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Critical 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.

In this paper we calculate and compare diagnosis and net premium rates for critical illness insurance using different models for the claim delay distribution (CDD). The choice of CDD affects the diagnosis rates and hence the net premium rates in two ways: through the estimation of missing dates of diagnosis and through the adjustment of the exposure to allow for claims diagnosed but not settled in the observation period. We consider two CDDs: a three-parameter Burr distribution and a lognormal distribution. Our conclusion, based on a single, but extensive, data set, is that net premium rates are not significantly affected by the choice of CDD.

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.

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.

Gui et al. (2006), in Part III of a series of papers, proposed a dynamic family history model of breast cancer (BC) and ovarian cancer in which the development of a family history was represented explicitly as a transition between states, and then applied this model to life insurance and critical illness insurance. In this study, we extend the model to income protection insurance. In this paper, Part IV of the series, we construct and parameterise a semi-Markov model for the life history of a woman with BC, in which events such as diagnosis, treatment, recovery and recurrence are incorporated. In Part V, we then show: (a) estimates of premium ratings depending on genotype or family history; and (b) the impact of adverse selection under various moratoria on the use of genetic information.

In Part IV we presented a comprehensive model of a life history of a woman at risk of breast cancer (BC), in which relevant events such as diagnosis, treatment, recovery and recurrence were represented explicitly, and corresponding transition intensities were estimated. In this part, we study some applications to income protection insurance (IPI) business. We calculate premiums based either on genetic test results or more practically on a family history of BC. We then extend the model into an IPI market model by incorporating rates of insurance-buying behaviour, in order to estimate the possible costs of adverse selection, in terms of increased premiums, under various moratoria on the use of genetic information.

This is the second 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. In this and the third paper we develop models for the mortality of IP claimants.We model this mortality as the sum of two components: a base, or background, 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 element which does depend on the current duration of the sickness. In this paper we discuss the modelling of the base mortality. In particular, we use data supplied by the Continuous Mortality Investigation relating to UK assured lives from 1975 (males) and 1983 (females) to 2003 to develop models of mortality which are functions of sex, age and calendar year. Such models are of interest in their own right, particularly at a time when expected future lifetimes are increasing.The modelling of the cause-specific component of the mortality model is discussed in Paper III.

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.

In 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.

Evaluating the risk of disorders in long-term insurance often relies on rates of onset estimated from quite small epidemiological studies. These estimates can carry considerable uncertainty, hence so may functions of them, such as a premium rate. In the case of genetic disorders, where it may be required to demonstrate the reliability of genetic information as a risk factor, such uncertainty may be material. Epidemiological studies publish their results in a variety of forms and it is rarely easy to estimate the sampling distribution of a premium rate without access to the original data. We found a large study of breast and ovarian cancer that cited relative risks of breast and ovarian cancer onset, with confidence intervals, in 10-year age groups. We obtained critical illness premium rates and their sampling distributions by parametric bootstrapping, and investigated the effect of possible patterns of sampling correlations. We found that this study provides ample statistical evidence that known BRCA1 or BRCA2 mutations, or a typical family history of breast or ovarian cancer, are reliable risk factors, but the sampling covariances of the relative risks could be important at some ages and terms. Studies that cite only standard errors of parameter estimates erect a small but awkward barrier between the models they describe, and some important actuarial questions.

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).

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.

Hereditary nonpolyposis colorectal cancer (HNPCC) is characterised by the familial aggregation of cancer of the colon and rectum (CRC). It may be caused by any of five mutations in DNA mismatch repair (MMR) genes or by non-genetic factors, such as life style. However, it accounts for only about 2% of CRC, which is a very common cancer. Previous actuarial models, of diseases with only genetic causes, assumed that a family history of the disease shows mutations to be present, but this is not true of HNPCC. This is a significant limitation, since the best information available to an underwriter (especially if the use of genetic test results is banned) is likely to be knowledge of a family history of CRC. We present a Markov model of CRC and HNPCC, which includes the presence of a family history of CRC as a state, and estimate its intensities allowing for MMR genotype. Using this we find the MMR mutation probabilities for an insurance applicant with a family history of CRC. Our model greatly simplifies the intensive computational burden of finding such probabilities by integrating over complex models of hidden family structure. We estimate the costs of critical illness insurance given the applicant's genotype or the presence of a family history. We then consider what the cost of adverse selection might be, if insurers are unable to use genetic tests or family history information. We also consider the effect of using alternative definitions of a family history in underwriting.

