A three-dimensional risk-return relationship based upon the inefficiency of a portfolio: derivation and implications

This paper derives a three-dimensional risk-return relationship employing a set of risky securities and a portfolio that is located inside the minimum-variance boundary of these securities. More specifically, it is proved that the inefficiency of a portfolio that has an expected return greater than the expected return of the global minimum variance portfolio, is a necessary and sufficient condition for expressing the expected return on any security under consideration as an exact linear function of its relative risk in that portfolio and an additional risk associated with moving inside the boundary portfolio set. Then two implications of the theoretical results are discussed, one related to the cost of equity capital and another related to past tests of the capital asset pricing model.