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 way to determine 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 constitutes the efficient frontier.

The efficient frontier can be better understood by plotting the opportunity set of portfolios, consisting of all possible portfolios in the expected-return standard-deviation space, with the expected return ploted 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 perfectly correlated assets.

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 with 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. 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.

An objective of modern portfolio theory is to find the set of all portfolios constituting the efficient frontier. The efficient set is 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 with a maximum expected return. Although the maximum return portfolio also has the highest risk of any portfolio on the efficient frontier, no other portfolio in the opportunity set has an equal or greater return but with lower risk.

Portfolio risk, 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 they are perfectly correlated, so risk is not reduced.

An efficient portfolio has the highest return for a given amount of risk. If the correlation coefficient = +1, 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 perfectly correlated assets. But risk can be reduced to 0 by combining assets with a correlation coefficient equal to −1. Portfolio risk will vary from the minimum risk of 2 assets with perfectly negative correlation to the maximum risk of perfectly correlated assets. Intermediate values of the correlation coefficient will have an intermediate risk.

Efficient Frontier with Short Sales Allowed

The efficient frontier can be extended beyond the maximum return portfolio by selling short. A short sale is selling a security that is not owned by borrowing it from someone else, then selling it, with the obligation of buying back the security to return it 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, risk also greatly increases. Additionally, extending the efficient frontier assumes that short selling has no special transaction costs, which does not reflect reality since margin must be posted and short-selling fees are charged, but it does allow at least a theoretical extension of the efficient frontier.