2
$\begingroup$

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?

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.