Optimal time of annuitization in the decumulation phase of a defined contribution pension scheme

In this paper, we consider the problem of finding the optimal time of annuitization for a retiree of a defined contribution pension scheme having the possibility of choosing her own investment and consumption strategy. We exploit the model introduced by Højgaard - Vigna (2010), who formulate the problem as a combined stochastic control and optimal stopping problem. They select a quadratic loss function that penalizes both the deviance of the running consumption rate from a desired consumption rate and the deviance of the final wealth at the time of annuitization from a desired target. We make extensive numerical investigations to address relevant issues such as optimal annuitization time, size of final annuity upon annuitization, extent of improvement when annuitization is not immediate and comparison between optimal annuitization and immediate annuitization. We find that the optimal annuitization time depends on personal factors such as the retiree's risk aversion and her subjective perception of remaining lifetime. It also depends on the financial market, via the Sharpe ratio of the risky asset. Optimal annuitization should occur a few years after retirement with high risk aversion, low Sharpe ratio and/or short remaining lifetime, and many years after retirement with low risk aversion, high Sharpe ratio and/or long remaining lifetime. Moreover, we show rigorously that with typical values of the model's parameters, a pension system where immediate annuitization is compulsory for all individuals is sub-optimal within this model. We measure the cost of sub-optimality in terms of loss of expected present value of consumption from retirement to death, and we find that the cost of sub-optimality, in relative terms, varies between 6% and 40%, depending on the risk aversion. This result gives an idea about the extent of loss in wealth suffered by a retiree who cannot choose programmed withdrawals, but is obliged to annuitize immediately on retirement all her wealth.