How to compute std dev of time series?

From: nomail1983 (nomail1983_at_hotmail.com)
Date: 12/27/04 

Date: 27 Dec 2004 00:04:51 -0800

Two questions ....

1. In general, how do we compute the std dev of a time
series (of periodic percentage change)?

2. Also, what is the "conventional" method for computing
the std dev of the periodic percentage change of
investments, if different from #1?

The two answers might be different due to "conventional
usage" by technical analysts. I don't know.

I intend to apply this to the time series of the periodic
percentage change in the S&P500 index, for example.

I thought this is a no-brainer. But I raise the question
because the results of my computation does not match
published numbers for an example time period.

According to one web site [1], the std dev of a time
series of percentage changes of EPS (earnings per share)
is computed as follows (is there a simpler expression?):

v = SUM( (LN(r[t]) - LN(gavg))^2, t=1,...,n ) / n
gsd = EXP(v^(1/2)) - 1

where gavg is the geometric mean of r[t], t=1,...,n.
r[t] is 1+g[t], where g[t] is the percentage change.

But a text on the mathematics of investment analysis [2]
shows an example where the std dev of the time series of
percentage changes of investment return rates is simply:

gsd = v^(1/2) - 1

What is the correct answer, either or both statistically
and according to "conventional usage" for investment
analysis?

I note that in [1], we are looking at percentage changes
of numbers that represent dollars, whereas in [2], we are
looking at percentage changes of numbers that represent
percentages. But if that makes a difference, I need some
help in understanding why. In my view, in both cases, we
are looking at percentage changes of numbers.

PS: Both sources have the same formula for the (geometric)
mean of the time series of percentage changes in return
rates, namely:

gavg = (r[1]*...*r[n])^(1/n) - 1
= EXP( SUM( LN(r[t]), t=1,...,n ) / n ) - 1

(gavg = (y[N]/y[0])^(1/N) when N = n -- e.g, when no
outlying r[t] are removed.)

However, the [2] text later computes the mean of an example
(the same example used for the std dev above) simply:

gavg = SUM( LN(r[t]), t=1,...,n ) / n ) - 1

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[1] http://www.thinkingapplied.com/means_folder/deceptive_means.htm

[2] Fabozzi, "Fixed income Math: Analytical & Statistical
Techniques", 3d ed, McGraw-Hill, 1997, ISBN 0-7863-1121-5.