# A Robust General Multivariate Chain Ladder Method

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. General Multivariate Chain Ladder Model

## 3. Seemingly Unrelated Regression

- ${\mathit{y}}_{k}^{\left(m\right)}={\left(\right)}^{{C}_{1,k+1}^{\left(m\right)}}\prime $ is the $n\left(k\right)$ vector of all observed losses at development period $k+1$ from triangle m;
- ${\mathit{X}}_{k}^{\left(m\right)}={({(1,{\mathit{C}}_{1,k}^{\prime})}^{\prime},\dots ,{(1,{\mathit{C}}_{n\left(k\right),k}^{\prime})}^{\prime})}^{\prime}$ is the $n\left(k\right)\times (M+1)$ matrix of the first $n\left(k\right)$ observations at development period k from each triangle, including the constant 1 for the intercept. Hence, ${\mathit{X}}_{k}^{\left(1\right)}=\dots ={\mathit{X}}_{k}^{\left(M\right)}$;
- ${\mathit{\beta}}_{k}^{\left(m\right)}={\left(\right)}^{{\beta}_{0,k}^{\left(m\right)}}\prime $ is the $M+1$ vector of development parameters of triangle m, including the intercept;
- ${\mathit{\epsilon}}_{k}^{\left(m\right)}={\left(\right)}^{{\u03f5}_{1,k}^{\left(m\right)}}\prime $ is the $n\left(k\right)$ vector of error terms of triangle m.

## 4. Robust GMCL Method

- $\rho $ is symmetric, twice continuously differentiable and satisfies $\rho \left(0\right)=0$;
- $\rho $ is strictly increasing on $[0,c]$ and constant on $[c,\infty [$ for some $c>0$.

## 5. Simulation Study

## 6. Real Data

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Table A1.**Development parameter estimates and empirical correlation estimates obtained from SCL-LS for a real insurance portfolio.

k | ${\widehat{\mathit{\beta}}}_{11}$ | ${\widehat{\mathit{\beta}}}_{22}$ | ${\widehat{\mathit{\beta}}}_{33}$ | ${\tilde{\mathit{\rho}}}_{12}$ | ${\tilde{\mathit{\rho}}}_{13}$ | ${\tilde{\mathit{\rho}}}_{23}$ |
---|---|---|---|---|---|---|

1 | 1.29 | 1.04 | 1.88 | 0.13 | 0.51 | 0.04 |

2 | 1.14 | 1.01 | 1.18 | −0.22 | −0.08 | 0.13 |

3 | 1.08 | 0.99 | 1.35 | 0.20 | −0.08 | −0.08 |

4 | 1.05 | 1.01 | 1.06 | 0.26 | −0.02 | −0.09 |

5 | 1.04 | 1.00 | 1.12 | 0.11 | −0.02 | 0.18 |

6 | 1.03 | 1.00 | 1.05 | −0.22 | −0.01 | 0.08 |

7 | 1.03 | 1.00 | 1.01 | −0.14 | −0.11 | 0.53 |

8 | 1.02 | 0.99 | 1.03 | 0.38 | 0.14 | 0.26 |

9 | 1.02 | 0.99 | 1.02 | 0.39 | 0.14 | 0.01 |

10 | 1.01 | 1.01 | 1.01 | 0.36 | −0.11 | 0.17 |

11 | 1.02 | 1.00 | 1.01 | −0.35 | −0.01 | −0.03 |

12 | 1.01 | 0.99 | 1.03 | 0.26 | 0.16 | 0.08 |

13 | 1.01 | 1.01 | 1.02 | −0.29 | −0.13 | −0.28 |

14 | 1.01 | 0.99 | 1.03 | 0.17 | 0.05 | −0.28 |

15 | 1.02 | 0.99 | 1.02 | 0.11 | −0.23 | −0.01 |

16 | 1.01 | 0.99 | 1.01 | 0.09 | 0.43 | 0.49 |

17 | 1.01 | 1.00 | 1.03 | −0.23 | −0.17 | 0.24 |

18 | 1.01 | 0.99 | 1.03 | −0.54 | −0.18 | −0.08 |

19 | 1.01 | 0.99 | 1.03 | 0.08 | −0.28 | 0.32 |

20 | 1.04 | 0.99 | 1.01 | −0.37 | −0.07 | −0.04 |

**Table A2.**Development parameter estimates and correlation estimates obtained from GMCL-FGLS for a real insurance portfolio.

