Managing a value-preserving portfolio over time

This paper discusses an intertemporally efficient value-preserving consumption plan for the intertemporal portfolio and consumption problem such that in each period a proportion of the portfolio value at time zero is consumed that equals the risk-adjusted portfolio rate of return in this period. The portfolio value of such a consumption plan remains constant over time and can hence be preserved. Value-preserving consumption plans were introduced by Hellwig. We use a martingale approach in a discrete-time, finite-state-space setting with dynamically incomplete markets and short-sale constraints to show that the value-preserving consumption plan is implemented by some kind of myopic expected log-utility maximization. If, however, leverage constraints (i.e. credit limits on the risk-free borrowing) are introduced the myopic policy does no longer induce value-preserving consumption plans. In this case a characterization of the equivalent (super-, sub-) martingale measure is found as the solution of a system of variational inequalities.