# On Comparison of Stochastic Reserving Methods with Bootstrapping

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Chain-Ladder Method as a Generalized Linear Model

- (1)
- incremental claim amounts ${C}_{ij}$ belong to the exponential family,
- (2)
- $E\left({C}_{ij}\right)={\mu}_{ij}$,
- (3)
- ${\eta}_{ij}=g\left({\mu}_{ij}\right)$, where $g(\xb7)$ is the link function,
- (4)
- linear predictor ${\eta}_{ij}=c+{\tilde{\alpha}}_{i}+{\tilde{\beta}}_{j}$ with an intercept c and factor effects ${\tilde{\alpha}}_{i}$ and ${\tilde{\beta}}_{j}$.

**Lemma**

**1.**

**Proof**

**of Lemma 1.**

## 3. The Bootstrap Technique

#### 3.1. Residuals

#### 3.2. Prediction Error and Confidence Limits

## 4. Case Study

- The over-dispersed Poisson model produces the highest estimated claim reserve, and the log-normal model produces the smallest estimated claim reserves. The figures of the gamma model are not that different from the ODP model.
- The standard errors of prediction are quite different and consequently the estimated upper limits. These differences tend to be greater especially on the first years, since estimations are based on few predictions. The highest prediction errors are produced by the log-normal model, and the lowest prediction errors were obtained by the over-dispersed Poisson model.
- With this particular dataset, the prediction errors are the lowest with the Anscombe residuals. Furthermore, no matter which residual of the two is used, the lowest prediction errors are obtained by using the zero correction with standardization.
- When comparing the two bootstrap procedures, we can conclude that using the (alternative) PPE method, the upper confidence limits for the total reserve are lower with each considered model.

## 5. Comparative Analysis with the Schedule P Database

#### 5.1. Schedule P Database

#### 5.2. Model Validation

#### 5.3. Results

- The over-dispersed Poisson model fits the data best. This confirms the results obtained in the previous section, where we obtained the smallest prediction errors precisely with the ODP model.
- The gamma model and the log-normal model are behaving rather similarly, but the log-normal model is fitting the data slightly better.
- The lowest values of the measures are obtained with the zero-corrected residuals and with the non-corrected residuals.
- The smallest score was obtained by the zero correction with the Pearson residuals.
- If comparing just the choice of residuals, we see that the Anscombe residuals perform more poorly than the Pearson residuals.

## 6. Discussion

- The large fluctuation of the values in the data substantiates the use of the over-dispersed Poisson model. The gamma model and the ODP model tend to give similar point estimates, whereas the log-normal model produces the smallest estimated claim reserves. Here, the expertise of an actuary would help to finalize a decision in model selection, depending on the company’s balance of hazard and conservatism.
- When the emphasis is on prediction errors, then the ODP model should be used for the lowest prediction errors. The log-normal model tends to give irrationally high errors.
- The choice of residuals matters in bootstrapping. The Pearson residual should be preferred, but in some cases (see Table 11), the Anscombe residual could be considered.
- The adjustment of residuals is not less important than the choice of the residual; the most precise predictions are obtained with either zero-corrected residuals or without any corrections.
- The proposed model validation and assessment ideas are generic and do not depend on a particular dataset, thus constituting a useful tool in reserve estimation.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Development Period j | |||||
---|---|---|---|---|---|

Year of Origin i | 1 | 2 | 3 | … | n |

1 | ${C}_{11}$ | ${C}_{12}$ | ${C}_{13}$ | … | ${C}_{1n}$ |

2 | ${C}_{21}$ | ${C}_{22}$ | … | ||

3 | ${C}_{31}$ | … | |||

⋮ | ⋮ | ||||

n | ${C}_{n1}$ |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|---|

2000 | 4,734,994 | 1,885,305 | 281,240 | 504,341 | 524,449 | 365,049 | 100,761 | 32,449 | 3697 | 56,901 |

2001 | 4,344,093 | 1,783,774 | 243,849 | 339,985 | 49,482 | 178,961 | 508,272 | 78,125 | 1022 | |

2002 | 5,288,867 | 1,795,855 | 303,246 | 351,320 | 316,038 | 33,501 | 88,774 | 31,102 | ||

2003 | 5,357,617 | 2,548,383 | 336,749 | 403,501 | 348,378 | 236,017 | 12,982 | |||

