Removing Biases in Computed Returns

Abstract

This paper presents a straightforward method for asymptotically removing the well-known upward bias in observed returns of equally-weighted portfolios. Our method removes all of the bias due to any random transient errors such as bid-ask bounce and allows for the estimation of short horizon returns. We apply our method to the CRSP equally-weighted monthly return indexes for the NYSE, Amex, and NASDAQ and show that the bias is cumulative. In particular, a NASDAQ index (with a base of 100 in 1973) grows to the level of 17,975 by 2006, but nearly half of the increase is due to cumulative bias. We also conduct a simulation in which we simulate true prices and set spreads according to a discrete pricing grid. True prices are then not necessarily at the midpoint of the spread. In the simulation we compare our method to calculating returns based on observed closing quote midpoints and find that the returns from our method are statistically indistinguishable from the (simulated) true returns. While the mid-quote method results in an improvement over using closing transaction prices, it still results in a statistically significant amount of upward bias. We demonstrate that applying our methodology results in a reversal of the relative performance of NASDAQ stocks versus NYSE stocks over a 25 year window.

Keywords: Unbiased market index, Bias in computed returns, Jensen's inequality, Asset pricing, Index construction

JEL Classification: G10, G12, C43