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We show that the rate of convergence of asymptotic expansions for solutions of SDEs is generally higher in the case of degenerate (or partial) diffusion compared to the elliptic case, i.e. it is higher when the Brownian motion directly acts only on some components of the diffusion. In the scalar case, this phenomenon was already observed in (Gobet and Miri 2014) using Malliavin calculus techniques. In this paper, we provide a general and detailed analysis by employing the recent study of intrinsic functional spaces related to hypoelliptic Kolmogorov operators in (Pagliarani et al. 2016). Relevant applications to finance are discussed, in particular in the study of path-dependent derivatives (e.g. Asian options) and in models incorporating dependence on past information.