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Modeling accounting year dependence in runoff triangles

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Abstract

Typically, non-life insurance claims data is studied in claims development triangles which display the two time axes accident years and development years. Most stochastic claims reserving models assume independence between different accident years. Therefore, such models fail to model claims inflation appropriately, because claims inflation acts on all accident years simultaneously. We introduce a Bayes chain ladder reserving model which enables us to model claims inflation. In this model we derive analytical formulas for the posterior distribution, the claims reserves and their prediction uncertainty.

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Authors and Affiliations

  1. RiskLab, Department of Mathematics, ETH Zurich, 8092 , Zurich, Switzerland

    Robert Salzmann & Mario V. Wüthrich

Authors
  1. Robert Salzmann
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  2. Mario V. Wüthrich
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Corresponding author

Correspondence to Robert Salzmann.

A. Appendix

A. Appendix

Proof of Theorem 2

For the proof of Theorem 2 we consider the moment generating function of \(\varvec{\xi }\). To do so, we choose \({\varvec{a}}=\left({{\varvec{a}}^{(1)}}^{\prime },\ldots ,{{\varvec{a}}^{(I+J)}}^{\prime }\right)^{\prime }\in \mathbb R ^{I(J+1)}\) with \({\varvec{a}}^{(k)}\in \mathbb R ^q\) for all \(k=1,\ldots ,I+J\). Then we obtain for the moment generating function of \(\varvec{\xi }\) (using the conditional independence of the \({\varvec{\xi }}^{(k)}\) in the fourth equation)

$$\begin{aligned} M_{{\varvec{\xi }}}({\varvec{a}})&= {\mathbb E }\left[\exp \left\{ {\varvec{a}}^{\prime }{\varvec{\xi }}\right\} \right] ={\mathbb E }\left[{\mathbb E }\left[\left.\exp \left\{ \sum _{k=1}^{I+J}{{\varvec{a}}^{(k)}}^{\prime }{\varvec{\xi }}^{(k)}\right\} \right|{\varvec{\mu }}\right]\right]\\&= {\mathbb E }\left[{\mathbb E }\left[\left.\prod_{k=1}^{I+J}\exp \left\{ {{\varvec{a}}^{(k)}}^{\prime }{\varvec{\xi }}^{(k)}\right\} \right|{\varvec{\mu }}\right]\right]={\mathbb E }\left[\prod_{k=1}^{I+J}\exp \left\{ {{\varvec{a}}^{(k)}}^{\prime }P_k{\varvec{\mu }}+\frac{1}{2}{{\varvec{a}}^{(k)}}^{\prime }\Sigma _k{{\varvec{a}}^{(k)}}\right\} \right]\\&= {\mathbb E }\left[\exp \left\{ \left(\sum _{k=1}^{I+J}{{\varvec{a}}^{(k)}}^{\prime }P_k\right){\varvec{\mu }}\right\} \right]\exp \left\{ \frac{1}{2}\sum _{k=1}^{I+J}{{\varvec{a}}^{(k)}}^{\prime }\Sigma _k{{\varvec{a}}^{(k)}}\right\} \\&= \exp \left\{ \sum _{k=1}^{I+J}{{\varvec{a}}^{(k)}}^{\prime }P_k{\varvec{\phi }}+\frac{1}{2}\left(\sum _{k=1}^{I+J}{{\varvec{a}}^{(k)}}^{\prime }P_k\right)T\left(\sum _{l=1}^{I+J}{{\varvec{a}}^{(l)}}^{\prime }P_l\right)^{\prime }+\frac{1}{2}\sum _{k=1}^{I+J}{{\varvec{a}}^{(k)}}^{\prime }\Sigma _k{{\varvec{a}}^{(k)}}\right\} \\&= \exp \left\{ \sum _{k=1}^{I+J}{{\varvec{a}}^{(k)}}^{\prime }\left(P_k{\varvec{\phi }}\right)+\frac{1}{2}\sum _{k,l=1}^{I+J}{{\varvec{a}}^{(k)}}^{\prime }\left(P_kTP_l^{\prime }+\Sigma _k1_{\{k=l\}}\right){{\varvec{a}}^{(l)}}\right\} \\&= \exp \left\{ {\varvec{a}}^{\prime }\varvec{\varphi }+\frac{1}{2}{\varvec{a}}^{\prime }\Upsilon {\varvec{a}}\right\} , \end{aligned}$$

