Optimising Income Drawdown in the Dual Space

One attractive objective for a pensioner using the income drawdown option is to minimise the deviation of the pension fund from a prescribed deterministic target. Typically, this problem is formulated as a linear-quadratic optimal control problem, which has the shortcoming that over-performance of the fund is penalised as much as underperformance. If one adopts an asymmetric terminal loss function then it is not clear how to solve the optimisation problem. However, the dual optimisation problem suggests a particular asymmetric utility function, which expresses the pensioner's preference for the annuity rate at compulsory annuitisation. The transformation between the state variable and its dual is quadratic under this utility function, so that conversion between spaces is straightforward. Using this technique, simulation of the optimal controls reveals the effect of asymmetric preferences on the optimal investment and consumption during income drawdown. One feature of the asymmetric objective is that it can be optimal not to make any risky investment at all, but if and when this occurs depends strongly on the target drawdown rate.