Stock Return Distribution(s) ... a continuation of CLT

After playing with the Central Limit Theorem I got to thinking that ...

>Thinking? That was your first mistake.
Pay attention.
Once upon a time I was asked by John Bollinger about the relationship between the Standard Deviation of daily stock returns and the Standard Deviation of stock prices over the past n days. When one speaks of the Standard Deviation (as it concerns stocks), one (usually) is referring to the SD of returns, not prices. However, Bollinger Bands considers the SD of prices ... which ain't usual.

Anyway, I tried to find a relationship and wrote a (lousy) tutorial on the subject.

Now I figure I should look at the daily returns over some n-day time period and calculate the total return over those n days and forget any mathematical gesticulation.

What I mean is this:

  • Suppose the daily returns for a month are r1, r2 ... r25 ... where we pretend that there are n = 25 market days in a month.
  • We calculate the total n-day gain: (1+r1)(1+r2)...(1+r25).
    By gain" I mean that $1 will become $(1+r1)(1+r2)...(1+r25) after n = 25 days.
Okay, so I do this (to see what the distribution of Total n-day Gains might look like):
  • I look at the daily returns for GE stock over the past 10 years. That's (about) 2500 daily returns.
  • I pick 25 successive returns at random and calculate the 25-day gain.
  • I repeat this ritual a jillion times and plot the distribution of these monthly gains.
  • I get something like Figure 1.
>A jillion times?
Well ... 5000 times, actually.
>I take it the red curve is one of them Normal distributions, eh?

Figure 1
Yes, but I'm not interested in the distribution. I just like to see that it looks like a "bell" ... not necessarily a "Normal" bell.    
If it's bell-shaped, we'll feel warm all over.

This is what I get (using GE as an example):

a Bell

It shows the possibilities for a $1K portfolio after n-days.
I tried 25-days then 50-days then 75-days. (That's like 1, 2 and 3 months worth of daily returns)
>How did GE perform?
Over that 10-year period? Like this:
The interesting thing (for me) is that I used all daily stock return info so that ...

>I would expect no less!
Well, gurus often do this:

  • Look at a jillion daily historical returns and extract a Mean and Standard Deviation
    ... and whatever other numbers seem tasty.
  • Discard the jillion returns, retain the magic numbers and construct some distribution.
  • Then use that distribution to predict future portfolios.
>If you use actual returns, then you don't have to assume some distribution, right?
Right ... but there ain't no interesting math involved.

10-years of GE
I remember getting all excited about Ito Calculus and writing an umpteen part tutorial.
It was really neat and, at the end, I could provide portfolio dsitributions T years into the future, like this
Of course, it's a log-norml distribution defined by a Mean and Standard Deviation and ...

>And it's smooooth, not your jagged distribution. I like it!
Well, predicting the future is a black art so you might as well use whatever makes you happy.
Using historical returns is like saying: "I expect the future to be something like the past."

The thing that I like about generating charts like Figure 1 is this: It's Exact!
I'm not claiming that it's some future distribution. I'm claiming that it's what actually happened in the past.

>Then you should use whatever makes you happy.
Thank you.

Ito predictions
Click on the picture to download ... and play:

You type in some yahoo stock symbol and click a download button.
Then you can play with the distribution(s) by picking:
n =the number of days (example: 25)   and   the number of n-day gains (example: 1000).

>Is it any good?
Yes, of course. It's entertaining, fascinating, educational, time consuming ...