The concept of an “efficient frontier” was developed by Harry Markowitz in the 1950s. The efficient frontier shows us the minimum risk (i.e. standard deviation) that can be achieved at each level of expected return for a given set of risky securities.
![]() A Matrix Based Example of Mean-Variance Optimization using Octave![]()
The efficient frontier can be plotted by making a scatter plot with the expected return values (cells P20:P50) on the y‐axis and the standard deviation values (cells R20:R50) on the horizontal axis. Creating efficient frontiers using excel. 1 0.06 1 0.3 0.3 0.079372 1 0.176666122 2.42961 1.558721 1 2 0.12 0.3 1 0.3 1.603166 1 3 0.03 0.3 0.3 1 -0.68254 1 To model the portfolio choice problem, I begin by highlighting the mean vector and giving it a name.
Of course, to calculate the efficient frontier, we need to have an estimate of the expected returns and the covariance matrix for the set of risky securities which will used to build the optimal portfolio. These parameters are difficult (impossible) to forecast, and the optimal portfolio calculation is extremely sensitive to these parameters. For this reason, an efficient frontier based portfolio is difficult to successfully implement in practice. However, a familiarity with the concept is still very useful and will help to develop intuition about diversification and the relationship between risk and return.
Calculating the Efficient Frontier
In this post, I’ll demonstrate how to calculate and plot the efficient frontier using the expected returns and covariance matrix for a set of securities.
In a future post, I’ll demonstrate how to calculate the security weights for various points on this efficient frontier using the two-fund separation theorem.
Calculations
In order to calculate the efficient frontier using n assets, we need two inputs. First, we need the expected returns of each asset. The vector of expected returns will be designated . The second input is the variance-covariance matrix for the n assets. This covariance matrix will be designated as . We also need a unity vector () with the same length as the vector .
Once we have this information, we can run the following calculations using a matrix based mathematical program such as Octave or Matlab.
Using these values, the variance () at each level of expected return () is given by this equation:
You can see from the equation, that the efficient frontier is a parabola in mean-variance space.
Using the standard deviation () rather than the variance, we have:
Example using Octave Script
As an example, lets consider four securities, A,B,C and D, with expected returns of 14%, 12%, 15%, and 7%. The expected return vector is:
The covariance matrix for our example is shown below. In practice, the historical covariance matrix can be calculated by reading the historical returns into Octave or Matlab and using the cov(X) command. Note that the diagonal of the matrix is the variance of our four securities. So, if we take the square root of the diagonal, we can calculate the standard deviation of each asset (13.6%, 14%, 20.27%, and 5%).
The example script for computing the efficient frontier from these inputs is shown at the end of this post. It can be modified for any number of assets by updating the expected return vector and the covariance matrix.
The plot of the efficient frontier for our four assets is shown here:
Derivation and References
Deriving the approach I have shown is beyond the scope of this post. However, for those who want to dive into the linear algebra, there are several excellent examples available online.
Derivation of Mean-Variance Frontier Equation using the Lagrangian (The Appendix B result is identical to what I show above, but the notation is a little different)
Octave Code:
This script will also work in Matlab, but I’ve chosen to use Octave since it is opensource and available for free.
Efficient Frontier DefinitionThe efficient frontier, also known as the portfolio frontier, is a set of ideal or optimal portfolios that are expected to give the highest return for a minimal level of return. This frontier is formed by plotting the on the y-axis and the as a measure of risk on the x-axis. It evinces the risk-and return trade-off of a portfolio. For building the frontier there are three important factors to be taken into consideration:. Expected Return,. Variance/ Standard Deviation as a measure of the variability of returns also known as risk and.
The of one asset’s return to that of another asset.This model was established by the American Economist in the year 1952. After that, he spent a few years on the research about the same which eventually led to him winning the Nobel Prize in 1990. Example of the Efficient FrontierLet us understand the construction of the efficient frontier with the help of a numerical example.
In this illustration, we have assumed that the portfolio consists of only two assets A1 and A2 for the sake of simplicity and easy understanding. We can in a similar fashion construct a portfolio for multiple assets and plot it to attain the frontier. In the above graph, any points outside to the frontier are inferior to the portfolio on the efficient frontier because they offer same return with higher risk or lesser return with the same amount of risk as those portfolios on the frontier.From the above graphical representation of efficient frontier, we can arrive at two logical conclusions:. It is where the optimal portfolios are. The efficient frontier is not a straight line. It is curved. It is concaved to the Y-axis. However, the efficient frontier would be a straight line if we are constructing it for a complete risk-free portfolio.
Assumptions of the Efficient Frontier Model. Investors are rational and have knowledge about all the facts of the markets. This assumption implies that all the investors are vigilant enough to understand the stock movements, predict returns and invest accordingly.
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