Continuous-time Portfolio Optimization for Absolute Return Funds

TL;DR

This paper develops a continuous-time portfolio optimization model for absolute return funds with constraints, solving the associated HJB equation numerically, and demonstrates the approach with artificial and empirical data, highlighting leverage necessity.

Contribution

It introduces a novel stochastic control framework for absolute return funds with specific constraints and proposes techniques for stable numerical solutions.

Findings

Leverage is necessary to reach target wealth levels.

Numerical stability techniques improve solution robustness.

Empirical data confirms model applicability.

Abstract

This paper investigates a continuous-time portfolio optimization problem with the following features: (i) a no-short selling constraint; (ii) a leverage constraint, that is, an upper limit for the sum of portfolio weights; and (iii) a performance criterion based on the lower mean square error between the investor's wealth and a predetermined target wealth level. Since the target level is defined by a deterministic function independent of market indices, it corresponds to the criterion of absolute return funds. The model is formulated using the stochastic control framework with explicit boundary conditions. The corresponding Hamilton-Jacobi-Bellman equation is solved numerically using the kernel-based collocation method. However, a straightforward implementation does not offer a stable and acceptable investment strategy; thus, some techniques to address this shortcoming are proposed. By applying the proposed methodology, two numerical results are obtained: one uses artificial data, and the other uses empirical data from Japanese organizations. There are two implications from the first result: how to stabilize the numerical solution, and a technique to circumvent the plummeting achievement rate close to the terminal time. The second result implies that leverage is inevitable to achieve the target level in the setting discussed in this paper.