Since Equation (1') is the necessary and sufficient condition for the optimal portfolio solution [omega]* within the mean-variance framework, and Equation (1') mathematically equals (1), the unique [omega] in Equation (1) must be the optimal mean-variance efficient portfolio [omega]*.

In addition, the object of Equation (3) is the optimal mean-variance efficient portfolio for n securities rather than the market equilibrium.

In other words, CAPM entails that the market portfolio is the optimal mean-variance efficient portfolio.

This implies that CAPM fails as shown in Proposition 5, if the market index does not exactly equal the optimal mean-variance efficient portfolio [[omega].

In other words, if the market index is not the optimal mean-variance efficient portfolio and is used to substitute for the optimal mean-variance market portfolio [[omega].

This means that every mean-variance efficient portfolio [[?

sigma]]))} where for a given level of the volatility [sigma], the mean-variance efficient portfolio [[?

sigma]] belonging to the efficient frontier are called mean-variance efficient portfolios.

Thus, our simulation creates a situation where the market is indeed a

mean-variance efficient portfolio and all securities earn true returns in accordance with their betas.

If so, you still want to solve the mean-variance problem of figure 1, and you still want a mean-variance efficient portfolio.

You can still achieve a mean-variance efficient portfolio just as in figure 1 by a combination of a money market fund and a single tangency portfolio, lying on the upper portion of the curved risky-asset frontier.

From a theoretical perspective, the mean-variance framework which underlies the Sharpe-Lintner capital asset pricing model (CAPM) is predicated on certain assumptions required for rational agents to hold

mean-variance efficient portfolios.