In Buchwalder et al. (2006) we revisited Mack's (1993) and Murphy's (1994) estimates for the mean square error of prediction (MSEP) of the chain ladder claims reserving method. This was done using a time series model for the chain ladder method. In this paper we extend the time series model to determine an estimate for the MSEP of a portfolio of N correlated run-off triangles. This estimate differs in the special case N = 2 from the estimate given by Braun (2004). We discuss the differences between the estimates.

This paper investigates the use of censored regression quantiles in the analysis of claim termination rates for income protection (IP) insurance. The paper demonstrates the importance of modeling quantiles given the growing interest of regulators and others in stochastic approaches to valuation of insurance liabilities and risk margins.

Insurance and annuity products covering several lives require the modelling of the joint distribution of future lifetimes. In the interest of simplifying calculations, it is common in practice to assume that the future lifetimes among a group of people are independent. However, extensive research over the past decades suggests otherwise. In this paper, a copula approach is used to model the dependence between lifetimes within a married couple using data from a large Canadian insurance company. As a novelty, the age difference and the gender of the elder partner are introduced as an argument of the dependence parameter. Maximum likelihood techniques are thus implemented for the parameter estimation. Not only do the results make clear that the correlation decreases with age difference, but also the dependence between the lifetimes is higher when husband is older than wife. A goodness-of-fit procedure is applied in order to assess the validity of the model. Finally, considering several annuity products available on the life insurance market, the paper concludes with practical illustrations.

In this paper, we study the optimal proportional reinsurance problem in a risk model with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component, and the criterion is to minimise the probability of drawdown, namely, the probability that the value of the surplus process reaches some fixed proportion of its maximum value to date. By the method of maximising the ratio of drift of a diffusion divided to its volatility squared, and the technique of stochastic control theory and the corresponding Hamilton–Jacobi–Bellman equation, we investigate the optimisation problem in two different cases. Furthermore, we constrain the reinsurance proportion in the interval [0,1] for each case, and derive the explicit expressions of the optimal proportional reinsurance strategy and the minimum probability of drawdown. Finally, some numerical examples are presented to show the impact of model parameters on the optimal results.

In this paper, we propose a model for the optimal premium pricing policy of an insurance company into a competitive environment using Dynamic Programming into a stochastic, discrete-time framework when the company is expected to drop part of the market. In our approach, the volume of business which is related to the past year experience, the average premium of the market, the company's premium which is a control function and a linear stochastic disturbance, have been considered. Consequently, maximizing the total expected linear discounted utility of the wealth over a finite time horizon, the optimal premium strategy is defined analytically and endogenously. Finally, considering two different strategies for the average premium of the market, the optimal premium policy for a company with an expected decreasing volume of business is derived and fully investigated. The results of this paper are further evaluated by using data from the Greek Automobile Insurance Industry.

Mortality 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.

Many recently introduced unit-linked life insurance policies contain provisions allowing policyholders to lapse the product. The problem of pricing this surrender option is difficult as it involves modelling lapse decisions which may be contingent on different factors. This paper develops a methodology which enables us to model lapse behaviour within a framework provided by developments in financial economics. Using marked point processes with stochastic intensities, we present an approach which accounts for changes in the lapse behaviour of policyholders due to different economic factors. As a result, the model produces more accurate financial values for insurance contracts contingent on financial markets. In the context of unit-linked policies, we illustrate the method by allowing the lapse decision to depend on the stochastic volatility of the underlying asset. Our simulation study indicates that there is a strong relation between the single premiums of these policies and the lapse behaviour.

Rapid population ageing in China has urged the need to understand health transitions of older Chinese to assist the development of social security programmes and financial products aimed at funding long-term care. In this paper, we develop a new flexible approach to modelling health transitions in a multi-state Markov model that allows for age effects, time trends and age-time interactions. The model is implemented in the generalised linear modelling framework. We apply the model to evaluate health transitions of Chinese elderly using individual-level panel data from the Chinese Longitudinal Healthy Longevity Survey for the period 1998–2012. Our results confirm that time trends and age–time interactions are important factors explaining health transitions in addition to the more commonly used age effects. We document that differences in disability and mortality rates continue to persist between urban and rural older Chinese. We also compute life expectancies and healthy life expectancies based on the proposed model as inputs for the development of aged care and financial services for older Chinese.

In this paper, we explore the use of an extensive list of Archimedean copulas in general and life insurance modelling. We consider not only the usual choices like the Clayton, Gumbel–Hougaard, and Frank copulas but also several others which have not drawn much attention in previous applications. First, we apply different copula functions to two general insurance data sets, co-modelling losses and allocated loss adjustment expenses, and also losses to building and contents. Second, we adopt these copulas for modelling the mortality trends of two neighbouring countries and calculate the market price of a mortality bond. Our results clearly show that the diversity of Archimedean copula structures gives much flexibility for modelling different kinds of data sets and that the copula and tail dependence assumption can have a significant impact on pricing and valuation. Moreover, we conduct a large simulation exercise to investigate further the caveats in copula selection. Finally, we examine a number of other estimation methods which have not been tested in previous insurance applications.

As the field of modelling mortality has grown in recent years, the number and importance of identifiability issues within mortality models has grown in parallel. This has led both to robustness problems and to difficulties in making projections of future mortality rates. In this paper, we present a comprehensive analysis of the identifiability issues in age/period mortality models in order to first understand them better and then to resolve them. To achieve this, we discuss how these identification issues arise, how to choose identification schemes which aid our demographic interpretation of the models and how to project the models so that our forecasts of the future do not depend upon the arbitrary choices used to identify the historical parameters estimated from historical data.

