In this paper, we derive exact series expressions for the conditional moments of a multivariate Cox-Ingersoll-Ross term structure model with a weak mean-reversion effect assumption. First, we construct the perturbation solution for the system of Riccati equations, which illustrates the conditional characteristic function of this term structure model. Second, we derive the power series expression for the perturbation solution. Via the power series solutions, we establish the exact series expressions for conditional moments. Furthermore, we demonstrate the error bounds for these perturbation approximations of the conditional moments. Numerical analysis indicates that our method is more efficient, accurate and superior to the Monte Carlo approach with the Euler scheme for computing the conditional moments.