Annuity Pricing Research Articles

Evolution of Coupled Lives' Dependency Across Generations and Pricing Impact

This paper studies the dependence between coupled lives - both within and across generations - and its effects on prices of reversionary annuities in the presence of longevity risk. Longevity risk is represented via a stochastic mortality intensity. Dependence is modelled through copula functions. We consider Archimedean single and multi-parameter copulas. We find that dependence decreases when passing from older generations to younger generations. Not only the level of dependence but also its features - as measured by the copula - change across generations: the best-fit Archimedean copula is not the same across generations. Moreover, for all the generations under exam the single-parameter copula is dominated by the two-parameter one. The independence assumption produces quantifiable mispricing of reversionary annuities. The misspecification of the copula produces different mispricing effects on different generations. The research is conducted using a well-known dataset of double life contracts.

How do Retirees Value Life Annuities? Evidence from Public Employees

Economists have long been puzzled by the low demand for life annuities. To shed new light on this puzzle, we study payout choices in the Oregon Public Employees Retirement System, where each retiree must choose between a lump sum and a life annuity. Notably, the average life annuity we study is better than actuarially fair when compared to the lump sum and 85% of retirees choose the life annuity. Whether and how retirees respond to variation in the value of life annuity payments depends crucially on the source of variation. We find strong evidence that demand responds to variation in retiree characteristics. In contrast, we find little evidence that demand responds to plausibly exogenous variation in annuity pricing, which is economically meaningful but less salient. Finally, we find robust evidence that demand for the lump sum increases with recent equity market returns and other salient measures of investor sentiment.

On Mean-Variance Selection of Payment Size and Price for Annuities

The paper studies problem of pricing of annuities under uncertainty in the future movements of market described via a diffusion stochastic model. In the the spirit of mean-variance pricing, the fair price of annuity and the size of payments are defined such that the risk for the annuitant and for annuity seller is minimal. Two versions of this problem are considered separately: the selection of the minimum risk annuity price given the size of the regular payments, and the selection of the minimum risk size of the regular payments given the amount invested into the annuity. The corresponding pricing formulas are obtained explicitly. It appears that these two problems have different solution.

Using fuzzy random variables in life annuities pricing

This paper develops life annuity pricing with stochastic representation of mortality and fuzzy quantification of interest rates. We show that modelling the present value of annuities with fuzzy random variables allows quantifying their expected price and risk resulting from the uncertainty sources considered. So, we firstly describe fuzzy random variables and define some associated measures: the mathematical expectation, the variance, distribution function and quantiles. Secondly, we show several ways to estimate the discount rates to price annuities. Subsequently, the present value of life annuities is modelled with fuzzy random variables. We finally show how an actuary can quantify the price and the risk of a portfolio of annuities when their present value is given by means of fuzzy random variables.

On the pricing of single premium variable annuities with periodic fees and periodic cost of insurance using option pricing techniques

The pricing of a series of products that combine insurance with investments, known as variable annuities, is considered. Given that there is a single premium instalment, then the death benefit at the time of death is equal to the maximum between the fund value and the sum assured. We have discussed in the past the problem in the case that there is a single premium instalment and the cost of insurance is collected at the beginning. We now move to examine the case that management fees need to be paid periodically via the cancellation of units and study the calculation of the charge that needs to be made by the insurer. We do not use standard actuarial techniques, but rather realise that the risk borne by the insurer resembles to the payoff of an option. We attempt to follow option valuation techniques in discrete time to find the insurance premium.

Pensions and genetics: can longevity genes be reliable risk factors for annuity pricing?

We consider a number of gene variants that have been found to affect longevity. Their effects have been modelled using Cox or logistic regressions, whose fitted parameters have simple asymptotic sampling distributions. The expected present value of a life annuity allowing for these genetic risk estimates inherits a sampling distribution, which can be found by simulation. We find that possibly significant uncertainty about annuity premiums may be overlooked if the standard errors of parameters estimated in medical studies are ignored by medical underwriters. Such considerations may play an important part when the acceptability of using a risk factor in underwriting is conditional on proof of its relevance and reliability. This is the current position in respect of genetic information in many countries, most prominently in the UK.

Compulsory and Voluntary Annuities Markets in the UK

This paper describes the operation of both the compulsory pension annuity and voluntary annuity markets in the UK. The paper reports on the movement of UK annuity price quotes in the voluntary market from 1957-2009, and in the pension annuity market from 1994 to 2009, and examines whether annuities were fairly priced over this period. The paper computes the money’s worth of annuities, and finds that on average the money’s worth in the voluntary market over the sample period for 65-year old males has been a very high 98%. In the larger compulsory pension annuity over a shorter sample period we estimate that the money’s worth for 65-year old males has been 89%, and for 65-year old females has been a similar but slightly larger 90%. Taking into account load factors associated with annuity contracts and in comparison with other financial and insurance products this implies that annuities are fairly priced. However the value of the money’s worth is sensitive to the assumptions made about life expectancy, and we explain the assumptions made about the appropriate life tables to apply to annuitants in these annuity markets. There is some evidence that money’s worth has fallen since 2002. We discuss a number of factors that could have effected the fall in money’s worth, including: changes in insurance regulation; changes in industrial concentration; an insurance cycle; pricing of mortality uncertainty and the growth in the impaired lives market.

