Why Reserve Method Selection Matters
A reserve estimate is not just a number produced by a formula. It reflects the data used and the assumptions that connect observed experience to ultimate losses. Paid chain ladder, incurred chain ladder, and expected-loss methods can therefore produce different estimates for the same portfolio. The useful question is not simply which estimate is largest, or what their average is, but why the estimates differ and which evidence is most relevant to the liabilities being measured.
This tutorial compares those methods for hypothetical accident-year data. It focuses on the unpaid loss reserve at an evaluation date, before any discounting. In practice, an actuary would also consider data quality, coverage and exposure consistency, claim handling, and whether the selected development assumptions are appropriate.
Three Methods, Three Sources of Information
The paid chain ladder projects cumulative paid losses using historical ratios of paid losses at successive development ages. It relies on the observed pattern of payment emergence. It can be sensitive to changes in settlement speed, payment practices, or the mix of claims.
The incurred chain ladder projects cumulative incurred losses, which combine paid losses and case reserves. It uses information from both payments and case-reserve estimates. It can be sensitive to changes in case reserving practices, including changes that raise or lower incurred amounts without changing expected ultimate losses.
The expected-loss method starts with an estimate of ultimate losses based on exposure or premium and an expected loss ratio. It does not require the immature accident year’s own loss development to be fully observed. It is particularly dependent on the reasonableness of the expected loss ratio and the premium basis.
For a basic chain ladder projection, assume the selected age-to-age factors are appropriate for the accident year being projected and that the cumulative amounts develop according to those factors. In a volume-weighted selection, the factor from age \(j\) to the next age is the sum of observed cumulative amounts at the next age divided by the sum of the corresponding amounts at age \(j\). The method is not appropriate without adjustment or further investigation when historical development patterns are not representative, data are inconsistent, or a material tail has been omitted.
Here \(\widehat{U}^{\,P}_i\) is projected ultimate on a paid basis and \(\widehat{R}^{\,P}_i\) is the paid chain ladder unpaid reserve. For incurred chain ladder, the incurred amount is projected to ultimate, but paid losses are still subtracted to obtain the unpaid reserve:
For an expected-loss method, let \(ELR_i\) be the selected expected loss ratio. The expected ultimate is \(EP_i ELR_i\), and the unpaid reserve is that amount less paid losses to date. This formulation assumes the expected loss ratio and earned premium appropriately represent the ultimate losses for the cohort.
Why the Chain Ladder Projection Works
The chain ladder projection follows directly from the meaning of the age-to-age factors. Suppose \(f_j\) is the selected multiplier from age \(j\) to age \(j+1\). A cumulative amount \(C_j\) at age \(j\) projects to \(C_jf_j\) at age \(j+1\). Projecting one more age multiplies that amount by \(f_{j+1}\), giving \(C_jf_jf_{j+1}\). Continuing to the stated ultimate age \(J\) gives:
This is a projection under the selected development pattern, not a guarantee that the pattern will hold. If development continues after \(J\), the projection must also include the tail factor. For paid chain ladder, \(C_j\) is paid losses; for incurred chain ladder, it is incurred losses. The calculation produces an ultimate estimate, not an unpaid reserve, until paid losses are subtracted.
Worked Comparison on a Hypothetical Portfolio
The following hypothetical data use development ages 12, 24, and 36 months. Assume 36 months is ultimate, with no additional tail development. Use volume-weighted age-to-age factors separately for paid and incurred losses. The data are in thousands of dollars.
| Accident year | Paid 12 | Paid 24 | Paid 36 | Incurred 12 | Incurred 24 | Incurred 36 |
|---|---|---|---|---|---|---|
| AY1 | 100 | 160 | 200 | 150 | 190 | 210 |
| AY2 | 120 | 180 | — | 170 | 205 | — |
| AY3 | 130 | — | — | 190 | — | — |
Worked Example: Comparing Paid and Incurred Chain Ladder
Estimate the unpaid reserves for AY2 and AY3 using paid chain ladder and incurred chain ladder. Exclude the fully developed AY1 from the immature-year reserve total. State the selected factors and show how each ultimate estimate becomes an unpaid reserve.
Method and assumptions. Use volume-weighted factors. Treat 36 months as ultimate, so the 24-to-36 factor completes the AY2 projection and the 12-to-24 and 24-to-36 factors complete the AY3 projection. No tail factor is required under this stated assumption.
