In this paper, we present an extension of the model proposed by Lin & Liu that uses the concept of physiological age to model the ageing process by using phase-type distributions to calculate the probability of death. We propose a finite-state Markov jump process to model the hypothetical ageing process in which it is possible the transition rates between non-consecutive physiological ages. Since the Markov process has only a single absorbing state, the death time follows a phase-type distribution. Thus, to build a mortality table the challenge is to estimate this matrix based on the records of the ageing process. Considering the nature of the data, we consider two cases: having continuous time information of the ageing process, and the more interesting and realistic case, having reports of the process just in determined times. If the ageing process is only observed at discrete time points we have a missing data problem, thus, we use a stochastic Expectation–Maximisation (SEM) algorithm to find the maximum likelihood estimator of the intensity matrix. And in order to do that, we build Markov bridges which are sampled using the Bisection method. The theory is illustrated by a simulation study and used to fit real data.

Chen et al. (2014), studied a discrete semi-Markov risk model that covers existing risk models such as the compound binomial model and the compound Markov binomial model. We consider their model and build numerical algorithms that provide approximations to the probability of ultimate ruin and the probability and severity of ruin in a continuous time two-state Markov-modulated risk model. We then study the finite time ruin probability for a discrete m-state model and show how we can approximate the density of the time of ruin in a continuous time Markov-modulated model with more than two states.

Over the last 20 years, the extent of defined benefit provision has declined substantially in the United Kingdom. Whilst most of the focus has been on deficits relating to past benefit accrual, the increasing cost of future benefit accrual is also important. There are two reasons for this. First, the change in the cost of defined benefit accrual represents the difference in the earnings for employees with membership of a defined benefit scheme and those with membership of a defined contribution scheme. Second, the current cost of defined benefit accrual gives an indication of the cost of an adequate pension. As such, it can be compared with levels of contribution to defined contribution schemes to determine whether these are adequate. I therefore look at how the cost of pensions has changed relative to the cost of non-pensions earnings. I also look at the main components of the change in pensions cost – those relating to benefits payable, discount rates and longevity – to analyse their relative importance. I find that the cost of employing a member of defined benefit pension scheme has consistently outpaced the cost of employing someone in a defined contribution arrangement. I also find that the current cost of accrual is significantly higher than the average level of payments to defined contribution schemes.

In this paper, we develop an extension to the Wilkie model, introducing share earnings and cover (earnings/dividends) as new variables, and deriving share dividends from them. Earnings are available from April 1962, but only for the Non-Financial index, and for the All-Share one only from 1992. We construct a Composite Earnings Index from these series. We then find a suitable annual time series model for changes in earnings, and then for cover, which is mean-reverting. We compare this new model with the original model, in which changes in dividends were modelled directly. We also investigate monthly data to give parameters for stochastic interpolation. We observe an unusual change in earnings over 2015–2016, consider the implications of this and show specimen simulations.

We investigate an actuarial approach to identifying the factors impacting government-funded maternal hospital costs in Australia, with a focus on women who experience adverse birth outcomes. We propose a two-phase modelling methodology that adopts actuarial methods from typical insurance claim cost modelling and extends to other statistical techniques to account for the large volume of covariates available for modelling. Specifically, Classification and Regression Trees and generalised linear mixed models are employed to analyse a data set that links longitudinal survey and administrative data from a large sample of women. The results show that adverse births are a statistically significant risk factor affecting maternal hospital costs in the antenatal and delivery periods. Other significant cost risk factors in the delivery period include mode of delivery, private health insurance status, diabetes, smoking status, area of residence and onset of labour. We demonstrate the efficacy of using actuarial techniques in non-traditional areas and highlight how the results can be used to inform public policy.

An usual reinsurance policy for insurance companies admits one or two layers of the payment deductions. Under optimality criterion of minimising the Conditional Tail Expectation (CTE) risk measure of the insurer’s total risk, this article generalises an optimal stop-loss reinsurance policy to an optimal multi-layer reinsurance policy. To achieve such optimal multi-layer reinsurance policy, this article starts from a given optimal stop-loss reinsurance policy f(⋅). In the first step, it cuts down the interval [0, ∞) into intervals [0, M1) and [M1, ∞). By shifting the origin of Cartesian coordinate system to (M1, f(M1)), and showing that under the CTE criteria $$f\left( x \right)I_{{[0,M_{{\rm 1}} )}} \left( x \right){\plus}\left( {f\left( {M_{{\rm 1}} } \right){\plus}f\left( {x{\minus}M_{{\rm 1}} } \right)} \right)I_{{[M_{{\rm 1}} ,{\rm }\infty)}} \left( x \right)$$ is, again, an optimal policy. This extension procedure can be repeated to obtain an optimal k-layer reinsurance policy. Finally, unknown parameters of the optimal multi-layer reinsurance policy are estimated using some additional appropriate criteria. Three simulation-based studies have been conducted to demonstrate: (1) the practical applications of our findings and (2) how one may employ other appropriate criteria to estimate unknown parameters of an optimal multi-layer contract. The multi-layer reinsurance policy, similar to the original stop-loss reinsurance policy is optimal, in a same sense. Moreover, it has some other optimal criteria which the original policy does not have. Under optimality criterion of minimising a general translative and monotone risk measure ρ(⋅) of either the insurer’s total risk or both the insurer’s and the reinsurer’s total risks, this article (in its discussion) also extends a given optimal reinsurance contract f(⋅) to a multi-layer and continuous reinsurance policy.

In this paper, the optimal safety loading that the reinsurer should set in the reinsurance pricing is studied, which is novel in the literature. It is first assumed that the insurer will choose the form of the reinsurance contract by following the results derived in Cai et al. Different optimality criteria from the reinsurer’s perspective are then studied, such as maximising the expectation of the profit, maximising the utility of the profit and minimising the value-at-risk of the reinsurer’s total loss. By applying the concept of comonotonicity, the problem in which the reinsurer is facing two risks with unknown dependency structure is also solved. Closed-form solutions are obtained when the underlying losses are zero-modified exponentially distributed. Finally, numerical examples are provided to illustrate the results derived.

This note proposes a practical way for modelling and projecting health insurance expenditures over short time horizons, based on observed historical data. The present study is motivated by a similar age structure generally observed for health insurance claim frequencies and yearly aggregate losses on the one hand and mortality on the other hand. As an application, the approach is illustrated for German historical inpatient costs provided by the Federal Financial Supervisory Authority. In particular, similarities and differences to mortality modelling are addressed.