Closed-form Solutions for an Explicit Modern Ideal Tontine with Bequest Motive

In this paper I extend the work of Bernhardt and Donnelly (2019) dealing with modern explicit tontines, as a way of providing income under a specified bequest motive, from a defined contribution pension pot. A key feature of the present paper is that it relaxes the assumption of fixed proportions invested in tontine and bequest accounts. In making the bequest proportion an additional control function I obtain, hitherto unavailable, closed-form solutions for the fractional consumption rate, wealth, bequest amount, and bequest proportion under a constant relative risk averse utility. I show that the optimal bequest proportion is the product of the optimum fractional consumption rate and an exponentiated bequest parameter. I show that under certain circumstances, such as a very high bequest motive, a life-cycle utility maximisation strategy will necessitate negative mortality credits analogous to a member paying life insurance premiums. Typical scenarios are explored using UK Office of National Statistics life tables.Comment: 21 pages, 6 figure

Methods for generating variates from probability distributions

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Diverse probabilistic results are used in the design of random univariate generators. General methods based on these are classified and relevant theoretical properties derived. This is followed by a comparative review of specific algorithms currently available for continuous and discrete univariate distributions. A need for a Zeta generator is established, and two new methods, based on inversion and rejection with a truncated Pareto envelope respectively are developed and compared. The paucity of algorithms for multivariate generation motivates a classification of general methods, and in particular, a new method involving envelope rejection with a novel target distribution is proposed. A new method for generating first passage times in a Wiener Process is constructed. This is based on the ratio of two random numbers, and its performance is compared to an existing method for generating inverse Gaussian variates. New "hybrid" algorithms for Poisson and Negative Binomial distributions are constructed, using an Alias implementation, together with a Geometric tail procedure. These are shown to be robust, exact and fast for a wide range of parameter values. Significant modifications are made to Atkinson's Poisson generator (PA), and the resulting algorithm shown to be complementary to the hybrid method. A new method for Von Mises generation via a comparison of random numbers follows, and its performance compared to that of Best and Fisher's Wrapped Cauchy rejection method. Finally new methods are proposed for sampling from distribution tails, using optimally designed Exponential envelopes. Timings are given for Gamma and Normal tails, and in the latter case the performance is shown to be significantly better than Marsaglia's tail generation procedure.Governors of Dundee College of Technolog

Metropolis Sampling

Monte Carlo (MC) sampling methods are widely applied in Bayesian inference, system simulation and optimization problems. The Markov Chain Monte Carlo (MCMC) algorithms are a well-known class of MC methods which generate a Markov chain with the desired invariant distribution. In this document, we focus on the Metropolis-Hastings (MH) sampler, which can be considered as the atom of the MCMC techniques, introducing the basic notions and different properties. We describe in details all the elements involved in the MH algorithm and the most relevant variants. Several improvements and recent extensions proposed in the literature are also briefly discussed, providing a quick but exhaustive overview of the current Metropolis-based sampling's world.Comment: Wiley StatsRef-Statistics Reference Online, 201

Bayesian inference for transportation origin-destination matrices: the Poisson-inverse Gaussian and other Poisson mixtures

Transportation origin–destination analysis is investigated through the use of Poisson mixtures by introducing covariate‐based models which incorporate different transport modelling phases and also allow for direct probabilistic inference on link traffic based on Bayesian predictions. Emphasis is placed on the Poisson–inverse Gaussian model as an alternative to the commonly used Poisson–gamma and Poisson–log‐normal models. We present a first full Bayesian formulation and demonstrate that the Poisson–inverse Gaussian model is particularly suited for origin–destination analysis because of its desirable marginal and hierarchical properties. In addition, the integrated nested Laplace approximation is considered as an alternative to Markov chain Monte Carlo sampling and the two methodologies are compared under specific modelling assumptions. The case‐study is based on 2001 Belgian census data and focuses on a large, sparsely distributed origin–destination matrix containing trip information for 308 Flemish municipalities

A review of multi-component maintenance models with economic dependence

In this paper we review the literature on multi-component maintenance models with economic dependence. The emphasis is on papers that appeared after 1991, but there is an overlap with Section 2 of the most recent review paper by Cho and Parlar (1991). We distinguish between stationary models, where a long-term stable situation is assumed, and dynamic models, which can take information into account that becomes available only on the short term. Within the stationary models we choose a classification scheme that is primarily based on the various options of grouping maintenance activities: grouping either corrective or preventive maintenance, or combining preventive-maintenance actions with corrective actions. As such, this classification links up with the possibilities for grouped maintenance activities that exist in practice

A stochastic differential equation approach to the analysis of the UK 2017 and 2019 general election polls

Human dynamics and sociophysics build on statistical models that can shed light on and add to our understanding of social phenomena. We propose a generative model based on a stochastic differential equation that enables us to model the opinion polls leading up to the UK 2017 and 2019 general elections, and to make predictions relating to the actual result of the elections. After a brief analysis of the time series of the poll results, we provide empirical evidence that the gamma distribution, which is often used in financial modelling, fits the marginal distribution of this time series. We demonstrate that the proposed poll-based forecasting model may improve upon predictions based solely on polls. The method uses the Euler-Maruyama method to simulate the time series, measuring the prediction error with the mean absolute error and the root mean square error, and as such could be used as part of a toolkit for forecasting elections

The impact of deferral and adverse selection on the actuarial fairness and cost neutrality of the UK state pension

Persons who have achieved UK state pension age (SPA) may defer their pension and instead receive an extra pension on termination of deferral. We define a scheme to be actuarially fair to a category of deferrer with agreed discount rate, when the expected net present value of pre-tax lifetime receipts is independent of the deferral period. After a review of the literature on deferral and early take-up of state pensions in the UK and other countries, this paper argues that the current UK scheme based upon a uniform accrual rate cannot be actuarially fair. Instead, we propose a scheme where the accrual rate is dependent upon deferral period, gender, SPA, deferrer’s discount rate, degree of pension uprating, and partnership status of the deferrer. Fair accrual rate curves are plotted for various scenarios and compared with the current uniform rates of 10.4% and 5.8% per annum that apply to those who attain SPA before 6 April and after 5 April 2016 respectively. A scheme that is actuarially fair will not be cost neutral to the Exchequer unless the discount rate is the same for both parties. In addition to this asymmetry, adverse selection will impact upon both actuarial fairness and cost to the Exchequer. Expressions are derived for the cost penalty to the Exchequer for attempting to achieve actuarial fairness both with and without an acknowledgement of adverse selection. Similarly, when the objective is to achieve cost neutrality for the Exchequer, expressions for the cost to the deferrer are obtained. Some numerical examples are given for various scenarios. The methodology should be applicable to public pensions in other countries, in order to inform fair policies for both early and postponed take-up of pensions