Immediate Annuity Pricing in the Presence of Unobserved Heterogeneity

Abstract

One of the acknowledged difficulties with pricing immediate annuities is that underwriting the annuitantis life is the exception rather than the rule. In the absence of underwriting, the price paid for a life-contingent annuity is the same for all sales at a given age. This exposes the market (insurance company and potential policyholder alike) to antiselection. The insurance company worries that only the healthiest people choose a life-contingent annuity and therefore adjust mortality accordingly. The potential policyholders worry that they are not being compensated for their relatively poor health and choose not to purchase what would otherwise be a very beneficial product.

This paper develops a model of underlying, unobserved health. Health is a state variable that follows a first-order Markov process. An individual reaches the state “death” either by accident from any health state or by progressively declining health state. Health state is one-dimensional, in the sense that health can either “improve” or “deteriorate” by moving farther from or close to the “death” state, respectively. The probability of death in a given year is a function of health state, not of age. Therefore, in this model a person is exactly as old as he or she feels.

I first demonstrate that a multistate, ageless Markov model can match the mortality patterns in the common annuity mortality tables. The model is extended to consider several types of mortality improvements: permanent through decreasing probability of deteriorating health, temporary through improved distribution of initial health state, and plateau through the effects of past health improvements.

I then construct an economic model of optimal policyholder behavior, assuming that the policyholder either knows his or her health state or has some limited information. the value of mortality risk transfer through purchasing a life-contingent annuity is estimated for each health state under various risk-aversion parameters. Given the economic model for optimal purchasing of annuities, the value of underwriting (limited information about policyholder health state) is demonstrated.     

“On the Merger of Two Companies,” Hans Gerber and Elias S. W. Shiu,July 2006

Abstract

Bankruptcy risk falls to pension plan participants if a plan sponsor fails when a defined benefit (DB) pension plan is underfunded. This article examines the incidence of that risk and how it changes when public policy provides a guarantee fund. Although government-based guarantee funds are in a unique position to provide pension protection, primarily because of the extent to which the risk of sponsor default is systematic in nature, a looming question is the extent to which such guarantees are exposed to moral hazard. The article focuses on that question using data from four Canadian provinces, including one (Ontario) that operates a guarantee fund for pensions. The findings show that plan assets per DB-plan participant increase with the earnings of workers and decrease with higher unemployment, and that level of assets also is moderated by the influence of taxes, with higher plan assets observed when and where tax rates are higher. Plans in Ontario had on average $20,035 less in asset value per participant, and Ontario plans covered by the guarantee fund had an average of $16,497 less per participant than other Canadian DB plans not backed by a guarantee fund. A separate model finds the presence of a guarantee fund to be one of a very small number of variables significant in explaining variability in the plans’ funded ratios. These empirical results are consistent with the existence of moral hazard.     

Determining the Optimum Guarantee Period for a One-Life Retirement Annuity

Abstract

We consider a risk averse retiree from a defined contribution plan who decides to purchase a onelife annuity with a guarantee period. Given the retiree has a bequest motive, we focus on the problem of determining the optimum length of the guarantee period. Assuming the retiree’s bequest function is proportional to his or her utility function, we determine necessary and/or sufficient conditions under which the retiree would choose an annuity with (i) no guarantee period, (ii) the maximum guarantee period, or (iii) an intermediate guarantee period.     

Relative Choice Models for Income Drawdown in a Defined Contribution Pension Scheme

Abstract

This paper extends a target-based model of income drawdown developed in Gerrard et al. (Insurance: Mathematics and Economics 35: 321–342 [2006]) (GHV) for the distribution phase of a defined contribution pension scheme. The optimal investment strategy of the pension fund and the optimal drawdown are found using linear-quadratic optimization, which minimizes the deviation of the fund and the drawdown from prescribed targets. The GHV model is modified by nondimensionalizing the loss function, so that there is a relative choice between outcomes.

