The process capability index Cpk has been widely used in manufacturing industry to provide numerical measures of process potential and performance. As noted by many quality control researchers and practitioners, Cpk is yield-based and is independent of the target T. This fails to account for process centering with symmetric tolerances, and presents an even greater problem with asymmetric tolerances. To overcome the problem, several generalizations of Cpk have been proposed to handle processes with asymmetric tolerances. Unfortunately, these generalizations understate or overstate the process capability in many cases, so reflect the process potential and performance inaccurately. In this paper, we first introduce a new index Cp"k, which is shown to be superior to the existing generalizations of Cpk. We then investigate the statistical properties of the natural estimator of Cp"k, assuming that the process is normally distributed.