k | ${\widehat{\mathit{\beta}}}_{01}$ | ${\widehat{\mathit{\beta}}}_{11}$ | ${\widehat{\mathit{\beta}}}_{21}$ | ${\widehat{\mathit{\beta}}}_{31}$ | ${\widehat{\mathit{\beta}}}_{02}$ | ${\widehat{\mathit{\beta}}}_{12}$ | ${\widehat{\mathit{\beta}}}_{22}$ | ${\widehat{\mathit{\beta}}}_{32}$ | ${\widehat{\mathit{\beta}}}_{03}$ | ${\widehat{\mathit{\beta}}}_{13}$ | ${\widehat{\mathit{\beta}}}_{23}$ | ${\widehat{\mathit{\beta}}}_{33}$ | ${\widehat{\mathit{\rho}}}_{12}$ | ${\widehat{\mathit{\rho}}}_{13}$ | ${\widehat{\mathit{\rho}}}_{23}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 23,397.72 | 1.14 | 0.02 | 0.93 | −11.47 | 0.08 | 1.00 | 0.83 | 21,694.08 | −0.02 | 0.00 | 1.22 | 0.20 | 0.50 | 0.03 |

2 | 15,223.35 | 1.09 | 0.01 | −0.15 | 20,020.27 | 0.12 | 0.95 | −0.23 | 1727.03 | 0.01 | 0.00 | 1.07 | −0.22 | −0.10 | 0.04 |

3 | 16,228.14 | 0.99 | 0.04 | −0.14 | 15,116.47 | −0.01 | 0.99 | −0.03 | −12,277.95 | 0.05 | −0.02 | 1.57 | 0.23 | 0.02 | −0.07 |

4 | 10,350.14 | 1.00 | 0.03 | −0.06 | 50,876.00 | −0.07 | 1.03 | −0.11 | 4182.92 | 0.00 | 0.00 | 1.00 | 0.23 | −0.19 | −0.23 |

5 | 1028.93 | 0.94 | 0.05 | −0.01 | −6957.99 | −0.05 | 1.04 | 0.01 | −1377.61 | 0.02 | −0.01 | 1.01 | −0.01 | 0.04 | 0.19 |

6 | 12,243.16 | 0.97 | 0.03 | −0.03 | 8286.35 | 0.06 | 0.98 | −0.36 | 10,968.80 | 0.00 | 0.00 | 0.97 | −0.29 | −0.01 | −0.01 |

7 | −3719.21 | 1.02 | 0.00 | 0.04 | −6260.32 | −0.13 | 1.07 | 0.00 | −379.22 | 0.00 | 0.00 | 1.00 | −0.22 | −0.05 | 0.62 |

8 | −755.07 | 1.03 | 0.00 | −0.01 | 5287.58 | −0.04 | 1.00 | 0.17 | −1120.14 | 0.00 | 0.00 | 1.01 | 0.41 | 0.19 | 0.45 |

9 | −11,302.41 | 1.05 | −0.01 | −0.07 | −4825.36 | 0.00 | 1.00 | −0.08 | 904.91 | 0.00 | 0.00 | 1.00 | 0.36 | 0.08 | −0.05 |

10 | 6920.22 | 0.97 | 0.03 | 0.01 | 37,848.84 | −0.15 | 1.09 | −0.06 | 502.78 | 0.00 | 0.00 | 1.00 | 0.17 | −0.02 | 0.24 |