2004 | 5,737,732 | 2,574,724 | 971,320 | 280,140 | 226,212 | 152,127 | ||||

2005 | 5,635,064 | 2,758,392 | 241,734 | 268,113 | 429,503 | |||||

2006 | 6,629,504 | 3,045,252 | 356,119 | 200,420 | ||||||

2007 | 6,824,829 | 2,669,579 | 166,400 | |||||||

2008 | 8,116,439 | 3,428,535 | ||||||||

2009 | 10,660,074 |

Year | Est. Reserve | Without Corrections | Zero-Correction | Zero-Correction & Stand. | |||
---|---|---|---|---|---|---|---|

SEP | Upper 95% | SEP | Upper 95% | SEP | Upper 95% | ||

2 | 50,796 | 90,377 | 199,466 | 89,795 | 198,509 | 69,858 | 165,713 |

3 | 57,837 | 97,791 | 218,702 | 97,051 | 217,485 | 74,943 | 181,117 |

4 | 120,029 | 135,467 | 342,872 | 135,571 | 343,043 | 115,039 | 309,268 |

5 | 348,993 | 220,918 | 712,403 | 225,328 | 719,658 | 207,318 | 690,031 |

6 | 552,215 | 271,860 | 999,425 | 270,829 | 997,728 | 259,708 | 979,434 |

7 | 1,024,516 | 374,459 | 1,640,501 | 381,686 | 1,652,389 | 361,525 | 1,619,225 |

8 | 1,406,290 | 441,811 | 2,133,069 | 444,127 | 2,136,879 | 421,587 | 2,099,800 |

9 | 2,283,616 | 576,547 | 3,232,037 | 578,861 | 3,235,843 | 549,794 | 3,188,028 |

10 | 7,560,816 | 1,264,024 | 9,640,136 | 1,249,066 | 9,615,530 | 979,054 | 9,171,360 |

Total | 13,405,108 | 1,944,083 | 16,603,125 | 1,944,997 | 16,604,628 | 1,603,405 | 16,042,710 |

PPE | 3,182,150 | 16,587,258 | 3,082,305 | 16,487,413 | 1,625,348 | 15,030,456 | |

PPE/SEP | 1.637 | 0.999 | 1.585 | 0.993 | 1.014 | 0.937 |

Year | Est. Reserve | Without Corrections | Zero-Correction | Zero-Correction & Stand. | |||
---|---|---|---|---|---|---|---|

SEP | Upper 95% | SEP | Upper 95% | SEP | Upper 95% | ||

2 | 50,796 | 85,484 | 191,418 | 87,509 | 194,748 | 69,751 | 165,536 |

3 | 57,837 | 91,878 | 208,975 | 94,369 | 213,073 | 75,139 | 181,440 |

4 | 120,029 | 129,886 | 333,690 | 133,400 | 339,471 | 114,032 | 307,612 |

5 | 348,993 | 211,456 | 696,839 | 216,857 | 705,723 | 200,673 | 679,101 |

6 | 552,215 | 260,652 | 980,988 | 262,344 | 983,771 | 250,894 | 964,937 |

7 | 1,024,516 | 357,010 | 1,611,798 | 364,678 | 1,624,412 | 347,598 | 1,596,314 |

8 | 1,406,290 | 415,877 | 2,090,406 | 421,084 | 2,098,972 | 406,742 | 2,075,380 |

9 | 2,283,616 | 542,459 | 3,175,961 | 544,458 | 3,179,249 | 522,033 | 3,142,361 |

10 | 7,560,816 | 1,124,403 | 9,410,459 | 1,109,368 | 9,385,726 | 934,009 | 9,097,261 |

Total | 13,405,108 | 1,727,161 | 16,246,288 | 1,758,340 | 16,297,578 | 1,469,680 | 15,822,732 |

PPE | 2,029,479 | 15,434,588 | 2,004,348 | 15,409,456 | 1,275,538 | 14,680,646 | |

PPE/SEP | 1.175 | 0.950 | 1.140 | 0.946 | 0.868 | 0.928 |

Year | Est. Reserve | Without Corrections | Zero-Correction | Zero-Correction & Stand. | |||
---|---|---|---|---|---|---|---|

