This paper shows how the well-known Bornhuetter–Ferguson claims-reserving method can be extended by applying fuzzy methods. The a priori information for the ultimate claims derives from market statistics, organisational data, etc. and might contain vagueness. Likewise, the parameters of the claims development pattern can be vague or are adapted, retrospectively, due to subjective judgement. With the help of fuzzy numbers we develop new predictors for the ultimate claims. Furthermore, we quantify the uncertainty of the ultimate claims for single and aggregated accident years.

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.

During the past decade, genetics research has allowed scientists and clinicians to explore the human genome in detail and reveal many thousands of common genetic variants associated with disease. Genetic risk scores, known as polygenic risk scores (PRSs), aggregate risk information from the most important genetic variants into a single score that describes an individual’s genetic predisposition to a given disease. This article reviews recent developments in the predictive utility of PRSs in relation to a person’s susceptibility to breast cancer and coronary artery disease. Prognostic models for these disorders are built using data from the UK Biobank, controlling for typical clinical and underwriting risk factors. Furthermore, we explore the possibility of adverse selection where genetic information about multifactorial disorders is available for insurance purchasers but not for underwriters. We demonstrate that prediction of multifactorial diseases, using PRSs, provides population risk information additional to that captured by normal underwriting risk factors. This research using the UK Biobank is in the public interest as it contributes to our understanding of predicting risk of disease in the population. Further research is imperative to understand how PRSs could cause adverse selection if consumers use this information to alter their insurance purchasing behaviour.

From 1870, in the United Kingdom, a generally liberal legal framework for life insurance existed which commanded wide support, particularly from the actuarial profession. Despite its apparent liberalism, however, it is likely that the regulatory framework impeded market entry, particularly of mutual companies. The liberal framework broke down in a number of incremental steps from 1946. This paper traces the development of U.K. life assurance regulation for 101 years from the mid-Victorian period, analyses contemporary reflections on the legal framework within the actuarial profession, and examines the appropriateness of the legal framework for achieving specific economic objectives for long-term insurance regulation.

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.

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.

Insurance regulation often dictates that insurers monitor their solvency risk in real time and take appropriate actions whenever the risk exceeds their tolerance level. Bayesian methods are appealing for prediction problems thanks to their ability to naturally incorporate both sample variability and parameter uncertainty into a predictive distribution. However, handling data arriving in real time requires a flexible non-parametric model, and the Monte Carlo methods necessary to evaluate the predictive distribution in such cases are not recursive and can be too expensive to rerun each time new data arrives. In this paper, we apply a recently developed alternative perspective on Bayesian prediction based on copulas. This approach facilitates recursive Bayesian prediction without computing a posterior, allowing insurers to perform real-time updating of risk measures to assess solvency risk, and providing them with a tool for carrying out dynamic risk management strategies in today’s “big data” era.

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.

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 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).

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 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.

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.

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.

There has been a growing interest among pension plan sponsors in envisioning how the mortality experience of their active and deferred members may turn out to be if a pandemic similar to the COVID-19 occurs in the future. To address their needs, we propose in this paper a stochastic model for simulating future mortality scenarios with COVID-alike effects. The proposed model encompasses three parameter levels. The first level includes parameters that capture the long-term pattern of mortality, whereas the second level contains parameters that gauge the excess age-specific mortality due to COVID-19. Parameters in the first and second levels are estimated using penalised quasi-likelihood maximisation method which was proposed for generalised linear mixed models. Finally, the third level includes parameters that draw on expert opinions concerning, for example, how likely a COVID-alike pandemic will occur in the future. We illustrate our proposed model with data from the United States and a range of expert opinions.

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.

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.

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.

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.

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.