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[1] More generally, if the asset returns obey an elliptical distribution, mean-variance analysis is consistent with expected-utility analysis. See Chamberlain (1983). The normal distribution is an example of an elliptical distribution. If the investor’s utility function can be approximated by a quadratic utility function, mean-variance analysis is also consistent with expected-utility analysis.
[2] For a justification of the Sharpe ratio from an analysis of an investor’s portfolio selection problem, see Bodie et al. (2005, p. 871).
[3] For the use of the Sharpe ratio in the second case, see Dowd (2000).
[4] For a justification of these measures from an analysis of an investor’s portfolio selection problem, see Breuer et al. (2004, pp. 379–396) or Scholz and Wilkens (2003).
[5] Eq. (2) is not identical to the definition originally suggested by Shadwick and Keating (2002) for Omega, but is based on an equivalent representation that can be interpreted more easily than the original definition. For a derivation of Eq. (2), see Kaplan and Knowles (2004).
[6] Defining the Calmar and the Sterling ratio, the maximum drawdown is preceded by a minus sign so that the denominator is positive and higher values for the denominator represent a higher risk.
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