LONGWAVE TRENDS:
DJIA long term "mean" trendline. POSTED AS REFERENCE 3/29/2004 UPDATED 8/5/04
The top chart, at the bottom of this page, shows the "exponential
regression" MEAN trend line for the Dow Jones Industrial Average from 1/2/1900
to the close on 3/29/2004.
As discussed in the introduction to the Longwave analysis, the
"x" axis independent variable is x= one trading day, since the data is from
a daily chart. As you can see, I ran an analysis on the exact same
data in all three of the statistical software programs we use at PTR. Those
are ORIGIN, DPLOT, and Microsoft Excel and they are all covered in the introduction
to the Longwave section of the Review, which can be accessed from the subscribers menu bar.
The top chart shows the three regression lines and also list
their equations and correlation coefficients (r2), near the top of that graphic.
All three "fitting" functions are the same, Ae^Bx, and all "initial
values" were set at A=28 and B=.00001 . The programs were then allowed to
vary both variables, and the curves and equation shown reflect the best fit
of this regression function.
As discussed in the LongWave introduction of the Review, the regression function
Ae^Bx is a excellent "two variable" stand-in for the compound interest formula,
which we contend is the underlying mathematical basis of the stock market.
As explained, the "default" exponential regression function in Dplot and
Origin cannot be used for stock analysis, because they are three variable
versions and do not replicate the action of compounding interest, which only
has two variables...the "starting value" and "rate" of return. While
the compound interest equation doesn't use the natural log coefficient, "e"
, as does most regression equations, that makes no difference what so even, since Ae^Bx will model Pv(1+i)^(t) exactly.
While it would be "nice" if we had one program that could
"spit out" the correct exponential mean trendline and equation without us
having to intervene in any way, that is rarely going to be the case. However, from experience,
I can tell you that MS-Excel comes about as close to that as you will get.
The only "snake pit" you have to watch out for in Excel is the "error" you
can get by using "dates" rather then one "x" for each "y," and this is also explained
in the introduction.
With MS-Excel, if your lowest "y" value (price) is well up off
zero, you can many times just let the program run on it's own without
setting a "starting value" (as a Y-intercept). However, more often then not,
you have to "play" with some starting values until you get the lowest r2
value with a curve that never goes negative and has a low value somewhere
close to the actual starting value of the price (or index value) for your
chart's first data record.
As you can see in the chart below, the three regression "mean trendlines"
are fairly close to each other, and from experience I can tell you they typical
have a close "relationship" to each other from one stock analysis to another. That is to say, ORIGIN usually
returns a little higher regression rate then MS-Excel and Excel usually returns
a little higher rate then DPLOT, which seems like it may have some small
error because it's always well below the other two programs. While
the MS-Excel program "says" it's curve had a correlation coefficient (r2=fit)
of .895, that seems a little high since Origin's r2 was only .57 and DPLOT's
was .65. This is also a typical relationship between the programs, but seems
a little "fishy" to me. From what I see, these programs are not comparing
apples to apples when it comes to the r2 values they return, but I have no
intentions of spending the time to find out which one is correct. In the
end, I selected the MS-Excel "mean,." (purple line), since it has a history
of being the better of the group and it also "visually" hit more "key"
changes in trend (CIT's or reversals) then Origin or DPLOT.
For the Dow longwave chart, below, the annually compounded rate of return from the MS-Excel "mean" line was
calculated as 4.75%. To do that, I looked at the plot data and found
the index value for record # 27,000 as 3481, then found the actual date
for record #27000 in the Excel file (which was 1/24/2002), and after that we
calculated the rate using 32 as the Pv, 3481 as the Fv, and T (time) as 101 years (1/24/2002-1/1/1900). That rate was
the 4.75% shown.
However, one must remember that a Longwave "mean" represents
an "average" rate of return, "as capital gains only," after many years of
historic data, and the current "mean" of all the data to date
may or may not accurately represent the correct Longwave rate well
into the future. As a matter of fact, based on our "best fit" from
our Longwave trend analysis, our Fibonacci trend, and our double regression
comparison to many other indexes and individual stocks, within this index,
we "firmly" believe that the Longwave regression "mean" for the Dow Jones
Industrial average (DJIA or INDU) will eventually be proven to be "very close
" to a centerline representing a 5.5%-5.76% rate of return on capital from
1896 to 2040. While our evidence to support this is fairly massive,
that value is not cast in stone either.
Just keep in mind this key tenant of long term investing: the
higher the "cash" dividends" a company pays the lower the capital gains and
the lower the long term regression mean, which is, or course, the "average
rate of return on capital excluding dividends." Furthermore, the U.S.
Government just made it more likely that the payout in dividends with be
higher and the payout as "capital gains rates" lower in the future, by halving
the tax rate on dividends.
For the PTR
Andrew J. Quiggly
Editor