Modeling longevity risks using a principal component approach: A comparison with existing stochastic mortality models

This research proposes a mortality model with an age shift to project future mortality using principal component analysis (PCA). Comparisons of the proposed PCA model with the well-known models—the Lee–Carter model, the age–period–cohort model ( Renshaw and Haberman, 2006), and the Cairns, Blake, and Dowd model—employ empirical studies of mortality data from six countries, two each from Asia, Europe, and North America. The mortality data come from the human mortality database and span the period 1970–2005. The proposed PCA model produces smaller prediction errors for almost all illustrated countries in its mean absolute percentage error. To demonstrate longevity risk in annuity pricing, we use the proposed PCA model to project future mortality rates and analyze the underestimated ratio of annuity price for whole life annuity and deferred whole life annuity product respectively. The effect of model risk on annuity pricing is also investigated by comparing the results from the proposed PCA model with those from the LC model. The findings can benefit actuaries in their efforts to deal with longevity risk in pricing and valuation.

On a Generalization of the World Bank Model of Retirement Savings: A Taxonomy of Systems with Two Cross-Subsidized Tiers

We develop a generalization of the World Bank (1994) model of forced saving for retirement. This broader model consists of two tiers of second pillar savings --- mandated and non-mandated (voluntary). Furthermore, the government can set two types of guarantees on the first (mandated) tier --- investment returns and annuity prices --- leading to possible cross-subsidization between the tiers. This has the potential to induce social redistribution, foster a liquid private market for life annuities, as well as obviate some of the investment risk and annuity price risk that retirees face. We formulate a quantitative model of financial flows within such a system, which explains the mechanism by which cross-subsidization occurs. Based on this analysis, a taxonomy of two-tiered retirement systems is presented, that is based on the choices that the government makes. Our modelling is inspired by the Swiss second pillar model, which serves as a motivation and illustration. The Swiss system features unique properties and remarkable outcomes. These include both high levels of savings and (unforced) annuitization, as well as a non-trivial level of internal cross-subsidies between the two tiers of the system. The analysis allows us to shed some light on some of the issues the Swiss system is currently facing.

Valuing medieval annuities: Were corrodies underpriced?

Medieval bishops condemned and restricted the sale of corrodies (a type of annuity), partly on the grounds of their perceived unprofitability. The available data on the profitability of corrodies is limited and little analysed, and the episcopal condemnation of corrodies has been adopted by modern researchers. After recognising the difficulties, this paper applies an annuity pricing model to study corrody pricing. Given various assumptions, contrary to the established view, it is argued that the sale of corrodies was financially profitable for institutions. Finally, some reasons are considered for the negative attitude of contemporary and historical opinion towards the sale of corrodies.

Longevity Risk and Annuity Pricing with the Lee-Carter Model

ABSTRACTSeveral important classes of liability are sensitive to the direction of future mortality trends, and this paper presents some recent developments in fitting smooth models to historical mortality-experience data. We demonstrate the impact these models have on mortality projections, and the resulting impact which these projections have on financial products. We base our work round the Lee-Carter family of models. We find that each model fit, while using the same data and staying within the Lee-Carter family, can change the direction of the mortality projections. The main focus of the paper is to demonstrate the impact of these projections on various financial calculations, and we provide a number of ways of quantifying, both graphically and numerically, the model risk in such calculations. We conclude that the impact of our modelling assumptions is financially material. In short, there is a need for awareness of model risk when assessing longevity-related liabilities, especially for annuities and pensions.

Longevity Risk and Annuity Pricing with the Lee–Carter Model. Discussion Held at the Faculty of Actuaries

Longevity Risk and Annuity Pricing with the Lee–Carter Model. Discussion Held at the Faculty of Actuaries

Pricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates

We consider the pricing of variable annuities with the Guaranteed Minimum Withdrawal Benefit (GMWB) under the Vasicek stochastic interest rates framework. The holder of the variable annuity contract pays an initial purchase payment to the insurance company, which is then invested in a portfolio of risky assets. Under the GMWB, the holder can withdraw a specified amount periodically over the term of the contract such that the return of the entire initial investment is guaranteed, regardless of the market performance of the underlying asset portfolio. The investors have the equity participation in the reference investment portfolio with protection on the downside risk.The guarantee is financed by paying annual proportional fees. Under the assumption of deterministic withdrawal rates, we develop the pricing formulation of the value function of a variable annuity with the GMWB. In particular, we derive analytic approximation solutions to the fair value of the GMWB under both equity and interest rate risks, obtaining both the lower and upper bound on the price functions. The pricing behavior of the embedded GMWB under various model parameter values is also examined.

The Distribution of Future Annuity Prices under Interest-Rate and Longevity Risks

This paper examines the impact of interest-rate risk and longevity risk on the distribution of annuity prices in the distant future. To so, the paper uses a computationally efficient algorithm that simulates the state variables out to the end of the horizon period and then uses a Taylor series approximation to compute approximate annuity prices at the end of that period. Illustrative results suggest that annuity prices are likely to rise considerably, but are also quite uncertain. These findings have some unpleasant implications for future pensioners and those who will have to look after them.

