Abstract: This article uses Principal Component Analysis to compute and extract the main factors for the financial risk of a portfolio, to determine the most dominating stock for each risk factor and for each portfolio and finally to compute the total risk of the portfolio. Firstly, each dataset is standardized and yields a new datasets. For each obtained dataset a covariance matrix is constructed from which the eigenvalues and eigenvectors are computed. The eigenvectors are linearly independent one to another and span a real vector space where the dimension is equal to the number of the original variables. They are also orthogonal and yield the principal risk components (pcs) also called principal risk axis, principal risk directions or main risk factors for the risk of the portfolios. They capture the maximum variance (risk) of the original dataset. Their number may even be reduced with minimum (negligible) loss of information and they constitute the new system of coordinates. Every principal component is a linear combination of the original variables (stock rate of returns). For each dataset, each financial transaction can be written as a linear combination of the eigenvectors. Since they are mutually orthogonal and linearly independent and that they capture the maximum variance of the original data, the risk of the portfolio is calculated by using the principal components, then they have been used to calculate the total risk of the portfolio which is a weighted sum of the variance explained by the principal components.
Keywords: Covariance matrix, eigenvalues and eigenvector, financial transaction, linear maps, principal component analysis, portfolio optimization, risk analysis, singular value decomposition, stock price, vector space