For the Parzen-Rosenblatt type density estimator for circular data we prove the existence of a minimiser, \( h_{\mathsf{MISE}}(f;K,n) \), of its exact mean integrated squared error. Under mild conditions we show that it is asymptotically equivalent to the bandwidth \( h_{\mathsf{AMISE}}(f;K,n) \) that minimises the leading terms of the mean integrated squared error expansion, and we obtain the order of convergence of the relative error \( h_{\mathsf{AMISE}}(f;K,n)/h_{\mathsf{MISE}}(f;K,n)-1 \). Some small and moderate sample-size comparisons between the two bandwidths are also presented when the underlying density is a mixture of von Mises densities.