  The inconsistency about the degree of geometrical frustration has been a long
issue in AV$_{2}$O$_{4}$ (A $\equiv$ Zn, Cd and Mg) compounds, which arises
from the two experimental results: (i) frustration indices and (ii) magnetic
moments. In the present study, we try to understand such inconsistency by using
{\it ab initio} electronic structure calculations. The orbital degrees of
freedom are found to play an important role in understanding the geometrically
frustrated magnetic behaviour of these compounds. The inclusion of the orbital
and spin angular momenta for calculating the frustration indices improves the
understanding about the degree of geometrical frustration in these compounds.
The calculated values of the frustration indices ($f$$_{\it J}$) are largest
for MgV$_{2}$O$_{4}$ and smallest for CdV$_{2}$O$_{4}$ for 3.3$\leq$ $U
\leq$5.3 eV. In this range of $U$, the calculated values of
$\Delta$M$_{2}$=M$_{\rm total}$-M$_{\rm exp}$ are largest for MgV$_{2}$O$_{4}$
and smallest for CdV$_{2}$O$_{4}$. Hence, the consistency about the degree of
geometrical frustration is achieved. The absolute values of the nearest
neighbour exchange coupling constant ({\it J$_{nn}$}) between V spins are found
to be largest for MgV$_{2}$O$_{4}$ and smallest for CdV$_{2}$O$_{4}$, which
indicate that the calculated absolute values of the Curie-Weiss temperature
($\varTheta$$_{CW}$)$_{\it J}$ are highest for MgV$_{2}$O$_{4}$ and smallest
for CdV$_{2}$O$_{4}$ for 3.3$\leq$ $U \leq$5.3 eV. In this range of $U$, the
magnetic transition temperature ($T$$_{N}$)$_{\it J}$ is found to be $\sim$150
K, $\sim$60 K and $\sim$22 K for MgV$_{2}$O$_{4}$, ZnV$_{2}$O$_{4}$ and
CdV$_{2}$O$_{4}$, respectively, which shows that the order of ($T$$_{N}$)$_{\it
J}$ is similar to that of ($T$$_{N}$)$_{\rm exp}$ for these compounds.
