Why the Loss Measure Matters
A credibility assessment depends on the experience measure used as its input. For claim losses, two common measures are paid losses and incurred losses. Paid losses reflect amounts already paid by a valuation date. Incurred losses, when defined on a paid-plus-case-reserve basis, also include estimates of amounts still expected to be paid on reported claims. The two measures can therefore produce different views of the same experience.
The distinction matters when the experience is used to estimate a loss rate or to form the observed component of a credibility-weighted estimate. Payment timing can make paid losses look small at an early valuation date. Case reserve estimates can make incurred losses higher or lower than paid losses, and can change as claim information develops. Neither measure should be treated as a direct, error-free observation of the ultimate cost of claims.
This tutorial focuses on comparing paid and paid-plus-case-reserve measures as credibility inputs. It assumes that the measures being compared refer to the same claims, exposure period, valuation date, coverage, and loss basis. It does not estimate losses that have not yet been reported.
Building Comparable Inputs
A loss total is often converted to a rate by dividing it by the exposure corresponding to that loss experience. For example, if losses relate to 800 earned exposures, the paid loss rate is paid losses divided by 800, and the case-incurred loss rate is incurred losses divided by 800. The units must be interpreted consistently: a rate of $100 per earned exposure is not $100 per claim unless each exposure represents exactly one claim.
Before comparing rates, check that the loss totals and exposures match. A paid total for one set of accident periods should not be divided by exposures from another period. Nor should case reserves be added if the reported incurred total already includes them. If allocated loss adjustment expenses are included in one measure, they should be included on a consistent basis in the other.
Here \(E\) is the exposure for the experience being measured. If the credibility method uses a weighted estimate \(Z X+(1-Z)M\), \(X\) must be the selected observed measure expressed on the same basis as the complement \(M\). The weight \(Z\) and the complement are assumed to have been selected separately; this tutorial examines how the choice of paid or incurred input changes \(X\).
Worked Example: Comparing Paid and Incurred Rates
A hypothetical insurer compares two groups, North and South, at the same valuation date. Each group has 800 earned exposures. For North, cumulative paid losses are $40,000 and case reserves are $30,000. For South, cumulative paid losses are $60,000 and case reserves are $20,000. Calculate both loss rates for each group and compare North’s credibility-weighted estimate under the two choices of input. The selected credibility weight is \(Z=0.4\), and the complement is $90 per exposure.
Step 1: Calculate paid rates. North’s paid rate is $40,000 divided by 800, or $50.00 per exposure. South’s paid rate is $60,000 divided by 800, or $75.00 per exposure.
Step 2: Calculate incurred totals and rates. North’s incurred losses are $40,000 plus $30,000, or $70,000. Its incurred rate is $70,000 divided by 800, or $87.50 per exposure. South’s incurred losses are $60,000 plus $20,000, or $80,000. Its incurred rate is $80,000 divided by 800, or $100.00 per exposure.
Step 3: Blend North’s paid rate with the complement. The paid-input estimate is \(0.4(50.00)+0.6(90.00)=20.00+54.00=\$74.00\) per exposure.
Step 4: Blend North’s incurred rate with the complement. The incurred-input estimate is \(0.4(87.50)+0.6(90.00)=35.00+54.00=\$89.00\) per exposure.
Check and interpretation. North’s incurred rate is below South’s: $87.50 is less than $100.00. Its incurred-input estimate, $89.00, is also greater than its paid-input estimate, $74.00, because the incurred input includes the $30,000 case reserve. The calculation avoids a common arithmetic error: $70,000 divided by 800 is $87.50, not $100.00. All displayed dollar rates are rounded to the nearest cent.
Why Paid and Incurred Can Differ
The relation between the two measures follows directly from the definition. For each reported claim, the case-incurred amount at a valuation date is the amount already paid plus the outstanding case reserve. Summing over the claims gives total incurred losses equal to total paid losses plus total case reserves. Dividing this equality by the same exposure converts each total into a per-exposure rate.