This is the third and last subpart of a long paper in which we consider stochastic interpolation for the Wilkie asset model, considering both Brownian bridges and Ornstein–Uhlenbeck (OU) bridges. In Part 3A, we developed certain properties for both these types of stochastic bridge, and in Part 3B we investigated retail prices and wages. In this paper, we investigate the remainder of many of our data series, relating to shares and interest rates. We conclude that, regardless of the form of the annual model, the monthly data within each year can be modelled by Brownian bridges, usually on the logarithm of the principal variable. But in no case is a simple Brownian bridge enough, and all series have their own peculiarities. Overall, however, our modelling produces simulations that are realistic in comparison with the known data. Many of our findings would apply to any similar model used for simulation over time. Our results have considerable importance for financial economics. We reconcile the conflict between the long-term mean-reverting modelling of Schiller and the short-term random walk modelling of Fama. This conclusion therefore has very wide significance.

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

We derive optimal portfolio choice patterns in retirement (ages 66–105) for a constant relative risk aversion utility maximising investor facing risky capital market returns, stochastic mortality risk, and income-reducing health shocks. Beyond the usual stocks and bonds, the individual can invest his assets in tontines. Tontines are cost-efficient financial contracts providing age-increasing, but volatile cash flows, generated through the pooling of mortality without guarantees, which can help to match increasing financing needs at old ages. We find that a tontine invested in the risk-free asset dominates stock investments for older investors without a bequest motive. However, with a bequest motive, it is optimal to replace the tontine investment over time with traditional financial assets. Our results indicate that early in retirement, a tontine is only an attractive investment option, if the tontine funds are invested in a risky asset. In this case, they crowd out stocks and risk-free bonds in the optimal portfolios of younger investors. Over time, the average optimal portfolio weight of tontines decreases. Introducing systematic mortality risks noticeably reduces the peak allocation to tontines.

We have investigated the performance of pension schemes of with-profit policies containing a guaranteed minimum rate of return and we have found that the price of the guarantee measured in terms of lost returns is enormous. We use simple simulations rather than complex pricing methods to illustrate that the price of an interest guarantee is high in pension products that are currently commercialised in the market. We have found that the customer loses up to about 0.75% yearly in the rate of return when an interest guarantee is purchased, compared to the return of an equivalent saving strategy with the same risk at the level 95%. This can explain why such arrangements are not widely popular. Our approach can be used to inform clients, who are not experts in modern financial models, the impact of paying for an interest guarantee.

The calculation of a fair premium is always a challenging topic in the real-world insurance applications. In this paper, a non-linear premium-reserve (P-R) model is presented and the premium is derived by minimising a quadratic performance criterion. The reserve is a stochastic equation, which includes an additive random non-linear function of the state, premium and not necessarily Gaussian noise, which is, however, independently distributed in time, provided only that the mean value and the covariance of the random function is 0 and a quadratic function of the state, premium and other parameters, respectively. In this quadratic representation of the covariance function, new parameters are implemented and enriched further by the previous linear models, such as the income insurance elasticity of demand, the number of insured and the inflation in addition to the company’s reputation. The quadratic utility function concerns the present value of the reserve. Interestingly, for the very first time, the derived optimal premium in a competitive market environment is also dependent on the company’s reserve among the other parameters. Finally, a numerical application illustrates the main findings of the paper.

We consider a compositional data analysis approach to forecasting the age distribution of death counts. Using the age-specific period life-table death counts in Australia obtained from the Human Mortality Database, the compositional data analysis approach produces more accurate 1- to 20-step-ahead point and interval forecasts than Lee–Carter method, Hyndman–Ullah method and two naïve random walk methods. The improved forecast accuracy of period life-table death counts is of great interest to demographers for estimating survival probabilities and life expectancy, and to actuaries for determining temporary annuity prices for various ages and maturities. Although we focus on temporary annuity prices, we consider long-term contracts that make the annuity almost lifetime, in particular when the age at entry is sufficiently high.

Models 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.

We use economic indicators to improve the prediction of the number of incurred but not recorded disability insurance claims, assuming that there is a link between the number of claims and the chosen economic indicators. We propose a Bayesian model where we model the claims development in three directions: along incurred periods, recording lag periods and calendar periods. A stochastic model of the economic indicators is incorporated into the calendar period development direction. Thus we allow for the impact of the economic environment on the number of claims. Applying the proposed model to data, we illustrate how the inclusion of economic indicators affects the prediction of the number of incurred but not recorded disability claims.

This research investigates the impact of deprivation on demographic inequalities in England and Wales among adults. Using demographic measures including the modal age at death, life expectancy, lifespan variation and mortality, it shows a negative correlation with deprivation as measured by the 2015 Index of Multiple Deprivation. Although it finds that life expectancy is increasing overall and the gap between men and women is lessening, improvements are slower paced in more deprived areas such that the gap between rich and poor is slowly worsening over time. Men are more adversely impacted by deprivation than women with the gap in period life expectancy at age 30 in 2015 between the top and bottom 1% of deprived neighbourhoods at 10.9 years for men and 8.4 years for women. Between 2001 and 2015 inequalities in male mortality rates at age 44 were 4.4 times greater in the most deprived 10% of neighbourhoods than those in the 10% least deprived and were much higher than in intervening deciles. The worst deprivation is concentrated in specific areas. For example, in 22 out of 326 English districts, 25% or more of neighbourhoods are in the most deprived 10% and in 5 districts it is 40% or above.