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Most portfolio allocation schemes rely on diversifiction -- many rely on naive calculations of correlation between assets like the S&P 500 (NYSEARCA:SPY) and fixed-income proxies such as, for example, iShares Core Total U.S. Bond Market ETF (NYSEARCA:AGG). However, without care, this calculation can be performed incorrectly and create misleading results.

Consider the following graph:

Click to enlarge images.

Our intuition around diversification is, quite simply, that these two assets don't exhibit any: They are going the same direction. But if you look at their daily returns, you get the following scatter plot:

Almost completely white noise. In fact, sample daily correlation in this case is 10%. How is this possible? This example is completely contrived: We generated 100 independent random normals with the same mean and variance and from them generate geometric brownian motions. No matter how we aggregate the returns, we can't get by the fact that they are generated independently, which by definition means that their correlation is zero.

Remember the definition of correlation:

We see that no matter the horizon we aggregate over, the values are always de-meaned. Correlation, therefore, is not a measure of how divergent the trends are, but rather how divergent the noise is, regardless of how large the trend is relative to the noise.

Consider another contrived example, but this time the means of the random normals have opposite signs:

Again, our intuition is that these assets "diversify" each other; the standard statistical measure of correlation says otherwise. (Excel says it is 8% -- with only 100 samples, we cannot reject the null hypothesis that it may actually be 0%.)

How do we handle this situation? We have to incorporate our mean into our noise measurement. This is pretty easy to do: Instead of using sample means, we simply assume mean returns are 0. This gives us much more sane results in line with what our intuitive definition of correlation is. So be careful using Excel and other tools with pre-built measures of correlation -- you might be getting values that don't make sense.

Business relationship disclosure: Newfound Research is a quantitative asset management firm. This article was written by Corey Hoffstein, co-founder and chief investment officer. We did not receive compensation for this article (other than from Seeking Alpha), and we have no business relationship with any company whose stock is mentioned in this article.

Source: Deceived By Correlations: A Quant Conundrum