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Prediction Uncertainty in the Bornhuetter-Ferguson Claims Reserving Method: Revisited

Published online by Cambridge University Press:  21 October 2010

D. H. Alai ,
M. Merz  and
M. V. Wüthrich
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Abstract

We revisit the stochastic model of Alai et al. (2009) for the Bornhuetter-Ferguson claims reserving method, Bornhuetter & Ferguson (1972). We derive an estimator of its conditional mean square error of prediction (MSEP) using an approach that is based on generalized linear models and maximum likelihood estimators for the model parameters. This approach leads to simple formulas, which can easily be implemented in a spreadsheet.

Keywords

Claims ReservingBornhuetter-FergusonOverdispersed Poisson DistributionChain Ladder MethodGeneralized Linear ModelsFisher Information MatrixConditional Mean Square Error of Prediction
Type
Papers
Information
Annals of Actuarial Science , Volume 5 , Issue 1 , March 2011 , pp. 7 - 17
Copyright
Copyright © Institute and Faculty of Actuaries 2010

1 Introduction

The prediction uncertainty in the Bornhuetter-Ferguson (BF) claims reserving method, Bornhuetter & Ferguson (Reference Bornhuetter and Ferguson1972), has recently been studied by several authors; see e.g. Mack (Reference Mack2008), Verrall (Reference Verrall2004) and Alai et al. (Reference Alai, Merz and Wüthrich2009). We revisit the model studied in Alai et al. (Reference Alai, Merz and Wüthrich2009). In the present paper we provide a different method of approximating the mean square error of prediction (MSEP), which substantially simplifies the formulas while preserving the accuracy established in the previous paper.

Alai et al. (Reference Alai, Merz and Wüthrich2009) maintain that in practice the chain ladder (CL) development pattern is used for calculating the BF reserves, and hence incorporate this into their model assumptions. This is done by assuming the data to be overdispersed Poisson distributed. This allows one to recreate the CL estimate of the development pattern; a result dating back to Hachemeister & Stanard (Reference Hachemeister and Stanard1975) and Mack (Reference Mack1991). This is different from the approach taken in Mack (Reference Mack2008), but closer to the implementation of practitioners. We furthermore maintain the necessary assumption that the initial estimates of the expected ultimate claims are independent of the data, an assumption that is the basis of the BF methodology. This independence assumption is certainly challenged in practice, both Mack (Reference Mack2008) and Schmidt & Zocher (Reference Schmidt and Zocher2008) suggest that estimates of the expected ultimate claims come from pricing, and it remains unclear how independent such information is from the data. We do not dwell on this issue at present and leave it for future consideration.

In this paper our main objective is to provide a far simpler method of implementing the results derived in Alai et al. (Reference Alai, Merz and Wüthrich2009). We direct the reader to the previous paper, as well as the works of Mack (Reference Mack2008), Neuhaus (Reference Neuhaus1992) and Schmidt & Zocher (Reference Schmidt and Zocher2008) for elaboration on the motivation of studying the BF method and when it should be applied in practice. Furthermore, we do not consider the claims inflation problem at present, a topic with recent developments made by Kuang et al. (Reference Kuang, Nielsen and Nielsen2008a,Reference Kuang, Nielsen and Nielsenb).

Organization of the paper. In Section 2 we provide the notation and data structure as well as the model considerations. In Section 3 we give a short review of the BF method. In Section 4 we give a simplified estimation procedure for the conditional MSEP in the BF method. Finally, in Section 5 we revisit the case study presented in Alai et al. (Reference Alai, Merz and Wüthrich2009) and compare our results with Mack (Reference Mack2008) and Verrall (Reference Verrall2004).

2 Data and Model

2.1 Setup

Let Xi,j denote the incremental claims of accident year i∈{0,1, … , I} and development year j∈{0,1, … , J}. We assume the data is given by a claims development triangle, i.e. I = J, and that after J development periods all claims are settled. At time I, we have observations )(]${\cal D}_{I} = \{ X_{{i,j}}, \ i + j\,{\leq}\,I\\ } $. We are interested in predicting the corresponding lower triangle {Xi,j, i + j > I, i ≤ I}. Furthermore, define Ci,j to be the cumulative claims of accident year i up to development year j. Hence,

\[C_{{i,j}} = \mathop{\sum}\limits_{k = 0}^j {X_{{i,k}} } .\eqno<?xpath string(ancestor::disp-formula/child::label)?>\]

2.2 Model Considerations

We adopt the overdispersed Poisson model presented in Alai et al. (Reference Alai, Merz and Wüthrich2009). Please refer to Section 4.1 of that paper for the density function, from which it is shown that the overdispersed Poisson belongs to the exponential dispersion family. The reader is also advised to see Kuang et al. (Reference Kuang, Nielsen and Nielsen2009) for further elaboration on the connection between the overdispersed Poisson model and the CL method using maximum likelihood estimators (MLEs), as well as Section 2.3 below.

