Stock Path Volatility
... and other thoughts, questions & musings
Motivated by comments of Taleb
Consider the chart in Fig. 1.
The big black dots are the monthly values of some stock (or portfolio).
The smaller blue dots follow the weekly values.

>Every 4 weeks gives a month? Are you kidding? It's 365/12 except leap years when ...?
Patience. I want to demonstrate something and we'll assume 4 weeks to a month and 12 months to a year ... hence 48 weeks per year.

If we calculate the mean and standard deviation of the weekly returns and the monthly returns, we get the numbers shown in Fig. 1. Note that the monthly mean is (roughly) 4 times the weekly mean (since there are 4 weeks per month) and the monthly standard deviations are (roughly) 2 times the weekly standard deviations (where 2 = SQRT[4]).

>Huh? Square root of 4?
Yeah, that has to do with the square root of time thing ... but don't worry about it. Just observe.

>And what's that "Mean Deviation" and, I presume, its standard deviation?
Uh ... yes.
SD[Dev] stand for the standard deviation of the deviations of returns from their mean.
We'll get to that later.

Figure 1
If we "annualize" the weekly values we'd multiply the returns by 48 and the standard deviation by SQRT[48], giving: annual return = 35% and annual SD = 24%.
To annualize the monthly values, we'd use 12 and SQRT[12] and get: annual return = 36% and annual SD = 26%.
Notice that ...

>They're pretty close, eh?
Yes. I find that's usually the case even if we change the numbers from 48 weeks to 52 or ...

>Aah, but how do you generate the weekly returns?
I download a years worth of weekly prices for some stock then randomize the weekly returns and ...

>Where's the spreadsheet?
Will you let me finish a sentence!
There's a spreadsheet to play with and it looks like this: (Click on the picture to download.)

In the above example, we're using GE stock.
Every time you press F9 you get a different collection of weekly (hence monthly) returns.
The charts show the weekly values of a $10K portfolio ... and the values every 4 weeks which we're calling a month.

>I notice that the big black dots coincide with the small blue dots every 4 weeks.
Clever observation.
So, although the portfolio arrives at the same final value, after a year (and every 4 weeks), the path taken affects the calculated values of annualized return and volatility.

>But they're close, eh?
Not always. Here's a few examples (using randomized weekly returns for MSFT):

>When you say "Annual Return" you really mean 48-week return, right?
Another clever observation.

Mean Variation

Consider the following:

  1. rk is a sequence of stock (or portfolio) returns, with k = 1, 2, ... n.
  2. m[r] = (1/n) (r1+r2+...+rn) = (1/n) Σ rk = the mean of the n returns.
  3. s2[r] = (1/n) Σ (rk - m[r])2 = mean-square deviation gives s[r], the standard deviation (or volatility) of the n returns.
  4. xk = | rk - m[r] | = the absolute value of the kth deviation (from the mean of the returns).
  5. M[x] = (1/n) (x1+x2+...+xn) = (1/n) Σ xk = the Mean of the n deviations.
  6. S2[x] = (1/n) Σ (xk - M[x])2 = mean-square deviation gives S[x], the standard deviation (or volatility) of the n deviations.
Now expand the sums like so:
[A]   s2[r] = (1/n) Σ (rk - m[r])2 = (1/n) Σ (rk2 - 2 m[r] rk + m[r]2)
= (1/n) Σrk2 - 2 m[r] (1/n) Σrk + (1/n)Σm[r]2
= (1/n) Σrk2 - 2 m[r] (1/n) Σrk + (1/n)m[r]2Σ(1)
= (1/n) Σrk2 - 2 m[r] m[r] + (1/n)m[r]2(n)
= (1/n) Σrk2 - m[r]2 = (the mean square) - (the square of the mean).
[B]   S2[x] = (1/n) Σxk2 - m[x]2

Note, however, that (1/n)Σxk2 = (1/n)Σ| rk - m[r] |2 = s2[r] (see #3 and #4, above.)

We conclude, from [A] and [B], that:
S2[x] + m[x]2 = s2[r]

>And that makes S[x] smaller than s[r], right?
Yes, the standard deviation of the deviations of the returns (that's S[r]) is smaller than the standard deviation of the returns themselves (that's s[r]).

>So what's this Mean Variation you're talking about?
  I forgot. It's M[x], the Mean of the deviations in the returns. (see #5, above.)

"Expected" Return

Suppose we have a bunch of historical returns (weekly, monthly, yearly, whatever).
We'd like to use that info to try to predict the next return: next week, month, year, whatever.
That'd give an "Expected" return. What would you use?

>I'd use whatever is the best predictor.
Define "best".

Suppose the returns have some (unknown) distribution. That is, f(r)dr is the probability of a return lying in the range r to r+dr.
In particular:
It's certain that the return will lie in   -∞ < r < ∞, so:
Suppose we consider all possible future returns (chosen from our assumed distribution f(r)).
The "best" choice for our expected return (we'll call it x) is the one which is closest to all possible future returns ... in some sense.
For some random future return r (chosen from f(r)), the distance from our "best" choice is | r - x|.
If we define "best" as that x-value which minimizes the mean deviation | r - x| we'd have:
Now differentiate with respect to x and set the derivative to 0 and ...

>Hey! I vaguely remember that from 1st year calculus when we used to ...
Pay attention!
Differentiating the right-side of [3] and setting the derivative to 0 gives:
In other words, the probability of lying in   -∞ < r < x is the same as the probability of lying in   x < r < ∞.
To put it differently, there as many returns, r, less than x as there are greater than x.
That means that x should be the MEDIAN (historical) return.

>So the Median is the best predictor for next year's return?
Define "best".

We might also define "best" as that x-return which minimizes the mean-square deviation. That is we choose x so that:
Again we set the derivative to 0 and get:
Do you recognize that first integral?

>You kidding?
It's the MEAN of the (historical) returns. How about the second integral?

>I thought that ... uh ... I saw that integral somewhere ...
Yes, it's [2], above and it's equal to 1.

>So the Mean is the best predictor for next year's return?
Define "best".

I should point out that, after exhausting exhaustive analysis of historical data, neither Mean nor Median seem to be a good predictor of near-future returns.