Continuous Mortality Investigation Research Articles

Seasonal mortality amongst UK occupational pension scheme members 2000–2016

Abstract Mortality at older ages varies by season, increasing the uncertainty associated with modelling and projecting mortality at older ages and ultimately contributing to pension providers’ overall risk. As the population ages, it becomes more important to understand variations in seasonal mortality between pensioners and to identify those most vulnerable to seasonal mortality differences. Using data from the Self-Administered Pension Schemes mortality investigation of the Continuous Mortality Investigation of the Institute and Faculty of Actuaries, UK, this paper investigates variations in seasonal mortality amongst members of UK occupational pension schemes over the period 2000–2016. Results are also compared with the corresponding population of England and Wales. For the oldest age groups (80+), which are most affected by seasonality, females are more vulnerable to seasonal differences in mortality for each pensioner group relative to males. Following a long-term decline in the winter-summer mortality gap the gap increased over the period, particularly for female pensioners and dependants. Seasonality remains a feature of UK mortality at older ages and risk management for pension schemes should consider seasonality when analysing overall mortality experience.

Constraints, the identifiability problem and the forecasting of mortality

AbstractModels of mortality often require constraints in order that parameters may be estimated uniquely. It is not difficult to find references in the literature to the “identifiability problem”, and papers often give arguments to justify the choice of particular constraint systems designed to deal with this problem. Many of these models are generalised linear models, and it is known that the fitted values (of mortality) in such models are identifiable, i.e., invariant with respect to the choice of constraint systems. We show that for a wide class of forecasting models, namely ARIMA $(p,\delta, q)$ models with a fitted mean and $\delta = 1$ or 2, identifiability extends to the forecast values of mortality; this extended identifiability continues to hold when some model terms are smoothed. The results are illustrated with data on UK males from the Office for National Statistics for the age-period model, the age-period-cohort model, the age-period-cohort-improvements model of the Continuous Mortality Investigation and the Lee–Carter model.

E-cigarettes: no smoke without fire?

Abstract Smoking was one of the biggest preventable killers of the 20th century, and it continues to cause the death of millions across the globe. The rapid growth of the e-cigarette market in the last 10 years and the claims that it is a safer form of smoking, and can help with smoking cessation, have led to questions being raised on their possible impact to society, the health of the population and the insurance industry. Recent media attention around the possible health implications of e-cigarette use has also ensured that this topic remains in the public eye. The e-cigarette working party was initiated by the Institute and Faculty of Actuaries’ Health and Care Research Sub-Committee in July 2016, with the primary objective of understanding the impact of e-cigarettes on life and health insurance. In this paper, we have looked at all areas of e-cigarette usage and how it relates to insurance in the UK market. In particular, we have covered the potential risks and benefits of switching to e-cigarettes, the results of studies that have been published, the potential impact on underwriting and claims processes and the potential impact on pricing (based on what modelling is possible with the data available). Research in this area is still in its infancy and data are not yet mature, which makes predicting the long-term impact of e-cigarette smoking extremely challenging, for example, there are no studies that directly measure the mortality or morbidity impact of long-term e-cigarette use and so we have had to consider studies that consider more immediate health impacts or look more simply at the constituents of the output of an e-cigarette and compare them to that of a cigarette. The data issue is further compounded by the findings of studies and the advice of national health authorities often being conflicting. For example, while National Health Service England has publicly stated that it supports the growth of e-cigarette usage as an aid to reduce traditional smoking behaviour, the US Food and Drug Administration has been much more vocal in highlighting the perceived dangers of this new form of smoking. Users’ behaviour also adds complexity, as dual use (using both e-cigarettes and cigarettes) is seen in a high percentage of users and relapse rates back to cigarette smoking are currently unknown. Having talked to a number of experts in the field, we have discovered that there is certainly not a common view on risk. We have heard from experts who have significant concerns but also to experts who do believe that e-cigarettes are far safer than tobacco. We have purposefully considered conflicting evidence and have consulted with various parties so we can present differing points of view, thereby ensuring a balanced, unbiased and fair picture of our findings is presented. The evidence we have reviewed does suggest that e-cigarettes are a safer alternative to traditional smoking, but not as safe as non-smoking. There are no large, peer-reviewed, long-term studies yet available to understand the true impact of a switch to e-cigarette use, so currently we are unable to say where on the risk spectrum between cigarette smoking and life-time non-smoking it lies. We do not yet understand if the benefits seen in the studies completed so far will reduce the risk in the long term or whether other health risks will come to light following more prolonged use and study. This, coupled with concerns with the high proportion of dual use of cigarettes and e-cigarettes, relapse rates and the recent growth in medical problems linked with e-cigarette use, means that we need to wait for experience to emerge fully before firm conclusions can be drawn. Although we have presented a view, it is vitally important that our industry continues to monitor developments in this area and fully considers what next steps and future actions may be required to ensure our position reflects the potential benefits and risks that e-cigarette use may bring. We feel that the time is right for a body such as the IFoA to analyse the feasibility of collecting the necessary data through the Continuous Mortality Investigation that would allow us to better analyse the experience that is emerging.

