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Claims reserving: chain ladder, BF and Mack
Chain ladder development factors, ultimate claims and IBNR from a pasted triangle, Bornhuetter-Ferguson reserves and Mack's standard errors.
Inputs
Results
Chain ladder reserve (IBNR)
18,680,856
Ultimate 53,038,946
Mack standard error
2,447,095
13.1% of the reserve
Reported to date
34,358,090
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Reported and ultimate claims by origin
ReportedUltimate
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Development factors
| From period | Age-to-age | To ultimate |
|---|---|---|
| 1 → 2 | 3.4906 | 14.4466 |
| 2 → 3 | 1.7473 | 4.1387 |
| 3 → 4 | 1.4574 | 2.3686 |
| 4 → 5 | 1.1739 | 1.6252 |
| 5 → 6 | 1.1038 | 1.3845 |
| 6 → 7 | 1.0863 | 1.2543 |
| 7 → 8 | 1.0539 | 1.1547 |
| 8 → 9 | 1.0766 | 1.0956 |
| 9 → 10 | 1.0177 | 1.0177 |
Reserves by origin
| Origin | Reported | CDF | Ultimate | Reserve | Mack SE | CV |
|---|---|---|---|---|---|---|
| 1 | 3,901,463 | 1.0000 | 3,901,463 | 0 | 0 | – |
| 2 | 5,339,085 | 1.0177 | 5,433,719 | 94,634 | 75,535 | 79.8% |
| 3 | 4,909,315 | 1.0956 | 5,378,826 | 469,511 | 121,699 | 25.9% |
| 4 | 4,588,268 | 1.1547 | 5,297,906 | 709,638 | 133,549 | 18.8% |
| 5 | 3,873,311 | 1.2543 | 4,858,200 | 984,889 | 261,406 | 26.5% |
| 6 | 3,691,712 | 1.3845 | 5,111,171 | 1,419,459 | 411,010 | 29.0% |
| 7 | 3,483,130 | 1.6252 | 5,660,771 | 2,177,641 | 558,317 | 25.6% |
| 8 | 2,864,498 | 2.3686 | 6,784,799 | 3,920,301 | 875,328 | 22.3% |
| 9 | 1,363,294 | 4.1387 | 5,642,266 | 4,278,972 | 971,258 | 22.7% |
| 10 | 344,014 | 14.4466 | 4,969,825 | 4,625,811 | 1,363,155 | 29.5% |
Methodology
Conventions
- The triangle is cumulative, oldest origin first. Development factors are volume-weighted (or simple averages) of the age-to-age ratios, and the tail factor applies beyond the last period.
- Bornhuetter-Ferguson takes the expected unreported share 1 − 1/CDF of an a priori ultimate (premium × expected loss ratio).
- Mack's standard errors cover process and parameter error of the chain ladder. The last σ² is extrapolated as Mack suggests, and the tail adds no error here.
References
- Mack, Distribution-free calculation of the standard error of chain ladder reserve estimates, ASTIN Bulletin 23 (1993).
- Friedland, Estimating Unpaid Claims Using Basic Techniques (CAS).
Formulas
- Development factor
- \hat f_k = \frac{\sum_i C_{i,k+1}}{\sum_i C_{i,k}}
- Ultimate and reserve
- \hat C_{i,\infty} = C_{i,\text{latest}} \prod_k \hat f_k \times \text{tail},\quad R_i = \hat C_{i,\infty} - C_{i,\text{latest}}
- Bornhuetter-Ferguson
- R_i^{BF} = P_i \cdot ELR \cdot (1 - 1/CDF_i)
- Mack
- \widehat{mse}(R_i) = \hat C_{i,n}^2 \sum_k \frac{\hat\sigma_k^2}{\hat f_k^2}\Big(\frac{1}{\hat C_{i,k}} + \frac{1}{\sum_j C_{j,k}}\Big)