Context: The initial-final mass relationship of white dwarfs, which is poorly constrained, is of paramount importance for different aspects of modern astrophysics. From an observational perspective, most of the studies up to now have been done using white dwarfs in open clusters. Aims: In order to improve the initial-final mass relationship, we explore the possibility of deriving a semi-empirical relation studying white dwarfs in common proper motion pairs. If these systems are comprised of a white dwarf and a FGK star, the total age and the metallicity of the progenitor of the white dwarf can be inferred from the detailed analysis of the companion. Methods: We have performed an exhaustive search for common proper motion pairs containing a DA white dwarf and a FGK star using the available literature and crossing the SIMBAD database with the Villanova White Dwarf Catalog. We have acquired long-slit spectra of the white dwarf members of the selected common proper motion pairs, as well as high resolution spectra of their companions. From these observations, a full analysis of the two members of each common proper motion pair leads to the initial and final masses of the white dwarfs. Results: These observations have allowed us to provide updated information for the white dwarfs, since some of them were misclassified. In the case of the DA white dwarfs, their atmospheric parameters, masses, and cooling times have been derived using appropriate white dwarf models and cooling sequences. From a detailed analysis of the FGK star spectra we have inferred the metallicity. Then, using either isochrones or X-ray luminosities we have obtained the main-sequence lifetime of the progenitors, and subsequently their initial masses. Conclusions: This work is the first one using common proper motion pairs to improve the initial-final mass relationship, and has also allowed us to cover the poorly explored low-mass domain. As in the case of studies based on white dwarfs in open clusters, the distribution of the semi-empirical data presents a large scatter, which is higher than the expected uncertainties in the derived values. This suggests that the initial-final mass relationship may not be a single-valued function. Based on observations obtained at: Calar Alto Observatory, Almería, Spain, el Roque de los Muchachos, Canary Islands, Spain, McDonald Observatory, Texas, USA, and Las Campanas Observatory, Chile. Tables [see full textsee full textsee full text], [see full textsee full textsee full text] and [see full textsee full textsee full text] are only available in electronic form at http://www.aanda.org