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All moving averages, from the simple to the complex, smooth time series data by some kind of averaging process. They differ in how they weigh the sample points that are averaged and in how well they adapt to changing conditions. The differences between moving averages arose from efforts to reduce lag and increase responsiveness. The most popular moving averages (equations below) are the simple moving average, the exponential moving average, and the front-weighted triangular moving average. Less popular is Chande s adaptive moving average (1992).
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ai =
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(2m + 1 - k) si-k] / ( J u (2m + 1 - k) ] Front-weighted triangular
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In the equations, aj represents the moving average at the i-th bar, si the i-th bar or data point of the original time series, m the period of the moving average, and c (normally set to 2 / (m + I)) is a coefficient that determines the effective period for the exponential moving average. The equations show that the moving averages differ in how the data points are weighted. In a simple moving average, all data points receive equal weight or emphasis. Exponential moving averages give more weight to recent points, with the weights decreasing exponentially with distance into the past. The front-weighted triangular moving average weighs the more recent points more heavily, but the weights decline in a linear fashion with time; TradeStation calls this a weighted moving average, a popular misnomer. Adaptive moving averages were developed to obtain a speedier response. The goal was to have the moving average adapt to current market behavior, much as Dolby noise reduction adapts to the level of sound in an audio signal: Smoothing increases when the market exhibits mostly noise and little movement (more noise attenuation during quiet periods), and smoothing declines (response quickens) during periods of more significant market activity (less noise suppression during loud
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passages). There are several adaptive moving averages. One that seems to work well was developed by Mark Jurik ( Another was VIDYA (Variable Index Dynamic Moving Average) developed by Chande. A recursive algorithm for the exponential moving average is as follows: For each bar, a coefficient (c) that determines the effective length (m) of the moving average is multiplied by the bar now being brought into the average and, to the result, is added 1.0 - c multiplied by the existing value of the moving average, yielding an updated value. The coefficient c is set to 2.0/(1.0 + m) where m is the desired length or period. Chande (1992) modified this algorithm by changing the coefficient (c) from a fixed number to a number determined by current market volatility, the market s loudness, as measured by the standard deviation of the prices over some number of past bars. Because the standard deviation can vary greatly between markets, and the measure of volatility needs to be relative, Chande divided the observed standard deviation on any bar by the average of the standard deviations over all bars on the S&P 500. For each bar, he recomputed the coefficient c in the recursive algorithm as 2.0 / (1.0 + m) multiplied by the relative volatility, thus creating a moving average with a length that dynamically responds to changes in market activity. We implemented an adaptive moving average based on VIDYA that does not require a fixed adjustment (in the form of an average of the standard deviations over all bars) to the standard deviation. Because markets can change dramatically in their average volatility over time, and do so in a way that is irrelevant to the adaptation of the moving average, a fixed normalization did not seem sound. We replaced the standard deviation divided by the normalizing factor (used by Chande) with a ratio of two measures of volatility: one shorter term and one longer term. The relative volatility required for adjusting c, and hence the period of the adaptive moving average, was obtained by dividing the shorter term volatility measure by the longer term volatility measure. The volatility measures were exponential moving averages of the squared differences between successive data points. The shorter moving average of squared deviations was set to a period of p, an adjustable parameter, while the period of the longer moving average was set top multiplied by four. If the longer term volatility is equal to the most recent volatility (i.e., if their ratio is 1). then the adaptive moving average behaves identically to a standard exponential moving average with period m; however, the effective period of the exponential moving average is monotonically reduced with an increasing ratio of short-term to long-term volatility, and increased with a declining ratio of short-term to long-term volatility. TYPES OF MOVING AVERAGE ENTRY MODELS A moving average entry model generates entries using simple relationships between a moving average and price, or between one moving average and another. Trend-following and countertrend models exist. The most popular models follow
