  We establish the existence of an integer degree for the natural projection
map from the space of parameterizations of asymptotically conical
self-expanders to the space of parameterizations of the asymptotic cones when
this map is proper. As an application we show that there is an open set in the
space of cones in the three-dimensional Euclidean space for which each cone in
the set has a strictly unstable self-expanding annuli asymptotic to it.
