# Macro vs. Micro Methods in Non-Life Claims Reserving (an Econometric Perspective)

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Macro and Micro Methods

- Those models neglect a lot of information that is available on a micro-level (per individual claim). Some additional covariates can be used, as well as exposure, etc. In most applications, not only is that information available, but usually, it has a valuable predictive power. To use that additional information, one cannot simply modify macro-level models, and it is necessary to change the general framework of the model. It becomes possible to emphasize large losses and to distinguish them from regular claims, to get more detailed information about future payments, etc.
- As discussed in this paper, macro-level models on aggregated data can be seen as models on clusters and not on individual observations, as we will do with micro-level models. In the context of macro-level models for loss reserving, [3] mention that prediction errors can be large, because of the small number of observations used in run-off triangles and the fact that clusters are usually not homogeneous. Quantifying uncertainty in claim reserving methods is not only important in actuarial practice and to assess accuracy of predictive models, it is also a regulatory issue. Finally, a small sample size can cause a lack of robustness and a risk of over-parametrization for macro-level models.

#### 1.2. Agenda

## 2. Clustering in Generalized Linear Mixed Models

#### 2.1. The Multiple Linear Regression Model

- (LRM1)
- no multicollinearity in the data matrix;
- (LRM2)
- exogeneity of the independent variables $\mathbb{E}\left(\right)open="["\; close="]">{\epsilon}_{i,g}|{\mathit{x}}_{g}$, $i=1,\dots ,{n}_{g}$, $g=1,\dots ,m$; and
- (LRM3)
- homoscedasticity and nonautocorrelation of error terms with $\text{Var}\left(\right)open="["\; close="]">{\epsilon}_{i,g}$.

**Proposition 1.**

- (i)
- ${\widehat{\mathit{a}}}_{OLS}={\widehat{\mathit{b}}}_{OLS}$ when weights ${n}_{g}$ are used in Model (2); and
- (ii)
- $\sum _{i,g}{\widehat{y}}_{i,g}=\sum _{g}{\widehat{y}}_{g}$ where ${y}_{g}={n}_{g}{\overline{y}}_{g}$.

**Proof.**

- (i)
- The ordinary least-squares estimator for $\mathit{a}$ - from Model (1)—is defined as$$\widehat{\mathit{a}}=\underset{\mathit{a}}{\text{argmin}}\left(\right)open="\{"\; close="\}">\sum _{i,g}{\left(\right)}^{{y}_{i,g}}2$$$$\widehat{\mathit{a}}=\underset{\mathit{a}}{\text{argmin}}\left(\right)open="\{"\; close="\}">\sum _{i,g}{\left(\right)}^{{y}_{i,g}}2.$$$$\begin{array}{c}\hfill \sum _{i,g}{\left(\right)}^{{y}_{i,g}}2\\ =\sum _{i,g}{({y}_{i,g}-{\overline{y}}_{g})}^{2}+{({\overline{y}}_{g}-{\mathit{x}}_{g}^{\mathsf{T}}\mathit{a})}^{2}\hfill \end{array}$$$$\widehat{\mathit{a}}=\underset{\mathit{a}}{\text{argmin}}\left(\right)open="\{"\; close="\}">\sum _{i,g}{({\overline{y}}_{g}-{\mathit{x}}_{g}^{\mathsf{T}}\mathit{a})}^{2}=\widehat{\mathit{b}}$$
- (ii)
- If we consider the sum of predicted values, observe that$$\sum _{i,g}{\widehat{y}}_{i,g}=\sum _{g}{n}_{g}{\mathit{x}}_{g}^{\mathsf{T}}\widehat{\mathit{a}}=\sum _{g}{n}_{g}\underset{{\widehat{\overline{y}}}_{g}}{\underbrace{{\mathit{x}}_{g}^{\mathsf{T}}\widehat{\mathit{b}}}}=\sum _{g}{\widehat{y}}_{g}$$

**Corollary 2.**

- (i)
- $\mathbb{E}\left[{\widehat{\mathit{A}}}_{OLS}\right]=\mathbb{E}\left[{\widehat{\mathit{B}}}_{OLS}\right]$ and $Var\left[{\widehat{\mathit{A}}}_{OLS}\right]=Var\left[{\widehat{\mathit{B}}}_{OLS}\right]$, when weights ${n}_{g}$ are used in Model (2); and
- (ii)
- $\mathbb{E}\left(\right)open="["\; close="]">\sum _{i,g}{\widehat{Y}}_{i,g}=\mathbb{E}\left(\right)open="["\; close="]">\sum _{g}{\widehat{Y}}_{g}$ and $Var\left(\right)open="["\; close="]">\sum _{i,g}{\widehat{Y}}_{i,g}=Var\left(\right)open="["\; close="]">\sum _{g}{\widehat{Y}}_{g}$.

