Table Of ContentStocks & Commodities V. 7:3 (82-85): Variable sensitivity stochastics by William Mason
Variable sensitivity stochastics
by William Mason
I
n this article, I am presenting elementary Statistical Analysis of Stocks and Indices (SASI) in an index,
three new indicators (SASITOP, SASIBOT and sigma limits) plus variable-sensitivity stochastics based
on statistical analysis. SASITOP is very similar to stochastics but uses plus and minus variance (sigma)
limits in place of the high and low over the time window. The data is modified for sharper sensitivity.
Because SASITOP and SASIBOT are the reciprocal of each other, I will concentrate only on SASITOP
in this article and apply it to the Technical Index which measures overall market breadth (see Stocks &
Commodities, January 1989) although it may be applied to any index or set of data.
To generate SASITOP:
1. Calculate the statistical sample variation s2. For a finite population, s2 is mathematically described by:
(cid:229) n (cid:230) (cid:229) n (cid:246) 2
n xi2-(cid:231)(cid:231) xi(cid:247)(cid:247)
Ł ł
s2 = i=1 i=1
n(n-1)
where X represents the values in your time series and n is number of observations
Don't panic. Use a spreadsheet like Lotus 1-2-3 which has a built in variance function. The Lotus 1-2-3
variance function has to be modified: s2m = s2[n/(n-1)] to account for a finite population (where n = the
time window). Some spreadsheets such as Excel have this modification built in to the variance (VAR)
function.
s = s2
2. Next calculate the standard deviation, S, which is the square root of the variation: m m
3. Calculate the index's 5-day average (AVG) for the data over the chosen time window.
4. Calculate the +3 sigma limit as: (3sm)+AVG; the -3 sigma limit is: AVG-(3s m).
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Stocks & Commodities V. 7:3 (82-85): Variable sensitivity stochastics by William Mason
Figure 1 depicts the – 3 sigma limits around an index. Value A is the difference between the last +3 sigma
value and the last index value, I. That is, A = +3 sigma -I and B=I- (-3 sigma).
The SASITOP value is: B‚ A. Just like moving average calculations, you continue sliding the time
window and calculating a new SASITOP value for each time increment.
Figure 2 is a plot of SASITOP on the Technical Index in which the SASITOP value has been adjusted to
fit the graph. When SASITOP turns in the region of 1,100-1,300 and heads south the investor should be
alert.
By looking at prior market data, the noise level can be estimated as in Figure 2. SASITOP signals an exit
when it breaks below the noise level (points A, B, C and D).
I found by empirical experimentation after the sigma limit calculations were complete that if the index
value, I, is replaced with a smoothed value, Is, the SASITOP value more or less approaches a constant if
the smoothing was approximately equal to one-half the time window. All of the SASITOP figures in this
article have this modification.
The overall SASITOP sensitivity is varied by first exponentially smoothing the index under evaluation.
SASITOP calculations are then done on the smoothed index.
Figure 2 is at a low sensitivity of 0.05 and Figure 3 is at a sensitivity of 1.0 (no smoothing) for very
active trading. The sensitivity can be set to accommodate anything from daily trading to
intermediate-term investing.
Figure 4 is a magnified view of Figure 3 showing the exit signals. When the SASITOP indicator turns
and forms a top (points A through H), this is the signal to exit or go short. At this sensitivity (1.0), you
are going to get knee-jerked fairly often (points F and G). If you had the courage to hold your position for
several days you would have been rewarded.
Exponential smoothing revisited
One exponential smoothing formula is:
Last + a (Ic - Last)
where Last is the previous exponential calculation and Ic is the current index value. The initial value for
Last is the current index value. Sensitivity is changed by changing the value of a (alpha). A useful range
for a is 0.05 to 1.0.
Variable-sensitivity stochastics
Stochastics is the analysis of an index or set of data relative to its range over a time window. Very simply
we find the range of the variable over the time window and then plot the current index value as a percent
of the range. Figure 5 illustrates the standard 5-day stochastic "slow" curves.
