- r\right) = \frac{\mu - r}{\sigma^{2}} > 0}$ since $\mu - r \mathbf{1} > 0$ was assumed. Then $$\begin{aligned}
& \left(\hat{\pi}_{1}(t ; v_{1})\right)_{i} > 0\ \Leftrightarrow\ V_{1}(t ; v_{1}) > 0, \\
& \left(\hat{\pi}_{1}(t ; v_{1})\right)_{i} < 0\ \Leftrightarrow\ V_{1}(t ; v_{1}) < 0.\end{aligned}$$ Even if the first part of the remark argues that ${V_{1}(t ; v_{1}) < 0}$ is a meaningful case, the conclusion ${\left(\hat{\pi}_{1}(t ; v_{1})\right)_{i} < 0}$ under ${\left(\Sigma^{-1} (\mu - r \mathbf{1})\right)_{i} > 0}$ sounds odd at a first glance. But when looking at the optimal exposure to risky asset $i$, one finds that $$\begin{aligned}
\left(\hat{\pi}_{1}(t ; v_{1}) V_{1}(t ; v_{1})\right)_{i} = {} & \frac{1}{1 - b(\tilde{t}_{1})} \left(\Sigma^{-1} (\mu - r \mathbf{1})\right)_{i} \left(V_{1}(t ; v_{1}) - F_{1}(t)\right),\end{aligned}$$ which, under ${\left(\Sigma^{-1} (\mu - r \mathbf{1})\right)_{i} > 0}$, is positive no matter if ${V_{1}(t ; v_{1}) < 0}$ or ${V_{1}(t ; v_{1}) > 0}$. Therefore, the amount of money invested in the risky assets is always positive. The opposite inequalities and conclusions for $\left(\hat{\pi}_{1}(t ; v_{1})\right)_{i}$ and $\left(\hat{\pi}_{1}(t ; v_{1}) V_{1}(t ; v_{1})\right)_{i}$ apply if ${\left(\Sigma^{-1} (\mu - r \mathbf{1})\right)_{i} < 0}$. In summary, the sign of the optimal exposure to the single risky assets is determined by 