# Modified Munich Chain-Ladder Method

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### Organization of the Paper

## 2. Chain-Ladder Models

**Assumption 1**

- (A1)
- We assume that the random vectors $({P}_{i,0},\dots ,{P}_{i,J},{I}_{i,0},\dots ,{I}_{i,J})$ have strictly positive components and are independent for different accident years $i=0,\dots ,J$.
- (A2)
- There exist parameters ${f}_{j}^{P},{f}_{j}^{I},{\sigma}_{j}^{P},{\sigma}_{j}^{I}>0$ such that for $0\le j\le J-1$ and $0\le i\le J$$$\begin{array}{c}\hfill \mathbb{E}\left(\right)open="["\; close="]">\left(\right)open\; close="|">{P}_{i,j+1}{\mathcal{B}}_{j}^{P}& ={f}_{j}^{P}\phantom{\rule{3.33333pt}{0ex}}{P}_{i,j}\end{array}\phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}& \mathrm{Var}\left(\right)open="("\; close=")">\left(\right)open\; close="|">{P}_{i,j+1}{\mathcal{B}}_{j}^{P}\hfill & ={({\sigma}_{j}^{P})}^{2}\phantom{\rule{3.33333pt}{0ex}}{P}_{i,j}^{2},$$

**PE**,

**PV**,

**IE**,

**IV**and

**PIU**in [1], except that we make a modification in the variance assumptions

**PV**and

**IV**. We make this change because it substantially simplifies our considerations (we come back to this in Remark 1 below). Assumption 1 states that cumulative payments ${({P}_{i,j})}_{i,j}$ and incurred claims ${({I}_{i,j})}_{i,j}$ fulfill the distribution-free chain-ladder model assumptions simultaneously. Our first aim is to show that there is a non-trivial stochastic model that fulfills the chain-ladder model assumptions simultaneously for cumulative payments and incurred claims. To this end, we define an explicit distributional model. The distributions are chosen such that the analysis becomes as simple as possible. We will see that assumption (A2) requires a sophisticated consideration.

**Assumption 2**

- (B1)
- We assume that the random vectors ${\Xi}_{i}$ are independent for different accident years $i=0,\dots ,J$.
- (B2)
- There exists a parameter vector $\mathit{\theta}={({\theta}_{0}^{P},\dots ,{\theta}_{J}^{P},{\theta}_{0}^{I},\dots ,{\theta}_{J}^{I})}^{\prime}\in {\mathbb{R}}^{2(J+1)}$ and a positive definite covariance matrix $\Sigma \in {\mathbb{R}}^{2(J+1)\times 2(J+1)}$ such that we have for $0\le i\le J$$${\Xi}_{i}\sim \mathcal{N}\left(\right)open="("\; close=")">\mathit{\theta},\Sigma .$$

**Lemma 1.**

**Proof.**

**Theorem 1.**

**Remark 1.**

**Lemma 2.**

**Proof.**

**Assumption 3**

- (C1)
- We assume that the random vectors ${\Xi}_{i}$ are independent for different accident years $i=0,\dots ,J$.
- (C2)
- There exists a parameter vector $\mathit{\theta}={({\theta}_{0}^{P},\dots ,{\theta}_{J}^{P},{\theta}_{0}^{I},\dots ,{\theta}_{J}^{I})}^{\prime}\in {\mathbb{R}}^{2(J+1)}$ and a matrix Σ of the form Equation (8) with positive definite Schur complements ${S}_{\left[J\right]}^{P}$ and ${S}_{\left[J\right]}^{I}$ such that we have for $0\le i\le J$$${\Xi}_{i}\sim \mathcal{N}\left(\right)open="("\; close=")">\mathit{\theta},\Sigma .$$

**Corollary 1.**

## 3. One-Step Ahead Prediction

**Lemma 3.**

**Proof.**

**Corollary 2**

**Example 1**

- Case $j=0$. We start the analysis for $j=0$, i.e., given information ${\mathcal{B}}_{0}$.

- Case $j=1$. This case is more involved. Set

## 4. Munich Chain-Ladder Model

**Assumption 4**

**Remark 2.**

**PQ**and

**IQ**of Quarg and Mack [1]. If we choose the log-link for Assumption 3 then the incurred-paid ratio ${Q}_{i,j}^{-1}$ is turned into a difference on the log scale, that is, $log({Q}_{i,j}^{-1})=log{I}_{i,j}-log{P}_{i,j}={\sum}_{l=0}^{j}{\xi}_{i,l}^{I}-{\sum}_{l=0}^{j}{\xi}_{i,l}^{P}$. The aim of this section is to analyze under which circumstances these Munich chain-ladder corrections lead to the optimal predictors provided in Corollary 2. Below we will see that the constants ${\lambda}^{P}$ and ${\lambda}^{I}$ are crucial, they measure the (positive) correlation between the cumulative payments and the incurred-paid ratio correction (and similarly for incurred claims), see also Section 2.2.2 in [1]. Moreover, ${\lambda}^{P}$ and ${\lambda}^{I}$ receive an explicit meaning in Theorem 3, below.