Model Assumptions 2.1 (Overdispersed Poisson Model)

  • • The incremental claims Xi,j are independent overdispersed Poisson distributed and there exist positive parameters γ 0, … ,γI, μ 0, … ,μI and φ > 0 with

    \[\displaylines{ \E\,[X_{{i,j}} ] = &amp; m_{{i,j}} = \mu _{i} \gamma _{j}, \cr \vskip3pt{\rm{Var}}\,(X_{{i,j}} ) = &amp; \phi \,m_{{i,j}}, \cr}\eqno<?xpath string(ancestor::disp-formula/child::label)?>\]
    and $\mathop{\sum}\nolimits_{j = 0}^I \,\gamma _{j} = 1.

  • • $\hskip-3pt\widehat{\\nu }_{k} $ are random variables that are unbiased estimators of the expected ultimate claim μk = E[Ck,I] for all k∈{0, … ,I}.

  • • Xi,j and $\widehat{\nu }_{k} $ are independent for all i,j,k.

Remarks 2.2:

  • • The exogenous estimator $\widehat{\nu }_{k} $ is a prior estimate of the expected ultimate claims E[Ck,I], which is used for the BF method; see also Section 2 in Mack (Reference Mack2008). In the work of Alai et al. (Reference Alai, Merz and Wüthrich2009)these estimators were assumed to be independent, we now adopt a more general assumption in which their dependence structure can be modelled.

  • • For MSEP considerations, an estimate of the uncertainty of the $\widehat{\nu }_{k} $ is required. Below, we assume that a prior variance estimate $\widehat{{{\rm{Var}}}}(\widehat{\nu }_{k} ) is given exogenously.

  • • For additional model interpretations we refer to Alai et al. (Reference Alai, Merz and Wüthrich2009).

2.3 Maximum Likelihood Estimators

Under Model Assumptions 2.1 the log-likelihood function for ${\cal D}_{I} $ is given by

\[\displaylines{\hskip-8pc l_{{{\cal D}_{I} }} (\mu _{i}, \gamma _{j}, \phi ) &amp;= \mathop{\sum}\limits_{{\mathop{i + j\leq I}\limits_\scale140%{j \lt I}}} {\left( {\frac{1}{\phi } \( X_{{i,j}} \log (\mu _{i} \gamma _{j} ){\rm{\, -\, }}\mu _{i} \gamma \_{j} ) + \log c(X_{{i,j}} ;\phi )} \right)} \cr \hskip5pc &amp; \quad \+ \left( {\frac{1}{\phi }\left( {X_{{0,I}} \log \left[ {\mu _{0} \left( \{1{\rm{ - }}\mathop{\sum}\limits_{n = 0}^{I{\rm{ - }}1} {\gamma _{n} } \} \right)} \right]{\rm{ - }}\mu _{0} \left( {1{\rm{ - }}\mathop{\sum}\limits_{n = 0}^{I{\rm{ - }}1} {\gamma _{n} } } \right) + \log c(X_{{0,I}} ;\phi \)} \right)} \right) , \cr}\eqno<?xpath string(ancestor::disp-formula/child::label)?>\]

where c(·,φ) is the suitable normalizing function. Notice that the substitution, $\gamma _{I} \= (1{\rm{ - }}\mathop{\sum}\nolimits_{n = 0}^{I{\rm{ - }}1} {\gamma _{n} \} )$ has been made in accordance with the constraint provided in Model Assumptions 2.1. The MLEs $\widehat{\mu }_{i}, \widehat{\gamma \}_{j} $ are found by taking the derivates with respect to μi, γj and setting the resulting equations equal to zero. They are given by,

(1)
\[\displaylines{ \widehat{\mu }_{0} = &amp; \mathop{\sum}\limits_{j = 0}^I {X_{{0,j}} }, \quad \ \cr \hskip13.1pc\widehat{\mu }_{i} \mathop{\sum}\limits_{j \= 0}^{I{\rm{ - }}i} {\widehat{\gamma }_{j} }= &amp; \mathop{\sum}\limits_{j \= 0}^{I{\rm{ - }}i} {X_{{i,j}} }, \quad \quad i \in \{ 1, \ldots, I\}, \\hskip8.6pc(1)\cr \hskip-3.6pc\widehat{\gamma }_{j} \left( {\mathop{\sum}\limits_{i = 1}^{I{\rm{ - }}j} {\widehat{\mu }_{i} } + X_{{0,I}} \frac{1}{{1{\rm{ - }}\mathop{\sum}\nolimits_{n = 0}^{I{\rm{ - }}1} {\widehat{\gamma }_{n} \} }}} \right) = &amp; \mathop{\sum}\limits_{i = 0}^{I{\rm{ - }}j} {X_{{i,j}} \}, \quad \quad j \in \{ 0, \ldots, I{\rm{ - }}1\} . \cr}\eqno <?xpath \string(ancestor::disp-formula/child::label)?>\]

