Equilibrium with investors using a diversity of deviation measures

It has been argued that investors who optimize their portfolios with attention paid only to mean and standard deviation will all end up choosing some multiple of a certain master fund portfolio. Justification for the capital asset pricing model of classical portfolio theory, which relates individual assets to such a master fund, has come from this direction in particular. Attempts have been made to provide solid mathematical support by showing that the imputed behavior of investors is a consequence of price equilibrium in a market in which assets are traded subject to budget constraints, and optimization is carried out with respect to utility functions that depend only on mean and standard deviation.In recent years, reliance on standard deviation has come under increasing criticism because of inconsistencies in its effect on portfolio preferences. One response has been to introduce generalized measures of deviation which lead to alternative master funds. The market implications of such extensions of theory have hitherto been unclear, but in this paper the existence of equilibrium is established in circumstances where nonstandard deviations are admitted. Equilibrium is guaranteed even when different investors use different measures of deviation and thereby end up with portfolios scaled from different master funds. Whether they employ the same measure or not, they may impose caps on deviation, which likewise may be different.