**Lemma 4.**

**Proof.**

**Example 2**

**Theorem 2.**

**Proof.**

**Theorem 3**

**Proof.**

**Remark 3.**

## 5. The Modified Munich Chain-Ladder Method

**Assumption 5**

- Conditionally, given parameter vector $\mathbf{\Theta}={({\Theta}_{0}^{P},\dots ,{\Theta}_{J}^{P},{\Theta}_{0}^{I},\dots ,{\Theta}_{J}^{I})}^{\prime}$, the random vectors ${\Xi}_{i}$ are independent for different accident years $i=0,\dots ,J$ with$${\left(\right)}_{{\mathbf{\Xi}}_{i}}\mathbf{\Theta}\sim \mathcal{N}\left(\right)open="("\; close=")">\mathbf{\Theta},\Sigma $$
- The parameter vector
**Θ**has prior distribution$$\mathbf{\Theta}\sim \mathcal{N}\left(\right)open="("\; close=")">\mathit{\theta},T,$$

**Lemma 5.**

**Θ**. However, we keep the hierarchy of parameters in order to obtain Bayesian parameter estimates for

**Θ**.

**ζ**by $n=2{(J+1)}^{2}+2(J+1)$. Choose $t,v\in \mathbb{N}$ with $t+v=n$. Denote by ${P}_{t}\in {\mathbb{R}}^{t\times n}$ and ${P}_{v}\in {\mathbb{R}}^{v\times n}$ the projections such that we obtain a disjoint decomposition of the components of

**ζ**

**ζ**. In complete analogy to Lemma 1 we have the following lemma.

**Lemma 6.**

**ζ**onto the components ${\xi}_{i,j}^{P}$ and ${\xi}_{i,j}^{I}$ with $i+j\le J$. These are exactly the components that generate information ${\mathcal{D}}_{J}$. Lemma 6 allows us to calculate the posterior distribution of ${\mathit{\zeta}}_{v}$, conditionally given ${\mathcal{D}}_{J}$. We split this calculation into two parts, one for parameter estimation and one for claims prediction. We consider therefore the following projection

**Θ**from the unobserved components ${\mathit{\zeta}}_{v}$.

**Corollary 3**

**Θ**is at time J given by

## 6. Claims Prediction and Prediction Uncertainty

**Theorem 4**

## 7. Example

**Table 1.**Sample standard deviations ${s}_{j}^{P}$ and ${s}_{j}^{I}$; posterior means ${\theta}_{j}^{(P),\mathrm{post}}$ and ${\theta}_{j}^{(I),\mathrm{post}}$ obtained from Corollary 3, see also Equation (21); and Hertig’s chain-ladder estimates ${\widehat{f}}_{j}^{P}$ and ${\widehat{f}}_{j}^{I}$ according to Equation (21).

a.y. i/d.y. j | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|

${\theta}_{j}^{(P),\mathrm{post}}$ | 7.2195 | 0.9163 | 0.1203 | 0.0296 | 0.0216 | 0.0205 | 0.0137 |

${s}_{j}^{P}$ | 0.4972 | 0.1600 | 0.0515 | 0.0069 | 0.0036 | 0.0101 | 0.0036 |

${\widehat{f}}_{j}^{P}$ | 1,573 | 2.5376 | 1.1296 | 1.0301 | 1.0219 | 1.0208 | 1.0138 |

${\theta}_{j}^{(I),\mathrm{post}}$ | 7.8404 | 0.5151 | 0.0137 | 0.0003 | 0.0115 | −0.0090 | −0.0037 |

${s}_{j}^{I}$ | 0.5182 | 0.1503 | 0.0406 | 0.0146 | 0.0022 | 0.0180 | 0.0022 |

${\widehat{f}}_{j}^{I}$ | 2,963 | 1.6959 | 1.0148 | 1.0004 | 1.0116 | 0.9912 | 0.9963 |

**Table 2.**Resulting reserves from the Hertig’s chain-ladder (HCL) method based on paid and incurred; from the log-normal Munich chain-ladder (LN–MCL) method based on paid and incurred; from the modified Munich chain-ladder (mMCL) paid method; the classical chain-ladder (CL) method based on paid and incurred (inc.); and the Quarg and Mack Munich chain-ladder (QM–MCL) method paid and incurred.

