Chaatit, Mascioni, and Rosenthal defined finite Baire index for a bounded real-valued function $f$ on a separable metric space, denoted by $i(f)$, and proved that for any bounded functions $f$ and $g$ of finite Baire index, $i(h)\leq i(f)+i(g)$, where $h$ is any of the functions $f+g$, $fg$, $f\vee g$, $f\wedge g$. In this paper, we prove that the result is optimal in the following sense : for each $n,k<\omega$, there exist functions $f,g$ such that $i(f)=n$, $i(g)=k$, and $i(h)=i(f)+i(g)$.