# Loss Reserving Estimation With Correlated Run-Off Triangles in a Quantile Longitudinal Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dependence (Correlation) Modeling in Loss Run-Off Triangles

#### 2.1. Correlation Within Claims Reserving Triangles

#### 2.2. Correlation Between Claims Reserving Triangles

#### 2.3. Correlation Within and Between Claims Reserving Triangles

#### 2.4. Why Quantile Regression Models with Correlated Run-off Triangles?

## 3. Preliminaries on Quantile Functions and on Quantile Regression

#### 3.1. Quantile Function

#### 3.2. Quantile Regression Estimation

## 4. Correlated Run-Off Triangles in a Quantile Longitudinal Model

#### 4.1. Quantile Regression with Longitudinal Data

#### 4.2. The Uniform Correlation Model

**Theorem**

**1.**

**Proof.**

**Proposition**

**1.**

**Proof.**

**Remark**

**1.**

#### 4.3. Parameters Estimation for QR Longitudinal Model

- Step 1. Produce some initial values $\tilde{\mathbf{\beta}}}^{0}={\widehat{\mathbf{\beta}}}_{I$, which have been obtained by the independence working model and $\mathsf{\Gamma}}^{0}={n}^{-1}{I}_{p$.
- Step 2. Given $\tilde{\mathbf{\beta}}}^{k-1$ and $\mathsf{\Gamma}}^{k-1$ from the $k-1$ step, update $\widehat{\delta}}^{k-1$, using the following equation:$$\begin{array}{c}\hfill {\widehat{\delta}}^{k-1}={\displaystyle \frac{{\sum}_{i=1}^{N}{\sum}_{k=1}^{{n}_{i}}{\sum}_{l\ne k}^{{n}_{i}}I[{\widehat{\u03f5}}_{ik}\le 0,{\widehat{\u03f5}}_{il}\le 0]}{{\sum}_{i=1}^{N}{n}_{i}({n}_{i}-1)}}.\end{array}$$
- Step 3. Update the estimation parameters $\tilde{\mathbf{\beta}}}^{k$ and the matrix $\mathsf{\Gamma}}^{k$ using the equations$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\tilde{\mathbf{\beta}}}^{k}={\tilde{\mathbf{\beta}}}^{k-1}+{\left\{{\tilde{D}}_{\theta}({\tilde{\mathbf{\beta}}}^{k-1},{\mathsf{\Gamma}}^{k-1})\right\}}^{-1}{\tilde{U}}_{\theta}({\tilde{\mathbf{\beta}}}^{k-1},{\mathsf{\Gamma}}^{k-1},{\widehat{\delta}}^{k-1}),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\mathsf{\Gamma}}^{k}={\tilde{D}}_{\theta}^{-1}({\tilde{\mathbf{\beta}}}^{k-1},{\mathsf{\Gamma}}^{k-1})V({\tilde{\mathbf{\beta}}}^{k-1},{\widehat{\delta}}^{k-1}){\tilde{D}}_{\theta}^{-1}({\tilde{\mathbf{\beta}}}^{k-1},{\mathsf{\Gamma}}^{k-1}).\hfill \end{array}$$
- Step 4. Repeat Steps 2 and 3 until convergence.

**Remark**

**2.**

## 5. Numerical Illustrations

#### 5.1. Numerical Example Based on Average Premium Per Exposure

**Remark**

**3.**

#### 5.2. Comparison Criteria

## 6. Risk Capital Requirement and Risk Margin

- Probability of sufficiency below 50% indicates that the technical provisions are set below the central estimate, which leads to an under-reserved position.
- Probability of sufficiency with values between 50% and 60% indicates that the technical provisions are approximately at the level of central estimate, which leads to weak prudence.
- For values of probability of sufficiency around 75%, the technical provisions are above the central estimate, which leads to adequate prudence.
- Finally, if the probability of sufficiency is above 75%, the technical provisions are enough to lead to strong prudence.

