An upper bound for the coefficient of performance (COP) of endoreversible refrigerators which depends only on the ratio $\ensuremath{\tau}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}T}_{c}/{T}_{h}$ between the cold and hot reservoir temperatures has been elusive to date. We address here this long standing problem by analyzing an endoreversible Carnot refrigerator that operates in conditions of maximum per-unit-time COP. Two novel results are obtained: (1) A long sought $\ensuremath{\tau}$-dependent upper bound for the COP of refrigerators. (2) A $\ensuremath{\tau}$-dependent optimum distribution of the heat conductances associated with the coupling between the refrigerant and the heat reservoirs. Moreover, when the method is applied to heat engines, the resulting optimum efficiency is even closer to real efficiencies than the well-known Curzon-Ahlborn result.