valuation basis and its components. For example, “paid indemnity plus paid ALAE through December 31” is more precise than “paid losses.” Similarly, “reported incurred indemnity and ALAE at December 31, including case outstanding amounts” identifies both the components and the valuation basis.
Indemnity represents the amounts paid or expected to be paid to resolve the covered claim. ALAE consists of expenses associated with particular claims, such as claim-specific legal or adjusting costs. ULAE is not assigned to individual claims in the same way; it is generally handled separately if the intended measure includes it. Data systems and reporting conventions can use labels differently, so the field definitions and data dictionary control. Do not infer a component’s inclusion from a column heading alone.
For each component \(C\) that follows the stated paid-plus-case definition, the reconciliation is:
Thus, if the measure includes both indemnity and ALAE, calculate the components separately and then add them. The corresponding paid measure is paid indemnity plus paid ALAE; the corresponding reported incurred measure is incurred indemnity plus incurred ALAE. If the intended basis is indemnity only, leave ALAE out of both measures rather than including it on just one side.
Why Incurred Equals Paid Plus Case Outstanding
The component identity follows from how a reported claim’s current estimated cost is represented. Amounts already paid are the portion settled by the valuation date. Case outstanding is the additional amount currently reserved for that same reported claim. Together, those amounts make up the claim’s reported incurred amount under this definition.
For example, identify indemnity paid and indemnity case outstanding separately from ALAE paid and ALAE case outstanding.
Add paid indemnity to case outstanding indemnity, and paid ALAE to case outstanding ALAE.
If the measure is indemnity plus ALAE, add those two incurred component totals. If it is indemnity only, do not add ALAE.
This identity does not say that reported incurred is the ultimate cost of all claims. It describes the amount paid plus case outstanding for reported claims under the stated fields and valuation date. An estimate that includes IBNR or other adjustments is a different measure and should be labelled accordingly.
Worked Example: Reconciling Paid and Incurred Components
A hypothetical commercial auto account has the following comparable fields for one experience period, all measured at the same valuation date. Paid indemnity is $480,000, incurred indemnity is $570,000, paid ALAE is $30,000, and incurred ALAE is $35,000. Earned premium on the intended basis is $1,500,000. Calculate the paid and reported incurred loss measures, reconcile the difference by component, and calculate both loss ratios.
Method: reconcile each component before combining them. The paid measure including indemnity and ALAE is:
The reported incurred measure including the same two components is:
The implied case outstanding amounts are obtained component by component. Outstanding indemnity is \(570{,}000-480{,}000=90{,}000\). Outstanding ALAE is \(35{,}000-30{,}000=5{,}000\). Their sum is \(90{,}000+5{,}000=95{,}000\), which reconciles the difference between the two totals: \(605{,}000-510{,}000=95{,}000\). This allocation assumes, as stated, that the paid and incurred fields are comparable and have the same scope.
Using the stated earned premium, the paid loss ratio is \(510{,}000/1{,}500{,}000=0.34\), or 34.000%. The reported incurred loss ratio is \(605{,}000/1{,}500{,}000=0.403333\ldots\), or 40.333% rounded to three decimal places as a percentage.
Check: Adding the $95,000 of case outstanding amounts to the $510,000 paid total gives $605,000, the reported incurred total. The common mistake is to say the $95,000 difference cannot be allocated by component; the supplied component fields show exactly $90,000 outstanding indemnity and $5,000 outstanding ALAE.
Choose the Basis That Fits the Data and Purpose
Paid and reported incurred are both valid descriptions of loss experience, but they answer different measurement questions at a valuation date. Paid data record actual payments made by that date. Reported incurred data also include case outstanding amounts for reported claims. The choice should reflect the intended indication basis and the available, well-defined data. A paid basis should not be selected merely because it avoids checking reserves; an incurred basis should not be selected merely because its total is larger.
Before calculating a loss input, record the following:
- Valuation basis: paid, reported incurred, or another explicitly defined measure.
- Claim-cost components: indemnity only, or indemnity plus ALAE; state any other included components separately.
