Before revision:  Their closed-form expressions are given by directional statisticsin a specific condition. Simulation analysis discloses the influence of market information, such as the number of stocks, on IC.
After revision:  Their closed-form expressions of the directional statistics' covariance matrix and IC's variance in a homoscedastic condition. Also, we solve the optimization of IC's maximum expectation and minimum variance. Simulation intuitively characterizes the standardized directional statistics and IC's p.d.f..