In the portfolio optimization page, we discussed how we can determine which portfolio has the best risk-return trade-off, as quantified by the Sharpe Ratio. We alternatively discussed, how an investor could optimize a portfolio given a specific risk or return value that they want to achieve. As part of either portfolio optimization process, we are required to calculate the expected return and risk (standard deviation) of a portfolio. Since we have two values, we can plot this portfolio as a single point, using risk as an x-value and return as a y-value.

What if an investor wants to determine what type of risk they can expect at any given return value, instead of just at one specific return value? To consider this problem, we can generate what is called the efficient frontier.  The efficient frontier is a graphical tool that helps an investor visualize all possible return and risk combinations. An generic example of this curve is shown below.

image source: https://en.wikipedia.org/wiki/File:Capital_Market_Line.png

The efficient frontier is an upper bounding set of portfolios, acting as an envelope to contain all possible portfolios. Everything above the efficient frontier is theoretically impossible to achieve. Everything along the curve is considered optimal, meaning that for a specific risk value, you cannot achieve a better return than the point along the curve. Everything below the curve is possible, but not optimal, meaning that it would be possible for an investor to improve their returns without taking on any additional risk. It is not recommended for an investor to use a portfolio that does not lie along the efficient frontier. The single portfolio with the best Sharpe Ratio, as discussed in the optimization problem formulation, is often called the Market Portfolio.

In order to construct the efficient frontier, the variation of the optimization problem formulation discussed on the portfolio optimization page is used. Then we break down all the possible return values into a finite and limited set. For example if the best possible expected return is 10%, we could break down the range from 0% to 10% into 100 evenly spaced values: 0%, 0.1%, 0.2%, … 9.8%, 9.9%, 10%. We can then solve an optimization problem for each return value, and plot all the results, using the risk value as an x-axis value and return as a y-axis value as previously described. The resulting curve is the efficient frontier. In MATLAB, the PortOpt() function can be used to assist in construction of this efficient frontier.

Next: Rebalancing