We propose a simple upper bound on trajectory deviations for an affine family of discrete-time systems under nonzero initial conditions subjected to bounded exogenous disturbances. It in-volves the design of a parametric quadratic Lyapunov function for the system. The apparatus of linear matrix inequalities and the method of invariant ellipsoids are used as technical tools. The original problem is reduced to a parametric semidefinite programming problem, which is easily solved numerically. Numerical simulation results demonstrate the relatively low conserv-atism of the upper bound. This paper continues the series of our previous publications on esti-mating trajectory deviations for linear continuous- and discrete-time systems with parametric uncertainty and exogenous disturbances. The results presented below can be extended to vari-ous robust formulations of the original problem and also the problem of minimizing trajectory deviations for an affine family of discrete-time control systems under exogenous disturbances via linear feedback.