The Effect of the Real Option to Transfer on the Value of Guaranteed Minimum Death Benefits

Variable annuity contracts frequently have many options and option‐like features embedded in the contracts. Some are obvious, such as guaranteed minimum death benefits (GMDBs), while others are less obviously option‐like. In this article, we consider the effect of the real option to transfer funds between fixed and variable accounts. If a GMDB rider is considered in isolation, it is sometimes in the policyholder's interest to transfer to the fixed fund if the fixed fund earns less than the variable fund in a risk‐neutral world. On the other hand, the option to transfer will not be used if the entire annuity and rider are considered together.     

Self-Annuitization and Ruin in Retirement

Abstract

At retirement, most individuals face a choice between voluntary annuitization and discretionary management of assets with systematic withdrawals for consumption purposes. Annuitization–buying a life annuity from an insurance company–assures a lifelong consumption stream that cannot be outlived, but it is at the expense of a complete loss of liquidity. On the other hand, discretionary management and consumption from assets–self-annuitization–preserves flexibility but with the distinct risk that a constant standard of living will not be maintainable.

In this paper we compute the lifetime and eventual probability of ruin (PoR) for an individual who wishes to consume a fixed periodic amount–a self-constructed annuity–from an initial endowment invested in a portfolio earning a stochastic (lognormal) rate of return. The lifetime PoR is the probability that net wealth will hit zero prior to a stochastic date of death. The eventual PoR is the probability that net wealth will ever hit zero for an infinitely lived individual.

We demonstrate that the probability of ruin can be represented as the probability that the stochastic present value (SPV) of consumption is greater than the initial investable wealth. The lifetime and eventual probabilities of ruin are then obtained by evaluating one minus the cumulative density function of the SPV at the initial wealth level. In that eventual case, we offer a precise analytical solution because the SPV is known to be a reciprocal gamma distribution. For the lifetime case, using the Gompertz law of mortality, we provide two approximations. Both involve “moment matching” techniques that are motivated by results in Arithmetic Asian option pricing theory. We verify the accuracy of these approximations using Monte Carlo simulations. Finally, a numerical case study is provided using Canadian mortality and capital market parameters. It appears that the lifetime probability of ruin–for a consumption rate that is equal to the life annuity payout–is at its lowest with a well-diversified portfolio.     

Price discovery in the U.S. stock and stock options markets: A portfolio approach

Option prices vary with not only the underlying asset price, but also volatilities and higher moments. In this paper, we use a portfolio of options to seclude the value change of the portfolio from the impact of volatility and higher moments. We apply this portfolio approach to the price discovery analysis in the U.S. stock and stock options markets. We find that the price discovery on the directional movement of the stock price mainly occurs in the stock market, more so now than before as an increasing proportion of options market makers adopt automated quoting algorithms. Nevertheless, the options market becomes more informative during periods of significant options trading activities. The informativeness of the options quotes increases further when the options trading activity generates net sell or buy pressure on the underlying stock price, even more so when the pressure is consistent with deviations between the stock and the options market quotes. JEL Classification C52, G10, G13, G14     

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.     

Asset Allocation and the Liquidity Premium for Illiquid Annuities

Academics and practitioners alike have developed numerous techniques for benchmarking investment returns to properly adjust seemingly high numbers for excessive levels of risk. The same, however, cannot be said for liquidity, or the lack thereof. This article develops a model for analyzing the ex ante liquidity premium demanded by the holder of an illiquid annuity. The annuity is an insurance product that is akin to a pension savings account with both an accumulation and decumulation phase. We compute the yield (spread) needed to compensate for the utility welfare loss, which is induced by the inability to rebalance and maintain an optimal portfolio when holding an annuity. Our analysis goes beyond the current literature, by focusing on the interaction between time horizon (both deterministic and stochastic), risk aversion, and preexisting portfolio holdings. More specifically, we derive a negative relationship between a greater level of individual risk aversion and the demanded liquidity premium. We also confirm that, ceteris paribus, the required liquidity premium is an increasing function of the holding period restriction, the subjective return from the market, and is quite sensitive to the individual's endowed (preexisting) portfolio.     

Bivariate Survival Models for Coupled Lives

This article presents alternative models for modeling the dependence of the time of deaths of coupled lives and applies these to a data set from a life annuity portfolio. The data indicates a very high positive correlation in the time of deaths between coupled lives. The analysis concludes that a mixed frailty copula model with Gompertz marginals fits the data very well. Another model that fits the data well is based on a generalized Frank copula.     

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.     

Bayesian multivariate regime-switching models and the impact of correlation structure misspecification in variable annuity pricing

ABSTRACT

We develop Bayesian multivariate regime-switching models for correlated assets, comparing three different ways to flexibly structure the correlation matrix. After developing the models, we examine their relative characteristics and performance, first in a straightforward asset simulation example, and later applied to a variable annuity product with guarantees. We find that the freedom allowed by the more flexible structures enables these models to more accurately reflect the actual asset dependence structure. We also show that the correlation structures inferred by the most commonly used (and simplest) model will result in significantly larger estimates of the cost of the annuity guarantees.     

The Poisson Log-Bilinear Lee-Carter Model

Abstract

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.     

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.     

Timing and diversification: A state-dependent asset allocation approach

Abstract

The influence of changing economic environment leads the distribution of stock market returns to be time-varying. A conditionally optimal investment hence requires a dynamic adjustment of asset allocation. In this context, this paper examines the improvement in portfolio performance by simulating portfolio strategies that are conditioned on the Markov regime switching behaviour of stock market returns. Including a memory effect eliminates the empirical shortcoming of discrete state models, namely that they produce a standard and an extreme state in stock returns. So far, this has prevented the regimes from being used as a valuable conditioning variable. Based on a discrete state indicator variable, is presented evidence of considerable performance improvement relative to the static model due to optimal shifting between aggressive and well diversified portfolio structures.     

