Annuity Contracts Research Articles

A generalization of the Kaplan–Meier estimator for analyzing bivariate mortality under right-censoring and left-truncation with applications in model-checking for survival copula models

In this paper we provide a new nonparametric estimator of the joint distribution of two lifetimes under random right censoring and left truncation which can be seen as a bivariate extension of the Kaplan–Meier estimator. We derive asymptotic results for this estimator, including uniform n1/2-consistency, and develop a general methodology for bivariate lifetime modeling, a critical issue in studying reversion conditions that are commonplace in defined benefit pensions and private annuity contracts. An application to goodness-of-fit for survival copula models is discussed. We show that the procedures that we use are consistent, and propose a bootstrap procedure based on our estimator to compute the critical values. The new technique that we propose is tested on the Canadian dataset initially studied by Frees et al. (1996).

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 rate 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 the analytic approximation solutions to the fair value of the GMWB under both equity and interest rate risks, obtaining both the lower and upper bounds on the price functions. The pricing behavior of the embedded GMWB under various model parameter values is also examined.

Stochastic control of annuity contracts under model misspecification

This article analyzes dynamic asset allocation for a defined-contribution pension fund whose benefits are paid in the form of annuities when a pension fund manager takes model misspecification into consideration. In this paper, model misspecification is determined from the level of ambiguity in asset returns, which follows Uppal and Wang’s (2003) framework. Optimal pre-retirement and post-retirement strategies under model misspecification will be used to illustrate the implications of asset and liability management constraint. We conclude that lower levels of ambiguity strengthen the “lifestyle strategy” effect based on our numecal applications. Numerical results demonstrate that when the overall level of ambiguity for an asset return disappears, our model will reduce to the model of Devolder et al., (2003).

Pricing European Options on Deferred Insurance Contracts

This paper considers the pricing of European call options written on pure endowment and deferred life annuity contracts, also known as guaranteed annuity options. These contracts provide a guaranteed value at the maturity of the option. The contract valuation is dependent on the stochastic interest rate and mortality processes. We assume single-factor stochastic square-root processes for both the interest rate and mortality intensity, with mortality being a time-inhomogeneous process. We then derive the pricing partial differential equation (PDE) and the corresponding transition density PDE for options written on the pure endowment and deferred annuity contracts. The general solution of the pricing PDE is derived as a function of the transition density function. We solve the transition density PDE by first transforming it to a system of characteristic PDEs using Laplace transform techniques and then applying the method of characteristics. Once an explicit expression for the density function is found, we then use sparse grid quadrature techniques to generate European call option prices on the pure endowment and deferred annuity contracts. This approach can easily be generalised to other contracts which are driven by similar stochastic processes presented in this paper. We test the sensitivity of the option prices by varying independent parameters in our model. As option maturity increases, the corresponding option prices significantly increase. The effect of mispricing the guaranteed annuity value is analysed, as is the benefit of replacing the whole-life annuity with a term annuity to remove volatility of the old age population.

Welfare Gains from Illiquid Annuities

In this article, we challenge the common thought that a liquid annuity contract is optimal. We indeed show, in a standard neoclassical framework, that another contract, which actually resembles much more to the contracts that are proposed by most insurance companies, may be preferred by rational individuals. According to this contract, the annuities are an illiquid asset and the premium is age independent. In an overlapping-generation economy, we show that liquid annuities are preferred only if the equilibrium is dynamically inefficient. Alternatively, an equilibrium displaying a positive demand for illiquid annuities is efficient. We conclude by showing how to implement an illiquid annuities market in an efficient economy.

