Molecular dynamics simulations have become a standard tool for the investigation of biological and soft matter systems. Water models serve as the basis of force fields used in molecular dynamics simulations of these systems. This article reports on an examination of the utility of a set of coarse-grained (CG) water models, with different resolutions, interaction potentials (Lennard-Jones, Morse), and cut-off distances. The relationships between the parameters under specific choices of the above options and the thermodynamic properties, such as density, surface tension, and compressibility, were found to fit simple mathematical equations. The limits of applicability of these CG water models were explored by checking the melting temperature. If a CG site is mapped to one or two real water molecules, a simple model with appropriate combinations of cut-off distances, functional forms, and parameters can be found to simultaneously match the experimental values of density, surface tension, and compressibility under ambient conditions. If more water molecules are included in a CG site, either the melting temperature approaches or surpasses room temperature, or the surface tension and compressibility cannot both be matched simultaneously. In striving for computational efficiency, it is still possible to find a simple CG water model with three water molecules contained in a CG bead that generates a liquid state of water with realistic values of density, surface tension and compressibility at ambient condition, but coarser models are not recommended.