  We consider two types of averaging of complex covariance matrices, a sample
mean (average) and the sample Fr\'echet mean. We analyse the performance of
these quantities as estimators for the true covariance matrix via `intrinsic'
versions of bias and mean square error, a methodology which takes account of
geometric structure. We derive simple expressions for the intrinsic bias in
both cases, and the simple average is seen to be preferable. The same is true
for the asymptotic Riemannian risk, and for the Riemannian risk itself in the
scalar case. Combined with a similar preference for the simple average using
non-intrinsic analysis, we conclude that the simple average is preferred
overall to the sample Fr\'echet mean in this context.
