Expanding the Frontier One Asset at a Time

We study the mean-variance optimization problem when investment opportunities are changing. We add a new risky asset to a set of n risky assets. An analytical relation between the original and the new minimum-variance frontiers is established. The two frontiers have a tangency point. We derive a new mutual fund theorem. All portfolios in the new minimum-variance set are portfolio combinations of three mutual funds: The two funds located on the original frontier and the third fund containing all assets. Analytical framework developed in the paper has implications for studies of testability of the mean-variance efficiency of a market portfolio (Roll critique). Implications for models of financial innovation are discussed.