When calculating the value at risk, you want to be able to do more than just simulate individual prices. To manage the risk in a portfolio, you have to be able to simulate all the prices of the securities involved. Since these prices are usually interdependent, the question is how to build such correlations into the simulation.

The correlated, standard normally-distributed random numbers

can be generated using linear transformation , where is a matrix and

a vector of uncorrelated, standard normally-distributed random numbers, that is

The matrix must be set up in such a way, that the following criteria are fulfilled when is multiplied by a vector :

You get the required matrix , by solving the equation

according to the elements of .

One procedure for calculating this matrix is the Cholesky distribution, where the Cholesky matrix is a lower, triangular matrix, provided you are assuming that the correlation matrix is a symmetrical matrix with a consistently positive sign. A correlation matrix or covariance matrix meets these criteria when the risk factors are different pair-wise and linearly independent.

Using the matrix , you then get the following recursive calculation rule for the elements in the matrix :

With the resulting matrix , the following then applies:

and

The advantage of the Cholesky distribution is that the multiplication requires few operations due to the lower triangular matrix.