A deep-deferred annuity is a deferred annuity where payments start very late in life, i.e. well after the normal retirement age. This annuity has received much attention lately as it was made accessible to 401(k) plans in the United States in 2014. By transferring the risk of outliving retirement savings at high ages to annuity providers, deep-deferred annuities provide annuitants with enhanced later-life financial security. However, the valuation of this annuity suffers from high uncertainty because the mortality data at high ages are sparse and possibly unreliable. In this paper, we use risk ratio to measure demographic risk in the valuation. Demographic risk is decomposed into the following four components: (1) mortality tail curve risk, (2) mortality improvement model risk, (3) parameter risk in mortality tail curves, and (4) parameter risk in mortality improvement rate models. Our quantitative analysis aims to provide insights into the development and risk management of deep-deferred annuities.

Variable annuities have become popular retirement and investment vehicles due to their attractive guarantee features. Nonetheless, managing the financial risks associated with the guarantees poses great challenges for insurers. One challenge is risk quantification, which involves frequent valuation of the guarantees. Insurers rely on the use of Monte Carlo simulation for valuation as the guarantees are too complicated to be valued by closed-form formulas. However, Monte Carlo simulation is computationally intensive. In this paper, we empirically explore the use of tree-based models for constructing metamodels for the valuation of the guarantees. In particular, we consider traditional regression trees, tree ensembles, and trees based on unbiased recursive partitioning. We compare the performance of tree-based models to that of existing models such as ordinary kriging and generalised beta of the second kind (GB2) regression. Our results show that tree-based models are efficient in producing accurate predictions and the gradient boosting method is considered the most superior in terms of prediction accuracy.

The financial system can be shown to be a complex adaptive system consisting primarily of a federation of systems and systems of systems. There are significant similarities between the characteristics of natural systems and financial systems suggesting that the type of analysis employed in understanding natural systems could have application in financial system analysis. Cladistics analysis has been used extensively for analysis of biological systems and has accordingly been used in the social sciences for some years but a rigorous justification for adopting the analysis has not been undertaken. This paper discusses the appropriateness of applying cladistics analysis to financial systems, and then considers the appropriate methodology to be adopted for analysis of different financial events.

Non-homogeneous renewal processes are not yet well established. One of the tools necessary for studying these processes is the non-homogeneous time convolution. Renewal theory has great relevance in general in economics and in particular in actuarial science, however, most actuarial problems are connected with the age of the insured person. The introduction of non-homogeneity in the renewal processes brings actuarial applications closer to the real world. This paper will define the non-homogeneous time convolutions and try to give order to the non-homogeneous renewal processes. The numerical aspects of these processes are then dealt with and a real data application to an aspect of motorcar insurance is proposed.

In this paper we analyse insurance claim frequency data using the bivariate negative binomial regression (BNBR) model. We use general insurance data on claims from simple third-party liability insurance and comprehensive insurance. We find that bivariate regression, with its capacity for modelling correlation between the two observed claim counts, provides both a superior fit and out-of-sample prediction compared with the more common practice of fitting univariate negative binomial regression models separately to each claim type. Noting the complexity of BNBR models and their potential for a large number of parameters, we explore the use of model shrinkage methodology, namely the least absolute shrinkage and selection operator (Lasso) and ridge regression. We find that models estimated using shrinkage methods outperform the ordinary likelihood-based models when being used to make predictions out-of-sample. We find that the Lasso performs better than ridge regression as a method of shrinkage.

In insurance risk theory, dividend and aggregate claim amount are of great research interest as they represent the insurance company’s payments to its shareholders and policyholders, respectively. Since the analyses of these two quantities are performed separately in the literature, the companion paper by Cheung et al. generalised the Gerber–Shiu expected discounted penalty function by further incorporating the moments of the aggregate discounted claims until ruin and the discounted dividends until ruin. While Cheung et al. considered the compound Poisson model with a dividend barrier in which ruin occurs almost surely, the present paper looks at this generalised Gerber–Shiu function under a threshold dividend strategy where the insurer has a positive survival probability. Because the Gerber–Shiu function is only defined for sample paths leading to ruin, we will additionally study the joint moments of the aggregate discounted claims and the discounted dividends without ruin occurring. Some explicit formulas are derived when the individual claim distribution follows a combination of exponentials. Numerical illustrations involving the correlation between aggregate discounted claims and discounted dividends are given. For the case where ruin occurs, we additionally compute the correlations between the time of ruin and the above two quantities.

