Moving average

In statistics, a moving average or rolling average is one of a family of similar techniques used to analyze time series data. It is applied in finance and especially in technical analysis. It can also be used as a generic smoothing operation, in which case the raw data need not be a timeseries.

A moving average series can be calculated for any time series. In finance it 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.

Prior moving average
A simple moving average (SMA) 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 $$p_{M}$$, $$p_{M-1}$$... $$p_{M-9}$$ then the formula is


 * $$\textit{SMA} = { p_{M} + p_{M-1} + \cdots + p_{M-9} \over 10 }$$

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,


 * $$\textit{SMA}_\mathrm{today} = \textit{SMA}_\mathrm{yesterday} - {p_{M-n+1} \over n} + {p_{M+1} \over n}$$

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 data point, simply from the nature of its smoothing. An SMA can lag to an undesirable extent, and can be disproportionately influenced by old data points dropping out of the average. This is addressed by giving extra weight to more recent data points, as in the weighted and exponential moving averages.

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.

Central moving average
For a number of applications it is advantageous to avoid the shifting induced by using only 'past' data. Hence a central moving average can be computed, using both 'past' and 'future' data. The 'future' data in this case are not predictions, but merely data obtained after the time at which the average is to be computed.

Weighted and exponential moving averages (see below) can also be computed centrally.

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.


 * $$\textit{WMA}_{M} = { n p_{M} + (n-1) p_{M-1} + \cdots + 2 p_{M-n+2} + p_{M-n+1} \over n + (n-1) + \cdots + 2 + 1}$$

When calculating the WMA across successive values, it can be noted the difference between the numerators of $$WMA_{M+1}$$ and $$WMA_{M}$$ is $$n p_{M+1} - p_{M} - \cdots - p_{M-n+1}$$. If we denote the sum $$p_{M} + \cdots + p_{M-n+1}$$ by $$Total_{M}$$, then
 * $$Total_{M+1} = Total_{M} + p_{M+1} - p_{M-n+1}$$


 * $$\textit{Numerator}_{M+1} = \textit{Numerator}_{M} + n p_{M+1} - Total_{M}$$


 * $$\textit{WMA}_{M+1} = { \textit{Numerator}_{M+1} \over n + (n-1) + \cdots + 2 + 1}$$

The denominator is a triangle number, and can be easily computed as $$\frac{n(n+1)}{2}.$$

The graph at the right shows how the weights decrease, from highest weight for the most recent data points, 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 older data point decreases exponentially, giving much more importance to recent observations while still not discarding older observations entirely. The graph at right shows an example of the weight decrease.

The degree of weighing decrease is expressed as a constant smoothing factor α, a number between 0 and 1. α may be expressed as a percentage, so a smoothing factor of 10% is equivalent to α=0.1. Alternatively, α may be expressed in terms of N time periods, where $$\alpha={2\over{N+1}}$$. For example, N=19 is equivalent to α=0.1.

The observation at a time period t is designated Yt, and the value of the EMA at any time period t is designated St. S1 is undefined. S2 may be initialized in a number of different ways, most commonly by setting S2 to Y1, though other techniques exist, such as setting S2 to an average of the first 4 or 5 observations. The prominence of the S2 initialization's effect on the resultant moving average depends on α; smaller α values make the choice of S2 relatively more important than larger α values, since a higher α discounts older observations faster.

The formula for calculating the EMA at time periods t≥2 is


 * $$S_{t} = \alpha \times Y_{t-1} + (1-\alpha) \times S_{t-1}$$

This formulation is according to Hunter (1986) ; an alternate approach by Roberts (1959) uses Yt in place of Yt-1 :


 * $$S_{t, alternate} = \alpha \times Y_{t} + (1-\alpha) \times S_{t-1}$$

This formula can also be expressed in technical analysis terms as follows, showing how the EMA steps towards the latest data point, but only by a proportion of the difference (each time):


 * $$\textit{EMA}_{\text{today}} = \textit{EMA}_{\text{yesterday}} + \alpha \times (\textit{price} - \textit{EMA}_{\text{yesterday}})$$

Expanding out $$\textit{EMA}_{\textit{yesterday}}$$ each time results in the following power series, showing how the weighting factor on each data point $$p_1$$, $$p_2$$, etc, decrease exponentially:


 * $$\textit{EMA} = { p_1 + (1-\alpha) p_2 + (1-\alpha)^2 p_3 + (1-\alpha)^3 p_4 + \cdots \over 1 + (1-\alpha) + (1-\alpha)^2 + (1-\alpha)^3 + \cdots }$$

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 specify the α factor. N is not a stopping point for the calculation in the way it is in an SMA or WMA. The first N data points in an EMA represent about 86% of the total weight in the calculation.

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. The weight omitted by stopping after k terms is
 * $$(1-\alpha)^k + (1-\alpha)^{k+1} + \cdots$$,

which is
 * $$(1-\alpha)^k \times (1 + (1-\alpha) + (1-\alpha)^2 \cdots)$$,

i.e. a fraction
 * $${(1-\alpha)^k \over \alpha}$$

out of the total weight.

For example, to have 99.9% of the weight,
 * $$k={ \log (0.001 \times \alpha) \over \log (1-\alpha)}$$

terms should be used. Since $$\log\,(1-\alpha)$$ approaches $$-2 \over N+1$$ as N increases, this simplifies to approximately
 * $$k=3.45\times(N+1)$$

for this example (99.9% weight).

Other weightings
Other weighting systems are used occasionally – for example, in share trading a volume weighting will weight each time period in proportion to its trading volume.

A further weighting, used by actuaries, is Spencer's 15-Point Moving Average (a central moving average).