Furthermore, we define $\widehat{\gamma }_{I} = 1{\rm{ \- }}\mathop{\sum}\nolimits_{n = 0}^{I{\rm{ - }}1} {\widehat{\gamma }_{n} \} $. The $\widehat{\mu }_{i}, \widehat{\gamma }_{j} $ can also be calculated with help from the well-known CL factors,

\[\widehat{\hskip-2pt f}_{j} = \frac{{\mathop{\sum}\nolimits_{i = 0}^{I{\rm{ - }}j{\rm{ - }}1} {C_{{i,j + 1}} } }}{\vskip3pt{\mathop{\sum}\nolimits_{i \= 0}^{I{\rm{ - }}j{\rm{ - }}1} {C_{{i,j}} } }};\eqno<?xpath string(ancestor::disp-formula/child::label)?>\]

see e.g. Corollary 2.18 and Remarks 2.19 in Wüthrich & Merz (Reference Wüthrich and Merz2008), i.e.

(2)
\[\widehat{\gamma }_{j} = \prod\limits_{k = j}^{I{\rm{ - }}1} {\frac{1}{{\widehat{f}_{k} }}} \left( {1{\rm{ \- }}\frac{1}{{\widehat{f}_{{j - 1}} }}} \right),\; \quad \widehat{\mu \}_{i} = C_{{i,I{\rm{ - }}i}} \widehat{f}_{{I{\rm{ - }}i}}\cdots \widehat{f}_{{I{\rm{ - }}1}} .\eqno<?xpath string(ancestor::disp-formula/child::label)?>\]

Although, as is clear in (1), φ has no influence on the parameter estimation of μi, γj, an estimate of φ is required to estimate the prediction uncertainty. As done in Alai et al. (Reference Alai, Merz and Wüthrich2009), we use Pearson residuals to estimate φ:

(3)
\[\widehat{\phi \} = \frac{1}{d}\mathop{\sum}\limits_{i + j\leq I} {\frac{{(X_{{i,j}} {\rm{ \- }}\widehat{m}_{{i,j}} )^{2} }}{{\widehat{m}_{{i,j}} }}}, \eqno<?xpath \string(ancestor::disp-formula/child::label)?>\]

where $d = {{{(I + 1)(I + 2)}}\over {2}}{\rm{ - }}2I{\rm{ - }}1$ is the degrees of freedom of the model and is the degrees of freedom of the model and $\widehat{m}_{{i,j}} = \widehat{\mu }_{i} \widehat{\gamma} _{j} $.

2.4 Asymptotic Properties of the MLE

As previously stated, the overdispersed Poisson model is a member of the exponential dispersion family. We use the proposition directly below which yields the asymptotic behaviour of the MLEs to quantify the parameter estimation uncertainty $\widehat{\gamma}_{j} {\rm{ \,-\, \}}\gamma _{j} $; see e.g. Lehmann (Reference Lehmann1983), Theorem 6.2.3.

Proposition 2.3 Assume X 1, … ,Xn are i.i.d. with density fζ(·) from the exponential dispersion family with parameters ζ=(ζ1, … ,ζm)T. Furthermore, $\widehat{\brzeta } = (\widehat{\zeta }_{1}, \ldots ,\widehat{\zeta \}_{m} )^{T} $is the MLE of ζ, then,

\[{\sqrt n} (\widehat{\brzeta }{\rm{ \,- \,}}\brzeta \)\mathop{\longrightarrow}^{{(d)}}{\cal N}({\bf 0},H(\brzeta )^{{{{ - }}1}} \),\quad{{as}} \ n \rightarrow \infty, \eqno<?xpath string(ancestor::disp-formula/child::label)?>\]

where we define the Fisher information matrix by H(ζ) = (h r,s)r,s =1,…,m with

\[\vskip16pt\scale 140%{ \h_{{r,s}} = H(\brzeta )_{{r,s}} = {\rm{ - }}E_{\brzeta } \left[\frac{{\partial ^{2} }}{{\partial \zeta _{r} \partial \zeta _{s} }}\log f_{\brzeta } \(X)\right].\eqno<?xpath string(ancestor::disp-formula/child::label)?>\]

Remark 2.4 One must be careful when considering asymptotic behaviour with respect to studying claims reserving triangles. We inherently limit ourselves to a finite dataset and hence introduce some error when using results from asymptotic theory. This issue was studied numerically, using the bootstrap method, in Section 7.3 of Wüthrich & Merz (Reference Wüthrich and Merz2008). There it was observed, using the same dataset we study in Section 5, that the bias and the estimation error can be estimated accurately under the asymptotic normal approximation.

