Two types of monthly water balance models at basin scale are used: PE models use precipitation and potential evapotranspiration (PET) as their observed input data, whereas P models need only precipitation. Calibration proceeds by comparing model runoff and observed runoff. Calibration is entirely automatic with the exclusion of subjective elements. All models differ only by their actual evapotranspiration equations. PE models from previous papers are generalized essentially by replacing the constant evapotranspiration parameter by a periodic one, thus increasing the number of parameters by two (a “parameter” is an unknown constant to be estimated, and which is a characteristic of the river basin to be described). P models use a periodic “driving force”, which is intended to represent periodicity of hydrological phenomena, normally originating in the (unavailable) PET time series. These eight PE models and three P models are then applied to 55 river basins in 10 countries with widely diverging climates and soil conditions. A marked improvement of model performance in about one third of the basins is due to the introduction of the above mentioned periodic functions. Even when PET data are available it is sometimes useful to consider P models. P models scarcely perform less well than PE models. An engineer, wanting to try out as few models as possible on a given river basin, can restrict his attention to the optimization of two or three models. The paper is an extension of a long effort towards monthly water balance models, and is believed to give a solution in most circumstances.