By T. Goodall
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Extra resources for Adequate Decision Rules for Portfolio Choice Problems
The concave curve is again the boundary of the feasible set of investments, if no ‘risk-free’ asset exists. To avoid confusion it will be called the ‘feasible set of all risky investments’. If a risk-free asset exists, the feasible set is constructed by drawing two lines emanating from the point (rf, 0), the risk–return combination of the risk-free asset. These lines must be a tangent to the border of the feasible set of all risky investments, because of the optimisation procedure. Tobin imposes the nonnegativity constraint xh ≥ 0 on all assets except for the risk-free one, which amounts to assuming that funds may be borrowed at a cost equal to the risk-free rate.
Markowitz’s µ–σ2 rule may be considered the mainstay of decision rules in portfolio choice theory, just as the EU principle may be considered the mainstay of decision theory. Although Markowitz was neither the ﬁrst nor the only one to treat portfolio choice problems as decisions under Knightian risk, his analysis had nevertheless the greatest inﬂuence on all subsequent modern portfolio and ﬁnance theories. Not a few cite his work as having laid the foundations of modern investment theory. Several reasons may be given for this overwhelming impact.
The likelihood of achieving the expected value in a single gamble or a sequence of gambles is not contemplated. Consequences of not achieving the expected value are not evaluated. The expected value is treated as if it will occur with certainty. It has thus been argued9 that the expected gain rule is suitable only for decision situations in which the gambles to choose among are played an inﬁnite number of times. In this case, long-run results may be derived from the law of large numbers. 10 It states that the mean of a random sample will converge in probability to the expected value of the population.