11 | 9660.89 | 0.95 | 0.04 | 0.00 | −27,830.17 | 0.09 | 0.96 | −0.05 | −438.20 | 0.00 | 0.00 | 1.02 | −0.26 | 0.20 | −0.04 |

12 | −16,214.89 | 1.01 | 0.01 | 0.00 | 8784.70 | −0.07 | 1.03 | 0.10 | −1370.21 | 0.01 | 0.00 | 1.00 | 0.20 | 0.14 | 0.16 |

13 | −18,821.47 | 1.00 | 0.02 | 0.01 | −30,184.25 | −0.08 | 1.07 | 0.08 | −1385.69 | 0.01 | 0.00 | 1.00 | −0.44 | 0.02 | −0.16 |

14 | −17,224.86 | 1.00 | 0.02 | 0.00 | 40,874.99 | −0.06 | 1.02 | −0.19 | −11,617.13 | 0.01 | 0.00 | 1.02 | 0.08 | 0.08 | −0.32 |

15 | −20,373.50 | 1.02 | 0.00 | 0.12 | −24,051.79 | 0.06 | 0.97 | −0.11 | −7141.82 | 0.00 | 0.00 | 1.01 | 0.20 | −0.21 | −0.12 |

16 | −2082.74 | 1.02 | 0.00 | −0.02 | 17,582.20 | 0.02 | 0.98 | −0.05 | 1397.56 | 0.00 | 0.00 | 1.00 | 0.05 | 0.36 | 0.43 |

17 | −44,523.11 | 1.04 | 0.00 | −0.02 | 61,268.64 | −0.04 | 1.00 | 0.03 | 2554.84 | 0.00 | 0.00 | 1.05 | −0.07 | −0.04 | −0.11 |

18 | −13,650.45 | 1.02 | 0.00 | −0.03 | −51,338.15 | 0.02 | 0.99 | 0.05 | −6862.07 | 0.01 | 0.00 | 1.04 | −0.56 | −0.13 | −0.10 |

19 | −37,910.90 | 1.01 | 0.01 | 0.06 | 4693.44 | 0.00 | 0.99 | 0.02 | −55,064.83 | 0.04 | 0.00 | 0.97 | 0.06 | −0.45 | 0.56 |

20 | 874,470.74 | 0.53 | 0.07 | −0.48 | −76,063.75 | 0.04 | 0.99 | 0.01 | −1304.77 | 0.00 | 0.00 | 1.00 | −0.31 | −0.18 | −0.03 |

**Table A3.**Development parameter estimates and correlation estimates obtained from GMCL-MM for a real insurance portfolio.

k | ${\widehat{\mathit{\beta}}}_{01}$ | ${\widehat{\mathit{\beta}}}_{11}$ | ${\widehat{\mathit{\beta}}}_{21}$ | ${\widehat{\mathit{\beta}}}_{31}$ | ${\widehat{\mathit{\beta}}}_{02}$ | ${\widehat{\mathit{\beta}}}_{12}$ | ${\widehat{\mathit{\beta}}}_{22}$ | ${\widehat{\mathit{\beta}}}_{32}$ | ${\widehat{\mathit{\beta}}}_{03}$ | ${\widehat{\mathit{\beta}}}_{13}$ | ${\widehat{\mathit{\beta}}}_{23}$ | ${\widehat{\mathit{\beta}}}_{33}$ | ${\widehat{\mathit{\rho}}}_{12}$ | ${\widehat{\mathit{\rho}}}_{13}$ | ${\widehat{\mathit{\rho}}}_{23}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 7820.38 | 1.15 | 0.02 | 1.16 | −3680.41 | 0.08 | 1.00 | 1.03 | 1717.07 | 0.01 | 0.00 | 1.11 | 0.24 | −0.31 | 0.06 |

2 | 12,144.56 | 1.09 | 0.01 | −0.11 | 16,619.03 | 0.13 | 0.95 | −0.19 | 873.94 | 0.01 | 0.00 | 1.06 | 0.20 | 0.11 | −0.10 |

