# A Likelihood Approach to Bornhuetter–Ferguson Analysis

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## Abstract

**:**

## 1. Introduction

## 2. The Bornhuetter–Ferguson Problem

#### 2.1. Data Structure

#### 2.2. Chain Ladder Method

#### 2.3. Bornhuetter–Ferguson Using Levels of Ultimates

#### 2.4. Bornhuetter–Ferguson Using Relative Ultimates

#### 2.5. Proposed Bornhuetter–Ferguson Reserves

## 3. Generalised Linear Model Framework

#### 3.1. Statistical Model

#### 3.2. The Chain Ladder

#### 3.3. Imposing External Information on the Relative Ultimates

**Theorem**

**1.**

**Theorem**

**2.**

#### 3.4. Implementation in GLM Software

#### 3.5. A Mixed Approach

#### 3.6. Pseudo Development Factors

#### 3.7. Chain Ladder Forecasts with the Mixed Approach

#### 3.8. Monotonicity

**Theorem**

**3.**

- (a)
- ${\Gamma}_{i}^{\u2020}>{G}_{i}$ for all $2\le i\le k$;
- (b)
- $\Delta {\widehat{\beta}}_{j}^{\u2020}>\Delta {\widehat{\beta}}_{j}$ for all $2\le j\le k$;
- (c)
- ${\widehat{\mu}}_{11}^{\u2020}<{\widehat{\mu}}_{11}$;
- (d)
- ${\tilde{Y}}_{ij}^{\u2021}>{\tilde{Y}}_{ij}^{\u2020}>{\tilde{Y}}_{ij}$ for all $i,j$ so that $k<i+j-1<2k$;
- (e)
- ${F}_{j}^{\u2020}>{F}_{j}$ for all $2\le j\le k$;
- (f)
- ${R}_{i}^{\u2021}>{R}_{i}^{\u2020}$ for all $2\le i\le k$;
- (g)
- ${R}_{i}^{\u2021}>{R}_{i}$ for all $2\le i\le k$.

## 4. Empirical Illustration

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proofs of Theorems

**Proof**

**of**

**Equation**

**.**Consider the Poisson model. The predictor is given in (24) and has the form ${\mu}_{ij}={\mu}_{11}+{\sum}_{\ell =2}^{i}\Delta {\alpha}_{\ell}+{\sum}_{\ell =2}^{j}\Delta {\beta}_{\ell}$, while the log likelihood is as in (25). This results in maximum likelihood estimators $\Delta {\widehat{\alpha}}_{i}$, $\Delta {\widehat{\beta}}_{j}$ and ${\widehat{\mu}}_{11}$ presented in (26)–(28), and in turn, the development factor ${F}_{j}$ is maximum likelihood estimator for ${\Phi}_{j}$ given in (32). When combining these we get the chain ladder forecast ${\tilde{Y}}_{ij}={R}_{i}({F}_{j}-1){\prod}_{\ell =k+2-i}^{j-1}{F}_{\ell}$ in (11).

**Proof**

**of**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**2.**

**Proof**

**of**

**Equation**

**.**First identity. Combine the forecasts; see (44).

**Proof**

**of**

**Equation**

**.**The point forecast is ${\tilde{Y}}_{ij}^{\u2020}=exp({\widehat{\mu}}_{11}^{\u2020}+{\sum}_{h=2}^{i}\Delta {\alpha}_{h}^{\u2020}+{\sum}_{h=2}^{j}\Delta {\widehat{\beta}}_{h}^{\u2020})$. Insert the expression for ${\widehat{\mu}}_{11}^{\u2020}$ from (47), for $\Delta {\alpha}_{i}^{\u2020}$ from (48) and $exp({\sum}_{h=2}^{j}\Delta {\widehat{\beta}}_{h}^{\u2020})=({F}_{j}^{\u2020}-1){\prod}_{\ell =2}^{j-1}{F}_{\ell}^{\u2020}$, which follows from (46), to get

**Proof**

**of**

**Equation**

**.**The point forecast is ${\tilde{Y}}_{ij}^{\u2021}=exp({\widehat{\mu}}_{11}+{\sum}_{h=2}^{i}\Delta {\alpha}_{h}^{\u2020}+{\sum}_{h=2}^{j}\Delta {\widehat{\beta}}_{h}),$ as given in (44). Insert the expression for ${\widehat{\mu}}_{11}$ from (28), the expression for $\Delta {\alpha}_{i}^{\u2020}$ from (51) and $exp({\sum}_{h=2}^{j}\Delta {\widehat{\beta}}_{h})=({F}_{j}-1){\prod}_{\ell =2}^{j-1}{F}_{\ell}$, which follows from (32) noting that ${F}_{j}={\widehat{\Phi}}_{j}$, to get

**Proof**

**of**

**Theorem**

**3.**

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2005 | 34,492,471 | 47,124,007 | 55,244,404 | 59,817,460 | 62,550,940 | 66,042,036 | 69,311,560 | 70,992,659 | 72,265,079 |

2006 | 39,467,733 | 54,003,286 | 61,349,336 | 69,986,825 | 76,412,887 | 81,768,759 | 86,684,598 | 90,726,054 | |

2007 | 38,928,855 | 57,087,550 | 65,905,902 | 77,128,507 | 84,158,380 | 92,436,441 | 97,838,371 | ||

2008 | 34,202,332 | 50,932,726 | 60,560,484 | 68,566,905 | 76,409,739 | 82,082,804 | |||

