Selection of the optimal portfolio depends on the risk-return tradeoff of available asset classes and the investor’s risk preferences.

Investment opportunity set

The investment opportunity set includes all portfolios which can be created using any possible combination of available assets. When the correlation coefficient between assets is not perfect, combining them in a portfolio reduces the portfolio’s risk-return trade-off. All such combinations form part of the investment opportunity set which is plotted on a graph with expected return on the y-axis and standard deviation on the x-axis.

As long as the correlation coefficient between the asset class already included in the portfolio and the new asset class is less than 1, the investment opportunity set will expand towards the y-axis (showing that higher return is earned per unit of risk).

Minimum-variance portfolios

Minimum-variance portfolios are portfolios included in the investment opportunity set which have the minimum variance (i.e. minimum risk) at any particular level of expected return. Any portfolios to the left of the minimum-variance portfolio (at the same expected return) are not attainable, and any portfolio to the right is not optimal.

All the minimum-variance portfolios when connected together form the minimum-variance frontier.

Global minimum variance portfolio

The point on the minimum-variance frontier which is closest to the y-axis (i.e. have the lowest risk) is called the global minimum-variance portfolio.

Efficient frontier of risky assets

Except for the global minimum variance portfolio, there are two minimum variance portfolios at the same risk level, one with a higher expected return and the other with a lower return. The portfolios on the curve below the global minimum-variance portfolio and to the right of the global minimum-variance portfolio have a lower expected return. A rational investor would prefer portfolios which occur on the minimum-variance frontier above the global minimum-variance portfolio. The part of the minimum-variance frontier represented by the curve that lies above and to the right of the global minimum-variance portfolio is referred to as the Markowitz efficient frontier.

The slope of the efficient frontier shows that as we move right from the global minimum-variance portfolio, the increase in risk for each additional unit of excess return is higher. In other words, we obtain decreasing increases in returns as we take on more risk.