A matrix optimization problem in finance

  A matrix optimization problem in finance   Description: in many situations, financial decisions are made based on a collection of pairwise correlations of various risks of a portfolio. these pairwise correlation values estimated empirically are then entered in a symmetric matrix M which could serve as a correlation/covariance matrix of a multivariate distribution. in general these original matrices are not necessarily positive definite and therefore cannot be used to construct valid multivariate distributions. so the main problem of interest here is to find the “closest" positive definite matrix P to the original matrix M and then for the optimal matrix M to obtain the probabilities for various portfolio scenarios. the problem is not obvious and in general cannot be obtained in closed form. the project will explore various situations where the student can obtain explicit optimal expressions for M. The knowledge about matrix algebra is essential here. Fl programming will be required in this project in order to solve the matrix problem using alternating projection method{ You can use the nearPD function from Fl). This code should be appropriate to be tested in real stock data. In my case , I would like to test a simple real data set of stocks(for example a 8x8 matrix of stocks from online) and to conclude about the optimal portfolio for an investor. Therefore the knowledge of portfolio theory is also useful. For this project I should use the weighted norm (1 .2) of the uploaded paper(page330). Of course the project should have a basic structure(summary,introducton,main project,concluding remarks,references,appendioes).