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.

The special tax treatment of United Kingdom pensions means that the decision on how to use pension assets is particularly involved. In particular, the ability to take up to 25% of pension assets as a tax-free cash lump sum at retirement, offers retirees opportunities to enhance their pension above that possible through the purchase of a compulsory purchase annuity (“CPA”). The tax-free cash lump sum can be used to buy a tax-efficient purchased life annuity (“PLA”), or in a phased retirement strategy. Income withdrawal can also be used to defer the purchase of an annuity until age 75 and, potentially, to generate a higher income. In this paper I compare the options available to retirees using stochastic modelling. I compare the expected excess pension and expected shortfall, both relative to the alternative risk-free pension available, to assess the various options. I find that if the maximum amount of tax-free cash is available to be used to enhance retirement income, then phased retirement offers the best risk/reward trade off. The advantage is greatest for higher-rate tax payers. As the level of tax-free cash falls, income withdrawal becomes more attractive to those wishing to take greater risks.

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.

In this paper we present methods and results for the estimation and modelling of the recovery intensity for Income Protection (IP) insurance claims, allowing for different causes of claim. We use UK data supplied by the Continuous Mortality Investigation relating to claims paid in the years 1975 to 2002, inclusive. Each claim is classified by one of 70 possible causes according to ICD8.

We group causes where appropriate, and then use the Cox model and generalised linear models to model the recovery intensity.

In two subsequent papers we complete our modelling of IP claim termination rates by discussing the modelling of the mortality of IP claimants.

There are two main reasons why it is useful to incorporate cause of sickness in the modelling of IP claim terminations:

(i) The cause of sickness will be known to the insurer for a claim in the course of payment. A reserve can be set more accurately for such a claim if a model of the termination rates appropriate for this cause is available.

(ii) Different causes of claim will become more or less significant over time. For example, tuberculosis may have been an important cause of sickness in the past, but is likely to be far less significant now; the swine flu pandemic starting in 2009 is likely to have a significant effect on observed aggregate claim termination rates, skewing them towards higher rates at shorter durations. Information about trends in morbidity, together with a model of termination rates by cause of claim, allows future aggregate claim termination rates to be predicted more accurately, reserves to be set at more appropriate levels and policies to be priced more accurately.

One of the covariates included in our models for recovery intensities is Calendar Year. Aggregate recovery intensities have been decreasing over the period considered, 1975 to 2002, and this is generally reflected in the models for recovery intensities by cause of sickness. However, when these intensities are projected for years beyond 2002, the results are not always plausible.

This is the second of three papers in which we present methods and results for the estimation and modelling of claim termination rates for Income Protection (IP) insurance, allowing for different causes of claim. In the first paper we discussed recoveries. In this and the third paper we develop models for the mortality of IP claimants.

We model this mortality as the sum of two components: a base, or background, mortality, which is a function of age and calendar year, but not of the specific cause of sickness or its current duration, and a cause-specific element which does depend on the current duration of the sickness. In this paper we discuss the modelling of the base mortality. In particular, we use data supplied by the Continuous Mortality Investigation relating to UK assured lives from 1975 (males) and 1983 (females) to 2003 to develop models of mortality which are functions of sex, age and calendar year. Such models are of interest in their own right, particularly at a time when expected future lifetimes are increasing.

The modelling of the cause-specific component of the mortality model is discussed in Paper III.

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.

This is the third of three papers in which we present methods and results for the estimation and modelling of claim termination rates for Income Protection (IP) insurance, allowing for different causes of claim. In the first paper we discussed recoveries. We model the mortality of IP claimants as the sum of two components: a base mortality, which is a function of age and calendar year, but not of the specific cause of sickness or its current duration, and a cause-specific component which does depend on the current duration of the sickness. The modelling of the base mortality, using data for UK assured lives, was discussed in the second paper. In this paper we discuss the modelling of the cause-specific component of the mortality of IP claimants. We use data supplied by the Continuous Mortality Investigation relating to IP claims paid in the years 1975 to 2002.

In the final section of this paper we present some numerical results for cause-specific claim annuity rates for current claims and aggregate claim termination rates based on the models developed in all three papers.

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.

This paper examines the chain-ladder technique using the recently developed theory for age-period-cohort models. The theory was set out by Kuang et al. (2008a), and we believe that it has some significant implications for claims reserving and the chain-ladder technique. This paper applies the age-period-cohort model using the over-dispersed Poisson framework, and examines a number of experiments in order to understand better how the chain-ladder technique deals with calendar year effects. The conclusions from these investigations are that the basic chain-ladder technique may have some fundamental difficulties in many circumstances. We would therefore recommend that it should be used with caution, and that the data are examined in detail before any projections are made. This has particular importance in the context of solvency calculations since the chain-ladder technique can impose some specific patterns into the projections.