Magnetocrystalline anisotropy, the microscopic origin of permanent magnetism, is often explained in terms of ferromagnets. However, the best performing permanent magnets based on rare earths and transition metals (RE-TM) are in fact ferrimagnets, consisting of a number of magnetic sublattices. Here we show how a naive calculation of the magnetocrystalline anisotropy of the classic RE-TM ferrimagnet GdCo$_5$ gives numbers which are too large at 0 K and exhibit the wrong temperature dependence. We solve this problem by introducing a first-principles approach to calculate temperature-dependent magnetization vs. field (FPMVB) curves, mirroring the experiments actually used to determine the anisotropy. We pair our calculations with measurements on a recently-grown single crystal of GdCo$_5$, and find excellent agreement. The FPMVB approach demonstrates a new level of sophistication in the use of first-principles calculations to understand RE-TM magnets.
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