Estimation error and bootstrapping in the chain-ladder model of Mack

Abstract

In 2006 there was quite some discussion on how to estimate the conditional estimation error in the chain-ladder (CL) model of Mack. Buchwalder, Bühlmann, Merz and Wüthrich (BBMW) (ASTIN Bull 36(2):521–542, 2006) proposed another estimator than the one derived by Mack (ASTIN Bull 23(2):213–225, 1993). These two estimators are also found in a broader context by new authors in recent papers. In the present paper we examine the theoretical properties of the two estimators and come to the conclusion that the BBMW estimator has some major deficiencies compared with the Mack estimator. It takes much less information of the observed triangle into account, the averaging is done over inappropriate sets and it does not properly fit to the Mack CL-model.

A Appendix further results

As in the previous sections we assume that $i>I-J+1.$ Further we assume that $\widehat{\sigma _{j}^{2}}$ is the usual unbiased estimator of $\sigma _{j}^{2}$ already presented in [9].

Theorem A.1

$$\begin{aligned}&E\left[ \left. EE_{i}^{*}\right| {\mathcal {B}}_{j_{i}}\right] =E\left[ \left. EE_{i}\right| {\mathcal {B}}_{j_{i}}\right] ,\nonumber \\&\quad \text {where }EE_{i}^{*},EE_{i}\text { are defined in }(4.19), (4.7). \end{aligned}$$

(A.1)

Proof

$$\begin{aligned} E\left[ \left. EE_{i}^{*}\right| {\mathcal {B}}_{j_{i}}\right]&= {\displaystyle \sum \limits _{j=j_{i}}^{J-1}} \left( E\left[ \left. \widehat{C_{i,j}}^{2}\frac{\sigma _{j}^{2}}{ {\displaystyle \sum \nolimits _{l=0}^{i_{j}-1}} C_{l,j}}\right| {\mathcal {B}}_{j_{i}}\right] {\displaystyle \prod \limits _{n=j+1}^{J-1}} f_{n}^{2}\right) \\&= {\displaystyle \sum \limits _{j=j_{i}}^{J-1}} \left( E\left[ \left. E\left[ \widehat{C_{i,j}}^{2}\left. \left( \widehat{f_{j}}-f_{j}\right) ^{2}\right| {\mathcal {B}}_{j}\right] \right| {\mathcal {B}}_{j_{i}}\right] {\displaystyle \prod \limits _{n=j+1}^{J-1}} f_{n}^{2}\right) \\&= {\displaystyle \sum \limits _{j=j_{i}}^{J-1}} \left( E\left[ \left. \widehat{C_{i,j}}^{2}\left( \widehat{f_{j}} -f_{j}\right) ^{2}\right| {\mathcal {B}}_{j_{i}}\right] {\displaystyle \prod \limits _{n=j+1}^{J-1}} f_{n}^{2}\right) \\&=E\left[ \left. EE_{i}\right| {\mathcal {B}}_{j_{i}}\right] . \end{aligned}$$

$\square$

Remarks

• Consider the right upper corner ${\mathcal {D}}_{I}^{U_{i}}$ defined in (4.20). The Mack pseudo-estimator is on average over all trajectories in ${\mathcal {D}}_{I}^{U_{i}}$ an unbiased estimator.

• Mack, Quarg, Braun [10] argue that the $\widehat{f_{j}}^{2}$ were negatively correlated due to (4.27) and that therefore the BBMW pseudo-estimator would overestimate on average over ${\mathcal {D}}_{I}^{U_{i}}$ the estimation error, that is

  $$\begin{aligned} E\left[ \left. EE_{i}^{**}\right| {\mathcal {B}}_{j_{i}}\right] >E\left[ \left. EE_{i}\right| {\mathcal {B}}_{j_{i}}\right] . \end{aligned}$$

  (A.2)

  However their reasoning is not correct. From (4.27), it does not follow that the $\widehat{f_{j}}^{2}$ are negatively correlated and that

  $$\begin{aligned} E\left[ \left. {\displaystyle \prod \limits _{k=j_{i}}^{j-1}} \widehat{f_{k}}^{2}\right| {\mathcal {B}}_{j_{i}}\right] < {\displaystyle \prod \limits _{k=j_{i}}^{j-1}} E\left[ \left. \widehat{f_{k}}^{2}\right| {\mathcal {B}}_{j_{i}}\right] . \end{aligned}$$

Theorem A.2

$$\begin{aligned}&E\left[ \left. {\widehat{EE}}_{i}^{\mathrm {BBMW}}\right| {\mathcal {B}} _{j_{i}}\right]>E\left[ \left. {\widehat{EE}}_{i}^{\mathrm {Mack}}\right| {\mathcal {B}}_{j_{i}}\right] >E\left[ \left. EE_{i}\right| {\mathcal {B}}_{j_{i}}\right] , \nonumber \\&\quad \text {where }{\widehat{EE}}_{i}^{\mathrm {BBMW}},{\widehat{EE}}_{i} ^{\mathrm {Mack}},EE_{i}\text { are given in }( 4.4) ,(4.1) ,(4.7). \end{aligned}$$

(A.3)

