= Solution
Let $M=\max_{p\in[0,1]}\mathbb E(1+pR)^r$ and choose an optimizer $p^*$. Existence follows from <continuity> and <compactness>: the integrands are dominated by a constant times $1+|R|$. Induction in the <Bellman equation> gives
$$
\boxed{J_n(x)=x^rM^{N-n},\qquad p_n=p^*\text{ is optimal at every period}.}
$$
Thus the maximal expected yield is $\boxed{M^N-1}$. Neither current wealth nor time changes the maximizing fraction; there may be several optimizers.
The right <derivative> at $p=0$ is $r\mathbb ER$, using the finite first moment and dominated differentiation near zero. If $\mathbb ER>0$, increasing $p$ slightly improves on $p=0$, so $\boxed{p^*>0}$. No explicit value of $p^*$ is needed.
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