In our context, a technical trading indicator can be considered as a combination of a specific technical trading rule with a particular moving average of prices. In two preceding blog posts we showed that there are many technical trading rules, as well as there are many popular types of moving averages. As a result, there exist a vast number of potential combinations of a specific trading rule with a specific moving average of prices. So far, the development in this field has consisted in proposing new ad-hoc trading rules and using more elaborate types of moving averages in the existing rules. Each new proposed rule (or moving average) appears on the surface as something unique. Often this new proposed rule (or moving average) is said to be better than its competitors; such a claim is usually supported by colorful narratives and anecdotal evidence.

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The existing situation in the field of market timing with moving averages is as follows. Technical traders are overwhelmed by the variety of choices between different trading indicators. Because traders do not really understand the response characteristics of the trading indicators they use, the selection of a trading indicator is made based mainly on intuition rather than any deeper analysis of commonalities and differences between miscellaneous choices for trading rules and moving averages. It would be no exaggeration to say that the existing situation resembles total chaos and mess from the perspective of a newcomer to this field. Therefore there is an urgent need to bring some order to the chaos in the field of market timing with moving averages.

In this blog post we uncover the anatomy of market timing rules with moving averages of prices. In particular, we are going to show that the computation of a technical trading indicator for every market timing rule can be interpreted as the computation of a weighted moving average of price changes over the averaging window. More formally, we will show that the computation of a technical trading indicator for every market timing rule can be written as

where, recall, Pt-i=Pt-i+1-Pt-i denotes the price change and  is the weight of the price change Pt-i in the computation of a weighted moving average of price changes. Therefore, despite a great variety of trading indicators that are computed seemingly differently at the first sight, the only real difference between the diverse trading indicators lies in the weighting function used to compute a moving average of price changes. This result allows us to study the computation of trading indicators in many market timing rules and analyze the commonalities and differences between the rules.

### Momentum Rule

The computation of the technical trading indicator for the Momentum rule can be re-written as follows:

Therefore,

where the mathematical symbol means “equivalence”. To see the equivalence of equations (1) and (2), observe that

(since n-1>0) and vice versa. Consequently, equation (2) allows us to re-interpret the computation of the technical indicator for the Momentum rule as the computation of an equally weighted moving average of price changes (where the weight of each price change equals 1/(n-1)).

### Price Minus Moving Average Rule

First of all, recall (from Part 1) the alternative representation of a general moving average:

where wj is the price weighting function and  is the price-change weighting function of a moving average. Therefore,

Consequently, the computation of the technical indicator for the Price Minus Moving Average rule can equivalently be interpreted as the computation of a weighted moving average of price changes.

In case all weights wj are strictly positive, the sequence of weights  is decreasing with increasing i

Therefore, in this case, regardless of the shape of the weighting function for prices wj, the weighting function  always over-weights the most recent price changes.

The closed-form expression for the computation of the technical indicator for the Price Minus Simple Moving Average rule is given by (we skip the details of the derivation)

The resulting formula suggests that we can alternatively interpret the computation of the technical indicator for the Price Minus Simple Moving Average rule as the computation of a Linearly Weighted Moving Average of price changes.

When the Exponential Moving Average is used in this rule, the closed-form expression for the computation of the technical indicator is given by

In words, the computation of the trading indicator for the Price Minus Exponential Moving Average rule is equivalent to the computation of an Exponential Moving Average of price changes.

For the sake of illustration, the figure below plots the shapes of the price change weighting functions in the Momentum (MOM) rule and three Price Minus Moving Average rules: Price Minus Simple Moving Average (P-SMA) rule, Price Minus Linear Moving Average (P-LMA) rule, and Price Minus Exponential Moving Average (P-EMA) rule. In all rules, the size of the averaging window equals n=30. Observe that in all but the Momentum rule the weighting function overweights the most recent price changes (note that Lag 1 denotes the most recent price change).

The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.

The interactive illustration of weighting in the Price Minus Moving Average rule can be found here.

### Moving Average Change of Direction Rule

The value of this technical trading indicator is based on the difference between the values of the same weighted moving average computed at times t and t-1 respectively. We assume that in the moving average the size of the averaging window equals n-1. The reason for this assumption is to ensure that the trading indicator is computed over the window of size n. The straightforward computation yields