Markowitz Without a Risk-Free Asset

Abstract

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.