  In this paper we provide the characterization of all finite-dimensional
Heath--Jarrow--Morton models that admit arbitrary initial yield curves. It is
well known that affine term structure models with time-dependent coefficients
(such as the Hull--White extension of the Vasicek short rate model) perfectly
fit any initial term structure. We find that such affine models are in fact the
only finite-factor term structure models with this property. We also show that
there is usually an invariant singular set of initial yield curves where the
affine term structure model becomes time-homogeneous. We also argue that other
than functional dependent volatility structures -- such as local state
dependent volatility structures -- cannot lead to finite-dimensional
realizations. Finally, our geometric point of view is illustrated by several
examples.
