Joint transmission (JT)-based Coordinated Multipoint (CoMP) systems achieve high performance gains by allowing full coordination among multiple cells, transforming unwanted intercell interference into useful signal power. In this paper, we present an analytical model to perform adaptive modulation for a typical JT CoMP system, consisting of three transmission points, under a target bit error rate (BER) constraint. Probability density functions of the signal-to-interference-plus-noise ratio (SINR) are derived for different JT CoMP schemes, and based on them, closed-form expressions for the average spectral efficiency (ASE) are obtained when adopting continuous-rate adaptive modulation. The study of ASE is also extended for the case of discrete-rate modulations, where the performance comparison of different practical quantized modulation schemes is carried out. Index Terms-Average spectral efficiency (ASE), continuous-rate adaptive modulation, Coordinated Multipoint (CoMP), discrete-rate adaptive modulation, joint transmission (JT).