We use the notation = (ζ 1, … ,ζ 2I +1)=(μ 0, … ,μI, γ 0, … ,γI −1) and for the corresponding MLE. Under Model Assumptions 2.1, we obtain for the components of the Fisher information matrix:

\[\displaylines {\hskip-2.5pc \h_{{i + 1, \,i + 1}} = &amp; \frac{{\mu _{i}^{{{\rm{ - }}1}} }}{\phi }\mathop{\sum}\limits_{j = 0}^{I{\rm{ - }}i} {\gamma _{j} }, \; \quad\quad\quad\quad\\quad\quad\quad\quad\quad i \in \{ 0, \ldots, I\}, \vskip18pt\cr \hskip-2.5pc \h_{{I + 2 + j,\,I + 2 + j}} = &amp; \frac{{\gamma _{j}^{{{\rm{ - }}1}} \}}{\phi }\mathop{\sum}\limits_{i = 0}^{I{\rm{ - }}j} {\mu _{i} } + \frac{{\mu _{0} }}{{\phi \left( {1{\rm{ - }}\mathop{\sum}\nolimits_{n = 0}^{I{\rm{ \- }}1} {\gamma _{n} } } \right)}}, \quad j \in \{ 0, \ldots, I - 1\}, \\vskip18pt\cr \hskip-0pc h_{{I + 2 + j,\,I + 2 + l}} = &amp; \frac{{\mu \_{0} }}{{\phi \left(1- {\mathop{\sum}\nolimits_{n = 0}^{I{\rm{ - }}1} \{\gamma _{n} } } \right)}}, \quad\quad\quad\quad\quad\quad j,l \in \{ \0, \ldots, I{\rm{ - }}1\}, \ j \ne l, \vskip22pt\cr \hskip-0.7pc h_{{i \+ 1, \, I + 2 + j}} = &amp; \frac{1}{\phi },\; \quad\quad\quad\quad\quad\quad\;\quad\quad i \in \{ 1, \ldots, I\}, \;j \in \{ 0, \ldots, I{\rm{ - }}i\}, \\cr \hskip 0.4pc h_{{I + 2 + j, \, i + 1}} = &amp; \frac{1}{\phi },\;\; \\quad\quad\quad\quad\quad\quad\quad\quad j \in \{ 0, \ldots, I{\rm{ - \}}1\}, \;i \in \{ 1, \ldots, I{\rm{ - }}j\} . \cr}\eqno<?xpath string(ancestor::disp-formula/child::label)?>\]

The remaining entries of the (2I+1)×(2I+1) matrix H(ζ) are zero. By replacing the parameters ζ and φ by their estimates given in (1) and (3), respectively, we obtain the estimated Fisher information matrix $H(\widehat{\brzeta },\widehat{\phi })$. The inverse of the estimated Fisher information matrix, $H(\widehat{\brzeta },\widehat{\phi })^{{{\rm{ - }}1}} $, contains, for our purposes, unnecessary information regarding the parameters μi. Therefore, we define the (I + 1)×(I × 1) matrix

\[{\cal \{G}} = (g_{{j,l}} )_{{j,l = 0, \ldots, I}}, \eqno<?xpath string(ancestor::disp-formula/child::label)?>\]

with

(4)
\[\vskip-10pt\displaylines{\hskip2.6pc g_{{j,l}} = &amp; \widehat{{\rm Cov}}\,(\widehat{\gamma }_{j}, \widehat{\gamma }_{l} ) = H(\widehat{\brzeta },\widehat{\phi \})_{{I + 2 + j,I + 2 + l}}^{{{\rm{ - }}1}}, \, \quad\quad\quad\quad j,l \\in \{ 0, \ldots, I{\rm{ - }}1\}, \cr \vskip12pt\hskip4.8pc g_{{j,I}} \= &amp; g_{{I,j}} = \widehat{{\rm Cov}}\,(\widehat{\gamma }_{j}, \widehat{\gamma }_{I} ) = \minus - \mathop{\sum}\limits_{m = 0}^{I{\rm{ - }}1} {H(\widehat{\brzeta },\widehat{\phi })_{{I + 2 + j,I + 2 + m}}^{{{\rm{ - }}1}} \}, \hskip10pt j \in \{ 0, \ldots, I{\rm{ - }}1\}, \hskip4.2pc(4)\cr \hskip-4.3pc \vskip18pt \quad g_{{I,I}} = &amp; \widehat{{\rm Var}}\,(\widehat{\gamma \}_{I} ) = \mathop{\sum}\limits_{{\mathop{{0\leq m\leq I{\rm{ - }}1}}\limits_\\scale140%{{0\leq n\leq I{\rm{ - }}1}}} } {H(\widehat{\brzeta },\widehat{\phi \})_{{I + 2 + m,I + 2 + n}}^{{{\rm{ - }}1}} } . \cr}\eqno<?xpath string(ancestor::disp-formula/child::label)?>\]

