Moving averages (MA) are one of the most accepted and simple to use tools available to the technical analysts. Moving Averages smooth a data series and make it simpler to spot trends. It is something that is extremely helpful in volatile markets (Ling, and Li). They also form the building blocks for so many other overlays and technical indicators. The moving average is a smoothing tool. Low and high prices are covered and the basic trend of the capital market is more accurately seen by averaging the price information. By its very nature the moving average line is always behind the market action.

A shorter period moving average i. e. of 3-5 days, would hug the price action more closely than a forty-day moving average. Shorter term MAs are more influenced by everyday shifts. MA demonstrates the average value of a security’s price in a time frame. The average price shifts upward or downward when the security’s price shifts. Advantages and Disadvantages of Moving Averages: The advantages of using MAs need to be weighed against the disadvantages. Moving averages are trend following, or lagging indicators that will always be one step behind. This is not necessarily a bad thing though.

After all, the trend is our friend and it is best to trade in the direction of the trend (Ling, and Li). MAs will help ensure that a trader is in line with the current trend. Markets, securities and stocks spend a great deal of time in trading ranges, which render MAs ineffective. Once in a trend, MAs will keep you in, but also give late signals. Do not expect to get out at the top and in at the bottom using MAs. As with most tools of technical analysis, MAs should not be used on their own, but in conjunction with other tools that complement them.

Using MAs to confirm other indicators and analysis can greatly enhance technical analysis. Types of Moving Averages: The most popular types of moving averages are simple or arithmetic, variable exponential, weighted and triangular moving averages. One can measure MAs on any data series comprising a security’s open, high, low, close, volume or any other indicator. The basic difference between MA variants is the weight which refers to the latest data. SMAs provide the same weight to all the prices. Triangular averages provide more weight to prices in the middle of the time period (Ling, and Li).

Exponential and weighted averages provide more weight to the recent prices. The different types of MAs are as follows: 1. Simple Moving Average (SMA) A simple moving average is shaped by calculating the average or mean price of a security over a specified number of periods. While it is possible to create moving averages from the High, the Low and the Open data points, most moving averages are created using the closing price. For example: a five day simple moving average is computed by adding the closing prices for last 5 days and dividing the total by 5.

= > 10+ 11 + 12 + 13 + 14 = 60 = > (60 / 5) =12 The calculation is repeated in the same manner for each price bar on the chart. The averages are then connected to form a smooth curving line or the moving average line. Continuing the example, if the next closing price in the average is 15, then this new period would be added and the oldest day, which is 10, would be dropped. The new five day simple moving average would be calculated as follows: => 11 + 12 + 13 + 14 +15 = 65 => (65 / 5) = 13 Over the last two days, the small moving average moved from 12 to 13.

As new days are further added, then the old days will be subtracted and the moving average will carry on moving over time. Day Daily Close 10-Day SMA 1 60. 33 2 59. 44 3 59. 38 4 59. 38 5 59. 22 6 58. 88 7 59. 55 8 59. 50 9 58. 66 10 59. 05 59. 34 11 57. 15 59. 02 12 57. 32 58. 81 13 57. 65 58. 64 14 56. 14 58. 31 15 55. 31 57. 92 16 55. 86 57. 62 17 54. 92 57. 16 18 53. 74 56. 58 19 54. 80 56. 19 20 54. 86 55. 78 In the afore-mentioned example, using closing prices from Eastman Kodak (EK), 10th day is the 1st day possible to calculate a ten day SMA.

As the calculation continues, the newest day is added and the oldest day is subtracted. The ten day small moving average for 11th day is calculated by adding the prices of 2nd day through 11th day and dividing by 10. The averaging process then moves on to the next day where the ten day small moving average for 12th day is calculated by adding the prices of 3rd day through 12th day and dividing by 10. The above chart is a plot that contains the data sequence in the table. The SMA begins on the 10th day and continues.

This diagram highlights the fact that all the moving averages are covering indicators and will always be behind the price. The price of Eastman Kodak Co. is showing is downwards trend, but the SMA, which is based on the previous 10-days of data, remains above the price. If the price were mounting, the small moving average would most likely be lower. SMA fits in the category of trend following indicators. When prices are trending, the moving averages work well. However, when prices are not trending, the moving averages can provide misleading signals. 2. Exponential Moving Average (EMA)

In order to lessen the lag in simple moving averages, technicians often make use of exponential moving averages which are also called exponentially weighted moving averages. EMA trims down the lag by applying more weight to the recent prices relative to older prices. The weighting applied to the latest price depends on the specified period of the moving average. The shorter the EMA period, the more will be the weight applied to the most recent price. For example: a ten-period EMA weighs the most recent price 18. 18% while a twenty-period EMA weighs the most recent price 9. 52%.

