# Overdispersed-Poisson Model in Claims Reserving: Closed Tool for One-Year Volatility in GLM Framework

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- (a)
- (b)
- Antonio and Beirlant (2008) GLMM (generalized linear mixed model) that allow for overcoming the hypothesis of independence among payments of claims occurring in the same generation but in different years—processed with stochastic simulation techniques;
- (c)
- Bjökwall et al. (2011) GLMs with smoothing effects;
- (d)
- Hudecovà and Pešta (2013) GEE (generalized estimating equations) implementation, where the connection among payments of the same accident year is made through a closed tool.

## 2. Claims Reserve Estimation

#### 2.1. Data Organization

#### 2.2. Chain Ladder Method: Basic Concept

#### 2.3. The Claims Development Result

## 3. Generalized Linear Models to Estimate the Claims Reserve

#### 3.1. GLM Models Structure

- the ${Y}_{ij}$ are stochastically independent;
- the density (or probability) function is in exponential family:$$f(y;{\theta}_{ij},\varphi )=exp\left\{\frac{{\omega}_{ij}}{\varphi}\left[y{\theta}_{ij}-b\left({\theta}_{ij}\right)\right]\right\}c(y;{\theta}_{ij},\varphi ),$$
- the moments can be generalized as follows:$$E\left[{Y}_{ij}\right]={g}^{-1}\left({x}_{ij}^{\top}\beta \right)={b}^{{}^{\prime}}\left({\theta}_{ij}\right)\phantom{\rule{1.em}{0ex}}and\phantom{\rule{1.em}{0ex}}Var\left[{Y}_{ij}\right]=\frac{\varphi}{{\omega}_{ij}}{b}^{{}^{\u2033}}\left({\theta}_{ij}\right)=\frac{\varphi}{{\omega}_{ij}}V\left({\mu}_{ij}\right),$$

**Remark**

**1.**

#### 3.2. Semi-Parametrical Models

#### 3.3. Elements for the Observed Data Goodness of Fit

## 4. The Claims Reserve Mean Square Error of Prediction

#### 4.1. The General Case

#### 4.2. GLMs Implementation in Claims Reserving

## 5. One-Year Volatility for the Claims Development Results

#### 5.1. Overall Accident Years Estimate

#### 5.2. One-Year Volatility for Accident Year

## 6. Numerical Investigation

- the coefficients of credibility alpha’s quantify the weight, in terms of influence, of the accident year in the next development factor calculation (decreasing for the more recent accident year);
- the q’s indicate the contribution to the overall volatility from the first development year to the latest development year and form a typical u-shape due to the level of the payment in the first year and to the small uncertainty for the oldest years for which residual payments relative to the still open claims over the ultimate cost are low;
- the r’s stand for the weight of the k-th development year parameter above the first k parameters decreasing with the increase of development year;
- the s’s involved directly in the accident year volatility is a function based on r interesting the next year development for this accident year plus the subsequent development year with credibility decreasing coefficients.

`R`software; the code is reported in vignettes 1, 2 and 3. The package

`ChainLadder`is requested and must be previously installed because it is not included in the default configuration. In order to replicate results in Table 3, Code 1 has to be run after the run-off triangle showed in Table 2 is uploaded and named

`Incremetal.Paid`. Code 2 computes the outcome of Table 7 and the

`R`Code 3 gives as outcome Table 4, Table 5 and Table 6.

Listing 1. Code for Table 3: Estimation GLM parameters. |

Listing 2. Code for Table 7: Estimation rMSEP ultimate. |

Listing 3. Code for Table 4, Table 5 and Table 6: Estimation rMSEP one-year. |

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BS | Bootstrap |

CDR | Claims Develpoment Result |

CT | Closed Tool |

GLM | Generalized Linear Model |

MSEP | Mean Square Error of Prediction |

rMSEP | $\sqrt{MSEP}$ |

USP | Undertaking Specific Parameters |

## References

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1 | Robert W. M. Wedderburn (1947–1975) could have become one of the most distinguished statistics experts of his time due to his early works in this field, but he died at the young age of 28 because of an anaphylactic shock caused by a wasp bite. |

2 | A useful reference about backtesting in relation to the use of the ultimate volatility of predictions is in Leong et al. (2014), winner of the Variance Prize. |

