Modern Portfolio Theory

Paolo Trenta
Dec 15, 2024
2 min read

The scope of this article is giving an introduction of the modern portfolio theory based on the work of the Nobel winning Professor Harry Markowitz. We are going to find the allocations weights in a 5-stock portfolio according to the theory proposed by the economist in its famous publication “Portfolio selection”.

Before delving in the practical portfolio construction, it important to introduce some useful concepts and statistical tools.

In the previously mentioned publication, Markowitz rejects the assumption that the investor’s goal is to maximize the discounted expected, or anticipated returns. Instead, he introduces the idea that the investor considers the expected returns as desirable and the variance of returns as undesirable. Thus, the goal of the model proposed is to consider the expected return and volatility of the assets in the investable universe to provide an optimal weight vector that indicates the best allocation for a given expected return or the best allocation for a given volatility.

In our analysis we are working under the assumption of a static model. So instead of considering the time series of the returns of a certain security, we will consider the “flow of returns” ri.

Some useful statistical elements:

It follows that the expected return of the portfolio is:

where Xi is the weight of the i-th portfolio element and ui is its expected return The variance of the portfolio is:

Now for the actual implementation. We choose 5 different stocks form 5 different sector to minimize the correlation between them, downloading the daily price time series for the last 5 years to have a satisfactory population of data. We calculate the logarithmic daily return as they allow us to consider the compounding effect. With the obtained data we can find the variance covariance matrix of the securities using the data analysis tool on Excel.

Done that we can move on to find the expected return and standard deviation of the securities. This is the main criticality of the model proposed by Markowitz as the expected returns are found using past data. A better idea would be to use the expected returns proposed by investment firms. At this point, using the excel solver we can find the allocation weighs that grants the highest Sharpe ratio, imposing that the sum of the weighs must be one and the weighs can’t be negative (so we do not have any short sales). The Sharpe ratio allows to take in consideration both risk and return or better the risk-adjusted return and it is defined as: