sults in the case the VMA rule. Section 4 contains the estimation results in the TVTP Markov-switching model, data description and the profitability analysis for six simple MA rules. The analysis for four types of trades is also presented. Finally, Section 5 presents conclusions.
3.2 Moving Average Trading Systems
Moving averages are common tools in technical analysis and have been widely used by invest-ment professionals around the world. Moving averages use a smoothing device, averaging asset prices over time, to verify the emergence of new upward or downward price movements. The most widely used moving averages, the Simple Moving Average (SMA), by definition is as follows:
where n is the fixed time interval over which the average is calculated and Pt−iis the asset price at time t − i. In the SMA, each price within the n periods is equally weighted based on the assumption that old prices are equally as relevant as more recent ones when they are used to forecast future price movement.
The mechanism of moving average systems to detect price movements stems from its char-acteristic, lagging behind the current price. In a rising market in which the direction of price movement is positive, the moving average will below the rising price line, whereas it will be above the falling price line when the price direction is negative. Therefore, when the current price crosses the moving average from below, it implies that the price direction reverses from downward to upward. In this situation, the best strategy, buying the asset (the long positions), will be recommended. On the contrary, when the current price crosses the moving average from above, sell signals (the short positions) will be generated because the market might be falling in the future. Similarly, the long-period moving average lags the short-period moving average. Thus, when the short-period moving average breaks above (below) the long-period moving average, a upward (downward) price movement is considered to be initiated.
The size of n determines the price movements detected are short-term or long-term. A SMA
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with smaller n is used to forecast short-term price movements and is called the shorter-period SMA rule. Due to the crossover trading mechanism, a shorter-period SMA rule tends to fol-low changes in underlying asset prices more closely and then generates trading signals quickly since it gives more recent prices higher weight in the average. In other words, as n is smaller, the weight of more recent prices in forming moving averages is greater and consequently the probability of the SMA rule to generate signals is higher.
3.2.1 Market Volatility in Moving Averages
Proposed by Chande (1992), the Variable Moving Average (VMA) rule is the moving average comprising the information of market volatility. It is built to motivate the SMA rule to be more responsive to the ever-changing market conditions on the premise that the SMA rule’s equally weighting to each price within the fixed n periods to detect future price movements is not suitable for all market conditions. For example, if the market price moves directionally (trending market), more recent prices are relatively important in moving averages to take better advantage of the price movements. In this situation, a SMA rule with smaller n (shorter-period or fast SMA) will be a better choice to get early signals. Conversely, if the price changes become directionless and noisier (ranging market), the best investment strategy is not to buy and sell any assets. Therefore, applying a SMA rule with larger n (longer-period or slow SMA) will avoid the usual whipsaw effect of over trading since the probability of generating trading signals is reduced by not giving more recent prices higher weight in moving averages. Compared to the SMA rule’s equally weighting, the central idea of the VMA rule is that in detecting future price movements, the relative importance between old prices and more recent ones in moving averages should not be equal; rather, it should depend on the market condition.
The VMA rule seeks to identify and adapt to ever-changing market conditions by use of the market volatility ratio (VR) as follows:
VRt≡ σnt
σreft , (15)
where σtn is the standard deviation of closing prices over past n periods at time t, while σreft is the reference standard deviation of closing prices over some period of time longer than n at
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time t. In essence the market volatility ratio measures the relative amplitude of price movements between the past n periods and the reference period. The limiting values of the market volatility ratio are VRt > 0. If the market price makes big moves in up or down direction, a reflection of recent good news or bad news, we expect that recent market prices will be more volatile than the reference period. Therefore, VRt > 1 may imply that the present market is trending.
Conversely, if the market price over the past n periods moves in a narrower range due to no influential or new information appearing on the market, VRt < 1, i.e. the ranging market. To sum up, the size of VRtmakes the judgement that whether the present market is active or not.
In order to adjust weighting that past prices have in moving averages according to mar-ket conditions, the VMA rule is constructed by revising the Exponential Moving Averages as follows:
VMAt= α∗tPt+ (1 − α∗t)VMAt−1, (16)
α∗t = αVRt, (17)
Nt∗ = 2
α∗t − 1. (18)
where VMAt is the current value of the Variable Moving Average; VMAt−1is the value of the Variable Moving Average in previous period; α∗t is the scaled smoothing parameter at time t, consists of a constant term α and the market volatility ratio, VRt; Ptis the closing price at time t; Nt∗ is the effective length of past prices used to calculate the moving average value at time t.7 Except for that fact that the smoothing parameter and effective length are time variable, the VMAtin Eq. (16) is mathematically identical to the conventional Exponential Moving Averages model.8
In the VMA, VRt acts as the automatically adjusting device for choosing more suitable effective length in constructing the moving average. As the market is trending, VRt will be
7Hutson (1984) shows that the relationship between N∗ and α∗ in Eq. (18) can be used to approximate the value of the exponential smoothing constant for an equivalent N∗-period SMA.
8As VRt= 1 for all t, the VMA will become the Exponential Moving Averages.
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larger than one. Therefore, the VMA will place more weight (larger α∗t) on the most recent price and adopt a shorter length of past prices (smaller Nt∗). As a result, the probability to take better advantage of the price movements will be increased since the gap between the VMA and the current price will be narrower. In contrast, as the market is ranging, VRt will be smaller than one. A greater weight is no longer given to the more recent data (smaller α∗t) and a longer length of past prices is consider (larger Nt∗). Consequently, the possibility to suffer losses from false tradings can be reduced. In summary, the information of market volatility makes the VMA to be a more responsive indicator to automatically become a shorter-period SMA or change into a longer-period one based on the whether the market is trending or ranging.