03.31.2023

## After examining the relationship between day-to-day volatility and yearly returns in the S&P 500, our conclusion might surprise you. Let's find out what the data reveals.

With one trading day left in the first quarter of 2023, the S&P 500 is just above 4,000. Market participants have been watching price dance around this level for almost a year, and many have been watching this psychological level since price first closed above it during April of 2021. Two years later, and day-to-day volatility is a large part of the market commentary. The assertion is, “we must see less day-to-day volatility if we are to expect a sustainable rally.” What follows is an investigation of this idea.

**A VISUAL INSPECTION**

Chart 1 below shows the S&P 500’s price action in the top panel from 1989 to present. The bottom panel shows the day-to-day volatility as measured in percentage change of closing prices. The bars are colored black when the day-to-day volatility is *normal*. For argument’s sake, we will consider day-to-day volatility to be *normal* when the close is less than 1% from the previous day’s close, up or down. The bars are highlighted in gray when the day-to-day volatility is *elevated.* We will consider day-to-day volatility to be *elevated* when the close is more than 1% from the previous day’s close, up or down.

##### Chart 1: S&P 500 Daily. 1988 - present. Daily volatility. Click to enlarge.

A casual look at Chart 1 acknowledges uptrends in the top panel generally have more black than gray bars in the bottom panel. When price moves sideways or down in the top panel, there are generally more gray than black bars in the bottom panel. Said differently, uptrends look to have day-to-day volatility that is more *normal* than *elevated *while sideways and falling trends look to have day-to-day volatility that is more *elevated* than *normal. *Looking back to 1950, where this data set begins, this same relationship between volatility and trend direction is evident. We are using S&P 500 data from www.Optuma.com.

**VOLATILITY & YEARLY PERFORMANCE**

Now that we have visual evidence of day-to-day volatility conditions changing with the direction of price's trend, we will look at the relationship of day-to-day volatility and yearly price returns.

Chart 2 below shows stacked columns in blue and orange. Each bar represents 1 year of trading. The blue part shows the number of day-to-day price changes where the close is more than 1% above yesterday’s close, while the orange part shows the number of day-to-day price changes where the close is more than 1% below yesterday’s close. The black numbers above the stacked bars show the total number of days with *elevated* volatility, blue days plus orange days.

The gray line plot above the stocked bars plots the S&P 500’s yearly price change. This is calculated from the closing price on the last day of the trading year to the closing price on the last day of the prior trading year. The exception is 1950 which uses the closing price of the last trading day and the first.

##### Chart 2: S&P 500 Elevated Volatility and Yearly Price Returns. Click to enlarge.

The rightmost bar on Chart 2 shows the S&P 500 finished 2022 with the third largest number, 122, of volatile days looking back to 1950, behind only 2008 with 134 and 2002 with 125. It turns out that these three years had some of the worst yearly returns on record. Considering all 73 years, the average number of trading days with *elevated* volatility is 52 per year. The average yearly return is 9.1%. See table 2a below fof the details.

After looking at the charts above, the frame or anchor of those largely volatile years and their negative returns makes it is logical to expect greater than average yearly returns when volatile days are less than average and less than average yearly returns when volatile days are more than average.

Looking at Chart 2:

1962, 1970, 1973, 1974, 1990, 2002, 2008, and 2022 stand out as years with more than average volatile days and less than average price returns as expected.

1954, 1963, 1964, 1967, 1972, 1979, 1989, 1995, 2006, 2013, 2014 and 2017 stand out as years with less than average volatile days and more than average price returns as expected.

There are also many years where our expected pattern does not hold.

1950, 1975, 1980, 1982, 1986, 1991, 1997, 1998, 1999, 2003, 2009 and 2020 stand out as years with more than average volatile days and more than average yearly returns.

We also see outliers in the opposite direction, with less than average volatile days and less than average yearly returns. See 1953, 1957, 1960, 1969, 1977, 1978, 1992, 1994, and 2005.

While it was a fun exercise to visually inspect Chart 2, the results were different from our expectations. We found years where the relationship we expected held, and we also found years where the relationship we expected did not hold.

For reference, below is table 2a, the data that populates Chart 2.