Applications of Forward Mortality Factor Models in Life Insurance Practice

Two of the most important challenges for the application of stochastic mortality models in life insurance practice are their complexity and their apparent incompatibility with classical life contingencies theory, which provides the backbone of insurers’ Electronic Data Processing systems. Forward Mortality Factor Models comprise one model class that overcomes these challenges. Relying on a simple model version that originates from a semi-parametric estimation based on British population mortality data, this paper demonstrates the merits of this model class by discussing several practically important example applications. In particular, we calculate the Economic Capital for a stylised life insurer, we present a closed-form solution for the value of a Guaranteed Annuity Option, and we derive the fair option fee for a Guaranteed Minimum Income Benefit within a Variable Annuity contract. Our numerical results illustrate the economic significance of systematic mortality risk.

Variable Lifetime Annuities: Can You Live Long Enough To Receive Fair Value?

<span>This article presents an analysis of variable lifetime annuities and quantifies the advantages and disadvantages associated with this type of instrument. Given recent long-term rates of return and current low inflation rates, variable annuity contracts provide an effective means of compensating for inflation. An individual only needs to invest a small portion of retirement funds in variable annuities to protest the entire portfolio against the risk of long-term inflation without the risk of having ones entire retirement income based on variable annuities.</span>

The Poisson Log-Bilinear Lee-Carter Model

Life insurance companies deal with two fundamental types of risks when issuing annuity contracts: financial risk and demographic risk. Recent work on the latter has focused on modeling the trend in mortality as a stochastic process. A popular method for modeling death rates is the Lee-Carter model. This methodology has become widely used, and various extensions and modifications have been proposed to obtain a broader interpretation and to capture the main features of the dynamics of mortality rates. In order to improve the measurement of uncertainty in survival probability estimates, in particular for older ages, the paper proposes an extension based on simulation procedures and on the bootstrap methodology. It aims to obtain more reliable and accurate mortality projections, based on the idea of obtaining an acceptable accuracy of the estimate by means of variance reducing techniques. In this way the forecasting procedure becomes more efficient. The longevity question constitutes a critical element in the solvency appraisal of pension annuities. The demographic models used for the cash flow distributions in a portfolio impact on the mathematical reserve and surplus calculations and affect the risk management choices for a pension plan. The paper extends the investigation of the impact of survival uncertainty for life annuity portfolios and for a guaranteed annuity option in the case where interest rates are stochastic. In a framework in which insurance companies need to use internal models for risk management purposes and for determining their solvency capital requirement, the authors consider the surplus value, calculated as the ratio between the market value of the projected assets to that of the liabilities, as a meaningful measure of the company’s financial position, expressing the degree to which the liabilities are covered by the assets.

A utility-based comparison of pension funds and life insurance companies under regulatory constraints

This paper compares two different types of annuity providers, i.e. defined benefit pension funds and life insurance companies. One of the key differences is that the residual risk in pension funds is collectively borne by the beneficiaries and the sponsor’s shareholders while in the case of life insurers it is borne by the external shareholders. First, this paper employs a contingent claim approach to evaluate the risk return tradeoff for annuitants. For that, we take into account the differences in contract specifications and in regulatory regimes. Second, a welfare analysis is conducted to examine whether a consumer with power utility experiences utility gains if she chooses a defined benefit plan or a life annuity contract over a defined contribution plan. We demonstrate that regulation can be designed to support a level playing field amongst different financial institutions.

Longevity-Indexed Life Annuities

This paper addresses the problem of the sharing of longevity risk between an annuity provider and a group of annuitants. An appropriate longevity index is designed in order to adapt the amount of the periodic payments in life annuity contracts. This accounts for unexpected longevity improvements experienced by a given reference population. The approach described in the present paper is in contrast with group self-annuitization, where annuitants bear their own risk. Here the annuitants bear only the nondiversifiable risk that the future mortality trend departs from that of the reference forecast. In that respect, the life annuities discussed in this paper are substitutes for reinsurance and securitization of longevity risk.