2009 | 35,657,409 | 52,397,264 | 59,849,582 | 66,698,806 | 72,724,524 | ||||

2010 | 25,404,394 | 37,040,589 | 42,371,049 | 50,709,319 | |||||

2011 | 21,268,516 | 31,311,410 | 35,973,015 | ||||||

2012 | 17,404,447 | 27,786,399 | |||||||

2013 | 17,676,374 |

2005 | 54,018,141 | 56,699,807 | 60,273,204 | 61,112,600 | 63,729,660 | 67,142,341 | 69,733,859 | 71,980,196 | 72,738,376 |

2006 | 68,706,483 | 70,534,436 | 70,254,136 | 75,919,965 | 77,900,147 | 83,401,774 | 88,690,144 | 92,171,660 | |

2007 | 64,613,205 | 72,600,950 | 76,163,387 | 82,388,057 | 87,424,383 | 96,246,891 | 102,854,340 | ||

2008 | 58,071,632 | 66,701,421 | 69,420,629 | 75,280,537 | 81,978,240 | 89,923,269 | |||

2009 | 60,368,719 | 67,868,349 | 72,528,239 | 80,726,223 | 85,339,588 | ||||

2010 | 47,282,519 | 56,488,940 | 60,896,832 | 65,900,623 | |||||

2011 | 49,905,225 | 54,801,141 | 60,026,903 | ||||||

2012 | 48,425,940 | 52,652,928 | |||||||

2013 | 47,449,977 |

$\mathbf{\Delta}{\widehat{\mathit{\alpha}}}_{\mathit{i}}$ | $\mathbf{\Delta}{\mathit{\alpha}}_{\mathit{i}}^{\u2020}$ | $\mathbf{\Delta}{\widehat{\mathit{\beta}}}_{\mathit{j}}$ | $\mathbf{\Delta}{\widehat{\mathit{\beta}}}_{\mathit{j}}^{\u2020}$ |
---|---|---|---|

0.24526809 | 0.247261682 | −0.80044252 | −0.76965582 |

0.11149938 | 0.145178053 | −0.68857388 | −0.65777806 |

−0.12057425 | −0.077312634 | 0.02370846 | 0.06137844 |

−0.04769497 | 0.027019249 | −0.32208939 | −0.29855013 |

−0.27637689 | −0.204202408 | −0.05908884 | −0.03399479 |

−0.21412347 | −0.018592530 | −0.22363447 | −0.20684905 |

−0.11353717 | −0.078902778 | −0.37786842 | −0.36440835 |

−0.08135422 | −0.005083078 | −0.68021278 | −0.67909386 |

${\widehat{\mu}}_{11}=17.18463300$ | ${\widehat{\mu}}_{11}^{\u2020}=17.00538277$ |

i,j | ${\mathit{R}}_{\mathit{i}}$ | ${\mathit{R}}_{\mathit{i}}^{\u2020}$ | ${\mathit{R}}_{\mathit{i}}^{\u2021}$ | ${\mathit{F}}_{\mathit{j}}$ | ${\mathit{F}}_{\mathit{j}}^{\u2020}$ |
---|---|---|---|---|---|

1 | 72,265,079 | 63,989,145 | 72,265,079 | ||

2 | 90,726,054 | 80,309,654 | 90,907,105 | 1.449130 | 1.463172 |

3 | 97,838,371 | 80,309,654 | 101,391,484 | 1.155676 | 1.163975 |

4 | 82,082,804 | 77,559,430 | 88,824,492 | 1.137937 | 1.149793 |

5 | 72,724,524 | 73,428,364 | 84,802,647 | 1.087838 | 1.096652 |

6 | 50,709,319 | 54,589,726 | 63,556,691 | 1.076112 | 1.085188 |

7 | 35,973,015 | 46,603,309 | 54,823,701 | 1.056555 | 1.063832 |

8 | 27,786,399 | 37,000,367 | 43,839,471 | 1.036684 | 1.041678 |

9 | 17,676,374 | 25,159,556 | 30,098,881 | 1.017923 | 1.020288 |

${\sum}_{\mathit{i}=1}^{\mathit{k}}{\mathit{V}}_{\mathit{i}}$ | External Valuation | ${\sum}_{\mathit{i}=1}^{\mathit{k}}{\mathit{V}}_{\mathit{i}}^{\u2020}$ | ${\sum}_{\mathit{i}=1}^{\mathit{k}}{\mathit{V}}_{\mathit{i}}^{\u2021}$ |
---|---|---|---|

110.1 | 137 | 149.1 | 156.6 |

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## Share and Cite

**MDPI and ACS Style**

Elpidorou, V.; Margraf, C.; Martínez-Miranda, M.D.; Nielsen, B.
A Likelihood Approach to Bornhuetter–Ferguson Analysis. *Risks* **2019**, *7*, 119.
https://doi.org/10.3390/risks7040119

**AMA Style**

Elpidorou V, Margraf C, Martínez-Miranda MD, Nielsen B.
A Likelihood Approach to Bornhuetter–Ferguson Analysis. *Risks*. 2019; 7(4):119.
https://doi.org/10.3390/risks7040119

**Chicago/Turabian Style**

Elpidorou, Valandis, Carolin Margraf, María Dolores Martínez-Miranda, and Bent Nielsen.
2019. "A Likelihood Approach to Bornhuetter–Ferguson Analysis" *Risks* 7, no. 4: 119.
https://doi.org/10.3390/risks7040119