Generalized Additive Models (GAMs) with age, period and cohort as possible covariates are used to predict future mortality improvements for the Irish population. The GAMs considered are the 1-dimensional age + period and age + cohort models and the 2-dimensional age-period and age-cohort models. In each case thin plate regression splines are used as the smoothing functions. The generalized additive models are compared with the P-Spline (Currie et al., 2004) and Lee-Carter (Lee & Carter, 1992) models included in version 1.0 of the Continuous Mortality Investigation (CMI) library of mortality projections. Using the Root Mean Square Error to assess the accuracy of future predictions, the GAMs outperform the P-Spline and Lee-Carter models over intervals of 25 and 35 years in the age range 60 to 90. The GAMs allow intuitively simple models of mortality to be specified whilst also providing the flexibility to model complex relationships between the covariates. The majority of morality improvements derived from the projections of future Irish mortality yield annuity values at ages 60, 65, 70 and 80 in 2007 in the range of annuity values calculated, assuming a 2 to 4 percent annual compound improvement in mortality rates for both males and females.

In most industrialised countries, one of the major societal challenges is the demographic change coming along with the ageing of the population. The increasing life expectancy observed over the last decades underlines the importance to find ways to appropriately cover the financial needs of the elderly. A particular issue arises in the area of health, where sufficient care must be provided to a growing number of dependent elderly in need of long-term care (LTC) services. In many markets, the offering of life insurance products incorporating care options and LTC insurance products is generally scarce. In our research, we therefore examine a life annuity product with an embedded care option potentially providing additional financial support to dependent persons. To evaluate the care option, we determine the minimum price that the annuity provider requires and the policyholder’s willingness to pay for the care option. For the latter, we employ individual utility functions taking account of the policyholder’s condition. We base our numerical study on recently developed transition probability data from Switzerland. Our findings give new and realistic insights into the nature and the utility of life annuity products proposing an embedded care option for tackling the financing of LTC needs.

This is the second in a series of three papers. In the first paper we describe a comprehensive stochastic model of an individual's lifetime that includes diagnosis with ischaemic heart disease and stroke and also the development of the major risk factors for these conditions. The third paper is devoted to some applications of this model.

In this paper we discuss the effect of BMI on diabetes, heart disease and mortality and we use data from the Framingham Heart Study and the Health Survey for England to develop stochastic models for changes in an individual's BMI. Some of these models incorporate time trends leading to increasing prevalence of obesity. We then use these models to investigate how future expected lifetime and future expected healthy lifetime depend on BMI.

Our conclusions are that if the prevalence of obesity increases, even to an extreme degree, then the prevalence of diabetes and hypertension in the population will increase, possibly to a significant extent, but the prevalence of heart disease and stroke will increase by a much smaller amount and the effect on expected future lifetime will be small.

This paper argues that most of the problems that actuaries have to deal with in the context of non-life insurance can be usefully cast in the framework of computational intelligence (a.k.a. artificial intelligence), the discipline that studies the design of agents which exhibit intelligent behaviour. Finding an adequate framework for actuarial problems has more than a simply theoretical interest: it also allows a knowledge transfer from the computational intelligence discipline to general insurance, wherever techniques have been developed for problems which are common to both contexts. This has already happened in the past (neural networks, clustering, data mining have all found applications to general insurance) but not systematically, with the result that many useful computational intelligence techniques such as sparsity-based regularisation schemes (a technique for feature selection) are virtually unknown to actuaries.

In this first of two papers, we will explore the role of statistical learning in actuarial modelling. We will show that risk costing, which is at the core of pricing, reserving and capital modelling, can be described as a supervised learning problem. Many activities involved in exploratory analysis, such as data mining or feature construction, can be described as unsupervised learning. A comparison of different computational intelligence methods will be carried out, and practical insurance applications (rating factor selection, IBNER analysis) will also be presented.

A method is presented for identifying an expected profitable set of customers, to offer them an additional insurance product, by estimating a customer specific latent risk profile, for the additional product, by using the customer specific available data for an existing insurance product of the specific customer. For the purpose, a multivariate credibility estimator is considered and we investigate the effect of assuming that one (of two) insurance products is inactive (without available claims information) when estimating the latent risk profile. Instead, available customer specific claims information from the active existing insurance product is used to estimate the risk profile and thereafter assess whether or not to include a specific customer in an expected profitable set of customers. The method is tested using a large real data set from a Danish insurance company and it is shown that sets of customers, with up to 36% less claims than a priori expected, are produced as a result of the method. It is therefore argued that the proposed method could be considered, by an insurance company, when cross-selling insurance products to existing customers.

This paper is concerned with presenting the Exponential-Lognormal (ELN) regression model as a competitive alternative to the Pareto, or Exponential-Inverse Gamma, regression model that has been used in a wide range of areas, including insurance ratemaking. This is the first time that the ELN regression model is used in a statistical or actuarial context. The main contribution of the study is that we illustrate how maximum likelihood estimation of the ELN regression model, which does not have a density in closed form, can be accomplished relatively easily via an Expectation-Maximisation type algorithm. A real data application based on motor insurance data is examined in order to emphasise the versatility of the proposed algorithm. Finally, assuming that the number of claims is distributed according to the classic Negative Binomial and Poisson-Inverse Gaussian regression models, both the a priori and a posteriori, or Bonus–Malus, premium rates resulting from the ELN regression model are calculated via the net premium principle and compared to those determined by the Pareto regression model that has been traditionally used for modelling claim sizes.

