Most real life decisions are taken without information about the probabilities associated with future outcomes; that is, ambiguity. Take, for example, choosing between different financial products. While for some of these products the probabilities associated with future rewards can be computed from past performance, these data are unlikely to be available for newly listed financial stock or when past performance is not a reliable indicator of the future. Research has shown that individuals have a tendency to prefer having to not having probabilistic information. In turn, it has been suggested that individuals are willing to pay to acquire ambiguity reducing information. This work explores factors that affect this pattern of results: endowment and reference dependence. We experimentally show that when individuals are endowed with ambiguous prospects they do not want to exchange them for unambiguous counterparts (unless there is a premium to do so), this invalidating the prediction that they are willing to pay to reduce ambiguity. The studies have important implications in the field of decision making under uncertainty, in the interface between psychology and economics.