UTILITY, RISK AVERSION, AND OPTIMALITY

UTILITY, RISK AVERSION, AND OPTIMALITY

UTILITY, RISK AVERSION, AND OPTIMALITY

UTILITY, RISK AVERSION, AND OPTIMALITY

The previous articles I have written on this subject have been based on the assumption that most investors like high rates of return but dislike risk. The definition of efficient portfolios follows from this. Given predictions about individual securities and their interrelationships, the efficient set is the same for all investors. Since investors' preferences forreturn vis-a-vis risk are likely to differ, we now need to discuss theways in which an investor chooses an optimum portfolio from those

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which are efficient. Optimization implies that something is maximized.We have said that the fact that almost all investors diversify indicatesthat they do not seek to maximize expected gain, rate of return, orwealth. The underlying principle that guides their behavior is the maximization of expected utility. Utility will be maximized when a givencombination of expected return (or wealth) and risk is preferred toall other combinations.

Markowitz provided the world with a way of analyzing data oneach security so as to specify all of the portfolios which were optimumor "efficient" in the sense that no other portfolio could be superiorin both dimensions. Specification of an efficient portfolio means indicating the proportion of the investor's assets that should be allocatedto each security included in his portfolio. Rational investors who arerisk averse would choose to hold one of the efficient portfolios. Thissection of the article discusses the problem that each investor facesin deciding which of the efficient portfolios is optimum for him.

It is now time to relate ideas about efficient portfolios to ideas aboututility functions.That is, utility increasesat a decreasing rate as wealth increases. Each additional (marginal)unit of wealth provides less of an increment in utility than the preceding unit. Thus, the curve is concave from below. It would be convenient if it could be shown that portfolios which are efficient in termsof their means and variances were portfolios which maximized theexpected utility of the investor. If that were true, all of the individualthings about portfolio management which seem plausible would beshown to be consistent with each other.

In beginning the task of attempted reconciliation between utilitytheory and efficient portfolios, it is unfortunately necessary to characterize the relationship between wealth and utility more explicitly. Allthat has been said so far is that the curve representing this relationshipis concave from below. A concave curve can be described by severalmathematical functions. It is not unusual to use some extremely simplequadratic equation such as the following:

whereU is the utility of a portfolio,Rpis the rate of return on aportfolio, anda, b andc are constants,b andc being positive, whose values will depend upon the preferences of the investor.

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If the utility function is represented by the foregoing equation, it can be shown that expected utility depends only on the mean and variance of the probability distribution of future returns.In otherwords, for investors whose utility functions are quadratic, the mean-variance approach to portfolio selection is valid. A portfolio whichis efficient in terms of the mean and variance will be one which maximizes expected utility.

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