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2006 | Mathematical, econometrical and computational methods in finance and insurance | 127--136
Tytuł artykułu

Coherent Risk Measures and Stochastic Dominance

Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The seminal Markowitz portfolio optimization model uses the variance as the risk measure in the mean-risk analysis. The mean-variance model is, in general, not consistent with stochastic dominance rules. Several other risk measures have been later considered thus creating the entire family of mean-risk (Markowitz type) models. Opposite to the mean-variance approach, for general random variables some consistency with the stochastic dominance rules was shown for the Gini's mean difference, for the mean absolute deviation and for many other LP solvable models as well. In this paper we introduce general conditions for risk measures sufficient to provide the SSD consistency of the corresponding models. Actually, we show that under simple and natural conditions on the risk measures they can be combined with the mean itself into the robust optimization criteria thus generating SSD consistent performances (safety) measures. The analysis is performed for general distributions but we also pay attention to special cases such as discrete or symmetric distributions. Recently, a class of coherent risk measures has been defined by means of several axioms. Again, the coherence has been shown for the MAD model and for some other LP computable measures. We too analyze when our conditions guarantee also the coherence of the corresponding performance functions. (fragment of text)
Twórcy
  • Warsaw University of Technology, Poland
  • Warsaw University of Technology, Poland
Bibliografia
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  • Artzner P., Delbaen F., Eber J.-M., Heath, D.: Coherent Measures of Risk. "Mathematical Finance" 1999, 9.
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  • Hardy G.H., Littlewood J.E., Polya G.: Inequalities. Cambridge University Press, Cambridge, MA 1934.
  • Konno H., Yamazaki H.: Mean-Absolute Deviation Portfolio Optimization Model and Its Application to Toltyo Stock Market. "Management Science" 1991,37.
  • Mansini R., Ogryczak W., Speranza M.G.: LP Solvable Models for Portfolio Optimization: A Classification and Computational Comparison. "IMA Journal of Management Mathematics" 2003, 14.
  • Markowitz H.M.: Portfolio Selection. "Journal of Finance" 1952, 7.
  • Markowitz H.M.: Portfolio Selection. Wiley, New York 1959.
  • Mueller A., Stoyan D.: Comparison Methods for Stochastic Models and Risks. Wiley, New York 2002.
  • Ogryczak W., Opolska-Rutkowska M.: On Mean Risk Models Consistent with Stochastic Dominance. Report ICCE, 04-08-2004.
  • Ogryczak W., Ruszczynski A.: From Stochastic Dominance to Mean-Risk Models: Semideviations as Risk Measures. "European Journal of Operational Research" 1999.
  • Ogryczak W., Ruszczynski A.: On Stochastic Dominance and Mean-Semi- deviation Models. "Mathematical Programming" 2001, 89.
  • Ogryczak W., Ruszczynski A.: Dual Stochastic Dominance and Related Mean-Risk Models. "SIAM J. Optimization" 2002, 13.
  • Porter R.B., Gaumnitz J.E.: Stochastic Dominance versus Mean-Variance Portfolio Analysis. "American Economic Review" 1972, 62.
  • Rothschild M., Stiglitz J.E.: Increasing Risk: I. A Definition. "Journal of Economic Theory" 1970, 2.
  • Stochastic Dominance: An Approach to Decision-Making under Risk. Eds. G.A Whitmore, M.C. Findlay. D.C.Heath. Lexington, MA 1978.
  • Yitzhaki S.: Stochastic Dominance, Mean Variance, and Gini's Mean Difference. "American Economic Review" 1982, 72.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
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