Valuing Equity-Indexed Annuities

Abstract

Equity-indexed annuities have generated a great deal of interest and excitement among both insurers and their customers since they were first introduced to the marketplace in early 1995. Because of the embedded options in these products, the insurers are presented with some challenging mathematical problems when it comes to the pricing and management of equity indexed annuities. This paper explores the pricing aspect of three of the most common product designs: the point-to-point, the cliquet, and the lookback. Based on certain assumptions, we are able to present the pricing formulas in closed form for the three product designs. The method of Esscher transforms is the fundamental tool for pricing such deferred annuities.     

Valuation of Equity-Linked Insurance and Annuity Products with Binomial Models

Abstract

In this paper we develop a valuation method for equity-linked insurance products. We assume that the premium information of term life insurances, pure endowment insurances, and endowment insurances at all maturities is obtainable within a company or from the insurance market. Using a method similar to that of Jarrow and Turnbull (1995), we derive three martingale probability measures associated with these basic insurance products. These measures are agedependent, include an adjustment for the mortality risk, and reproduce the premiums of the respective insurance products. We then extend the martingale measures to include the financial market information using copulas and use them to evaluate equity-linked insurance contracts and equity-indexed annuities in particular. This is different from the traditional approach under which diversification of mortality risk is assumed. A detailed numerical analysis is performed for various existing equity-indexed annuities in the North American market.     

Pricing Annuity Guarantees Under a Regime-Switching Model

Abstract

We consider the pricing problem of equity-linked annuities and variable annuities under a regimeswitching model when the dynamic of the market value of a reference asset is driven by a generalized geometric Brownian motion model with regime switching. In particular, we assume that regime switching over time according to a continuous-time Markov chain with a finite number state space representing economy states. We use the Esscher transform to determine an equivalent martingale measure for fair valuation in the incomplete market setting. The paper is complemented with some numerical examples to highlight the implications of our model on pricing these guarantees.     

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

Abstract

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.     

Factors Affecting Retirement Mortality

Abstract

As many countries consider mandatory individual retirement accounts as their answer to a secure social security system, the question arises as to whether all workers can get true “market value” annuities when they retire. It is clear today that private-sector life annuities are priced assuming that the applicant is healthy—very healthy. Very little underwriting or risk classification now exists in the individual annuity marketplace. However, if a large percentage of the population were looking to annuitize their social security accounts upon retirement, there would be strong pressure for more risk classes in the annuity-pricing structure.

Even without the advent of individual accounts for social security, the authors of this paper feel there may be real market opportunities for more risk classification in the individual annuity market and the offering of “impaired life annuities.” Given that this pressure does or might soon exist, this paper reviews 45 recent research papers that look at factors that affect mortality after retirement. In particular, factors that seem to be important in predicting retirement mortality include age, gender, race and ethnicity, education, income, occupation, marital status, religion, health behaviors, smoking, alcohol, and obesity. for each factor, this paper gives highlights relative to the named factor of the impact expected from that variable as described in the 45 reviewed research papers.

The authors believe there is a wealth of information contained in the summaries that follow, and it is our sincere hope that this paper will cause an increased interest in a more broadly based risk classification structure for individual annuities.

Summaries of the 45 papers can be found at www.soa.org/sections/farm/farm.html.     

Optimal Risk Classification with an Application to Substandard Annuities

Abstract

Substandard annuities pay higher pensions to individuals with impaired health and thus require special underwriting of applicants. Although such risk classification can substantially increase a company's profitability, these products are uncommon except for the well-established U.K. market. In this paper we comprehensively analyze this issue and make several contributions to the literature. First, we describe enhanced, impaired life, and care annuities, and then we discuss the underwriting process and underwriting risk related thereto. Second, we propose a theoretical model to determine the optimal profit-maximizing risk classification system for substandard annuities. Based on the model framework and for given price-demand dependencies, we formally show the effect of classification costs and costs of underwriting risk on profitability for insurers. Risk classes are distinguished by the average mortality of contained insureds, whereby mortality heterogeneity is included by means of a frailty model. Third, we discuss key aspects regarding a practical implementation of our model as well as possible market entry barriers for substandard annuity providers.     

Oversikt av utländska aktuarietidskrifter

Abstract

Under Makehami's law, life annuities are expressed by incomplete gamma functions.     

