Methodology

The technical rule examined in this chapter is the popular and well documented moving average rule. A moving average is an average of observations of the level of the index over several consecutive time periods. The objective is to smooth out seasonal variations (volatility) in the data. The standard moving average rule, which utilizes the price line and the moving average of price, generates buy/sell signals on which the investor trades. This strategy is expressed as buying (or selling) when the short-period moving average rises above (or falls below) the long-period moving average. Thus buy and sell signals are generated by crossovers of a long moving average (calculated over L days) by a short moving average (S days, S < L). The buy signal is generated when the short-period moving average moves higher than the long-period moving average:

[∑ _{( )}⁄ ] [∑ _{( )}⁄ ] (5.1)

Where Pt is the price at time t and λ is the length of the moving average. Sell signals are

generated when the inequality is reversed:

[∑ _{( )}⁄ ] [∑ _{( )}⁄ ] (5.2)

A x% band is included to reduce the number of signals by eliminating “whiplash” signals when the short and long period moving averages are close27. The most popular moving average rule in the literature is the (1,200), where the short period is one day and the long period is 200 days. However for completeness, the three most popular variations of the rule are used: (1,50), (1,150) and (1,200). The shorter the size of the moving average, the closer it follows the market, and the longer the size of the moving average, the more it smoothes market fluctuations. Thus a rule with S = 1 is very responsive, that is, whenever the actual returns rises above (below) the moving average, the signal is to buy (sell).

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The moving average rule is examined over the full sample for each market to examine if the rule has been successful over the full period28 and these results are compared to the results of seminal paper by BLL to determine whether the returns of the rules have decreased since their examination of them. To investigate further how returns have fared since the BLL publication, the moving average rule is examined since 1987 (the end of BLL’s sample period) for all markets. Also to examine how the moving average rules have behaved over time, five yearly subsamples of buy-sell returns are calculated and plotted over time as before. Similar to previous chapters, a dotted polynomial trendline is included to smooth the picture of how the anomalies have behaved over time. Again, the suggested classification of return behaviour in Chapter Two is used to categorize the trading rules behaviour as in the previous chapters.

The fact that the moving average rule has been successful for such a long period of time suggests that the rule is picking up some intrinsic property of the market. This property was unknown to investors in general before the publication of the BLL paper but since its publication, investors may have known about the success of the rule and may have begun to implement the rule into their investment strategy. If many investors follow the moving average rule, it will force more buying (selling) pressure on the price when a buy (sell) signal is generated. Thus prices will go up (down) more (less) than they would have done before the publication of BLL due to the increased volume of trading caused by this rule. This means that the stock will become overvalued (undervalued) relative to what it would previous have been due to the high (low) buying pressure at the start of the buy period (which is beyond what was causing the phenomenon on the first place) which may result in the rule not doing so well in the future and explain why the rule after the BLL publication is not very successful.

Figure 5.1 presents the effect of investors knowing about the success of the moving average and trading on it. Initially it is assumed that the value of the stock before the rule is known to investors is zero for simplicity reasons (along the x-axis). However when investors begin trading on the moving average rule and a buy signal is generated, the increased volume of buying causes the stock’s value to increase to X1

beyond what it should be. The same can be said when a sell signal is generated and the increased volume of selling causes the stock’s

28

Atanasova and Hudson (2010) examined the DJIA up to March 2009 while this study investigates up to December 2009.

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value to fall to X2 less than it should be. Thus it is clear to see from Figure 5.1 that if many investors trade on the moving average rule, the value of the stock when a signal is created is distorted. This may cause the moving average rules predictability to diminish.

Informed investors may realise this and begin to anticipate the next day’s signal take advantage of the overvaluing/undervaluing of the price. To investigate this possibility, perfectly anticipated and imperfectly anticipated moving average rules are proposed and examined. These two rules anticipate the next day’s signal and trade that signal today to take advantage of the overvalued/undervalued price. The perfectly anticipated moving average rule perfectly predicts the signal for the following day through the moving average rule. This is often quite possible since the long run moving average is often not close to the current price and so it is fairly certain what the following days signal is going to be. The imperfectly anticipated moving average rule incorporates the fact that investors will not always be 100% confident what the next day’s signal is going to be. This rule is the same as the perfectly anticipated moving average rule except when the current price is very close to the long run moving average, a neutral signal is created like before. Bands of 0.25%, 0.50% and 1% are used, similar to before to create these neutral signals. That is, if the short run moving average is within the long run moving average by 0.25%, 0.5% and 1% the investor is faced with a neutral signal. Instead of not trading, when the investor is faced with a neutral signal they choose the current days signal and trade on that. That is, if the investor is uncertain what the following days signal is going to be, instead of predicting it they use the current days signal. These two rules are examined in section 5.4 of this chapter.

Figure 5.1: The effect on the value of the stock when investors

know about the moving average rule and begin to trade on it.

Buy Signal Sell Signal

X1

X2

Time Value of the Stock (normalised

to be flat before the moving average rule is widely known to investors)

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An important question to ask when dealing with any technical rule is whether an investor can use them to gain returns greater than the market. Thus the degree to which investors can earn profits that beat the buy-and-hold strategy using two simple trading strategies are analysed. This chapter considers two simple trading strategies, which are also used in the previous chapter for calendar anomalies.

This study prefers simple trading strategies to complicated strategies since calendar anomalies are straightforward to understand and thus it should be relatively simple to make profits from them. Many studies use a trading strategy that invests in the risk-free asset if they are not in the market. Even though this may give a more equivalent risk to the buy-and- hold strategy since the investor is always in some market, investing in the risk-free rate may be costly and time consuming since investors may only be out of the market for one or two days. Since this thesis uses data from the US since 1897, from the UK since 1935 and from Japan since 1951, risk-free rate data was not available for the full sample and so is ignored in these trading strategies. Since the investor does not invest in risk-free assets when they are out of the market in either of the trading strategies examined, the overall returns for the trading strategy will be less than if the investor had invested in the risk-free asset, making it more difficult for these rules to gain returns greater than the buy-and-hold strategy than if investment in the risk-free asset was conducted for every sell signal, thus the figures generated are conservative. Nevertheless, a “double to out” trading strategy, which has broadly the same risk as the buy-and-hold strategy is studied, as well as a simple trading strategy which does not have comparable risk.

The first trading strategy adopted is similar to Fifield et al (2005; 2008) and is as follows. The investor is initially assumed to hold a buy position and upon the first buy signal, the trader buys and holds until a sell signal is generated. Upon this sell signal, the trader sells and goes out of the market until the next buy signal. Upon the last sell signal, it is assumed that the investor liquidates his position. At the end of the sample period, the profit from the different trading rules are calculated and compared with the profit from the naïve buy-and- hold strategy. The profits from this strategies are calculated net of transaction costs (transaction costs taken from Ratner and Leal 1999 for the US and Japan, and Hudson et al

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1996 for the UK29). The trading strategy evaluated here differs from those in the majority of the previous papers. For example, this rule assumes that the investor has a limited amount of wealth that is invested in full at each buy (sell) transaction. That is, this rule assumes that the investor can only sell after a buy transaction (and buy only after a sell transaction) whereas other studies assume that the investor has an unlimited amount of wealth and can implement multiple buys or sells after each price change. The strategy examined here can therefore be characterised as prudent, and as satisfying the risk-averse nature of many investors (Fifield et al 2005).

The second trading strategy examined follows the “double or out” rule suggested by Bessembinder and Chan (1998). An investor who conducts the previous simple trading strategy faces a lot less risk than an investor who conducted the buy-and-hold strategy. This is because they are out of the market for a considerable period of time and avoid the risk associated with being in the market all of the time. Acknowledging this fact, a slightly modified version of the “double or out” trading strategy suggested by Bessembinder and Chan (1998) is applied to the various moving average rules previously examined. If a neutral signal is generated there is an investment in the index. If a buy day is indicated the investment in the index is doubled whereas, if a sell day is indicated, the funds are invested in cash thus giving broadly similar risk to a buy-and-hold strategy. Bessembinder and Chan (1998) invest in the daily risk-free rate when a sell signal is generated but since no risk-free rates are available for long periods of the data examined, the investor invests in cash with no return when a sell signal is generated. The profits from this strategy are also calculated net of transaction costs (transaction costs taken from Ratner and Leal 1999 for the US and Japan, and Hudson et al 1996 for the UK). These two trading strategies are conducted to determine if simple trading on the calendar anomalies can beat the buy-and-gold strategy for each index.

One issue with this type of trading rule is that the issue of stocks indices were not easily tradable until the 1980s when futures on indices and exchange traded funds (ETFs) became available to investors. However this thesis assumes, similar to most other studies examining the profitability of trading rules, that investors can trade on the stock indices easily without any extra cost incurred.

29 _{Although these transaction costs are accurate for the period in which they were calculated, they do not}

correspond to the costs faced in the total sample examined in this thesis. Nevertheless with no data available for the full sample, these transaction costs are employed.

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