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Efficient Portfolio Selection for Pareto-Lévy Investments**

Published online by Cambridge University Press:  19 October 2009

Paul A. Samuelson
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Extract

The Markowitz analysis of efficient portfolio selection, which can be interpreted as solving the quadratic-programming problem of minimizing the variance of a normal variate subject to each prescribed mean value, easily can be generalized (in the special case of independently distributed investments) to the concave-programming problem of minimizing the “dispersion” of a stable Pareto-Lévy variate subject to each prescribed mean value. Some further generalizations involving interdependent distributions will also be presented here.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 2 , Issue 2 , June 1967 , pp. 107 - 122
Copyright
Copyright © School of Business Administration, University of Washington 1967

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References

1 Mandelbrot, B., “New Methods in Statistical Economics”, Journal of Political Economy, Vol. 61, (October, 1963), pp. 421–440Google Scholar; “The Variation of Certain Speculative Prices”, Journal of Business, Vol. 36, (October 1963), pp. 394–419Google Scholar; with F. Zarnfaller, “Five Place Tables of Certain Stable Distributions”, Research note, Thomas J. Watson Research Center, Yorktown Heights, N.Y., December 31, 1959; Fama, E. F., “Mandelbrot and the Stable Paretian Hypothesis”, Journal of Business, Vol. 36, (October 1963), pp. 420–429Google Scholar; “The Behavior of Stock-Market Prices”, Journal of Business of the University of Chicago, Vol. 38, No. 1, (01 1965), pp. 34–105Google Scholar; Fama gives a useful bibliography with reference to Levy, P., Calcul des Probabilitiés, (Paris, Gauthier-Villars, 1925Google Scholar), and Gnedenko, B. V. and Kolmogorov, A. N., Limit Distributions for Sums of Independent Random Variables, (Addison-Wesley, 1954).Google Scholar

Actually, the Mandelbrot—Fama data refer to Pareto—LéVy distribution of the logarithms of price changes, not of arithmetic price changes. If logPt+1 − logPt is Pareto-Lévy with 1<α<2, Pt+1 −pt has no finite mean and this would then be true of the investments (Xl,…, Xn), which would not themselves follow a Pareto—Lévy distribution. Similarly, if logPt+1 − logPt follows the normal distribution, Xi will be log—normal not normal and the Markowitz methods cannot be applied without serious modifications. Here I waive these difficulties, with due warning. For some progress in this matter, see Lintner, J., “Valuation of Risk Assets”, Review of Economics and Statistics, Vol. 47, 1965, pp. 13–37Google Scholar, and “Optimum Dividends and Uncertainty”, Quarterly Journal of Economics, Vol. 78, 1964, pp. 49–95Google Scholar and unpublished appendix.

2 Samuelson, P.A., “General Proof That Diversification Pays”, Journal of Financial and Quantitative Analysis, Vol. II, No. 1 (March, 1967), pp. 1–13.CrossRefGoogle Scholar

3 This model turns out to coincide with the cited Sharpe—Fama model if Y represents a business—cycle component common to most stocks.