Ben Zehnwirth

Claims reserving, state-space models and the Kalman filter

1.1. This paper describes a consistent and justifiable means of establishing adequate claims provisions in General Insurance. The topic has created widespread interest amongst actuaries, accountants and regulatory authorities. The issue of adequate provisions is of utmost importance to policyholders, whose justifiable claims must be paid, insurance companies who must be able to satisfy shareholders and make proper assessments of premiums, and regulatory authorities who must be satisfied that adequate provision has been made for all liabilities.

A Generalization of the Kalman Filter for Models with State-Dependent Observation Variance

Abstract The Kalman filter is generalized to cover state-space models in which the variance of the observation error depends on the state vector. Derivations of the filter yielding minimum mean squared error linear estimators and associated error covariance matrices are obtained from two differing viewpoints: linear Bayes theory and Gauss—Markov theory. The results are applied to a model for which {y t: t = 1, 2, …, n} follow a Poisson distribution with corresponding intensities {θt: t = 1, 2, …,n} that are assumed to follow an autoregressive process of order 1, namely . The steady-state generalized Kalman filter algorithm in the case for which ρ = 1 gives a generalization of exponential smoothing for a Poisson process with time-varying intensity.

Credibility theory and the Kalman filter

Abstract Following Mehra (1975) we indicate how some of the well known credibility models may be formulated as Kalman filters. The formulation yields recursive premium forecasts including recursive predictions errors which are of importance to practitioners.

Linear Filtering and Recursive Credibility Estimation

Recursive credibility estimation is discussed from the viewpoint of linear filtering theory. A conjunction of geometric interpretation and the innovation approach leads to general algorithms not developed before. Moreover, covariance characterizations considered by other researchers drop our elegantly as a result of geometric considerations. Examples are presented of Kalman type filters valid for non-Gaussian measurements.

An international comparison of economic leading indicators of telecommunications traffic

Abstract The purpose of this paper is to draw international comparisons of the coherence of indexes of leading economic indicators with selected telecommunications traffic series. The traffic series under consideration are total Australian telephone outgoing and U.S. outgoing telephone to Australia with data consisting of monthly observations spanning the period 1970–1983. The response of the telecommunications traffic to these indexes is analysed using cross-spectral techniques. Additionally, a dynamic regression forecasting model for Australian traffic is estimated using the Australian index as an explanatory variable. In comparison to an ARIMA model for the telecommunications data this model reduces post-sample MSE by 19 percent.

The impact of economic cycles on the demand for international telecommunications in Australia

Abstract Recently an Australian Index of Leading Indicators has been constructed which is designed to consistently anticipate economic cycles. It is shown in this investigative paper, using cross spectral techniques, that there is a strong association between international telecommunications traffic and this leading index. The findings provide evidence in support of the hypothesis that telecommunications traffic responds to fluctuations in aggregate economic activity. Therefore the results suggest the appropriacy of the inclusion of the index as a putative explanatory variable in the construction of a forecasting model for telecommunications traffic.

Comments on Taylor's see-saw approach to claims reserving

Abstract Taylor (1981) introduces the See-Saw (SS) model for claims reserving in order to make allowance for speed of finalization. The model is applied to live data given in Taylor (1981) and there is prima facie evidence to suggest that it does well, especially in the light of comparisons of actual versus expected payments for each payment year. The purpose of the present paper is to demonstrate that from the point of view of operational forecasting, which is the object of the claims reserving exercise, the fitting of the linear SS can be improved upon. Moreover, we employ the SS as a vehicle for indicating the kind of validation tests that ought to be carried out once the parameters of a proposed model have been estimated from the data. Essentially, we indicate how the properties of the residuals may be used for diagnostic checking of the model. Many researchers involved in the claims reserving area are of the view that the data are extremely noisy especially if the model put forward only explains a small proportion of the total variation. Why not test whether this is the case? It turns out that the particular linear SS used by Taylor does not explain the signal accurately so that the balance of the variation is not entirely due to noise.

Nonparametric linear bayes estimation of survival curves from incomplete observations

A linear Bayes estimator of a survival curve is derived.The estimator has a relatively simple interpretation as a Kaplan-Meier estimator based on an augemented data base - prior information plus sampling information.It is Bayes if the prior is a Dirichlet process, and otherwise an approximation to the Bayes rule against any prior.

Credibility: Estimation of Structural Parameters

Invited Lectures given at the Nato Advanced Study Institute on Insurance Premiums, Katholique Universiteit Leuven, July 18–31, 1983.

Hachemeister's Bayesian regression model revisited

Abstract Hachemeisters (1975) regression model is discussed from a number of viewpoints including credibility theory, Gauss-Markov theory and the Kalman filter. The last formulation facilitates recursive premium formulae including forecast errors. Recursive estimation of structural parameters is also briefly mentioned and an illustration involving a data set is presented. The paper basically serves to unify a number of recently developed ideas rather than present new results.