This identity explains two different influences. First, when a payment is made on a claim, cumulative paid losses rise. If the payment replaces an equal amount of case reserve and the reserve estimate otherwise stays unchanged, incurred losses remain the same: the paid amount rises by \(a\), while the reserve falls by \(a\), so \((P+a)+(C-a)=P+C\). Second, if a case reserve is revised, incurred losses change even if no payment is made. A reserve increase of \(b\) raises incurred losses by \(b\); a decrease of \(b\) lowers them by \(b\).
The first influence makes paid experience sensitive to when cash payments occur relative to the valuation date. The second makes incurred experience sensitive to the estimates recorded for outstanding claims. In practice, a payment may not replace a reserve dollar for dollar, because the reserve may be revised at the same time. The identity still holds for the paid and reserve amounts actually recorded at each valuation date.
Worked Example: Payment Timing and a Fixed Claim Set
A hypothetical portfolio has 400 earned exposures. At the first valuation date, it has $36,000 paid and $24,000 in case reserves. At a later valuation date, assume that payments have increased to $50,000 and case reserves have decreased to $10,000, with no other changes to the total case-incurred estimate. Compare the paid and incurred rates at the two dates. Then calculate credibility-weighted estimates using \(Z=0.4\) and a complement of $130 per exposure.
First valuation date. Paid losses are $36,000, so the paid rate is $36,000 divided by 400, or $90.00 per exposure. Incurred losses are $36,000 plus $24,000, or $60,000. The incurred rate is $60,000 divided by 400, or $150.00 per exposure.
Later valuation date. The paid rate is $50,000 divided by 400, or $125.00 per exposure. Incurred losses are $50,000 plus $10,000, again $60,000. The incurred rate remains $60,000 divided by 400, or $150.00 per exposure.
Credibility-weighted estimates at the first date. Using the paid rate gives \(0.4(90.00)+0.6(130.00)=36.00+78.00=\$114.00\) per exposure. Using the incurred rate gives \(0.4(150.00)+0.6(130.00)=60.00+78.00=\$138.00\) per exposure.
Credibility-weighted estimates at the later date. The paid-input estimate is \(0.4(125.00)+0.6(130.00)=50.00+78.00=\$128.00\) per exposure. The incurred-input estimate remains \(0.4(150.00)+0.6(130.00)=\$138.00\) per exposure.
Check and interpretation. Paid losses increased by $14,000, while case reserves decreased by the same amount; therefore, the incurred total stayed at $60,000. The paid-input estimate rose from $114.00 to $128.00, but the incurred-input estimate stayed at $138.00. This example isolates the effect of payment timing; actual reserve revisions can also change incurred losses. The calculation assumes the exposure base and complement remain unchanged. Rates are shown to the nearest cent.
Case Reserves Are Estimates, Not Payments
Case reserves make incurred losses more informative than paid losses about the estimated cost of reported claims at a valuation date, but they do not turn incurred losses into certain ultimate losses. A reserve may reflect the adjuster’s current assessment of claim facts, coverage, and expected settlement cost. It can be revised as additional information becomes available. As a result, incurred experience may move even when paid experience does not.
Differences in case-reserving practices can also affect comparisons across groups. For example, one group may have higher case reserves because its reported claims are more severe, because its claims have different characteristics, or because its reserving practices differ. The observed data alone do not establish which explanation is responsible. A credibility calculation can be completed using the selected input, but the result should be interpreted in light of how that input was produced.
Worked Example: Different Case Reserves, Same Paid Rate
Two hypothetical groups each have 500 earned exposures. Group A has $25,000 paid losses and $15,000 in case reserves. Group B has $25,000 paid losses and $35,000 in case reserves. For this calculation, assume the groups’ loss and exposure definitions are otherwise comparable. Using \(Z=0.5\) and a complement of $90 per exposure, calculate the paid and incurred credibility-weighted estimates for each group.
Step 1: Calculate paid rates. For each group, $25,000 divided by 500 is $50.00 per exposure. Thus both paid-input estimates are \(0.5(50.00)+0.5(90.00)=25.00+45.00=\$70.00\) per exposure.
Step 2: Calculate incurred rates. Group A has incurred losses of $25,000 plus $15,000, or $40,000. Its incurred rate is $40,000 divided by 500, or $80.00 per exposure. Group B has incurred losses of $25,000 plus $35,000, or $60,000. Its incurred rate is $60,000 divided by 500, or $120.00 per exposure.