We construct a stochastic model of an individual's lifetime that includes diagnosis with ischaemic heart disease and stroke and also the development of the major risk factors for these conditions: hypercholesterolaemia, hypertension, diabetes and obesity. Smoking, another major risk factor, is treated deterministically. Mathematically, the model is a continuous time, finite state space Markov process, with the individual's age playing the rôle of time. The model is parameterised using data from the Framingham Heart Study, with parameter values adjusted so that the model is appropriate for UK conditions in the early 21st century. The model has been designed so that it can be used to quantify the effects of:

(i) trends, in particular increasing prevalence of obesity.

(ii) changes in behaviour, in particular smoking patterns, and

(iii) treatments, in particular statins for hypercholesterolaemia.

These applications are covered in two accompanying papers.

This is the third in a series of three papers. In the first paper we describe a comprehensive stochastic model of an individual's lifetime that includes diagnosis with ischaemic heart disease and stroke and also the development of the major risk factors for these conditions. The second paper discusses in some detail models for changes in body mass index (BMI) and also the effects of these changes, in particular the current trend towards increasing prevalence of obesity, on diabetes, cardiovascular diseases and expected future lifetime. This paper is devoted to the following applications of the model described in the first paper:

(a) quantifying the effects of smoking and of changes in smoking habits, and,

(b) quantifying the effects of treatment with statins (drugs designed to lower cholesterol).

The COVID-19 pandemic interrupts the relatively steady trend of improving longevity observed in many countries over the last decades. We claim that this needs to be addressed explicitly in many mortality modelling applications, for example, in the life insurance industry. To support this position, we provide a descriptive analysis of the mortality development of several countries up to and including the year 2020. Furthermore, we perform an empirical and theoretical investigation of the impact a mortality jump has on the parameters, forecasts and implied present values of the popular Lee–Carter mortality model. We find that COVID-19 has resulted in substantial mortality shocks in many countries. We show that such shocks have a large impact on point and interval forecasts of death rates and, consequently, on the valuation of mortality-related insurance products. We obtain similar findings under the Cairns–Blake–Dowd mortality model, which demonstrates that the effects caused by COVID-19 show up in a variety of models. Finally, we provide an overview of approaches to handle extreme mortality events such as the COVID-19 pandemic in mortality modelling.

The Poisson process is an essential building block to move up to complicated counting processes, such as the Cox (“doubly stochastic Poisson”) process, the Hawkes (“self-exciting”) process, exponentially decaying shot-noise Poisson (simply “shot-noise Poisson”) process and the dynamic contagion process. The Cox process provides flexibility by letting the intensity not only depending on time but also allowing it to be a stochastic process. The Hawkes process has self-exciting property and clustering effects. Shot-noise Poisson process is an extension of the Poisson process, where it is capable of displaying the frequency, magnitude and time period needed to determine the effect of points. The dynamic contagion process is a point process, where its intensity generalises the Hawkes process and Cox process with exponentially decaying shot-noise intensity. To facilitate the usage of these processes in practice, we revisit the distributional properties of the Poisson, Cox, Hawkes, shot-noise Poisson and dynamic contagion process and their compound processes. We provide simulation algorithms for these processes, which would be useful to statistical analysis, further business applications and research. As an application of the compound processes, numerical comparisons of value-at-risk and tail conditional expectation are made.

The uncertainty regarding financial returns and the life expectancy, joint to the reduced social security benefits, increasingly expose individuals to the risk of outliving their post-retirement assets. However, the demand for longevity guarantees remains low, due to high costs. The providers, on their side, may be reluctant to offer non-adjustable longevity guarantees, as the risk is long term and difficult to predict. It is therefore convenient to reconsider the design of longevity guarantees. In particular, a participating structure, providing a link to some longevity experience, could allow a sharing of losses, and possibly profits, resulting in a reduction of the cost of the retained guarantee. The literature review suggests a number of alternatives to define a longevity-linking arrangement, but the topic is not yet completely explored. It is useful, in particular, to have a common framework, under which the various solutions can be interpreted and compared, also with a view to the trade-off between the retained risk and the cost of the guarantee. Developing a general structure describing longevity-linked post-retirement benefits is the main purpose of this paper. Allowing for aggregate longevity risk, we then examine suitable solutions for insurance products.

This paper provides a stochastic model, consistent with Solvency II and the Delegated Regulation, to quantify the capital requirement for demographic risk. In particular, we present a framework that models idiosyncratic and trend risks exploiting a risk theory approach in which results are obtained analytically. We apply the model to non-participating policies and quantify the Solvency Capital Requirement for the aforementioned risks in different time horizons.