Calculation of EMA is much harder than calculation of SMA. The important thing to remember here is that the EMA puts more weight on the recent prices. It will react faster to recent price changes than a SMA. Exponential Moving Average Calculation It can be specified in two ways: 1. Percent based EMA: It has a percentage as it’s single parameter 2. Period-based EMA: It has a parameter that represents the duration of the EMA Formula for EMA: ? EMA (current) = (Price-current – EMA-previous x Multiplier) + EMA-previous For a percentage-based EMA, Multiplier is equal to the EMA specified percentage.

For a period-based EMA, Multiplier is equal to 2 / (1+N); where N is the specified number of periods. For example, a ten-period EMA Multiplier is calculated in the following fashion: ? (2 / (Time periods + 1) ) ? (2 / (10 + 1) ) ? 0. 1818 or 18. 18% This means that a ten-period EMA is equivalent to 18. 18% EMA. Below is a table with the results of an EMA calculation for EK. For the first period’s EMA, the SMA was used as the previous period’s EMA. From period-11 onward, the previous period’s EMA was used. The calculation in period-11 breaks down as follows: ? (C – P) = (57.

15 – 59. 439) = -2. 289 ? (C – P) x K = -2. 289 x . 181818 = -0. 4162 ? ( (C – P) x K) + P = -0. 4162 + 59. 439 = 59. 023 EMA Period (N): 10 Smoothing Constant (K): 0. 1818 Period Date (C) Previous Period’s EMA (P) 10-Day EMA (X) 1 9-Nov-99 $ 60. 33 2 10-Nov-99 $ 59. 44 3 11-Nov-99 $ 59. 38 4 12-Nov-99 $ 59. 38 5 13-Nov-99 $ 59. 22 6 14-Nov-99 $ 58. 88 7 15-Nov-99 $ 59. 55 8 16-Nov-99 $ 59. 50 9 17-Nov-99 $ 58. 66 10 18-Nov-99 $ 59. 05 $ 59. 439 11 19-Nov-99 $ 57. 15 $ 59. 439 $ 59. 023 12 20-Nov-99 $ 57. 32 $ 59. 023 $ 58. 713 13 21-Nov-99 $ 57. 65 $ 58. 713 $ 58. 520

14 22-Nov-99 $ 56. 14 $ 58. 520 $ 58. 087 15 23-Nov-99 $ 55. 31 $ 58. 087 $ 57. 582 16 24-Nov-99 $ 55. 86 $ 57. 582 $ 57. 269 17 25-Nov-99 $ 54. 92 $ 57. 269 $ 56. 842 18 26-Nov-99 $ 53. 74 $ 56. 842 $ 56. 278 19 27-Nov-99 $ 54. 80 $ 56. 278 $ 56. 009 20 28-Nov-99 $ 54. 86 $ 56. 009 $ 55. 800 Every previous closing price in the data set is used in the calculation of each exponential moving average that makes up the EMA line. The impact of older data points reduces over time, it never fully disappears. This is true regardless of the exponential moving average’s specified period.

The effects of older data lessen speedily for shorter exponential moving averages than for longer ones but they never completely disappear. 3. Double Exponential Moving Average (DEMA) The Double Exponential Moving Average or DEMA consists of a single EMA and a double EMA. Its main preference is that it provides a reduced amount of delays than if the 2 moving averages had been used apart. The DEMA is calculated as follows: => (2 * n-day EMA) – (n-day EMA of EMA) Where EMA = exponential moving average When price crosses the MA and increases, a continuing upward trend can be predicted.

So as far as we see, it’s possible to use the DEMA in the same way as the SMA or EMA. 4. Triple Exponential Moving Average (TEMA, or TRIX) The Triple Exponential Moving Average (TEMA) is a unique combination of a single EMA, a DEMA, and a TEMA that provides less lag than any of those 3 individually. It can be used instead of traditional MAs for smoothing price data or other indicators. As well as a momentum indicator, TEMA is an oscillator which follows over sold and over bought markets. When TEMA or TRIX is used as a momentum indicator, a negative value suggests momentum is lessening while a positive value suggests escalating momentum.

Some analysts think that the TRIX crossing above the 0-line is a purchase signal and a closing below the 0-line is a sell signal. Difference between price and TRIX can also show important market turning points. TRIX filtration of market noise and its propensity to be a leading and not a lagging indicator are 2 advantages of TRIX over other trend indicators. Insignificant cycles are excluded by using triple exponential smoothing. It is able to lead a market as it computes the difference between each bar’s smoothed versions of the price information.

TRIX is best used together with another market timing indicator so as to decrease false signals when used as a leading indicator. The MA Technical Indicator shows the mean instrument price value for a certain period of time. When one calculates the MA, one averages out the instrument price for this time period. As the price changes, its MA either increases, or decreases. Here are the SMA, EMA, SMMA, and LWMA on the chart: 5. Adaptive Moving Average (AMA) To make a good signal, MA should be smooth but should not lag too much behind the price.