#### Risk Margin

**Remark**

**4.**

## 7. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Abdallah, Anas, Jean-Philippe Boucher, and Hélène Cossette. 2015. Modeling Dependence between Loss Triangles with Hierarchical Archimedean Copulas. ASTIN Bulletin 45: 577–99. [Google Scholar] [CrossRef] [Green Version]
- Ajne, Björn. 1994. Additivity of chain–ladder projections. ASTIN Bulletin 24: 313–8. [Google Scholar] [CrossRef] [Green Version]
- Antonio, Katrien, and Jan Beirlant. 2006. Actuarial statistics with generalized linear mixed models. Insurance: Mathematics and Economics 75: 643–76. [Google Scholar] [CrossRef]
- Avanzi, Benjamin, Greg Taylor, Phuong Anh Vu, and Bernard Wong. 2016a. Stochastic loss reserving with dependence: A flexible multivariate Tweedie approach. Insurance: Mathematics and Economics 71: 63–78. [Google Scholar] [CrossRef] [Green Version]
- Avanzi, Benjamin, Greg Taylor, and Bernard Wong. 2016b. Correlations between insurance lines of business: An illusion or a real phenomenon? Some methodological considerations. ASTIN Bulletin 46: 225–63. [Google Scholar] [CrossRef] [Green Version]
- Avanzi, Benjamin, Greg Taylor, and Bernard Wong. 2018. Common Shock Models for Claim Arrays. ASTIN Bulletin 48: 1109–36. [Google Scholar] [CrossRef] [Green Version]
- Barnett, Glen, and Ben Zehnwirth. 2000. Best estimates for reserves. Proceedings of the Casualty Actuarial Society 87: 245–321. [Google Scholar]
- Bartlett, M. S. 1951. An inverse matrix adjustment arising in discriminant analysis. Annals of Mathematical Statistics 22: 107–11. [Google Scholar] [CrossRef]
- Bermúdez, Lluis, Antoni Ferri, and Montserrat Guille. 2013. A Correlation Sensitivity Analysis for non-life underwriting risk module SCR. ASTIN Bulletin 43: 21–37. [Google Scholar] [CrossRef] [Green Version]
- Braun, Christian. 2004. The prediction error of the chain ladder method applied to correlated run-off triangles. ASTIN Bulletin 34: 399–423. [Google Scholar] [CrossRef] [Green Version]
- Brown, Bruce Maxwell, and You-Gan Wang. 2005. Standard errors and covariance matrices for smoothed rank estimators. Biometrika 92: 149–58. [Google Scholar] [CrossRef]
- Buchinsky, Moshe. 1998. Recent advances in regression models: A practical guideline for empirical research. The Journal of Human Resources 33: 88–126. [Google Scholar] [CrossRef]
- CEIOPS. 2009. Advice for Level 2 Implementing Measures on Solvency II: Technical Provisions, Article 86 (d). Frankfurt: CEIOPS. [Google Scholar]
- Chan, Jeniffer S. K., S. T. Boris Choy, and Udi E. Makov. 2008. Dynamic and robust models for loss reserves using generalized-t distribution. ASTIN Bulletin 38: 207–30. [Google Scholar] [CrossRef] [Green Version]
- Chen, Li, Lee-Jen Wei, and Michael I. Parzen. 2004. Quantile regression for correlated observations. In Proceedings of the Second Seattle Symposium in Biostatistics. Lecture Notes in Statistics. New York: Springer, vol. 179, pp. 51–69. [Google Scholar]
- Christofides, S. 1990. Regression models based on log-incremental payments. In Claims Reserving Manual 2. London: Institute of Actuaries. [Google Scholar]
- Clark, David R. 2006. Variance and covariance due to inflation. CAS Forum 11: 61–95. [Google Scholar]
- Dal Moro, Eric, and Yuriy Krvavych. 2017. Probability of sufficiency of solvency II reserve risk margins: Practical approximations. ASTIN Bulletin 47: 737–85. [Google Scholar] [CrossRef]
- De Jong, Piet. 2012. Modeling Dependence between Loss Triangles. North American Actuarial Journal 16: 74–86. [Google Scholar] [CrossRef]
- Diggle, Peter J., Patrick J. Heagerty, Kung-Yee Liang, and Scott L. Zeger. 2002. Analysis of Longitudinal Data. New York: Oxford University Press. [Google Scholar]
- Dong, Alice X. D., Jennifer S. K. Chan, and Gareth W. Peters. 2015. Risk margin quantile function via parametric and non-parametric bayesian approaches. ASTIN Bulletin 45: 503–50. [Google Scholar] [CrossRef]
- Fu, Liya, and You-Gan Wang. 2012. Quantile regression for longitudinal data with a working correlation model. Computational Statistics & Data Analysis 56: 2526–38. [Google Scholar]
- Gesmann, M., D. Murphy, Y. Zhang, A. Carrato, M. Wuthrich, F. Concina, and E. Dal Moro. 2018. ChainLadder: Statistical Methods and Models for Claims Reserving in General Insurance. R package version 0.2.9. Available online: https://CRAN.R-project.org/package=ChainLadder (accessed on 27 May 2019).
- Gilchrist, Warren G. 2000. Statistical Modelling with Quantile Functions. London: Chapman & Hall. [Google Scholar]
- Harrison, David A., and Charles L. Hulin. 1989. Investigations of absenteeism: Using event-history models to study the absence-taking process. Journal of Applied Psychology 74: 300–16. [Google Scholar] [CrossRef]
- Holmberg, Randall D. 1994. Correlation and the measurement of loss reserve variability. CAS Forum 1: 247–78. [Google Scholar]
- Hubert, Mia, Tim Verdonck, and Özlem Yorulmaz. 2017. Fast robust SUR with economical and actuarial applications. Statistical Analysis and Data Mining 10: 77–88. [Google Scholar] [CrossRef]
- Hudecová, Šárka, and Michal Pešta. 2013. Modeling dependencies in claims reserving with GEE. Insurance: Mathematics and Economics 53: 786–94. [Google Scholar] [CrossRef] [Green Version]
- Hyndman, Rob J., and Anne B. Koehler. 2006. Another look at measures of forecast accuracy. International Journal of Forecasting 22: 679–88. [Google Scholar] [CrossRef] [Green Version]
- Jung, Sin-Ho. 1996. Quasi-likelihood for median regression models. Journal of the American Statistical Association 91: 251–7. [Google Scholar] [CrossRef]
- Kim, Sungil, and Heeyoung Kim. 2016. A new metric of absolute percentage error for intermittent demand forecasts. International Journal of Forecasting 32: 669–79. [Google Scholar] [CrossRef]
- Koenker, Roger, and Gilbert Basset. 1982. Robust Tests for Heteroscedasticity Based on Regression Quantiles. Econometrica 50: 43–61. [Google Scholar] [CrossRef] [Green Version]
- Koenker, Roger, and Vasco D’Orey. 1994. A remark on algorithm AS229: Computing dual regression quantiles and regression rank scores. Applied Statistics 43: 410–4. [Google Scholar] [CrossRef]
