Abstract. After compactification on a thermal circle, BPS black holes in dimension $D$ yield BPS instantons in dimension $D-1$. Thus, BPS saturated couplings in the low energy effective action in dimension $D-1$ conveniently encode BPS black hole degeneracies. I will demonstrate this connection in the case of $R^4$ and $\nabla^4 R^4$ couplings in string vacua with 32 supercharges, and $F^4, \nabla^2 F^4$ couplings in string vacua with 16 supercharges, all of which can be deternined exactly using constraints from U-duality and supersymmetry. In particular, the $\nabla^2 F^4$ couplings provide a new point of view on the celebrated DVV formula counting BPS dyons in $D=4$.