The prediction of adequate claims reserves is a major subject in actuarial practice and science. Due to their simplicity, the chain ladder (CL) and Bornhuetter–Ferguson (BF) methods are the most commonly used claims reserving methods in practice. However, in contrast to the CL method, no estimator for the conditional mean square error of prediction (MSEP) of the ultimate claim has been derived in the BF method until now, and as such, this paper aims to fill that gap. This will be done in the framework of generalized linear models (GLM) using the (overdispersed) Poisson model motivation for the use of CL factor estimates in the estimation of the claims development pattern.

We examine the data and techniques underlying the estimation of mortality rates at older ages in Ireland since 1950. Previous attempts to elucidate the level and trends in mortality at advanced ages in Ireland have been frustrated by significant non-random biases arising from age exaggeration and age heaping, together with a lack of correspondence, growing with increasing age, between the exposed-to-risk estimated from census data and the death count from registration data. Applying the method of extinct generations, we re-estimate crude mortality rates and report the somewhat unexpected result that mortality rates were lower, and did not increase as steeply with age, than those recorded in the official Irish Life Tables. The reestimated crude rates show, for both sexes, a very slight decrease in mortality rates between the 1950s and 1980s up to age 90 years, with no improvement discernible at older ages. Improvements at advanced ages in Ireland have lagged behind those in England and Wales and other developed countries over the same period. The companion paper, Mortality in Ireland at Advanced Ages, 1950–2006: Part 2: Graduated Rates, Whelan (2009), graduates the crude rates and extends the method of extinct generations to estimate mortality rates of more recent, still surviving, generations.

We graduate the Irish mortality experience from 1950 to 2003 by mathematical formulae from ages 75 years and upwards. The shape of the mortality curve at advanced ages is shown to be different to that recorded in the official tables, with the curve best fitted with Kannisto's version of Perks's Law. Mortality rates show only a modest trend of improvement in the early decades, below improvements in other developed countries. We evaluate the various approaches suggested to date to extend the method of extinct generations so mortality rates for non-extinct generations can be estimated. It is shown that the key advantage of this method is not in correcting for age misstatements but in achieving a close correspondence between death counts and the exposed to risk. This insight allows a rather straightforward approach to estimating the mortality of non-extinct generations. Applying the approach, we show that there has been an acceleration in the rate of improvement in more recent decades, but secular improvements in Irish mortality at advanced ages still lag behind those of England and Wales.

It has long been known that maximum likelihood estimation in a Poisson model reproduces the chain-ladder technique. We revisit this model. A new canonical parametrisation is proposed to circumvent the inherent identification problem in the parametrisation. The maximum likelihood estimators for the canonical parameter are simple, interpretable and easy to derive. The boundary problem where all observations in one particular development year or on particular underwriting year is zero is also analysed.

A.A.S. 3, I/II, 127-185 (2008)

Stochastic Actuarial Modelling of a Defined-Benefit Social Security Pension Scheme: An Analytical Approach