A stochastic implementation of the APCI model for mortality projections

Abstract The Age-Period-Cohort-Improvement (APCI) model is a new addition to the canon of mortality forecasting models. It was introduced by Continuous Mortality Investigation as a means of parameterising a deterministic targeting model for forecasting, but this paper shows how it can be implemented as a fully stochastic model. We demonstrate a number of interesting features about the APCI model, including which parameters to smooth and how much better the model fits to the data compared to some other, related models. However, this better fit also sometimes results in higher value-at-risk (VaR)-style capital requirements for insurers, and we explore why this is by looking at the density of the VaR simulations.

Basis risk modelling: a cointegration-based approach

ABSTRACTIn this paper we propose a multivariate approach for forecasting pairwise mortality rates of related populations. The need for joint modelling of mortality rates is analysed using a causality test. We show that for the datasets considered, the inclusion of national mortality information enhances predictions on its subpopulations. The investigated approach links national population mortality to that of a subset population, using an econometric model that captures a long-term relationship between the two mortality dynamics. This model does not focus on the correlation between the mortality rates of the two populations, but rather their long-term behaviour, which suggests that the two times series cannot wander off in opposite directions for long before mean reverting, which is consistent with biological reasoning. The model can additionally capture short-term adjustments in the mortality dynamics of the two populations. An empirical comparison of the forecast of one-year death probabilities for policyholders is performed using both a classical factor-based model and the proposed approach. The robustness of the model is tested on mortality rate data for England and Wales, alongside the Continuous Mortality Investigation assured lives dataset, representing the subpopulation.

Mortality forecasting using a modified CMI Mortality Projections Model for China II: cities, towns and counties

AbstractIn this paper, we conduct the study of long-term age-sex-specific mortality forecasting for subpopulations in different areas of China: cities, towns and counties. We use a modified CMI (Continuous Mortality Investigation) Mortality Projections Model, which has been discussed in Huang & Browne (Paper I), for modelling purposes. From the historical experience, we find that people in cities have lower mortality rates and higher mortality improvement rates than people in towns and counties for most ages. If this trend continues, the mortality of different areas will diverge further in the future. From the projection results, we find that there will be significant mortality and life expectancy differences between cities, towns and counties for both males and females. Sensitivity analysis for long-term rates of mortality improvement and the speed of convergence from “initial” to “long-term” rates of mortality improvement are conducted. Uncertainties are attached to the central estimates to overcome the limitation of the original CMI approach from which only deterministic results can be obtained.

