I’ll begin with a familiar image:
That plot shows the closing values of the S&P 500 index from 1990 until today. It’s a useful representation — at a glance, you can tell when the market rose and fell. That said, it does have some problems: we’re looking at absolute movements in the index, when we really care about relative changes. We don’t want the early 1990s to be dwarfed by the 2000s, so let’s take a look at daily logarithmic returns:
Much better! I find this plot extremely interesting. Each point shows the logarithmic returns from holding the index close-to-close until the next trading day. The return volatility appears to vary dramatically: the mid-1990s and mid-2000s were calm; 2008 was extremely turbulent. I didn’t see that in the first plot, but now it’s quite clear.
Let’s continue with a plot of the 200-day running mean of logarithmic returns. If you bought the index on a given day, what average logarithmic returns would you have seen across the next 200 trading days? Look below:
Are the S&P 500 returns i.i.d.? At a glance, I would say no, since the mean and variance both appear to change across time. I’m sure this can be tested statistically, but I would much prefer to look at these plots!
Here’s my code; suggestions and comments welcome.
# Yahoo finance url for S&P 500 data str