Besides going over the course syllabus during the first day of class on Tuesday, August 27, 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 7,465 daily observations on these two variables, spanning the time period from January 3, 1990 through August 16, 2019. When we fit a regression line through this scatter diagram, we obtain the following equation:

,

where corresponds to the daily return on the SP500 index and corresponds to the daily return on the VIX index. The slope of this line (-0.1129) 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, 48.87%) can be statistically “explained” by changes in volatility, and the correlation between and comes out to -0.699. While a correlation of -0.699 does not imply that and will always move in opposite directions, it does indicate that this will be the case more often than not. Indeed, closing daily returns on and during this period moved inversely 78.43% of the time.

You can see how the relationship between the SP500 and VIX evolves prospectively by entering http://finance.yahoo.com/quotes/^GSPC,^VIX into your web browser’s address field.