Merton portfolio allocation problem proportions/weights >1 or <0?

Question

In the classical Merton portfolio problem, lets assume:

$$ dX_t \, = \, \frac{\pi_t X_t}{S_t} S_t(\mu dt +\sigma dW_t) = \pi_t X_t (\mu dt +\sigma dW_t) $$

ie: zero interest rates for simplicity.

We get HJB eqn:

$$\frac{\partial V}{\partial t} + \sup_{\pi \in \mathcal{A} } \left( \pi x \mu \frac{\partial V}{\partial x} + \frac{1}{2} \pi^2 \sigma^2 x^2 \frac{\partial^2 V}{\partial x^2} \right) = 0$$

I calculate the optimal weight for the risky asset to be $$\pi^* = \frac{\mu}{x \sigma^2 \alpha} $$

This is of course constant, as Merton points out. My question is:

Obviously the sum of weights corresponding to proportion of wealth invested in each asset must equal 1, but can weights be >1 for a certain asset or <0 in the framework of this classic model?

eg: mu = 0.05, sigma = 0.2 and risk aversion alpha =1 leads to proportion in risky asset of 1.25? Is this valid?

Answer 1

Your statement should be correct, the weights into the risky asset are not bounded between $0$ and $1$. Essentially, by setting $r=0$ you omit the term which shows that your weights always sum up to one, simply by choosing the weight for the risk-free asset to be $1-\pi^*$. In other words, obtaining $\pi^*>1$ simply implies you go short in the risk-free asset. Risk seeking investors (low $\alpha$) will choose to leverage by lending risk-free and putting that money into the stock.