It is well known since many years ago that for the linear time-varying system e = Ae + B(t)theta, theta = -B(t)(inverted perpendicular)e with A Hurwitz, and B(t) bounded and globally Lipschitz, it is necessary and sufficient for global exponential stability, that B(t) satisfy the so-called persistency of excitation condition. In this note, we revisit this question and provide explicit bounds for the convergence rate and on the overshoot of the transient behaviour of the solutions e(t), theta(t) as functions of the richness of B(.). We believe that the result that we present is useful since knowing convergence rates aids in the construction of converse Lyapunov functions. Moreover, the type of systems that we study here appear for instance in model reference adaptive control (MRAC). (C) 2004 Elsevier Ltd. All rights reserved.