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One way to adapt to the market is to obtain an overview of market conditions in recent history. I do this for MACD in my 3D program by plotting the two EMAs as the independent X and Y coordinates and profit in the Z dimension as the dependent variable. When profit is calculated for all combinations of the independent EMAs, a three-dimensional surface of profitability results (Figure 6).

Figure 6 is interesting from several perspectives. First, the surface is relatively smooth, so this means the profitability is relatively insensitive to variations in the EMA parameters. The implication is that MACD is robust, and minor variations in market conditions will not severely affect profitability for trading in the immediate future for the selected EMAs. The second interesting aspect of Figure 6 is that the maximum profit occurs in the "inverse" domain where the first EMA is smaller than the second EMA.

Cycle analysis has helped provide a rationale as to why inverse MACD parameters produce profitable trading results in some market conditions.

Figure 7 shows the trades formed in the February through April time frame using the inverse parameter combination. In this case, the buy/sell signals are almost always early relative to the cyclic highs and lows. The early signals allow us to accommodate the necessary one-day lag in making an entry after the signal has been received. However, MACD cannot be applied universally this way. For example, note the January sell signal (second bar from the left edge of the chart) is dead wrong. As calculated, this 4/12-day EMA parameter set is valid only in the immediate past relative to the right-hand side of the bar chart.

Note, also, that the MACD is somewhat out of phase with the cyclic component of price (Figure 7). When the price reaches a peak, the MACD is at a valley, and vice versa. The signal is relatively smooth because it is now being calculated using the longer EMA. We need to return to phasor analysis to fully understand why this set of MACD parameters has given good entry signals.

The first four-day EMA attenuates a 12-day cycle to about 80% of its unaveraged amplitude and introduces a phase lag of about 40 degrees. The second 12-day EMA attenuates the 12-day cycle to about 31% of full amplitude and introduces a 72-degree phase lag (Figure 8). When we invert the first (four-day) EMA and perform the vector addition, we see that the resulting MACD lags the price function by 203 degrees. Viewed another way, the resulting MACD leads the next cycle by 157 degrees (203+157=360 degrees). The MACD signal is attenuated to 31% of the MACD value and is delayed by 72 degrees. Thus, the buy/sell signals lag the MACD by about 90 degrees; so the buy/sell signals lead the next cycle by about 67 degrees (157-90=67). This extra lead is just about right to compensate for the one-day lag in making the entry after we get the signal.

OVERALL, WE CONCLUDE

Although somewhat complex, cycle analysis has helped provide a rationale as to why inverse MACD parameters can produce profitable trading results in some market conditions.

When an overview perspective is taken, new twists on such old familiar indicators as MACD can be found. More important to your trading, you can look at several indicators in the same market to find the one that is the most robust and least sensitive to market variations. You can formulate your trading strategy in such a manner. You will tend to use trend-following indicators such as MACD and double moving averages in trending markets and oscillators such as RSI and stochastics in sideways markets. Parabolic stop and reverse (SAR) is applicable to many market conditions, depending on the value of acceleration factor you use. You can find unusual parameter combinations for all these indicators when you view the profits from an overview perspective in three dimensions.

FIGURE 4: Moving averages introduce time (phase) lag and reduce amplitude of higher-frequency components. Full-cycle EMAs introduce about 72 degrees of phase lag and compress cycle to about 31% of unaveraged amplitude; a half-cycle introduces about 54 degrees of lag and reduces cycles about 57% of the unaveraged amplitude.

FIGURE 5: By reversing the direction of the longer EMA and then performing vector addition, the resulting vector lags the original price function about 35 degrees. This vector can be thought of as the detrended synthetic price because it is near the price in phase.

FIGURE 6: Plotting the two EMAs as the independent X and Y coordinates and the profit in the Z dimension as the dependent variable produces this three-dimensional graph. The maximum profit occurs when the first EMA is shorter than the second EMA.

FIGURE 7: Using the four- and 12-day parameters produce early signals relative to the cyclic highs and lows. Note that in early January the sell signal was dead wrong. This underscores the important point that this particular set of parameters is only valid in the near past.

FIGURE 8: The four-day EMA attenuates a 12-day cycle to about 80% of its unaveraged amplitude and introduces a phase lag of about 40 degrees. The 12-day EMA attenuates the 12-day cycle to about 31% of the full amplitude and introduces a 72-degree phase lag. Performing vector addition after inverting the four-day EMA results in the new vector (MACD line) that lags the price by 203 degrees. Viewed another way, the resulting MACD leads the next cycle by 157 degrees.