Harry Markowitz is the U.S. economist who devised Modern Portfolio Theory in —and fundamentally changed the way that people and institutions invest. Markowitz used math to quantify diversification and is cited as an early adopter of the concept that mathematical models could be applied to investing. MPT was developed by economist Harry Markowitz in the s; his theories surround the importance of portfolios, risk, diversification, and the connections. VEGAS ODDS DENVER BRONCOS

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One might ask how to select an optimal portfolio. Basically, from portfolios with the same return, we pick the one with the lowest risk. On the other hand, from portfolios with the same risk level, we choose the one with the highest rate of return. The efficient portfolios The Efficient Frontier is a hyperbola representing portfolios with all the different combinations of assets that result into efficient portfolios i.

Risk is depicted on the X-axis and return is depicted on the Y-axis. The area inside the efficient frontier but not directly on the frontier represents either individual assets or all of their non-optimal combinations. Tangency portfolio, the red point in the picture above, is the so-called optimal portfolio that realizes the highest possible Sharpe ratio.

As we move from this point either to the right or to the left on the frontier, the Sharpe ratio, or in other words, the excess return-to-risk, will be lower. The point where the hyperbola changes from convex to concave is where the minimum variance portfolio green point in the picture lies. For a given level of volatility, there also exists a so-called maximum return portfolio orange point in the picture , which, as the name suggests, maximizes the return given the level of volatility.

This line is tangent to the efficient frontier exactly at the Maximum Sharpe portfolio point. Calculation When it comes to the math, an efficient frontier can be calculated explicitly, i. Once you start imposing more constraints on asset weights, an optimization procedure needs to be used for calculation of the frontier and the frontier may deviate from the original hyperbola to different curves, given the new constraints.

Unlike some quadratic optimizers, this method works well even if the number of assets N is much greater than the number of observations T. The main idea of CLA is based on a few simple steps. Firstly, you start with the asset with the greatest return, at the upper right side of the Efficient Frontier.

Then you follow the Efficient Frontier to the left by looking for the next best asset to be added in or removed one by one. The portfolios on one critical line, including the corner points, contain same pairs of assets. The only thing that changes are the weights. The set of all critical lines and corner points build up the Efficient Frontier, beginning at the upper right point to the Minimum Variance solution at the far left.

Markowitz Model in Practice We decided to investigate the Markowitz model also in practice. A 1-year calculation period from 8. We use the fPortfolio package in R to calculate the efficient frontier. As always, the x-axis shows the annualized values of volatility, and the y-axis shows the annualized values of return.

We may observe several interesting phenomena. Firstly, all risky assets are located in the far-right part of the chart, due to their high volatility. Secondly, the displayed part of the efficient frontier, as well as all three displayed portfolios do exhibit much lower volatility than risky assets. The following chart shows different asset weights of these portfolios at discrete steps, starting from the southernmost point on the frontier and ending at the northernmost one. We may observe that, e.

Every point on this curve corresponds to a particular portfolio of weights between the assets. If we keep going with this, we can draw new curves from any point on the first curve to any point on the second curve. This means that by introducing a third asset, we now have an entire region of possible portfolios.

Each portfolio that we can build out of the three assets is a point within this region. In short, the only portfolios that a rational investor would want to choose are those that are on the edge of this frontier. This is known as the efficient frontier. As Investopedia defines: The efficient frontier is the set of optimal portfolios that offer the highest expected return for a defined level of risk or the lowest risk for a given level of expected return.

Although this is a simple concept in theory, choosing a portfolio that is on the efficient frontier is much harder in practice. It's important to note that there is actually only one portfolio on the efficient frontier, which is the one with the highest expected return and lowest volatility. We'll discuss this point in more detail below.

In this section, we'll look at how we can actually plot the efficient frontier. To do so, first we'll discuss the necessary theory and then put this into an optimizer that will tell us where the efficient frontier lies.

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Stock Market Analysis \u0026 Markowitz Efficient Frontier on Python - Python # 11

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