3 | The key to understanding how we have calculated these derivatives is that these residuals fluctuate around zero, the first exactly on average and the second only with the condition of a potential bias in the maximum likelihood estimate. Thus, the replacement of the empirical and the estimated figure with the hypothetical unknown value—respectively for $\xi $ and for $\zeta $—is possible. |

$\mathit{i}\phantom{\rule{0.277778em}{0ex}}/\phantom{\rule{0.277778em}{0ex}}\mathit{j}$ | 0 | 1 | ⋯ | j | ⋯ | J |
---|---|---|---|---|---|---|

1 | ${Y}_{10}$ | ${Y}_{11}$ | ⋯ | ${Y}_{1j}$ | ⋯ | ${Y}_{1J}$ |

2 | ${Y}_{20}$ | ${Y}_{21}$ | ⋯ | |||

⋮ | ⋮ | |||||

i | ${Y}_{i0}$ | ${Y}_{ij}$ | ||||

⋮ | ⋮ | |||||

I | ${Y}_{I0}$ |

i/j | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 22,603 | 39,938 | 35,073 | 25,549 | 20,031 | 17,593 | 14,930 | 15,004 | 10,319 | 8240 | 8104 | 6020 | 19,145 |

2 | 22,382 | 41,502 | 26,508 | 19,734 | 18,715 | 13,983 | 12,885 | 16,371 | 7921 | 7204 | 4428 | 12,897 | |

3 | 25,355 | 45,707 | 33,062 | 24,232 | 16,765 | 13,180 | 11,639 | 8864 | 9994 | 6044 | 3954 | ||

4 | 26,830 | 52,347 | 37,324 | 23,590 | 18,248 | 13,895 | 13,142 | 11,119 | 9429 | 5057 | |||

5 | 26,868 | 62,313 | 33,772 | 22,925 | 16,341 | 12,419 | 12,646 | 9459 | 6658 | ||||

6 | 28,470 | 56,097 | 41,672 | 24,843 | 22,818 | 18,787 | 16,947 | 14,942 | |||||

7 | 26,170 | 55,362 | 39,026 | 26,817 | 22,881 | 19,663 | 19,395 | ||||||

8 | 24,101 | 58,520 | 38,749 | 22,449 | 16,008 | 12,506 | |||||||

9 | 22,714 | 48,707 | 28,970 | 18,798 | 13,369 | ||||||||

10 | 19,973 | 38,262 | 23,298 | 14,819 | |||||||||

11 | 17,252 | 36,994 | 24,361 | ||||||||||

12 | 17,591 | 30,074 | |||||||||||

13 | 16,907 |

**Table 3.**Estimation of GLM parameter using the data in Table 2.

Parameter | Estimate | Std. Error | Parameter | Estimate | Std. Error |
---|---|---|---|---|---|

$\widehat{c}$ | 10.1263 | 0.0572 | |||

${\widehat{a}}_{2}$ | −0.0883 | 0.0620 | ${\widehat{b}}_{1}$ | 0.7024 | 0.0468 |

${\widehat{a}}_{3}$ | −0.0715 | 0.0629 | ${\widehat{b}}_{2}$ | 0.3132 | 0.0513 |

${\widehat{a}}_{4}$ | 0.0155 | 0.0620 | ${\widehat{b}}_{3}$ | −0.0972 | 0.0579 |

${\widehat{a}}_{5}$ | 0.0126 | 0.0628 | ${\widehat{b}}_{4}$ | −0.3241 | 0.0635 |

${\widehat{a}}_{6}$ | 0.1579 | 0.0614 | ${\widehat{b}}_{5}$ | −0.5254 | 0.0703 |

${\widehat{a}}_{7}$ | 0.1551 | 0.0627 | ${\widehat{b}}_{6}$ | −0.5737 | 0.0753 |

${\widehat{a}}_{8}$ | 0.0425 | 0.0662 | ${\widehat{b}}_{7}$ | −0.6904 | 0.0843 |

${\widehat{a}}_{9}$ | −0.1261 | 0.0716 | ${\widehat{b}}_{8}$ | −1.0112 | 0.1051 |

${\widehat{a}}_{10}$ | −0.3171 | 0.0795 | ${\widehat{b}}_{9}$ | −1.2910 | 0.1317 |

${\widehat{a}}_{11}$ | −0.3326 | 0.0858 | ${\widehat{b}}_{10}$ | −1.4622 | 0.1643 |

${\widehat{a}}_{12}$ | −0.4592 | 0.1044 | ${\widehat{b}}_{11}$ | −-0.9285 | 0.1553 |