- Koenker, Roger. 2005. Quantile Regression. Cambridge: University Press. [Google Scholar]
- Koenker, Roger. 2018. Quantreg: Quantile Regression. R Package Version 5.36. Available online: https://CRAN.R-project.org/package=quantreg (accessed on 13 December 2019).
- Kremer, Erhard. 2005. The correlated chain-ladder method for reserving in case of correlated claims developments. Blatter DGVFM 27: 315–22. [Google Scholar] [CrossRef]
- Kuang, Di, Bent Nielsen, and Jens Perch Nielsen. 2011. Forecasting in an Extended Chain-Ladder–Type Model. The Journal of Risk and Insurance 78: 345–59. [Google Scholar] [CrossRef] [Green Version]
- Mack, Thomas. 1993. Distribution-free calculation of the standard error of chain-ladder reserve estimates. ASTIN Bulletin 23: 213–25. [Google Scholar] [CrossRef] [Green Version]
- Merz, Michael, and Mario Wüthrich. 2008a. Prediction error of the chain ladder reserving method applied to correlated run off trapezoids. Annals of Actuarial Science 2: 25–50. [Google Scholar] [CrossRef] [Green Version]
- Merz, Michael, and Mario Wüthrich. 2008b. Prediction error of the multivariate chain ladder reserving method. North American Actuarial Journal 12: 175–97. [Google Scholar] [CrossRef]
- Merz, Michael, Mario Wüthrich, and Enkelejd Hashorva. 2012. Dependence modelling in multivariate claims run-off triangles. Annals of Actuarial Science 7: 3–25. [Google Scholar] [CrossRef] [Green Version]
- Pang, Lei, Wenbin Lu, and Huixia Judy Wang. 2012. Variance estimation in censored quantile regression via induced smoothing. Computational Statistics and Data Analysis 56: 785–96. [Google Scholar] [CrossRef] [Green Version]
- Parzen, M. I., L. J. Wei, and Z. Ying. 1994. A resampling method based on pivotal estimating functions. Biometrika 81: 341–50. [Google Scholar] [CrossRef]
- Peremans, Kris, Stefan Van Aelst, and Tim Verdonck. 2018. A robust general multivariate chain ladder method. Risks 6: 108. [Google Scholar] [CrossRef] [Green Version]
- Pešta, Michal, and Ostap Okhrin. 2014. Conditional least squares and copulae in claims reserving for a single line of business. Insurance: Mathematics and Economics 56: 28–37. [Google Scholar] [CrossRef] [Green Version]
- Pitt, David G. W. 2006. Regression quantile analysis of claim termination rates for income protection insurance. Annals of Actuarial Science 1: 345–57. [Google Scholar] [CrossRef] [Green Version]
- Pontius, Robert Gilmore, Olufunmilayo Thontteh, and Hao Chen. 2008. Components of information for multiple resolution comparison between maps that share a real variable. Environmental Ecological Statistics 15: 111–42. [Google Scholar] [CrossRef]
- Pröhl, Carsten, and Klaus D. Schmidt. 2005. Multivariate Chain-Ladder. Dresdner Schriften zur Versicherungsmathematik. Available online: https://www.math.tu-dresden.de/sto/schmidt/dsvm/dsvm2005-3.pdf (accessed on 13 December 2019).
- Quarg, Gerhard, and Thomas Mack. 2004. Munich chain ladder. Blatter DGVFM 4: 597–630. [Google Scholar] [CrossRef]
- Radtke, Michael, Klaus D. Schmidt, and Anja Schnaus. 2012. Handbook on Loss Reserving. EAA Lecture Notes. European Actuarial Academy. Berlin: Springer. [Google Scholar]
- Schmidt, Klaus D. 2006. Optimal and additive loss reserving for dependent lines of business. Paper presented at 2006 CAS Casualty Loss Reserve Seminar, Atlanta, GA, USA, September 11–12; pp. 319–51. [Google Scholar]
- Searle, Shayle R., George Casella, and Charles E. McCulloch. 1992. Variance Components. New York: John Wiley and Sons. [Google Scholar]
- Shi, Peng, and Edward Frees. 2011. Dependent loss reserving using copulas. ASTIN Bulletin 41: 449–86. [Google Scholar]
- Shi, Peng, Sanjib Basu, and Glenn G. Meyers. 2012. A Bayesian log-normal model for multivariate loss reserving. North American Actuarial Journal 16: 29–51. [Google Scholar] [CrossRef]
- Stoner, Julie A., and Brian G. Leroux. 2002. Analysis of clustered data: A combined estimating equations approach. Biometrika 89: 567–78. [Google Scholar] [CrossRef]
- Taylor, Greg, and Grainne Mcguire. 2005. Synchronous bootstrapping of seemingly unrelated regressions. Paper presented at the 36th International ASTIN Colloquium, ETH, Zurich, September 4–7. [Google Scholar]
- Taylor, Greg, and Grainne Mcguire. 2007. A synchronous bootstrap to account for dependencies between lines of business in the estimation of loss reserve prediction error. North American Actuarial Journal 11: 70–88. [Google Scholar] [CrossRef]
- Taylor, Gregory. 2000. Loss Reserving—An Actuarial Perspective. Norwell: Kluwer Academic Publishers. [Google Scholar]
- Wang, You-Gan, Quanxi Shao, and Min Zhu. 2009. Quantile regression without the curse of unsmoothness. Computational Statistics and Data Analysis 53: 3696–705. [Google Scholar] [CrossRef]
- Willmott, Cort, and Kenji Matsuura. 2006. On the use of dimensioned measures of error to evaluate the performance of spatial interpolators. International Journal of Geographic Information Science 20: 89–102. [Google Scholar] [CrossRef]
- Wüthrich, Mario, and Michael Merz. 2008. Stochastic Claims Reserving Methods in Insurance. Hoboken: Wiley Finance. [Google Scholar]
- Wüthrich, Mario V. 2010. Accounting Year Effects Modeling in the Stochastic Chain Ladder Reserving Method. North American Actuarial Journal 14: 235–55. [Google Scholar] [CrossRef]
- Yin, Guosheng, and Jianwen Cai. 2005. Quantile regression models with multivariate failure time data. Biometrics 61: 151–61. [Google Scholar] [CrossRef]
- Zehnwirth, Ben, and Glen Barnett. 2001. Reserving for multiple excess layers. Paper presented at the ASTIN Colloquium at the ASTIN Colloquium, Washington, DC, USA, July 8–11. [Google Scholar]
- Zhang, Yanwei, Vanja Dukic, and James Guszcza. 2012. A Bayesian non-linear model for forecasting insurance loss payments. Journal Royal Statistical Society A 175: 637–56. [Google Scholar] [CrossRef]
- Zhang, Yanwei. 2010. A general multivariate chain ladder model. Insurance: Mathematics and Economics 46: 588–99. [Google Scholar] [CrossRef]

