Experts and pundits sometimes cite a high Chicago Board Options Exchange (CBOE) Volatility Index (VIX), the options-implied volatility of the S&P 500 Index, as contrarian indication of investor panic and therefore of pending U.S. stock market strength. Conversely, they cite a low VIX as indication of complacency and pending market weakness. However, a more nuanced conventional wisdom considers both very high VIX and very low VIX as favorable for future stock market returns. Does evidence support belief in either version of conventional wisdom? To check, we relate the level of VIX to S&P 500 Index returns over the next 5, 10, 21, 63 and 126 trading days. Using daily and monthly closes for VIX and for the S&P 500 Index over the period January 1990 through July 2014 (296 months), *we find that:*

VIX data for three stock market trading days are missing. For these three days, we assume that VIX has the same closing value as the previous trading day.

The following chart compares the level of the S&P 500 Index to VIX over the available sample period. The relationship appears to mostly inverse, with VIX spiking at pronounced market bottoms. However, it is not obvious that the level of VIX reliably predicts stock market returns.

The average level of VIX over the entire sample period is 20.04. Average levels for the first and second halves of the sample period are 19.51 and 20.57, respectively, supporting some belief that the series has a stable average of about 20. However, existence of a few persistent regimes makes it difficult to decide exactly what level is high or low at any given time.

For greater precision, we apply some statistical tests.

The next chart employs daily data to calculate correlations between VIX and S&P 500 Index future returns over the selected horizons for the entire sample period and the decades of the 1990s, 2000s and 2010s (partial). Results for the entire sample period, with positive correlations between VIX and future returns, are slightly favorable to conventional wisdom. However, correlations vary considerably across subperiods, relatively strong during the 1990s and 2010s (partial) but much weaker during the 2000s. Inconsistency across subperiods undermine belief in reliable relationships.

Return intervals used to generate this chart are overlapping, potentially biasing the statistics. For additional insight, we look at future return intervals separately and winnow data accordingly to eliminate measurement overlap.

The following scatter plot relates S&P 500 Index 5-day future return to VIX level over the entire sample period, with the data winnowed such that no 5-day return intervals overlap. The Pearson correlation for the two series is 0.04 and the R-squared statistic 0.002, indicating practically no linear relationship.

R-squared statistics for 10-day, 21-day, 63-day and 126-day future returns are 0.000, 0.000, 0.008 and 0.006, respectively. In other words, the level of VIX explains less than 1% of the variation in S&P 500 Index returns across all horizons. Results generally do not support belief that the level of VIX predicts stock market returns.

To check for non-linearity in the relationship, we look at average future returns by ranges of VIX.

The next chart summarizes average S&P 500 Index 5-day future return by ranked tenth (decile) of VIX, both for the entire sample period and for two equal subperiods (winnowed data). The number of observations for each decile is about 124 for the entire period and 62 for subperiods. Lack of systematic progressions across deciles and inconsistencies between subperiods undermine belief in a reliable relationship between 5-day future stock market returns and VIX.

However, the relationship between VIX and variability (standard deviation) of 5-day future returns is generally systematic and consistent. A low (high) VIX indicates low (high) variability of 5-day future returns.

What happens if we apply this systematic variability as a risk adjustment?

The next chart summarizes average S&P 500 Index risk-adjusted 5-day future return (average return divided by standard deviation of 5-day future returns) by decile of VIX for the entire sample period and two equal subperiods (winnowed data). Again, lack of systematic progressions across deciles and inconsistencies between subperiods undermine belief in a reliable relationship between risk-adjusted 5-day future stock market returns and VIX.

Might the power of VIX to predict risk-adjusted stock market returns emerge for a longer return measurement interval?

The next four charts summarizes average S&P 500 Index risk-adjusted future returns at horizons of 10, 21, 63 and 126 trading days by range of VIX for the entire sample period and, except for the longest interval, two equal subperiods (all based on winnowed data). The number of observations decreases with measurement interval, so we reduce the number of ranges of VIX to five (quintiles) for 10-day and 21-day measurement intervals and to three (terciles) for 63-day and 126-day measurement intervals. The number of independent 126-day measurement intervals is so small (18 per tercile) that we do not consider subperiods.

The progression to longer measurement intervals generally supports belief in the nuanced conventional wisdom that both very high VIX and very low VIX as favorable for (risk-adjusted) future stock market returns.

However:

- Return sample/subsample sizes decrease as measurement interval lengthens.
- Analyses are in-sample. An investor operating in real time may have experienced different results, especially because of the persistence of VIX in a few regimes.

For another perspective, we look systematically at the lead-lag relationship between VIX and stock market returns.

The final chart shows correlations between monthly S&P 500 Index returns and two measures of volatility for lead-lag relationships ranging from stock market returns lead volatility by 12 months (-12) to volatility leads stock market returns by 12 months (12) over the entire sample period. We focus on monthly returns because VIX is designed to look ahead one month. The two measures of volatility are:

- Implied volatility (VIX), measured at the end of each month.
- Realized volatility, measured as the standard deviation of daily S&P 500 Index returns during the 21 trading days (about one month) preceding the end of each month.

Results suggest that:

- The coincident relationship (0) between both measures of volatility and returns is negative. When stocks rise (fall) during a month, volatility measures tend to fall (rise).
- Returns lead volatility measures with negative relationships, most notably at a one-month horizon (-1).
- VIX perhaps leads returns weakly with a positive relationship (high VIX indicates strong returns) at horizons of two to six months, but the indication is not compelling in magnitude or consistency.
- VIX and realized volatility relate to future returns in modestly different ways. The common peak in correlation at six months (6) could be a result of randomness.

Note that this approach assumes a linear relationship between volatility and return, thereby limiting conclusiveness. For example, a closer look shows that most of the power of monthly return to predict VIX the next month derives from just a handful of extreme observations.

In summary, *evidence from simple tests offers a little support (at intermediate-term horizons with consequently small samples/subsamples) for belief in the evolved conventional wisdom that both very high and very low levels of VIX are favorable for future stock market performance.*

Considerations relevant to trading on a VIX signal include:

- The above tests involve testing different parameter settings (the return interval) on the same data set. Other parameter settings and variations on the VIX indicator may work better, but the more variations tested, the stronger the data snooping bias (role of luck) in extreme results.
- As noted, the above tests have an in-sample perspective, in that they look across entire sample periods or subperiods and thereby use data unavailable in real time.
- Especially in the presence of apparently multi-year VIX regimes (making it difficult to define “high and “low” levels in real time), the 24.5-year sample period is modest for testing of VIX as a stock return predictor based solely on past data.
- Use of a rolling window of historical data to determine whether VIX is high or low requires selection of parameter values (window length, thresholds for high and low) that invite optimization and attendant data snooping bias.