Master funds in portfolio analysis with general deviation measures

Generalized measures of deviation are considered as substitutes for standard deviation in a framework like that of classical portfolio theory for coping with the uncertainty inherent in achieving rates of return beyond the risk-free rate. Such measures, derived for example from conditional value-at-risk and its variants, can reflect the different attitudes of different classes of investors. They lead nonetheless to generalized one-fund theorems in which a more customized version of portfolio optimization is the aim, rather than the idea that a single “master fund” might arise from market equilibrium and serve the interests of all investors.The results that are obtained cover discrete distributions along with continuous distributions. They are applicable therefore to portfolios involving derivatives, which create jumps in distribution functions at specific gain or loss values, well as to financial models involving finitely many scenarios. Furthermore, they deal rigorously with issues that come up at that level of generality, but have not received adequate attention, including possible lack of differentiability of the deviation expression with respect to the portfolio weights, and the potential nonuniqueness of optimal weights.The results also address in detail the phenomenon that if the risk-free rate lies above a certain threshold, the usually envisioned master fund must be replaced by one of alternative type, representing a “net short position” instead of a “net long position” in the risky instruments. For nonsymmetric deviation measures, the second type need not just be the reverse of the first type, and there can sometimes even be an interval for the risk-free rate in which no master fund of either type exists. A notion of basic fund, in place of master fund, is brought in to get around this difficulty and serve as a single guide to optimality regardless of such circumstances.