This work investigates the downlink performance of a multi-cell massive multiple-input multiple-output (MIMO) system that employs one-bit analog-to-digital converters (ADCs) and digital-to-analog converters (DACs) in the receiving and transmitting radio frequency (RF) chains at each base station (BS) in order to reduce the power consumption. We utilize Bussgang decomposition to derive the minimum mean squared error (MMSE) channel estimates at each BS based on the quantized received uplink training signals, and the asymptotic closed-form expressions of the achievable downlink rates under one-bit quantized zero-forcing (ZF) precoding implemented using the estimated channels. The derived expressions explicitly show the impact of quantization noise, thermal noise, pilot contamination, and interference, and are utilized to study the number of additional antennas needed at each BS of the one-bit MIMO system to perform as well as the conventional MIMO system. Numerical results verify our analysis, and reveal that despite needing more antennas to achieve the same sum average rate, the one-bit massive MIMO system is more energy-efficient than the conventional system, especially at high sampling frequencies.