Let $\Sigma$ be a complete fan, $\Psi$ a virtual support function on $\Sigma$ and $(L,s)$ the corresponding toric line bundle and section. Let $\|\cdot\|$ be a toric algebraic metric on $L^{{\text{\rm an}}}$. Then the function $\psi_{\|\cdot\|}$ is rational piecewise affine. If moreover $\psi_{\|\cdot\|}$ is concave, the toric algebraic metric $\|\cdot\|$ is semipositive and it comes from a toric model.