## Volatility

In 2006 Ang et al documented the ‘low volatility anomaly’ for individual stocks (*Ang et al. High idiosyncratic volatility and low returns: International and further US evidence. Journal of Financial Economics, volume 91, p1-23*). The finding that low volatility stocks tend to outperform the broad market, was later confirmed in several publications.

In this section we will examine if, for ETF’s, low volatility periods are correlated with (high) returns. Even if this would be the case, in order to build a strategy on a possible temporal volatility anomaly, we need to find out if periods with low or high volatility are followed by periods with high or low returns.

Again our analysis will be mostly based on SPY, the S&P500 tracker, but we will also check if other major-index-ETF’s show analogous behavior.

**Volatility and return in the same time periods**

For SPY we have data starting from January 29 1993. These data we used to calculate Pearson’s correlation coefficient for one-year periods, for one-month periods and for one-week periods.

The value of Pearson’s correlation coefficient can range from +1 to +1 where +1 indicates a perfect positive relationship, -1 indicates a perfect negative relationship, and a 0 indicates no relationship exists.

When the value of the correlation coefficient is between 0 and +/- 0.3 we assume a small or no correlation; a value between +/-0.3 en +/-0.7 is a moderate correlation. Values higher than +/-0.7 indicate a strong correlation.

If we calculate the value of the correlation between volatility and return in the same period for all one-year periods since january 1993, we find a value of -0.47. So we can say **a moderate negative correlation exists between volatility and return in the same one-year period:** higher volatility is moderately correlated with lower returns. For one-month periods and one-week periods the value is -0.19 and +0.08 respectively. So apperently no real correlation exists between volatility and return in the same one-month and one-week periods.

To learn more about the moderate negative correlation between volatility and return in the same one-year period we calculated the distribution of returns in years with the highest and lowest volatility.

Figure 1 and 2 show the distribution of returns of SPY in one-year periods when volatility is higher than the 90th percentile and lower than the 10th percentile respectively.

In one-year periods when volatility is above the 90th percentile we see the mean and median returns are both negative. Actually in almost 60% of these one-year periods returns are negative.

When volatility is under the 10th percentile, we see a completely different picture. Mean and median returns are both 17.8%. This is much higher than the mean return (CAGR) of SPY for the complete 28-year period, which is 10.44%. Actually none of these one-year periods with volatility under the 10th percentile, has negative returns.

We can conclude that, at least in one-year periods with the most extreme volatilities, the low-volatility anomaly really exists: periods with the lowest volatility have better returns than periods with the highest volatility. Moreover, returns of low volatility periods are much better than the mean return (CAGR) in the complete period for which we have data (January 1993 till now).

**Returns in relation to volatility in previous periods**

If we want to build a strategy upon the relation between volatility and returns, we need to examine the relationship between volatility in the actual period and returns in the following period.

Again for one-month periods and one-week periods we find no correlation between volatility in the actual period and return in the next period (correlation coefficient 0.17 and -0.003 respectively).

Overall we find the same results for one-year periods: no overall correlation between volatility in one year and return in the next year (correlation coefficient 0.06). But as for returns in the same year, we will take a closer look at the years with highest (above the 90th percentile) and lowest volatility (below the 10th percentile). The results are shown in figure 3 and 4.

Figure 3 shows returns in the years following a year with low volatility (< P10). We see returns in these years are excellent: mean return 17.4%, median return 18.9%. Negative results are rare: P10 results is positive: 1.98%. These results are comparable with the returns in the years with low volatility themselves. So, we can say for low volatility years, returns are excellent in the same year and in the next year as well. This finding is consistent with the momentum anomaly on which most of our strategies are based (SMI, DMI, CCPI).

Figure 4 is more surprising. The returns in years following a year with high volatility (> P90) are shown. These returns are even better than the returns in years following low-volatility years: mean return is 27.1%, median return is 24.4%. Only one year with a negative return (-3.3%). This is completely different from the returns in the high-volatility years themselves (figure 1). So here, instead of the effects of the momentum anomaly, we see a trend reversal. High-volatility years have low returns but the next year has fantastic returns.

Since years with high volatility have bad returns, we need to examine whether the trend reversal in returns after high-volatility years is specific for high volatility. So the question we want to answer is: do we see the same trend reversal in all years with bad returns. But first we’ll look at the returns in years following high-return years. Figure 5 and six show the results.

In figure 5 we see that years with high returns are followed by years with excellent returns: mean return 18.7%, median return 19.7%; no years with negative returns. The momentum anomaly at work.

As illustrated in figure 6, years with low returns are followed by years with average returns: mean return 12.3%; the third percentile of returns is still negative: -0.7%. So for returns in years following low-return years we see a regression to the mean , but no real trend reversal, as we see after high-volatility years.