The Impact of Disability Insurance on a Portfolio of Life Insurances

The aim of this article is to study the impact of disability insurance on an insurer's risk situation for a portfolio that also consists of annuity and term life contracts. We provide a model framework using discrete time nonhomogeneous bivariate Markov renewal processes and in a simulation study focus on diversification benefits as well as potential natural hedging effects (risk-minimizing or risk-immunizing portfolio compositions) that may arise within the portfolio because of the different types of biometric risks. Our analyses emphasize that disability insurances are a less efficient tool to hedge shocks to mortality and that their high sensitivity toward shocks to disability risks cannot be easily counterbalanced by other life insurance products. However, the addition of disability insurance can still considerably lower the overall company risk.     

Discussion of the Papers By Robert L. Brown and Thomas Trauth

Abstract

We find the minimum probability of lifetime ruin of an investor who can invest in a market with a risky and a riskless asset and who can purchase a deferred life annuity. Although we let the admissible set of strategies of annuity purchasing process be the set of increasing adapted processes, we find that the individual will not buy a deferred life annuity unless she can cover all her consumption via the annuity and have enough wealth left over to sustain her until the end of the deferral period.     

Hedging Longevity Risk When Interest Rates are Uncertain

Abstract

This paper proposes an asset liability management strategy to hedge the aggregate risk of annuity providers under the assumption that both the interest rate and mortality rate are stochastic. We assume that annuity providers can invest in longevity bonds, long-term coupon bonds, and shortterm zero-coupon bonds to immunize themselves from the risks of the annuity for the equity holders subject to a required profit. We demonstrate that the optimal allocation strategy can lead to the lowest risk under different yield curves and mortality rate assumptions. The longevity bond can also be regarded as an effective hedging vehicle that significantly reduces the aggregate risk of the annuity providers.     

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.     

Variable Annuities with VIX-Linked Fee Structure under a Heston-Type Stochastic Volatility Model

The Chicago Board of Options Exchange (CBOE) advocates linking variable annuity (VA) fees to its trademark VIX index in a recent white paper. It claims that the VIX-linked fee structure has several advantages over the traditional fixed percentage fee structure. However, the evidence presented is largely based on nonparametric extrapolation of historical data on market prices. Our work lays out a theoretical basis with a parametric model to analyze the impact of the VIX-linked fee structure and to verify some claims from the CBOE. In a Heston-type stochastic volatility setting, we jointly model the dynamics of an equity index (underlying the value of VA policyholders’ accounts) and the VIX index. In this framework, we price a guaranteed minimum maturity benefit with VIX-linked fees. Through numerical examples, we show that the VIX-linked fee reduces the sensitivity of the insurer's liability to market volatility when compared to a VA with the traditional fixed fee rate.     

Auswirkungen von Enhanced Annuities auf den Bestand eines Versicherers

The value for money of a standard annuity is the higher, the longer the life expectancy of an insured, and therefore it is only acceptable for persons with an above average life expectancy. The discrepancy is intensified by tax regulations that favor lifelong annuity payments opposed to a lump sum. This discrimination of impaired insureds could be prevented if so-called enhanced annuities were offered, i.e. products where the annuity paid is the larger, the lower the person’s life expectancy. The article presents a quantitative comparison of the risk profile of insurance companies offering standard annuity contracts compared to enhanced annuities and an analysis of the impact of adverse selection on a standard insurer. By definition of individual mortality rates a heterogeneous insurance portfolio is specified. Besides we model the individual underwriting of enhanced annuities. A Monte Carlo Simulation provides results to compare the profit/loss situation of a portfolio of traditional annuity products and a portfolio of enhanced annuities with individual underwriting of different quality and to assess the impact of selection effects.     

Markovian Approaches to Joint-Life Mortality

Abstract

Many insurance products provide benefits that are contingent on the combined survival status of two lives. To value such benefits accurately, we require a statistical model for the impact of the survivorship of one life on another. In this paper we first set up two models, one Markov and one semi-Markov, to model the dependence between the lifetimes of a husband and wife. From the models we can measure the extent of three types of dependence: (1) the instantaneous dependence due to a catastrophic event that affect both lives, (2) the short-term impact of spousal death, and (3) the long-term association between lifetimes. Then we apply the models to a set of jointlife and last-survivor annuity data from a large Canadian insurance company. Given the fitted models, we study the impact of dependence on annuity values and examine the potential inaccuracy in pricing if we assume lifetimes are independent. Finally, we compare our Markovian models with two copula models considered in previous research on modeling joint-life mortality.     

Mortality regimes and longevity risk in a life annuity portfolio

This paper explores the presence of changes of trends or jumps in French mortality from 1947 to 2007, and assesses their implications on the longevity risk management of a life annuity portfolio. We accomplish this by extending the Poisson log-bilinear regression developed by Brouhns et al. (2002) with a regime-switching model. Estimation results show that French mortality is characterized by two distinct regimes. One refers to a strong uncertainty state, which corresponds to the longevity conditions observed during the decade following World War II. The second regime is related to the low volatility of longevity improvements observed during the last 30 years. We use these results to analyze the impact of mortality regimes on the longevity risk management of a life annuity portfolio. Simulation results suggest that the changes of trends in the mortality process have some implications for longevity risk management.     

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.