Why Merton Portfolio Weights Can Exceed One
Summary
The document considers whether the optimal risky-asset weight in the classical Merton portfolio problem must lie between zero and one. In the setup described, the investor allocates a fraction of wealth to a risky asset, with zero interest rates assumed for simplicity. The answer explains that the risky weight is not inherently bounded by those limits.
A weight above one means the investor has borrowed at the risk-free rate to invest more than current wealth in the risky asset; the risk-free asset’s weight is then negative, so the total portfolio weights still sum to one. The example in the question illustrates that the calculated risky allocation can exceed one. This interpretation depends on the model permitting borrowing and short positions. The brief answer does not address real-world constraints, borrowing costs, or how the allocation changes when those assumptions are relaxed.
Key ideas
- The Merton model does not inherently constrain the risky-asset weight to the interval from zero to one.
- A risky weight above one corresponds to a negative allocation to the risk-free asset.
- The risk-free and risky weights together still sum to one.
- The leverage interpretation assumes borrowing and short positions are allowed.
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# Merton portfolio allocation problem proportions/weights >1 or <0?
# Merton portfolio allocation problem proportions/weights >1 or <0?
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 by Stefan Voigt (score 2, accepted)
https://quant.stackexchange.com/a/28052
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.