$$\begin{aligned}
\left(\hat{\pi}_{1}(t ; v_{1}) V_{1}(t ; v_{1})\right)_{i} > 0\ \Leftrightarrow\ \left(\Sigma^{-1} (\mu - r \mathbf{1})\right)_{i} > 0.\end{aligned}$$ Thus, ${\left(\hat{\pi}_{1}(t ; v_{1}) V_{1}(t ; v_{1})\right)_{i} > 0}$ is possible although it might be ${\left(\hat{\pi}_{1}(t ; v_{1})\right)_{i} < 0}$.

Finally, let ${\left(\Sigma^{-1} (\mu - r \mathbf{1})\right)_{i} > 0}$ for all ${i \in \left\{1, \hdots, N\right\}}$. When ${V_{1}(t ; v_{1}) < 0}$, the optimal exposure to the risk-free asset is negative because $$\begin{aligned}
\underbrace{V_{1}(t ; v_{1})}_{< 0} \left(1 - \underbrace{\hat{\pi}_{1}(t ; v_{1})' \mathbf{1}}_{< 0}\right) < V_{1}(t ; v_{1}) < 0.\end{aligned}$$ This in turn implies that in case of ${V_{1}(t ; v_{1}) < 0}$, the investor takes leverage by borrowing from the risk-free account to achieve her investment goals. Leverage at this point can make sense as future income provides some security; note that ${V_{1}(t ; v_{1}) < 0}$ immediately implies that the time $t$ value of accumulated future income exceeds the expected value of consumption.

Some more properties of $\hat{\pi}_{1}(t ; v_{1})$ can be found analytically as follows. The first and second derivative of $\left(\hat{\pi}_{1}(t ; v_{1})\right)_{i}$, $i = 1, \hdots, N$, with respect to wealth $V_{1}(t ; v_{1})$ are $$\begin{aligned}
\frac{\partial}{\partial V_{1}(t ; v_{1})} \left(\hat{\pi}_{1}(t ; v_{1})\right)_{i} = {} & \frac{1}{1 - 