Randomly generating portfolio-selection covariance matrices with specified distributional characteristics

In portfolio selection, there is often the need for procedures to generate “realistic” covariance matrices for security returns, for example to test and benchmark optimization algorithms. For application in portfolio optimization, such a procedure should allow the entries in the matrices to have distributional characteristics which we would consider “realistic” for security returns. Deriving motivation from the fact that a covariance matrix can be viewed as stemming from a matrix of factor loadings, a procedure is developed for the random generation of covariance matrices (a) whose off-diagonal (covariance) entries possess a pre-specified expected value and standard deviation and (b) whose main diagonal (variance) entries possess a likely different pre-specified expected value and standard deviation. The paper concludes with a discussion about the futility one would likely encounter if one simply tried to invent a valid covariance matrix in the absence of a procedure such as in this paper.