3 | 23,528.36 | 1.00 | 0.03 | −0.20 | 22,422.99 | 0.00 | 0.98 | −0.10 | 1918.65 | 0.01 | 0.00 | 0.99 | 0.08 | 0.30 | 0.10 |

4 | 8438.94 | 1.01 | 0.02 | −0.04 | 891.69 | 0.06 | 0.97 | −0.11 | 4896.14 | 0.00 | 0.00 | 1.00 | 0.03 | 0.00 | 0.21 |

5 | −2355.67 | 0.98 | 0.03 | −0.02 | −30,886.96 | −0.06 | 1.05 | 0.04 | 1715.40 | −0.01 | 0.00 | 1.03 | −0.03 | −0.21 | −0.04 |

6 | 8351.98 | 0.97 | 0.04 | −0.04 | 9538.34 | 0.07 | 0.97 | −0.33 | −209.96 | 0.00 | 0.00 | 1.00 | −0.29 | −0.20 | −0.08 |

7 | −2873.28 | 1.02 | 0.00 | 0.03 | −4771.62 | −0.12 | 1.07 | 0.02 | −243.36 | 0.00 | 0.00 | 1.00 | −0.23 | −0.17 | 0.64 |

8 | −806.41 | 1.00 | 0.01 | 0.01 | 821.12 | −0.06 | 1.02 | 0.13 | −1135.19 | 0.00 | 0.00 | 1.01 | 0.06 | 0.09 | 0.32 |

9 | −6931.74 | 1.03 | 0.00 | −0.03 | 1925.54 | −0.03 | 1.01 | −0.02 | 1272.45 | 0.00 | 0.00 | 1.00 | −0.19 | 0.21 | 0.02 |

10 | 8446.18 | 0.97 | 0.02 | 0.00 | 13,573.18 | 0.00 | 0.99 | −0.06 | 44.18 | 0.00 | 0.00 | 1.00 | −0.46 | −0.05 | −0.17 |

11 | −1481.68 | 0.98 | 0.03 | 0.00 | −3558.47 | 0.04 | 0.97 | 0.04 | −588.16 | 0.00 | 0.00 | 1.02 | −0.02 | 0.15 | −0.03 |

12 | −19,036.01 | 1.01 | 0.01 | 0.00 | 10,657.18 | −0.07 | 1.03 | 0.08 | −1020.05 | 0.00 | 0.00 | 1.00 | 0.13 | 0.77 | 0.10 |

13 | −17,979.52 | 1.03 | 0.00 | −0.02 | 21,175.00 | −0.07 | 1.03 | 0.08 | −1469.87 | 0.01 | 0.00 | 1.00 | 0.23 | −0.25 | −0.18 |

14 | −6110.32 | 1.01 | 0.00 | 0.00 | −21,779.08 | 0.02 | 1.00 | −0.08 | −4066.28 | 0.00 | 0.00 | 1.00 | −0.34 | 0.66 | 0.16 |

15 | −2628.61 | 0.99 | 0.00 | 0.15 | −20,629.80 | 0.05 | 0.97 | −0.10 | 219.13 | 0.00 | 0.00 | 1.01 | −0.07 | −0.50 | −0.11 |

16 | 621.54 | 1.02 | 0.00 | −0.03 | −42,626.20 | 0.03 | 1.00 | −0.05 | −2510.24 | 0.00 | 0.00 | 0.99 | −0.59 | 0.79 | −0.22 |

17 | −39,374.59 | 1.04 | 0.00 | −0.10 | 70,972.58 | −0.07 | 1.00 | 0.25 | 2017.96 | 0.00 | 0.00 | 1.00 | 0.15 | −0.12 | 0.60 |

18 | 25,424.10 | 0.98 | 0.01 | −0.02 | 101,648.10 | −0.11 | 1.03 | 0.09 | −25,270.86 | 0.02 | −0.01 | 1.02 | 0.12 | 0.09 | −0.97 |

