Beta bias in low-priced stocks due to trading price rounding

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Virginia Polytechnic Institute and State University

Stocks and similar securities are normally traded in prices which are integral multiples of one-eiqhth of a dollar (a few are traded in one-sixteenths of a dollar). This price constraint may introduce a bias in the estimates of beta for low-priced securities, and the purpose of this dissertation is to examine the bias introduced from this source.

The research methodology briefly consists of constructing a price series for a hypothetical stock by computing"true" prices from an assumed"true" beta and alpha, the series of returns generated from a market index, and a random disturbance term. The constructed price series is rounded to the nearest one-eighth of a dollar and an"observed" beta for this rounded price series is calculated.

The"observed” beta is compared to the"true" beta to observe the degree of bias. Replications are made which differ in their randomly chosen starting point in the market index series; and the experiment is repeated for various"true" betas and alphas within the range of interest, for different intervals between price observations, and for different initial prices.

Chapter I provides an introduction to the study. In Chapter II the relevant literature for this study is reviewed. The first part includes previous studies of the one-eighth effect and the intervalling effect, while the second part of the chapter focuses on the composition and characteristics of common market indexes. The analytical considerations are discussed in Chapter III. The price generating mechanism and the constraint placed upon it by one eighth price rounding are explicitly stated. Alternative rounding procedures are presented and their implications discussed. In the next section the characteristics of the rounding functions are discussed. Finally, expressions for the amount of bias in beta estimates introduced by the one eighth price rounding are derived for both logarithmic returns and holding period (arithmetic) returns.

In Chapter IV the methodology used to secure the results presented in Chapter V is reviewed. The simulation itself is discussed as well as the statistical and ad hoc procedures used to evaluate the results. The results presented in the next chapter also include the results pertinent to two ancillary issues discussed in Chapter IV, namely, how many replications are needed and how reproducible are the results. Chapter VI summarizes the findings, draws a conclusion, and suggests extensions of the study.