Insurers’ access to genetic test results is often restricted and the only genetic information that might be collected during underwriting in some countries is family history. Previous studies have included family history in a simple way but only for diseases which have no cause other than gene mutations, because then the event ‘affected parent’ contributes all possible information short of a genetic test result. We construct a model of breast cancer (BC) and ovarian cancer (OC) — common diseases with rare genetic variants — in which the development of a family history is represented explicitly as a transition between states, hence as part of the applicant's own life history. This allows the impact of a moratorium to be modelled. We then apply this family history model to life insurance in a semi-Markov framework and to critical illness (CI) insurance in a Markov framework to: (a) estimate premium ratings depending on genotype or family history; and (b) model the potential cost of adverse selection.

We consider a classical surplus process where the insurer can choose a different level of reinsurance at the start of each year. We assume the insurer’s objective is to minimise the probability of ruin up to some given time horizon, either in discrete or continuous time. We develop formulae for ruin probabilities under the optimal reinsurance strategy, i.e. the optimal retention each year as the surplus changes and the period until the time horizon shortens. For our compound Poisson process, it is not feasible to evaluate these formulae, and hence determine the optimal strategies, in any but the simplest cases. We show how we can determine the optimal strategies by approximating the (compound Poisson) aggregate claims distributions by translated gamma distributions, and, alternatively, by approximating the compound Poisson process by a translated gamma process.

In this paper we discuss the numerical calculation of finite time ruin probabilities for two particular insurance risk models. The first model allows for the investment at a fixed rate of interest of the surplus whenever this is above a given level. This is related to a model studied by Embrechts and Schmidli [Embrechts, P., Schmidli, H., 1994. Ruin estimation for a general insurance risk model. Adv. Appl. Probability 26 (2), 404–422] and by Schmidli [Schmidli, H., 1994a. Corrected diffusion approximations for a risk process with the possibility of borrowing and investment. Schweizerische Vereinigung der Versicherungsmathematiker. Mitteilungen (1), 71–82; Schmidli, H., 1994b. Diffusion approximations for a risk process with the possibility of borrowing and investment. Commun. Stat. Stochastic Models 10 (2), 365–388]. Our second model is the classical risk model but with the insurer’s premium rate depending on the level of the surplus. In our final section, we discuss the extension of the these models to allow for the parameters to change over time in a deterministic way. Our methodology for calculating finite time ruin probabilities is to bound the surplus process by discrete-time Markov chains; the average of the bounds gives an approximation to the ruin probability. This approach was used by the authors in a previous paper, Cardoso and Waters [Cardoso, R.M.R., Waters, H.R., 2003. Recursive calculation of finite time ruin probabilities under interest force. Insurance Math. Econ. 33 (3), 659–676], which considered a risk process with interest earned on the surplus.

In this paper we present many investigations into the results of simulating the process of hedging a vanilla option at discrete times. We consider mainly a ‘maxi’ option (paying Max(A, B)), though calls, puts and ‘minis’ are also considered. We show the sensitivity of the variability of the hedging error to the actual investment strategy adopted, and to the many ways in which the simulated real world can diverge from the assumed option pricing model. We show how prudential reserves can be calculated, using conditional tail expectations, and how net premiums or fair values (which we present as the same) can be calculated, allowing for the necessary prudential reserves. We use two bond models, the very simple Black-Scholes one and a less unrealistic one. We also use the Wilkie model as an even more realistic real-world model, allowing for many complications in it to make it more realistic. We make observations on the important difference between real-world models and option pricing models, and emphasise the latter as the way of getting hedging quantities, and not just option prices.

In Part I we constructed a model for the development of coronary heart disease (CHD) or stroke that either incorporates, or includes pathways through, the major risk factors of interest when underwriting for critical illness (CI) insurance. In Part II we extend this model to include other critical illnesses, for example, cancers and kidney failure, and describe some applications of the model. In particular, we discuss CI premium ratings for applicants with combinations of some or all of high body mass index, smoking, high blood pressure, high cholesterol, and diabetes. We also consider the possible effect on CI premium ratings of genetic conditions that increase the likelihood of high blood pressure, high cholesterol, diabetes, CHD event, or stroke.