- Valuation date and population: confirm that the fields cover the same experience period, claims, and cutoff.
- Exclusions and additions: identify whether IBNR, ULAE, or other adjustments are excluded or separately included.
When data fields are available for both paid and incurred, use the version that matches the chosen basis throughout the calculation. If only one basis is available for part of the data, do not silently fill the gap using a different basis. First establish whether a comparable field can be obtained or whether a clearly defined common basis can be constructed. If not, the inputs are not yet a consistent total.
Worked Example: Keeping a Two-Year Measure Consistent
A hypothetical equipment insurer is preparing a two-year loss ratio input on a paid indemnity-plus-ALAE basis. In Year 1, paid indemnity is $300,000, paid ALAE is $20,000, incurred indemnity is $360,000, and incurred ALAE is $25,000. In Year 2, the corresponding amounts are $420,000, $30,000, $500,000, and $35,000. Earned premium is $1,000,000 in Year 1 and $1,200,000 in Year 2. Calculate the combined paid-basis loss ratio and explain why substituting Year 2 incurred losses would be inconsistent.
Method: apply the requested paid basis to both years. Year 1 paid losses including ALAE are \(300{,}000+20{,}000=320{,}000\). Year 2 paid losses including ALAE are \(420{,}000+30{,}000=450{,}000\). Combined paid losses are \(320{,}000+450{,}000=770{,}000\), and combined earned premium is \(1{,}000{,}000+1{,}200{,}000=2{,}200{,}000\).
For comparison, Year 1 reported incurred losses including ALAE are \(360{,}000+25{,}000=385{,}000\), and Year 2 reported incurred losses including ALAE are \(500{,}000+35{,}000=535{,}000\). The combined incurred total is \(385{,}000+535{,}000=920{,}000\); on the same combined premium its loss ratio is \(920{,}000/2{,}200{,}000=41.818\%\), rounded to three decimal places.
A tempting but incorrect mixed numerator is Year 1 paid losses plus Year 2 incurred losses: \(320{,}000+535{,}000=855{,}000\). Dividing by $2,200,000 gives 38.864%, but that result is neither a two-year paid-basis ratio nor a two-year incurred-basis ratio. It mixes valuation bases. The requested result is 35.000%; the corresponding consistent incurred result, if wanted, is 41.818%.
Check: The common denominator is $2,200,000 in both complete calculations, and each numerator consistently uses either paid or reported incurred amounts for both years. The choice of paid versus incurred changes the measured numerator; it does not justify switching bases partway through the experience.
Do Not Treat Every “Incurred” Field as the Same
A data extract may report “incurred” using a definition other than paid plus case outstanding. It may incorporate adjustments, include selected estimates for claims not yet reported, or omit a component such as ALAE. The standard identity in this tutorial applies only when the paid and incurred fields have comparable scope and the incurred field is defined as paid plus case outstanding for the component in question. Confirm the definition before using the identity.
Likewise, adding IBNR or ULAE changes the measure. IBNR estimates relate to claims not yet reported (and, in some conventions, further development on reported claims); they are not case outstanding amounts on the reported claims in the definition above. ULAE is also not the same as claim-specific ALAE. If either is included in a stated data measure, identify it explicitly rather than describing the result simply as reported incurred indemnity and ALAE.
A component table makes these distinctions easier to audit:
| Measure | Indemnity | ALAE | Other amounts |
|---|---|---|---|
| Paid indemnity only | Paid | Excluded | Excluded unless stated |
| Paid indemnity plus ALAE | Paid | Paid | Excluded unless stated |
| Reported incurred indemnity plus ALAE | Paid plus case outstanding | Paid plus case outstanding | IBNR and ULAE excluded unless stated |
Worked Example: Separating Case Reserves From Other Amounts
A hypothetical specialty insurer’s valuation file gives paid indemnity of $720,000 and case outstanding indemnity of $180,000. It also gives paid ALAE of $42,000 and case outstanding ALAE of $18,000. A separate schedule estimates $95,000 of IBNR indemnity and $8,000 of IBNR ALAE. The requested measure is reported incurred indemnity plus ALAE. Calculate that measure, calculate the paid measure on the same component scope, and identify what is outside the requested measure.