A non-parametric test based on nested L-statistics and designed to compare the riskiness of portfolios was introduced by Brazauskas et al. (2007). Its asymptotic and small-sample properties were primarily explored for independent portfolios, though independence is not a required condition for the test to work. In this paper, we investigate how performance of the test changes when insurance portfolios are dependent. To achieve that goal, we perform a simulation study where we consider three different risk measures: conditional tail expectation, proportional hazards transform, and mean. Further, three portfolios are generated from exponential, Pareto, and lognormal distributions, and their interdependence is modelled with the three-dimensional t and Gaussian copulas. It is found that the presence of strong positive dependence (comonotonicity) makes the test very liberal for all the risk measures under consideration. For types of dependence that are more common in an insurance environment, the effect of dependence is less dramatic but the results are mixed, i.e., they depend on the chosen risk measure, sample size, and even on the test’s significance level. Finally, we illustrate how to incorporate such findings into sensitivity analysis of the decisions. The risks we analyse represent tornado damages in different regions of the United States from 1890 to 1999.

This paper is intended to illustrate the adaptive control approach in insurance. A zone-adaptive control strategy harmonising the requirements of principles of solvency and equity is considered in the simplistic framework of a diffusion multiperiodic risk model. Other works by the author set similar adaptive control strategies in a more realistic Poisson-exponential multiperiodic risk model. There is much scope for further generalisations.

We consider various aspects of longevity trend risk viewed through the prism of a finite time window. We show the broad equivalence of value-at-risk (VaR) capital requirements at a p-value of 99.5% to conditional tail expectations (CTEs) at 99%. We also show how deferred annuities have higher risk, which can require double the solvency capital of equivalently aged immediate anuities. However, results vary considerably with the choice of model and so longevity trend-risk capital can only be determined through consideration of multiple models to inform actuarial judgement. This model risk is even starker when trying to value longevity derivatives. We briefly discuss the importance of using smoothed models and describe two methods to considerably shorten VaR and CTE run times.

Already home to 23% of the global elderly population, China will experience further demographic change in the coming decades. To address the consequences of population ageing, the Chinese government is implementing major social insurance reforms and promotes the development of private insurance markets. We aim to inform these initiatives by developing a new method to project healthy life expectancy (HLE) in different regions. HLE is an important population health measure which is increasingly used in the actuarial literature. Our new approach relies on publicly available data from the Global Burden of Disease Study for life expectancy and HLE for 139 countries. We use the model to estimate HLE at birth in 2015 for 31 province-level regions in China for both males and females. We discuss the implications of our results for planned increases in the retirement age in China and for long-term care insurance pricing.

This paper explores the characteristics of 2,141 operational risk events amongst European (EU) and US banks over the period 2008–2014. We have analysed the operational risk events using a method originating in biology for the study of interrelatedness of characteristics in a complex adaptive system. The methodology, called cladistics, provides insights into the relationships between characteristics of operational risk events in banks that is not available from the traditional statistical analysis. We have used cladistics to explore if there are consistent patterns of operational risk characteristics across banks in single and different geographic zones. One significant pattern emerged which indicates there are key, stable characteristics across both geographic zones and across banks in each zone. The results identify the characteristics that could then be managed by the banks to reduce operational risk losses. We also have analysed separately the characteristics of operational risk events for “big” banks and extreme events and these results indicate that big banks and small banks have similar key operational risk characteristics, but the characteristics of extreme operational risk events are different to those for the non-extreme events.

Human longevity is changing, but at what rate? Insurance claims are increasing, but at what rate? Are the trends that we glean from data true or illusionary? The shocking fact is that true trends might be quite different from those that we actually see from visualised data. Indeed, in some situations the upward trends (e.g. inflation) may even look decreasing (e.g. deflation). In this paper, we discuss this “trends in disguise” phenomenon in detail and offer a way for estimating true trends.

This paper proposes a discrete-time risk model that has a certain type of correlation between premiums and claim amounts. It is motivated by the well-known bonus-malus system (also known as the no claims discount) in the car insurance industry. Such a system penalises policyholders at fault in accidents by surcharges, and rewards claim-free years by discounts. For simplicity, only up to three levels of premium are considered in this paper and recursive formulae are derived to calculate the ultimate ruin probabilities. Explicit expressions of ruin probabilities are obtained in a simplified case. The impact of the proposed correlation between premiums and claims on ruin probabilities is examined through numerical examples. In the end, the joint probability of ruin and deficit at ruin is also considered.

If the general level of house prices falls a long way, policymakers may introduce new policies which seek to support prices. This paper considers the effect of such interventions on the valuation of no-negative-equity guarantees (NNEG) in equity release mortgages. I model interventions by a reflecting barrier expressed as a fraction of the current level of house prices. Reflection at the barrier is instantaneous, so the no-arbitrage property is preserved, and hence risk-neutral valuation of NNEG is possible. The reflecting barrier can alternatively be justified as a representation of the different economic nature of the underlying housing (and particularly freehold land) assets in NNEG valuations, compared with the underlying equity assets in many other option valuations.