${\widehat{a}}_{13}$ | −0.3909 | 0.1660 | ${\widehat{b}}_{12}$ | −0.2665 | 0.1573 |

k | $13-\mathit{k}$ | ${\widehat{\mathit{\alpha}}}_{\mathit{k}}^{\left(13\right)}$ | ${\widehat{\mathit{q}}}_{\mathit{k}+1}$ | ${\widehat{\mathit{\mu}}}_{\mathit{t}-\mathit{k},\mathit{k}+1}$ | ${\widehat{\mathit{r}}}_{\mathit{k}+1}^{\left(13\right)}$ |
---|---|---|---|---|---|

0 | 13 | 0.0569 | 0.0415 | 34127.94 | 0.6687 |

1 | 12 | 0.0563 | 0.0192 | 21598.78 | 0.3118 |

2 | 11 | 0.0677 | 0.0127 | 16260.70 | 0.1714 |

3 | 10 | 0.0738 | 0.0097 | 13162.94 | 0.1202 |

4 | 9 | 0.0965 | 0.0094 | 13026.95 | 0.0895 |

5 | 8 | 0.1264 | 0.0108 | 14693.99 | 0.0786 |

6 | 7 | 0.1619 | 0.0115 | 14633.21 | 0.0653 |

7 | 6 | 0.1937 | 0.0096 | 10647.17 | 0.0453 |

8 | 5 | 0.2077 | 0.0075 | 6959.96 | 0.0331 |

9 | 4 | 0.2630 | 0.0078 | 5882.08 | 0.0271 |

10 | 3 | 0.3271 | 0.0158 | 9194.30 | 0.0442 |

11 | 2 | 0.4779 | 0.0412 | 17527.56 | 0.0789 |

${\widehat{\mathit{s}}}_{2}$ | ${\widehat{\mathit{s}}}_{3}$ | ${\widehat{\mathit{s}}}_{4}$ | ${\widehat{\mathit{s}}}_{5}$ | ${\widehat{\mathit{s}}}_{6}$ | ${\widehat{\mathit{s}}}_{7}$ | ${\widehat{\mathit{s}}}_{8}$ | ${\widehat{\mathit{s}}}_{9}$ | ${\widehat{\mathit{s}}}_{10}$ | ${\widehat{\mathit{s}}}_{11}$ | ${\widehat{\mathit{s}}}_{12}$ | ${\widehat{\mathit{s}}}_{13}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

0.0789 | 0.0442 | 0.0271 | 0.0331 | 0.0453 | 0.0653 | 0.0786 | 0.0895 | 0.1202 | 0.1714 | 0.3118 | 0.6687 |

0.0377 | 0.0145 | 0.0071 | 0.0069 | 0.0088 | 0.0106 | 0.0099 | 0.0086 | 0.0089 | 0.0116 | 0.0176 | |

0.0377 | 0.0145 | 0.0071 | 0.0069 | 0.0088 | 0.0106 | 0.0099 | 0.0086 | 0.0089 | 0.0116 | ||

0.0377 | 0.0145 | 0.0071 | 0.0069 | 0.0088 | 0.0106 | 0.0099 | 0.0086 | 0.0089 | |||

0.0377 | 0.0145 | 0.0071 | 0.0069 | 0.0088 | 0.0106 | 0.0099 | 0.0086 | ||||

0.0377 | 0.0145 | 0.0071 | 0.0069 | 0.0088 | 0.0106 | 0.0099 | |||||

0.0377 | 0.0145 | 0.0071 | 0.0069 | 0.0088 | 0.0106 | ||||||

0.0377 | 0.0145 | 0.0071 | 0.0069 | 0.0088 | |||||||

0.0377 | 0.0145 | 0.0071 | 0.0069 | ||||||||

0.0377 | 0.0145 | 0.0071 | |||||||||

0.0377 | 0.0145 | ||||||||||

0.0377 |

a.y. | $\widehat{\mathit{rMSEP}}\left({\mathit{CDR}}_{{\mathit{i}}^{\ast},\mathit{t}+1}\right)$ | $\mathsf{\Delta}$% | $\mathit{\sigma}$% | $\mathsf{\Delta}$ | $\frac{\widehat{\mathit{rMSEP}}\left({\mathit{CDR}}_{{\mathit{i}}^{\ast},\mathit{t}+1}\right)}{\widehat{\mathit{rMSEP}}\left({\widehat{\mathit{R}}}_{\mathit{i}}\right)}$ | |||
---|---|---|---|---|---|---|---|---|

