Moshe A. Milevsky

Mortality derivatives and the option to annuitise

Most US-based insurance companies offer holders of their tax-sheltered savings plans (VAs), the long-term option to annuitise their policy at a pre-determined rate over a pre-specified period of time. Currently, there is approximately one trillion dollars invested in such policies, with guaranteed annuitisation rates, in addition to any guaranteed minimum death benefit. The insurance company has essentially granted the policyholder an option on wo underlying stochastic variables; future interest rates and future mortality rates. Although the (put) option on interest rates is obvious, the (put) option on mortality rates is not. Motivated by this product, this paper attempts to value (options on) mortality-contingent claims, by stochastically modelling the future hazard-plus-interest rate. Heuristically, we treat the underlying life annuity as a defaultable coupon-bearing bond, where the default occurs at the exogenous time of death. From an actuarial perspective, rather than considering the force of mortality (hazard rate) at time t for a person now age x, as a number μx(t), we view it as a random variable forward ratẽμx(t), whose expectation is the force of mortality in the classical sense ( μx(t) = E[μ̃x(t)]). Our main qualitative observation is that both mortality and interest rate risk can be hedged, and the option to annuitise can be priced by locating a replicating portfolio involving insurance, annuities and default-free bonds. We provide both a discrete and continuous-time pricing framework.

Asian options, the sum of lognormals, and the reciprocal gamma distribution

Arithmetic Asian options are difficult to price and hedge as they do not have closed-form analytic solutions. The main theoretical reason for this difficulty is that the payoff depends on the finite sum of correlated lognormal variables, which is not lognormal and for which there is no recognizable probability density function. We use elementary techniques to derive the probability density function of the infinite sum of correlated lognormal random variables and show that it is reciprocal gamma distributed, under suitable parameter restrictions. A random variable is reciprocal gamma distributed if its inverse is gamma distributed. We use this result to approximate the finite sum of correlated lognormal variables and then value arithmetic Asian options using the reciprocal gamma distribution as the state-price density function. We thus obtain a closed-form analytic expression for the value of an arithmetic Asian option, where the cumulative density function of the gamma distribution, G(d) in our formula, plays the exact same role as N(d) in the Black-Scholes/Merton formula. In addition to being theoretically justified and exact in the limit, we compare our method against other algorithms in the literature and show our method is quicker, at least as accurate, and, in our opinion, more intuitive and pedagogically appealing than any previously published result, especially when applied to high yielding currency options.

Real Longevity Insurance with a Deductible: Introduction to Advanced-Life Delayed Annuities

Abstract This paper explores the financial properties of a concept product called an advanced-life delayed annuity (ALDA). The ALDA is a variant of a pure deferred annuity contract that is acquired by installments, adjusted for consumer price inflation, and pays off toward the end of the human life cycle. The ALDA concept is aimed at the growing population of North Americans without access to a traditional defined benefit (DB) pension plan and the implicit longevity insurance that a DB plan contains. I show that under quite reasonable pricing assumptions, a consumer can invest or allocate

Self-Annuitization and Ruin in Retirement

1 per month, while saving for retirement, and receive between

A CLOSED-FORM APPROXIMATION FOR VALUING BASKET OPTIONS

20 and

Asset Allocation and Annuity-Purchase Strategies to Minimize the Probability of Financial Ruin

40 per month in benefits, assuming the deductible in this insurance policy is set high enough. The ALDA concept might go a long way in mitigating the psychological barrier to voluntary lump-sum annuitization.

Asset Allocation Via The Conditional First Exit Time or How To Avoid Outliving Your Money

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.

A Sustainable Spending Rate without Simulation

The no-arbitrage valuation .f basket options is complicated by the fact that the sum of lognormal random variables is not lognormal. This problem is shared with arithmetic Asian options as well. Various ad hoc approximation techniques have been proposed, none of them very satisfactory or accurate. In this article we suggest using the rec&rocal gamma distribution as an approximation for the state-price density (SPD) function .f the underlying basket stochastic variable. This, in turn, allows us to obtain a closed-jorm expression for the price of a basket option. The technique, when compared against a simple lognormal approximation, perjorms at its best when the correlation structure of the underlying basket exhibits a spec

Optimal Annuitization Policies

c decaying pattern. As a by-product, we introduce a formal approach for assessing the goodness of _fit of candidate distributions for approximating the SPD. Finally, we present a numerical example in which we apply our formula to value (G-7) index-linked guaranteed investment certificates, which can be decomposed into a zero-coupon bond and a basket option.

Optimal asset allocation in life annuities: a note

In this paper, we derive the optimal investment and annuitization strategies for a retiree whose objective is to minimize the probability of lifetime ruin, namely the probability that a fixed consumption strategy will lead to zero wealth while the individual is still alive. Recent papers in the insurance economics literature have examined utility-maximizing annuitization strategies. Others in the probability, finance, and risk management literature have derived shortfall-minimizing investment and hedging strategies given a limited amount of initial capital. This paper brings the two strands of research together. Our model pre-supposes a retiree who does not currently have sufficient wealth to purchase a life annuity that will yield her exogenously desired fixed consumption level. She seeks the asset allocation and annuitization strategy that will minimize the probability of lifetime ruin. We demonstrate that because of the binary nature of the investors goal, she will not annuitize any of her wealth until she can fully cover her desired consumption with a life annuity. We derive a variational inequality that governs the ruin probability and the optimal strategies, and we demonstrate that the problem can be recast as a related optimal stopping problem which yields a free-boundary problem that is more tractable. We numerically calculate the ruin probability and optimal strategies and examine how they change as we vary the mortality assumption and parameters of the financial model. Moreover, for the special case of exponential future lifetime, we solve the (dual) problem explicitly. As a byproduct of our calculations, we are able to quantify the reduction in lifetime ruin probability that comes from being able to manage the investment portfolio dynamically and purchase annuities.