The modern portfolio theory of Markowitz (1952) is a rich source of intuition and also the basis for many practical decisions. Mean-variance agents may differ with respect to the degree they are willing to trade off mean against variance but all will choose from the set of efficient portfolios, those which maximize mean given a constraint on variance. Moreover, under certain conditions, the mean-variance model of portfolio selection leads to two-fund separation (Tobin, 1958), i.e., all agents hold a combination of the same portfolio of risky assets combined with the risk-free asset. Two-fund separation greatly simplifies the advice one should give to a heterogenous set of agents since the proportion of risky assets in the optimal portfolio is independent from agent’s risk aversion. Moreover, it implies a simple asset pricing structure in which a single risk factor explains the rewards agents get in equilibrium.
We derive sufficient conditions for two-fund separation in a general reward-risk model, where agents’ preferences are assumed to be increasing functions of a reward measure and decreasing functions of a risk measure. We show that two-fund separation holds if reward and risk measures can be transformed by means of strictly increasing functions into positive homogeneous, translation invariant or translation equivariant functionals. In this case, the efficient frontier is a straight line in the transformed reward-risk diagram.
Several reward and risk measures introduced in the literature satisfy the conditions for two-fund separation. Mean and variance, semi-variance (Markowitz, 1959), lower partial moments (Bawa and Lindenberg, 1977, Fishburn, 1977 and Harlow and Rao, 1989), the Gini measure (Yitzhaki, 1982), general deviation measures (Rockafellar et al., 2006), etc. Since many of these measures are defined based only on few principles of rationality, it follows that two-fund separation is a common property to many rational mean-risk models, including even those which are consistent with second order stochastic dominance (De Giorgi, 2005). This result is surprising because strong conditions on agents’ utility functions are needed in order to obtain two-fund separation within expected utility theory (Cass and Stiglitz, 1970).
In Section 2 we introduce the general reward-risk model and derive our main result. Examples are discussed in Section 3. All proofs are given in Appendix A.
