Optimize portfolio of non-normal binary return assets

[arocker]

I am facing t = 1,..T investment periods where each period I have x$ to invest. Suppose each period I can build a portfolio from thousands of assets (some are uncorrelated whilst some are highly correlated) for which I know the exact probability distribution so I can calculate expected values and covariance matrices exactly (no estimation error). Cash (zero return zero risk) can be included as an asset too.

The return distributions of these assets are non-normal. In particular, for one period, they return a large positive value (>100%) with a small probability (eg. 5% - 15%) and return -100% otherwise (i.e. binary outcomes). Each period I am faced with a different set of assets with similar, but never identical, characteristics.

So each period I have to choose what weight to place on each asset, assuming no short selling and no leverage i.e. (0% <= Wi <= 100%) for each asset i, Sum(Wi)=100%.

My question is, given the above, what is the best way to optimize such a portfolio each period? Ideally I would like an efficient frontier which gives me the least risky portfolio for each level of expected return but a single point portfolio will also do. Probably the best definition of risk (other suggestions are welcomed) is the magnitude or likelihood of sub-zero returns. But it is important the risk measure takes into account the dependence between the assets.

Some possibilities with my concerns:

• Mean-variance optimization - Return distributions are heavily non-normal
• Mean-semivariance - Is non-normality an issue?
• Geometric mean (Kelly) optimization - Does it take into account the correlation between assets?
• Any other optimization techniques?

 

[yetanotherquant]

for which I know the exact probability distribution so I can calculate expected values and covariance matrices exactly (no estimation error).

Dream further! :P
I suggest you play around with a simple R-script from this post
Stripping down the robo-advisors: sparrow-brains inside — letYourMoneyGrow.com - Serving Retail Investors which shows how difficult (and virtrually impossible) it is to estimate the genuine market parameters from the limited observed sample.

But if - for whatever reason - this assumption holds true - then yes, Kelly will (asymptotically) beat any other approach. And yes, it does take correlation into account.
Have a look at my paper: Kelly Criterion for Multivariate Portfolios: A Model-Free Approach by Vasily Nekrasov :: SSRN
You have high absolute value of returns (upto +/-100%, so my analytical approximation can work poorly). But my Monte-Carlo Grope Method should work well, and having thousands of assets is not a big obstacle for it.

 

[arocker]

Dream further! :P
I suggest you play around with a simple R-script from this post
Stripping down the robo-advisors: sparrow-brains inside — letYourMoneyGrow.com - Serving Retail Investors which shows how difficult (and virtrually impossible) it is to estimate the genuine market parameters from the limited observed sample.

But if - for whatever reason - this assumption holds true - then yes, Kelly will (asymptotically) beat any other approach. And yes, it does take correlation into account.
Have a look at my paper: Kelly Criterion for Multivariate Portfolios: A Model-Free Approach by Vasily Nekrasov :: SSRN
You have high absolute value of returns (upto +/-100%, so my analytical approximation can work poorly). But my Monte-Carlo Grope Method should work well, and having thousands of assets is not a big obstacle for it.

Thanks for the advice and interesting paper. Just wondering, since in my case I know the probability distribution, can't I just calculate the E[ln(1+r)] exactly without using any approximations? If I maximise the criterion directly I struggle to see where the dependence structure between the assets is taken into account (of course I see it with the Taylor approximation). Any light on this?

 

[yetanotherquant]

You are welcome.

Just wondering, since in my case I know the probability distribution, can't I just calculate the E[ln(1+r)] exactly without using any approximations?

Well, if you know the joint distribution on your returns, then you may do without approximations.
Look in Breiman, L. (1961). Optimal gambling systems for favourable games. In Fourth Berkeley Symposium on Mathematical Statistics and Probability, volume 1. Univ. Calif. Press, Berkeley, CA. as far as I remember, he considered this case.

P.S.
I am really wondering what a special case you have - exactly knowing the parameters and the type of distribution - one can only dream about it in praxis.
So I am very curious (please PM me if you don't want to disclose the details to a wide public).