Step 3: Calculate incurred-input estimates. For Group A, the estimate is \(0.5(80.00)+0.5(90.00)=40.00+45.00=\$85.00\) per exposure. For Group B, it is \(0.5(120.00)+0.5(90.00)=60.00+45.00=\$105.00\) per exposure.
Check and interpretation. The paid-input estimates agree because the groups have the same paid losses and exposures. The incurred-input estimates differ by $20.00 per exposure, exactly half the $40.00 difference in their incurred rates because \(Z=0.5\). This result is a calculation, not proof that Group B’s underlying loss cost is higher: the reserve difference may reflect claim differences, reserving practices, or both. Rates are shown to the nearest cent.
Choosing and Communicating the Input
There is no universal rule that paid or incurred losses are always the correct credibility input. The selection should reflect the question being estimated and the nature of the available data. Paid losses provide a record of amounts actually paid by the valuation date, but can be strongly affected by payment timing. Case-incurred losses incorporate estimates on reported claims, but can be affected by reserve adequacy and reserving practices.
A sound comparison makes the selected basis explicit. State whether the input is paid or paid plus case reserves, the valuation date, the exposure denominator, and which expense or loss components are included. If two estimates are shown, calculate each from its own clearly defined numerator while holding the other comparison assumptions fixed. This makes it possible to identify whether the difference comes from payments, case reserves, or a change in the exposure basis.
Specify the claim set, experience period, valuation date, exposure measure, and included loss components.
Use cumulative paid losses for a paid input, or add outstanding case reserves once for a paid-plus-case-reserve incurred input.
Divide the selected loss total by matching exposures, then apply the chosen credibility weight and complement on a consistent basis.
Relate changes to payment timing or reserve estimates without assuming that either observed measure equals ultimate losses.
Common Mistakes
- Using the wrong denominator. Divide losses by exposures corresponding to those losses, not by a claim count or an unrelated exposure total.
- Calling paid losses incurred losses. Under the definition used here, incurred losses include case reserves as well as paid amounts.
- Adding reserves twice. If a supplied incurred total already includes case reserves, do not add them again.
- Assuming incurred losses cannot change without payment. A reserve revision can change incurred losses even when cumulative paid losses are unchanged.
- Miscomputing a rate. For example, $70,000 divided by 800 is $87.50 per exposure, not $100.00. Keep the numerator and denominator visible until the division is checked.
- Treating a credibility result as a data-quality correction. Weighting an observed input with a complement does not resolve inconsistencies in its loss definition or reserve basis.
Check Your Understanding
For each problem, assume the stated loss and exposure definitions are consistent. Use the given paid-plus-case-reserve definition of incurred losses.
- A hypothetical book has 300 earned exposures, $18,000 paid losses, and $12,000 case reserves. Find the paid and incurred rates per exposure. (Answer: paid $60.00; incurred $100.00 per exposure)
- A portfolio has 600 exposures, $54,000 paid losses, and $18,000 case reserves. What are the paid and incurred rates? (Answer: paid $90.00; incurred $120.00 per exposure)
- At one valuation date, a hypothetical group has 200 exposures, $32,000 paid, and $8,000 in case reserves. At a later date, paid losses are $38,000 and case reserves are $2,000. Calculate the paid and incurred rates at both dates. (Answer: first date paid $160.00 and incurred $200.00; later date paid $190.00 and incurred $200.00 per exposure)
- A group has 500 exposures, $42,000 paid losses, and $18,000 case reserves. If \(Z=0.25\) and the complement is $100 per exposure, calculate the incurred-input credibility-weighted estimate. (Answer: incurred rate $120.00; estimate $105.00 per exposure)
- A hypothetical segment has 1,000 exposures, $80,000 paid losses, and $20,000 case reserves. With \(Z=0.3\) and a complement of $110 per exposure, calculate the paid-input and incurred-input credibility-weighted estimates, and their difference. (Answer: paid-input estimate $101.00; incurred-input estimate $107.00; difference $6.00 per exposure)