##### Table 2a: The Data Behind Chart 2. Click to enlarge.

Using data from Table 2a above, Table 2b below takes a formulaic look beyond our visual inspection. It quantifies the observations based on meeting or failing to meet our expectations, that years with less than average volatility should have greater than average returns and years with greater than average volatility should have less than average returns.

##### Table 2b: Expected and Unexpected Results. Click to enlarge.

Table 2b exposes that 55% of observations, 40 of 73, act as expected. 45% of observations, 33 of 73, do not. While the results lean towards our expectations, 55% is only marginally better than a 50/50 chance of using day-to-day volatility to predict yearly price returns.

**WHAT DO THE STATISTICS TELL US? **

We have done a visual inspection of day-to-day volatility versus yearly returns in the S&P 500. Our visual inspection led to the idea that we might expect greater returns when day-to-day volatility is less than average and to expect less than average returns when day-to-day volatility is more than average, but we noticed many instances of this expected relationship not being true. A formulaic approach to the data reveals how weak that ideal relationship actually is. Only 55% of observations are as expected. Can we discover anything else about this relationship using inferential statistics?

Below is Chart 3. It again plots the day-to-day volatility versus the S&P 500’s yearly returns. This time the observations are plotted as a scatter chart with a linear regression trend line. This *line-of-best-fit* reveals the direction and strength of the correlation between our two variables, day-to-day volatility and yearly returns. The number of *elevated* volatility days is the independent variable and so plotted on the X-axis. The S&P 500’s yearly returns is the dependent variable and so plotted on the Y-axis. Note, we have changed the total number of elevated volatility days to a percentage using the number of volatile days divided by 252, the average number of trading days per year. Comparing percentage change to percentage change is a best practice for regression and correlation calculations.

##### Chart 3: S&P 500. Percentage of volatile trading versus yearly returns. Click to enlarge.

The first thing we see is the blue linear regression trend line traveling from the top left to the bottom right. This tells us there is a negative correlation. A negative correlation is a relationship between two variables such that one increases while the other decreases. In our case, as the independent variable on the X-axis increases, the % of trading days that have *elevated* volatility, the dependent variable on the Y-axis, SPX’s yearly returns, decreases. Looking at the position of the observations, the blue dots, around the regression line, we see observations that are far away from the regression line of *best fit*. This tells us that the data does not *fit* around the line very well. The *fit* could be described as loose with outliers.

To quantify this relationship, we look to the r2 (read r-squared) value printed the top of Chart 3. R2 is called the Coefficient of Determination. It measures the strength of the linear relationship between our two variables. R2 = .1335, or roughly 13.35% of the variation in SPX’s yearly change can be explained by the percent of trading days with *elevated* volatility. Perhaps more familiar is the Correlation Coefficient *r. *This is the square root of r*2*. The square root of .1335 is .36. An *r *value, or correlation coefficient, of .36 tells us there is meaningful, but low, co-movement between the percentage of *elevated* volatility days and SPX’s yearly returns. This validates the deduction we made from Table 2b and learning the expected relationship only held 55% of the time.

For any statistically minded readers, please know our regression analysis returns a p-value of 0.001482 or .1482%. Rounding, we get a p-value of .15%. .15% is less than the accepted 5% *significance level* we need to prove statistical significance between these two variables. Even though our regression analysis is valid, the weak correlation makes using the regression equation to predict yearly returns unhelpful.

**CONCLUSION**

Looking once more to Table 2b, we find above average returns on below average volatility 33% of the time and above average returns on above average volatility 22% of the time. This means that while it is more likely to have a sustained rally should day-to-day volatility subside, there is also a good chance of a sustained rally if the day-to-day volatility does not subside. Less day-to-day volatility is more of a want than a need for a sustainable rally in the S&P 500.

This has been part 1 of the investigation. Part 2 will look into the relationship between 1st quarter volatility and 2nd quarter returns. Perhaps shortening the time horizon and increasing the observations will allow the regression equation to better predict returns based on volatility.

As always, thank you for reading. Please know, this is an open dialogue. This is a living document. Being human, I am imperfect. Please disagree. Pushback and discussion is welcomed and encouraged. Send your feedback to __Louis@eastcoastcharts.com__

This article is for educational and informational purposes only. The author may or may not have a position in the securities mentioned. __Read our full disclaimer here.__

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