Optimal portfolio - dubious weights

[Ken Peters]

Hi guys! I was wondering if you could help me figure something out. I'm simulating stock returns and are trying to build the optimal portfolio based on these returns but can't seem to get it right.

What i have is the beta of 8 stocks (given to me), the non-systematic risk of the same stocks as well as the market risk premium and volatility of market portfolio. I have assumed 10 year government bond as the risk free rate. I have calculated the covariance matrix based on the numbers that i currently have and applied eigensystem to get the desired correlation of my simulated returns. However, when i try to maximize Sharpe to get the optimal portfolio only two or maybe three stocks have positive weights (no short sale). This doesn't seem right. Am i doing something wrong or is this a problem when you use fabricated inputs (beta, non-systematic risk, etc) and not real values. I think i have read somewhere that this might happen if you work with fictional data because some of the fictional assets are just -better-. Thoughts? Is there something i can do to improve this? Other methods i can use? I guess i could use some sort of shrinkage estimator on the covariance matrix but the covariance matrix of my simulated returns are very close to the real covariance matrix (based on beta, etc).

I realise that this might be a little bit basic for you but i would really appreciate some help. Thank you


Here are the numbers if anyone wants to look at it:
Beta:
0.38 , 0.75 , 0.94 , 1.25 , 1.56 , 1.56 , 1.75 , 2.00
Non-systematic:
3.75 % , 5.00 % , 1.88 % , 12.50 % , 25.00 % , 18.75 % , 23.75 % , 27.00 %

Market risk premium of 5.00 % and volatility of market 15 %. I have assumed risk free rate of 2.00 %.

 

[Yike Lu]

I believe this is a commonly known problem of mean-variance optimization -- it has a tendency to go to corner cases. You're going to have to add some type of "diversity" metric or shrinkage to the utility.

 

[yetanotherquant]

Is there something i can do to improve this?

Oh yes! First of all do not blindly rely on computation (and model behind) an do engages common sense!
Given your 8 stocks, do they all come from different economy sectors?!
Have you took into account that the stocks in a diversified portfolio grow more or less independent but in crisis time the fall together (keyword for modeling: Clayton copula)?
Last but not least look as I do it: Somewhat better than DUCKS

Other methods i can use?.

Kelly Criterion for Multivariate Portfolios: A Model-Free Approach by Vasily Nekrasov :: SSRN
(my paper, the problem you mentioned is discussed as well)