Mortality forecasting using a modified Continuous Mortality Investigation Mortality Projections Model for China I: methodology and country-level results

AbstractIn this paper, we project future mortality rates for actuarial use with Chinese data using a modified Continuous Mortality Investigation (CMI) Mortality Projections Model. The model adopts a convergence structure from “initial” to “long-term” rates of mortality improvement as the process of projection. The initial rates of mortality improvement are derived using two-dimensional P-spline methodology. Given the short history of Chinese data, the long-term rates of mortality improvement are determined by borrowing information from international experience. K-means clustering with dynamic time warping distance is used to classify populations, which is novel in the actuarial mortality research field. The original CMI approach is deterministic, however, in this paper we make it stochastic using techniques outlined by Koller and described by Browne et al. Comparing our results with a pure extrapolative approach, we find that the CMI Mortality Projections Model is more suitable for long-term projections for China.

Model and parameter uncertainty in critical illness insurance ‐ Abstract of the Edinburgh Discussion

AbstractCritical illness (CI) insurance involves cover that pays out on the diagnosis of an illness which is deemed to be critical. Estimation and graduation of CI insurance claim rates has been challenging, partly because of the diagnosis of the insured event often being unclear or not recorded. This introduces additional uncertainties in the evaluation of claim rates. In this work, which was funded by an Institute and Faculty of Actuaries research grant, we have addressed the issue of model and parameter uncertainty in claim rate estimation, when the date of diagnosis is missing, aiming at obtaining graduated rates that can be applied to estimate the future cash flow of CI policies and determine insurers’ liability more accurately. Better understanding of uncertainty in rate graduation and pricing is important for insurers, not least because of future changes in the interpretation of the “definition” of an illness or advances in medical practice leading to more efficient diagnosis and treatment.Estimation of claim rates was addressed in this project using a parametric Poisson model for the number of claims, which accounts for claims that have not been settled by the end of the observation. This was achieved by considering probabilities of the distribution of the delay period between diagnosis of insured illness and settlement of the corresponding claim. We have developed appropriate generalised-linear-type models (including lognormal, Burr, generalised gamma and generalised beta) to investigate this delay, using data supplied by the Continuous Mortality Investigation (1999–2005). The analysis includes various claim risk factors (e.g. gender, benefit amount, policy duration and so on) and was performed under a Bayesian framework, using Markov chain Monte Carlo estimation techniques.Based on this experience, our analysis suggests that estimates of the claim delay distribution are model sensitive, but claim rates and pricing are not. Under the best claim delay distribution fitting, the following risk factors are considered important for predictive purposes: policy duration, selling office, benefit type, benefit amount, policy type and cause of claim.

Using bootstrapping to incorporate model error for risk-neutral pricing of longevity risk

Where mortality projection is concerned, it is essential to quantify the extent of the prediction error. This is especially important in light of the aggravating risk of longevity and as a result the increasing demand for longevity-linked products. In the literature so far, only parameter error and process error have been considered jointly while the issue of model error has yet been systematically studied. In this paper, we propose a method to account for process error, parameter error and model error in an integrated manner by modifying the semi-parametric bootstrapping technique. We apply the method to two data sets from the Continuous Mortality Investigation (CMI) and use the simulated scenarios to price the q -forward contracts via the maximum entropy approach. We find that model selection has a significant impact on the risk-neutral valuation results and thus it is crucial to incorporate model error in mortality projection.

Modelling critical illness claim diagnosis rates II: results

This is Paper II in a series of two papers. In Paper I we developed a methodology for estimating and graduating Critical Illness (CI) insurance diagnosis rates. In this paper we use data from the UK for 1999–2005 supplied by the Continuous Mortality Investigation (CMI) to illustrate our methodology by deriving and discussing all causes and cause specific critical illness diagnosis rates.

