Approximating the Expectation and Variance of the Square Root of the Quadratic Form in Normal Random Variables

Extensive research has been conducted on the properties of quadratic forms in normal random variables, represented by the symbol Q here. However, analytical results on the square root of the quadratic form, √Q, are limited. The square root of the quadratic form is, for example, used in finance to calculate a portfolio's tracking error. Here we combine analytical reasoning and numerical simulations to evaluate the accuracy of truncated Taylor approximations for the expected value and variance of the random variable √Q.We show that an adjusted second-order Taylor expansion of the expected value of √Q has a relative error bound of about 3.8%. The second-order Taylor expansion is simple to calculate as it depends on the entries of the covariance matrix.The truncated Taylor approximations for the variance of √Q are more imprecise. An adjusted indirect second-order Taylor expansion fares the best, with a relative error bound of about 16.6%. It is also a function of the entries of the covariance matrix