On Portfolio Separation Theorems with Heterogeneous Beliefs and Attitudes towards Risk

The early work of Tobin (1958) showed that portfolio allocation decisions can be reduced to a two stage process: first decide the relative allocation of assets across the risky assets, and second decide how to divide total wealth between the risky assets and the safe asset. This so called twofund separation relies on special assumptions on either returns or preferences. Tobin (1958) analyzed portfolio demand in a mean-variance setting. We revisit the fund separation in settings that allow not only for heterogeneity of preferences for higher order moments, but also for heterogeneity of beliefs among agents. To handle the various sources of heterogeneity, beliefs, and preferences, we follow the framework of Samuelson (1970) and its recent generalization by Chabi-Yo, Leisen, and Renault (2006). This generic approach allows us to derive, for risks that are infinitely small, optimal shares of wealth invested in each security that coincide with those of a Mean-Variance-Skewness-Kurtosis optimizing agent. Besides the standard Sharpe-Lintner CAPM mutual fund separation we obtain additional mutual funds called beliefs portfolios, pertaining to heterogeneity of beliefs, a skewness portfolio similar to Kraus and Litzenberger (1976), beliefs about skewness portfolios with design quite similar to beliefs portfolios, a kurtosis portfolio, and finally portfolio heterogeneity of the preferences for skewness across investors in the economy as well as its covariation with heterogeneity of beliefs. These last two mutual funds are called cross-co-skewness portfolio and cross-co-skewness-beliefs portfolios. Under various circumstances related to return distribution characteristics, cross-agent heterogeneity and market incompleteness, some of these portfolios disappear.