Continuing increases in life expectancy beyond previously-held limits have brought to the fore the critical importance of mortality forecasting. Significant developments in mortality forecasting since 1980 are reviewed under three broad approaches: expectation, extrapolation and explanation. Expectation is not generally a good basis for mortality forecasting, as it is subjective; expert expectations are invariably conservative. Explanation is restricted to certain causes of death with known determinants. Decomposition by cause of death poses problems associated with the lack of independence among causes and data difficulties. Most developments have been in extrapolative forecasting, and make use of statistical methods rather than models developed primarily for age-specific graduation. Methods using two-factor models (age-period or age-cohort) have been most successful. The two-factor Lee–Carter method, and, in particular, its variants, have been successful in terms of accuracy, while recent advances have improved the estimation of forecast uncertainty. Regression-based (GLM) methods have been less successful, due to nonlinearities in time. Three-factor methods are more recent; the Lee–Carter age-period-cohort model appears promising. Specialised software has been developed and made available. Research needs include further comparative evaluations of methods in terms of the accuracy of the point forecast and its uncertainty, encompassing a wide range of mortality situations.

The Lee–Carter (LC) model is a basic approach to forecasting mortality rates of a single population. Although extensions of the LC model to forecasting rates for multiple populations have recently been proposed, the structure of these extended models is hard to justify and the models are often difficult to calibrate, relying on customised optimisation schemes. Based on the paradigm of representation learning, we extend the LCmodel to multiple populations using neural networks, which automatically select an optimal model structure. We fit this model to mortality rates since 1950 for all countries in the Human Mortality Database and observe that the out-of-sample forecasting performance of the model is highly competitive.

This paper extends the methods introduced in England & Verrall (2002), and shows how predictive distributions of outstanding liabilities in general insurance can be obtained using bootstrap or Bayesian techniques for clearly defined statistical models. A general procedure for bootstrapping is described, by extending the methods introduced in England & Verrall (1999), England (2002) and Pinheiro et al. (2003). The analogous Bayesian estimation procedure is implemented using Markov-chain Monte Carlo methods, where the models are constructed as Bayesian generalised linear models using the approach described by Dellaportas & Smith (1993). In particular, this paper describes a way of obtaining a predictive distribution from recursive claims reserving models, including the well known model introduced by Mack (1993). Mack's model is useful, since it can be used with data sets which exhibit negative incremental amounts. The techniques are illustrated with examples, and the resulting predictive distributions from both the bootstrap and Bayesian methods are compared.

This paper presents pricing innovations to German car insurance. The purpose is to provide an effective approach to adapting actuarial pricing decision to incorporate telematic data, which differs substantially from established tariff criteria in complexity and volume. A vehicle mobility model and a real-world sample of driving profiles form the input into the analysis. We propose an allocation of the driving profiles based on velocity and acceleration parameters to specific driving styles for evaluating the driving behaviour to subsequently enable discounts or surcharges on the premiums to obtain usage-based insurance premiums. The result is highly relevant for actuaries, who calculate the tariffs, but also for managers, as they have to make a pricing decision.

This paper examines various forms of individual claim model for the purpose of loss reserving, with emphasis on the prediction error associated with the reserve. Each form of model is calibrated against a single extensive data set, and then used to generate a forecast of loss reserve and an estimate of its prediction error.

The basis of this is a model of the “paids” type, in which the sizes of strictly positive individual finalised claims are expressed in terms of a small number of covariates, most of which are in some way functions of time. Such models can be found in the literature.

The purpose of the current paper is to extend these to individual claim “incurreds” models. These are constructed by the inclusion of case estimates in the model's conditioning information. This form of model is found to involve rather more complexity in its structure.

For the particular data set considered here, this did not yield any direct improvement in prediction error. However, a blending of the paids and incurreds models did so.

This paper builds on the two-factor mortality model known as the Cairns-Blake-Dowd (CBD) model, which is used to project future mortality. It is shown that these two factors do not follow a random walk, as proposed in the original model, but that each should instead be modelled as a random fluctuation around a trend, the trend changing periodically. The paper uses statistical techniques to determine the points at which there are statistically significant changes in each trend. The frequency of change in each trend is then used to project the frequency of future changes, and the sizes of historical changes are used to project the sizes of future changes. The results are then presented as fan charts, and used to estimate the range of possible future outcomes for period life expectancies. These projections show that modelling mortality rates in this way leaves much greater uncertainty over future life expectancy in the long term.