The first equation of (4) gives an estimator for the covariances between the MLEs $\widehat{\gamma }_{j} \,{\rm{and}}\ \widehat{\gamma }_{l} \$, whereas the last two equations of (4) incorporate the MLE $\widehat{\gamma }_{I} = 1{\rm{ - }}\mathop{\sum}\nolimits_{n = 0}^{I{\rm{ - }}1} {\widehat{\gamma }_{n} } $.

3 The Bornhuetter-Ferguson Method

In practice, the BF predictor, which dates back to Bornhuetter & Ferguson (Reference Bornhuetter and Ferguson1972), relies on the data for the development pattern γj and on external data or expert opinion for the expected ultimate claims E[Ci,I]. The ultimate claim Ci,I of accident year i under Model Assumptions 2.1 using the BF method, given ${\cal D}_{I}$, is predicted by

(5)
\[\widehat{C}_{{i,I}}^{{BF}} = C_{{i,I{\rm{ - }}i}} + \widehat{\nu \}_{i} \mathop{\sum}\limits_{j \gt I{\rm{ - }}i} {\widehat{\gamma }_{j} \}, \eqno<?xpath string(ancestor::disp-formula/child::label)?>\]

where $\widehat{\nu }_{i} $ are the MLEs produced in Section 2.3 and $\widehat{\nu }_{i} $ is an exogenous prior estimator for the expected ultimate claim E[Ci,I] introduced in Model Assumptions 2.1.

Note that we define the BF predictor with the CL development pattern $\widehat{\gamma }_{j} $, which is the approach used in practice; see equation (2). A different approach for the estimation of the development pattern γj is given in Mack (2008), we further discuss this in the case study in Section 5.

4 The MSEP of the Bornhuetter-Ferguson Method

We begin by considering the (conditional) MSEP of the BF predictor $\widehat{C}_{{i,I}}^{{BF}} $ for single accident years i∈{1, … ,I}. From (5.5) in Alai et al. (Reference Alai, Merz and Wüthrich2009) we have

(6)
\[\displaylines{ \hskip-15pc{\rm{msep}}_{{C_{{i,I}} \|{\cal D}_{I} }} (\widehat{C}_{{i,I}}^{{BF}} ) = &amp; E \,[(\widehat{C}_{{i,I}}^{{BF}} {\rm{ - }}C_{{i,I}} )^{2} |{\cal D}_{I} ] \cr \hskip9.95pc= \&amp; \mathop{\sum}\limits_{j \gt I{\rm{ - }}i} {{\rm{Var}}\,(X_{{i,j}} \)} + \left( {\mathop{\sum}\limits_{\,j \gt I{\rm{ - }}i} {\widehat{\gamma \}_{j} } } \right)^{2} {\rm{Var}}\,(\widehat{\nu }_{i} ) + \mu _{i}^{2} \\left( {\mathop{\sum}\limits_{\,j \gt I{\rm{ - }}i} {\widehat{\gamma }_{j} \} {\rm{ \, - }}\mathop{\sum}\limits_{j \gt I{\rm{ - }}i} {\gamma _{j} \} } \right)^{2} . \hskip1.7pc(6)\cr}\eqno <?xpath string(ancestor::disp-formula/child::label)?>\]

The first term on the right-hand side of equation (6) is the (conditional) process variance, it represents the stochastic movement of the Xi,j, the inherent uncertainty from our model assumptions. The latter two terms form the (conditional) estimation error; these terms constitute the uncertainty in the prediction of the prior estimate $\widehat{\nu }_{i} and the MLEs \widehat{\gamma }_{j} $. The first two terms on the right-hand side of equation (6) can be estimated by replacing unknowns with their estimates; see e.g. Sections 5.1.1 and 5.1.2 in Alai et al. (Reference Alai, Merz and Wüthrich2009). The last term, however, if tackled this way would equal zero. The standard approach, see England & Verrall (Reference England and Verrall2002), is to estimate

\[\left(\; {\mathop{\sum}\limits_{j \gt I{\rm{ - }}i} {(\widehat{\gamma }_{j} \{\rm{ - }}\gamma _{j} )} } \right)^{2} \eqno<?xpath string(ancestor::disp-formula/child::label)?>\]

