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Investment Portfolios
Since investors like to increase their expected wealth and like toavoid risk or uncertainty, it is possible to imagine different combinations of expected gain and risk which are valued equally by an investor. That is, an investor will be willing to assume greater risk, if heachieves greater expected wealth.
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The individual investor is now conceptually prepared to select theoptimum portfolio from those constituting the efficient set. The optimum portfolio (i.e., the one which maximizes expected utility) is the one at the point of tangency between the efficient frontier and anindifference curve. In images it can be seen that the investorcan do no better than choose the portfolio at pointA on the efficientfrontier, since no other portfolio is on as high an indifference.Another escape is to say that concavity does not necessarily implythat the relationship is quadratic and that other equations can preservethe concavity without ever implying a maximum value from whichutility will decline as wealth increases. The difficulty with these othercurves is that efficiency in terms of the mean and variance of a portfolio does not necessarily imply maximization of expected utility.Markowitz has shown, however, that many utility functions can bereasonably approximated by the quadratic.
A different line of criticism has been advanced by Arditti andothers.They argue that investors may be interested in characteristicsof distributions of rates of return additional to the mean and variance.In particular, they argue that skewness may be of importance. Thatis, if the rates of return on the portfolios have the same mean andvariance, but different skewness, investors may prefer the distribution which is more skewed to the right.One is not excused from reaching tentative conclusions simply because the theoretical development of a field is still rudimentary.
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Aconclusion which is consistent with much that has been observed in the real world and which is satisfying theoretically is the one withwhich we started: namely, that portfolios which are efficient in termsof their means and variances necessarily maximize expected utilitywhich can be represented by a quadratic equation. Markowitz, perhaps, does the best job of showing that his efficient portfolios are veryclose to optimum or come very close to maximizing expected utility,even if things other than the mean and variance of the distributionsof returns make a difference to or affect the expected utility of inves tors.Even if the investor is concerned about the magnitude of theexpected loss, the maximum expected loss, the probability of a loss,or other attributes of the distribution, the portfolios selected accordingto those criteria will be very similar to portfolios selected accordingto their means and variances.
