Stochastic annuitization and asset allocation under uncertain lifetime

Abstract

In this thesis, we investigate a pensioner’s gains from access to annuities. We observe the optimal asset allocation and annuitization strategies for a pensioner whose retiring age is 65, with an individual pension wealth at retirement and with a guaranteed income from social security during the retirement period. We also observe with particular personal risk preferences towards risk, with a certain amount to buy more annuities after retirement and with certain limitations on pensioner’s asset allocation and annuitization strategies. The pensioner’s objective is to maximize the utility drawn from consumption during retirement with a Constant Relative Risk Aversion utility function. We develop and solve two main models on stochastic volatility for the pensioner who receives an income after retirement from life annuities and investment performance which are under the Constant Elasticity of Variance model and Heston’s Model. We start with the model proposed by Milevsky and Young in 2007 under the Geometric Brownian Motion model and address using the change variable technique. We extend the model under stochastic volatility and solve it using the combination of Legendre transform, dual theory and change variable technique. By adopting the Legendre transform, dual theory and change variable approaches, the explicit solution for optimal investment, consumption and annuitization strategies is derived for the power utility. We use a numerical example to investigate the influence of life annuities and model parameters on the optimal strategy. The results show that the optimal strategy depends on model parameters and the presence of life annuities in the model affects the pensioner’s decisions regarding optimal investment and annuity income level strategies for a period after retirement under the stochastic market price of risk. Besides, an annuity income plays a role in altering the consumption rates for all levels of risk aversion.