a.y. i | HCL | LN-MCL | mMCL | CL | QM-MCL | ||||
---|---|---|---|---|---|---|---|---|---|

paid | inc. | paid | inc. | paid | paid | inc. | paid | inc. | |

1 | 32 | 97 | 35 | 95 | 16 | 32 | 97 | 35 | 96 |

2 | 157 | 92 | 92 | 147 | 115 | 158 | 88 | 103 | 135 |

3 | 337 | 286 | 262 | 346 | 375 | 332 | 276 | 269 | 326 |

4 | 416 | 201 | 289 | 330 | 382 | 408 | 191 | 289 | 302 |

5 | 925 | 459 | 656 | 688 | 906 | 924 | 466 | 646 | 655 |

6 | 4,339 | 6,594 | 5,395 | 5,534 | 5,130 | 4,084 | 6,385 | 5,505 | 5,606 |

total | 6’205 | 7’730 | 6’729 | 7’140 | 6’924 | 5’938 | 7’503 | 6’847 | 7’120 |

**Figure 1.**(lhs) Incurred-paid residuals obtained from ${Q}_{i,j}^{-1}={I}_{i,j}/{P}_{i,j}$, see Remark 2, versus claims payments residuals obtained from ${P}_{i,j+1}$, straight line has slope ${\widehat{\lambda}}^{P}=49\%$; (rhs) paid-incurred residuals obtained from ${Q}_{i,j}={P}_{i,j}/{I}_{i,j}$ versus incurred claims residuals obtained from ${I}_{i,j+1}$, straight line has slope ${\widehat{\lambda}}^{I}=45\%$.

**Table 3.**Resulting reserves and square-rooted conditional mean square error of prediction of the different chain-ladder methods.

^{*}is calculated from Equation (23).

Reserves | msep^{1/2} | |
---|---|---|

Hertig’s chain-ladder HCL paid | 6,205 | 1,249 |

Hertig’s chain-ladder HCL incurred | 7,730 | 1,565 |

log-normal Munich chain-ladder LN–MCL paid | 6,729 | 1,224^{*} |

log-normal Munich chain-ladder LN–MCL incurred | 7,140 | 1,673^{*} |

modified Munich chain-ladder mMCL paid | 6,924 | 1,208 |

modified Munich chain-ladder mMCL incurred | 7,730 | 1,565 |

classical chain-ladder CL paid | 5,938 | 994 |

classical chain-ladder CL incurred | 7,503 | 995 |

Quarg-Mack Munich chain-ladder QM–MCL paid | 6,847 | n/a |

Quarg-Mack Munich chain-ladder QM–MCL incurred | 7,120 | n/a |

## 8. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix

## A. Data of Quarg and Mack [1]

a.y. i/d.y. j | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|

0 | 576 | 1,804 | 1,970 | 2,024 | 2,074 | 2,102 | 2,131 |

1 | 866 | 1,948 | 2,162 | 2,232 | 2,284 | 2,348 | |

2 | 1,412 | 3,758 | 4,252 | 4,416 | 4,494 | ||

3 | 2,286 | 5,292 | 5,724 | 5,850 | |||

4 | 1,868 | 3,778 | 4,648 | ||||

5 | 1,442 | 4,010 | |||||

6 | 2,044 |

a.y. i/d.y. j | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|

0 | 978 | 2,104 | 2,134 | 2,144 | 2,174 | 2,182 | 2’174 |

1 | 1,844 | 2,552 | 2,466 | 2,480 | 2,508 | 2,454 | |

2 | 2,904 | 4,354 | 4,698 | 4,600 | 4,644 | ||

3 | 3,502 | 5,958 | 6,070 | 6,142 | |||

4 | 2,812 | 4,882 | 4,852 | ||||

5 | 2,642 | 4,406 | |||||

6 | 5,022 |

## B. Inverse Matrix Σ_{[1]}

## References

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**MDPI and ACS Style**

Merz, M.; Wüthrich, M.V.
Modified Munich Chain-Ladder Method. *Risks* **2015**, *3*, 624-646.
https://doi.org/10.3390/risks3040624

**AMA Style**

Merz M, Wüthrich MV.
Modified Munich Chain-Ladder Method. *Risks*. 2015; 3(4):624-646.
https://doi.org/10.3390/risks3040624

**Chicago/Turabian Style**

Merz, Michael, and Mario V. Wüthrich.
2015. "Modified Munich Chain-Ladder Method" *Risks* 3, no. 4: 624-646.
https://doi.org/10.3390/risks3040624