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Wealthy People and Fat Tails: An Explanation for the Lévy Distribution of Stock Returns

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eScholarship - University of California
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Abstract

Although the Levy (stable-Paretian) distribution of stock returns was first observed by Mandelbrot 35 years ago, an explanation for this phenomenon has not yet been found. Several extensive studies have recently shown that short-term rates of return on stock indices and on single stocks are distributed according to the truncated Levy distribution. An apparently unrelated but well-documented fact is that wealth is distributed according to the Pareto-law distribution at high wealth levels. In this paper we suggest that the Levy distribution of rates of return originates from the Pareto-law wealth distribution among investors. We present a model which assumes i) a Pareto distribution of wealth and ii) that the effect an investor has on the price of a stock is proportional (in a ctochastic sense) to the investor’s wealth. This model generates a truncated Levy distribution of stock returns. This result is robust to many variations of the basic model. The model leads to the prediction that the parameter Al of the Levy rate of return distribution should be equal to the Pareto constant Aw of the Pareto wealth distribution. Empirical evidence from the U.S., the U.K. and France reveals a striking agreement between these a-priori unrelated parameters (U.S.: Al=1.39, Aw=1.35; U.K. Al=1.12, ax=1.06; France: al=1.82, Aw=1.83).

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