What is Markowitz Modern Portfolio Theory (MPT)?
In the world of finance, one of the fundamental goals when making investments is to achieve the highest return at a specific level of risk. However, accomplishing this goal can be a complex task. This is where Harry Markowitz’s “Modern Portfolio Theory” comes into play. This theory seeks to address this problem by teaching investors how to optimize their portfolios and simplify the risk-return trade-off.
Fundamentals of Modern Portfolio Theory
Modern Portfolio Theory (MPT) was introduced in 1952 with the publication of the article titled “The Mathematical Principles of Portfolio Selection” by Harry Markowitz. This theory encompasses the fundamental concepts that investors should consider when constructing or adjusting their portfolios. These core concepts are risk and return. According to MPT, investors are assumed to be rational and risk averse. Therefore, the most ideal portfolio for an investor is one that minimizes risk while maximizing return. The primary objective here is to determine the optimal portfolio among assets with different expected returns and risks.
As mentioned above, since the primary objective is to minimize risk and maximize return, MPT makes a recommendation for risk reduction: portfolio diversification. According to MPT, including assets with negative correlation, especially in the same portfolio, reduces risk. The fundamental idea behind this claim is that, in the event of price fluctuations, including an asset with falling prices alongside another asset with rising prices in the portfolio minimizes the overall loss. In other words, two assets with price correlation in the negative direction dampen each other’s price fluctuations and protect the overall portfolio from sharp swings.
On the other hand, according to the ‘rational’ investor (or MPT), when there are two portfolios assumed to have equal risk, the one with the highest expected return should be selected. The total expected return of the portfolio is calculated based on the weighted average of the expected returns of each asset in the portfolio. The fundamental idea here is to use a weighted average for a more sensible measurement since the asset with a higher weight will have a greater impact on the total return due to its price movements.
As you can see, to construct a portfolio in line with MPT, it is necessary to express the two fundamental concepts, risk and return, in some mathematical form. In the later sections of the article, we will delve into these two topics and how they can be calculated in more detail.
As seen in the graph above, the X-axis represents variance, which is the measure of risk, while the Y-axis represents the mean, which is the expected return. The yellow dots represent individual assets, and the blue curve represents the efficient frontier, which forms the boundary of individual assets. Additionally, the straight gray line shown in the graph represents the Capital Allocation Line (CAL). The CAL curve is constructed using combinations of all market risk assets and risk-free assets. It is worth noting that the CAL does not start from the point (0,0) because there are assets in the market, such as interest deposits, with almost negligible risk, and due to the opportunity cost, an investor’s baseline scenario is always the interest yield. In other words, all investment instruments initially compete with interest. An investment is considered only if it offers a higher expected return than interest. This situation can be briefly considered as an opportunity cost.
Returning to the graph, the point where the CAL curve intersects with the efficiency frontier represents the risk-return ratio of the most ideal portfolio for the investor. This intersection point can be considered as a target, and a portfolio allocation is created from individual assets to match the risk-return ratio of the Orange point. According to MPT, this portfolio symbolizes the most optimal portfolio under the current market conditions.
Now, having understood how a portfolio is optimized according to MPT, let’s take a look at how the expected return and risk of a portfolio are calculated.
Expected Return
As mentioned earlier, the total expected return of a portfolio is calculated based on the weighted average of the expected returns of individual assets. For example, suppose we have two stocks: Stock X, with a current price of $20 and an expected future price of $30, and Stock Y, with a current price of $40 and an expected future price of $50. The expected return for each of these two stocks is calculated as follows:
So, for X stock;
And for Y stock;
These are the individual expected returns for each stock. Let’s say the investor has invested $250 in Stock X and $750 in Stock Y out of a total capital of $1000. In this case, the total expected return for this portfolio would be:
Portfolio Risk
According to Modern Portfolio Theory (MPT), portfolio risk (or portfolio variance) represents situations where it is difficult to predict the expected return of an investment. For example, the stock market is volatile, and the value of a stock can suddenly rise or fall. These fluctuations represent risk. Now, before we delve into how to calculate risk, let’s revisit some basic statistics. In statistics, variance is equal to the square of the standard deviation and it measures how much data points deviate from the mean. As mentioned earlier, according to MPT, assets with sharp price fluctuations are considered to have higher risk. Therefore, the larger (more deviation from the mean) the variance of a portfolio, the higher the risk it carries. Now, let’s quickly take a look at how to calculate portfolio risk, which is portfolio variance:
By using the formula above, we essentially calculate the risk of a portfolio.
Finally, by calculating the expected returns and risks of the constructed portfolios, we create the Risk-Return graph mentioned at the beginning. The efficiency frontier is formed, and the optimal point where it intersects with the CAL is determined.
To sum up, while Modern Portfolio Theory has several limitations and criticized assumptions, I believe it contains a lot of educational content in terms of fundamental finance knowledge. Therefore, I have attempted to convey the underlying logic of the theory to you with a slightly more mathematical approach. In my upcoming writings, while continuing my theoretical education from here, I also aim to provide Python implementations of these theories.
Thank you for reading this far and have a nice day!
References
Ayan, T. Y., & Akay, A. (2013). Estimation-Based Portfolio Optimization: An Alternative Approach to Risk and Expected Return Concepts in Modern Portfolio Theory. Dumlupınar University Journal of Social Sciences, EYİ 2013 Special Issue, 1–14.
Ercan M. K. (2009). “Modern Portfolio Theory.” Retrieved August 1, 2012, from http://www.makelecioglu.com/sitebuilder/MAK/mpt.ppt
Markowitz, H. M. (1952). “Portfolio Selection”. The Journal of Finance, Vol. 7, №1. (Mar., 1952), pp. 77–91.
