10.1 Introduction

Uses Bayesian techniques to allow incorporation of expert opinion and also maintain the integrity of the prediction error

  • i.e. how expert opinion from sources other than the specific data set under consideration can be incorporated into the predictive distributions of the reserves

  • Take into account prior knowledge in setting reserves

    e.g. Adjusting data to reflect changes in benefits or claims handling, or select L-year average LDFs

  • Calculating prediction error using prior knowledge

  • Use Bayesian to take into account our a-priori and the strength of that a-priori (credibility)

Paper focus on use of a-priori knowledge for LDF selection and BF Method

Development of MCMC techniques has made Bayesian methods much easier

  • Makes it easier to find the posterior distributions for future observations (defining the Bayesian model is always easy)

  • MCMC breaks down the simulation process into a number of simulations that are easy to carry out

    \(\therefore\) This solves the common Bayesian problem of difficulty in finding the posterior distribution (as they can be multidimensional)

  • MCMC doesn’t simulate all the parameters at once, it use the conditional distribution of each parameter, given all the others

    \(\therefore\) Reducing the simulation to a univariate distribution

  • Markov chain is formed because each parameter is considered in turn, and it is a simulation-based method

    \(\therefore\) MCMC

10.1.1 Notation

For a \(n \times n\) triangle

Claims

  • Incremental claims for AY \(i\) and age \(j\): \(c_{ij}\)

\[\{ c_{ij} \: : \: j=1,...,n-i+1 \: ; \: i = 1,...,n \}\]

  • Cumulative claims

\[D_{ij} = \sum_{k=1}^j c_{ik}\]

  • Ultimate losses

\[D_{in} = \sum_{k=1}^n c_{ik}\]

Remark. We only consider forecasting losses up to the latest development year \(n\)

  • It is possible to extend this to allow a tail factor but not in this paper

Row Parameters

  • Expected ultimate losses for AY \(i\): \(x_i\)

\[x_i = \mathrm{E}[D_{in}]\]

Column Parameters

  • Expected % reported in each period: \(y_j\)

\[\sum y_i = 1\]

  • Expected reported to date: \(p_j\)

\[p_j = \sum_{k=1}^j y_k\]

  • Expected development from \(D_{ij-1}\) to \(D_{ij}\):

\[\lambda_j = \dfrac{p_j}{p_{j-1}}\]

\[\{\lambda_j \: : \: j = 2 ,...,n\}\]

  • Weighted average LDFs: \(\hat{\lambda}_j\)

\[\hat{\lambda}_j = \dfrac{\sum_{i=1}^{n-j+1} D_{ij}}{\sum_{i=1}^{n-j+1} D_{ij-1}}\]