 Solution

For expected final capital, the Bellman equation is linear in wealth, with one-period factor $\mathbb{E}(1+pR)=1+p\mathbb{E}R$. Therefore 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 $\mathbb{E}R>0$, 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 $\mathbb{E}R\le 0$, Jensen inequality gives $\mathbb{E}(1+pR{)}^r\le (1+p\mathbb{E}R{)}^r\le 1$, so $p=0$ is optimal. Thus the root objective can favor partial investment over the all-in policy.