We can conclude that trend reversal in returns is specific for years following high volatility years and is not a consequence of the lower returns, we see in the high-volatility years.

**Does Trend Reversal in returns after high-volatility years also show up in other indexes?**

The figures below show the distribution of returns after high volatility years for other indexes.

Being The New Belgian Dentist, we start with the BEL 20 index (tickersymbol ^BFX). For BEL 20 we have data starting from april 9 1991. For the complete 30-year period up to now, the annualised return (CAGR) for BEL 20 is 4.27%; annualised volatility is 18.44%.

Figure 7 shows the distribution of BEL 20 returns in years following years with volatility above the 90th percentile.

For BEL 20, as for SPY, we find excellent returns in years following high-volatility years. Mean return is 23.1% (much higher than the mean return for the complete period, which is 4.27%). Median return is 23.40%. And we see only a few periods with negative returns: P10 for returns is 4.61%; minimum return is -18%.

Figure 8 shows the results of the same excercise for iShares MSCI World ETF (tickersymbol URTH). For URTH we have data since January 12 2012. Mean annualised return for URTH in the complete period is 12.69%, mean volatility is 17.26%. Returns in one-year periods after years with high volatility, again are fantastic. Mean return is 27.68%. No negative returns; minimum return is 1.52%. Of course the returns are this high because we only have data since 2012; so the two big bear markets of the 21th century are avoided. But even than, the mean return for these years following high-volatility years is much higher than the mean return for the complete period: 27.68% versus 12.69%.

For iShares Emerging Markets ETF (tickersymbol EEM) we have data starting from april 14 2003. The results are shown in figure 9. Annualised return for the complete period is 10.67%; annualised volatility is 28.73%. Again the distribution of returns in years following high-volatility years is better than the distribution of overall returns. Mean return is 32.1%. There are years with important negative returns: P10 return is -17.48%. But P20 return already is +13%.

Figure 10 illustrates the results for the DAX Performance Index (tickersymbol ^GDAXI). For the DAX Performance Index available data go back to December 30 1987. Annualised return for the complete period up to now is 8.48%, annualised volatility is 22,22%. Again mean and median returns are much better in years following high-volatility years (26.14% and 24% respectively). There are years with important negative returns, the worst being -18.17%. but P10 return is already 8.23%.

Figure 11: results for CAC 40 (tickersymbol ^FCHI). Data available from March 1 1990. Annualised return for the complete period is 4.24%; annualised volatility is 21.72%. Mean return in years after high volatility years is 17.74%. Again we see years with negative returns; the worst return in these years after high-volatility years is -9.3%. P10 return is 0.9%.

For the Nasdaq Composite (tickersymbol ^IXIC) data are available starting from february 5 1971. Annualised return for the complete period is 10.34%, annualised volatility is 19.82%. Figure 12 shows Nasdaq Composite doesnot show the trend reversal of returns in the years following high volatility years. Mean return in these years is 7.53%. The worst return is -62.75% and even P40 return is negative: -11%. Figure 13 shows even worse results for the tracker QQQ, for which our data go back to March 10 1999. Annualised return for QQQ for the complete period is 9.86%, annualised volatility is 27.69%. The mean return in years following high-volatility years is -34.94%. Actually the P90 return is still very negative: -17.39%.

**Volatility and returns conclusions**

Overall we see bad returns in years with highest volatility (above 90th percentile) and good returns in years with lowest volatility (under 10th percentile).

In the years following the lowest-volatility years we see excellent returns.

But the most remarkable observation is a trend reversal in return in years following highest-volatility years. The returns in these years are even better than the returns in years following low-volatility years. This trend reversal effect is pervasive in all indexes we examined, with one notorious exception: for Nasdaq Composite (and even more so for QQQ, for which available data go back less far) years following highest-volatility years have terrible returns: actually it is difficult to have worse returns than we see for QQQ in these years.

Since the overall correlation between volatility and return in the subsequent years is moderate at best, and the strongest effects are seen in the extreme-volatility years (<P10 and >P90), we cannot use volatility alone to design an investing strategy. If we would for example choose to invest only in years following highest-volatility years, we would be invested in only 1 year out of 10. Even if returns in these single years would be fantastic, the returns over 10 years (in which we would be invested only in one year) would be at best very moderate.

But this doesnot mean the observation of trend reversal of returns in years after high-volatility years, is worthless. For investors it is good to know, that in years following years with very high volatility, returns are very often far better than average. This can help investors overcome their fear after these scary years. Of course, investors should be aware that there is one terrible exception to this rule: Nasdaq/QQQ.