# Individual Loss Reserving Using a Gradient Boosting-Based Approach

^{*}

^{†}

## Abstract

**:**

## 1. Introduction and Motivation

## 2. Loss Reserving

- If ${T}_{1}^{(k)}<{t}^{*}<{T}_{2}^{(k)}$, the accident has happened but has not yet been reported to the insurer. It is therefore called an “incurred but not reported” (IBNR), claim. For one of those claims, the insurer does not have specific information about the accident, but can use policyholder and external information to estimate the reserve.
- If ${T}_{2}^{(k)}<{t}^{*}<{T}_{3}^{(k)}$, the accident has been reported to the insurer but is still not settled, which means the insurer expects to make additional payments to the insured. It is therefore called a “reported but not settled” (RBNS), claim. For one such claim, the historical information as well as policyholder and external information can be used to estimate the reserve.
- If ${t}^{*}>{T}_{3}^{(k)}$, the claim is classified as settled, or S, and the insurer does not expect to make more payments.

## 3. Models for Loss Reserving

#### 3.1. Bootstrap Mack’s Model and Generalized Linear Models for Loss Reserving

#### 3.2. Gradient Boosting for Loss Reserving

`R`programming language in conjunction with

`caret`and

`xgboost`libraries.

`caret`is a powerful package used to train and to validate a wide range of statistical models including XGBoost algorithm.

- 1.
- The simplest solution is to train the model on data ${\mathcal{D}}_{\mathcal{T}}$ where only settled claims (or non-censored claims) are included. Hence, the response is known for all claims. However, this leads to a selection bias because claims that are already settled at ${t}^{*}$ tend to have shorter developments, and claims with shorter development tend to have lower total paid amounts. Consequently, the model is almost exclusively trained on simple claims with low training responses, which leads to underestimation of the total paid amounts for new claims. Furthermore, a significant proportion of the claims are removed from the analysis, which causes a loss of information. We will analyze this bias further in Section 4 (see model B).
- 2.
- In Lopez et al. (2016), a different and interesting approach is proposed: in order to correct the selection bias induced by the presence of censored data, a strategy called “inverse probability of censoring weighting” (IPCW) is implemented, which involves assigning weights to observations to offset the lack of complete observations in the sample. The weights are determined using the Kaplan-Meier estimator of the censoring distribution, and a modified CART algorithm is used to make the predictions.
- 3.
- A third approach is to develop claims that are still open at ${t}^{*}$ using parameters from a classical approach such as Mack’s or the GLM model. We discuss this idea in more detail in Section 4 (see model C and model D).

- we can assume that they are static, which can lead to a bias in the predictions obtained (see model E in Section 4); or
- we can, for each of these variables, (1) adjust a dynamic model, (2) obtain a prediction of the complete trajectory, and (3) use the algorithm conditionally to the realization of this trajectory. Moreover, there may be dependence between these variables, which would warrant a multivariate approach.

## 4. Analyses

#### 4.1. Data

#### 4.2. Training of XGBoost Models

#### 4.3. Learning of Prediction Function

Algorithm 1: Obtaining ${\widehat{f}}_{A}$ with least square TreeBoost. |

#### 4.4. Results

- Mack’s model, for which we present results obtained with the bootstrap approach developed by England and Verrall (2002), based on both quasi-Poisson and gamma distributions; and
- generalized linear models for which we present results obtained using a logarithmic link function and a variance function $\mathcal{V}(\mu )=\varphi {\mu}^{p}$ with $p=1$ (quasi-Poisson), $p=2$ (gamma), and $1<p<2$ (Tweedie).

- individual generalized linear models (see Section 3.1), for which we present results obtained using a logarithmic link function and three variance functions: $\mathcal{V}(\mu )=\mu $ (Poisson) and $\mathcal{V}(\mu )=\varphi {\mu}^{p}$ with $p=1$ (quasi-Poisson) and $\mathcal{V}(\mu )=\varphi {\mu}^{p}$ with $1<p<2$ (Tweedie); and
- XGBoost models (models A, B, C, D and E) described in Section 4.2.

