The Returns Generation Process, Returns Variance, and the Effect of Thinness in Securities Markets

Abstract

THIS PAPER USES A DEMAND FORMULATION to link the advent of idiosyncratic tenders and aggregate demand shifts to variations in security prices, and hence to the generation of returns. For this purpose: (a) compound Poisson processes are used to describe the manner in which shifts in the market demand curve to hold a security are generated, and (b) demand elasticity is used to translate idiosyncratic demand shifts into a returns dimension. Our use of compound Poisson processes to model the temporally discrete arrival of idiosyncratic tenders is consistent with Garman (1976). Further, our treatment of the returns generation process allows for upward drift in security prices, and thus is consistent with the submartingale model of efficient capital markets [see, e.g., Fama (1970)]. Our analysis does not focus upon incomplete equilibria, and thus is distinguished from Copeland's (1976) model of time-distributed price-volume adjustments under sequential arrival of information.The model of the returns generation process is used to obtain the expected value and variance of security returns. This is done first for returns based on bid/ask price quotations (which we call "quotation returns") and then for returns based on transaction prices (which we call "transaction returns"). Although we abstract from the bid-ask spread, a separate analysis of transaction returns is needed because demand shifts need not trigger transactions, and hence closing transaction prices, unlike closing quotes, need not reflect all current information.Our interest in the moments of returns distributions is focused primarily on variance,' and on the existence of a relationship between variance and the value of shares outstanding for individual securities. We consider the market value of shares outstanding for a security to be an (inverse) proxy measure of thinness in the , as well as Hans Stoll, a referee for this Journal, for their helpful comments on earlier drafts of this paper.1. Total returns variance rather than market model beta was modeled for two reasons. First, for investors with small portfolios, both beta and residual variance (and hence total variance) are relevant to portfolio selection; Blume, Crockett, and Friend (1974) have shown that most individual investors' portfolios contain relatively few stocks. Second, empirical tests of the capital asset pricing model have raised doubt about the proposition that stock returns are a function of beta alone (and thus have questioned the corollary proposition that beta is the only component of total variance that matters in portfolio selection); see Douglas (1969), Lintner [as cited in Douglas (1969)], Miller and Scholes (1972), and Black, Jensen, and Scholes (1972). 150The Journal of Finance market for that security, with a smaller value indicating a thinner market.2 It will be shown that the generation of idiosyncratic buy and sell tenders can have its greatest impact on thin markets, and that, cet. par., this leads to increased returns variance.There has been relatively little formal analysi...

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