by the unconditional expectation

Neglecting that MLEs have a possible bias term, see Remark 2.4, we make the following approximation:

\[\mathop{\sum}\limits_{{\mathop{{j \gt I{\rm{ - }}i}}\limits_\scale140%{{l \gt I{\rm{ - }}i}}} } {E\,[(\,\widehat{\gamma }_{j} {\rm{ \,- \,}}\gamma _{j} )(\,\widehat{\gamma }_{l} {\rm{ \- }}\gamma _{l} )]} \approx \mathop{\sum}\limits_{{\mathop{{j \gt I{\rm{ \\,-\, }}i}}\limits_\scale140%{{l \gt I{\rm{ \,-\, }}i}}} } {{\rm{Cov}}\,(\,\widehat{\gamma }_{j} {\rm{ \,-\, }}\widehat{\gamma }_{l} )} .\eqno<?xpath \string(ancestor::disp-formula/child::label)?>\]

We now deviate from Alai et al. (Reference Alai, Merz and Wüthrich2009) and directly use ${\cal G} $, given by equations (4) to estimate the covariance terms. Hence, an estimate of the MSEP in the BF method for single accident year i is given by:

Estimator 4.1 (MSEP for the BF method, single accident year) Under Model Assumptions 2.1 an estimator for the (conditional) MSEP for a single accident year i∈{1,…,I} is given by

\[\widehat{{{\rm{msep}}}}_{{C_{{i,I}} |{\cal D}_{I} }} (\widehat{C}_{{i,I}}^{{BF}} ) = \mathop{\sum}\limits_{j \\gt I{\rm{ \,-\, }}i} {\widehat{\phi }\widehat{\nu }_{i} \widehat{\gamma \}_{j} } + \left(\, {\mathop{\sum}\limits_{j \gt I{\rm{ \,-\, }}i} {\widehat{\\gamma }_{j} } } \right)^{2} \widehat{{{\rm{Var}}}}\,(\,\widehat{\nu }_{i} \) + \widehat{\nu }_{i}^{2} \mathop{\sum}\limits_{{\mathop{{j \gt I{\rm{ \\,-\, }}i}}\limits_{\scale140%{l \gt I{\rm{ \,-\, }}i}}} } {g_{{j,l}} \} .\eqno<?xpath string(ancestor::disp-formula/child::label)?>\]

Remark 4.2 If we compare the above estimator to equation (5.30) in Alai et al. (Reference Alai, Merz and Wüthrich2009)we observe that the first two terms on the right-hand side are identical. However, the last term, i.e. the uncertainty in $\widehat{\gamma }_{j} $, has substantially simplified and can be easily calculated in a spreadsheet environment.

For multiple accident years the (conditional) MSEP is defined as follows:

\[\vskip10pt\displaylines{ {\pre}{{{\rm{msep}}}}_{{\mathop{\sum}\nolimits_{i \= 1}^I {C_{{i,I}} |{\cal D}_{I} } }} \left( {\mathop{\sum}\limits_{i = \1}^I {\widehat{C}_{{i,I}}^{{BF}} } } \right) = &amp; E\left[ {\left. {\left( \{\mathop{\sum}\limits_{i = 1}^I {\widehat{C}_{{i,I}}^{{BF}} } {\rm{ \,-\, \}}\mathop{\sum}\limits_{i = 1}^I {C_{{i,I}} } } \right)^{2} } \right|{\cal \D}_{I} } \right] \ \ \qquad\qquad\qquad\qquad\qquad\cr \qquad\qquad\qquad\qquad\qquad= &amp; \mathop{\sum}\limits_{i = 1}^I {{\rm{msep}}_{{C_{{i,I}} \|{\cal D}_{I} }} } (\widehat{C}_{{i,I}}^{{BF}} ) + 2\mathop{\sum}\limits_{i \\lt k} {\mu _{i} \mu _{k} } \mathop{\sum}\limits_{{\mathop{{j \gt I - \i}}\limits_{\scale140%{l \gt I - k}}} } {(\,\widehat{\gamma }_{j} {\rm{ \\,-\, }}\gamma _{j} )(\widehat{\gamma }_{l} {\rm{ \,-\, }}\gamma _{l} \)} \cr &amp; \qquad\qquad\qquad+ 2\mathop{\sum}\limits_{i \lt k} {\left(\, \{\mathop{\sum}\limits_{j \gt I{\rm{ \,-\, }}i} {\widehat{\gamma }_{j} \} } \right)} \left(\, {\mathop{\sum}\limits_{l \gt I{\rm{ \,-\, }}k} {\widehat{\gamma }_{l} } } \right){\rm{Cov}}\,(\,\widehat{\nu }_{i}, \widehat{\nu \}_{k} ). \cr}\eqno<?xpath string(ancestor::disp-formula/child::label)?>\]

Remark 4.3 Here we obtain an additional term compared to Alai et al. (Reference Alai, Merz and Wüthrich2009) since we allow the exogenous estimators $\widehat{\nu \}_{i} $ to depend on one another, which is a natural assumption if one estimates $\widehat{\nu }_{i} $ from the time series $\widehat{\nu }_{l}, l\leq i$. One can now make use of a variety of methods to estimate the covariances, methods that imply decaying positive correlation are recommended; see e.g. Mack Reference Mack(2008).

