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Real data: Allstate Ins Co Grp β Workers' Compensation (paid basis, amounts in thousands of USD)
Source: Meyers, G.G. and Shi, P. (2011). The Retrospective Testing of Stochastic Loss Reserve Models. CAS E-Forum, using data pulled from NAIC Schedule P Parts 1-4, 1988-1997. https://www.casact.org/publications-research/research/research-resources/loss-reserving-data-pulled-naic-schedule-p
Development, reserves and loss ratios by accident year
| AY | Earned premium | Latest paid | Age | CDF | CL ultimate | BF ultimate | CL IBNR | Ult. loss ratio | Actual ultimate | CL hindsight error |
|---|---|---|---|---|---|---|---|---|---|---|
| 1988 | 394,742 | 325,322 | 10 | 1.000 | 325,322 | 325,322 | 0 | 82.4% | 325,322 | 0.0% |
| 1989 | 374,252 | 273,873 | 9 | 1.011 | 276,864 | 277,149 | 2,991 | 74.0% | 277,574 | -0.3% |
| 1990 | 280,320 | 256,788 | 8 | 1.047 | 268,961 | 267,070 | 12,173 | 95.9% | 263,000 | 2.3% |
| 1991 | 313,982 | 239,195 | 7 | 1.080 | 258,402 | 258,110 | 19,207 | 82.3% | 248,319 | 4.1% |
| 1992 | 252,698 | 159,496 | 6 | 1.130 | 180,151 | 182,977 | 20,655 | 71.3% | 168,844 | 6.7% |
| 1993 | 201,055 | 87,215 | 5 | 1.196 | 104,286 | 113,889 | 17,071 | 51.9% | 90,686 | 15.0% |
| 1994 | 174,381 | 91,077 | 4 | 1.307 | 119,003 | 124,243 | 27,926 | 68.2% | 94,730 | 25.6% |
| 1995 | 146,366 | 87,311 | 3 | 1.514 | 132,157 | 127,565 | 44,846 | 90.3% | 91,161 | 45.0% |
| 1996 | 93,294 | 44,916 | 2 | 2.025 | 90,948 | 83,185 | 46,032 | 97.5% | 49,255 | 84.6% |
| 1997 | 7,651 | 691 | 1 | 4.501 | 3,110 | 5,514 | 2,419 | 40.7% | 2,909 | 6.9% |
| Total | 2,238,741 | 1,565,884 | 1,759,204 | 1,765,024 | 193,320 | 78.6% | 1,611,800 | 9.1% |
Selected volume-weighted age-to-age factors: 2.2230, 1.3377, 1.1584, 1.0927, 1.0586, 1.0455, 1.0314, 1.0361, 1.0109 (tail 1.000).
Reserve risk
| Measure | Value |
|---|---|
| Mack total standard error | 58,633 |
| Mack CV of total reserve | 30.3% |
| ODP bootstrap mean reserve | 195,949 |
| ODP bootstrap s.d. | 42,833 |
| 75th / 95th / 99th percentile | 221,978 / 270,136 / 302,665 |
| VaR 99.5% / TVaR 99% | 310,739 / 323,642 |
Incremental-cell severity (lognormal MLE on 55 positive cells): , , mean 33,316, CV 1.51.
Key formulas: ; ; ; .
Observations
- Volume-weighted age-to-age factors: 2.223, 1.338, 1.158, 1.093β¦ β slow early development typical of a long-tailed line.
- Chain-ladder IBNR on the paid basis is 193,320 (thousands of USD), 12.3% of the latest diagonal.
- Premium-weighted ultimate loss ratio 78.6%; by year it ranges 40.7%β97.5%.
- Mack standard error of total reserve 58,633 (CV 30.3%); ODP bootstrap 95th percentile 270,136 vs mean 195,949.
- Hindsight test: chain ladder over-estimated the realized ultimate by 9.1% (actual data through age 10 is known for every year).
- BornhuetterβFerguson (ELR 81.0%) is 0.3% above chain ladder in total β the gap sits in the immature years.
- Incremental paid amounts fit a lognormal with Ο = 1.09 (CV 1.51), i.e. heavy-tailed cell values.