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There is a serious misunderstanding of the Analysis of the Fixed Ratio and
other pyramid schemes. So lets start with some basics.
All the information concerning a trading scheme is contained in its
Equity curve ( see Steady Profits with Ensemble Trading, Futures Magazine,
August, 1999). Subtracting the value of the Equity curve on December 31 ( or
the last day of the year) from the value for the previous year yields the
annual return of the trading scheme. When you do this for all 13 years
presented in Mark Johnston's data and average the results you get the
average yearly return for the three pyramid schemes.You can calculate the
standard error of the average yearly return with S.E. = Std.Dev/Sqr(13). The
Std. Deviation
is calculated from each of the 13 annual sample returns. The results are:
Average Std. Error
Average - S.E.
Fixed Ratio $39,364 $436,
-$397,146
Spear $145,155
$894,344 -$719,189
Unequal F.R. $124, 906
$1,035,078 -$910,172.
It is obvious that when the standard error exceeds the average by 6
to 11 times, it is quite likely that the true returns of all these schemes
are negative.
None of these formulas have any proven profitbility. All
of them are highly speculative schemes which produce tremendous swings in
annual
profitability. The swings will produce enormous losses and break any one
foolish enough to trade them.
A note about the dependent t test. Because Mark computed equity curves
for each scheme on the SAME commodity data, they can be compared using a
more sensitive statistical test called the dependent t test. Subtract the
yearly returns of each pair and compute the standard deviation of the
difference over the 13 differences representated in Mark's data. This test
as applied here has been used widely in the drug industry to test the safety
and efficicacy of new drugs. ( see Futures Magazine, January 1999).
The test is approved by the FDA. I have used
the test widely in my legal work. I have had recognized experts
present similar t test data to juries in cases I have tried.
The test as applied here has never been challenged before or even
questioned. ( See e.g. Applied Multivariate Statistical Analysis, James
Stevens, or any other standard first year text book.). The present
criticisms of the
test as applied are completely mistaken.
The result of the t test is that the difference betweeen the annual
returns for all 3 pairs of schemes cannot be distinquished from zero.
There is no difference in the profitability of the schemes. The standard
deviation of the yearly
differences between the schemes is 16 to 54 times the average yearly
difference. With such a huge standard deviation no one can ever say with
confidence that the differences are anything but zero.
A comment about Mark's example. It is simply not valid because :
1) the trading schemes and the analysis deal with random statistical
data. His example contains non-random trending data without any error.
2) Statistical analysis deals with flucating data, Mark's data does
not flucate.
3) As pointed out, you don't need a t test to determine that none of
these schemes has any proven profitability. Mark's data is contrived to
represent a non-flucating profitability and is completely unrealistic.
4) Mark presented us with 13 independent statistical samples for the
13 years of annual results for the pyramid schemes. Valid statisical
analysis can be performed with 13 samples. But Mark's contrived example
containes only a single sample.You can't compute standard deviations with a
single sample. Valid statistical analysis cannot be performed on one sample.
( See any first year course in statistics.)zx
In summary, pyramid schemes are dangerous allusions that will result
in bankrupting anyone who actually attempts to trade them over a time. The
analysis of Mark Johnston's computer diversions are only useful to
demonstrate valid statistical analysis of real trading data. The simple
conclusion:
if the stardard deviation of the returns is enormous, the system is
dangerous
to trade and its profitability is unproven. It is ill-advised to trade
any system with a standard deviation that exceeds the average profitbility
by more than 10 times. Don't do it, you and your bank account will regret
it.
Paul H. Lasky
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