Analysis of relations between covariance weights and corresponding maximum Eigen values

Abstract

This paper studies on how to get an idea through maximum eigen values, when allocating weights to covariance matrix. The eigen density distribution with respect to the largest eigen value is analysed. This study will help to determine the fluctuation of the eigen distribution with respect to allocated weight of the covariance matrix. This can be used to develop the classic portfolio asset allocation model by adding investors‘ ideas as parameters or weights. The maximum eigen value is 2.24 and the corresponding weight is 1.6, the minimum eigen value is 0.24 and the corresponding weight is 0.9. There are two peaks of the eigen values at 0.62 and 2.24 respectively 0.5 and 1.6 of weights. Two minimum points identified with corresponding eigen values are 0.43 and 0.24 respectively 0.2 and 0.9 of weights. For comparison the density function is plotted with Q = 3.22 and variance 0.85: this theoretical value was obtained assuming that the matrix is purely random except for its highest eigen value. The fact that the lower edge of the density is strictly positive (Except for the Q = 1); then there are no eigen values between 0 and the minimum eigen value.