Paid factors. The 12-to-24 factor is \((160+180)/(100+120)=340/220=1.54545\). The 24-to-36 factor is \(200/160=1.25\).
For AY2, paid ultimate is \(180(1.25)=225\). Paid to date is 180, so the unpaid reserve is \(225-180=45\).
For AY3, the cumulative factor from 12 months to ultimate is \(1.54545(1.25)=1.93182\). Paid ultimate is \(130(1.93182)=251.136\). Subtracting paid to date gives an unpaid reserve of \(251.136-130=121.136\).
The paid chain ladder reserve for AY2 and AY3 is \(45+121.136=166.136\), or 166.14 thousand rounded to two decimals.
Incurred factors. The 12-to-24 factor is \((190+205)/(150+170)=395/320=1.234375\). The 24-to-36 factor is \(210/190=1.105263\).
For AY2, incurred ultimate is \(205(1.105263)=226.579\). AY2 paid losses are 180, so the unpaid reserve is \(226.579-180=46.579\).
For AY3, the cumulative factor is \(1.234375(1.105263)=1.364309\). Incurred ultimate is \(190(1.364309)=259.219\). Subtracting paid losses of 130 gives an unpaid reserve of \(259.219-130=129.219\).
The incurred chain ladder reserve for AY2 and AY3 is \(46.579+129.219=175.798\), or 175.80 thousand rounded to two decimals. The result exceeds the paid estimate by \(175.798-166.136=9.662\) thousand, or 9.66 thousand rounded to two decimals. That difference is a consequence of the distinct data bases and selected development patterns; it is not evidence that either method is automatically correct. A common mistake is to subtract incurred losses, rather than paid losses, from projected incurred ultimate when calculating the unpaid loss reserve.
Expected Loss as an Independent Reference
Worked Example: Adding an Expected-Loss Estimate
For the same hypothetical immature years, suppose earned premium is 260 thousand for AY2 and 300 thousand for AY3. A selected expected loss ratio is 0.80 for each year. Paid losses are 180 thousand for AY2 and 130 thousand for AY3. Calculate expected-loss reserves and compare them with the two chain ladder totals.
Method and assumptions. Expected ultimate losses equal earned premium multiplied by the selected expected loss ratio. The unpaid reserve equals expected ultimate less paid losses to date. Assume the premium is on a consistent basis with the loss ratio.
For AY2, expected ultimate is \(260(0.80)=208\) thousand, and the unpaid reserve is \(208-180=28\) thousand.
For AY3, expected ultimate is \(300(0.80)=240\) thousand, and the unpaid reserve is \(240-130=110\) thousand.
The expected-loss reserve total is \(28+110=138\) thousand. The three estimates are therefore 166.14 thousand from paid chain ladder, 175.80 thousand from incurred chain ladder, and 138.00 thousand from the expected-loss method. The expected-loss estimate is lower because the selected expected ultimate losses are lower than the chain ladder projections, not because the methods are averaging different views of the same observed triangle.
Common mistake avoided. The expected ultimate is not the reserve: paid losses must be deducted. For example, reporting AY3’s 240 thousand expected ultimate as its reserve would overstate the unpaid amount by the 130 thousand already paid.
Interpreting Differences and Testing Sensitivity
The estimates above rely on different evidence. Paid chain ladder depends on payment emergence. Incurred chain ladder also depends on the development and consistency of case reserves. The expected-loss method depends on premium and the selected expected loss ratio. A method-selection discussion should connect differences to these specific dependencies and investigate whether they are plausible for the portfolio.
Sensitivity is useful for understanding how much a result depends on an assumption. For the expected-loss method in the example, the combined premium is \(260+300=560\) thousand. A change in the expected loss ratio by 0.05 changes expected ultimate losses, and thus the reserve total, by \(560(0.05)=28\) thousand, provided paid losses and premium are unchanged. A selected loss ratio of 0.75 gives a reserve of \(560(0.75)-310=110\) thousand; a ratio of 0.85 gives \(560(0.85)-310=166\) thousand. This range describes the effect of that assumption change; it does not by itself establish that either alternative is reasonable.