Pricing and hedging guaranteed minimum withdrawal benefits under a general Lévy framework using the COS method

This paper extends the Fourier-cosine (COS) method to the pricing and hedging of variable annuities embedded with guaranteed minimum withdrawal benefit (GMWB) riders. The COS method facilitates efficient computation of prices and hedge ratios of the GMWB riders when the underlying fund dynamics evolve under the influence of the general class of Lévy processes. Formulae are derived to value the contract at each withdrawal date using a backward recursive dynamic programming algorithm. Numerical comparisons are performed with results presented in Bacinello et al. [Scand. Actuar. J., 2014, 1–20], and Luo and Shevchenko [Int. J. Financ. Eng., 2014, 2, 1–24], to confirm the accuracy of the method. The efficiency of the proposed method is assessed by making comparisons with the approach presented in Bacinello et al. [op. cit.]. We find that the COS method presents highly accurate results with notably fast computational times. The valuation framework forms the basis for GMWB hedging. A local risk minimisation approach to hedging intra-withdrawal date risks is developed. A variety of risk measures are considered for minimisation in the general Lévy framework. While the second moment and variance have been considered in existing literature, we show that the Value-at-Risk (VaR) may also be of interest as a risk measure to minimise risk in variable annuities portfolios.     

Estimating International Adverse Selection in Annuities

Abstract

It is well known that purchasers of annuities have lower mortality than the general population. Less widely known is the quantitative extent of this adverse selection and how it varies across countries. This paper proposes and applies several methods for comparing alternative mortality tables and illustrates their impact on annuity valuation for men and women in the U.S. and the U.K. Our results indicate that the relatively lower mortality among older Americans who purchase annuities is equivalent to using a discount rate that is 50–100 basis points below the U.K. rate for compulsory annuitants or 10–20 basis points lower than the U.K. rate for voluntary annuitants. We then draw on the mortality experience of over half a billion lives to estimate mortality differentials due to varying degrees of adverse selection controlling for country, gender, and an allowance for mortality improvements. Results show that adverse selection associated with the purchase of individual annuities reduces mortality rates by at least 25% in the international context. We also find that the system of mortality tables used to value Japanese annuities is quite distinct from international norms.?     

Author’s Reply: Statistical Independence and Fractional Age Assumptions - Discussion by Cecil Nesbitt; Elias S. W. Shiu; Serena Tiong,January 1997

Abstract

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.     

Equity-Indexed Life Insurance: Pricing and Reserving Using the Principle of Equivalent Utility

Abstract

The author applies the principle of equivalent utility to price and reserve equity-indexed life insurance. Young and Zariphopoulou (2002a, b) extended this principle to price insurance products in a dynamic framework. However, in those papers, the insurance risks were independent of the risky asset in the financial market. By contrast, the death benefit for equity-indexed life insurance is a function of a risky asset; therefore, this paper further extends the principle of equivalent utility. In a second extension, the author applies the principle of equivalent utility to calculate reserves, as introduced by Gerber (1976). In a related paper, Moore and Young (2002) price equity-indexed pure endowments, the building blocks of equity-indexed life annuities.     

Mortality-Indexed Annuities Managing Longevity Risk Via Product Design

Abstract

Longevity risk has become a major challenge for governments, individuals, and annuity providers in most countries. In its aggregate form, the systematic risk of changes to general mortality patterns, it has the potential for causing large cumulative losses for insurers. Since obvious risk management tools, such as (re)insurance or hedging, are less suited for managing an annuity provider’s exposure to this risk, we propose a type of life annuity with benefits contingent on actual mortality experience.

Similar adaptations to conventional product design exist with investment-linked annuities, and a role model for long-term contracts contingent on actual cost experience can be found in German private health insurance. By effectively sharing systematic longevity risk with policyholders, insurers may avoid cumulative losses.

Policyholders also gain in comparison with a comparable conventional annuity product: Using a Monte Carlo simulation, we identify a significant upside potential for policyholders while downside risk is limited.     

Author’s Reply: Principal Applications of Bayesian Methods in Actuarial Science: A Perspective - Discussion by Marjorie A. Rosenberg; M. Mendoza; Francisco José VáZquez-Polo; James C. Hickman; Donald A. Jones; Enrique de Alba; David P. M. Scollnik

Abstract

This paper considers the pricing of equity-indexed annuities (EIAs). Traditionally, the values of the guarantees embedded in these contracts are priced by modeling the underlying index fund while keeping the interest rates constant. The assumption of constant interest rates becomes unrealistic in pricing and hedging the EIAs since the embedded guarantees are often of much longer maturity. To solve this problem, the authors propose an economic model that has the flexibility of modeling the underlying index fund as well as the interest rates. Some popular EIAs are illustrated to assess the implication of the proposed model.     

A Hermite-spline model of post-retirement mortality

ABSTRACT

We present a model for post-retirement mortality where differentials automatically reduce with increasing age, but without the fitted mortality rates for subgroups crossing over. Selection effects are catered for, as are age-modulated time trends and seasonal variation in mortality. Central to the model are Hermite splines, which permit parsimonious modelling of complex risk factors in even modest-sized portfolios. The model is therefore suitable for the stand-alone analysis of experience data for reinsurance, bulk annuities and longevity swaps. We also illustrate the contrast between the statistical significance of a risk factor and its financial significance and discuss reasons why one might include risk factors like season that are not directly financially significant.     

