When do two- or three-fund separation theorems hold?

We show that when asset returns satisfy a location-scale property (possibly conditionally as e.g. for a multivariate generalized hyperbolic distribution) and the investor has law-invariant and increasing preferences, the optimal investment portfolio always exhibits two-fund or three-fund separation. As a consequence, we recover many of the three-fund (and two-fund) separation theorems that have been derived in the literature under very specific assumptions on preferences or distributions. These are thus merely special cases of the general characterization result for optimal portfolios that we provide.