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ETFs and Powerball - relative measurements in an absolute world

Does buying a Powerball ticket increase your odds of winning Powerball? On a relative basis, Yes, on an absolute basis, No.

The odds of winning the Powerball jackpot with a single ticket is approximately 0.0000003% - as close a value to zero as anyone would bother to calculate.


By any practical measure, one’s likelihood of winning Powerball is not materially changed with the purchase of a ticket. However, the relative difference in the likelihood of winning with-versus-without a ticket is infinitely higher - a number that’s almost zero (0.0000003%) over a number that is actually zero (the probability of winning without a ticket).

While relative-vs-absolute measurements on a Powerball scale are unlikely to confuse ETF investors, two of the most popular measures in ETFs have a similar relative-vs-absolute property.

Return Correlations Are a Relative Measure.

Researchers, journalists, traders and marketers all look to summarize complex risk and return comparisons with a simple statistic, and correlation is a favorite.  Common wisdom is that return correlations approaching 1 indicate a good proxy, return correlations approaching -1 indicate a good hedge, and return correlations approaching 0 indicate the lack of a relationship (which may be beneficial for diversification).

While there are different flavors of correlation calculations, they all do basically the same thing; they measure the relative movements of a data series around their respective average values (further normalized by their own volatilities such that values land between -1 and +1) – correlation formulas are prized for their relative simplicity, but with simplicity comes the need for careful post-calculation interpretation.

When we think of low return correlations in ETFs, we think of scatterplot clouds of seemingly unrelated returns. Alternatively, when we think of high return correlations (e.g. in excess of 0.90), we think of nicely aligned and coincidentally trending returns and similar ending values; it’s easy to forget that the individual averages can be terrible proxies for each data series, and that the correlation result is a scaling and normalization within the individual volatilities.

The figure below shows 10 periods for two hypothetical assets – one asset rises by 8.8%, and the other asset declines almost 18%, and yet the correlation of returns is over 0.95 (the data points have been included for those wishing to reproduce the results – remember to use returns and not values). Even halfway through the data series, the rising asset is up over 14%, and the declining asset is down more than 1.5% - and yet the correlation approaches 1.0. So while the correlation estimator did what it’s supposed to do (measure the respective coincidental movements around the respective average values), if we failed to examine the data, we would be tempted to conclude that these assets would have similarly trending returns and similar final values. Because correlation measures relative dispersion around the respective means (rather than a measure of absolute trending), results can be misleading and counterintuitive.




Return Volatility is a Relative Measure of Dispersion.

High return volatility is generally associated with assets which have the potential for sizable losses, and low return volatility is generally associated with assets which may be immune from sizable losses. Return volatility is a summary statistic which is limited to telling us about the consistency of returns (by measuring the relative deviations around the average return). Importantly, it does not provide a useful indication about relative direction of those returns or the magnitude of that direction.

It’s not uncommon to see comparative asset discussions center around return volatilities. But without other information relating to relative return details, volatility doesn’t provide much in the way of an absolute comparison.


Consider a series in which we wagered on an appearance of teams in the NCAA Final Four.  A series of wagers placed on Columbia would have a lower volatility than the series of wagers on Duke, but that doesn’t ensure that it was the better choice. The Columbia bet is not only centered around zero, it also doesn’t deviate from zero – low volatility, but a (consistently) terrible result.

In Summary.

Be sure to look at the data, and be particularly careful where averages are concerned. Be sure to translate relative value comparisons into absolute measures. And be sure to know when a summary statistic is presenting a relative measure over averages.

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