where \(\varvec{\varphi }\) and \(\Upsilon \) are defined in Theorem 2. Therefore, the unconditional distribution of \({\varvec{\xi }}\) is multivariate normal with parameters \(\varvec{\varphi }\) and \(\Upsilon \). This completes the proof.\(\square \)

Proof of Theorem 3

Formula (4.3) and Corollary 1 imply that

$$\begin{aligned} \widehat{C}_{i,J}&= {\mathbb E }\left[\left.C_{i,J}\right|{\varvec{\xi }}_{{\mathcal D}_I}\right]=C_{i,I-i}\sum _{s=1}^{2^{J-(I-i)}}{\mathbb E }\left[\left.\exp \left\{ {\varvec{e}}_{i,s}^{\prime }{\varvec{\xi }}_{{\mathcal D}_I^c}\right\} \right|{\varvec{\xi }}_{{\mathcal D}_I}\right]\\&= C_{i,I-i}\sum _{s=1}^{2^{J-(I-i)}}\exp \left\{ {\varvec{e}}_{i,s}^{\prime }\varvec{\varphi }^{post}+\frac{1}{2}{\varvec{e}}_{i,s}^{\prime }\Upsilon ^{post}{\varvec{e}}_{i,s}\right\} , \end{aligned}$$

where the last equation is a direct consequence of log-normal distributions.\(\square \)

Proof of Theorem 4

By applying Corollary 1 and using the notation introduced in Theorem 3 we obtain for the covariance terms in formula (4.4)

$$\begin{aligned} {\rm Cov}\left(\!\left.\!C_{i,J}\!,\!C_{l,J}\!\right|\!{\mathcal D}_I\!\right)&= C_{i,I-i}C_{l,I-l}\sum _{s=1}^{2^{J-(I-i)}}\sum _{u=1}^{2^{J-(I-l)}}\!{\rm Cov}\!\left(\!\left.\exp \left\{ {\varvec{e}}_{i,s}^{\prime }{\varvec{\xi }}_{{\mathcal D}_{I}^{c}}\right\} ,\exp \left\{ {\varvec{e}}_{l,u}^{\prime }{\varvec{\xi }}_{{\mathcal D}_{I}^{c}}\right\} \!\right|\!{\varvec{\xi }}_{{\mathcal D}_I}\!\right)\\&= C_{i,I-i}C_{l,I-l}\sum _{s=1}^{2^{J-(I-i)}}\sum _{u=1}^{2^{J-(I-l)}}\exp \left\{ {\varvec{e}}_{i,s}^{\prime }\varvec{\varphi }^{post}+\frac{1}{2}{\varvec{e}}_{i,s}^{\prime }\Upsilon ^{post}{\varvec{e}}_{i,s}\right\} \\&\times \exp \left\{ {\varvec{e}}_{l,u}^{\prime }\varvec{\varphi }^{post}+\frac{1}{2}{\varvec{e}}_{l,u}^{\prime }\Upsilon ^{post}{\varvec{e}}_{l,u}\right\} \left(\exp \left\{ {\varvec{e}}_{i,s}^{\prime }\Upsilon ^{post}{\varvec{e}}_{l,u}\right\} -1\right). \end{aligned}$$