Pay-how-you-drive (PHYD) or usage-based (UB) systems for automobile insurance provide actuaries with behavioural risk factors, such as the time of the day, average speeds and other driving habits. These data are collected while the contract is in force with the help of telematic devices installed in the vehicle. They thus fall in the category of a posteriori information that becomes available after contract initiation. For this reason, they must be included in the actuarial pricing by means of credibility updating mechanisms instead of being incorporated in the score as ordinary a priori observable features. This paper proposes the use of multivariate mixed models to describe the joint dynamics of telematics data and claim frequencies. Future premiums, incorporating past experience can then be determined using the predictive distribution of claim characteristics given past history. This approach allows the actuary to deal with the variety of situations encountered in insurance practice, ranging from new drivers without telematics record to contracts with different seniority and drivers using their vehicle to different extent, generating varied volumes of telematics data.

This paper synthesizes and extends the literature on multivariate two-part regression modelling, with an emphasis on actuarial applications. To illustrate the modelling, we use data from the US Medical Expenditure Panel Survey to explore expenditures that come in two parts. In the first part, zero expenditures correspond to no payments for health care services during a year. For the second part, a positive expenditure corresponds to the payment amount, a measure of utilization. Expenditures are multivariate, the five components being (i) office-based, (ii) hospital outpatient, (iii) emergency room, (iv) hospital inpatient, and (v) home health expenditures. Not surprisingly, there is a high degree of association among expenditure types and so we utilize models that account for these associations. These models include multivariate binary regressions for the payment type and generalized linear models with Gaussian copulas for payment amounts.

As anticipated, the strong associations among expenditure types allow us to establish significant model differences on an in-sample basis. Despite these strong associations, we find that commonly used statistical measures perform similarly on a held-out validation sample. In contrast, out-of-sample risk measures used by actuaries reveal differences in the association among expenditure types.

A central issue in claims reserving is the modelling of appropriate dependence structures. Most classical models cannot cope with this task. We define a multivariate log-normal model that allows to model both, dependence between different sub-portfolios and dependence within sub-portfolios such as claims inflation. In this model we derive closed form solutions for claims reserves and the corresponding prediction uncertainty.

Reverse mortgages provide a mechanism for seniors to release the equity that has been built up in their home. At termination, the mortgagors are usually guaranteed to owe no more than the value of their property. The value of the reverse mortgage guarantee is heavily dependent on the maturity or termination date, which is uncertain. In this paper, we model reverse mortgage terminations using a semi-Markov multiple state model which incorporates three different modes of exit: death, entrance into a long-term care facility, and voluntary prepayment. We apply the proposed model specifically to develop the valuation formulas for roll-up mortgages in the UK and Home Equity Conversion Mortgages (HECMs) in the USA. We examine the significance of each mode of termination by valuing the contracts allowing progressively for each mode. On the basis of our model and assumptions, we find that both health related terminations and voluntary (non-health related) terminations significantly impact the contract value. In addition we analyze the premium structure for US reverse mortgage insurance, and demonstrate that premiums appear to be too high for some borrowers, and substantial cross-subsidies may result.

We consider a classical surplus process modified by the payment of dividends when the insurer's surplus exceeds a threshold. We use a probabilistic argument to obtain general expressions for the expected present value of dividend payments, and show how these expressions can be applied for certain individual claim amount distributions. We then consider the question of maximising the expected present value of dividend payments subject to a constraint on the insurer's ruin probability.

In this paper, we propose a multivariate Hawkes framework for modelling and predicting cyber attacks frequency. The inference is based on a public data set containing features of data breaches targeting the US industry. As a main output of this paper, we demonstrate the ability of Hawkes models to capture self-excitation and interactions of data breaches depending on their type and targets. In this setting, we detail prediction results providing the full joint distribution of future cyber attacks times of occurrence. In addition, we show that a non-instantaneous excitation in the multivariate Hawkes model, which is not the classical framework of the exponential kernel, better fits with our data. In an insurance framework, this study allows to determine quantiles for number of attacks, useful for an internal model, as well as the frequency component for a data breach guarantee.

We discuss risk diversification in multivariate regularly varying models and provide explicit formulas for Value-at-Risk asymptotics in this case. These results allow us to study the influence of the portfolio weights, the overall loss severity, and the tail dependence structure on large portfolio losses. We outline sufficient conditions for the sub- and superadditivity of the asymptotic portfolio risk in multivariate regularly varying models and discuss the case when these conditions are not satisfied. We provide several examples to illustrate the resulting variety of diversification effects and the crucial impact of the tail dependence structure in infinite mean models. These examples show that infinite means in multivariate regularly varying models do not necessarily imply negative diversification effects. This implication is true if there is no loss-gain compensation in the tails, but not in general. Depending on the loss-gain compensation, asymptotic portfolio risk can be subadditive, superadditive, or neither.

