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Generalized concavity of a function in portfolio theory

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Summary

The analysis of portfolio selection and capital market equilibrium problems involving multivariate lognormal distributions is impeded because this distribution is not closed under addition. To overcome this difficulty it is convenient to use the approximation that this sum is lognormal. Simulation and empirical tests and theoretical results lend support to the appropriateness of the approximation. This paper develops the necessary theory to determine whether such approximating functions are concave or at least quasiconcave and thus their analysis can proceed by standard methods. The characterizations, which involve explicit algebraic criteria related to the parameter values of the random variables and utility function, are evaluated using monthly and yearly stock price data.

Zusammenfassung

Die Analyse von Portfolio Selection Problemen sowie von Gleichgewichts-problemen wird beim Vorliegen logarithmisch-normaler Verteilungen dadurch erschwert, daß diese Klasse von Verteilungen nicht abgeschlossen gegenüber der Addition ist. Das Ersatzmodell, das man dadurch erhält, daß die Summe als logarithmisch-normalverteilt unterstellt wird, erweist sich jedoch als gute Approximation des ursprünglichen Problems, wie empirische sowie theoretische Ergebnisse zeigen. In der vorliegenden Arbeit wird untersucht, unter welchen Bedingungen die approximierende Zielfunktion konkav oder wenigstens quasikonkav ist, und daher das Ersatzproblem mit einem Standardverfahren der konkaven Programmierung gelöst werden kann. Es werden notwendige und hinreichende Bedingungen für die Quasikonkavität der approximierenden Funktion für verschiedene Parameterwerte der Verteilung und der Nutzenfunktion abgeleitet. Die Verwendung der erhaltenen Kriterien, die leicht nachzuprüfen sind, wird am Beispiel monatlicher und jährlicher Daten illustriert.

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Authors and Affiliations

  1. University of Alberta, Edmonton, Alberta, Canada

    S. Schaible

  2. University of British Columbia, Vancouver, British Columbia, Canada

    W. T. Ziemba

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  1. S. Schaible
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  2. W. T. Ziemba
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Schaible, S., Ziemba, W.T. Generalized concavity of a function in portfolio theory. Zeitschrift für Operations Research 29, 161–186 (1985). https://doi.org/10.1007/BF01920307

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  • DOI: https://doi.org/10.1007/BF01920307

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