Portfolio Weight Sum and Negative Weights

Question

I'm calculating the weights of 10 securities in a portfolio for a course project, with the objective of maximizing the sharpe ratio. I'm getting both positive and negative results for weights. The course guide says that negative weights mean that the optimal portfolio contemplates short selling. The results looks like the image.

I have doubts in the interpretation of these results. What exactly does a negative weight means for the portfolio assets, and how it benefits from short selling with those weights? Also, the positive sum of weights is larger than 1. The model is restricted so the sum of all weights is equal to 1, so the sum of positive and negative weights is equal to 1. By taking the course test, the answers seems to be correct, but I still dont understand the logic behind the results. How could be positive weights greater than 1, and what is the logic behind it?

Answer 1

Weights larger than 1 would mean if you have 100 000 USD you invest more (by taking credit or using futures where you only post margin). For negative weights it works similarly. The question is: are you allowed to do this in the contractual setting that you are in? In portfolio optimization it is crucial to define constraints on the weights such that the result can be invested. If you want only non-negative weights, then you have to constrain them to be non-negative (which is often the default for variables in an optimization program). Otherwise you will just get any weights. The solver does not know that $-300\%$ is a bad weight :) Box contraints of the form $l \le w \le u$. are common where there is a lower and an upper limit on each weight. Usually you want $\sum_i w_i = 1$. It gets much more involved if you use the really interesting constraints:

• At most $K$ asset have weight different from zero (cardinality).
• If a weight is greater zero, then it has to be greater than some minimal investment.
• turn-over constraints.

And some more are useful (and needed) in practice in order to really be able to invest the resulting portfolio. You can search for the paper "Portfolio Selection: How to Integrate Complex Constraints" and you get a flavor of what I mean.