Kristen S. Moore

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

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.

Pricing equity-linked pure endowments via the principle of equivalent utility

Abstract We consider a pure endowment contract whose life contingent payout is linked to the performance of a risky stock or index. Because of the additional mortality risk, the market is incomplete; thus, a fundamental assumption of the Black–Scholes theory is violated. We price this contract via the principle of equivalent utility and demonstrate that, under the assumption of exponential utility, the indifference price solves a nonlinear Black–Scholes equation; the nonlinear term reflects the mortality risk and exponential risk preferences in our model. We discuss qualitative and quantitative properties of the premium, including analytical upper and lower bounds.

Optimal and Simple, Nearly Optimal Rules for Minimizing the Probability Of Financial Ruin in Retirement

Abstract The increasing risk of poverty in retirement has been well documented; it is projected that current and future retirees’ living expenses will significantly exceed their savings and income. In this paper, we consider a retiree who does not have sufficient wealth and income to fund her future expenses, and we seek the asset allocation that minimizes the probability of financial ruin during her lifetime. Building on the work of Young (2004) and Milevsky, Moore, and Young (2006), under general mortality assumptions, we derive a variational inequality that governs the ruin probability and optimal asset allocation. We explore the qualitative properties of the ruin robability and optimal strategy, present a numerical method for their estimation, and examine their sensitivity to changes in model parameters for specific examples. We then present an easy-to-implement allocation rule and demonstrate via simulation that it yields nearly optimal ruin probability, even under discrete portfolio rebalancing.

Optimal Design of a Perpetual Equity-Indexed Annuity

Abstract We find the participation rate, guaranteed death benefit, guaranteed surrender benefit, and initial and maintenance fees that most appeal to a buyer of a perpetual equity-indexed annuity (EIA) from the standpoint of maximizing the buyer’s expected discounted utility of wealth at death, also called bequest, while still allowing the issuer of the EIA to (at least) break even on the basis of the expected discounted value of the issuer’s payout. In calculating the buyer’s expected utility, we use the physical probability faced by the buyer. However, in calculating the expected value of the issuer’s payout, we use a type of risk-neutral probability by assuming that the issuer sells many independent policies. We demonstrate our method with an illustrative numerical example.

Minimizing the Probability of Lifetime Ruin under Random Consumption

Abstract We determine the optimal investment strategy in a financial market for an individual whose random consumption is correlated with the price of a risky asset. Bayraktar and Young consider this problem and show that the minimum probability of lifetime ruin is the unique convex, smooth solution of its corresponding Hamilton-Jacobi-Bellman equation. In this paper we focus on determining the probability of lifetime ruin and the corresponding optimal investment strategy. We obtain approximations for the probability of lifetime ruin for small values of certain parameters and demonstrate numerically that they are reasonable ones. We also obtain numerical results in cases for which those parameters are not small.

Existence and uniqueness theorems for singular anisotropic quasilinear elliptic boundary value problems

On bounded domains Ω ⊂ R2 we consider the anisotropic problems u−auxx + u−buyy = p(x, y) in Ω with a, b > 1 and u = ∞ on ∂Ω and uuxx+uuyy+q(x, y) = 0 in Ω with c, d ≥ 0 and u = 0 on ∂Ω. Moreover, we generalize these boundary value problems to space-dimensions n > 2. Under geometric conditions on Ω and monotonicity assumption on 0 < p, q ∈ Cα(Ω) we prove existence and uniqueness of positive solutions.

Balancing Income and Bequest Goals in a DB/DC Hybrid Pension Plan

Defined benefit (DB) plans can provide guaranteed income for life; however, there is no potential for wealth accumulation. Moreover, most DB plans offer little or no death benefit. But defined contribution (DC) plans offer the potential for wealth accumulation; participants might retire quite comfortably and leave a generous bequest for their heirs. However, since the participant bears all of the investment and longevity risk in a DC plan, she also faces the possibility of outliving her accumulated wealth. In this article, we examine hybrids (combinations) of DB and DC plans. We simulate investment returns and the time of death, and we measure the hybrid plans’ performance relative to income and bequest goals. Through this analysis, we quantify the trade-offs between the income security of a DB plan and the potential for wealth accumulation in a DC plan, addressing the question, “How much income security will I forfeit by focusing more on wealth accumulation?” and vice versa. In addition, we suggest allocations between DB and DC that perform particularly well relative to given metrics.

Minimizing the Probability of Lifetime Ruin When Shocks Might Occur: Perturbation Analysis

We determine the optimal investment strategy to minimize the probability of an individual’s lifetime ruin when the underlying model parameters are subject to a shock. Specifically, we consider two possibilities: (1) changes in the individual’s net consumption and mortality rate and (2) changes in the parameters of the financial market. We assume that these rates might change once at a random time. Changes in an individual’s net consumption and mortality rate occur when the individual experiences an accident or other unexpected life event, while changes in the financial market occur due to shifts in the economy or in the political climate. We apply perturbation analysis to approximate the probability of lifetime ruin and the corresponding optimal investment strategy for small changes in the model parameters and observe numerically that these approximations are reasonable ones, even when the changes are not small.