Proof

$$\begin{aligned} E\left[ \left. {\widehat{EE}}_{i}^{\mathrm {BBMW}}\right| {\mathcal {B}} _{j_{i}}\right]&= {\displaystyle \sum \limits _{j=j_{i}}^{J-1}} E\left[ \widetilde{C_{i,j}}^{2}\frac{\widehat{\sigma _{j}^{2}}}{ {\displaystyle \sum \nolimits _{l=0}^{i_{j}-1}} C_{l,j}}\left( {\displaystyle \prod \nolimits _{n=j+1}^{J-1}} \widehat{f_{n}}^{2}\right) {\mathcal {B}}_{j_{i}}\right] \\&> {\displaystyle \sum \limits _{j=j_{i}}^{J-1}} E\left[ \left. \widehat{C_{i,j}}^{2}\frac{\widehat{\sigma _{j}^{2}}}{ {\displaystyle \sum \nolimits _{l=0}^{i_{j}-1}} C_{l,j}}\left( {\displaystyle \prod \nolimits _{n=j+1}^{J-1}} \widehat{f_{n}}^{2}\right) \right| {\mathcal {B}}_{j_{i}}\right] \\&=E\left[ \left. EE_{i}^{\mathrm {Mack}}\right| {\mathcal {B}}_{j_{i} }\right] .\\ E\left[ \left. {\widehat{EE}}_{i}^{\mathrm {Mack}}\right| {\mathcal {B}} _{j_{i}}\right]&= {\displaystyle \sum \limits _{j=j_{i}}^{J-1}} E\left[ \left. E\left[ \left. \widehat{C_{i,j}}^{2}\frac{\widehat{\sigma _{j}^{2}}}{ {\displaystyle \sum \nolimits _{l=0}^{i_{j}-1}} C_{l,j}}\left( {\displaystyle \prod \nolimits _{n=j+1}^{J-1}} \widehat{f_{n}}^{2}\right) \right| {\mathcal {B}}_{j+1}\right] \right| {\mathcal {B}}_{j_{i}}\right] \\&> {\displaystyle \sum \limits _{j=j_{i}}^{J-1}} E\left[ \left. \widehat{C_{i,j}}^{2}\frac{\widehat{\sigma _{j}^{2}}}{ {\displaystyle \sum \nolimits _{l=0}^{i_{j}-1}} C_{l,j}}E\left[ \left. \left( {\displaystyle \prod \nolimits _{n=j+1}^{J-1}} \widehat{f_{n}}\right) \right| {\mathcal {B}}_{j+1}\right] ^{2}\right| {\mathcal {B}}_{j_{i}}\right] \\&= {\displaystyle \sum \limits _{j=j_{i}}^{J-1}} \left\{ E\left[ \left. \widehat{C_{i,j}}^{2}\frac{\widehat{\sigma _{j}^{2}} }{ {\displaystyle \sum \nolimits _{l=0}^{i_{j}-1}} C_{l,j}}\right| {\mathcal {B}}_{j_{i}}\right] {\displaystyle \prod \nolimits _{n=j+1}^{J-1}} f_{n}^{2}\right\} \\&= {\displaystyle \sum \limits _{j=j_{i}}^{J-1}} \left\{ E\left[ \left. E\left[ \left. \widehat{C_{i,j}}^{2}\frac{\widehat{\sigma _{j}^{2}}}{ {\displaystyle \sum \nolimits _{l=0}^{i_{j}-1}} C_{l,j}}\right| {\mathcal {B}}_{j}\right] \right| {\mathcal {B}}_{j_{i} }\right] {\displaystyle \prod \nolimits _{n=j+1}^{J-1}} f_{n}^{2}\right\} \\&= {\displaystyle \sum \limits _{j=j_{i}}^{J-1}} \left\{ E\left[ \left. \widehat{C_{i,j}}^{2}\frac{\sigma _{j}^{2}}{ {\displaystyle \sum \nolimits _{l=0}^{i_{j}-1}} C_{l,j}}\right| {\mathcal {B}}_{j_{i}}\right] {\displaystyle \prod \nolimits _{n=j+1}^{J-1}} f_{n}^{2}\right\} \\&= {\displaystyle \sum \limits _{j=j_{i}}^{J-1}} \left\{ E\left[ \left. E\left[ \widehat{C_{i,j}}^{2}\left. \left( \widehat{f_{j}}-f_{j}\right) ^{2}\right| {\mathcal {B}}_{j}\right] \right| {\mathcal {B}}_{j_{i}}\right] {\displaystyle \prod \nolimits _{n=j+1}^{J-1}} f_{n}^{2}\right\} \\&=E\left[ \left. EE_{i}\right| {\mathcal {B}}_{j_{i}}\right] , \end{aligned}$$

where in the third equation we have used (3.9) and in the third equation that $E\left[ \left. \widehat{\sigma _{j}^{2}}\right| {\mathcal {B}}_{j}\right] =\sigma _{j}^{2}.$ $\square$

Remark From Theorem A.2 we see that, conditional on ${\mathcal {B}} _{j_{i}},$ both estimators are positively biased, but that ${\widehat{EE}} _{i}^{\mathrm {Mack}}$ has a smaller bias than ${\widehat{EE}}_{i}^{\mathrm {BBMW}}$.