In my last article, I promised to review a portfolio that was extremely non-normal. The motivation for doing so was to illustrate how a portfolio optimizer based on mean-variance theory could be used to improve the performance of a portfolio that did not comply with the assumptions under which an optimizer would be effective.

The 8 stocks we will review in this article are highly volatile stocks. Seven out of the 8 stocks tested non-normal (or more precisely, rejected for normality at the 5% significance level). The stocks that went into the universe were Archer Daniels Midland Co (NYSE:ADM), Corning Inc (NYSE:GLW), Newmont Mining Corp (NYSE:NEM), Nokia Corp (NYSE:NOK), Telefonica SA (NYSE:TEF), Teva Pharmaceutical Industries Ltd (NASDAQ:TEVA), TOTAL SA (NYSE:TOT), and Yahoo! Inc (NASDAQ:YHOO).

The distribution of the stocks across the volatility-return spectrum is shown by the blue and green stars in the figure below where blue denotes non-normality while green denotes that the assumption of normality has not been statistically rejected.

The x-axis measures volatility which is defined by the standard deviation of the expected returns. The expected returns are the averages of historical daily-moving annualized returns. The frontier is unique to the universe of securities which has been used in its calculation. In other words, you can find a unique frontier for any universe of securities you can dream of.

**Instruction Manual Not Included**

It's easy to use any optimizer that you can download from the web. Plug in the 3 required inputs that comprise expected returns, volatilities, and correlations and an efficient frontier will magically appear. But in order to use it correctly, you have to test for normality -- something I have talked about in previous articles.

If the returns are indeed normally distributed, the optimization results will be correct -- at that point of time. Going forward, if you deploy a buy-and-hold strategy you are making the added assumption that returns will *remain* normally distributed.

In practice, many portfolios suffer from non-stationarity which is a condition where the mean and variance change with time. In such situations, reversion-to-the-mean, which normality suggests, may not occur -- at least not in the manner it should.

Only when normality is the order of the day, both at the time of analysis and it continues into the future, will a buy-and-hold strategy that depends on reversion-to-the-mean work.

**Formulaic Rebalancing**

Taking into account the possibility of non-normal behavior in the context of what has just been discussed, we have the following performance of the 8 stocks using a formulaic rebalancing strategy. This strategy was formulaic in that it automatically determined the appropriate formulas to use for the inputs going into the optimizer as well as the kind of rebalancing procedure to adopt during each review point.

The next two figures show the growth in a portfolio of $100 (for ease of interpretation we use $100) where the blue bars use formulaic rebalancing while the green bars use a buy-and-hold strategy. The first figure starts at 26 January 2005 and ends 5 January 2009 while the second figure starts at 26 June 2008 and ends 5 June 2012.

**Conclusion**

The point of this exercise is not to see if formulaic rebalancing yields the highest returns. Rather, it is to see if it out-performs the traditional buy & hold strategy for a portfolio that is non-normal and non-stationary in times of boom and bust. Note that taxes, commissions, and the use of a risk-free proxy are not included in the calculations. The exercise is not offered as a rigorous proof but as an observation on the potential of such an approach.

The portfolio above was reviewed (but not necessarily re-balanced) on a monthly basis. At every single point of review, the formulaic rebalancing approach out-performed the simple buy-and-hold strategy, despite the fact the portfolio was from a single asset class.

The formulaic rebalancing strategy out-performed during times of boom and resulted in less downside when the market was in a "bearish" state. That's what optimization is supposed to do.

But while an optimizer can be a powerful tool that employs the science of mean-variance optimization theory, there is an art in knowing how to use it effectively.

**Disclosure: **I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.