Dynamic optimal asset allocation with optimal stopping
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We develop a model of optimal consumption, labor and portfolio choice with endogenous retirement for an individual's life-cycle decisions. Explicit solutions for finite horizon are derived both for an individual with power utility and for an individual with log utility. There are two distinct phases in the life-cycle, the first being accumulation phase and the second being retirement phase. The individual simultaneously chooses consumption, labor, portfolio and whether to retire so as to maximize the expected utility. We show that the dynamic budget constraint can be reduced to a static budget constraint. For this static optimization problem involving both stochastic optimal control and optimal stopping, we use the convex duality approach to transform it to a pure optimal stopping problem. The value function can be characterized using early exercise premium representation which depends on the optimal retirement boundary. We show that immediate retirement is optimal when a state variable hits the boundary. We derive the backward recursive equation of the boundary parameterized by a multiplier which itself satisfies a nonlinear equation from the static budget constraint. The optimal wealth and the optimal portfolio are derived and they depend on the retirement boundary and the derivative of this boundary with respect to the multiplier. A numerical algorithm is developed for computation of the solutions. We analyze the properties and the structures of the optimal policies and also prove that retirement is optimal when the financial wealth crosses its boundary of which we derive an explicit form. We study next a model of optimal dividend-contribution, portfolio and liquidation from the viewpoint of a defined benefit pension fund. The sponsor faces a stream of intermediate liability and a terminal liability. The optimization problem stops at the optimal liquidation date rather than continues for the second phase. Preference of the sponsor is defined over net cash flow (dividends or contributions) depending on whether outlay from the fund is higher than or lower than the liability. We analyze the behavior of the optimal policies and identify in the optimal portfolio the hedges against fluctuations in the intermediate liability and in the terminal liability.