By substituting \(\beta _{m,n}^{post}=\exp \left\{ {\varvec{e}}_{m,n}^{\prime }\varvec{\varphi }^{post}+\frac{1}{2}{\varvec{e}}_{m,n}^{\prime }\Upsilon ^{post}{\varvec{e}}_{m,n}\right\} \) we obtain for the conditional MSEP

$$\begin{aligned}&\sum _{I-J<i,l\le I}C_{i,I-i}C_{l,I-l}\sum _{s=1}^{2^{J-(I-i)}}\sum _{u=1}^{2^{J-(I-l)}}\beta _{i,s}^{post}\beta _{l,u}^{post}\left(\exp \left\{ {\varvec{e}}_{i,s}^{\prime }\Upsilon ^{post}{\varvec{e}}_{l,u}\right\} -1\right)\\&\quad =\sum _{I-J<i,l\le I}\widehat{C}_{i,J}\widehat{C}_{l,J} \sum _{s=1}^{2^{J-(I-i)}}\sum _{u=1}^{2^{J-(I-l)}}\frac{\beta _{i,s}^{post}}{\sum _{v=1}^{2^{J-(I-i)}}\beta _{i,v}^{post}}\frac{\beta _{l,u}^{post}}{\sum _{w=1}^{2^{J-(I-l)}}\beta _{l,w}^{post}}\left(\exp \left\{ {\varvec{e}}_{i,s}^{\prime }\Upsilon ^{post}{\varvec{e}}_{l,u}\right\} -1\right). \end{aligned}$$

In the last equation we multiply and divide each term \(C_{m,I-m}\) by \(\sum _{n=1}^{2^{J-(I-m)}}\beta _{m,n}^{post}\) and set, see Theorem 3,

$$\begin{aligned} \widehat{C}_{m,J}=C_{m,I-m}\sum _{n=1}^{2^{J-(I-m)}}\beta _{m,n}^{post}. \end{aligned}$$

Substituting \(\omega _{m,n}^{post}\) as defined in Theorem 4 provides the claim.\(\square \)

Proof of Corollary 2

For the coefficient of variation of the ultimate claim we obtain

$$\begin{aligned}&{\rm Vco}\left(\left.C_J\right|{\mathcal D}_I\right)\\&=\frac{\left[\sum _{I-J<i,l\le I}\widehat{C}_{i,J}\widehat{C}_{l,J}\sum _{s=1}^{2^{J-(I-i)}}\sum _{u=1}^{2^{J-(I-l)}}\omega _{i,s}^{post}\omega _{l,u}^{post}\left(\exp \left\{ {\varvec{e}}_{i,s}^{\prime }\Upsilon ^{post}{\varvec{e}}_{l,u}\right\} \!-\!1\right)\right]^{1/2}}{\sum _{I-J<i\le I}\widehat{C}_{i,J}}\\&\le \frac{\left[\sum _{I-J<i,l\le I}\widehat{C}_{i,J}\widehat{C}_{l,J}\sum _{s=1}^{2^{J-(I-i)}}\sum _{u=1}^{2^{J-(I-l)}}\omega _{i,s}^{post}\omega _{l,u}^{post}\left(v^{{\rm max}}_{i,l}\!-\!1\right)\right]^{1/2}}{\sum _{I-J<i\le I}\widehat{C}_{i,J}}\\&=\frac{\left[\sum _{I-J<i,l\le I}\widehat{C}_{i,J}\widehat{C}_{l,J}\left(v^{{\rm max}}_{i,l}\!-\!1\right)\right]^{1/2}}{\widehat{C}_J}. \end{aligned}$$

The last step follows because \(\sum _{n=1}^{2^{J-\!(\!I-m\!)}}\!\omega _{m,n}^{post}\!=\!1\). The second upper bound for \({\rm Vco}\!\left(\!\left.C_J\!\right|\!{\mathcal D}_I\!\right)\) follows directly from

$$\begin{aligned} {\rm Vco}\left(\left.C_J\right|{\mathcal D}_I\right)\le \frac{\left[\sum _{I-J<i,l\le I}\widehat{C}_{i,J}\widehat{C}_{l,J}\left(v^{{\rm max}}_{i,l}\!-\!1\right)\right]^{1/2}}{\widehat{C}_J}\le (v^{{\rm max}}-1)^{1/2}. \end{aligned}$$

\(\square \)  

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Salzmann, R., Wüthrich, M.V. Modeling accounting year dependence in runoff triangles. Eur. Actuar. J. 2, 227–242 (2012). https://doi.org/10.1007/s13385-012-0055-3

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