AMAs adjust their period or weightings in order to meet this objective. Taking a simple method of forming an AMA, by giving the current point a weighting of 1, and then lessen the weightings for each of the previous points every time there is a change in the price, but not in direction. Thus when the price is highly erratic, the AMA defaults to a SMA, while if the price is going in one direction, it can be described as a weighted moving average (WMA) with an exponential decline in weights. The AMA gives signals in much the same way as the WMA, but should be much quicker in identifying the end of a trend.

A buy signal is generated when the price goes higher than the AMA, while a sell signal is generated when the price stays under the AMA. 6. Linearly Weighted Moving Average (LWMA) It is a type of MA that assigns a higher weighting to recent price data than does the common SMA. LWMA is calculated by taking each of the closing prices over a given time period and multiplying them by its certain position in the data series. Once the position of the time periods have been accounted for they are added together and the divided by the sum of the number of time periods.

For example, in a 15-day LWMA, today’s closing price is multiplied by 15 and the yesterday’s closing price is multiplied by 14, and so on until day 1 in the period’s range is reached. These results are then added together and divided by the sum of the multipliers. ? 15 + 14 + 13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 120 The LWMA was one of the first responses to placing a greater importance on recent data. The popularity of LWMA has been diminished by the EMA, but none the less it still proves to be very useful. 7. Smoothed Moving Average (SMMA)

The first value of this smoothed moving average (SMMA) is calculated as the simple moving average (SMA): ? SUM1 = SUM (CLOSE, N) ? SMMA1 = SUM1/N The second and succeeding MAs are calculated according to this formula: ? SMMA(i) = (SUM1-SMMA1+CLOSE(i))/N Where SUM1 is the total sum of closing prices for N periods; SMMA1 is the smoothed moving average of the first bar; SMMA (i) is the smoothed moving average of the current bar except for the 1st one; CLOSE (i) is the current closing price; N is the smoothing period. 8. Linear Weighted Moving Average (LWMA)

In the case of weighted moving average, the newest data is of extra value than more early data. LWMA is calculated by multiplying each one of the closing prices within the considered series, by a certain weight coefficient. LWMA = SUM (Close (i) * i, N)/SUM (i, N) Where: SUM (i, N) is the total sum of weight coefficients. Simple Versus Exponential From far afield, it would appear that the distinction between an EMA and a SMA is minimal. The EMA is consistently closer to the actual price and on average; the EMA is 3/8 of a point closer to the actual price than the SMA.

Period EMA Absolute Difference SMA Absolute Difference 10 0. 39 0. 29 11 1. 87 1. 87 12 1. 39 1. 49 13 0. 87 0. 99 14 1. 95 2. 17 15 2. 27 2. 61 16 1. 41 1. 76 17 1. 92 2. 24 18 2. 54 2. 84 19 1. 21 1. 39 20 0. 94 0. 92 Average Difference 1. 52 1. 69 From 10th day to 20th day, the EMA was closer to the price than the SMA 8 out of 11 times. The average absolute difference between the EMA and the current price was 1. 52 and the SMA had an average absolute difference of 1. 69. This means that on average, the EMA was 1. 52 point above or below the current price and the SMA was 1.

69 points above or below the current price. When Kodak stopped declining and started to trade flat, the SMA kept on declining. During this period, the SMA was closer to the real price than the EMA. The EMA began to level out with the actual price, and remain more away. This was because the actual price started to level out. Due to this lag, the SMA continued to decline and nearly touched the actual price on Dec 13. Conclusion: MAs can be effective tools to identify and confirm trend, identify resistance and support levels, and develop trading systems (Ling, and Li).

However, investors and traders should learn to identify securities that are suitable for analysis with MAs and how this analysis should be applied. Usually, an assessment can be made with a visual examination of the price chart, but occasionally it will require a more detailed approach. The Average Directional Index (ADI), is one tool that can help identify securities that are trending and those that are not. Works Cited Benjamin, Michael A. , Robert A. Rigby, and D. Mikis Stasinopoulos. “Generalized Autoregressive Moving Average Models.

” Journal of the American Statistical Association 98. 461 (2003): 214+. Ling, Shiqing, and W. K. Li. “On Fractionally Integrated Autoregressive Moving-Average Time Series Models with Conditional Heteroscedasticity. ” Journal of the American Statistical Association 92. 439 (1997): 1184+. Runger, George C. , and Sharad S. Prabhu. “A Markov Chain Model for the Multivariate Exponentially Weighted Moving Averages Control Chart. ” Journal of the American Statistical Association 91. 436 (1996): 1701+.