Accident | Development Year j | ||||||
---|---|---|---|---|---|---|---|

Yearr | 1 | 2 | $\cdots$ | j | $\cdots$ | I$-$1 | I |

1 | $Y}_{11$ | $Y}_{12$ | $\cdots$ | $Y}_{1j$ | $\cdots$ | $Y}_{1,I-1$ | $Y}_{1I$ |

2 | $Y}_{21$ | $Y}_{22$ | $\cdots$ | $Y}_{2j$ | $\cdots$ | $Y}_{2,I-1$ | |

$\vdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | ||

r | $Y}_{r1$ | $\cdots$ | $\cdots$ | $Y}_{r,I+1-r$ | |||

$\vdots$ | $\cdots$ | $\cdots$ | $\cdots$ | ||||

I | $Y}_{I1$ |

Subject | Observation | Response | Covariates | ||
---|---|---|---|---|---|

1 | 1 | $y}_{11$ | $x}_{111$ | $\cdots$ | $x}_{11p$ |

1 | 2 | $y}_{12$ | $x}_{121$ | $\cdots$ | $x}_{12p$ |

$\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |

1 | $n}_{1$ | $y}_{1{n}_{1}$ | $x}_{1{n}_{1}1$ | $\cdots$ | $x}_{1{n}_{1}p$ |

$\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |

$\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |

N | 1 | $y}_{N1$ | $x}_{N11$ | $\cdots$ | $x}_{N1p$ |

N | 2 | $y}_{N2$ | $x}_{N21$ | $\cdots$ | $x}_{N2p$ |

$\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ |

N | $n}_{N$ | $y}_{N{n}_{N}$ | $x}_{N{n}_{N}1$ | $\cdots$ | $x}_{N{n}_{N}p$ |

Accident | Development Year | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Year | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Premium |

2007 | 58,134 | 162,688 | 101,105 | 100,964 | 61,591 | 71,009 | 34,024 | 2746 | 646 | 10,190 | 1,051,637 |

2008 | 51,437 | 197,139 | 120,641 | 74,807 | 76,771 | 77,276 | 39,070 | 4396 | 13,809 | 1,190,965 | |

2009 | 57,906 | 116,191 | 143,953 | 103,883 | 70,760 | 177,194 | 35,341 | 6088 | 1,327,568 | ||

2010 | 40,352 | 121,837 | 88,389 | 320,429 | 75,127 | 70,190 | 63723 | 1,418,348 | |||

2011 | 82,227 | 279,591 | 151,260 | 230,293 | 82,378 | 47,315 | 1,504,056 | ||||

2012 | 196,417 | 119,755 | 228,499 | 99,894 | 44,266 | 1,580,233 | |||||

2013 | 67,161 | 107,098 | 198,252 | 75,172 | 1,619,382 | ||||||

2014 | 78,293 | 141,865 | 106,150 | 1,727,540 | |||||||

2015 | 74,472 | 118,886 | 1,820,104 | ||||||||

2016 | 43,281 | 1,883,017 |

Accident | Development Year | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Year | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Premium |

