Dynamics of a Well-Diversified Equity Index

Abstract

The paper derives a parsimonious model for the long-term dynamics of a well-diversified stock index, the S&P500. The index is modeled as growth optimal portfolio. Its normalized value evolves, in some market time, as a square root process. The derivative of market time is a linear function of the squared derivative of a smoothed proxy of the single driving Brownian motion. The model explains the feedback effects from index moves typically observed for monthly and daily S&P500 values. It is highly tractable, permits almost exact simulation and leads beyond classical assumptions in fi ance.

Keywords: long-term index model, growth optimal portfolio, square root process, market time, leverage effect puzzle, benchmark approach

JEL Classification: G10, C10, C15