Betting on Death and Capital Markets in Retirement: A Shortfall Risk Analysis of Life Annuities Versus Phased Withdrawal Plans

How might retirees consider deploying the retirement assets accumulated in a defined contribution pension plan? One possibility would be to purchase an immediate annuity. Another approach, called the strategy in the literature, would have the retiree invest his funds and then withdraw some portion of the account annually. Using this second tactic, the withdrawal rate might be determined according to a fixed benefit level payable until the retiree dies or the funds run out, or it could be set using a variable formula, where the retiree withdraws funds according to a rule linked to life expectancy. Using a range of data consistent with the German experience, we evaluate several alternative designs for phased withdrawal strategies, allowing for endogenous asset allocation patterns, and also allowing the worker to make decisions both about when to retire and when to switch to an annuity. We show that one particular phased withdrawal rule is appealing since it offers relatively low expected shortfall risk, good expected payouts for the retiree during his life, and some bequest potential for the heirs. We also find that unisex mortality tables if used for annuity pricing can make women's expected shortfalls higher, expected benefits higher, and bequests lower under a phased withdrawal program. Finally, we show that delayed annuitization can be appealing since it provides higher expected benefits with lower expected shortfalls, at the cost of somewhat lower anticipated bequests.

Valuing Medieval Annuities: Were Corrodies Underpriced?

Medieval bishops condemned and restricted the sale of corrodies (a type of annuity), partly on the grounds of their perceived unprofitability. The available data on the profitability of corrodies is limited and little analysed, and the episcopal condemnation of corrodies has been adopted by modern researchers. After recognising the difficulties, this paper applies an annuity pricing model to study corrody pricing. Given various assumptions, contrary to the established view, it is shown that the sale of corrodies was profitable for institutions. Finally, some reasons are considered for the negative attitude of medieval bishops towards the sale of corrodies.

Mortality projections and unisex pricing of annuities in the UK

PurposeIn the UK, there is a strong government commitment to the compulsory use of annuities to manage the “decumulation” of assets in defined contribution pension schemes. Almost all annuity rates are determined by reference to the gender of the individual involved. This has the implication that females receive a lower pension for a given size of pension fund. It is arguable that this situation represents a clear case of sex discrimination and moral, legal‐ and policy‐based arguments can be made for and against this view. The purpose of this paper is to review these arguments in the light of emerging evidence about longevity.Design/methodology/approachThe paper outlines the nature of the UK annuity market and the associated methods of annuity pricing, details the difficulties of predicting longevity and discusses the economic implications of a move to unisex annuity rates.FindingsA number of recent trends are weakening the financial and statistical arguments against introducing unisex annuity rates. The life expectancy of males and females is converging, the use of annuity pricing factors other than gender is increasingly common and it has become clear that there is great uncertainty in mortality projections.Practical implicationsStatistical and financial arguments that gender should be a primary factor for costing annuities should be accorded less weight than in the past.Originality/valueThe paper offers an evaluation of the merits of unisex annuity rates in the light of recent evidence about longevity.

Annuities and Aggregate Mortality Uncertainty

This paper explores the effect of aggregate mortality risk on the pricing of annuities. It uses a two-period model; in the second period people face a constant but initially unknown risk of death. Old people can either carry the aggregate emortality risk for themselves or buy annuities which are sold by young people. A market-clearing price for such annuities is established. It is found that old people would, given the choice, decide to carry a considerable part of aggregate mortality risk for themselves.

Indifference pricing of pure endowments and life annuities under stochastic hazard and interest rates

We study indifference pricing of mortality contingent claims in a fully stochastic model. We assume both stochastic interest rates and stochastic hazard rates governing the population mortality. In this setting we compute the indifference price charged by an insurer that uses exponential utility and sells k contingent claims to k independent but homogeneous individuals. Throughout we focus on the examples of pure endowments and temporary life annuities. We begin with a continuous-time model where we derive the linear pdes satisfied by the indifference prices and carry out extensive comparative statics. In particular, we show that the price-per-risk grows as more contracts are sold. We then also provide a more flexible discrete-time analog that permits general hazard rate dynamics. In the latter case we construct a simulation-based algorithm for pricing general mortality-contingent claims and illustrate with a numerical example.

Optimum and Risk‐Class Pricing of Annuities

When information on longevity (survival functions) is unknown early in life, individuals have an interest in insuring themselves against moving into different 'risk-classes' as their life expectancy is revealed. The First-Best allocation involves transfers across states of nature. With symmetric information, competitive equilibrium separates different risk classes and cannot provide such transfers because insurance firms are unable to precommit. When utility is invariant to risk-class realisation, the optimum entails uniform consumption and optimum retirement age independent of risk-class and an optimum social security scheme is superior to competitive equilibrium. When preferences depend on risk-class, welfare ranking of systems becomes indeterminate. Much work has been undertaken by economists to understand and rationalise the 'annuity puzzle': the fact that the markets for private voluntarily purchased annuities