Optimal consumption and sale strategies for a risk averse agent

TL;DR

This paper analyzes an optimal consumption and sale strategy for a risk-averse agent in a simplified model where the agent can only sell, not buy, a risky asset with transaction costs, leading to new analytical solutions and surprising insights.

Contribution

It introduces novel solution techniques for a simplified asset sale problem, enabling detailed analysis of optimal strategies and their sensitivities.

Findings

Consumption rates are not monotone in asset return.

Certainty equivalents are not monotone in risk aversion.

New analytical solutions for the sale-only risky asset problem.

Abstract

In this article we consider a special case of an optimal consumption/optimal portfolio problem first studied by Constantinides and Magill and by Davis and Norman, in which an agent with constant relative risk aversion seeks to maximise expected discounted utility of consumption over the infinite horizon, in a model comprising a risk-free asset and a risky asset with proportional transaction costs. The special case that we consider is that the cost of purchases of the risky asset is infinite, or equivalently the risky asset can only be sold and not bought. In this special setting new solution techniques are available, and we can make considerable progress towards an analytical solution. This means we are able to consider the comparative statics of the problem. There are some surprising conclusions, such as consumption rates are not monotone increasing in the return of the asset, nor are the certainty equivalent values of the risky positions monotone in the risk aversion.