**Proof.**

- (i)
- Let$$\begin{array}{c}\hfill \mathbb{E}\left(\right)open="["\; close="]">{\widehat{\mathit{B}}}_{OLS}\\ ={\left(\right)}^{\overline{\mathit{x}}}-1\overline{\mathit{x}}\mathbf{1}{\mathbf{1}}^{\mathsf{T}}\mathbb{E}\left(\right)open="["\; close="]">\overline{\mathit{Y}}\hfill \end{array}$$$$\begin{array}{cc}& \text{Var}\left(\right)open="["\; close="]">{\widehat{\mathit{B}}}_{OLS}\hfill \end{array}$$
- (ii)
- Let$$\begin{array}{c}\hfill \mathbb{E}\left(\right)open="["\; close="]">\sum _{g}{\widehat{Y}}_{g}\\ =\mathbb{E}\left(\right)open="["\; close="]">{\mathbf{1}}_{m}\mathbf{1}{\mathbf{1}}^{\mathsf{T}}\widehat{\overline{\mathit{Y}}}=\mathbb{E}\left(\right)open="["\; close="]">{\mathbf{1}}_{m}\mathbf{1}{\mathbf{1}}^{\mathsf{T}}{\overline{\mathit{x}}}^{\mathsf{T}}\widehat{\mathit{B}}\hfill \end{array}$$

#### 2.2. The Quasi-Poisson Regression

**Proposition 3.**

**Proof.**

- (i)
- Maximum likelihood estimator of $\mathit{a}$ is the solution of$$\begin{array}{c}\hfill \sum _{i,g}\left(\right)open="("\; close=")">\frac{{y}_{i,g}-exp\left[{\mathit{x}}_{g}^{\mathsf{T}}\mathit{a}\right]}{{\phi}_{\text{micro}}}{\mathit{x}}_{g}\\ =\mathbf{0}\hfill \end{array}$$$$\begin{array}{c}\hfill \sum _{i,g}\left(\right)open="("\; close=")">{y}_{i,g}-exp\left[{\mathit{x}}_{g}^{\mathsf{T}}\mathit{a}\right]{\mathit{x}}_{g}\\ =\mathbf{0}\hfill \end{array}$$$$\begin{array}{c}\hfill \sum _{g}\left(\right)open="("\; close=")">{y}_{g}-{n}_{g}exp\left[{\mathit{x}}_{g}^{\mathsf{T}}\mathit{b}\right]{\mathit{x}}_{g}\\ =0\hfill \end{array}$$$$\begin{array}{c}\hfill \sum _{i,g}\left(\right)open="("\; close=")">{y}_{i,g}-exp\left[{\mathit{x}}_{g}^{\mathsf{T}}\mathit{b}\right]{\mathit{x}}_{g}\\ =0\hfill \end{array}$$
- (ii)
- The sum of predicted values is$$\begin{array}{cc}\hfill \sum _{i,g}{\widehat{y}}_{i,g}& =\sum _{g}{n}_{g}{\widehat{\lambda}}_{i,g}=\sum _{g}{n}_{g}exp\left[{\mathit{x}}_{g}^{\mathsf{T}}\widehat{\mathit{a}}\right]=\sum _{g}{n}_{g}exp\left[{\mathit{x}}_{g}^{\mathsf{T}}\widehat{\mathit{b}}\right]\hfill \\ & =\sum _{g}exp[{\mathit{x}}_{g}^{\mathsf{T}}\widehat{\mathit{b}}+log\left({n}_{g}\right)]=\sum _{g}{\widehat{\lambda}}_{g}^{*}=\sum _{g}{\widehat{y}}_{g}\hfill \end{array}$$

**Corollary 4.**

- (i)
- $\mathbb{E}\left[{\widehat{\mathit{A}}}_{MLE}\right]=\mathbb{E}\left[{\widehat{\mathit{B}}}_{MLE}\right]$ and $Var\left[{\widehat{\mathit{A}}}_{MLE}\right]=Var\left[{\widehat{\mathit{B}}}_{MLE}\right]$, when n goes to infinity; and
- (ii)
- $\mathbb{E}\left(\right)open="["\; close="]">\sum _{i,g}{\widehat{Y}}_{i,g}=\mathbb{E}\left(\right)open="["\; close="]">\sum _{g}{\widehat{Y}}_{g}$ and $Var\left(\right)open="["\; close="]">\sum _{i,g}{\widehat{Y}}_{i,g}=Var\left(\right)open="["\; close="]">\sum _{g}{\widehat{Y}}_{g}$, when n goes to infinity.