Optimal Annuitization Policies

Abstract

At, or about, the age of retirement, most individuals must decide what additional fraction of their marketable wealth, if any, should be annuitized. Annuitization means purchasing a nonrefundable life annuity from an insurance company, which then guarantees a lifelong consumption stream that cannot be outlived. The decision of whether or not to annuitize additional liquid assets is a difficult one, since it is clearly irreversible and can prove costly in hindsight. Obviously, for a large group of people, the bulk of financial wealth is forcefully annuitized, for example, company pensions and social security. For others, especially as it pertains to personal pension plans, such as 401(k), 403(b), and IRA plans as well as variable annuity contracts, there is much discretion in the matter.

The purpose of this paper is to focus on the question of when and if to annuitize. Specifically, my objective is to provide practical advice aimed at individual retirees and their advisors. My main conclusions are as follows:

? Annuitization of assets provides unique and valuable longevity insurance and should be actively encouraged at higher ages. Standard microeconomic utility-based arguments indicate that consumers would be willing to pay a substantial “loading” in order to gain access to a life annuity.

? The large adverse selection costs associated with life annuities, which range from 10% to 20%, might serve as a strong deterrent to full annuitization.

? Retirees with a (strong) bequest motive might be inclined to self-annuitize during the early stages of retirement. Indeed, it appears that most individuals—faced with expensive annuity products—can effectively “beat” the rate of return from a fixed immediate annuity until age 75?80. I call this strategy consume term and invest the difference.

? Variable immediate annuities (VIAs) combine equity market participation together with longevity insurance. This financial product is currently underutilized (and not available in certain jurisdictions) and can only grow in popularity.     

A Computationally Efficient Algorithm for Estimating the Distribution of Future Annuity Values Under Interest-Rate and Longevity Risks

Abstract

This paper proposes a computationally efficient algorithm for quantifying the impact of interestrate risk and longevity risk on the distribution of annuity values in the distant future. The algorithm simulates the state variables out to the end of the horizon period and then uses a Taylor series approximation to compute approximate annuity values at the end of that period, thereby avoiding a computationally expensive “simulation-within-simulation” problem. Illustrative results suggest that annuity values are likely to rise considerably but are also quite uncertain. These findings have some unpleasant implications both for defined contribution pension plans and for defined benefit plan sponsors considering using annuities to hedge their exposure to these risks at some point in the future.     

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.     

Efficient Post-Retirement Asset Allocation

Abstract

To examine post-retirement asset allocation, an extension to the classic Markowitz risk-return framework is suggested. Assuming that retirees make constant (real dollar) annual withdrawals from their portfolios, reward and risk measures are defined to be the mean and standard deviation of wealth remaining at end of life. Asset returns and time of death are both treated as random variables. Assuming constant lifetime asset allocation, the risk and reward measures can be evaluated analytically, and an efficient frontier can be determined. Life annuities can be used to extend the left-hand (low-risk) side of the efficient frontier. The desired level of wealth at end of life can be used to choose a desirable portfolio on the efficient frontier. The desirable portfolio strongly depends on the withdrawal rate. It is suggested (although not proven) that asset allocations strategies that vary with age do not add efficiency in this model, and asset allocation strategies that vary with wealth can add efficiency.     

Private Rentenversicherung, Besteuerung und adverse Selektion

Adverse selection is commonly regarded as an important explanation for the limited size of voluntary annuity markets. Annuitants tend to be longer-lived than the population at large, thus making annuities too expensive for average individuals. Because German tax law discrimates between annuities and other forms of investment, privileged taxation of annuities might compensate for the cost of adverse selection. A major change in the taxation of annuities has been passed recently: From 2005 on, premium payments for deferred life annuities will be tax-deductible and annuities paid out will be fully taxable, if the annuitant cannot opt for an endowment. Premiums for other contracts are not tax-deductible and annuities are partly taxable. Unlike today, endowment payments won’t be tax free, anymore. We evaluate the cost of theoretical, fairly priced contracts for annuitants and non-annuitants using the money’s worth ratio. Therefore, the money’s worth concept is extended to the case of deferred annuities. Our calculations suggest that, under current legislation, average individuals have no incentive to annuitize. In contrast, under new legislation, the tax advantage of the deferred annuity without endowment option more than compensates for the cost of adverse selection. Single premium annuities will remain too expensive for average men, but may become advantageous for average women in some cases.     

Optimal rate classification for enhanced annuities

For a given premium, enhanced annuities pay higher pensions to policyholders with impaired health. Even though risk classification is a common concept in the insurance sector and should allow insurers to increase their profitability, enhanced annuities are rarely offered outside of the United Kingdom. The paper provides a general method of determining an optimal risk classification system for enhanced annuities that will maximize an insurance company’s profits. The cost of risk classification, as well as that incurred when insureds are assigned to inappropriate risk classes (a chief component of underwriting risk), are explicitly considered.