## 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 8×8
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).