**Proof.**

- (i)
- A classical result of asymptotic theory for maximum likelihood estimators indicates that, under mild regularity conditions, $\mathbb{E}\left(\right)open="["\; close="]">{\widehat{\mathit{A}}}_{MLE}$ and $\mathbb{E}\left(\right)open="["\; close="]">{\widehat{\mathit{B}}}_{MLE}$ as $n\to \infty $. Since $\mathit{a}=\mathit{b}$, we have $\mathbb{E}\left(\right)open="["\; close="]">{\widehat{\mathit{B}}}_{MLE}$ when $n\to \infty $. For Model (7), the Fisher information matrix is $\mathit{I}\left(\mathit{A}\right)=\mathit{x}\mathit{W}{\mathit{x}}^{\mathsf{T}}$ and, when $n\to \infty $, $\text{Var}\left(\right)open="["\; close="]">\widehat{\mathit{A}}$, where $\mathit{W}=\text{diag}(({\lambda}_{1}/{n}_{1}){\mathbf{1}}_{{n}_{1}},\dots ,({\lambda}_{m}/{n}_{m}){\mathbf{1}}_{{n}_{m}})$. For Model (10), we have $\mathit{I}\left(\mathit{B}\right)=\overline{\mathit{x}}\mathbf{1}\mathit{W}{\mathbf{1}}^{\mathsf{T}}{\overline{\mathit{x}}}^{\mathsf{T}}=\mathit{x}\mathit{W}{\mathit{x}}^{\mathsf{T}}$ and, when $n\to \infty $, $\text{Var}\left(\right)open="["\; close="]">\widehat{\mathit{B}}$.
- (ii)
- By using a similar argument, we have when n goes to infinity$$\begin{array}{c}\hfill \mathbb{E}\left(\right)open="["\; close="]">\sum _{g}{\widehat{Y}}_{g}\\ =\mathbb{E}\left(\right)open="["\; close="]">{\mathbf{1}}_{m}\mathbf{1}{\mathbf{1}}^{\mathsf{T}}\widehat{\overline{\mathit{Y}}}={\mathbf{1}}_{m}\mathbf{1}{\mathbf{1}}^{\mathsf{T}}{M}_{\widehat{\mathit{B}}}\left(\right)open="("\; close=")">{\overline{\mathit{x}}}^{\mathsf{T}}\hfill \end{array}$$

**Corollary 5.**

- (i)
- $\mathbb{E}\left[{\widehat{\mathit{A}}}_{QLE}\right]=\mathbb{E}\left[{\widehat{\mathit{B}}}_{QLE}\right]$ but $Var\left[{\widehat{\mathit{A}}}_{QLE}\right]\ne Var\left[{\widehat{\mathit{B}}}_{QLE}\right]$, when n goes to infinity; and
- (ii)
- $\mathbb{E}\left(\right)open="["\; close="]">\sum _{i,g}{\widehat{Y}}_{i,g}=\mathbb{E}\left(\right)open="["\; close="]">\sum _{g}{\widehat{Y}}_{g}$ but $Var\left(\right)open="["\; close="]">\sum _{i,g}{\widehat{Y}}_{i,g}\ne Var\left(\right)open="["\; close="]">\sum _{g}{\widehat{Y}}_{g}$, when n goes to infinity.

**Proof.**

- (i)
- The property that variances are not equal is a direct consequence of classical results from the theory of generalized linear models (see [23]), since the covariance matrices of estimators are given by$$\begin{array}{c}\hfill \text{Var}\left(\right)open="["\; close="]">\widehat{\mathit{B}}\\ \to {\widehat{\phi}}_{macro}{\left(\right)}^{\mathit{x}}-1\hfill \end{array}$$$$\begin{array}{c}\hfill \text{Var}\left(\right)open="["\; close="]">\widehat{\mathit{A}}\\ \to {\widehat{\phi}}_{macro}{\left(\right)}^{\mathit{x}}-1\hfill \end{array}$$
- (ii)
- Since the MLE and the QLE share the same asymptotic distribution (see [23]), the proof is similar to Corollary 4(ii).

