Choosing the optimal annuitization time post-retirement

Abstract

In the context of decision making for retirees of a defined contribution pension scheme in the de-cumulation phase, we formulate and solve a problem of finding the optimal time of annuitization for a retiree having the possibility of choosing her own investment and consumption strategy. We formulate the problem as a combined stochastic control and optimal stopping problem. As criterion for the optimization we select a 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 find closed form solutions for the problem and show the existence of three possible types of solutions depending on the free parameters of the problem. In numerical applications we find the optimal wealth that triggers annuitization, compare it with the desired target and investigate its dependence on both parameters of the financial market and parameters linked to the risk attitude of the retiree. Simulations of the behaviour of the risky asset seem to show that under typical situations optimal annuitization should occur a few years after retirement.

Income drawdown option with minimum guarantee

“…Introducing the possibility of choosing the annuitization time would increase the difficulty of the problem substantially. The problem of finding the optimal investment-consumption couple as well as optimal annuitization time has been treated rigorously as a combined stochastic control and optimal stopping problem in [Milevsky, Moore & Young, 2006], [Milevsky & Young, 2007] and [Gerrard, Højgaard & Vigna, 2012]. However, none of these papers treats the case with constraints on the investment strategy and the state variable.…”

“…Therefore, we use a known procedure from portfolio optimization, which allows us to transform the original equation into a nicer looking dual one. This procedure has been used, e.g., in [Elie & Touzi, 2008], [Gao, 2008], [Gerrard, Højgaard & Vigna, 2012], [Milevsky, Moore & Young, 2006], [Milevsky & Young, 2007] and [Xiao , Zhai & Qin, 2007]. In all such papers the dual equation is always linear.…”

“…The three degrees of freedom of the retiree (amount of consumption, investment allocation, and time of annuitization), together with the important issue of ruin possibility, have been investigated in the actuarial and financial literature in many papers. Among others we recall [Albrecht & Maurer, 2002], [Blake, Cairns & Dowd, 2003], [Di Giacinto & Vigna, 2012], [Gerrard, Haberman & Vigna, 2006], [Gerrard, Højgaard & Vigna, 2012] [Milevsky, 2001], [Milevsky, Moore & Young, 2006], [Milevsky & Young, 2007].…”

“…2 In [Gerrard, Haberman & Vigna, 2004], the only control variable is the investment strategy, in [Gerrard, Haberman & Vigna, 2006] the control variables are the investment and the consumption policies, while in [Gerrard, Højgaard & Vigna, 2012] the retiree is allowed to choose the annuitization time, together with the investment-consumption policies.…”

“…We choose the framework of [Gerrard, Haberman & Vigna, 2004], [Gerrard, Haberman & Vigna, 2006], and [Gerrard, Højgaard & Vigna, 2012] taking a quadratic target-based loss function. Indeed our approach could be used also to treat different objective functions.…”