2007 | 63,078 | 143,002 | 144,235 | 75,007 | 60,775 | 70,804 | 27,508 | 4757 | 3172 | 6385 | 1,633,833 |

2008 | 65,567 | 177,292 | 107,870 | 137,305 | 72,741 | 68,708 | 102,864 | 4335 | 6107 | 1,675,707 | |

2009 | 87,394 | 146,346 | 158,876 | 199,846 | 53,161 | 72,764 | 42,915 | 10,898 | 1,636,855 | ||

2010 | 70,017 | 153,893 | 119,028 | 93,771 | 49,600 | 185,689 | 28,331 | 1,689,715 | |||

2011 | 104,638 | 186,326 | 335,477 | 136,857 | 87,941 | 69,248 | 1,649,386 | ||||

2012 | 76,390 | 190,629 | 192,606 | 121,704 | 66,297 | 1,712,587 | |||||

2013 | 58,620 | 184,557 | 135,174 | 118,180 | 2,105,361 | ||||||

2014 | 87,845 | 166,511 | 145,385 | 2,265,432 | |||||||

2015 | 53,616 | 152,751 | 1,976,188 | ||||||||

2016 | 62,904 | 1,351,719 |

Company | 0–1 | 1–2 | 2–3 | 3–4 | 4–5 | 5—6 | 6-7 | 7–8 | 8–9 |
---|---|---|---|---|---|---|---|---|---|

A | 2.9324 | 1.6060 | 1.3737 | 1.1265 | 1.1491 | 1.0677 | 1.0068 | 1.0117 | 1.0171 |

B | 3.2502 | 1.6822 | 1.3042 | 1.1188 | 1.1541 | 1.0782 | 1.0096 | 1.0069 | 1.0107 |

Accident | Development Year | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

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

2007 | 118 | 272 | 241 | 169 | 103 | 106 | 71 | 4 | 3 | 5 |

2008 | 134 | 287 | 235 | 192 | 121 | 117 | 84 | 14 | 13 | |

2009 | 129 | 254 | 267 | 193 | 110 | 136 | 80 | 16 | ||

2010 | 87 | 255 | 218 | 152 | 111 | 86 | 52 | |||

2011 | 148 | 285 | 277 | 212 | 127 | 94 | ||||

2012 | 108 | 255 | 227 | 185 | 103 | |||||

2013 | 129 | 215 | 220 | 150 | ||||||

2014 | 128 | 277 | 234 | |||||||

2015 | 122 | 236 | ||||||||

2016 | 94 |

Accident | Development Year | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

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

2007 | 139 | 286 | 276 | 170 | 137 | 140 | 74 | 15 | 8 | 6 |

2008 | 143 | 337 | 258 | 224 | 158 | 158 | 90 | 20 | 13 | |

2009 | 151 | 273 | 310 | 239 | 145 | 135 | 81 | 11 | ||

2010 | 138 | 285 | 273 | 182 | 122 | 127 | 70 | |||

2011 | 161 | 372 | 349 | 282 | 185 | 129 | ||||

2012 | 131 | 327 | 297 | 237 | 150 | |||||

2013 | 144 | 345 | 284 | 222 | ||||||

2014 | 146 | 337 | 295 | |||||||

2015 | 130 | 301 | ||||||||

2016 | 155 |

Accident | Quantile 50% | Quantile 60% | Quantile 75% | Quantile 90% | Quantile 95% | Quantile 99.5% | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Year | Reserves | Ultimate | Reserves | Ultimate | Reserves | Ultimate | Reserves | Ultimate | Reserves | Ultimate | Reserves | Ultimate |

2007 | 0 | 603,097 | 0 | 603,097 | 0 | 603,097 | 0 | 603,097 | 0 | 603,097 | 0 | 603,097 |

2008 | 11,702 | 667,049 | 11,496 | 666,843 | 12,348 | 667,695 | 12,348 | 667,695 | 12,348 | 667,695 | 12,348 | 667,695 |

2009 | 25,522 | 736,838 | 25,770 | 737,085 | 30,734 | 742,050 | 30,734 | 742,050 | 30,734 | 742,050 | 30,734 | 742,050 |

2010 | 30,123 | 810,171 | 32,114 | 812,161 | 48,436 | 828,484 | 48,436 | 828,484 | 61,841 | 841,888 | 61,841 | 841,888 |

2011 | 84,962 | 958,027 | 86,226 | 959,291 | 102,917 | 975,981 | 102,917 | 975,981 | 102,917 | 975,981 | 102,917 | 975,981 |

2012 | 144,793 | 833,624 | 150,033 | 838,863 | 295,824 | 984,655 | 485,528 | 1,174,359 | 485,528 | 1,174,359 | 485,528 | 1,174,359 |

2013 | 218,293 | 665,976 | 223,617 | 671,300 | 222,374 | 670,057 | 485,688 | 933,370 | 485,688 | 933,370 | 485,688 | 933,370 |