Worked Example: Isolating Incurred-Data Sensitivity
Suppose a review of AY3 case reserving increases its 12-month incurred amount from 190 to 210 thousand, while paid losses remain 130 thousand. To isolate the effect of that change, hold the selected incurred chain ladder factors from the comparison example fixed. Recalculate the AY3 and total incurred chain ladder reserves, and explain which other methods change under this controlled comparison.
Method and assumption. This is a one-input sensitivity: the selected incurred factors are held fixed, and only the AY3 incurred starting amount changes. The factor to ultimate remains \(1.364309\).
The revised AY3 incurred ultimate is \(210(1.364309)=286.505\) thousand. Its revised unpaid reserve is \(286.505-130=156.505\) thousand.
AY2’s incurred reserve remains 46.579 thousand under the stated fixed-factor assumption. The revised total incurred chain ladder reserve is \(46.579+156.505=203.084\) thousand, or 203.08 thousand rounded to two decimals. The increase from the prior incurred total is \(203.084-175.798=27.286\) thousand, or 27.29 thousand rounded to two decimals. Paid chain ladder and the expected-loss estimate do not change in this controlled comparison because their inputs have not changed.
In a full analysis, an actuary would also consider whether the revised incurred observation affects the selected factors. Holding factors fixed isolates the direct effect of the changed starting amount; it does not claim that a complete recalculation would leave all factors unchanged. The common trap is to call the entire 20-thousand increase in incurred data a 20-thousand reserve increase. Under the fixed-factor projection, the increase is multiplied by the factor to ultimate, producing a 27.29-thousand change.
Selecting a Method Without Mechanical Averaging
The best-supported estimate depends on the circumstances. A practical comparison asks what each method assumes, whether its data are credible and consistent, and whether the observed differences have a coherent explanation. The estimates should not be averaged merely because they disagree. An unexplained difference is a reason to investigate, not a reason to conceal the disagreement in a midpoint.
Check that accident-year definitions, evaluation dates, coverage, premium, paid losses, and incurred losses are comparable. A mismatch can create differences unrelated to method performance.
For paid chain ladder, examine payment emergence; for incurred chain ladder, examine both emergence and case-reserve consistency; for the expected-loss method, examine the premium basis and expected loss ratio.
Ask whether changes in settlement speed, case reserving, or expected loss assumptions could explain the direction and size of the differences. Avoid treating an explanation as established unless the data support it.
State the method or methods relied upon, the reasons for that reliance, and the material assumptions. If methods are combined, explain the rationale and weights rather than using an unexplained arithmetic average.
This comparison is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus. In professional reserve reporting, the same discipline helps distinguish an estimate’s numerical result from the assumptions that make it credible.
Check Your Understanding
For each hypothetical problem, identify the method being used and keep ultimate losses distinct from unpaid reserves.
- A paid triangle has a cumulative paid amount of 90 thousand at 12 months. The selected 12-to-24 factor is 1.40 and the 24-to-36 factor is 1.20; 36 months is ultimate and there is no tail. Calculate the 12-month paid chain ladder ultimate and unpaid reserve. (Answer: ultimate 151.20 thousand; unpaid reserve 61.20 thousand.)
- A cohort has earned premium of 400 thousand, an expected loss ratio of 0.65, and paid losses to date of 180 thousand. Calculate its expected ultimate and expected-loss unpaid reserve. (Answer: ultimate 260 thousand; unpaid reserve 80 thousand.)
- A cohort has incurred losses of 250 thousand at the evaluation age, paid losses of 190 thousand, and a selected incurred age-to-ultimate factor of 1.08. Calculate incurred chain ladder ultimate and unpaid reserve. (Answer: ultimate 270 thousand; unpaid reserve 80 thousand.)
- For the same cohort, paid chain ladder estimates ultimate at 300 thousand, incurred chain ladder estimates ultimate at 325 thousand, and the expected-loss ultimate is 315 thousand. Paid losses are 210 thousand. Calculate each unpaid reserve and identify the largest; do not average them. (Answer: paid chain ladder 90 thousand; incurred chain ladder 115 thousand; expected-loss 105 thousand; incurred chain ladder is largest.)
- A portfolio has earned premium of 700 thousand and paid losses of 260 thousand. With an expected loss ratio of 0.72, calculate the expected-loss reserve. Then calculate the change in that reserve if the selected ratio increases by 0.03, holding premium and paid losses fixed. (Answer: initial reserve 244 thousand; reserve increases by 21 thousand to 265 thousand.)