Correlation matrices have many applications, particularly in marketing and financial economics - such as in risk management, option pricing and to forecast demand for a group of products in order to realize savings by properly managing inventories, etc.

Various methods have been proposed by different authors to solve the nearest correlation matrix problem by majorization, hypersphere decomposition, semi-definite programming, or geometric programming, etc. We propose to obtain the nearest valid correlation matrix by the Differential Evaluation/Repulsive Particle Swarm method of global optimization.

We may draw some conclusions from our exercise. First, the "nearest correlation matrix problem" may be solved satisfactorily by the evolutionary algorithm like the Differential Evolution or Repulsive Particle Swarm method. Secondly, these methods are easily amenable to choice of the norm to minimize. Absolute, Frobenius or Chebyshev norm may easily be used. Thirdly, the "complete the correlation matrix problem" can be solved (in a limited sense) by these methods. Fourthly, one may easily opt for weighted norm or un-weighted norm minimization. Fifthly, minimization of absolute norm to obtain nearest correlation matrices appears to give better results.

In solving the nearest correlation matrix problem the resulting valid correlation matrices are often near-singular and thus they are on the borderline of semi-negativity. One finds difficulty in rounding off their elements even at 6^th or 7^th places after decimal, without running the risk of making the rounded off matrix negative definite. Such matrices are, therefore, difficult to handle. It is possible to obtain more robust positive definite valid correlation matrices by constraining the determinant (the product of eigenvalues) of the resulting correlation matrix to take on a value significantly larger than zero. But this can be done only at the cost of a compromise on the criterion of "nearness." The method proposed by us does it very well.

How to Use the Program(s)

The program(s) is (are) written in FORTRAN 77. Hence, one must have a FORTRAN compiler to use it (them). FORCE (a FORTRAN 77 compiler) may be downloaded free from http://www.guilherme.tk (there going to Downloads and clicking on UK against Force208.exe) and installed on a computer. After installing FORCE on the computer, click and copy-paste the *.txt version there in FORCE's editor and save giving an appropriate name to the program file (suggested : nearcormat-de, for example). Once saved, it takes the name nearcormat-de.f. This version can simply be fouble-clicked to be accepted by FORCE.

One has also to make a data file, such as mycor.txt. This will be only the invalid correlation matrix. for example,

1.0	0.9	0.7
0.9	1.0	0.3
0.7	0.3	1.0

Note that the program and the data file should be stored in the same directory (folder). After this the program nearcormat-de.f may be double clicked to run. Then follow the instructions in the program pop up.