19 | −42,462.66 | 1.02 | 0.02 | −0.11 | 74,563.74 | −0.04 | 1.01 | −0.13 | 4055.82 | 0.00 | 0.00 | 1.02 | 0.83 | −0.89 | −0.99 |

20 | −23,405.29 | 1.01 | 0.00 | 0.03 | −61,530.32 | 0.04 | 0.99 | 0.00 | 2593.46 | 0.00 | 0.00 | 1.00 | 0.21 | 0.52 | −0.08 |

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**Figure 1.**Cumulative and incremental claims for a pair of dependent run-off triangles. Development periods are on the horizontal axis, accident periods are on the vertical axis. The bar plot represents a color code indicating the magnitude of the numbers.

**Figure 2.**RMSEP estimates of ${\widehat{C}}_{I,2}^{\left(1\right)}$ obtained from SCL-LS, GMCL-FGLS and GMCL-MM as a function of I for the restricted, general and outlier settings.

**Figure 4.**RMSEP estimates of ${\widehat{C}}_{I,2}^{\left(1\right)}$ obtained from SCL-LS, GMCL-FGLS and GMCL-MM as a function of the outlier distance d.

**Figure 5.**RMSEP estimates of ${\widehat{C}}_{I,15}^{\left(1\right)}$ obtained from SCL-LS, GMCL-FGLS and GMCL-MM as a function of I for the general setting.

**Figure 6.**Cumulative run-off triangles (divided by 100,000) of a real insurance portfolio. Development periods are on the horizontal axis, accident periods are on the vertical axis. The bar plot represents a color code indicating the magnitude of the numbers.

**Figure 7.**Weights obtained from GMCL-MM for a real insurance portfolio. Each row corresponds to an accident trimester used in the fitting procedure. Each columns represents a SUR model.

Accident | Development Period k | ||||||
---|---|---|---|---|---|---|---|

Period$\mathit{i}$ | 1 | 2 | $\mathit{k}$ | $\mathit{I}-\mathbf{1}$ | $\mathit{I}$ | ||

1 | |||||||

2 | ${C}_{i,k}^{\left(m\right)}$ | ||||||

(observed) | |||||||

i | |||||||

${C}_{i,k}^{\left(m\right)}$ | |||||||

$I-1$ | (predicted) | ||||||

I |

**Table 2.**Total reserve estimates for all run-off triangles of a real insurance portfolio obtained from SCL-LS, GMCL-FGLS and GMCL-MM.

Method | Run-Off Triangle | ||
---|---|---|---|

MTPL Paid | MTPL Incurred | GTPL Paid | |

SCL-LS | 1,924,001 | −654,695 | 386,949 |

GMCL-FGLS | 12,198,112 | −1,175,336 | −670,116 |

GMCL-MM | 167,221 | 1,043,591 | −128,463 |

**Table 3.**MSEP for the last diagonal of all run-off triangles (and totals) of a real insurance portfolio obtained from SCL-LS, GMCL-FGLS and GMCL-MM.

Method | Run-Off Triangle | Total | ||
---|---|---|---|---|

MTPL Paid | MTPL Incurred | GTPL Paid | ||

SCL-LS | 0.024 | 0.021 | 0.142 | 0.187 |

GMCL-FGLS | 0.032 | 0.057 | 0.337 | 0.426 |

GMCL-MM | 0.024 | 0.040 | 0.076 | 0.140 |

© 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**

Peremans, K.; Van Aelst, S.; Verdonck, T.
A Robust General Multivariate Chain Ladder Method. *Risks* **2018**, *6*, 108.
https://doi.org/10.3390/risks6040108

**AMA Style**

Peremans K, Van Aelst S, Verdonck T.
A Robust General Multivariate Chain Ladder Method. *Risks*. 2018; 6(4):108.
https://doi.org/10.3390/risks6040108

**Chicago/Turabian Style**

Peremans, Kris, Stefan Van Aelst, and Tim Verdonck.
2018. "A Robust General Multivariate Chain Ladder Method" *Risks* 6, no. 4: 108.
https://doi.org/10.3390/risks6040108