Variance of random quadratic forms: A trace-based approach with applications in portfolio optimization

Random quadratic forms are fundamental to numerous disciplines such as physics, engineering, statistics, and econometrics. In finance, they play an important role in quantifying the estimation risk inherent in mean-variance portfolio optimisation. This paper introduces new properties for matrix traces of inverse Wishart matrices and uses them to derive new, closed-form expressions for the variance and covariance of quadratic forms involving the plug-in estimator of the constrained minimum-variance portfolio. Our results not only provide key analytical insights for measuring estimation risk in portfolio optimisation, but are also derived using matrix trace identities applicable to a wide range of practical problems beyond finance.