### First and Second Derivatives of the SPX

November 23, 2009
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Here is another off the wall post that will produce a few more interesting

observations. Please see my last post **Recipe for Disaster** to see where this train of thought came from and I will be linking some conclusions together between this post and that one.

First, what the hell is a derivative?

Maybe you are familiar with this concept on calculus and maybe not. Here is the real background if you are interested (**http://en.wikipedia.org/wiki/Derivative**), but here is the 1 paragraph version that gives you an understanding for this post.

A function is a description of how a dependent variable (lets call it **y**) moves with respect to an independent variable (lets call it **x**). A straight line is a very simple example: y = slope*x + intercept. Or a parabola: y = x^2. A derivative tells you **the instantaneous rate of change** of that function as x changes. It is also the slope of the function, but rate of change is the key concept. A straight line is always changing at a constant value, hence its derivative is a constant number. The parabola has a negative slope for x < 0, has a slope of zero at x = 0, and has a positive slope for x > 0. The second derivative tells you the rate of change of the first derivative.

Okay, that was a bit abstract. So lets use a physical example. Position (or displacement) as a dependent variable tells you where you are with respect to time, the independent variable. The first derivative will tell you the rate of change of position with respect to time. This is the velocity, or speed. Everybody is familiar with this concept. The second derivative will tell you the rate of change of velocity with respect to time. This is the acceleration. Another familiar concept.

Now lets apply this concept to the stock market. I will specifically be using the S&P 500 here. The price is analogous to the position, and we will look at the first and second derivatives of price, to help us to understand the behavior of how the price is changing with time.