In Part I we construct a model for the development of coronary heart disease (CHD) or stroke that either incorporates, or includes pathways through, the major risk factors of interest when underwriting for critical illness insurance. Our main purpose is to develop a model that could be used to assess the impact on insurance underwriting of genetic information relevant to CHD and/or stroke. Our model is parameterized using data from the Framingham Heart Study in the United States. In Part II we extend this model to include other critical illnesses, for example, cancers and kidney failure, and describe some applications of the model.

The text of these papers, together with the abstract of the discussion held by the Faculty of Actuaries on 15 March 2004, are printed in British Actuarial Journal, 10, IV, 685-898.

The main objective of this paper is to offer a detailed analysis of mortality change in the United Kingdom at the beginning the 21st century. Starting from an exploration of 20th century mortality trends, focusing in particular on the 1990s, underlying forces driving trends in longevity are discussed. These include the ‘cohort effect’ and the ‘ageing of mortality improvement’. International mortality statistics and trends are also analysed. The pace of medical advances is discussed, with specific focus on research into the ageing process and a potential treatment for cardiovascular disease. The paper also discusses the potential threat from infectious diseases.The analysis of underlying trends suggests that life expectancy in retirement in the U.K. is likely to increase rapidly in the early part of the 21st century. Some scientists are also claiming that we will be seeing the fruits of anti-ageing research within just a few decades.A core theme of the paper is that future projections should be grounded in as good an understanding of the past as possible. Different methods for projecting future rates of mortality are discussed, and it is noted that emphasis should be placed on the uncertainty surrounding projections.The financial impact of using different assumptions for future mortality is explored. Significant differences in the cost of an annuity or pension arise from the use of the various projection bases.Life assurance companies have already declared significant losses as a result of strengthening reserves on annuity portfolios. Taken together, future increases in life expectancy, increasing awareness of the risk of providing longevity insurance, changes in legislation and shortages in market capacity and capital, may well lead to worsening annuity rates.It is difficult to assess the precise impact of future changes in life expectancy on final salary pension schemes. There is a lack of readily available information on the mortality assumptions being used in practice. It is therefore suggested that more disclosure in this area would be helpful. Employers sponsoring final salary schemes are making promises to their employees that extend up to 70 or 80 years into the future. Actuaries should be clear in spelling out to employers and trustees the nature of the risks behind the promises they are making. Future scheme design should reflect the possibility of substantial increases in life expectancy.An over-riding implication of the anticipated increases in life expectancy is that people will remain in work for longer in the future. The age at which people retire will inevitably have to increase, and this trend will necessarily drive changes in all aspects of our society. As actuaries we have a vital role in helping to inform the wider debate.

We consider a situation originally discussed by De Finetti (1957) in which a surplus process is modified by the introduction of a constant dividend barrier. We extend some known results relating to the distribution of the present value of dividend payments until ruin in the classical risk model and show how a discrete time risk model can be used to provide approximations when analytic results are unavailable. We extend the analysis by allowing the process to continue after ruin.

In this paper, we consider a classical insurance surplus process affected by a constant interest force. We present numerical algorithms for the calculation of finite time ruin probabilities using a discrete time Markov chain to approximate the risk process. Based on this method, upper and lower bounds are also obtained.

We present a Markov model of breast cancer (BC) and ovarian cancer (OC) and estimate its transition intensities, mainly using United Kingdom population data. In the case of BC and OC, we estimate intensities according to BRCA1 and BRCA2 genotype. We use this to estimate the probabilities that an applicant for insurance has a BRCA1 or BRCA2 mutation, given complete or incomplete knowledge of her family history of BC and OC. Life (and other) insurance underwriters typically have incomplete knowledge of family history, for example no information on the number of healthy relatives. We show how these probabilities depend strongly on estimates of the mutation frequencies and penetrances, and conclude that it may not be appropriate to apply risk estimates based on studies of high-risk families to other groups.

We present a model of breast cancer (BC) and ovarian cancer (OC) and other events that would lead to a claim under a Critical Illness (CI) insurance policy, and estimate its transition intensities, mainly using United Kingdom population data. We use this to estimate the costs of CI insurance in the presence of a family history of BC or OC, using the probabilities from Part I of carrying a BRCA1 or BRCA2 mutation, given the family history. In practice, the family history may not include all relevant facts; we look at the range of costs depending on what is known. We show the effect of lower penetrance than is observed in high-risk families. Finally, we consider what the cost of adverse selection might be, were insurers unable to use genetic test or family history information.