Method: include paid and case outstanding amounts for the two requested components. Reported incurred indemnity is \(720{,}000+180{,}000=900{,}000\). Reported incurred ALAE is \(42{,}000+18{,}000=60{,}000\). Therefore reported incurred indemnity plus ALAE is \(900{,}000+60{,}000=960{,}000\).
The paid measure on the same component scope is \(720{,}000+42{,}000=762{,}000\). The difference is \(960{,}000-762{,}000=198{,}000\), which checks against total case outstanding amounts: \(180{,}000+18{,}000=198{,}000\).
The $95,000 IBNR indemnity and $8,000 IBNR ALAE are outside the requested reported incurred measure, for a total of $103,000 excluded from that measure. They must not be added while still labelling the result reported incurred under the stated definition. A different requested measure could include them, but it would need a different, explicit label.
Check: Paid plus case outstanding is $762,000 plus $198,000, or $960,000. The separate IBNR schedule does not alter this reconciliation. The common mistake is to add IBNR automatically whenever a file contains it, even when the requested measure is reported incurred.
Common Mistakes and Practical Checks
- Using “losses” without defining it. State whether the measure is paid or reported incurred and whether it includes indemnity, ALAE, or other components.
- Adding ALAE on only one side. If a measure includes indemnity plus ALAE, include the corresponding ALAE fields consistently in the paid or incurred total.
- Mixing paid and incurred periods. A total containing paid losses for one period and incurred losses for another is not a consistent paid or incurred measure.
- Calling reported incurred ultimate. Paid plus case outstanding is not automatically an estimate including IBNR or all future claim development.
- Assigning a total difference without component evidence. When comparable paid and incurred fields are supplied by component, calculate each difference separately. If the fields are not comparable or the component detail is missing, do not guess the allocation.
A brief audit before calculating an indication input can prevent all of these errors: read the field definitions, confirm the valuation date and experience scope, reconcile paid to incurred by component where possible, and write the selected measure in words. The calculation is then transparent to anyone reviewing the numerator.
Check Your Understanding
For each problem, identify the specified basis and keep the component scope consistent.
- A hypothetical account has paid indemnity of $210,000, paid ALAE of $15,000, case outstanding indemnity of $40,000, and case outstanding ALAE of $5,000. Calculate paid indemnity plus ALAE and reported incurred indemnity plus ALAE. (Answer: Paid = $225,000; reported incurred = $270,000.)
- A valuation file reports paid indemnity of $380,000, paid ALAE of $22,000, incurred indemnity of $455,000, and incurred ALAE of $31,000. Assuming comparable paid-plus-case fields, calculate the paid and incurred totals including ALAE, and reconcile the difference by component. (Answer: Paid = $402,000; incurred = $486,000; outstanding indemnity = $75,000 and outstanding ALAE = $9,000, totaling $84,000.)
- A hypothetical insurer selects a paid indemnity-only basis. For a period, paid indemnity is $260,000 and paid ALAE is $18,000; earned premium is $800,000. Calculate the loss ratio on the selected basis and state why the ALAE amount is excluded. (Answer: $260,000 / $800,000 = 32.500%; ALAE is excluded because the selected basis is indemnity only.)
- For two hypothetical periods, paid indemnity-plus-ALAE totals are $275,000 and $350,000, while incurred indemnity-plus-ALAE totals are $330,000 and $420,000. Earned premiums are $900,000 and $1,100,000. Calculate the combined incurred-basis loss ratio. (Answer: ($330,000 + $420,000) / ($900,000 + $1,100,000) = 37.500%.)
- A reported incurred indemnity-plus-ALAE measure is $640,000. A separate schedule shows $55,000 of IBNR indemnity and $5,000 of IBNR ALAE. What is the reported incurred measure, and what would the total be if those IBNR estimates were deliberately added as a separate extension? (Answer: Reported incurred remains $640,000; with the separately identified $60,000 IBNR extension, the total is $700,000.)