By Ilya Kipnis
Happy new year. This post will be a quick one covering the relationship between the simple moving average and time series momentum. The implication is that one can potentially derive better time series momentum indicators than the classical one applied in so many papers.
Okay, so the main idea for this post is quite simple:
I’m sure we’re all familiar with classical momentum. That is, the price now compared to the price however long ago (3 months, 10 months, 12 months, etc.). E.G. P(now) – P(10)
And I’m sure everyone is familiar with the simple moving average indicator, as well. E.G. SMA(10).
Well, as it turns out, these two quantities are actually related.
It turns out, if instead of expressing momentum as the difference of two numbers, it is expressed as the sum of returns, it can be written (for a 10 month momentum) as:
MOM_10 = return of this month + return of last month + return of 2 months ago + … + return of 9 months ago, for a total of 10 months in our little example.
This can be written as MOM_10 = (P(0) – P(1)) + (P(1) – P(2)) + … + (P(9) – P(10)). (Each difference within parentheses denotes one month’s worth of returns.)
Which can then be rewritten by associative arithmetic as: (P(0) + P(1) + … + P(9)) – (P(1) + P(2) + … + P(10)).
In other words, momentum — aka the difference between two prices, can be rewritten as the difference between two cumulative sums of prices. And what is a simple moving average? Simply a cumulative sum of prices divided by however many prices summed over.
Here’s some R code to demonstrate.
require(quantmod)
require(TTR)
require(PerformanceAnalytics)
getSymbols('SPY', from = '1990-01-01')
monthlySPY
With the resulting number of times these two signals are equal:
[1] 267
In short, every time.
Now, what exactly is the punchline of this little example? ...read more
Source:: r-bloggers.com