In this paper, we study the effect of channel aging on the uplink and downlink performance of an FDD massive MIMO system, as the system dimension increases. Since the training duration scales linearly with the number of transmit dimensions, channel estimates become increasingly outdated in the communication phase, leading to performance degradation. To quantify this degradation, we first derive bounds on the mean squared channel estimation error. We use the bounds to derive deterministic equivalents of the receive SINRs, which yields a lower bound on the achievable uplink and downlink spectral efficiencies. For the uplink, we consider maximal ratio combining and MMSE detectors, while for the downlink, we consider matched filter and regularized zero forcing precoders. We show that the effect of channel aging can be mitigated by optimally choosing the frame duration. It is found that using all the base station antennas can lead to negligibly small achievable rates in high user mobility scenarios. Finally, numerical results are presented to validate the accuracy of our expressions and illustrate the dependence of the performance on the system dimension and channel aging parameters.