Correlation Matrix & Portfolio Variance



What is the correlation matrix & portfolio variance?


A correlation matrix is presented in a table that displays the correlation coefficients between data sets. Every random variable (Xi) in the table is associated with each value in the table (Xj). It is going to allow you to see which pairs have the highest correlation.


The correlation matrix is nothing more than a table of correlations. Pearson's correlation coefficient, which compares two interval or ratio variables, is the most commonly used correlation coefficient. However, there are several options depending on the sort of data you wish to correlate.


For this, we need first to create multiple portfolios with different weights reflecting different capital allocations to each stock and calculate the standard deviation of each of the resulting portfolios and then choose the one with the lowest risk.


Portfolio variance is a statistical way to measure the dispersion degree of returns of a portfolio. In modern investment, it is a widely accepted concept that considers the aggregate of the actual returns of a given portfolio by considering a specific period. Portfolio variance is determined by taking the standard deviation of each asset in the portfolio and multiplying it by the correlation of the securities in the portfolio.


So, the correlation matrix & portfolio variance both are statistical measures.


Portfolio variance matrix formula


An essential feature of portfolio variance is that its value is a weighted mixture of each asset's variances corrected for covariances. This signifies that the total portfolio variance is lower than a simple weighted average of the variances of the portfolio's stocks.


  • Portfolio variance matrix formula:

Portfolio Variance = w12σ12 + w22σ22 + 2w1w2Cov


  • w1 = the first asset's portfolio weight

  • w2 = the second asset's portfolio weight

  • 1 = the first asset's standard deviation

  • 2 = the second asset's standard deviation

  • Cov1,2 = the covariance of two assets, denoted by p(1,2)12, where p(1,2) is the correlation coefficient between the two assets.


The terms in the Portfolio variance matrix formula expand exponentially as the number of assets in the portfolio grows. A three-asset portfolio, for example, contains six terms in the variance calculation, but a five-asset portfolio has 15.


Correlation Matrix Calculator


This correlation matrix calculator would construct a correlation matrix for a collection of samples you supply. Please enter two or more samples in the field.


The Formula of the Correlation Matrix:

  • Correlation of two stocks = Covariance of (StockA, StockB) / (Standard Deviation of Stock A x Standard Deviation of Stock B)

A Correlation Matrix is a table that easily organizes the pairwise correlations between numerous variables in the form of a matrix. To understand how to use the correlation matrix calculator, you must first understand how to compute Pearson's correlation because the correlation matrix is the matrix of all possible pairings of variables' correlations.


Correlation Matrix Calculation Example


The example below depicts the correlation coefficients between many factors connected to education.

  • So every cell in the table indicates the relation between independent variables. The highlighted cell below, for example, reveals that the correlation between "hours spent studying" and "exam score" is 0.82, indicating that they're substantially positively associated. Higher exam scores are closely associated with more hours spent studying.

  • And, as seen in the highlighted cell below, the correlation between "hours spent studying" and "hours spent sleeping" is -0.22, indicating that they are weakly negatively associated. Less time spent sleeping is related to more time spent learning.

  • And the highlighted cell below reveals that the correlation between "hours spent sleeping" and "IQ score" is 0.06, indicating that they are essentially unrelated. There is very little correlation between the number of hours a student sleeps and their IQ score.

  • Also, in this Correlation Matrix Calculation Example, correlation coefficients along the table's diagonal are all equal to one since each variable is completely associated with itself. These cells are useless for interpretation.


Portfolio Variance Formula With Correlation


Portfolio variance calculation is achieved with the multiplication of the squared weight of each security by its associated variance. Then you will need to add twice the weighted average weight multiplied by the covariance of all individual security pairs.


It is important to note that covariance and correlation are mathematically connected. The following is how the connection is expressed:


Ρ1,2 = Cov1,2 / σ1σ2


Understanding the link between covariance and correlation, we can rewrite the calculation for portfolio variance as follows:


In the portfolio, the lower correlations of two returns of assets show a lower portfolio risk and, therefore, higher diversification benefits. The same applies to vice versa,


Portfolio Variance = w12σ12 + w22σ22 + 2w1w2Cov1,2


In the portfolio variance formula with correlation, It is important to note that when calculating the variance for a portfolio with many assets, you must compute the factor 2wiwjCovi.j (or 2wiwji,j,ij) for each possible pair of assets in the portfolio.

If you want to measure the portfolio's risk using portfolio variance and portfolio standard deviation, you must first establish the weights of each stock in the portfolio. This is accomplished by dividing the entire value of each stock by the total value of the portfolio.

  • WA = Rs. 100 / Rs.100+Rs. 85+Rs. 65 = 0.4

  • WB= Rs.85 / Rs.100+Rs. 85+Rs. 65 = 0.34

  • WC= Rs. 65 / Rs.100+Rs. 85+Rs. 65 = 0.26

Furthermore, you must understand the relationship between each pair of stocks. His calculations reveal the following correlations:


ΡA,B, = 0.1

ΡB,C = 0.2

ΡA,C = -0.8


The portfolio variance may therefore be computed as follows:

Portfolio Variance,

= 0.42+0.12+0.342 x 0.252 +0.262 x 0.42 + 2 x 0.4 x 0.34 x 0.1 x 0.1 x 0.25 + 2 x 0.4 x 0.26 x (-0.8) x 0.1 x 0.4 + 2 x 0.34 x 0.26 x 0.2 x 0.25 x 0.4

= 0.0172


Portfolio Std Deviation = √0.0172 = 0.1312 = 13.12%


Summary


As per the above information and calculation of portfolio variance and correlation matrix with the right, you will learn more about it as you practice it more.

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