#### 2.3. Poisson Regression with Random Effect

## 3. Clustering and Loss Reserving Models

#### 3.1. The Quasi-Poisson Model for Reserves

#### 3.1.1. Construction

**Proposition 6.**

**Proof.**

#### 3.1.2. Illustration and Discussion

`R`, using packages

`ChainLadder`and

`gtools`. The final reserve amount obtained from the Mack’s model [2] is $28,655,773.

- simulate the number of payments for each cluster assuming ${N}_{g}\sim \mathcal{P}\left(\theta \right)$, $g=1,\dots ,m$;
- for each cluster, simulate a $({n}_{g}\times 1)$ vector of proportions assuming ${\omega}_{g}={\left[\begin{array}{ccc}{\omega}_{1}& \dots & {\omega}_{{n}_{g}}\end{array}\right]}^{\mathsf{T}}\sim \text{Dirichlet}\phantom{\rule{3.33333pt}{0ex}}\left(\mathbf{1}\right)$, $g=1,\dots ,m$;
- for each cluster, define$$\begin{array}{cc}\hfill \left[\begin{array}{c}{Y}_{1,g}\\ \vdots \\ {Y}_{{n}_{g},g}\end{array}\right]& ={\omega}_{g}{Y}_{g},\phantom{\rule{2.em}{0ex}}g=1,\dots ,m\hfill \end{array}$$
- adjust
**Model C**and**Model D**; and - calculate the best estimate and the MSEP of the reserve by using Proposition 6.

**Model A**and

**C**), results are similar, which is consistent with Corollary 4. For micro-level models, convergence of $\sqrt{MSEP}$ towards (11622) is fast. For quasi-Poisson regression (

**Model B**and

**D**), expected values are equal and Figure 1 shows $\sqrt{MSEP}$ as a function of the expected total number of payments for the portfolio. Above a certain level, (close to 3400 here), accuracy of the “micro” approach exceeds the “macro”. Again, those results are consistent with Corollary 5. Here, we consider that the expected number of payments by cluster $(\theta $) is constant but it would also be possible to consider a mixture model where $\left(\right)open="("\; close=")">{N}_{g}|{\Theta}_{g}$, $g=1,\dots ,m$, and ${\Theta}_{g}\sim \text{Gamma}(\alpha ,\beta )$. This modification does not change the conclusions. Finally, a comparison of estimated MSEP for both Poisson and quasi-Poisson models confirms the presence of over-dispersion in the data.

**Model E**) and with a strongly correlated covariate (

**Model F**). Following a similar procedure, we obtain results presented in the bottom part of Table 3 and in Figure 2.

**Model B**), for several reasons,

- (i)
- impossible to compute that average without individual data;
- (ii)
- discrete explanatory variables used in the micro-level model; and
- (iii)
- since claims reserve model have a predictive motivation, it is risky to project the value of an aggregated variable on future clusters.

**Model D**and

**E**are very close. As claimed by Proposition 6 and Equation (14), an explanatory variable highly correlated with the response variable will decrease the value of $\sqrt{MSEP}$, and lowers the threshold above which the micro-level model is more accurate than the macro-level one.

**Model B**) with maximum likelihood estimators leads to the same reserves as the chain-ladder algorithm and the Mack’s model (see [31]), assuming the clusters exposure, for $(i,j)\in \mathcal{K}$, is one. To obtain similar results with a quasi-Poisson micro-level model (

**Model D**), a similar assumption is necessary: exposure of each claim within cluster $(i,j)$ is $1/{n}_{i,j}$. That assumption implies, on a micro level, that predicted individual payments ${\widehat{Y}}_{ij}^{\left(k\right)}$ are proportional to $1/{n}_{ij}$. That assumption has unfortunately no foundation.

**Model C**and

**D**), payments related to the same claim, in two different clusters are supposed to be non-correlated. As discussed in the previous Section, it is possible to include dependencies among payments for a given claim using a Poisson regression with random effects.

#### 3.2. The Mixed Poisson Model for Reserves

#### 3.2.1. Construction

**model G**) are

#### 3.2.2. Illustration and Discussion

- 1-3.
- see previous section;
- 4.
- for each accident year, allocate randomly the source (t) of each payment;
- 5.
- fit
**model G**; and - 6.
- compute the best estimate and the MSEP of the reserve.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Square root of the mean square error of prediction obtained for

**Model D**

**(solid**line) and

**Model B**(

**broken**line) from simulated values for increasing expected number of payments for the portfolio.

**Figure 2.**Mean square error of prediction ($\pm 2\sigma $) obtained from simulated values as a function of the expected number of payments for

**Model E**(

**red**lines) and

**Model F**(

**blue**lines). For comparison purposes, the MSEP obtained for the

**Model D**(

**solid black**line) and the

**Model B**(

**broken black**line) are added.

**Figure 3.**Observed data (circles) with conditional predictions (

**red**lines) and unconditional ones (

**blue**lines) from

**Model G**with $\theta =10$.

**Figure 4.**Predictions with the quasi-Poisson macro-level model (

**strong black**line), with conditional predictions (

**red**lines) and unconditional ones (

**blue**lines) from

**Model G**with $\theta =10$.

**Table 1.**Quasi-Poisson macro- and micro-level models for reserve ($i,j=1,\dots ,I$). All clusters and all payments are independent.

