Simulation of Diversified Portfolios in a Continuous Financial Market

Abstract

The paper analyzes the simulated long-term behavior of well diversified portfolios in continuous financial markets. It focuses on the equi-weighted index and the market portfolio. The paper illustrates that the equally weighted portfolio constitutes a good proxy of the growth optimal portfolio, which maximizes expected logarithmic utility. The multi-asset market models considered include the Black-Scholes model, the Heston model, the ARCH diffusion model, the geometric Ornstein-Uhlenbeck volatility model and a multi-asset version of the minimal market model. All these models are simulated exactly or almost exactly over an extremely long period of time to analyze the long term growth of the respective portfolios. The paper illustrates the robustness of the diversification phenomenon when approximating the growth optimal portfolio by the equi-weighted index. Significant outperformance in the long run of the market capitalization weighted portfolio by the equi-weighted index is documented for different market models. Under the multi-asset minimal market model the equi-weighted index outperforms remarkably the market portfolio. In this case the benchmarked market portfolio is a strict supermartingale, whereas the benchmarked equi-weighted index is a martingale. Equal value weighting overcomes the strict supermartingale property that the benchmarked market portfolio inherits from its strict supermartingale constituents under this model.

Keywords: growth optimal portfolio, diversification theorem, diversified portfolios, market portfolio, equi-weighted index, almost exact simulation, minimal market model