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db2100
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Joined: August 12th, 2017, 5:43 pm

Portfolio optimization with heavily non-normal asset returns

August 12th, 2017, 6:31 pm

There are N investment periods where each period I can allocate X$ to an investment portfolio. The universe of assets to choose from is composed of thousands of assets which returns solely depend on the outcome of a set of independent events A={A1, ..., Ak}. Each asset can depend on any subset of A, hence any two assets can be independent (if no events from A are shared) or dependent (if one or more of the events from A affect both assets). I can estimate the probabilities of the events in A occurring fairly accurately and so can derive the dependence structure between any two assets. In particular, I can compute expected returns and the variance-covariance matrix with little estimation error. 

The assets themselves display a non-normal distribution of returns in a binary fashion. Each period the return can be large and positive (>100%) with a small probability of occurring or it can be -100% with a large probability of occurring. See two examples below. As you can see, for these assets it holds that the higher the positive return, the less frequent it is (and so the more frequent the -100% returns are).

Distribution.jpg

Distribution2.jpg

My problem is choosing each period how to allocate X among all the different assets, assuming no short selling and no leverage. Also, each period I am faced with a different set of assets with similar, but never identical, characteristics and so the optimization should probably be single-period. I could maximize using MVO but as you can see the return distributions are clearly not normal. Any suggestions on how to go about optimizing such a portfolio are welcomed. Thanks.
 
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Alan
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Re: Portfolio optimization with heavily non-normal asset returns

August 13th, 2017, 6:06 pm

It's a classic portfolio selection problem. You just need to choose a utility function [$]U(1 + R)[$] for the single period portfolio returns. If the return generating process is really nailed down, some people like [$]U(1+R) = \log(1+R)[$] -- the so-called Kelly criterion --  because of some attractive long-run properties. In general, stock market investors are known to be more risk-averse than Kelly investors, so a broader class of popular choices are power utility:

(*) [$]U(1 + R) = (1+R)^{\gamma}[$].

Here [$]\gamma = 0[$] corresponds to Kelly, but choose [$]\gamma < 0[$] to be more risk-averse. For example, studies have shown the 'typical' investor may have, very roughly,  [$]\gamma \approx -2[$].

You will definitely need to include cash as one of the assets. Allowing an allocation to cash is the simplest way to avoid eventual bankruptcy from the -100% returns, and allows the optimization problem to be well-posed. The amount to allocate to cash will be one of the outputs of the optimizer. 

It sounds like you have access to a mean-variance optimizer. If so, it can be used in a fast and iterative fashion to maximize the expected value of (*) with linear constraints on the weights. I discussed that in an old article, which I have attached. The same idea is mentioned in the Kelly link.

Lewis.SimpleAlgorithmPortfolioSelection.zip
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