2014 | 376,255 | 702,563 | 396,658 | 722,966 | 439,704 | 766,012 | 439,675 | 765,983 | 400,741 | 727,049 | 400,741 | 727,049 |

2015 | 433,878 | 627,236 | 514,291 | 707,648 | 547,762 | 741,119 | 519,185 | 712,542 | 482,151 | 675,508 | 482,151 | 675,508 |

2016 | 364,632 | 407,912 | 377,312 | 420,592 | 439,462 | 482,742 | 396,155 | 439,435 | 374,632 | 417,912 | 374,632 | 417,912 |

Total | 1,690,161 | 7,012,491 | 1,817,516 | 7,139,846 | 2,139,562 | 7,461,893 | 2,520,666 | 7,842,996 | 2,436,579 | 7,758,909 | 2,436,579 | 7,758,909 |

LR | 46.37% | 47.21% | 49.34% | 51.86% | 51.31% | 51.31% |

Accident | Quantile 50% | Quantile 60% | Quantile 75% | Quantile 90% | Quantile 95% | Quantile 99.5% | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Year | Reserves | Ultimate | Reserves | Ultimate | Reserves | Ultimate | Reserves | Ultimate | Reserves | Ultimate | Reserves | Ultimate |

2007 | 0 | 598,722 | 0 | 598,722 | 0 | 598,722 | 0 | 598,722 | 0 | 598,722 | 0 | 598,722 |

2008 | 7915 | 750,705 | 8760 | 751,550 | 9338 | 752,128 | 7642 | 750,431 | 7642 | 750,431 | 7642 | 750,431 |

2009 | 15,633 | 787,833 | 15,012 | 787,213 | 14,631 | 786,831 | 20,011 | 792,211 | 20,011 | 792,211 | 20,011 | 792,211 |

2010 | 16,638 | 716,969 | 16,780 | 717,111 | 20,452 | 720,783 | 19,694 | 720,025 | 19,694 | 720,025 | 19,694 | 720,025 |

2011 | 64,164 | 984,651 | 77,249 | 997,735 | 156,940 | 1,077,426 | 232,413 | 1,152,900 | 232,413 | 1,152,900 | 232,413 | 1,152,900 |

2012 | 124,174 | 771,801 | 131,694 | 779,320 | 212,699 | 860,326 | 348,719 | 996,345 | 348,719 | 996,345 | 348,719 | 996,345 |

2013 | 184,956 | 681,488 | 196,298 | 692,830 | 270,111 | 766,642 | 413,334 | 909,865 | 413,334 | 909,865 | 413,334 | 909,865 |

2014 | 326,036 | 725,777 | 308,932 | 708,673 | 422,205 | 821,946 | 621,763 | 1,021,504 | 621,763 | 1,021,504 | 621,763 | 1,021,504 |

2015 | 398,134 | 604,501 | 382,772 | 589,139 | 477,907 | 684,274 | 614,736 | 821,104 | 614,736 | 821,104 | 614,736 | 821,104 |

2016 | 514,302 | 577,206 | 499,960 | 562,864 | 588,777 | 651,681 | 742,201 | 805,105 | 742,201 | 805,105 | 742,201 | 805,105 |

Total | 1,651,953 | 7,199,653 | 1,637,457 | 7,185,156 | 2,173,060 | 7,720,759 | 3,020,513 | 8,568,213 | 3,020,513 | 8,568,213 | 3,020,513 | 8,568,213 |

LR | 40.68% | 40.60% | 43.63% | 48.42% | 48.42% | 48.42% |

Accident | Quantile 50% | Quantile 60% | Quantile 75% | Quantile 90% | Quantile 95% | Quantile 99.5% | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Year | Reserves | Ultimate | Reserves | Ultimate | Reserves | Ultimate | Reserves | Ultimate | Reserves | Ultimate | Reserves | Ultimate |

2007 | 0 | 603,097 | 0 | 603,097 | 0 | 603,097 | 0 | 603,097 | 0 | 603,097 | 0 | 603,097 |

2008 | 10,465 | 665,812 | 13,163 | 668,509 | 14,614 | 669,961 | 14,998 | 670,344 | 16,354 | 671,701 | 14,284 | 669,631 |

2009 | 15,332 | 726,647 | 19,519 | 730,834 | 24,491 | 735,807 | 25,753 | 737,069 | 27,823 | 739,139 | 24,671 | 735,987 |

2010 | 18,773 | 798,820 | 24,548 | 804,595 | 30,472 | 810,519 | 33,114 | 813,161 | 36,000 | 816,047 | 31,594 | 811,641 |

2011 | 109,292 | 982,356 | 146,947 | 1,020,012 | 162,394 | 1,035,459 | 280,876 | 1,153,941 | 270,861 | 1,143,926 | 271,168 | 1,144,233 |

2012 | 142,717 | 831,548 | 195,409 | 884,240 | 227,356 | 916,187 | 388,819 | 1,077,650 | 534,912 | 1,223,743 | 535,184 | 1,224,015 |

2013 | 121,615 | 569,298 | 164,628 | 612,311 | 205,183 | 652,866 | 313,448 | 761,131 | 303,832 | 751,515 | 304,086 | 751,769 |

