Complexity as a Means of Portfolio Diversification
A portfolio is a collection of securities: stocks, bonds or treasury bills. The calculation of the expected return of a portfolio is straightforward. The corresponding standard deviation can be used as a measure of risk. In order to compute the standard deviation it is necessary to resort to the concept of covariance – a measure of relationship between the returns of two securities. A fundamental objective in selecting securities for a portfolio is diversification. By diversifying, an investor can reduce the total risk of his portfolio. The Markowitz model, developed in the 1950s, quantifies the portfolio diversification procedure and is the basis of the Modern Portfolio Theory (MPT). Higher diversification means lower gains but also lower risks. Thus, an objective in portfolio diversification is to obtain a low standard deviation, while at the same achieving the highest possible expected return. At this point, one must trade-off between acceptable risk and expected return on investment. The MPT at this point takes the investor to the Efficient Frontier on the so-called feasible set of portfolios. This is done using quadratic programming techniques, yielding the best composition (weights) of the portfolio for a given risk tolerance and expected returns.
One of the difficulties with implementing the Markowitz method in portfolio selection is the effort in computing the variances of the portfolios. One has to compute the estimates for the covariances between all securities. For n securities, the number of covariances is n(n-1)/2. Hence the idea of a single index model, developed by W. Sharpe. The single index model relates the returns on a security to the percent change of a common index (e.g. DJIA, S&P 500, etc.) and is based on the famous “beta” parameter, which measures systematic risk. In order to implement the single index approach one must make a pretty critical assumption: all securities are related to each other only through the common index. This, of course, reduces dramatically the amount of computation that is needed to estimate the variance of the portfolio. However, the (unknown) effects of this assumption (plus all the other assumptions that are already in the process) are there. Clearly, increasing the number of securities in a portfolio reduces the overall risk but never below the systematic risk, no matter how diversified the portfolio is.
All this is known stuff. But where does complexity come into the picture? Complexity offers a phenomenal mechanism to rank portfolios. Yes, we can rank portfolios using complexity as a measure of diversification instead of covariances. Everyone knows (really?) that the assumption of Gaussianity is not quite that realistic (life often follows a Power Law, which, unlike the Gaussian distribution, allows extreme events to take place). The methods used in OntoSpace to compute the complexity of a portfolio are model-fee methods: they do not need assumptions or simplifications. In numerous experiments we have found that portfolios with higher covariance (global risk) are also more complex.
However, complexity is a far superior measure of diversification than covariance. First of all it takes into account the way the securities co-vary but without making assumptions of Gaussianity. In other words, the true nature of the structure of the dependencies is taken into account. We do not use statistics to derive this information. Instead, we use our proprietary algorithm which treats data as images, more or less just like the human eye would do. Secondly, we bring entropy into the picture. Entropy is a superior measure of variability than standard deviation. In fact, a standard deviation tells us how far (on average) data lies away from the mean. This is OK if the data has a nice Gaussian (uni-modal) distribution. But what if the data scatters around two most-likely values (bi-modal distribution). Where is the mean? What is the significance of a mean value in such cases? Entropy reflects the nature of variability better than standard deviation because it looks at the entire distribution, not just the mean.
Complexity can effectively help establish an innovative and effective mechanism for portfolio design, ranking and diversification. Our on-line service is an ideal tool for a quick complexity- check of a set of portfolios. If you have more than one candidate portfolio, upload each, obtain the corresponding complexities and then make your choice. Whenever available, the simplest solution that works is the best one.