Estimator 4.4 (MSEP for the BF method, aggregated accident years) Under Model Assumptions 2.1 an estimator for the (conditional) MSEP for aggregated accident years is given by

\[\displaylines {\hskip-1pc {\pre}{{\widehat{{{\rm{msep}}}}}}_{{\mathop{\sum}\nolimits_{i = 1}^I {C_{{i,I}} |{\cal D}_{I} \} }} \left( {\mathop{\sum}\limits_{i = 1}^I {\widehat{C}_{{i,I}}^{{BF}} \} } \right) = &amp; \mathop{\sum}\limits_{i = 1}^I {\widehat{{{\rm{msep}}}}_{{C_{{i,I}} |{\cal D}_{I} }} } (\widehat{C}_{{i,I}}^{{BF}} ) + 2\mathop{\sum}\\limits_{i \lt k} {\widehat{\nu }_{i} \widehat{\nu }_{k} } \mathop{\sum}\limits_{{\mathop{{j \gt I{\rm{ \,-\, }}i}}\limits_\scale140%{{l \gt I{\rm{ \\,-\, }}k}}} } {g_{{j,l}} } \cr \hskip9.5pc &amp; + 2\mathop{\sum}\limits_{i \\lt k} {\left(\, {\mathop{\sum}\limits_{j \gt I{\rm{ \,-\, }}i} {\widehat{\gamma }_{j} } } \right)\left(\, {\mathop{\sum}\limits_{l \gt I{\rm{ \,-\, \}}k} {\widehat{\gamma }_{l} } } \right)} \widehat{{{\rm{Cov}}}}(\,\widehat{\nu }_{i}, \widehat{\nu }_{k} ). \cr}\eqno<?xpath string(ancestor::disp-formula/child::label)?>\]

Remark 4.5 The above estimator should be compared with equation (5.35) in Alai et al. (Reference Alai, Merz and Wüthrich2009). Again, the additional term present due to our more general assumptions on the exogenous estimators of the expected ultimate claims. On the other hand, the second term on the right-hand side of Estimator 4.4 has a much simpler form compared with Alai et al. (Reference Alai, Merz and Wüthrich2009).

5 Case Study

We utilize the dataset {Xi,j:i + j ≤ I} provided in Alai et al. (Reference Alai, Merz and Wüthrich2009), which is shown in Table 1. We assume given external estimates $\widehat{\nu }_{i} $ of the ultimate claims, presented in Table 2. Furthermore, we assume the uncertainty of these estimates to be given by a coefficient of variation of 5% and assume they are uncorrelated. Hence,

\[\vskip-12pt\widehat{{{\rm{Var}}}}\,(\widehat{\nu }_{i} ) = \widehat{\nu }_{i}^{2} (0.05)^{2} .\eqno<?xpath string(ancestor::disp-formula/child::label)?>\]

Table 1 Observed incremental claims Xi,j.

Table 2 Prior estimates for the expected ultimate claims.

Using equation (3), we obtain for the dispersion parameter φ, the estimate $\widehat{\phi } = 14,\!714$.

We demonstrate the numerical results in Table 3. Note that they are the same as the results presented in Alai et al. (Reference Alai, Merz and Wüthrich2009), but the implementation is much simpler now. We compare the results in Table 3 to those from Mack (Reference Mack2008). We start by calculating the development pattern using equation (3) in Mack (Reference Mack2008). We normalize these results such that the pattern sums to one. Note that the normalization is necessary due to the fact that the prior estimates $\widehat{\nu }_{i} $ are rather conservative (as mentioned in Wüthrich & Merz (Reference Wüthrich and Merz2008), Example 2.11).

Table 3 Reserve and uncertainty results for single and aggregated accident years using the method in Alai et al. (Reference Alai, Merz and Wüthrich2009).

