Why Simple Moving Averages Move Less Than Stock Prices

Most traders use simple moving averages as part of their trading techniques. They buy when the stock price is above a certain moving average, or sell when it is below. Some traders also look at moving average crossovers for buy and sell signals. All of this information is found in the plethora of trading books available. Some are better than others, but we'll discuss that later.

Now go and look at your favorite stock chart, and plot several moving averages, 5, 10, 20, 50, 100, 200, etc. If you look at the percent increase and percent decrease for the stock price on a given day and compare it to the moving average, you will notice the moving average moves less than the stock price;i.e., it has a smaller percent increase/decrease. This is one of the main reasons to use moving averages, because they give you a better picture of what the stock price is doing by removing some of the "noise".

This should all be somewhat intuitively true, but why is true mathematically?  Let's start first with the following.

Define X_m = (x_1 + x_2 + ... + x_m)/m to be the arithmetic mean of the closing price of the last m trading days. This is the m-day Simple Moving Average. Once a new day passes, X_m changes, more precisely,
x_1 becomes x_2
x_2 becomes x_3
.
.
.
x_m becomes x_(m + 1),

where x_(m + 1) is the last days closing price.


Now if you remember any statistics, X_m is a random variable, therefore it actually has a standard deviation.

Now let SIGMA = standard deviation of the closing prices of the stock.

Then the standard deviation of X_m is  SIGMA/Sqrt(m).

Note that SIGMA is always smaller than(or equal to in the case where m = 1) than SIGMA/sqrt(m). More precisely

SIGMA <= SIGMA/Sqrt(m), always, no matter what, even if it snows in Puerto Rico.

This means that simple moving averages, on average, move less than stock prices.

Oh and as far as books, the best one I have read is linked below. Yes it is expensive but if you are going to buy a book on TA, buy this one, it is worth owning. The methods described in this book work, the problem is that most people don't follow those methods.