While prices do have a tendency to trend every so often, these trends do not seem to recur with definite regularity. Moreover, the magnitude of the price move in a trend varies over time, and no two trends are exact replications. Although the existence of trends cannot be denied, there is an annoying randomness as regards their magnitude and periodicity. This randomness is the Achilles heel of conventional mechanical systems based on fixed, market-invariant parameters, since it is virtually impossible for them to capture trends in a timely fashion consistently. Consequently, a fixed lag system may perform exceptionally in one time period, but can be woefully inadequate during a slightly different market environment (Balsara 1992).
A fixed-parameter
mechanical system implicitly expects market conditions to adapt to its
invariant and market insensitive logic. A flexible-parameter system, on
the other hand, seeks to modify its parameters to adapt to changes in market
conditions. This paper seeks to develop a flexible alternative to the conventional
fixed-parameter Bollinger band system. While it might be wishful thinking
to conjecture that such a system would dominate its fixed-parameter counterpart
in terms of profitability, it is reasonably safe to hypothesize that a
flexible system would be more consistent in its performance.
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The
Bollinger band system defines standard deviation bands around a moving
average of prices over a fixed lag length. This moving average defines
the market’s pivot point at a given point in time. The average gives equal
weight to each and every data point in the data set without regard to the
directionality of the data. Consequently, as is evident from Appendix
I, it is possible for two data sets to be very different in terms of
their directionality and yet share the same average or pivot point. Whereas
data series 1 has a clear upward trend, data series 2 has no clear trend.
Notice that both data series share the same average, leading to the same
exit or reversal price. Given the upward trend of data series 1, prices
will have to move lower quite a bit more as compared to data series 2 simply
to reach the same pivot point. Trend switches are therefore likely to be
more difficult in a trending market as opposed to a non-directional market.
To offset this, an up trending market (data series 1) ought to have a higher
pivot as compared to a sideways market (data series 2). This is a major
weakness of the conventional fixed-parameter Bollinger band system (Kaufman
1995).
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Total "Up" days - Total "Down" days
In an up-trending market, when the momentum index is positive, the pivot point of the current trend ought to be higher than the moving average, somewhere near the high end of the trading range (Ruggiero 1996; Ruggiero 1998). This will allow for a faster reversal from a buy to a sell, when the up trend reverses. Accordingly, a flexible pivot point is defined as the n period moving average of prices plus some fraction of the difference between the n period highest price and the n period moving average. The fraction in question is given by the current n period momentum of the market. Therefore, the stronger the directionality of the up trend, the closer the pivot point is to the highest high over the past n periods. The formula for the n period flexible pivot in an up trending market, when the momentum is positive, is as under:
In the extreme case, when the momentum is +1, the flexible pivot would be exactly equal to the highest high price over the past n periods.
In a down trending market, when the momentum index is negative, the pivot point of the trend ought to be lower than the average over the past n periods, closer to the lowest low over the past n periods. This will allow for a quicker reversal from a sell to a buy when the down trend reverses. A flexible pivot point is defined as the n period moving average of prices minus some fraction of the difference between the moving average and the n period lowest low. Again, the fraction is given by the current momentum of the market. The stronger the directionality of the down trend, the closer the pivot point is to the lowest low over the past n periods. The formula for the flexible pivot point in a down trending market, when the momentum is negative, is:
In the extreme case, when the momentum is -1, the flexible pivot would be exactly equal to the lowest low price over the past n periods.
To check for statistically significant differences in profits resulting from the two approaches over the entire time period, we will use a T-test of differences in the means of paired two-samples with the null hypothesis that there is no difference between the methods. A calculated t-value in excess of the critical value shows a significant difference between the two approaches. To check for consistency over shorter time periods, we propose to divide the sample into two approximately equal sub-periods, January 1987 to December 1992, and January 1993 to July 1998. A similar T-test will be used to test for differences between the two methods during each of the two sub-periods. Again, the null hypothesis is that there is no difference between the two methods for any given time period. Assuming that the normality assumptions behind the T-test are not entirely satisfied, we will also employ a non-parametric Wilcoxon signed-rank test of the differences in the profit performance of the two approaches.
To
check for consistency of performance under the two approaches, we will
compute the coefficient of variation of completed trade profits for each
commodity for both the fixed and flexible approaches over the period January
1987 to July 1998. Next, we will use a T-test and the Wilcoxon signed-rank
test to check for significant differences between the coefficients of variation
for the two approaches, with the null hypothesis that there is no difference
between them.
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The results of the Wilcoxon signed-rank test are given in Table I(b). The z-statistic for differences in profits for the two approaches over the entire sample period is 2.2014, which is well beyond the critical z value of 1.645 at the 5 percent level of significance. This suggests that the flexible method returns profits that are statistically significantly higher than those provided by the fixed approach. The same result is observed for each of the two sub-periods, with a z-statistic of 1.9917 for both the 1987/1992 and 1993/1998 periods. Hence we can say that the flexible approach results in higher profits as compared to the fixed approach.
The results of the T-test for paired-sample differences in the coefficient of variation for both methods are given in Table II(a). The t-value for differences in the coefficient of variation for the two approaches over the entire sample period is 2.8496, signifying a statistically significantly difference. The coefficient of variation is higher in case of the fixed pivot approach as compared to the flexible pivot approach. However, the t-value for differences in the coefficient of variation for both methods drops to a statistically insignificant 1.6462 for the 1987/1992 sub-period and is once again statistically insignificant at -0.9437 for the 1993/1998 sub-period. Although this could be due to the rather limited number of trades executed in each of the sub-periods, the results prevent us from emphatically concluding that the flexible approach results in a lower coefficient of variation as compared to the fixed approach.
The
results of the Wilcoxon signed-rank test are summarized in Table
II(b). The z-statistic for differences in the coefficient
of variation over the entire sample period is 2.2014, once again suggesting
a statistically significant difference in the coefficient of variation
for the two approaches over the entire sample period. However, the z-statistic
is statistically insignificant at 1.3628 and 0.1048 for the 1987/1992 and
1993/1998 sub-periods. This reiterates our earlier finding that we cannot
conclusively say that the flexible approach results in a lower coefficient
of variation as compared to the fixed approach.
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