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The Gini Methodology pp 387–408Cite as

The Mean-Gini Portfolio and the Pricing of Capital Assets

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Part of the book series: Springer Series in Statistics ((SSS,volume 272))

Abstract

Since its development by Markowitz (1952, 1970), the mean-variance (MV) model for portfolio selection has become the standard tool by which risky financial assets are allocated. MV has gained a prominent place in finance because of its conceptual simplicity and ease of computation. Many authors, however, have challenged the model’s assumptions, primarily the normality of the probability distributions of the assets’ returns or the quadraticity of the preferences. MV validity has been reasserted by Levy and Markowitz (1979) and by Kroll, Levy, and Markowitz (1984), who showed that MV faithfully approximates expected utility.

This chapter is based on Shalit and Yitzhaki (2005, 2010).

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Notes

  1. 1.

    When applied to empirical data, the problem is one of a piece-wise linear optimization. See Okunev (1991) and Okunev and Dillon (1988) for a linear programming solution.

  2. 2.

    To be precise, for each covariance in the MV framework we substitute a Gini correlation multiplied by the appropriate Gini.

  3. 3.

    Actually, it is possible to have an equilibrium under different expectations. As will be seen later, the adjustment needed is that given the different expectations, the marginal rates of substitution between two assets are the same between all assets and all investors.

  4. 4.

    Under a MV framework, both Harris (1980) and Nielsen (1990) use the Edgeworth box to model capital market equilibrium, the first by analyzing the trade-off between risk and return, and the second by characterizing allocation risk.

  5. 5.

    To illustrate this issue, assume that X is uniformly distributed between [0, 1] while Y is uniformly distributed between [1,000, 2,000]. Clearly, all investors prefer Y over X, but both of them are included in the efficient MV set. Consequently, relying on MV to analyze portfolios may produce efficient portfolios that are inconsistent with expected utility theory.

  6. 6.

    Only the shares allocated to risky assets are shown on the axes. The share of the risk-free asset determines the location of the “budget constraint”; the farther it is from the origin, the lower the share of risk-free asset in total wealth is.

  7. 7.

    Although investors are homogeneous in the way they perceive risk, they can be heterogeneous in the way they price risk, as reflected by the risk-free-to-market portfolio ratios.

  8. 8.

    Homogeneity of risk perception implies that all MG investors have the same ν, or that all investors are MV investors. See property ix of δ.

  9. 9.

    If returns are multivariate normal, heterogeneity is reduced to homogeneity and the standard MV result is obtained.

  10. 10.

    This result is derived from the homogeneity property of the isoquants. The contract curve cannot cross the diagonal, as it can only be the diagonal itself or lie on one side of it.

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Authors and Affiliations

  1. Department of Economics, The Hebrew University, Mount Scopus, Jerusalem, Israel

    Shlomo Yitzhaki

  2. Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel

    Edna Schechtman

Authors
  1. Shlomo Yitzhaki
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  2. Edna Schechtman
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Yitzhaki, S., Schechtman, E. (2013). The Mean-Gini Portfolio and the Pricing of Capital Assets. In: The Gini Methodology. Springer Series in Statistics, vol 272. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4720-7_18

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