The efficient index hypothesis and its implications in the BSM model

TL;DR

This paper explores the behavior of a tradable index under the BSM model, showing that efficiency constraints imply its growth rate is close to r + σ², offering insights into the equity premium puzzle.

Contribution

It demonstrates that the efficiency of a tradable index imposes strict growth rate restrictions, providing a simple partial explanation for the equity premium puzzle.

Findings

Index growth rate approximates r + σ² under efficiency.

Efficiency constraints limit the index's outperforming strategies.

Provides a simple mathematical perspective on the equity premium puzzle.

Abstract

This note studies the behavior of an index I_t which is assumed to be a tradable security, to satisfy the BSM model dI_t/I_t = \mu dt + \sigma dW_t, and to be efficient in the following sense: we do not expect a prespecified trading strategy whose value is almost surely always nonnegative to outperform the index greatly. The efficiency of the index imposes severe restrictions on its growth rate; in particular, for a long investment horizon we should have \mu\approx r+\sigma^2, where r is the interest rate. This provides another partial solution to the equity premium puzzle. All our mathematical results are extremely simple.