# Optimize portfolio of non-normal binary return assets

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?
• You might find that the Quaranta, Zaffaroni 2009 paper "Efficient and robust portfolio optimization in the multivariate Generalized Hyperbolic framework" discusses some of your concerns and has references for further digging – jaamor Jan 8 '17 at 3:08

For a single period, I would consider scenario optimisation: simulate your assets' returns (which you can do since you know their statistical properties, including correlations), and in this way create a large number of scenarios. Collect these scenarios in a matrix, R, say, with scenarios in rows, assets in columns.
The portfolio returns for a given weight vector w are easily computed as Rw.