Abstract
It is known that the decision to purchase an annuity may be associated to an optimal stopping problem. However, little is known about optimal strategies if the mortality force is a generic function of time and the subjective life expectancy of the investor differs from the objective one adopted by insurance companies to price annuities. In this paper, we address this problem by considering an individual who invests in a fund and has the option to convert the fund’s value into an annuity at any time. We formulate the problem as a real option and perform a detailed probabilistic study of the optimal stopping boundary. Due to the generic time-dependence of the mortality force, our optimal stopping problem requires new solution methods to deal with nonmonotonic optimal boundaries.
Highlights
In an ageing world, an accurate management of retirement wealth is crucial for financial well-being
It is important for working individuals to carefully consider the existing offer of financial and insurance products designed for retirement, beyond the state pension. This offer includes for example occupational pension funds and taxadvantaged retirement accounts (e.g. Individual Retirement Account (US)). Most of these products rely on annuities to turn retirement wealth into guaranteed lifetime retirement income
Life annuities provide a lifelong stream of guaranteed income in exchange for a premium
Summary
An accurate management of retirement wealth is crucial for financial well-being. Assuming a constant force of mortality and CRRA utility, Stabile [18] analytically solves a time-homogeneous optimal stopping problem He proves that if the individual has the same degree of risk aversion before and after the annuitisation, an annuity is purchased either immediately or never (the so-called now-or-never policy). The mortality force is a time-dependent, deterministic function and the individual aims at maximising the market value of future cashflows before and after annuitisation. Due to the generic time-dependence of the mortality force, it is not possible to establish any monotonicity of the mapping t → b(t) It is well known in optimal stopping and free boundary theory that monotonicity of b is the key to a rigorous study of the regularity of the boundary (e.g. continuity) and of the value function (e.g. continuous differentiability).
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