$owl$

"Change in the Rate of Change"

~Being a stock & options trader (investor in some cases), I listen to many talks business professionals give on TV & read data in the printed media.

~One expression that has become popular recently is the 2nd derivative. They use this to indicate a trend in economic conditions measuring a given quantity (like a stock price, interest rates, unemployment, etc.).

~To understand this, one must understand the 1st derivative & what it measures in calculus. So, let's review that. More mathematical detail in given on this web page: See The Derivative.

~The 1st derivative of a function has various interpretations, both geometric & in general terms. Geometrically, it measures the slope of the tangent line drawn to the curve at a specified point. For a non-linear curve, there are many different tangent lines with various slopes. As you move along a non-linear curve, each point has its own tangent line with its own slope. So, the derivative is a way of describing a function at an isolated point (by means of the slope of its tangent line). A Very powerful tool & gives rise to solutions of many dynamic problems. For the first time we are able to isolated a function at an instant (point) & describe or modify its behavior.

~In general terms, it measures the instantaneous rate of change of the function with respect to the independent variable, usually x. This is equivalent to above geometrical description, since the slope of a tangent line at a point measures the instantaneous rate of change of y with respect to x at a specific point. Since slopes change as we move along the curve, this rate of change gives us information on the behavior of y at each point along the curve. It describes how the quantity y changes as x changes along the curve.

~If the 1st derivative is negative, y will decrease as x increases. Graphically speaking, the graph of the function will fall from left to right. If the 1st derivative is positive, y will increase as x increases & the graph of the function will rise from left to right.

~In the business world, the function could be a number of popular quantities. Stock price is probably the most popular. The independent variable is usually time.

~So, if the 1st derivative of stock price wrt (with respect to) time is positive, the price is on the upswing. Similarly, a negative 1st derivative, means the stock price is falling.

~Now for the 2nd derivative. It measures the instantaneous rate of change of the 1st derivative. It gives us the same information about the 1st derivative as the 1st derivative gave about the function y. It will tell us if the first derivative is increasing or decreasing.

~When the 1st derivative increases, the slopes of the tangent lines (geometric interpretation of the derivative) are increasing. This will shape the graph of the function being measured in a very distinct way. It will be CONCAVE UP. Like a U. Be careful here, the graph could also look like a partial backwards C, just make sure no vertical lines cut it more than once (destroys the function property).

In this case the 2nd derivative is POSITIVE.

~When the 1st derivative is decreasing, the slopes of the tangent lines are decreasing. This forces the curve to become CONCAVE DOWN. Like an inverted U. Also, be careful here, the graph could look like a piece of a C, just make sure the piece does not have vertical lines cutting it more than once (destroys the function property).
In this case the 2nd derivative is NEGATIVE.

~One way to remember this is the following:

2nd derivative positive: Concave up----Curve holds water.

2nd derivative negative: Concave down-----Curve spills water.

~The important feature in business is the fact that the curve can be falling and still become concave up. The point where the concavity changes is called the INFECTION POINT. This is the point where the TREND REVERSAL BEGINS. The curve is still falling (i.e., stock price is still on the decline, but the rate of decline has reversed). The tendency after this point is for the rate of decline to become less & less until the curve finally hits bottom & swings upward.

~The 2nd derivative can also give a negative trend reversal, as well. In that case, we would go from a concave up situation to a concave down (but still increasing curve) to a high point, then to a decline in the curve.

~In economics, if y measures OUTPUT of a business, and x measures INPUT ( a quantity invested in the business, i.e, dollars, workers, equipment, etc.), this infection point (a change from concave up to concave down) is referred to as "The point of diminishing returns". At this point of trend reversal, less & less output is gained from more input.

~Important case in business: Let's take the case where the stock price (could be any other quantity of interest) is rising and the shape of the curve is concave down. This would indicate a negative trend for the quantity. However, as it rises the curve could reach an inflection point (reversal in concavity) at a point at the very top of it's concave down shape and become concave up there afterwards. This would indicate a "breakout" situation for the quantity and a positive trend for that quantity. This type of inflection point is of popular focus in the business world. The inflection point here is high and gives a "flat point" on the curve (i.e., the first derivative is also equal to zero).

~So, in summary, when a business commentator mentions the 2nd derivative, it usually refers to this trend reversal. Keep in mind, that a quantity being measured can still be falling & its graph has become concave up (positive trend reversal) or the quantity could still be rising & its graph has become concave down (negative trend reversal). What many do not mention (and should) is the sign of the reversal. In other words, they should say, "We have reached a point where the 2nd derivative has become positive". This would indicate a positive trend reversal or "We have reached a point where the 2nd derivative is negative", indicating a negative trend reversal.