Merton’s portfolio problem: A Monte Carlo study

Abstract

A classical problem in continuous time finance is Merton’s portfolio problem. The problem was formulated and solved in 1969 by economist and Nobel prize winner Robert C. Merton. At each instant of time, a risk averse investor has to decide how much of his wealth to consume, and how much to invest in a portfolio consisting of a safe and a risky asset, in order to maximize expected utility. In this thesis we study the simplified problem with no consumption. The risky asset is assumed to be a stock evolving in a random fashion, and the safe asset is assumed to be an interest earning bank deposit. Under certain assumptions on the dynamics of both the safe and the risky asset, it can be shown that the optimal strategy is to invest a constant fraction of wealth in the risky asset while leaving the rest to be invested in the safe asset. The solution, also known as the Merton fraction, is derived in a continuous time framework where trading and rebalancing of the portfolio occurs instantaneously. By Monte Carlo simulations we investigate the performance of the solution in the more realistic discrete time setting for different rebalancing frequencies. Results derived in the continuous time framework serve as a benchmark for comparison with the results obtained from Monte Carlo simulations. In particular we measure how sub-optimal the discrete time rebalancing strategies are compared to the continuous time rebalancing strategy, in terms of maximizing expected utility of terminal wealth. We also look at different measures of risk and return for the respective strategies. Gradually the original problem is modified to better resemble the real world financial market. In the first modified problem, the safe bank deposit is replaced by a zero coupon bond whose value depends on the short rate driven by a stochastic differential equation. Towards the end, the problem is further modified to include a stochastic volatility model for the stock price process. Can the Merton fraction be generalized to take this into account, and how well does this strategy perform in the highly stochastic environment? The results of our simulations suggest that the new strategy is somewhat robust to small-medium levels of noise in both the interest rate and the volatility model.