The saturation properties of neutron-rich matter are investigated in a relativistic mean-field formalism using two accurately calibrated models: NL3 and FSUGold. The saturation properties density, binding energy per nucleon, and incompressibility coefficient are calculated as a function of the neutron-proton asymmetry α≡(N-Z)/A to all orders in α. Good agreement (at the 10% level or better) is found between these numerical calculations and analytic expansions that are given in terms of a handful of bulk parameters determined at saturation density. Using insights developed from the analytic approach and a general expression for the incompressibility coefficient of infinite neutron-rich matter, i.e., K0(α)=K0+Kτα2+ , we construct a hybrid model with values for K0 and Kτ as suggested by recent experimental findings. Whereas the hybrid model provides a better description of the measured distribution of isoscalar monopole strength in the Sn isotopes relative to both NL3 and FSUGold, it significantly underestimates the distribution of strength in 208Pb. Thus, we conclude that the incompressibility coefficient of neutron-rich matter remains an important open problem.