= Solution
For expected final capital, the <Bellman equation> is linear in wealth, with one-period factor $\mathbb E(1+pR)=1+p\mathbb ER$. Therefore \b[invest everything when the mean return is positive, nothing when it is negative, and any fraction when it is zero]. For $N=1$ this agrees with the yield objective.
For $N>1$, the root objective is concave and penalizes losses, so its optimal fraction need not be $1$ even when the mean is positive. For example, if $\mathbb P(R=-1)>0$ and $\mathbb ER>0$, the left <derivative> of $\mathbb E(1+pR)^r$ at $1$ is $-\infty$, while its right <derivative> at $0$ is positive; the optimum lies strictly between $0$ and $1$. If $\mathbb ER\leq0$, <Jensen inequality> gives $\mathbb E(1+pR)^r\leq(1+p\mathbb ER)^r\leq1$, so $p=0$ is optimal. Thus \b[the root objective can favor partial investment over the all-in policy].
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