We study the distribution of the time to ruin in the classical risk model. We consider some methods of calculating this distribution, in particular by using algorithms to calculate finite time ruin probabilities. We also discuss calculation of the moments of this distribution.

Algorithms for the calculation of the distribution of the aggregate claims from a life insurance portfolio have been derived by Kornya (1983), Hipp (1986) and De Pril (1986 and 1989). All these authors considered the distribution of the aggregate claims over a single period. In this paper we derive algorithms for the calculation of the joint distribution of the aggregate claims from a life portfolio over several periods.

We consider a classical surplus process modified by the action of a constant force of interest. We derive recursive algorithms for the calculation of the probability of ruin in finite time. We also discuss the numerical evaluation of the probability of ultimate ruin using methods proposed by De Vylder [De Vylder, F., 1996. Advanced Risk Theory. Editions de l’Universite de Bruxelles.] and Sundt and Teugels [Sundt, B., Teugels, J.L., 1995. Insurance: Mathematics and Economics 16, 7–22]. Finally, we consider the problem of recovery from ruin.

(2007). “On the Class of Erlang Mixtures with Risk Theoretic Applications”, Gordon E. Willmot and Jae-Kyung Woo, April 2007. North American Actuarial Journal: Vol. 11, No. 2, pp. 115-117.

The problem of determining optimal retention levels for a non-life portfolio consisting of a number of independent sub-portfolios was first discussed by de Finetti (1940). He considered retention levels as optimal if they minimised the variance of the insurer's profit from the portfolio subject to the constraint of a fixed level of expected profit. In this paper we consider a similar problem, changing the criterion for optimality to minimising the probability of ruin, either in discrete or continuous time. We investigate this problem through a series of case studies based on a real portfolio.

We study the effect of reinsurance on the probability of ultimate ruin in the classical surplus process and consider a retention level as optimal if it minimises the ruin probability. We show that optimal retention levels can be found when the reinsurer's premium loading depends on the retention level. We also show that when the aggregate claims process is approximated by a translated Gamma process, very good approximations to both optimal retention levels and ruin probabilities can be obtained. Finally, we discuss the effect of reinsurance on the probability of ruin in finite time.

In this paper we present a method for the numerical evaluation of the ruin probability in continuous and finite time for a classical risk process where the premium can change from year to year. A major consideration in the development of this methodology is that it should be easily applicable to large portfolios. Our method is based on the simulation of the annual aggregate claims and then on the calculation of the ruin probability for a given surplus at the start and at the end of each year. We calculate the within-year ruin probability assuming a translated gamma distribution approximation for aggregate claim amounts.We illustrate our method by studying the case where the premium at the start of each year is a function of the surplus level at that time or at an earlier time.

In this paper we present a stable recursive algorithm for the calculation of the probability of ultimate ruin in the classical risk model. We also present stable recursive algorithms for the calculation of the joint and marginal distributions of the surplus prior to ruin and the severity of ruin. In addition we present bounds for these distributions.

This must-have manual provides solutions to all exercises in Dickson, Hardy and Waters' Actuarial Mathematics for Life Contingent Risks, the groundbreaking text on the modern mathematics of life insurance that is the required reading for the SOA Exam MLC and also covers more or less the whole syllabus for the UK Subject CT5 exam. The more than 150 exercises are designed to teach skills in simulation and projection through computational practice, and the solutions are written to give insight as well as exam preparation. Companion spreadsheets are available for free download to show implementation of computational methods.

1.1 The insurer will calculate the premium for a term or whole life insurance policy assuming that the policyholder is in relatively good health; otherwise, if the insurer assumed that all purchasers were unhealthy, the cost of insurance would be prohibitive to those customers who are healthy. The assumption then is that claims will be relatively rare in the first few years of insurance, especially since most policies are sold to lives in their 30s and 40s. This means that the price is too low for a life who is very unwell, for whom the risk of a claim shortly after purchase might be 10 or 100 times greater than for a healthy life. The insurer therefore needs evidence that the purchaser is in good health, to avoid the risk that insurance is bought too cheaply by lives who have a much higher probability of claim. The objective of underwriting is to produce a relatively homogeneous insured population when policies are issued. The risk that the policyholder purchases the insurance because they are aware that their individual risk is greater than that of the insured population used to calculate the premium, is an example of adverse selection risk. Underwriting is a way of reducing the impact of adverse selection for life insurance. Adverse selection for an annuity purchaser works in the other direction – a life might buy an annuity if they considered their mortality was lighter than the general population.