4. Markowitz Without a Risk-Free Asset

In this chapter we solve the Markowitz problem of finding the investment portfolios which, for a given level of expected return, present the minimum risk. The assumption is made that the returns of the assets (and consequently of the portfolios) follow a Gaussian distribution, and the risk is defined as the standard deviation of the returns. Except in the case where all the risky assets have the same returns, the solution portfolios \(\mathcal {F}\) of this mean-variance optimisation problem define a hyperbola when representing in a plane the set \(\mathcal {F}(\sigma ,m)\) of their standard deviations and expected returns. This hyperbola also determines the limit of all the investment portfolios that can be built. Its upper side \(\mathcal {F}^+(\sigma ,m)\) corresponds to the efficient portfolios and is called the efficient frontier , while its lower side \(\mathcal {F}^-(\sigma ,m)\) is called the inefficient frontier . The two fund theorem demonstrated here proves that, when taking any pair of distinct portfolios from \(\mathcal {F}\), any other portfolio from \(\mathcal {F}\) can be constructed through an allocation between these two portfolios. As a consequence, when two optimal portfolios are found, the subsequent problem of finding other optimal portfolios is just a problem of allocation between these two funds.