A multi-asset investment and consumption problem with transaction costs

TL;DR

This paper analyzes a multi-asset investment and consumption problem with transaction costs, focusing on a simplified case with costs only for a single illiquid asset, and derives the optimal trading strategy and properties of the problem.

Contribution

It introduces a tractable approach to a complex multi-asset problem by reducing it to a boundary value problem for a differential equation, revealing key properties of the solution.

Findings

Optimal trading occurs only when the illiquid asset's portfolio fraction exits a fixed interval.

The problem's well-posedness depends on the behavior of specific algebraic functions.

The HJB equation can be transformed into a boundary value problem for a first-order differential equation.

Abstract

In this article we study a multi-asset version of the Merton investment and consumption problem with proportional transaction costs. In general it is difficult to make analytical progress towards a solution in such problems, but we specialise to a case where transaction costs are zero except for sales and purchases of a single asset which we call the illiquid asset. Assuming agents have CRRA utilities and asset prices follow exponential Brownian motions we show that the underlying HJB equation can be transformed into a boundary value problem for a first order differential equation. The optimal strategy is to trade the illiquid asset only when the fraction of the total portfolio value invested in this asset falls outside a fixed interval. Important properties of the multi-asset problem (including when the problem is well-posed, ill-posed, or well-posed only for large transaction costs) can be inferred from the behaviours of a quadratic function of a single variable and another algebraic function.