A generalization of the mutual fund theorem
This paper is concerned with the problem of designing a portfolio of risky and riskless assets so as to maximize an investors utility from consumption and terminal wealth. A central result which considerably simplifies the analysis of this problem is the mutual fund theorem, also called the separation theorem. It states that, under certain assumptions, investor's allocation decision can be separated into two stages. In the first stage, an efficient portfolio of risky assets is determined, the so called tangent portfolio. In the second stage, the investor decides the allocation between this tangent portfolio and a riskless asset. A generalization of the continuous time mutual fund theorem is given, with no assumptions made on the investors utility functions for consumption and final wealth, except that they are time-additive and non-decreasing. The results are given for complete as well as certain incomplete markets. Optimal investment strategies that are known only for complete markets with a single risky asset, are automatically extended to complete and incomplete markets with multiple risky assets. An important feature of this paper is that the results are proved without resorting to the sophisticated mathematics of dynamic programming, martingales and duality theory, which are often employed in the study of continuous time portfolio theory. Instead, only simple properties of diffusion processes are used, together with linear algebra.
scientific article; zbMATH DE number 1336681