There are several purposes in regard to trading that Moving Averages (MA) serve. A Moving Average is a good way to gauge momentum as well as to confirm trends, and define areas of support and resistance. Also, Moving Averages form the basis of several other well-known technical analysis tools such as Bollinger Bands and the MACD. One of the most important roles that the Moving Averages play in trading is a form of lagging (or reactive) indicator, which plots average asset prices over time. The main idea is to smooth the curve of points given by the close prices.
There are lots of types of Moving Averages which all take the same basic premise and add a variation. Most notable are the Simple Moving Average (SMA), which is the simple average of a security over a defined number of time periods, and the exponential moving average (EMA), which gives bigger weight to more recent prices exponentially. Also, widely used moving averages are the Linear Weighted Moving Average (LWMA), which gives bigger weight to more recent prices linearly, and Kaufman Adaptive Moving Average
(KAMA), which was created by Perry J. Kaufman (1995) and takes into consideration not only the direction, but also the market volatility.
2.1 The Simple Moving Average (SMA)
The Simple Moving Average (SMA) is arguably one of the oldest and the most popular of all technical indicators. Its rule is one of the most widely used technical trading rules by investors to analyze the stock and eventually strive to outperform the stock market. Technically, SMA is represented by the equation:
In equation 1 the number of periods included in the average is marked as argument n, the relative position of the period currently being considered within the total number of periods is marked as argument k, and is the price at time t. SMA is also known as just “moving average”, “weighted moving average (WMA)” or as an “arithmetic average” because each price in the data series is equally weighted. In other words, Simple Moving Average (SMA) is calculated when you divide the sum of the prices for the past n days by the number of days (n days). There are no weighting factors applied to any of the data points, but one of the possibilities for improving SMA is weighting more heavily the market’s most recent behavior and so incurred other moving averages.
2.2 The Linear Weighted Moving Average (LWMA)
Linear Weighted Moving Average (LWMA) is used to smooth out price and volume fluctuations ("noise") that can confuse interpretation to emphasize the direction of a trend in a very specific way. Similarly to the EMA, the main idea of the Linear Weighted Moving Average (LWMA) is to apply more weight to the most recent price data, but there is a difference in weighting prices. In LWMA there is a less contrast between the weighting of earlier and more recent prices than in the EMA. The popularity of this moving average has been diminished by the exponential moving average, but none the less it still proves to be very useful. LWMA is given by the formula:
where k is the relative position of the period currently being considered within the total number of periods, n is the number of periods included in the average, is the value of Linear Weighted Moving Average (LWMA) at time k, is the price at time k-n+j. From this formula we can see that the oldest price included in the calculation receives a weighting of 1, the next oldest price receives a weighting of 2, and the next oldest price receives a weighting of 3, all the way up to the most recent price which receives a weighting of n (n is a number of periods).
The downside to using a weighted moving average is that the specific resulting average line could make it more difficult to discern a market trend from a fluctuation. For this reason, some traders prefer to place both a SMA and a LWMA on the same price chart. Though the linear WMA is more sensitive to trend changes than the simple moving average, it is less sensitive than the exponential moving average. Some traders find this method more relevant for trend determination, especially in a fast-moving market. LWMA is also called just “Weighted Moving Average (WMA)”.
2.3 The Exponential Moving Average (EMA)
As well as for all moving averages, the basis for Exponential Moving Average (EMA) is Simple Moving Average (SMA). There are two main differences between EMA and SMA:
calculation method,
the way that prices are weighted.
Calculation of the EMA is a little more complex than the calculation of the SMA, but in the age of computers it is facilitated considerably. EMA is represented by the equation:
where k is the relative position of the period currently being considered within the total number of periods, and are the values of Exponential Moving Average at time k and k-1, respectively, is the price at time k, and is weighting multiplier.
For the construction of the EMA we have to calculate weighted multiplier given by the formula:
where n is the number of periods included in the average. Weighting multiplier is also known as “smoothing factor”, and limiting values of weighting multiplier are .
Obviously, EMA has to start somewhere and for calculation of the first EMA ( , where n is the number of periods) we can use price value at that time, simple moving average, some other type of moving average, etc. According to Boylan and Johnston (2003), the best solution is to use an adequate SMA ( ). That's because EMA needs a previous value and thus we need to start somewhere.
Essentially, EMA is a weighted moving average. The theory is that more recent prices are considered to be more important than older prices. So, the weighting is such that the recent days' prices are given more weight than older prices and that’s the reason why EMA reacts faster to recent price changes than a SMA. The Exponential Moving Average (EMA) is also known as "Exponentially Weighted Moving Average (EWMA)".
2.4 Kaufman Adaptive Moving Average (KAMA)
One of the most used types of all adaptive moving averages is Kaufman Adaptive Moving Average (KAMA), which was created by Perry Kaufman (1995) and first presented in his book Smarter Trading. He wanted to overcome two key shortcomings of different smoothing algorithms for financial time series:
accidental price leaps can result in the appearance of false trend signals, smoothing leads to the unavoidable lag in predicting the trends.
Adaptive Moving Average (KAMA) is represented by the equation:
The formula for KAMA is similar to the formula for the EMA, but the difference is that instead of weighting multiplier Kaufman proposed smoothing constant . In equation 5 the current (at time t) value of the KAMA is marked as argument , value of the KAMA in the previous period (at time t-1) is marked as argument , is the price at time t, and is smoothing constant.
Smoothing constant is calculated every day as:
where sc is scaled smoothing constant, calculated as:
In equation 7 fast sc and slow sc are represented by the same equation as weighting multiplier in equation 3. Kaufman (1995) realized that weighting multiplier should be adjusted both for fast and slow trend speed, and because of that he introduced fast and slow smoothing constants. He suggested that the number of days in the fastest and slowest applicable Simple Moving Average should be 2 and 30 days, respectively. For that matter, fast and slow smoothing constants will be 0.6667 and 0.0645, respectively. Of course, these parameters are usually optimized and adapted to the market under study.
given in equation 7 is Efficiency Ratio at time t. Kaufman (1995) wanted to make a mechanism that senses market speed and “chopines” and gives feedback to the moving average. Based on this moving average adjust the speed of its smoothing. Efficiency Ratio is represented by the equation:
where is Efficiency Ratio at time t, ismomentum or the n-day change in price, and is volatility or the sum of the absolute value of daily price changes. is calculated as:
where and are prices at time t, and t-n, respectively. is calculated as:
where and Are prices at time t, and t-i, respectively.
In other words, Efficiency Ratio measures the strength of the trend. ER divides the net price movement by the total price movement. The first and one of the most important steps in the construction of the KAMA is an estimation of the Efficiency Ratio. The limiting values of the efficiency ratio are . When markets are very noisy for the current amount of direction and movements of the market are in a larger range . The opposite of that, when prices are highly directional and changing of price contain relatively little noise
.
Essence of trend-following methods adapt to different market conditions, and that is the starting idea of all adaptive moving averages. KAMA seeks to adapt its behavior, according to the combination of market direction and speed by use of the Efficiency Ratio (ER). Kaufman (1995) noticed four characteristic types of market for research and the development of KAMA: runaway markets (with very fast speeds), fast markets (with fast speeds), congested markets (with very slow speeds) and middle-trends with some volatility (slightly faster sometimes). He based his adaptive moving average on a theory that slow (i.e. Long-length) moving average will be preferred when markets are ranging and fast (i.e. Short-length) will be preferred when market prices are trending.