By T. Goodall
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Extra resources for Adequate Decision Rules for Portfolio Choice Problems
As soon as a decision principle is deﬁned such that it becomes a special case of the EU principle, inﬁnitely many repeats are necessary to support it. This point also applies to the most renowned decision rule used in portfolio choice theory, Markowitz’s (1952) µ–σ2 rule, which has been forced by Markowitz and many other authors to submit itself under the EU principle. 4 MARKOWITZ’S µ–σ2 RULE µ–σ2 rules fall into the category of ‘classical’ decision rules. Classical decision rules do not utilise all the information that is provided by the distributions of the random variables related to a gamble.
22 The EU principle was designed by von Neumann and Morgenstern (1944). 23 The EU principle assigns a preference index to all gambles according to ( ) ∞ Ψ(G ) = ψ E[Φ] = ∫ ϕ( u( r )) dF( r ) −∞ where u(r) is again a function assigning utilities to the gamble’s possible results. ϕ(u) is a strictly monotonically increasing function that is deﬁned up to a linear transformation with a positive slope coefficient. It may be interpreted as capturing something like the individual’s attitude towards entire gambles and their perceived ‘risk’.
Bernoulli’s rule thus replaces a one-to-one utility function u(r) = r by a logarithmic utility function, which in the general case can be written as u(r) = b⋅ln(r), where ln is the natural logarithm, and where b serves as a coefficient characterising the individual. A logarithmic utility function attributes diminishing marginal utilities to money wealth, which suffices to assure us that the expected value of the utilities, the ‘moral expectation’ as G. Cramér calls it, is finite. Applying de l’Hôpital’s rule will easily conﬁrm this.
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