SEP | Upper 95% | SEP | Upper 95% | SEP | Upper 95% | ||

2 | 50,012 | 38,160 | 112,785 | 39,008 | 114,180 | 30,965 | 100,950 |

3 | 37,119 | 26,904 | 81,376 | 27,827 | 82,895 | 22,081 | 73,442 |

4 | 93,433 | 48,396 | 173,045 | 49,667 | 175,135 | 44,190 | 166,126 |

5 | 332,152 | 159,500 | 594,530 | 162,956 | 600,215 | 161,306 | 597,501 |

6 | 454,013 | 193,496 | 772,314 | 197,412 | 778,756 | 198,066 | 779,831 |

7 | 782,169 | 329,614 | 1,324,384 | 324,932 | 1,316,682 | 324,997 | 1,316,788 |

8 | 1,031,664 | 423,941 | 1,729,046 | 438,924 | 1,753,693 | 429,318 | 1,737,892 |

9 | 2,090,955 | 945,444 | 3,646,210 | 974,441 | 3,693,911 | 879,649 | 3,537,977 |

10 | 7,270,705 | 4,520,261 | 14,706,534 | 4,810,081 | 15,183,288 | 3,060,442 | 12,305,132 |

Total | 12,142,220 | 4,692,325 | 19,861,095 | 5,038,337 | 20,430,285 | 3,356,603 | 17,663,832 |

PPE | 7,586,523 | 19,728,743 | 6,993,945 | 19,136,166 | 2,839,397 | 14,981,617 | |

PPE/SEP | 1.617 | 0.993 | 1.388 | 0.937 | 0.846 | 0.848 |

Year | Est. Reserve | Without Corrections | Zero-Correction | Zero-Correction & Stand. | |||
---|---|---|---|---|---|---|---|

SEP | Upper 95% | SEP | Upper 95% | SEP | Upper 95% | ||

2 | 42,904 | 52,003 | 128,449 | 51,823 | 128,152 | 15,151 | 67,827 |

3 | 36,824 | 39,224 | 101,347 | 47,523 | 114,999 | 13,957 | 59,783 |

4 | 80,170 | 57,605 | 174,930 | 63,622 | 184,828 | 31,949 | 132,726 |

5 | 215,413 | 107,603 | 391,661 | 120,549 | 413,716 | 94,383 | 370,673 |

6 | 351,163 | 166,083 | 624,369 | 172,026 | 634,146 | 162,592 | 618,626 |

7 | 600,400 | 290,000 | 1,077,450 | 288,431 | 1,074,868 | 281,236 | 1,063,033 |

8 | 819,029 | 422,285 | 1,513,687 | 431,693 | 1,529,163 | 406,269 | 1,487,341 |

9 | 1,790,227 | 1,254,931 | 3,854,588 | 1,437,501 | 4,154,916 | 1,092,042 | 3,586,636 |

10 | 6,871,745 | 8,625,252 | 21,060,284 | 10,534,588 | 24,201,142 | 2,188,477 | 10,471,789 |

Total | 10,807,874 | 8,751,120 | 25,203,481 | 10,741,298 | 28,477,309 | 2,696,996 | 15,244,432 |

PPE | 6,591,299 | 17,399,173 | 6,084,858 | 16,892,732 | 3,153,855 | 13,961,729 | |

PPE/SEP | 0.753 | 0.690 | 0.566 | 0.593 | 1.169 | 0.916 |

Year | Est. Reserve | Without Corrections | Zero-Correction | Zero-Correction & Stand. | |||
---|---|---|---|---|---|---|---|

SEP | Upper 95% | SEP | Upper 95% | SEP | Upper 95% | ||

2 | 42,904 | 34,201 | 99,164 | 34,575 | 99,779 | 11,889 | 62,461 |

3 | 36,824 | 26,572 | 80,534 | 31,038 | 87,881 | 10,885 | 54,729 |

4 | 80,170 | 39,629 | 145,359 | 43,036 | 150,964 | 24,827 | 121,010 |

5 | 215,413 | 79,247 | 345,774 | 86,722 | 358,152 | 72,523 | 334,713 |

6 | 351,163 | 124,969 | 556,737 | 128,456 | 562,473 | 124,209 | 555,486 |

7 | 600,400 | 221,705 | 965,104 | 220,991 | 963,930 | 217,118 | 957,559 |

8 | 819,029 | 321,006 | 1,347,083 | 328,293 | 1,359,070 | 311,412 | 1,331,301 |

9 | 1,790,227 | 924,401 | 3,310,866 | 1,014,638 | 3,459,306 | 804,123 | 3,113,009 |

