The efficient index hypothesis and its implications in the BSM model

This article studies the behavior of an index It which is assumed to be a tradable security, to satisfy the BSM model dIt/It = μdt + σdWt, 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 appreciation rate; in particular, for a long investment horizon we should have μ ≈ r + σ2, where r is the interest rate. This provides another partial solution to the equity premium puzzle. All our mathematical results are extremely simple.