In Table 4 we compare the cumulative development pattern (referred to as $\widehat{z\,}_{j}^{\ast} $ in Mack (Reference Mack2008)) with the cumulative development pattern obtained using the method of Alai et al. (Reference Alai, Merz and Wüthrich2009)(referred to as $\widehat{\beta }_{j} $). Also shown in Table 4 are the standard errors calculated for the cumulative development patterns using the respective methods. In Alai et al. (Reference Alai, Merz and Wüthrich2009) the approximated standard errors of the development pattern were compared with empirical standard errors obtained from simulation. The simulation study consisted of 10,000 simulated run-off triangles under the assumption of overdispersed Poisson data. The results from this simulation study shows the accuracy of the approximation, which is maintained here. Therefore, although a different, much simpler, method of approximation is utilized in this paper, it preserves the accuracy established in the previous paper.

Table 4 Cumulative development pattern, a comparison.

Remark 5.1 The distinction is made between estimates of the development pattern $\widehat{\gamma }_{j} $ and of the cumulative development pattern $\widehat{\beta }_{j} $; the latter being defined as follows:

\[\widehat{\beta }_{j} \= \mathop{\sum}\limits_{k = 0}^j {\widehat{\gamma }_{k} }, \; \quad {\rm{for}}\ j \in \{ 0, \ldots, I\} .\eqno<?xpath string(ancestor::disp-formula/child::label)?>\]

Table 4 indicates a slower decrease of the uncertainty in our approach.

In Table 5 we provide the $\widehat{s}_{j}^{2} \$ calculated using equation (4) in Mack (Reference Mack2008). The role of the $\widehat{s}_{j}^{2} $ are comparable to that of $\widehat{\phi }$. The difference originates from the fact that the $\widehat{\phi }$ depend on the development year j, whereas $\widehat{\phi }$ does not.

Table 5 $\widehat{s}_{j}^{2} \$ calculated from equation (3) in Mack (Reference Mack2008).

Finally, we apply the same coefficient of variation to determine the standard error of the ultimates using the method in Mack (Reference Mack2008), namely 5%. Table 6 provides the MSEP results under the method described in Mack (Reference Mack2008). It should be compared to Table 3, which provides the results under the method described in Alai et al. (Reference Alai, Merz and Wüthrich2009) and in this paper.

Table 6 Reserve and uncertainty results for single and aggregated accident years using the method in Mack (Reference Mack2008).

As becomes clear from comparing Tables 3 and 6, one main difference between the two methods lies in the estimated process variance. It is evident that this difference originates in the model assumptions with respect to the structure of the variance of the incremental claims. Alai et al. (Reference Alai, Merz and Wüthrich2009) assume

\[\{\rm{Var}}\,(X_{{i,j}} ) = \phi \,m_{{i,j}}, \eqno<?xpath string(ancestor::disp-formula/child::label)?>\]

whereas Mack (Reference Mack2008) assumes

\[{\rm{Var}}\,(X_{{i,j}} ) = s_{j}^{2} \,m_{{i,j}} .\eqno<?xpath string(ancestor::disp-formula/child::label)?>\]

Table 5 shows the volatility of the sj 2, which heavily impacts the process variance. A similar picture is obtained for the parameter standard deviation, in contrast to the prior standard deviation, which almost perfectly coincide.

Finally, in Table 7, we present the MSEP results for the distribution-free CL method, see Mack (Reference Mack1993), as well as the MSEP results from Verrall (Reference Verrall2004). To obtain the (conditional) MSEP in the distribution-free CL method, we use the approach described in Buchwalder et al. (Reference Buchwalder, Bühlmann, Merz and Wüthrich2006). The calculations for the Bayesian negative binomial approach presented in Verrall (Reference Verrall2004) were performed using WinBugs, we ran 20,000 iterations, discarding the first 10,000. Although in no way conclusive, the overall approach of Alai et al. (Reference Alai, Merz and Wüthrich2009) is more in line with the CL MSEP figures.

Table 7 Aggregate reserve and uncertainty results for the CL method, the BF approach of Alai et al. (Reference Alai, Merz and Wüthrich2009), the BF approach of Mack (Reference Mack2008), and the BF approach of Verrall (Reference Verrall2004).

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Figure 0

Table 1 Observed incremental claims Xi,j.

Figure 1

Table 2 Prior estimates for the expected ultimate claims.

Figure 2

Table 3 Reserve and uncertainty results for single and aggregated accident years using the method in Alai et al. (2009).

Figure 3

Table 4 Cumulative development pattern, a comparison.

Figure 4

Table 5 $\widehat{s}_{j}^{2} \$ calculated from equation (3) in Mack (2008).

Figure 5

Table 6 Reserve and uncertainty results for single and aggregated accident years using the method in Mack (2008).

Figure 6

Table 7 Aggregate reserve and uncertainty results for the CL method, the BF approach of Alai et al. (2009), the BF approach of Mack (2008), and the BF approach of Verrall (2004).