Moving average (finance)
A moving average, in finance and especially in technical analysis, is one of a family of similar statistical techniques used to analyze time series data.
A moving average series can be calculated for any time series, but is most often applied to stock prices, returns or trading volumes. Moving averages are used to smooth out short-term fluctuations, thus highlighting longer-term trends or cycles. The threshold between short-term and long-term depends on the application, and the parameters of the moving average will be set accordingly.
Mathematically, each of these moving averages is an example of a convolution. These averages are also similar to the low-pass filters used in signal processing.
Simple moving average
A simple moving average is the unweighted mean of the previous n data points. For example, a 10-day simple moving average of closing price is the mean of the previous 10 days' closing prices. If those prices are p1 to pn then the formula is
When calculating successive values, a new value comes into the sum and an old value drops out, meaning a full summation each time is unnecessary,
In technical analysis there are various popular values for n, like 10 days, 40 days, or 200 days. The period selected depends on the kind of movement one is concentrating on, such as short, intermediate, or long term. In any case moving average levels are interpreted as support in a rising market, or resistance in a falling market.
In all cases a moving average lags behind the latest price action, simply from the nature of its smoothing. An SMA can lag to an undesirable extent, and can be influenced too much by old prices dropping out of the average. This is addressed by giving extra weight to recent prices, as in the WMA and EMA below.
One characteristic of the SMA is that if the data has a periodic fluctuation, then applying an SMA of that period will eliminate that variation (the average always containing one complete cycle). But a perfectly regular cycle is rarely encountered in economics or finance.
Weighted moving average
A weighted average is any average that has multiplying factors to give different weights to different data points. But in technical analysis a weighted moving average (WMA) has the specific meaning of weights which decrease arithmetically. In an n-day WMA the latest day has weight n, the second latest n-1, etc, down to zero.
When calculating the WMA across successive values, it can be noted an amount p2 to pn + 1 drops out of the numerator each day. The WMA can thus be calculated starting with the above formula but then stepping successively with just additions and subtractions, not a full set of multiplications,
- Totaltoday = Totalyesterday + p1 −- pn + 1
- Numeratortoday = Numeratoryesterday + np1 −- Totalyesterday
The denominator, incidentally, is a triangle number, and equals
The graph at the right shows how the weights decrease, from highest weight for the most recent days, down to zero. It can be compared to the weights in the exponential moving average which follows.
Exponential moving average
An exponential moving average (EMA), sometimes also called an exponentially weighted moving average (EWMA), applies weighting factors which decrease exponentially. The weighting for each day decreases by a factor, or percentage, on the one before it. The graph at right shows an example of the decrease.
There are two ways to express the decrease, both result in a smoothing factor αa. Firstly as a percentage so 10% is α=0.1. Or alternately as N periods where  , so for instance N=19 is equivalent to the 10%. In either case the formula for calculating successive days is
Which can also be rewritten as follows, showing how the EMA steps towards the latest price, but only by a proportion of the difference (each time),
Expanding out EMAyesterday each time results in the following power series, showing how the weighting factor on each price p1, p2, etc, decrease exponentially,
In theory this is an infinite sum, but because 1-α is less than 1, the terms become smaller and smaller, and can be ignored once small enough. The denominator approaches 1/α, and that value can be used instead of adding up the powers, provided one is using enough terms that the omitted portion is negligible.
The N periods in an N-day EMA only specifies the αa factor. It isn't a stopping point for the calculation in the way N is in an SMA or WMA. The first N days in an EMA do represent about 86% of the total weight in the calculation though.
The power formula above gives a starting value for a particular day, after which the successive days formula shown first can be applied.
The question of how far back to go for an initial value depends, in the worst case, on the data. If there are huge p price values in old data then they'll have an effect on the total even if their weighting is very small. If one assumes prices don't vary too wildly then just the weighting can be considered, and work out how much weight is omitted by stopping after say k terms. This is  , which is  , ie. a fraction (1 − α)k out of the total weight.
Thus if the aim was to have 99.9% of the weight then  many terms should be used. And what's more it can be shown  approaches  as N increases, so this simplifies to (roughly)  for this example 99.9% weight.
J. Welles Wilder
Noted technical analyst J. Welles Wilder uses a different form for specifying the period of an EMA. For say 14 days he writes
So α=1/N rather than α=2/(N+1) as described above. The calculation and properties are all the same, it's just a different reckoning of the rate of smoothing. Clearly care must be taken with which is intended. A conversion can be easily made, for instance 14-days from Wilder is equivalent to 27-days in the above (conversion 2N-1).
Other weightings
Other weighting systems are used occasionally – for example, a volume weighting will weight each time period in proportion to its trading volume.
There are weighting systems designed using a combination of moving averages: The DEMA indicator (and TEMA indicator (Triple Exponential Moving Average) are unique composites of a single exponential moving average, a double exponential moving average, and in the latter case a triple exponential moving average that provides less lag than either of the three components individually. They were originally introduced January 1994 by Patrick Mulloy.
The TRIX indicator uses a triple-EMA in its calculation. This ends up as just a certain set of weights on past data, and a set quite different to a plain EMA actually.
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