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Geometric Mean Approximations of Individual Security and Portfolio Performance*

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  • Young, William E.
  • Trent, Robert H.

Abstract

The objectives of this paper are to derive the relationship of the geometric mean of a distribution of positive values to the conventional first four moments — arithmetic mean, variance, absolute skewness, and absolute kurtosis — and to empirically evaluate certain approximations involving these four moments for estimating the geometric means of monthly and annual holding period returns for individual stocks and for portfolios. The geometric mean is shown to be positively related to the arithmetic mean and absolute skewness and negatively related to variance and absolute kurtosis. In the case of a normal distribution a very good approximation to the geometric mean is revealed to be a function of just the arithmetic mean and variance. Additionally, empirical evidence indicates that even though a number of the monthly and annual distributions deviate significantly from normality, the approximation involving only the mean and variance produces quite accurate estimates of the geometric means of these distributions.

Suggested Citation

  • Young, William E. & Trent, Robert H., 1969. "Geometric Mean Approximations of Individual Security and Portfolio Performance*," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 4(2), pages 179-199, June.
    • Handle: RePEc:cup:jfinqa:v:4:y:1969:i:02:p:179-199_01
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    Cited by:

    1. Marcus Berliant & Axel H. Watanabe, 2018. "A scale‐free transportation network explains the city‐size distribution," Quantitative Economics, Econometric Society, vol. 9(3), pages 1419-1451, November.
    2. Charles-Cadogan, G., 2018. "Losses loom larger than gains and reference dependent preferences in Bernoulli’s utility function," Journal of Economic Behavior & Organization, Elsevier, vol. 154(C), pages 220-237.
    3. Murad J. Antia & Richard L. Meyer, 1984. "The Growth Optimal Capital Structure: Manager Versus Shareholder Objectives," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 7(3), pages 259-267, September.
    4. Antonio Díaz & Carlos Esparcia, 2021. "Dynamic optimal portfolio choice under time-varying risk aversion," International Economics, CEPII research center, issue 166, pages 1-22.
    5. Robert Becker, 2012. "The Variance Drain and Jensen's Inequality," Caepr Working Papers 2012-004, Center for Applied Economics and Policy Research, Economics Department, Indiana University Bloomington.
    6. Markowitz, Harry, 2014. "Mean–variance approximations to expected utility," European Journal of Operational Research, Elsevier, vol. 234(2), pages 346-355.
    7. Locatelli, Giorgio & Mancini, Mauro, 2011. "Large and small baseload power plants: Drivers to define the optimal portfolios," Energy Policy, Elsevier, vol. 39(12), pages 7762-7775.
    8. Robert H. Trent & Robert S. Kemp, 1984. "The Writings Of Henry A. Latané: A Compilation And Analysis," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 7(2), pages 161-174, June.
    9. David E. Upton, 1982. "Single-Period Mean-Variance In A Multiperiod Context," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 5(1), pages 55-68, March.
    10. Gregor Dorfleitner & Mai Nguyen, 2017. "A new approach for optimizing responsible investments dependently on the initial wealth," Journal of Asset Management, Palgrave Macmillan, vol. 18(2), pages 81-98, March.
    11. Diamond, Harvey & Gelles, Gregory, 1999. "Gaussian approximation of expected utility," Economics Letters, Elsevier, vol. 64(3), pages 301-307, September.
    12. Cristinca FULGA, 2017. "Integrated Decision Support System for Portfolio Selection with Enhanced Behavioral Content," ECONOMIC COMPUTATION AND ECONOMIC CYBERNETICS STUDIES AND RESEARCH, Faculty of Economic Cybernetics, Statistics and Informatics, vol. 51(3), pages 127-142.
    13. Robert Becker, 2012. "The Variance Drain and Jensen's Inequality," CAEPR Working Papers 2012-004, Center for Applied Economics and Policy Research, Department of Economics, Indiana University Bloomington.
    14. Yong, Luo & Bo, Zhu & Yong, Tang, 2013. "Dynamic optimal capital growth with risk constraints," Economic Modelling, Elsevier, vol. 30(C), pages 586-594.
    15. Soumalya Mukhopadhyay & Arnab Hazra & Amiya Ranjan Bhowmick & Sabyasachi Bhattacharya, 2016. "On comparison of relative growth rates under different environmental conditions with application to biological data," METRON, Springer;Sapienza Università di Roma, vol. 74(3), pages 311-337, December.
    16. Harry M. Markowitz, 2002. "Efficient Portfolios, Sparse Matrices, and Entities: A Retrospective," Operations Research, INFORMS, vol. 50(1), pages 154-160, February.
    17. Jack Clark Francis & Ge Zhang, 2024. "Investing in US Timberland Companies," JRFM, MDPI, vol. 17(6), pages 1-13, May.
    18. Sonntag, Dominik, 2018. "Die Theorie der fairen geometrischen Rendite [The Theory of Fair Geometric Returns]," MPRA Paper 87082, University Library of Munich, Germany.

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