Standard deviations are a popular and often useful measure of dispersion. To be sure, a standard deviation is merely the most likely deviation from the mean. It also doesn’t take into account the shape of the probability distribution function (this is done better using, for example, entropy, which is a more versatile measure of dispersion).
Standard deviations, however, may be ‘adjusted’ to take into account an interesting aspect of data, namely complexity. Let’s see an example. Say you have a portfolio of 28 stocks, all of which are independent (i.e. uncorrelated). In such a case the complexity map of the portfolio is as the one below.
Clearly, just like in the previous case, one can calculate the standard deviations of all stocks one by one. However, in the first case all stocks were uncorrelated, here some of them are. These two cases are obviously different, in particular from a structural point of view. The question now is this: why not use the information in the Complexity Profile to ‘adjust’ standard deviations by adding a correction originating from complexity? Clearly, a stock that is heavily correlated to other stocks in a portfolio could be more ‘dangerous’ than an uncorrelated one. Evidently, it is the job of covariance to express this:
Covariance(i,j) = Correlation(i,j) x STD(i) x STD(j)
Adjusted STD = (1 + Complexity contribution) x STD
Portfolio complexity, which is a factor that is neglected while analyzing or designing a portfolio (a covariance matrix is a poor substitute) ‘increases’ standard deviations, illustrating eloquently the concept of complexity-induced risk.