Modelling critical illness claim diagnosis rates I: methodology

In a series of two papers, this paper and the one by Ozkok et al. (Modelling critical illness claim diagnosis rates II: results), we develop statistical models to be used as a framework for estimating, and graduating, Critical Illness (CI) insurance diagnosis rates. We use UK data for 1999–2005 supplied by the Continuous Mortality Investigation (CMI) to illustrate their use. In this paper, we set out the basic methodology. In particular, we set out some models, we describe the data available to us and we discuss the statistical distribution of estimators proposed for CI diagnosis inception rates. A feature of CI insurance is the delay, on average about 6 months but in some cases much longer, between the diagnosis of an illness and the settlement of the subsequent claim. Modelling this delay, the so-called Claim Delay Distribution, is a necessary first step in the estimation of the claim diagnosis rates and this is discussed in the present paper. In the subsequent paper, we derive and discuss diagnosis rates for CI claims from ‘all causes’ and also from specific causes.

Survival Analysis of Left Truncated Income Protection Insurance Data

Abstract One of the main characteristics of Income Protection Insurance (IPI) claim duration data, which has not been considered in the actuarial literature on the topic, is left-truncation. Claimants that are observed are those whose sickness durations are longer than the deferred periods specified in the policies, and hence left-truncation exists in these data. This paper investigates a series of conditional mixture models when applying survival analysis to model sickness durations of IPI claimants, and examines the consequence of treating the IPI data with lengthy deferred periods as complete data and therefore ignoring the left-truncation by fitting the corresponding unconditional distributions. It also quantifies the extent of the bias in the resulting parameter estimates when ignoring the left-truncation in the data. Using the UK Continuous Mortality Investigation (CMI) sickness duration data, some well-fitting survival model results are estimated. It is demonstrated that ignoring the left-truncation in certain IPI data can lead to substantially different statistical estimates. We therefore suggest taking left-truncation into account by fitting conditional mixture distributions to IPI data. Furthermore, the best fitting model is extended by introducing a number of covariates into the conditional part to do regression analysis.

Bayesian modelling of the time delay between diagnosis and settlement for Critical Illness Insurance using a Burr generalised-linear-type model

We discuss Bayesian modelling of the delay between dates of diagnosis and settlement of claims in Critical Illness Insurance using a Burr distribution. The data are supplied by the UK Continuous Mortality Investigation and relate to claims settled in the years 1999–2005. There are non-recorded dates of diagnosis and settlement and these are included in the analysis as missing values using their posterior predictive distribution and MCMC methodology. The possible factors affecting the delay (age, sex, smoker status, policy type, benefit amount, etc.) are investigated under a Bayesian approach. A 3-parameter Burr generalised-linear-type model is fitted, where the covariates are linked to the mean of the distribution. Variable selection using Bayesian methodology to obtain the best model with different prior distribution setups for the parameters is also applied. In particular, Gibbs variable selection methods are considered, and results are confirmed using exact marginal likelihood findings and related Laplace approximations. For comparison purposes, a lognormal model is also considered.

Longevity Hedging 101

Basis risk is an important consideration when hedging longevity risk with instruments based on longevity indices, since the longevity experience of the hedged exposure may differ from that of the index. As a result, any decision to execute an index-based hedge requires a framework for (1) developing an informed understanding of the basis risk, (2) appropriately calibrating the hedging instrument, and (3) evaluating hedge effectiveness. We describe such a framework and apply it to a U.K. case study, which compares the population of assured lives from the Continuous Mortality Investigation with the England and Wales national population. The framework is founded on an analysis of historical experience data, together with an appreciation of the contextual relationship between the two related populations in social, economic, and demographic terms. Despite the different demographic profiles, the case study provides evidence of stable long-term relationships between the mortality experiences of the two populations. This suggests the important result that high levels of hedge effectiveness should be achievable with appropriately calibrated, static, index-based longevity hedges. Indeed, this is borne out in detailed calculations of hedge effectiveness for a hypothetical pension portfolio where the basis risk is based on the case study. A robustness check involving populations from the United States yields similar results.