2014 | 143,793 | 470,101 | 216,128 | 542,436 | 274,426 | 600,734 | 369,152 | 695,460 | 398,023 | 724,331 | 624,747 | 951,055 |

2015 | 156,935 | 350,292 | 245,965 | 439,323 | 255,528 | 448,885 | 319,659 | 513,016 | 354,969 | 548,327 | 355,450 | 548,807 |

2016 | 108,851 | 152,131 | 205,266 | 248,546 | 205,286 | 248,567 | 211,902 | 255,182 | 211,842 | 255,123 | 211,977 | 255,257 |

Total | 827,771 | 6,150,102 | 1,231,572 | 6,553,903 | 1,399,751 | 6,722,081 | 1,957,720 | 7,280,050 | 2,154,617 | 7,476,947 | 2,373,161 | 7,695,491 |

LR | 40.67% | 43.34% | 44.45% | 48.14% | 49.44% | 50.89% |

Accident | Quantile 50% | Quantile 60% | Quantile 75% | Quantile 90% | Quantile 95% | Quantile 99.5% | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Year | Reserves | Ultimate | Reserves | Ultimate | Reserves | Ultimate | Reserves | Ultimate | Reserves | Ultimate | Reserves | Ultimate |

2007 | 0 | 598,722 | 0 | 598,722 | 0 | 598,722 | 0 | 598,722 | 0 | 598,722 | 0 | 598,722 |

2008 | 14,812 | 757,602 | 19,390 | 762,179 | 21,292 | 764,082 | 21,818 | 764,608 | 23,573 | 766,363 | 20,865 | 763,655 |

2009 | 23,811 | 796,012 | 31,465 | 803,665 | 38,726 | 810,927 | 40,529 | 812,730 | 43,425 | 815,625 | 38,967 | 811,168 |

2010 | 31,032 | 731,363 | 42,019 | 742,349 | 51,128 | 751,459 | 54,931 | 755,262 | 59,120 | 759,451 | 52,655 | 752,985 |

2011 | 146,138 | 1,066,624 | 203,353 | 1,123,839 | 221,691 | 1,142,178 | 370,226 | 1,290,712 | 358,282 | 1,278,769 | 358,674 | 1,279,161 |

2012 | 220,915 | 868,541 | 313,593 | 961,219 | 357,766 | 1,005,392 | 589,509 | 1,237,135 | 806,340 | 1,453,966 | 806,747 | 1,454,373 |

2013 | 197,552 | 694,083 | 279,346 | 775,877 | 337,958 | 834,489 | 497,873 | 994,404 | 484,759 | 981,290 | 485,153 | 981,684 |

2014 | 217,454 | 617,195 | 342,324 | 742,065 | 418,633 | 818,374 | 541,538 | 941,279 | 583,882 | 983,622 | 899,021 | 1,298,762 |

2015 | 243,640 | 450,007 | 397,190 | 603,557 | 410,815 | 617,182 | 490,441 | 696,808 | 541,561 | 747,928 | 542,253 | 748,621 |

2016 | 189,118 | 252,022 | 356,630 | 419,534 | 356,665 | 419,569 | 368,159 | 431,063 | 368,056 | 430,960 | 368,290 | 431,194 |

Total | 1,284,472 | 6,832,172 | 1,985,308 | 7,533,008 | 2,214,675 | 7,762,375 | 2,975,023 | 8,522,722 | 3,268,999 | 8,816,698 | 3,572,625 | 9,120,325 |

LR | 38.61% | 42.57% | 43.86% | 48.16% | 49.82% | 51.54% |

**Table 12.**Estimated reserves using Individual Quantile Regression (IQR) and the Longitudinal Quantile Regression (LQR).

Quantile 50% | Quantile 60% | Quantile 75% | Quantile 90% | Quantile 95% | Quantile 99.5% | |
---|---|---|---|---|---|---|

Company A IQR | 1,690,161 | 1,817,516 | 2,139,562 | 2,520,666 | 2,436,579 | 2,436,579 |

Company B IQR | 1,651,953 | 1,637,457 | 2,173,060 | 3,020,513 | 3,020,513 | 3,020,513 |

Company A LQR | 827,771 | 1,231,572 | 1,399,751 | 1,957,720 | 2,154,617 | 2,373,161 |

Company B LQR | 1,284,472 | 1,985,308 | 2,214,675 | 2,975,023 | 3,268,999 | 3,572,625 |

sumIQR-sumLQR | 1,229,871 | 238,092 | 698,196 | 608,436 | 33,477 | $-$488,694 |

Root Mean Square Error (RMSE) | Percentage Total (PT) | |||||
---|---|---|---|---|---|---|

Quantile | Company A | Company B | Longitudinal | Company A | Company B | Longitudinal |

50% | 457.52 | 248.60 | 398.07 | 82.78 | 90.37 | 84.02 |

60% | 455.12 | 244.40 | 389.59 | 90.26 | 93.66 | 93.32 |

75% | 511.97 | 263.43 | 388.36 | 123.37 | 106.04 | 106.02 |

90% | 730.79 | 466.38 | 536.24 | 158.87 | 139.59 | 151.32 |

95% | 693.67 | 466.38 | 762.02 | 158.87 | 139.59 | 185.72 |

99.5% | 730.79 | 466.38 | 789.86 | 158.87 | 139.59 | 186.01 |

Mean Absolute Error (MAE) | Mean Absolute Percentage Error (MAPE) | |||||
---|---|---|---|---|---|---|

