We explore the implications of a common market and academic practice which is known as ‘freezing the drift’ when dealing with swap interest rate dynamics. In mathematical terms this can be better understood as imposing a low variance martingale (LVM) assumption. We look into the LVM assumption implications, both on the shape and dynamics for default-free yield curves. We show that the LVM assumption is equivalent to consider future yield curves are nothing but deterministic translations of the initial curve. For the particular case of the Nelson–Siegel yield curve calibration, we show the LVM assumption requires a deterministic parameter's evolution and, thus, imposes the need to constantly recalibrate the model. Finally, based upon European Central Bank historical data on evolution of the default-free Euro area yield curve, we illustrate periods in which the LVM may be applicable and others in which is not.