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George Arrington in the June 1991 STOCKS & COMMODITIES discussed a variable-length simple moving average in which the number of days changed by discrete integers. The length is increased or decreased by an integral number of days based on the magnitude of price changes. He argued that this approach does not work well with exponential moving averages. A closer look at the smoothing constant t suggests that it could be a continuous variable (index), thus allowing the use of fractional time periods. For example, we can write t as
(2) t = k ? V
where k is a numerical constant such as 0.15 and V is a dimensionless market-related variable, such as a ratio of the standard deviation of the market's closing prices over two different periods. The smoothing constant is simply a mechanism for incrementing the old value of the moving average to a new value, and a fractional value of t less than 1 prevents instability. Varying the smoothing index corresponds to taking larger or smaller portions of the latest data to update the moving average.

Any indicator may be used to connect the index t to the nature of the market's price changes. For example, the midpoint oscillator, %M, may be used as a measure of the market and inserted as V in equation 2. By design, the moving average will move faster as prices approach an overbought or oversold condition. Alternately, market momentum indicators (say, the 26-week price rate of change) may be used so that changes occur more quickly as prices change rapidly, slowing down as prices stabilize. This ability to quicken or slow down gives these variable-index exponential averages their dynamism.

DEFINING VIDYA AND RAVI

Specifically, I constructed a variable-index dynamic moving average (VIDYA) connecting the smoothing index to the market's volatility as follows:

Here, the subscripts d and d-1 denote the new and old time period, C is the closing price at the end of period d, s (sigma) is the standard deviation of the market's prices over the past n periods, s is a reference standard deviation of the market over some period of time longer than n, and k is a numerical constant. The reference standard deviation could also be an arbitrary value to obtain the desired degree of smoothing.

From an investor's viewpoint, a "long" VIDYA can be defined with k=0.078, corresponding roughly to a 25-week exponential moving average. A 13-week standard deviation is used to adapt to market volatility. A reference standard deviation of 6 is used, which represents a 10-year average for weekly Standard & Poor's 500 data. More precisely, the long VIDYA is given by

(4) VIDYALd = 0.078 ? s13 / 6 ? Cd + (1 - 0.078 ? s13/6) ? VIDYALd-1

A "short" dynamic average is also defined with k = 0.15, roughly equal to a 12-week exponential moving average. The standard deviation of closing prices is calculated over 10 weeks. The value of the reference standard deviation is set at 4.

(5) VIDYASd = 0.15 ? s10/4 ? Cd + (1-0.15 ? s10/4) ? VIDYASd-1

To clarify the dynamism of these averages, I tabulated the 13-week standard deviation of the S&P 500 weekly close during three market periods. Also shown is the effective smoothing index using equation 4. Then I estimated the effective length of the equivalent simple moving average using the well-known formula for the smoothing constant of an exponential moving average (2/(n+1)), where n is the length of the equivalent simple moving average.

The dynamic range of VIDYAL is about a factor of 10 (i.e.,30.75 3.18), since it adjusts all the way from a rather long to a very short moving average based on market volatility. As market volatility increases, the effective length decreases. Even greater dynamic range is possible, as long as the factor (1-t) is positive. Clearly, VIDYAL is superbly responsive to the market. VIDYAS also exhibits a similar dynamism. Now, we define the rapid adaptive variance indicator (RAVI ), defined as

(6) RAVId = VIDYASd - VIDYALd

where the long and short dynamic averages are as defined in equations 4 and 5.