Components | Macro | Micro |
---|---|---|

Exp. value | $\mathbb{E}\left(\right)open="["\; close="]">{Y}_{i,j}$ | $\mathbb{E}\left(\right)open="["\; close="]">{Y}_{i,j}^{\left(k\right)}$ |

Inv. link func. | ${\lambda}_{i,j}=exp\left[{\mathit{x}}_{i,j}^{\mathsf{T}}\mathit{b}\right]$ | ${\lambda}_{i,j}=exp[{\mathit{x}}_{i,j}^{\mathsf{T}}\mathit{a}+log(1/{n}_{i,j})]$ |

$\phantom{{\lambda}_{i,j}}=exp[{b}_{i}+{b}_{I+j}]$ | $\phantom{{\lambda}_{i,j}}=exp[{a}_{i}+{a}_{I+j}+log(1/{n}_{i,j})]$ | |

with ${b}_{I+1}=0$ | with ${a}_{I+1}=0$ | |

Variance | $\text{Var}\left(\right)open="["\; close="]">{Y}_{i,j}$ | $\text{Var}\left(\right)open="["\; close="]">{Y}_{i,j}^{\left(k\right)}$ |

Pred. value | ${\widehat{Y}}_{i,j}=exp[{\widehat{b}}_{i}+{\widehat{b}}_{I+j}]$ | ${\widehat{Y}}_{i,j}^{\left(k\right)}=exp[{\widehat{a}}_{i}+{\widehat{a}}_{I+j}+log(1/{n}_{i,j})]$ |

Known values | ${\mathcal{Y}}_{macro}$ | ${\mathcal{Y}}_{micro}$ |

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

1 | 3511 | 3215 | 2266 | 1712 | 1059 | 587 | 340 |

2 | 4001 | 3702 | 2278 | 1180 | 956 | 629 | – |

3 | 4355 | 3932 | 1946 | 1522 | 1238 | – | – |

4 | 4295 | 3455 | 2023 | 1320 | – | – | – |

5 | 4150 | 3747 | 2320 | – | – | – | – |

6 | 5102 | 4548 | – | – | – | – | – |

7 | 6283 | – | – | – | – | – | – |

Method | $\mathbb{E}\left[\mathbf{Reserve}\right]$ | $\sqrt{\mathit{MSEP}}$ |
---|---|---|

Mack’s model | 28655773 | 1417267 |

Poisson reg. | ||

Model A | 28655773 | 11622 |

Model C | 28655773 | 11622 |

quasi-Poisson reg. | ||

Model B | 28655773 | 1708196 |

Model D | 28655773 | see Figure 1 |

quasi-Poisson reg. | ||

Model E ($\rho \approx 0$) | 28657364 | see Figure 2 |

Model F ($\rho \approx 0.8$) | 20514566 | see Figure 2 |

Modèle | $\mathbb{E}\left[\mathbf{Reserve}\right]$ | $\sqrt{\mathrm{Var}\left(\mathrm{Reserve}\right)}$ |
---|---|---|

coll. quasi-Pois. | 28656423 | 1708216 |

mixed Poisson non-cond. | 27930624 | 3297401 |

mixed Poisson cond. | 25972947 | 2280902 |

© 2016 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/).

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Charpentier, A.; Pigeon, M.
Macro *vs.* Micro Methods in Non-Life Claims Reserving (an Econometric Perspective). *Risks* **2016**, *4*, 12.
https://doi.org/10.3390/risks4020012

**AMA Style**

Charpentier A, Pigeon M.
Macro *vs.* Micro Methods in Non-Life Claims Reserving (an Econometric Perspective). *Risks*. 2016; 4(2):12.
https://doi.org/10.3390/risks4020012

**Chicago/Turabian Style**

Charpentier, Arthur, and Mathieu Pigeon.
2016. "Macro *vs.* Micro Methods in Non-Life Claims Reserving (an Econometric Perspective)" *Risks* 4, no. 2: 12.
https://doi.org/10.3390/risks4020012