## 5. Conclusions

- (1)
- the censored nature of the data could strongly bias the loss reserving process; and
- (2)
- the use of a micro-level model based solely on generalized linear models could be unstable for loss reserving but an approach combining a macro-level (or a micro-level) model for the artificial completion of open claims and a micro-level gradient-boosting model represents an interesting approach for an insurance company.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Structure of the Dataset

Policy Number | Claim Number | Party | File Number | Date | … |
---|---|---|---|---|---|

P100000 | C234534 | 1 | F0000001 | 31 March 2004 | … |

P100000 | C234534 | 1 | F0000001 | 30 June 2004 | … |

P100000 | C234534 | 1 | F0000001 | 30 September 2004 | … |

… | … | … | … | … | … |

P100000 | C234534 | 2 | F0000002 | 31 March 2004 | … |

P100000 | C234534 | 2 | F0000002 | 30 June 2004 | … |

P100000 | C234534 | 2 | F0000002 | 30 September 2004 | … |

… | … | … | … | … | … |

P100034 | C563454 | 1 | F0000140 | 31 March 2004 | … |

P100034 | C563454 | 1 | F0000140 | 30 June 2004 | … |

P100034 | C563454 | 1 | F0000140 | 30 September 2004 | … |

… | … | … | … | … | … |

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**Figure 2.**(

**a**) Status of claims of incurred amounts on 31 December 2016; (

**b**) Means and standard deviations of incurred amounts on 31 December 2016.

**Figure 3.**Comparison of predictive distributions (incurred but not reported (IBNR) + reported but not settled (RBNS)) for collective models.

Model | Response Variable (${\widehat{\mathit{C}}}_{{\mathit{T}}_{3}}$) | Covariates | Usable in Practice? |
---|---|---|---|

Model A | $\{{C}_{{T}_{3}}\}$ | ${\mathit{x}}_{7}$ | No |

Model B | ${\{{C}_{7}^{(k)}\}}_{k\in {\mathcal{T}}_{B}}$, ${\mathcal{T}}_{B}=\{k\in \mathcal{T}:{T}_{3}^{(k)}<7\}$ | ${\mathit{x}}_{7}$ | Yes |

Model C | closed: $\{{C}_{7}\}$ | ${\mathit{x}}_{7}$ | Yes |

open: $\{{\widehat{\lambda}}_{j}^{c}{C}_{7}\}$ ($\widehat{\lambda}$ from bootstrap) | ${\mathit{x}}_{7}$ | ||

Model D | all: $\{{C}_{7}+{\sum}_{t=8}^{13}{\widehat{Y}}_{t}\}$ (with ${\widehat{Y}}_{t}={q}_{{\mathit{Y}}_{t}}({\kappa}_{D})$) | ${\mathit{x}}_{7}$ | Yes |

Model E | closed: $\{{C}_{7}\}$ | ${\mathit{x}}_{13}$ | No |

open: $\{{\widehat{\lambda}}_{j}^{c}{C}_{7}\}$ ($\widehat{\lambda}$ from bootstrap) | ${\mathit{x}}_{13}$ |

Development Year | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

Accident year | |||||||

2004 | 79 | 102 | 66 | 49 | 57 | 48 | 37 |

2005 | 83 | 128 | 84 | 55 | 52 | 41 | · |

2006 | 91 | 138 | 69 | 49 | 38 | · | · |

2007 | 111 | 155 | 98 | 61 | · | · | · |

2008 | 100 | 178 | 99 | · | · | · | · |

2009 | 137 | 251 | · | · | · | · | · |

2010 | 155 | · | · | · | · | · | · |

Development Year | 1 | 2 | 3 | 4 | 5 | 6 | 7 | $8+$ |
---|---|---|---|---|---|---|---|---|

Accident year | ||||||||

2004 | 34 | 41 | 23 | 13 | 14 | 14 | 9 | 7 |

2005 | 37 | 60 | 36 | 29 | 45 | 21 | 20 | 24 |

2006 | 41 | 64 | 34 | 23 | 21 | 14 | 4 | 21 |

2007 | 46 | 67 | 40 | 37 | 15 | 18 | 3 | 13 |

2008 | 46 | 82 | 39 | 42 | 16 | 11 | 15 | 33 |

2009 | 54 | 109 | 62 | 51 | 31 | 36 | 11 | 2 |

2010 | 66 | 93 | 47 | 45 | 16 | 16 | 9 | ? |

**Table 4.**Prediction results (incurred but not reported (IBNR) + reported but not settled (RBNS)) for collective approaches.

