On the Existence of the Graded Exponent for Finite Dimensional $\mathbb{Z}_p$-graded Algebras Let $F$ be an algebraically closed field of characteristic zero, and let $A$ be an associative unitary $F$-algebra graded by a group of prime order. We prove that if $A$ is finite dimensional then the graded exponent of $A$ exists and is an integer. Keywords:exponent, polynomial identities, graded algebrasCategories:16R50, 16R10, 16W50