Rapid advances in artificial intelligence (AI) and machine learning are creating products and services with the potential not only to change the environment in which actuaries operate but also to provide new opportunities within actuarial science. These advances are based on a modern approach to designing, fitting and applying neural networks, generally referred to as “Deep Learning.” This paper investigates how actuarial science may adapt and evolve in the coming years to incorporate these new techniques and methodologies. Part 1 of this paper provides background on machine learning and deep learning, as well as an heuristic for where actuaries might benefit from applying these techniques. Part 2 of the paper then surveys emerging applications of AI in actuarial science, with examples from mortality modelling, claims reserving, non-life pricing and telematics. For some of the examples, code has been provided on GitHub so that the interested reader can experiment with these techniques for themselves. Part 2 concludes with an outlook on the potential for actuaries to integrate deep learning into their activities. Finally, a supplementary appendix discusses further resources providing more in-depth background on machine learning and deep learning.

This paper explores and develops alternative statistical representations and estimation approaches for dynamic mortality models. The framework we adopt is to reinterpret popular mortality models such as the Lee–Carter class of models in a general state-space modelling methodology, which allows modelling, estimation and forecasting of mortality under a unified framework. We propose alternative model identification constraints which are more suited to statistical inference in filtering and parameter estimation. We then develop a class of Bayesian state-space models which incorporate a priori beliefs about the mortality model characteristics as well as for more flexible and appropriate assumptions relating to heteroscedasticity that present in observed mortality data. To study long-term mortality dynamics, we introduce stochastic volatility to the period effect. The estimation of the resulting stochastic volatility model of mortality is performed using a recent class of Monte Carlo procedure known as the class of particle Markov chain Monte Carlo methods. We illustrate the framework using Danish male mortality data, and show that incorporating heteroscedasticity and stochastic volatility markedly improves model fit despite an increase of model complexity. Forecasting properties of the enhanced models are examined with long-term and short-term calibration periods on the reconstruction of life tables.

We compare quantitatively six simulation strategies for mortality projection with the Poisson Lee–Carter model. We test these strategies on New Zealand mortality data and discuss the simulated results of the mortality index, death rates, and life expectancy.

Rapid advances in artificial intelligence (AI) and machine learning are creating products and services with the potential not only to change the environment in which actuaries operate, but also to provide new opportunities within actuarial science. These advances are based on a modern approach to designing, fitting and applying neural networks, generally referred to as “Deep Learning”. This paper investigates how actuarial science may adapt and evolve in the coming years to incorporate these new techniques and methodologies. Part 1 of this paper provides background on machine learning and deep learning, as well as an heuristic for where actuaries might benefit from applying these techniques. Part 2 of the paper then surveys emerging applications of AI in actuarial science, with examples from mortality modelling, claims reserving, non-life pricing and telematics. For some of the examples, code has been provided on GitHub so that the interested reader can experiment with these techniques for themselves. Part 2 concludes with an outlook on the potential for actuaries to integrate deep learning into their activities. Finally, a supplementary appendix discusses further resources providing more in-depth background on machine learning and deep learning.

The analysis of causal mortality provides rich insight into changes in mortality trends that are hidden in population-level data. Therefore, we develop and apply a multinomial logistic framework to model causal mortality. We use internationally classified cause-of-death categories and data obtained from the World Health Organization. Inherent dependence amongst the competing causes is accounted for in the framework, which also allows us to investigate the effects of improvements in, or the elimination of, cause-specific mortality. This has applications to scenario-based forecasting often used to assess the impact of changes in mortality. The multinomial model is shown to be more conservative than commonly used approaches based on the force of mortality. We use the model to demonstrate the impact of cause-elimination on aggregate mortality using residual life expectancy and apply the model to a French case study.

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.

We consider an insurer who has a fixed amount of funds allocated as the initial surplus for a risk portfolio, so that the probability of ultimate ruin for this portfolio is at a known level. We consider the question of whether the insurer can reduce this ultimate ruin probability by allocating part of the initial funds to the purchase of a reinsurance contract. This reinsurance contract would restore the insurer's surplus to a positive level k every time the surplus fell between 0 and k. The insurer's objective is to choose the level k that minimizes the ultimate ruin probability. Using different examples of reinsurance premium calculation and claim size distribution we show that this objective can be achieved, often with a substantial reduction in the ultimate ruin probability from the situation when there is no reinsurance. We also show that by purchasing reinsurance the insurer can release funds for other purposes without altering its ultimate ruin probability.