Model | Assessment | $\mathbf{E}\phantom{\rule{-0.166667em}{0ex}}\left[\mathbf{Res}.\right]$ | $\sqrt{\mathbf{Var}\phantom{\rule{-0.166667em}{0ex}}\left[\mathbf{Res}.\right]}$ | ${\mathit{q}}_{0.95}$ | ${\mathit{q}}_{0.99}$ |
---|---|---|---|---|---|

Bootstrap Mack | out-of-sample | 76,795,136 | 7,080,826 | 89,086,213 | 95,063,184 |

(quasi-Poisson) | in-sample | 75,019,768 | 8,830,631 | 90,242,398 | 97,954,554 |

Bootstrap Mack | out-of-sample | 76,803,753 | 7,170,529 | 89,133,141 | 95,269,308 |

(Gamma) | in-sample | 75,004,053 | 8,842,412 | 90,500,323 | 98,371,607 |

GLM | out-of-sample | 75,706,046 | 2,969,877 | 80,655,890 | 82,696,002 |

(Quasi-Poisson) | in-sample | 74,778,091 | 3,084,216 | 79,922,183 | 81,996,425 |

GLM | out-of-sample | 73,518,411 | 2,263,714 | 77,276,416 | 78,907,812 |

(Gamma) | in-sample | 71,277,218 | 3,595,958 | 77,343,035 | 80,204,504 |

GLM | out-of-sample | 75,688,916 | 2,205,003 | 79,317,520 | 80,871,729 |

(Tweedie) | in-sample | 74,706,050 | 2,197,659 | 78,260,722 | 79,790,056 |

Model | Assessment | $\mathbf{E}\phantom{\rule{-0.166667em}{0ex}}\left[\mathbf{Res}.\right]$ | $\sqrt{\mathbf{Var}\phantom{\rule{-0.166667em}{0ex}}\left[\mathbf{Res}.\right]}$ | ${\mathit{q}}_{0.95}$ | ${\mathit{q}}_{0.99}$ | |
---|---|---|---|---|---|---|

Poisson | out-of-sample | 86,411,734 | 9007 | 86,426,520 | 86,431,211 | |

in-sample | 75,611,203 | 8655 | 75,625,348 | 75,631,190 | ||

Quasi-Poisson | out-of-sample | 86,379,296 | 894,853 | 87,815,685 | 88,309,697 | |

in-sample | 75,606,230 | 814,608 | 76,984,768 | 77,433,248 | ||

Tweedie | out-of-sample | 84,693,529 | 2,119,280 | 88,135,187 | 90,011,542 | |

in-sample | 70,906,225 | 1,994,004 | 74,098,686 | 75,851,991 |

Model | $\mathbf{E}\phantom{\rule{-0.166667em}{0ex}}\left[\mathbf{Res}.\right]$ | $\sqrt{\mathbf{Var}\phantom{\rule{-0.166667em}{0ex}}\left[\mathbf{Res}.\right]}$ | ${\mathit{q}}_{0.95}$ | ${\mathit{q}}_{0.99}$ |
---|---|---|---|---|

Model A | 73,204,299 | 3,742,971 | 79,329,916 | 82,453,032 |

Model B | 14,339,470 | 6,723,608 | 25,757,061 | 30,643,369 |

Model C | 67,655,960 | 2,411,739 | 71,708,313 | 73,762,242 |

Model D | 68,313,731 | 4,176,418 | 75,408,868 | 78,517,966 |

Model E | 67,772,822 | 2,387,476 | 71,722,744 | 73,540,516 |

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**MDPI and ACS Style**

Duval, F.; Pigeon, M.
Individual Loss Reserving Using a Gradient Boosting-Based Approach. *Risks* **2019**, *7*, 79.
https://doi.org/10.3390/risks7030079

**AMA Style**

Duval F, Pigeon M.
Individual Loss Reserving Using a Gradient Boosting-Based Approach. *Risks*. 2019; 7(3):79.
https://doi.org/10.3390/risks7030079

**Chicago/Turabian Style**

Duval, Francis, and Mathieu Pigeon.
2019. "Individual Loss Reserving Using a Gradient Boosting-Based Approach" *Risks* 7, no. 3: 79.
https://doi.org/10.3390/risks7030079