I. IntroductionLOGNORMALITY OF ASSET RETURNS is a popular assumption in investigations of investor behavior. Empirical observation, however, has disclosed that daily returns are consistently more leptokurtic (are more peaked and have fatter tails) than lognormality indicates. Two responses to these empirical findings have evolved. The first was that the stable paretian' (SP) class of distribution, with a-characteristic <2, is better suited to the description of asset returns [7,9,18,20,23]. More recently the description of asset returns as a subordinated stochastic process (SSP) has developed [4, 6, 19, 21, 22, 24].The use of a daily period in investigations of the nature of the return distribution contrasts sharply with the use of monthly [3], quarterly [5], and annual [4, 15] returns in investigations of investor behavior. Several authors [2, 12, 13, 16] have noted that the nature of the return distribution may change as the period length changes. Section II and the appendix discuss this disparity and point out that the asymptotic tendency under the SSP approach is lognormality. Alternatively, the SP distribution reproduces itself asymptotically. Thus, the observed distributions of asset returns over longer period lengths have relevance for the SP/SSP controversy.The present paper, then has two purposes. The first is a direct empirical investigation of the suitability of the lognormal assumptions for returns to common stocks over monthly, quarterly, and annual periods. The second is to comment on the SP and SSP approaches by comparing the empirical observations to the asymptotic implications. The tests and comparisons are reported in Section III. Our results support the use of the lognormal assumption for the period lengths studied. We also conclude that the asymptotic tendencies of the return distribution are in agreement with the implications of the SSP approach.
II. Asymptotic Implication of SP and SSPThe traditional justification of lognormality is based on a multiplicative version * The authors would like to thank Peter K. Clark for his insightful comments. ' While the description "stable" is often used in the literature, the distribution referred to is actually ln-stable, since the variable investigated is usually the ln of returns (price changes). Unfortunately, the literature often fails to discriminate between the two variables. See, for instance, Officer [20] p. 811, note 13.
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