Efficient Frontier of Portfolios

Investors prefer greater returns over lesser returns, and lower risk over greater risk. If 2 portfolios have the same risk, but one has a higher expected return, then any investor would obviously choose the portfolio with the higher return, regardless of their own individual aversion to risk. An efficient portfolio offers the highest return for a given risk. Such a portfolio is said to be Markowitz efficient, named after Harry Markowitz, who, in 1952, developed a means of determining the efficient frontier by considering the expected returns and risks of each security, and the covariance of each security with every other security in the portfolio. The set of efficient portfolios over different levels of risk constitute the efficient frontier.

The efficient frontier can be better understood by plotting the opportunity set of portfolios, consisting of all portfolios that are possible, in the expected-return standard-deviation space, with the expected return represented by the vertical axis and the standard deviation, representing risk, by the horizontal axis. Some combination of assets forms a straight line, such as the combination of assets that are perfectly correlated.

The Efficient Frontier of a Riskless Asset and a Risky Asset Constitutes the Capital Market Line

Capital allocation is the allocation of funds between risky assets and riskless assets. A portfolio consisting of a riskless asset and a risky asset is a straight line. Because the riskless asset has no variance, the risk of the portfolio increases proportionately to the weighting of the risky asset. This is the capital allocation line and represents the efficient frontier for a portfolio consisting of a riskless asset and a risky asset. (Of course, it is also the investment opportunity set.) At one end of the capital allocation line is a portfolio consisting only of the risk-free asset, which, by definition, has no risk, such as a portfolio of T-bills, so its expected return = the risk-free rate. As the proportion of the risky asset increases, the risk increases until the portfolio consists only of the risky asset, in which case the risk of the portfolio is that of the asset itself.

By combining the portfolio that has the highest return on the efficient frontier with a riskless asset, the greatest range of expected returns can be achieved, allowing any investor, regardless of risk aversion, to select the combination that best suits them.

If only lending is involved, such as buying T-bills, then the maximum return is achieved by investing all funds in the risky asset. If borrowing is allowed, then the straight line is extended beyond the full investment in the risky asset, by borrowing money to buy the risky asset.

The Efficient Frontier of Risky Assets

Diversification lowers risk by combining assets with different coefficients of correlation among the assets composing the portfolio. However, a portfolio consisting only of risky assets always has some risk, since most assets have some correlation because they are all subject to systemic risk, caused by macroeconomic factors that affect virtually all assets.

An objective of modern portfolio theory is to find the set of all portfolios that constitute the efficient frontier. The efficient set consists of all portfolios that lie between the global minimum variance portfolio and the maximum return portfolio, which is the efficient frontier. The minimum variance portfolio is the portfolio on the efficient frontier with minimal risk; the maximum return portfolio is a portfolio on the efficient frontier that has a maximum expected return. Although the maximum return portfolio also has the highest risk of any portfolio on the efficient frontier, there is no other portfolio in the opportunity set that has an equal or greater return, but with lower risk.

Assets can be individual securities or entire portfolios. The expected return of a portfolio = the expected return of each of its assets multiplied by the weighting of that asset.

However, the risk of a portfolio, which is measured by the standard deviation of the expected return, is not a weighted average. A fundament of portfolio theory is that a combination of assets will have lower risk than the weighted proportion of the individual assets unless the assets have a coefficient of correlation of +1, meaning that they are perfectly correlated, in which case there would be no difference.

An efficient portfolio has the highest return for a given amount of risk. If the correlation coefficient = +1, then the risk and return of the portfolio is simply the linear combinations of the risk and return of each asset. Thus, different combinations of the 2 assets will yield a straight line in the expected-return variance space. This is the same as a portfolio consisting of a risk-free asset and a risky asset. Since the risk-free asset has a variance of 0, the expected return and variance of the portfolio will be proportional to the weighting of the risky asset.

Therefore, risk is not reduced by adding assets that are perfectly correlated. On the other hand, risk can be reduced to 0 by combining assets with a correlation coefficient equal to -1. If 2 assets have a perfectly negative correlation, then there will be some combination of the 2 that will reduce risk to 0. The risk of a portfolio will vary from the risk posed by 2 assets with perfectly negative correlation to the maximum risk when the assets are perfectly correlated. Intermediate values of the correlation coefficient will have lower risk than for the perfectly correlated assets.

This graph plots the expected return versus risk, as measured by the standard deviation, for varying proportions of 2 assets with different coefficients of correlation (σ) between the 2 assets of -1, -0.5, 0, +0.5, and +1. Asset A has an expected return and risk of 14% and 6%; Asset B, 8% and 3%. Note that the perfectly correlated assets form a straight line: risk is only reduced by increasing the proportion of the less risky asset. The risk of this portfolio can be less than the least risky asset if the 2 assets have a lower correlation. Where σ = -1, there is a proportion of the 2 assets that reduces the risk to 0.
The plots of 2 risky assets showing the relationship between the expected return versus risk for assets with a coefficients of correlation of -1, -0.5, 0, +0.5, and +1.

Efficient Frontier with Short Sales Allowed

The efficient frontier can be extended beyond the maximum return portfolio by selling short. A short sale is the ability to sell a security that is not owned by borrowing it from someone else, then selling it, with the obligation of buying back the security so that it can be returned to the original owner.

The proceeds of the short sale can be used to purchase securities with a higher expected return. Theoretically, short sales can increase possible returns infinitely, since any number of stocks can be sold short to purchase higher-yielding securities. However, the risk would also greatly increase. Additionally, this extension is based on the assumption that there are no special transaction costs in short selling, which does not reflect reality, since margin must be posted, but it does allow at least a theoretical extension of the efficient frontier.