The retirement systems in many developed countries have been increasingly moving from defined benefit towards defined contribution system. In defined contribution systems, financial and longevity risks are shifted from pension providers to retirees. In this paper, we use a probabilistic approach to analyse the uncertainty associated with superannuation accumulation and decumulation. We apply an economic scenario generator called the Simulation of Uncertainty for Pension Analysis (SUPA) model to project uncertain future financial and economic variables. This multi-factor stochastic investment model, based on the Monte Carlo method, allows us to obtain the probability distribution of possible outcomes regarding the superannuation accumulation and decumulation phases, such as relevant percentiles. We present two examples to demonstrate the implementation of the SUPA model for the uncertainties during both phases under the current superannuation and Age Pension policy, and test two superannuation policy reforms suggested by the Grattan Institute.

Simulations of a model pension scheme are run with stochastic economic and demographic factors, with an aim to investigate the impact of these factors on movements in funding ratio and average contribution rates. These impacts are analysed by running regressions of movements in funding ratio and average contribution rates against the economic and demographic factors. It is found that, for a typical scheme closed to new entrants and a balanced asset allocation including equity investment, the mismatch between discount rate movements and investment returns is by far the biggest predictor of funding ratio movements, with average contribution rates affected more by events in a few individual years rather than averaged over an entire simulation. Where the scheme invests to cash-flow match liabilities, mortality improvement becomes the most significant predictor of funding ratio movements, although mortality improvement still has little impact on average contribution rates.

Mutations in the BRCA1 and BRCA2 genes confer very high risk of breast cancer (BC), but only account for about 25% of the observed familial clustering of BC. Antoniou et al. (2002) proposed a model which included the BRCA1 and BRCA2 genes, and a polygenic component which acted multiplicatively on the rate of onset of BC. We use this model to find premium rates for critical illness insurance: (a) given knowledge of an applicant's polygenotype; and (b) given knowledge of a family history of BC or ovarian cancer. We find that the polygenic component causes large variation in premium rates even among non-mutation carriers, therefore affecting the whole population. In some cases the polygenic contribution is protective enough to reduce or remove the additional risk of a BRCA1/2 mutation, leading to cases where it will be advantageous to disclose genetic test results which are adverse in absolute terms. Premiums based on family history are lower than those found in an earlier study which attributed all genetic BC risk to the BRCA1/2 genes.

This is the third of three papers in which we present methods and results for the estimation and modelling of claim termination rates for Income Protection (IP) insurance, allowing for different causes of claim. In the first paper we discussed recoveries. We model the mortality of IP claimants as the sum of two components: a base mortality, which is a function of age and calendar year, but not of the specific cause of sickness or its current duration, and a cause-specific component which does depend on the current duration of the sickness. The modelling of the base mortality, using data for UK assured lives, was discussed in the second paper. In this paper we discuss the modelling of the cause-specific component of the mortality of IP claimants. We use data supplied by the Continuous Mortality Investigation relating to IP claims paid in the years 1975 to 2002.

In the final section of this paper we present some numerical results for cause-specific claim annuity rates for current claims and aggregate claim termination rates based on the models developed in all three papers.

Different mortality rates for different socio-economic groups within a population have been consistently reported throughout the years. In this study, we aim to exploit data from multiple public sources, including highly detailed cause-of-death data from the United States Centers for Disease Control and Prevention, to explore the mortality gap between the better and worse off in the US during the period 1989–2015, using education as a proxy.

Among the systems in place in different countries for the protection of the population against the long-term contingencies of old-age (or retirement), disability and death (or survivorship), defined-benefit social security pension schemes, i.e. social insurance pension schemes, by far predominate, despite the recent trend towards defined-contribution arrangements in social security reforms. Actuarial valuations of these schemes, unlike other branches of insurance, continue to be carried out almost exclusively on traditional, deterministic lines. Stochastic applications in this area, which have been restricted mainly to occasional special studies, have relied on the simulation technique. This paper develops an analytical model for the stochastic actuarial valuation of a social insurance pension scheme. Formulae are developed for the expected values, variances and covariances of and among the benefit expenditure and salary bill projections and their discounted values, allowing for stochastic variation in three key input factors, i.e., mortality, new entrant intake, and interest (net of salary escalation). Each deterministic output of the valuation is thus supplemented with a confidence interval, that is, a range with an attached probability. The treatment covers the premiums under the different possible financial systems for these schemes, which differ from the funding methods of private pensions, as well as the testing of the level of the Fund ratio when the future contributions schedule is pre-determined. Although it is based on a relatively simplified approach and refers only to retirement pensions, with full adjustment in line with salary escalation, the paper brings out the stochastic features of pension scheme projections and illustrates a comprehensive stochastic valuation. It is hoped that the paper will stimulate interest in further research, both of a theoretical and a practical nature, and lead to progressively increasing recourse to stochastic methods in social insurance pension scheme valuations.

