# On the relationship between the S&P 500 and the CBOE Volatility Index (VIX)

Besides going over the course syllabus during the first day of class on Tuesday, January 15, we will also discuss a particularly important “real world” example of financial risk. Specifically, we will look at the relationship between stock market returns (as indicated by daily percentage changes in the SP500 stock market index) and stock market volatility (as indicated by daily percentage changes in the CBOE Volatility Index (VIX)):

As indicated by this graph from page 21 of the lecture note for the first day of class, daily percentage changes on closing prices for VIX and the SP500 are strongly negatively correlated. In the graph above, the y-axis variable is the daily return on the SP500, whereas the x-axis variable is the daily return on the VIX. The blue points represent 7,311 daily observations on these two variables, spanning the time period from January 2, 1990 through January 7, 2019. When we fit a regression line through this scatter diagram, we obtain the following equation:

${R_{SP500}} = 0.0588 - 0.1139{R_{VIX}}$,

where ${R_{SP500}}$ corresponds to the daily return on the SP500 index and ${R_{VIX}}$ corresponds to the daily return on the VIX index. The slope of this line (-0.1139) indicates that on average, daily VIX returns during this time period were inversely related to the daily return on the SP500; i.e., when volatility as measured by VIX went down (up), then the stock market return as indicated by SP500 typically went up (down). Nearly half of the variation in the stock market return during this time period (specifically, 48.73%) can be statistically “explained” by changes in volatility, and the correlation between ${R_{SP500}}$ and ${R_{VIX}}$ comes out to -0.696. While a correlation of -0.698 does not imply that ${R_{SP500}}$ and ${R_{VIX}}$ will always move in opposite directions, it does indicate that this will be the case more often than not. Indeed, closing daily returns on ${R_{SP500}}$ and ${R_{VIX}}$ during this period moved inversely 78.4% of the time.