Let $f\colon X \rightarrow Y$ be a proper morphism with $Y$ Noetherian (and even affine, if you wish), and let $\mathscr{A} = \bigoplus_{n \ge 0} \mathscr{A}_n$ be a quasi-coherent graded $\mathscr{O}_X$-algebra of finite type. Is the quasi-coherent graded $\mathscr{O}_Y$-algebra $f_* \mathscr{A}$ of finite type?

An analogue of this for graded modules is EGA III, 3.3.1. Perhaps the version for algebras that I am asking about can somehow be deduced, but I could't see how. I'd be happy to accept any precise reference (as well as any proof or counterexample, of course) as an answer.