10 | 6,871,745 | 5,677,582 | 16,211,367 | 6,631,999 | 17,781,383 | 1,712,612 | 9,688,991 |

Total | 10,807,874 | 5,782,386 | 20,319,898 | 6,793,931 | 21,983,890 | 2,082,248 | 14,233,171 |

PPE | 5,934,922 | 16,742,796 | 5,279,051 | 16,086,925 | 2,693,775 | 13,501,649 | |

PPE/SEP | 1.026 | 0.824 | 0.0.777 | 0.731 | 1.294 | 0.949 |

Type of Adjustment | Poisson Model | Log-Normal Model | Gamma Model | ||
---|---|---|---|---|---|

Pearson | Anscombe | Pearson | Anscombe | Anscombe | |

Without corrections | 2281 | 1132 | 0 | 0 | 0 |

Zero-correction | 2314 | 1172 | 0 | 0 | 0 |

Zero-correction & Stand. | 2207 | 912 | 0 | 0 | 0 |

**Table 9.**The upper confidence limits for the total reserve and for the mean of the total reserve by the Poisson model and Pearson residuals.

The Quantile | The Adjustment Type | |||
---|---|---|---|---|

Without Corrections | Zero-Correction | Zero-Correction & Stand. | ||

90% | Reserve | 16,189,247 | 16,284,477 | 15,903,554 |

Mean | 15,893,535 | 15,894,705 | 15,457,467 | |

95% | Reserve | 17,046,324 | 17,712,940 | 17,141,555 |

Mean | 16,603,125 | 16,604,628 | 16,042,710 | |

99% | Reserve | 18,476,692 | 18,570,018 | 17,712,940 |

Mean | 17,934,822 | 17,936,951 | 17,141,042 |

**Table 10.**The upper confidence limits for the total reserve and for the mean of the total reserve by the Poisson model and the Anscombe residuals.

The Quantile | The Adjustment Type | |||
---|---|---|---|---|

Without Corrections | Zero-Correction | Zero-Correction & Stand. | ||

90% | Reserve | 15,903,554 | 16,189,247 | 15,617,861 |

Mean | 15,615,874 | 15,655,784 | 15,286,299 | |

95% | Reserve | 17,427,248 | 17,332,017 | 16,760,632 |

Mean | 16,246,288 | 16,297,578 | 15,822,732 | |

99% | Reserve | 18,094,816 | 18,285,278 | 17,143,460 |

Mean | 17,429,394 | 17,502,041 | 16,829,463 |

**Table 11.**Model validation using the Dawid–Sebastiani scoring rule (DSS). The three lowest scores are indicated in bold.

Model | Residual | Type of Adjustment | ||
---|---|---|---|---|

Without Corrections | 0’s Corrected | 0’s Corrected & Stand. | ||

Poisson | Pearson | 36.7 | 33.9 | 396.8 |

Anscombe | 46.2 | 44.2 | 598.4 | |

Gamma | Pearson | 112.3 | 106.6 | 256.9 |

Anscombe | 168.7 | 159.1 | 401.6 | |

Log-normal | Pearson | 103.1 | 98.2 | 251.6 |

Anscombe | 158.0 | 149.9 | 383.7 |

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**MDPI and ACS Style**

Tee, L.; Käärik, M.; Viin, R.
On Comparison of Stochastic Reserving Methods with Bootstrapping. *Risks* **2017**, *5*, 2.
https://doi.org/10.3390/risks5010002

**AMA Style**

Tee L, Käärik M, Viin R.
On Comparison of Stochastic Reserving Methods with Bootstrapping. *Risks*. 2017; 5(1):2.
https://doi.org/10.3390/risks5010002

**Chicago/Turabian Style**

Tee, Liivika, Meelis Käärik, and Rauno Viin.
2017. "On Comparison of Stochastic Reserving Methods with Bootstrapping" *Risks* 5, no. 1: 2.
https://doi.org/10.3390/risks5010002