Portfolio Selection in the Mean-Variance Model: A Note

By Lars Tyge Nielsen

Journal of Finance 42 (1987), 1371-1376.


A little-recognized feature of the mean-variance portfolio-selection model is that induced preferences for asset holdings are not necessarily monotone; more of an asset (or portfolio) is not necessarily better, even if the asset (or portfolio) has positive expected return. The model predicts that an investor typically wants only a limited number of shares of an asset, and beyond that the increase in mean return from costlessly getting more of an asset is not sufficient to compensate for the increased risk. This is reasonable if the model describes a market for contracts with unlimited liability where total returns may be negative (e.g., they may be normally distributed). It is unreasonable, however, if the model is used to describe a stock market with limited liability. In that case, it is desirable to impose conditions that ensure that there is no upper limit to how much of an asset (or portfolio) the investor wants. The sharpest possible conditions are identified in Section I.

Since portfolio preferences are not monotone, global satiation is possible. Again, this is unrealistic when liability is limited. Satiation is bothersome also because it may lead to nonexistence of equilibrium in Black’s [4] CAPM without a riskless asset, as demonstrated in Nielsen [9]. (Uniqueness of equilibrium in CAPM is investigated in Nielsen [10]). Section I identifies the exact conditions under which global satiation does or does not occur. The conditions compare the maximum ratio of mean to standard deviation of total return available from the assets with the investor’s limiting degree of risk aversion (as measured by the slope of his or her indifference curves) at high levels of risk.

An optimal portfolio may or may not exist in the portfolio-selection problem, depending on the asymptotic slopes of the efficient frontier and the investor’s indifference curves. Section II identifies the exact conditions.

Mean-variance behavior can arise from expected utility maximization based on normal distributions. In that case, the conditions for nonsatiation and for existence of optimal portfolios can be stated in terms of the von Neumann-Morgenstern utility function. Section III relates the relevant properties of the utility function for standard deviation and mean to properties of the von Neumann-Morgenstern utility function.

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