# 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. t

**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. r programming will be required in this project in order to solve the matrix problem using alternating projection method. this code should be provided and it must be appropriate to be tested in real stock data. 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,appendices).**

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