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.

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.

This paper studies estimation of the conditional mean squared error of prediction, conditional on what is known at the time of prediction. The particular problem considered is the assessment of actuarial reserving methods given data in the form of run-off triangles (trapezoids), where the use of prediction assessment based on out-of-sample performance is not an option. The prediction assessment principle advocated here can be viewed as a generalisation of Akaike’s final prediction error. A direct application of this simple principle in the setting of a data-generating process given in terms of a sequence of general linear models yields an estimator of the conditional mean squared error of prediction that can be computed explicitly for a wide range of models within this model class. Mack’s distribution-free chain ladder model and the corresponding estimator of the prediction error for the ultimate claim amount are shown to be a special case. It is demonstrated that the prediction assessment principle easily applies to quite different data-generating processes and results in estimators that have been studied in the literature.

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.

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.

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.

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.

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.

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.

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.

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 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.

In 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.

Spatial analysis ranges from simple univariate descriptive statistics to complex multivariate analyses and is typically used to investigate spatial patterns or to identify spatially linked consumer behaviours in insurance. This paper investigates if the incorporation of publicly available spatially linked demographic census data at population level is useful in modelling customers’ lapse behaviour (i.e. stopping payment of premiums) in life insurance policies, based on data provided by an insurance company in Ireland. From the insurance company’s perspective, identifying and assessing such lapsing risks in advance permit engagement to prevent such incidents, saving money by re-evaluating customer acquisition channels and improving capital reserve calculation and preparation. Incorporating spatial analysis in lapse modelling is expected to improve lapse prediction. Therefore, a hybrid approach to lapse prediction is proposed – spatial clustering using census data is used to reveal the underlying spatial structure of customers of the Irish life insurer, in conjunction with traditional statistical models for lapse prediction based on the company data. The primary contribution of this work is to consider the spatial characteristics of customers for life insurance lapse behaviour, via the integration of reliable government provided census demographics, which has not been considered previously in actuarial literature. Company decision-makers can use the insights gleaned from this analysis to identify customer subsets to target with personalized promotions to reduce lapse rates, and to reduce overall company risk.

Equity linked products are popular in many countries. These contracts generally provide a guaranteed return combined with some participation in the market performance, often computed using a complicated formula. Hedging these contracts often presents a real challenge for insurers in particular during a financial crisis. In this paper, we explain how insurers can benefit from selling a pool of different contracts with different sensitivities to certain key variables to reduce risk exposure. We show how they can diversify their menu of policy designs to stabilize the market value of their liabilities against changes in the market volatility and against estimation error in the volatility parameter. We illustrate the methodology with specific examples of equity annuity contracts with opposite sensitivities to vega risk.

Home equity release products have been promoted as a potential solution to residential long-term care costs for the elderly. Lower than expected utilisation of home equity release loans has prompted efforts to better model and price the no-negative-equity-guarantee (NNEG) built into the contracts, but loan rates are still widely perceived by homeowners as being unattractive. We propose the introduction of a new adjustable rate loan based on a regional house price index, with the NNEG being borne by a specially created intermediary. The proposed approach allows us to directly address and separately price the basis risk between individual house price returns and index returns. In addition, it offers the opportunity to create securities based on residential real estate that would be attractive to a wider class of investors. The alternative risk-sharing mechanism creates a more transparent and simple pricing structure for the loans. We then use house sales data to demonstrate the approach. We find in our sample that it would be possible to make higher loans than seen in previous literature using standard roll-up contracts. In the most favourable scenario for our simulations, the maximum loan is 89% of the appraised home value if the loan is advanced as a lump sum and 95% if the loan is advanced in instalments.