Quantile | Company A | Company B | Combined | Company A | Company B | Combined |

50% | 272.30 | 158.31 | 212.53 | 38.00% | 27.86% | 31.99% |

60% | 274.38 | 155.00 | 210.89 | 39.13% | 27.19% | 32.60% |

75% | 375.69 | 178.86 | 233.63 | 62.06% | 33.69% | 39.49% |

90% | 521.73 | 336.46 | 441.93 | 93.37% | 66.41% | 84.37% |

95% | 520.07 | 336.46 | 610.00 | 92.54% | 66.41% | 83.07% |

99.5% | 522.96 | 336.46 | 614.50 | 93.53% | 66.41% | 84.38% |

Company A | Company B | |||||
---|---|---|---|---|---|---|

Accident Year | SCR | Capital Charge 6% | Discounted Capital Charge | SCR | Capital Charge 6% | Discounted Capital Charge |

1% Discount Rate | (1% Discount Rate) | |||||

1 | 340,874 | 20,452 | 20,452 | 383,015 | 22,981 | 22,981 |

2 | 231,100 | 13,866 | 13,594 | 287,630 | 17,258 | 16,919 |

3 | 93,717 | 5623 | 5405 | 259,313 | 15,559 | 14,955 |

4 | 52,555 | 3153 | 2971 | 208,157 | 12,489 | 11,769 |

5 | 40,749 | 2445 | 2259 | 170,761 | 10,246 | 9465 |

6 | 55,963 | 3358 | 3041 | 57,434 | 3446 | 3121 |

7 | 5585 | 335 | 298 | 2310 | 139 | 123 |

8 | 3663 | 220 | 191 | 60 | 4 | 3 |

9 | 1672 | 100 | 86 | 0 | 0 | 0 |

Total | 825,879 | 49553 | RM = 48,297 | 1,368,680 | 82,121 | RM = 79,337 |

Company A | Company B | |||||
---|---|---|---|---|---|---|

Accident Year | SCR | Capital Charge 6% | Discounted Capital Charge | SCR | Capital Charge 6% | Discounted Capital Charge |

1% Discount Rate | (1% Discount Rate) | |||||

1 | 389,957 | 23,397 | 23,397 | 511,493 | 30,690 | 30,690 |

2 | 321,754 | 19,305 | 19,114 | 436,219 | 26,173 | 25,914 |

3 | 244,764 | 14,686 | 14,396 | 368,081 | 22,085 | 21,650 |

4 | 118,911 | 7135 | 6925 | 162,896 | 9774 | 9486 |

5 | 73,566 | 4414 | 4242 | 116,999 | 7020 | 6746 |

6 | 12,598 | 756 | 719 | 19,949 | 1197 | 1139 |

7 | 1699 | 102 | 96 | 2103 | 126 | 119 |

8 | 554 | 33 | 31 | 550 | 33 | 31 |

9 | 0 | 0 | 0 | 0 | 0 | 0 |

Total | 1,163,802 | 69,828 | RM =68,921 | 1,618,289 | 97,097 | RM = 95,774 |

Company A | Company B | |||||
---|---|---|---|---|---|---|

Accident Year | SCR | Capital Charge 6% | Discounted Capital Charge | SCR | Capital Charge 6% | Discounted Capital Charge |

1% Discount Rate | (1% Discount Rate) | |||||

1 | 2,79,592 | 16775 | 16775 | 369922 | 22195 | 12797 |

2 | 245258 | 14715 | 14570 | 362541 | 21752 | 11567 |

3 | 209005 | 12540 | 12293 | 287763 | 17266 | 9251 |

4 | 168286 | 10097 | 9800 | 213236 | 12794 | 7042 |

5 | 147734 | 8864 | 8518 | 215192 | 12912 | 6485 |

6 | 93559 | 5614 | 5341 | 142384 | 8543 | 4099 |

7 | 63681 | 3821 | 3599 | 80768 | 4846 | 2289 |

8 | 56266 | 3376 | 3149 | 75132 | 4508 | 1842 |

9 | 37758 | 2265 | 2092 | 69887 | 4193 | 1515 |

Total | 1301139 | 78068 | RM = 76138 | 1816825 | 109009 | RM = 56886 |

© 2020 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**

Badounas, I.; Pitselis, G.
Loss Reserving Estimation With Correlated Run-Off Triangles in a Quantile Longitudinal Model. *Risks* **2020**, *8*, 14.
https://doi.org/10.3390/risks8010014

**AMA Style**

Badounas I, Pitselis G.
Loss Reserving Estimation With Correlated Run-Off Triangles in a Quantile Longitudinal Model. *Risks*. 2020; 8(1):14.
https://doi.org/10.3390/risks8010014

**Chicago/Turabian Style**

Badounas, Ioannis, and Georgios Pitselis.
2020. "Loss Reserving Estimation With Correlated Run-Off Triangles in a Quantile Longitudinal Model" *Risks* 8, no. 1: 14.
https://doi.org/10.3390/risks8010014