In this paper, we present a Markov chain Monte Carlo (MCMC) simulation algorithm for estimating parameters in the kernel density estimation of bivariate insurance claim data via transformations. Our data set consists of two types of auto insurance claim costs and exhibits a high-level of skewness in the marginal empirical distributions. Therefore, the kernel density estimator based on original data does not perform well. However, the density of the original data can be estimated through estimating the density of the transformed data using kernels. It is well known that the performance of a kernel density estimator is mainly determined by the bandwidth, and only in a minor way by the kernel. In the current literature, there have been some developments in the area of estimating densities based on transformed data, where bandwidth selection usually depends on pre-determined transformation parameters. Moreover, in the bivariate situation, the transformation parameters were estimated for each dimension individually. We use a Bayesian sampling algorithm and present a Metropolis-Hastings sampling procedure to sample the bandwidth and transformation parameters from their posterior density. Our contribution is to estimate the bandwidths and transformation parameters simultaneously within a Metropolis-Hastings sampling procedure. Moreover, we demonstrate that the correlation between the two dimensions is better captured through the bivariate density estimator based on transformed data.

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.

Mortality volatility is crucially important to many aspects of index-based longevity hedging, including instrument pricing, hedge calibration and hedge performance evaluation. This paper sets out to develop a deeper understanding of mortality volatility and its implications on index-based longevity hedging. First, we study the potential asymmetry in mortality volatility by considering a wide range of generalised autoregressive conditional heteroskedasticity (GARCH)-type models that permit the volatility of mortality improvement to respond differently to positive and negative mortality shocks. We then investigate how the asymmetry of mortality volatility may impact index-based longevity hedging solutions by developing an extended longevity Greeks framework, which encompasses longevity Greeks for a wider range of GARCH-type models, an improved version of longevity vega, and a new longevity Greek known as “dynamic Delta”. Our theoretical work is complemented by two real-data illustrations, the results of which suggest that the effectiveness of an index-based longevity hedge could be significantly impaired if the asymmetry in mortality volatility is not taken into account when the hedge is calibrated.

We consider an insurance entity endowed with an initial capital and a surplus process modelled as a Brownian motion with drift. It is assumed that the company seeks to maximise the cumulated value of expected discounted dividends, which are declared or paid in a foreign currency. The currency fluctuation is modelled as a Lévy process. We consider both cases: restricted and unrestricted dividend payments. It turns out that the value function and the optimal strategy can be calculated explicitly.

AI has had many summers and winters. Proponents have overpromised, and there has been hype and disappointment. In recent years, however, we have watched with awe, surprise, and hope at the successes: Better than human capabilities of image-recognition; winning at Go; useful chatbots that seem to understand your needs; recommendation algorithms harvesting the wisdom of crowds. And with this success comes the spectre of danger. Machine behaviours that embed the worst of human prejudice and biases; techniques trying to exploit human weaknesses to skew elections or prompt self-harming behaviours. Are we seeing a perfect storm of social media, sensor technologies, new algorithms and edge computing? With this backdrop: is AI coming of age?

We analyse ruin probabilities for an insurance risk process with a more generalised dependence structure compared to the one introduced in Constantinescu et al. (2016). In this paper, we assume that a random threshold window is generated every time after a claim occurs. By comparing the previous inter-claim time with the threshold window, the distributions of the current threshold window and the inter-arrival time are determined. Furthermore, the statuses for the previous and current inter-arrival times give rise to the current claim size distribution as well. Like Constantinescu et al. (2016), we first identify the embedded Markov additive process where all the randomness takes a general form. Inspired by the Erlangisation technique, the key message of this paper is to analyse such risk process using a Markov fluid flow model where the underlying random variables follow phase-type distributions. This would further allow us to approximate the fixed observation windows by Erlang random variables. Then ruin probabilities under the process with Erlang(n) observation windows are proved to be Erlangian approximations for those related to the process with fixed threshold windows at the limit. An exact form of the limit can be obtained whose application will be illustrated further by a numerical example.

The claims development result (CDR) is the difference between the best estimate predictions of the ultimate claim in 2 successive years. With best estimate reserves it is often argued that CDR’s in consecutive years should fluctuate randomly around zero. However, in practice one frequently observes that CDR’s in a given line of business have the same sign over several consecutive years. We show that this is a phenomenon which is not unusual and to be expected in situations of change. Moreover, we show how situations of change can adequately be described by a model, taking into account the evolving external information.