  In this paper, we study when positivity conditions of vector bundles are
preserved by extension. We prove that an extension of a big (resp.
pseudo-effective) line bundle by an ample (resp. a nef) vector bundle is big
(resp. pseudo-effective). We also show that an extension of an ample line
bundle by a big line bundle is not necessarily pseudo-effective. In particular,
this implies that an almost nef vector bundle is not necessarily
pseudo-effective.