BS | CT | BS | CT | BS | CT | |||

1 | 0 | 0 | - | - | - | - | - | - |

2 | 3888 | 3870 | −0.46% | 22.12% | 22.08% | −0.04% | 100.00% | 100.00% |

3 | 3238 | 3234 | −0.12% | 11.96% | 11.97% | 0.01% | 68.54% | 68.52% |

4 | 3083 | 3073 | −0.32% | 8.70% | 8.69% | −0.01% | 56.59% | 56.47% |

5 | 3242 | 3233 | −0.28% | 7.67% | 7.66% | −0.01% | 54.97% | 54.98% |

6 | 3980 | 3969 | −0.28% | 6.68% | 6.67% | −0.01% | 55.91% | 55.72% |

7 | 4477 | 4473 | −0.09% | 6.05% | 6.05% | 0.00% | 56.53% | 56.43% |

8 | 4494 | 4490 | −0.09% | 5.56% | 5.56% | 0.00% | 54.46% | 54.53% |

9 | 4319 | 4333 | 0.32% | 5.31% | 5.33% | 0.02% | 52.08% | 52.24% |

10 | 4535 | 4538 | 0.07% | 5.64% | 5.65% | 0.01% | 53.53% | 53.50% |

11 | 5705 | 5691 | −0.25% | 5.98% | 5.97% | −0.01% | 57.11% | 56.98% |

12 | 8364 | 8341 | −0.27% | 7.91% | 7.90% | −0.01% | 67.22% | 67.34% |

13 | 21,651 | 21,616 | −0.16% | 14.69% | 14.69% | 0.00% | 86.09% | 86.17% |

Tot | 38,603 | 38,578 | −0.06% | 4.56% | 4.56% | 0.00% | 73.09% | 73.18% |

a.y. | ${\widehat{\mathit{R}}}_{\mathit{i}}$ | $\mathsf{\Delta}$% | $\widehat{\mathit{rMSEP}}\left({\widehat{\mathit{R}}}_{\mathit{i}}\right)$ | $\mathsf{\Delta}$% | ||
---|---|---|---|---|---|---|

BS | CT | BS | CT | |||

1 | 0 | 0 | 0 | 0 | - | - |

2 | 17,573 | 17,528 | −0.26% | 3888 | 3870 | −0.46% |

3 | 27,068 | 27,018 | −0.18% | 4724 | 4720 | −0.08% |

4 | 35,429 | 35,356 | −0.21% | 5448 | 5442 | −0.11% |

5 | 42,295 | 42,212 | −0.20% | 5898 | 5880 | −0.31% |

6 | 59,560 | 59,463 | −0.16% | 7118 | 7123 | 0.07% |

7 | 74,021 | 73,930 | −0.12% | 7920 | 7926 | 0.08% |

8 | 80,879 | 80,752 | −0.16% | 8252 | 8234 | −0.22% |

9 | 81,354 | 81,245 | −0.13% | 8293 | 8295 | 0.02% |

10 | 80,401 | 80,285 | −0.14% | 8472 | 8483 | 0.13% |

11 | 95,412 | 95,309 | −0.11% | 9989 | 9988 | −0.01% |

12 | 105,715 | 105,579 | −0.13% | 12,443 | 12,386 | −0.46% |

13 | 147,336 | 147,172 | −0.11% | 25,149 | 25,085 | −0.25% |

Tot | 847,041 | 845,851 | −0.14% | 52,813 | 52,714 | −0.19% |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Strascia, S.C.; Tripodi, A.
Overdispersed-Poisson Model in Claims Reserving: Closed Tool for One-Year Volatility in GLM Framework. *Risks* **2018**, *6*, 139.
https://doi.org/10.3390/risks6040139

**AMA Style**

Strascia SC, Tripodi A.
Overdispersed-Poisson Model in Claims Reserving: Closed Tool for One-Year Volatility in GLM Framework. *Risks*. 2018; 6(4):139.
https://doi.org/10.3390/risks6040139

**Chicago/Turabian Style**

Strascia, Stefano Cavastracci, and Agostino Tripodi.
2018. "Overdispersed-Poisson Model in Claims Reserving: Closed Tool for One-Year Volatility in GLM Framework" *Risks* 6, no. 4: 139.
https://doi.org/10.3390/risks6040139