Momentum: Measurement?

Question

With the table below as an example, what is the best way to measure relative momentum, comparing EACH DAY TO THE PREVIOUS DAY? Each of the numbers in the table represents the number of stocks up minus the number of stocks down on the New York Stock Exchange each day. I have been doing it thus far by simply measuring percent change and then removing the negative signs. On a related issue, my numbers don't look correct the last two days. For instance, is a move from 7 to -749 really a greater percentage change than from -749 to 1611. The latter seems like a much larger move percentage wise, but according to the excel formula =(a1-a2)/abs(a2), it's not.

8/2: 1611/ 315.09%
8/1: -749/ -10800.00%
7/31: 7/ -99.65%
7/28: 2020/ 541.05%
7/27: -458

Answer 1

If is curious that you are asking this question in Physics / Mathematics and not Finance (I checked). So I will assume the intent of the question is to see if the physical notion of momentum provides any insight into financial concept of momentum.

Momentum is the propensity of a something to move in a specific direction. It consist of inertial mass, which is the degree to which the entity resists change times a velocity giving the direction and degree of movement. From Newton's First Law [1] we know that it takes a external force to change either direction or amount of momentum. That is the physical concept of momentum. With me so far? If it seems geeky - remember you did asked a physicist's perspective.

What is the financial equivalent? In finance it is defined as - the acceleration of a security's price or volume[2]. Physically accelerations are the result of forces acting on an object which changes momentum. So the real question is whether the force will persist to keep the change continuing in a given direction, or once the force has been applied - because of inertia - the trend will persist for awhile until another financial "force" moves it in another direction. So momentum trading is attempting to discover a financial "force" (acceleration) and tracking the momentum (propensity in moving the same direction) resulting from that force.

So what you are looking for are changes that indicate a trend that you believe will persist in a direction for awhile. Lots of luck. But I will leave that as your exercise. The question is how to measure what you have.

Let a1(d) be the number of stocks up in a day and a2(d) be the number of stocks down in a day. Let A(d) be the total number of stocks traded for the day. Then the following would be measures that can be derived from these numbers:

DA(d) = a1(d) + a2(d) : the number of stocks that changed either up or down in a day.

A(d) - DA(d) : the number of stocks that had no change for the day.

dA(d) = a1(d) - a2(d) : the direction and degree of change of the changes in stocks up or down. In physics this would be an acceleration so in finance this is a momentum. Negative indicates downward change and positive an upward change.

To make this number have meaning from day to day and remove the fluctuations of both number of stocks traded and number of stocks changing each day - I would normalize these using either:

M1(d) = dA(d) / DA(d) = (a1(d) - a2(d)) / (a1(d) + a2(d)) : which removes side way move of the stock market (i.e. more sensitive to changes), or

M2(d) = dA(d) / A = (a1(d) - a2(d)) / A : which allows side way moves that dampens the degree of change.

Both of these have the property of changing between -1 and 1 and evenly measuring both up and downward trends. 1 would indicate all stocks traded moved up, and -1 would indicate all stocks moved downward.

Now M1(d) and M2(d) can be used to characterize changes from day to day. Since financial "forces" are highly random (uncorrelated) and the non-random parts are highly non-linear, these numbers must be combined using a great deal of weighting and smoothing to extract predictable trends. M1 and M2 plus other measures that you develop using this same approach are now discrete time series of numbers that are likely correlated to each other and respond differently to external forces. Much of the variation is random and hence noise - you are trying to find the proverbial signal in this noise. But that represents a whole other set of mathematical questions and methods.

Hopefully this helps in answering your original question.