A Gravity Model of Mortality Rates for Two Related Populations

The mortality rate dynamics between two related but different-sized populations are modeled consistently using a new stochastic mortality model that we call the gravity model. The larger population is modeled independently, and the smaller population is modeled in terms of spreads (or deviations) relative to the evolution of the former, but the spreads in the period and cohort effects between the larger and smaller populations depend on gravity or spread reversion parameters for the two effects. The larger the two gravity parameters, the more strongly the smaller population’s mortality rates move in line with those of the larger population in the long run. This is important where it is believed that the mortality rates between related populations should not diverge over time on grounds of biological reasonableness. The model is illustrated using an extension of the Age-Period-Cohort model and mortality rate data for English and Welsh males representing a large population and the Continuous Mortality Investigation assured male lives representing a smaller related population.

Smoothing dispersed counts with applications to mortality data

AbstractMortality data are often classified by age at death and year of death. This classification results in a heterogeneous risk set and this can cause problems for the estimation and forecasting of mortality. In the modelling of such data, we replace the classical assumption that the numbers of claims follow the Poisson distribution with the weaker assumption that the numbers of claims have a variance proportional to the mean. The constant of proportionality is known as the dispersion parameter and it enables us to allow for heterogeneity; in the case of insurance data the dispersion parameter also allows for the presence of duplicates in a portfolio. We use both the quasi-likelihood and the extended quasi-likelihood to estimate models for the smoothing and forecasting of mortality tables jointly with smooth estimates of the dispersion parameters. We present three main applications of our method: first, we show how taking account of dispersion reduces the volatility of a forecast of a mortality table; second, we smooth mortality data by amounts, ie, when deaths are amounts claimed and exposed-to-risk are sums assured; third, we present a joint model for mortality by lives and by amounts with the property that forecasts by lives and by amounts are consistent. Our methods are illustrated with data from the Continuous Mortality Investigation.

Mortality Projections using Generalized Additive Models with applications to annuity values for the Irish population

AbstractGeneralized 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.

Assessing the Impact of Suicide Exclusion Periods on Life Insurance

Assessing the Impact of Suicide Exclusion Periods on Life Insurance

Modelling Income Protection Claim Termination Rates by Cause of Sickness I: Recoveries

ABSTRACTIn this paper we present methods and results for the estimation and modelling of the recovery intensity for Income Protection (IP) insurance claims, allowing for different causes of claim. We use UK data supplied by the Continuous Mortality Investigation relating to claims paid in the years 1975 to 2002, inclusive. Each claim is classified by one of 70 possible causes according to ICD8.We group causes where appropriate, and then use the Cox model and generalised linear models to model the recovery intensity.In two subsequent papers we complete our modelling of IP claim termination rates by discussing the modelling of the mortality of IP claimants.There are two main reasons why it is useful to incorporate cause of sickness in the modelling of IP claim terminations:(i) The cause of sickness will be known to the insurer for a claim in the course of payment. A reserve can be set more accurately for such a claim if a model of the termination rates appropriate for this cause is available.(ii) Different causes of claim will become more or less significant over time. For example, tuberculosis may have been an important cause of sickness in the past, but is likely to be far less significant now; the swine flu pandemic starting in 2009 is likely to have a significant effect on observed aggregate claim termination rates, skewing them towards higher rates at shorter durations. Information about trends in morbidity, together with a model of termination rates by cause of claim, allows future aggregate claim termination rates to be predicted more accurately, reserves to be set at more appropriate levels and policies to be priced more accurately.One of the covariates included in our models for recovery intensities is Calendar Year. Aggregate recovery intensities have been decreasing over the period considered, 1975 to 2002, and this is generally reflected in the models for recovery intensities by cause of sickness. However, when these intensities are projected for years beyond 2002, the results are not always plausible.

Modelling Income Protection Claim Termination Rates by Cause of Sickness III: Mortality

ABSTRACTThis 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.