The Valuation of Guaranteed Lifelong Withdrawal Benefit Options in Variable Annuity Contracts and the Impact of Mortality Risk

In light of the growing importance of the variable annuities market, in this paper we introduce a theoretical model for the pricing and valuation of guaranteed lifelong withdrawal benefit (GLWB) options embedded in variable annuity products. As the name suggests, this option offers a lifelong withdrawal guarantee; therefore, there is no limit on the total amount that is withdrawn over the term of the policy because if the account value becomes zero while the insured is still alive, he or she continues to receive the guaranteed amount annually until death. Any remaining account value at the time of death is paid to the beneficiary as a death benefit. We offer a specific framework to value the GLWB option in a market-consistent manner under the hypothesis of a static withdrawal strategy, according to which the withdrawal amount is always equal to the guaranteed amount. The valuation approach is based on the decomposition of the product into living and death benefits. The model makes use of the standard no-arbitrage models of mathematical finance, which extend the Black-Scholes framework to insurance contracts, assuming the fund follows a geometric Brownian motion and the insurance fee is paid, on an ongoing basis, as a proportion of the assets. We develop a sensitivity analysis, which shows how the value of the product varies with the key parameters, including the age of the policyholder at the inception of the contract, the guaranteed rate, the risk-free rate, and the fund volatility. We calculate the fair fee, using Monte Carlo simulations under different scenarios. We give special attention to the impact of mortality risk on the value of the option, using a flexible model of mortality dynamics, which allows for the possible perturbations by mortality shock of the standard mortality tables used by practitioners. Moreover, we evaluate the introduction of roll-up and step-up options and the effect of the decision to delay withdrawing. Empirical analyses are performed, and numerical results are provided.

An application of comonotonicity theory in a stochastic life annuity framework

A life annuity contract is an insurance instrument which pays pre-scheduled living benefits conditional on the survival of the annuitant. In order to manage the risk borne by annuity providers, one needs to take into account all sources of uncertainty that affect the value of future obligations under the contract. In this paper, we define the concept of annuity rate as the conditional expected present value random variable of future payments of the annuity, given the future dynamics of its risk factors. The annuity rate deals with the non-diversifiable systematic risk contained in the life annuity contract, and it involves mortality risk as well as investment risk. While it is plausible to assume that there is no correlation between the two risks, each affects the annuity rate through a combination of dependent random variables. In order to understand the probabilistic profile of the annuity rate, we apply comonotonicity theory to approximate its quantile function. We also derive accurate upper and lower bounds for prediction intervals for annuity rates. We use the Lee–Carter model for mortality risk and the Vasicek model for the term structure of interest rates with an annually renewable fixed-income investment policy. Different investment strategies can be handled using this framework.

Market Discipline in the Individual Annuity Market

AbstractTheoretical expectations related to market discipline generally suggest a positive relationship between firm financial strength and price. We examine market discipline in the individual annuity market by measuring annuity contract yields during the accumulation phase and find that, among other results, firm financial strength is positively related to yield (i.e., negatively related to price). We argue that this apparent anomaly can be viewed as a form of market discipline itself, for at least four related reasons, the foremost reason being that in order to compete in the asset accumulation market, an insurer has an incentive to provide a track record of historically strong credited interest rates within the annuity. In addition, the credited interest rates within an annuity are only revealed ex post over time, thus diminishing consumer ability to impose traditional market discipline relating firm financial strength and price, and also enabling financially weaker insurers to impose higher ex post prices in the form of lower realized annuity yields.

Level Dependent Annuities: Defaults of Multiple Degrees

Motivated by the risk of stopped debt coupon payments from a leveraged company in iB01nancial distress, we value a level dependent annuity contract where the annuity rate depends on the value of an underlying asset-process. The range of possible values of the asset is divided into a iB01nite number of regions. The annuity rate is constant within each region. Such annuities are common in models of debt with credit risk in iB01nancial economics. Suspension of debt service under the US Chapter 11 provisions is one well-known real-world example. We present closed-form formulas for the market value of multi-level annuities contracts when the market value of the underlying asset is assumed to follow a geometric Brownian motion.

