The Moving Average Convergence Divergence (MACD) is a trading indicator used in technical analysis of securities prices, created by Gerald Appel in the late 1970s. It is designed to reveal changes in the strength, direction, momentum, and duration of a trend in a stock's price. The MACD indicator (or "oscillator") is a collection of three time series calculated from historical price data, most often the closing price. These three series are: the MACD series proper, the "signal" or "average" series, and the "divergence" series which is the difference between the two. The MACD series is the difference between a "fast" (short period) exponential moving average (EMA), and a "slow" (longer period) EMA of the price series. The average series is an EMA of the MACD series itself. The MACD indicator thus depends on three time parameters, namely the time constants of the three EMAs. The notation "MACD(a,b,c)" usually denotes the indicator where the MACD series is the difference of EMAs with characteristic times a and b, and the average series is an EMA of the MACD series with characteristic time c. These parameters are usually measured in days. The most commonly used values are 12, 26, and 9 days, that is, MACD(12,26,9). As true with most of the technical indicators, MACD also finds its period settings from the old days when technical analysis used to be mainly based on the daily charts. The reason was the lack of the modern trading platforms which show the changing prices every moment. As the working week used to be 6-days, the period settings of (12, 26, 9) represent 2 weeks, 1 month and one and a half week. Now when the trading weeks have only 5 days, possibilities of changing the period settings cannot be overruled. However, it is always better to stick to the period settings which are used by the majority of traders as the buying and selling decisions based on the standard settings further push the prices in that direction. Although the MACD and average series are discrete values in nature, they are customarily displayed as continuous lines in a plot whose horizontal axis is time, whereas the divergence is shown as a bar chart (often called a histogram).
A fast EMA responds more quickly than a slow EMA to recent changes in a stock's price. By comparing EMAs of different periods, the MACD series can indicate changes in the trend of a stock. It is claimed that the divergence series can reveal subtle shifts in the stock's trend. Since the MACD is based on moving averages, it is a lagging indicator. As a future metric of price trends, the MACD is less useful for stocks that are not trending (trading in a range) or are trading with unpredictable price action. Hence the trends will already be completed or almost done by the time MACD shows the trend.
Terminology Over the years, elements of the MACD have become known by multiple and often over-loaded terms. The common definitions of particularly overloaded terms are: Divergence:
As the D in MACD, "divergence" refers to the two underlying moving averages drifting apart, while "convergence" refers to the two underlying moving averages coming towards each other. Gerald Appel referred to a "divergence" as the situation where the MACD line does not conform to the price movement, e.g. a price low is not accompanied by a low of the MACD. Thomas Asprey dubbed the difference between the MACD and its signal line the "divergence" series. In practice, definition number 2 above is often preferred. Histogram:
Gerald Appel referred to bar graph plots of the basic MACD time series as "histogram". In Appel's Histogram the height of the bar corresponds to the MACD value for a particular point in time. The difference between the MACD and its Signal line is often plotted as a bar chart and called a "histogram". In practice, definition number 2 above is often preferred.
Formula The formula for the MACD line is based on two exponential moving averages of the close prices, usually with the periods of 12 and 26:
MACD line = EMA 12 − EMA 26 {\displaystyle {\text{MACD line}}={\text{EMA}}_{12}-{\text{EMA}}_{26}}
The signal line is then built as the exponential moving average of the MACD line:
Signal line = EMA 9 ( MACD line ) {\displaystyle {\text{Signal line}}={\text{EMA}}_{9}({\text{MACD line}})}
Mathematical interpretation In signal processing terms, the MACD series is a filtered measure of the derivative of the input (price) series with respect to time. (The derivative is called "velocity" in technical stock analysis.) MACD estimates the derivative as if it were calculated and then filtered by the two low-pass filters in tandem, multiplied by a "gain" equal to the difference in their time constants. It also can be seen to approximate the derivative as if it were calculated and then filtered by a single low pass exponential filter (EMA) with time constant equal to the sum of time constants of the two filters, multiplied by the same gain. So, for the standard MACD filter time constants of 12 and 26 days, the MACD derivative estimate is filtered approximately by the equivalent of a low-pass EMA filter of 38 days. The time derivative estimate (per day) is the MACD value divided by 14. The average series is also a derivative estimate, with an additional low-pass filter in tandem for further smoothing (and additional lag). The difference between the MACD series and the average series (the divergence series) represents a measure of the second derivative of price with respect to time ("acceleration" in technical stock analysis). This estimate has the additional lag of the signal filter and an additional gain factor equal to the signal filter constant.
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