# Computing the Nearest Correlation Matrix---A Problem from Finance

Higham, Nicholas J. (2002) Computing the Nearest Correlation Matrix---A Problem from Finance. IMA Journal of Numerical Analysis, 22 (3). pp. 329-343. ISSN 0272-4979

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## Abstract

Given a symmetric matrix what is the nearest correlation matrix, that is, the nearest symmetric positive semidefinite matrix with unit diagonal? This problem arises in the finance industry, where the correlations are between stocks. For distance measured in two weighted Frobenius norms we characterize the solution using convex analysis. We show how the modified alternating projections method can be used to compute the solution for the more commonly used of the weighted Frobenius norms. In the finance application the original matrix has many zero or negative eigenvalues; we show that for a certain class of weights the nearest correlation matrix has correspondingly many zero eigenvalues and that this fact can be exploited in the computation.

Item Type: Article correlation matrix, positive semidefinite matrix, nearness problem, convex analysis, weighted Frobenius norm, alternating projections method, semidefinite programming MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theoryMSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis Nick Higham 02 May 2006 20 Oct 2017 14:12 http://eprints.maths.manchester.ac.uk/id/eprint/232

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