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Fat Tails and VIX Day of Week Returns

I confirm the findings in a published paper and quant blog about robust VIX average daily increase on Mondays and average daily decrease on Fridays. I demonstrate that these averages, although seemingly robust, cannot be used in inferences about market returns because due to fat tails in VIX returns they are random variables.

Eran Ravin in his excellent quant blog (see reference at the end) confirmed the findings in an academic paper (also referenced below) about the robust VIX average daily increase on Mondays and average daily decrease on Fridays. One claim in the paper is that:

Consistent with decreasing sentiment on Monday I document a strong and robust 2.16% average daily increase in VIX on Mondays. On Fridays, VIX experiences an average daily decrease of nearly 70 basis points. While decreasing sentiment is associated with increases in VIX, it is also associated with a “flight to safety,” and therefore theory predicts that a decrease in sentiment will be associated with increasing returns for Treasuries.

Let us start with verifying these claims using our methodology. Below is a chart that shows Monday (day of week = 1) and Friday (day of week = 5) returns and associated hypothetical equity curves.  The daily returns calculation period is from 01/03/1990 to 04/26/2019.

Average Monday return is 2.16% and average Friday return is -0.73%, and these are in agreement with the blog and academic paper referenced. The important observation here is that the growth in hypothetical equity from investing in VIX on Mondays (buy at the previous close and exit at the close of Monday) shows robust  increase while on the other hand investing in VIX on Fridays ( buy at the previous close and exit at the close of Friday) shows a robust decrease in equity.

Do we have something here? Apparently there is nothing here to get excited about. The main reason is small samples from fat-tailed distributions. We have 7386 returns and this is a small number to make conclusions about returns generated by fat-tailed stochastic processes. We need a huge number of observations, probably more than a trillion to make any statistically significant conclusions. Take a look below at the VIX daily returns frequency distribution:

The frequency distribution excess Kurtosis is 17.7. There is a +115.60% daily return that occurred on February 5, 2018, at the right tail, which is not an outlier by the way.  This type of low probability returns can occur any time. The following day, February 6, 2018,  the XIV ETP lost about 95% of its value and was terminated as some may  remember.

The important consideration here is that the average of +2.16% for Mondays is not robust but a random variable. But let us look only at the distribution of Monday returns below:

Excess Kurtosis is much higher at 39.31, as expected. The population average is unknown and variance is probably not defined.  There is no way to make any valid inferences based on averages from limited samples from this kind of distributions. Averages converge to distribution mean, or expectation, only at the limit of sufficient samples. The sample here is very small for making any valid claims.

Enter the practitioner world

Let us put statistics aside for now and look at the practitioner side. After all, I am a practitioner. If rising VIX is truly associated with decreasing sentiment and falling VIX with increasing sentiment, then, as the paper author claims, this should be reflected in S&P 500 total return (SPY).

From the above SPY daily chart we may see that Monday and Friday returns since inception are statistically indistinguishable. Despite the huge divergence in VIX average returns on Monday and Fridays, there is no effect on SPY returns; win rate and equity growth are about the same.

What about TLT, the bond ETF? Below is the relevant chart.

The opposite of what the paper author claims has occurred in TLT at least after 2011: Monday returns are flattish to decreasing but Friday returns are increasing.

But this is not all: let us look at ZIV below, the inverse XIV ETF. This ETF should increase in value when VIX falls and decrease when it rises. We would expect then to see the equity curve from holding ZIV on Mondays to decrease and that from holding it of Fridays to increase.

Surprisingly, the opposite effect is also observed for ZIV since inception and Monday returns grow faster than Friday returns. The stochasticity is evident from the equity returns for Fridays: there is a decrease in equity curve  in 2015 and 2016.

I also looked at next day returns to maybe identify any effects from rebalancing but the result were not significant either.

So what is going on? Below are possible reasons from the discrepancy:

  • Fat tails
  • Randomness
  • Selective perception

Although it appears that VIX returns are higher on Mondays and lower of Fridays, on the average, averages are misleading. Traders and investors do not make decisions based on longer-term averages but based on patterns evaluated in short-term price action context, news and a host of other parameters and strategies. Averages can be misleading, especially when they are from limited samples taken from fat-tail stochastic processes.

References

Eran Ravin’s blog article

Academic paper

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