Optimal Investment with Herd Behaviour Using Rational Decision Decomposition
Huisheng Wang, H. Vicky Zhao
TL;DR
This paper extends the classical Merton optimal investment problem by incorporating herd behaviour between a follower and a leading expert. It introduces the average deviation with exponential decay and solves the resulting stochastic control problem using a variational method, yielding an analytical follower strategy that is a convex mix of the two agents' rational decisions. The authors formalize rational decision decomposition and define an investment opinion that evolves by a differential equation, enabling quantitative analysis of how herd strength, initial wealth, excess return, and volatility shape decisions. Numerical experiments on real stock data corroborate the theoretical findings, showing how higher herd influence pulls the follower toward the leader while the decay rate shapes the temporal emphasis of deviations. Overall, the work provides a rigorous framework for understanding and quantifying herd effects in dynamic investment settings with practical implications for guidance mechanisms in financial decision-making.
Abstract
In this paper, we study the optimal investment problem considering the herd behaviour between two agents, including one leading expert and one following agent whose decisions are influenced by those of the leading expert. In the objective functional of the optimal investment problem, we introduce the average deviation term to measure the distance between the two agents' decisions and use the variational method to find its analytical solution. To theoretically analyze the impact of the following agent's herd behaviour on his/her decision, we decompose his/her optimal decision into a convex linear combination of the two agents' rational decisions, which we call the rational decision decomposition. Furthermore, we define the weight function in the rational decision decomposition as the following agent's investment opinion to measure the preference of his/her own rational decision over that of the leading expert. We use the investment opinion to quantitatively analyze the impact of the herd behaviour, the following agent's initial wealth, the excess return, and the volatility of the risky asset on the optimal decision. We validate our analyses through numerical experiments on real stock data. This study is crucial to understanding investors' herd behaviour in decision-making and designing effective mechanisms to guide their decisions.