  We present a theoretical study on interfacial magnetocrystalline anisotropy
for Fe/MgAl$_2$O$_4$. This system has a very small lattice mismatch at the
interface and therefore is suitable for realizing a fully coherent
ferromagnet/oxide interface for magnetic tunnel junctions. On the basis of
density functional theory, we calculate the interfacial anisotropy constant
$K_{\rm i}$ and show that this system has interfacial perpendicular magnetic
anisotropy (PMA) with $K_{\rm i} \approx 1.2\,{\rm mJ/m^2}$, which is a little
bit smaller than that of Fe/MgO ($K_{\rm i} \approx$ 1.5--1.7$\,{\rm mJ/m^2}$).
Second-order perturbation analysis with respect to the spin-orbit interaction
clarifies that the difference in $K_{\rm i}$ between Fe/MgAl$_2$O$_4$ and
Fe/MgO originates from the difference in contributions from spin-flip
scattering terms at the interface. We propose that the insertion of tungsten
layers into the interface of Fe/MgAl$_2$O$_4$ is a promising way to obtain huge
interfacial PMA with $K_{\rm i} \gtrsim 3\,{\rm mJ/m^2}$.