Using this model, three classes of target are studied. Endogenous deterministic targets are suggested from the form of the optimal controls, while exogenous deterministic targets can be stated without knowledge of the optimization problem. The third class of stochastic targets is similar to recent annuity products, which incorporate investment risk. Each scheme represents a trade-off between investment risk and return, and this is illustrated by numerical simulation with reference to a canonical example. A particularly attractive form of income drawdown is given by an implied rate of return target. This yields a reasonable investment strategy and a robust consumption profile with age. In addition, it can be easily explained to pension scheme members.     

Voluntary Termination of Life Insurance Policies

Abstract

In this paper, one error-correction model (ECM) that is able to avoid the problem of producing noise within traditional multiple cointegration vectors has been employed to explore the dynamics of surrender behavior. The evidence shows that both the emergency fund hypothesis and interest rate hypothesis are sustained in the short run as well as in the long run. A unique cointegration relationship within the surrender dynamics has been validated. In addition, a new hypothesis test that stresses the competition for the withdrawal of life insurance policy cash values has also been conducted. Such a crowding-out effect between policy loans and policy surrenders might be attributed to the motivation that keeps a life policy in force, the existence of surrender charges, and the automatic premium loan provision.     

Optimal Reinsurance and Investment for a Jump Diffusion Risk Process under the CEV Model

Abstract

We consider an optimal reinsurance-investment problem of an insurer whose surplus process follows a jump-diffusion model. In our model the insurer transfers part of the risk due to insurance claims via a proportional reinsurance and invests the surplus in a “simplified” financial market consisting of a risk-free asset and a risky asset. The dynamics of the risky asset are governed by a constant elasticity of variance model to incorporate conditional heteroscedasticity. The objective of the insurer is to choose an optimal reinsurance-investment strategy so as to maximize the expected exponential utility of terminal wealth. We investigate the problem using the Hamilton-Jacobi-Bellman dynamic programming approach. Explicit forms for the optimal reinsuranceinvestment strategy and the corresponding value function are obtained. Numerical examples are provided to illustrate how the optimal investment-reinsurance policy changes when the model parameters vary.     

Mortality Measurement at Advanced Ages

Abstract

Accurate estimates of mortality at advanced ages are essential to improving forecasts of mortality and the population size of the oldest old age group. However, estimation of hazard rates at extremely old ages poses serious challenges to researchers: (1) The observed mortality deceleration may be at least partially an artifact of mixing different birth cohorts with different mortality (heterogeneity effect); (2) standard assumptions of hazard rate estimates may be invalid when risk of death is extremely high at old ages and (3) ages of very old people may be exaggerated. One way of obtaining estimates of mortality at extreme ages is to pool together international records of persons surviving to extreme ages with subsequent efforts of strict age validation. This approach helps researchers to resolve the third of the above-mentioned problems but does not resolve the first two problems because of inevitable data heterogeneity when data for people belonging to different birth cohorts and countries are pooled together. In this paper we propose an alternative approach, which gives an opportunity to resolve the first two problems by compiling data for more homogeneous single-year birth cohorts with hazard rates measured at narrow (monthly) age intervals. Possible ways of resolving the third problem of hazard rate estimation are elaborated. This approach is based on data from the Social Security Administration Death Master File (DMF). Some birth cohorts covered by DMF could be studied by the method of extinct generations. Availability of month of birth and month of death information provides a unique opportunity to obtain hazard rate estimates for every month of age. Study of several single-year extinct birth cohorts shows that mortality trajectory at advanced ages follows the Gompertz law up to the ages 102–105 years without a noticeable deceleration. Earlier reports of mortality deceleration (deviation of mortality from the Gompertz law) at ages below 100 appear to be artifacts of mixing together several birth cohorts with different mortality levels and using cross-sectional instead of cohort data. Age exaggeration and crude assumptions applied to mortality estimates at advanced ages may also contribute to mortality underestimation at very advanced ages.