Inconsistent Investment and Consumption Problems

Abstract

In a traditional Black-Scholes market we develop a verification theorem for a general class of investment and consumption problems, where the standard dynamic programming principle does not hold. The theorem is an extension of the standard Hamilton-Jacobi-Bellman equation in the form of a system of non-linear differential equations. We derive the optimal investment and consumption strategy for a mean-variance investor. In the case of constant risk aversion it turns out, as it does for a portfolio investor (without consumption), that the optimal amount of money to invest in stocks is independent of wealth. The optimal consumption strategy is given as a deterministic bang-bang strategy. In order to have a more realistic model we allow the risk aversion to depend dynamically on current wealth and time. Using the verification theorem we give a detailed analysis of the problem. It turns out that the optimal amount of money to invest in stocks is given by an affine function of wealth and the optimal consumption strategy is again given as a deterministic bang-bang strategy. We also calculate the optimal investment and consumption strategy for a mean-standard deviation investor. It turns out that it is optimal to take no risk at all.