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the trend and lag the market. Conversely, countertrend moving average models anticipate reversals and lead, or coincide with, the market. This is not to imply that countertrend systems trade better than trend-following ones. Consistently entering trends, even if late in their course, may be a more reliable way to make money than anticipating reversals that only sometimes occur when expected. Because of the need for a standard exit, and because no serious trader would trade without the protection of money management stops, simple moving average models that are always in the market are not tested. This kind of moving average reversal model will, however, be fairly well-approximated when fast moving averages are used, causing reversal signals to occur before the standard exit closes out the trades. Trend-following entries may be generated in many ways using moving averages. One simple model is the ntoving average C~SSDWE The trader buys when prices cross above the moving average and sells when they cross below. Instead of waiting for the raw prices to cross a moving average, the trader can wait for a faster moving average to cross a slower one: A buy signal occurs when the faster moving average crosses above the slower moving average, and a sell is signalled when the crossover is in the other direction. Smoothing the raw price series with a fast moving average reduces spurious crossovers and, consequently, minimizes whipsaws. Moving averages can also be used to generate countemend entries. Stock prices often react to a moving average line as they would to the forces of support or resistance; this forms the basis of one countertrend entry model. The rules are to buy when prices touch or penetrate the moving average from above, and to sell when penetration is from below. Prices should bounce off the moving average, reversing direction. Countertrend entries can also be achieved by responding in a contrary manner to a standard crossover: A long position is taken in response to prices crossing below a moving average, and a short position taken when prices cross above. Being contrarydoing the opposite of what seems right +ften works when trading: It can be profitable to sell into demand and buy when prices drop in the face of heavy selling. Since moving averages lag the market, by the time a traditional buy signal is given, the market may be just about to reverse direction, making it time to sell, and vice versa. Using a moving average in a counter&end model based on support and resistance is not original. Alexander (1993) discussed retracement to moving average sup port after a crossover as one way to set up an entry. Tilley s (1998) discussion of a two parameter moving average model, which uses the idea of support and resistance to trade mutual funds, is also relevant. Finally, Sweeney (1998) described the use of an end-of-day moving average to define inuaday levels of support and resistance. CHARACTERISTICS OF MOVING AVERAGE ENTRIES A trend-following moving average entry is like a breakout: Intuitively appealing and certain to get the trader aboard any major trend: it is also a traditional, readily
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available approach that is easy to understand and implement, even in a spreadsheet. However, as with most trend-following methods, moving averages lag the market, i.e., the trader is late entering into any move. Faster moving averages can reduce lag or delay, but at the expense of more numerous whipsaws. A countertrend moving average entry gets one into the market when others are getting out, before a new trend begins. This means better fills, better entry prices, and greater potential profits. Lag is not an issue in countertrend systems. The danger, however, is entering too early, before the market slows down and turns around. When trading a countertrend model, a good risk-limiting exit strategy is essential; one cannot wait for the system to generate an entry in the opposite direction. Some countertrend models have strong logical appeal, such as when they employ the concept of support and resistance. ORDERS USED TO EFFECT ENTRIES Entries based on moving averages may be effected with stops, limits, or market orders, While a particular entry order may work especially well with a particular model, any entry may be used with any model. Sometimes the entry order can form part of the entry signal or model. A basic crossover system can use a stop order priced at tomorrow s expected moving average value. To avoid intraday whipsaws, only a buy stop or a sell stop (not both) is issued for the next day. If the close is above, a sell stop is posted; if below, a buy stop. Entry orders have their own advantages and disadvantages. A market order will never miss a signalled entry. A stop order will never miss a significant trend (in a trend-following model), and entry will never occur without contirmation by movement in favor of the trade; the disadvantages are greater slippage and less favorable entry prices. A limit order will get the best price and minimize transaction costs, but important trends may be missed while waiting for a retracement to the limit price. In countertrend models, a limit order may occasionally worsen the entry price: The entry order may be filled at the limit price, rather than at a price determined by the negative slippage that sometimes occurs when the market moves against the trade at the time of entry! TEST METHODOLOGY In all tests that follow, the standard portfolio is used. The number of contracts to buy or sell on entry, in any market, at any time, was chosen to approximate the dollar volatility of two S&P 500 contracts at the end of 1998. Exits are the standard ones. All tests are performed using the C-Trader toolkit. The portfolios, exit strategies, and test platform are identical to those used previously, making all results comparable. The tests are divided into trend-following and counter&end ones. They were run using a script containing instructions to set parameters, run
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