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The CAPM and Stochastic Dominance

  • Haim Levy
Part of the Studies in Risk and Uncertainty book series (SIRU, volume 12)

Abstract

The M-V and stochastic dominance paradigms represent two distinct branches of expected utility, each implying a different technique for portfolio investment selection. Each paradigm has its pros and cons. The advantage of the M-V approach is that it provides a method for determining the optimal diversification among risky assets which is necessary in establishing the Capital Asset Pricing Model (CAPM) (see Sharpe [1964]1 and Lintner [1965]2). The main disadvantage of the M-V paradigm is that it relies on the assumption of normal distribution of returns, an assumption not needed for SD rules. The normality assumption is obviously inappropriate for assets traded in the stock market because asset prices cannot drop below zero (−100% rate of return) whereas the normal distribution is unbounded. As the equilibrium risk-return relationship implied by the CAPM has very important results, it has been developed under other frameworks which do not assume normality. Levy (1977)3 has shown that technically the CAPM holds even if all possible mixes of distributions are log-normal (bounded from below by zero). However, with discrete time models, a new problem emerges: if x and y are lognormally distributed, a portfolio z, where z = α x + (1 − α) y, will no longer distribute lognormally. Merton (1973)4 assumes continuous-time portfolio revisions and shows that under this assumption, the terminal wealth will be lognormally distributed and the CAPM will hold in each single instantaneous period. By employing the continuous portfolio revision, the nagging problem of additivity of lognormal distributions, which characterizes discrete models disappears. However, the disadvantage of the continuous time model is that the CAPM result breaks down if minor transaction costs, no matter how small, are incorporated.

Keywords

Risky Asset Stochastic Dominance Efficient Frontier Single Period Capital Asset Price Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 4.
    Merton, R.C., “An Intertemporal Capital Asset Pricing Model,” Econometrica, 41, September 1973, pp. 867–887.CrossRefGoogle Scholar
  2. 5.
    Levy, H., “Stochastic Dominance Among Log-Normal Prospects,” International Economic Review, 14, 1973, pp. 601–614.CrossRefGoogle Scholar
  3. 6.
    Levy, H. and Samuelson, P., “The Capital Asset Pricing Model with Diverse Holding Periods,” Management Science, 38, 1992, pp. 1529–1540.CrossRefGoogle Scholar
  4. 13.
    Ohlson, J.A. and W.T. Ziemba, “Portfolio Selection in a Lognormal Market when the Investor has a Power Utility Function,” Journal of Financial and Quantitative Analysis, 11, 1976, pp. 51–57.CrossRefGoogle Scholar
  5. 14.
    Dexter, A.S., J.S.W. Yu and W.T. Ziemba, “Portfolio Selection in a Lognormal Market when the Investor has a Power Utility Function: Computational Results,” Stochastic Programming, Dempster, M.A., (ed.), Academic Press, London, 1980.Google Scholar
  6. 15.
    Merton, R.C., Continuous-Time Finance, Basil Blackwell, Cambridge, 1990.Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Haim Levy
    • 1
  1. 1.The Hebrew University of JerusalemIsrael

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