CM in MA

Measure, Don't Model: Probability, VIX and Bad Math

Good premise. An article that covers a great deal of ground. I'll pick up just one narrow thread, that being calculations using the VIX. Unfortunately, this article seems to perpetuate a myth surrounding the VIX. Specifically, that the VIX predicts an expected return or range of returns. While that may be somewhat true, there's a bit more to it. And that bit more calls into question the calculations and conclusions of the article. Let's leave aside whether the VIX does, in fact, predict anything and deal with what it is supposed to predict.

First, this from the CBOE website: the VIX is a measure of expected volatility calculated as 100 times the square root of the expected 30-day variance (var) of the S&P 500 rate of return. The variance is annualized and VIX expresses volatility in percentage points.

To translate the CBOE definition: the VIX is the expected first standard deviation of the annual S&P 500 rate of return. So, to try to use the VIX for anything, one must first have an "S&P rate of return" for the time period under consideration. That's a big problem since it leads to relying on mean returns, and we know that market returns are lumpy. Since the VIX is already annualized, here's what it tells us if we assume the widely used 8% annual return on the S&P. For a VIX of 25, the expected annual return will be between -17% and +33% at the first standard deviation. At the second standard deviation, it would be between -42% and +58%.

It's easy to calculate the first (or any other) standard deviation of the expected return given any VIX value and time period. The article shows correctly how to do this. For example, at VIX=25 for 30 calendar days, the first standard deviation is 7.17%. But that standard deviation, as well as second, third, etc. deviations, has to be applied to the expected 30-calendar day return. The equation in the article assumes this value to be zero. Perhaps that's ok for very short periods of time (a few days), but at 30-calendar days that is a dangerous assumption. It certainly matters when evaluating the accuracy of 30-calendar day predictions.

All that said, caution is warranted when using means and standard deviations for expected market returns!