Implementing Markowitz asset allocation model

Introduction

Markowitz’s work is widely regarded as a pioneer work in financial economics and corporate finance due to its theoretical foundations and applicability in the financial sector. Harry Markowitz received the Nobel Prize in 1990 for his contributions to these disciplines, which he outlined in his 1952 article “Portfolio Selection” published in The Journal of Finance. His major work established the foundation for what is now commonly referred to as “Modern Portfolio Theory” (MPT).

To find the portfolio’s minimal variance, the Markowitz model uses a constrained optimization strategy. The goal of the Markowitz model is to take into account 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 level of expected return or the best allocation for a given level of volatility. The expected return, volatility (standard deviation of expected return), and the variance-covariance matrix to reflect the co-movement of each asset in the overall portfolio design are the major inputs for this portfolio allocation model for an n-asset portfolio. We’ll go over how to use this complex method to find the best portfolio weights in the next sections.

Mathematical foundations

The investment universe is composed of N assets characterized by their expected returns μ and variance-covariance matrix V. For a given level of expected return μP, the Markowitz model gives the composition of the optimal portfolio. The vector of weights of the optimal portfolio is given by the following formula:

With the following notations:

• wP = vector of asset weights of the portfolio
• μP = desired level of expected return
• e = identity vector
• μ = vector of expected returns
• V = variance-covariance matrix of returns
• V-1 = inverse of the variance-covariance matrix
• t = transpose operation for vectors and matrices

A, B and C are intermediate parameters computed below:

The variance of the optimal portfolio is computed as follows

To calculate the standard deviation of the optimal portfolio, we take the square root of the variance.

For a more in-depth insight on this topic, you can refer to the following link for more insights on the implementation of the Markowitz portfolio allocation. You can download the Excel spreadsheets that complement the explanations offered in this article.

Academic research

Petters, A. O., and Dong, X. 2016. An Introduction to Mathematical Finance and Applications. Springer Undergraduate Texts in Mathematics and Technology.

Markowitz, H., 1952. Portfolio Selection. The Journal of Finance, 7(1): 77-91.

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