Standard stochastics analysis (such as on the Dow Jones Industrial Average) uses the theoretical high and
theoretical low over the time window to calculate the range. The raw curve (called the %K line) is
smoothed to produce the %D line and the %D line is smoothed to produce what I call the %E line. (See
"Using Stochastics," S&C, July 1987.) The curves can theoretically vary between 0% and 100%, but
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Stocks & Commodities V. 7:3 (82-85): Variable sensitivity stochastics by William Mason
typically vary between 30% and 70%. Crossing the %D and %E lines at the upper and lower extremes
signals exit and entry.
For SASI stochastics, I make use of the – 3 sigma limits. In Figure 6, I used the last sigma limits and the
last index value to calculate %K. The last sigma values are used because they already statistically
represent the data variation over the time window. The calculations are:
A = I - (-3 sigma limit)
B = +3 sigma limit - (-3 sigma limit)
%K = 100(A ‚ B)
The %D line is a 0.35 exponential smoothing of %K line and %E line is a 0.35 exponential smoothing of
the %D line. These values were picked empirically. Sensitivity is varied as for SASITOP by
exponentially smoothing the index under evaluation before doing statistical calculations.
At short time windows and moderately low sensitivity (0.08) an interesting effect occurs: When a market
trend is under way the %D and %E lines will flatten out at the 70% or 30% level and remain there for the
duration of the trend.
This effect makes it relatively easy to stay with a trend for the full duration and enter or exit within a few
days of major turning points. An empirical trigger value can be picked based on your individual trading
preference. When the %D line crosses this level it is the entry or exit signal.
Figure 7 shows SASI variable sensitivity stochastics based on ±3 sigma limits and operating on the
Technical Index. The exponential smoothing sensitivity is set low at 0.05 to emphasize the effect. A
trigger level for exit at 69% and a trigger level for entry at 31% is depicted. Note how using this
technique would have kept you either in (long) or out (short) for the lion's share of the major trends.
The flattening effect of the %D and %E lines diminishes as the time window is increased. Experience
suggests that time windows of five or six days are best for this flattening effect.
Another benefit of the overall sigma limit technique is that it makes it very easy to set up a spreadsheet
like Lotus 1-2-3 so that any index or set of data can be quickly evaluated over a range of sensitivities.
Sigma limits
Figures 8 and 9, are plots of the ±3 sigma limits at sensitivities of 0.05 and 0.10. The exponentially
smoothed index is between the sigma limits, which act as dynamic channel lines.
There are several important features to note on these charts: At the lower sensitivity (Figure 8), the upper
and lower sigma limits will move parallel to each other while a trend is in effect and will not deviate
significantly during minor market moves. (This is the reason that stochastics flatten out).
When the sigma limits pinch together, as at points A, B, C and D in Figure 8, it means the market is
making a significant move one direction or the other. It is up to the individual to use all the technical
analysis techniques at his/her disposal plus common sense to determine which way the market will break.
After a three-month run-up of 25% it is not too hard to guess that it might be a top or, conversely, after a
two-month correction of 10% that a bottom has been established.
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At the lower sensitivity, the upper and lower sigma limits will
move parallel to each other while a trend is in effect and will not
deviate significantly during minor market moves.
Another slightly more analytical technique is to watch which sigma limit turns first. As a general
empirical, but not infallible, rule at market tops the upper sigma limit will turn down before the lower
limit and at market bottoms the lower sigma limit will turn up while the upper limit is still moving down,
causing the pinching effect.
Sensitivity can be varied depending on personal investment strategies. At sensitivities of 0.10 or greater
the chances of being knee-jerked are increased. Figure 9 at a sensitivity of 0.10 shows the pinching effect
on minor market moves (points A and B).
Figure 10 is a magnified view of Figure 9 showing the pinching of sigma limits at the August 25, 1987
market top. At this sensitivity, the effect lags the top by five days.
Bill Mason, (503) 625-5384 is a retired engineering manager from Tektronix, Inc. with a background in
math and physics. He is currently president, engineer, chief cook and bottle washer for SASI Software
Corp.
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FIGURE 8:
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FIGURE 10:
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