Level-Dependent Annuities: Defaults of Multiple Degrees

AbstractMotivated by the effect on valuation of stopped or reduced debt coupon payments from a company in financial distress, we value a level-dependent annuity contract where the annuity rate depends on the value of an underlying asset process. The range of possible values of this asset is divided into a finite number of regions, with constant annuity rates within each region. We present closed-form formulas for the market value of level-dependent annuities contracts when the market value of the underlying asset is assumed to follow a geometric Brownian motion. Such annuities occur naturally in models of debt with credit risk in financial economics. Our results are applied for valuing both corporate debt with suspended interest payments under the U.S. Chapter 11 provisions and loans with contractual level-dependent interest rates.

Efficient Pricing of Ratchet Equity-Indexed Annuities in a Variance-Gamma Economy

In this paper we propose a new method for approximating the price of arithmetic Asian options in a Variance-Gamma (VG) economy, which is then applied to the problem of pricing equityindexed annuity contracts. The proposed procedure is an extension to the case of a VG-based model of the moment-matching method developed by Turnbull and Wakeman and Levy for the pricing of this class of path-dependent options in the traditional Black-Scholes setting. The accuracy of the approximation is analyzed against RQMC estimates for the case of ratchet equityindexed annuities with index averaging.

Valuation of equity-indexed annuity under stochastic mortality and interest rate

An equity-indexed annuity (EIA) contract offers a proportional participation in the return on a specified equity index, in addition to a guaranteed return on the single premium. In this paper, we discuss the valuation of equity-indexed annuities under stochastic mortality and interest rate which are assumed to be dependent on each other. Employing the method of change of measure, we present the pricing formulas in closed form for the most common product designs: the point-to-point and the annual reset. Finally, we conduct several numerical experiments, in which we analyze the relationship between some parameters and the pricing of EIAs.

An extended CEV model and the Legendre transform–dual–asymptotic solutions for annuity contracts

This paper develops an extended constant elasticity of variance (E-CEV) model to overcome the shortcomings of the general CEV model. Under the E-CEV model, we study the optimal investment strategy before and after retirement in a defined contribution pension plan where benefits are paid by annuity. By applying the Legendre transform, dual theory and an asymptotic expansion approach, we respectively derive two asymptotic strategies for a CRRA and CARA utility functions in two different periods. Furthermore, we find that each asymptotic strategy can be decomposed into an optimal zero-order strategy and a perturbation strategy. The optimal zero-order strategy denotes an investment strategy where the current volatility is just equal to the mean level of the volatility, whereas the perturbation strategy provides an approximation solution to hedge the slow varying nature of the current volatility deviating from mean level. Finally, we find that the optimal zero-order strategy under given conditions will reduce to the results of Devolder et al. (2003), Xiao et al. (2007) and Gao (2009), respectively.

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.

Optimal Mandates and The Welfare Cost of Asymmetric Information: Evidence from The U.K. Annuity Market.

Much of the extensive empirical literature on insurance markets has focused on whether adverse selection can be detected. Once detected, however, there has been little attempt to quantify its welfare cost, or to assess whether and what potential government interventions may reduce these costs. To do so, we develop a model of annuity contract choice and estimate it using data from the U.K. annuity market. The model allows for private information about mortality risk as well as heterogeneity in preferences over different contract options. We focus on the choice of length of guarantee among individuals who are required to buy annuities. The results suggest that asymmetric information along the guarantee margin reduces welfare relative to a first best symmetric information benchmark by about £127 million per year, or about 2 percent of annuitized wealth. We also find that by requiring that individuals choose the longest guarantee period allowed, mandates could achieve the first-best allocation. However, we estimate that other mandated guarantee lengths would have detrimental effects on welfare. Since determining the optimal mandate is empirically difficult, our findings suggest that achieving welfare gains through mandatory social insurance may be harder in practice than simple theory may suggest.