Testing the efficient market hypothesis in the foreign exchange market with the Bollinger Bands indicator

Testing the efficient market hypothesis in the foreign

exchange market with the Bollinger Bands indicator

Bachelor Thesis in Economics and Finance

University of Amsterdam

Faculty of Economics and Business

Author:

Lukas Snapstys

Student number:

10436073

Date:

June 29, 2016

Field:

Finance, trading

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Abstract

The technical analysis is a widespread analytical tool which uses historic data to predict securities' future price movements. The weak-form efficient market hypothesis states that all available historic information such as past prices and volume is already priced in the security and therefore it is impossible to outperform the market and other investors thereby challenging the use of technical analysis. This paper analyses the ability of the Bollinger Band technical indicator to generate positive risk-adjusted returns in the foreign exchange market represented by four major currency pairs EUR/USD, GBP/USD, USD/CHF and USD/JPY over the period of 1995-2015. The back-test and walk-forward optimization results of the research show that it was impossible to generate risk-adjusted returns during the specified period. However, no definite conclusion can be drawn about the efficiency of the foreign exchange market only from this research.

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Statement of Originality

This document is written by Lukas Snapstys who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Table of Contents

1. Introduction ... 4

2. Literature review ... 5

3. Theoretical Framework ... 7

3.1 Efficient and Adaptive Market Hypothesis ... 7

3.2 Behavioral Finance ... 8

3.3 Technical analysis ... 8

3.4 The Bollinger Bands ... 9

4. Research Methodology ... 10

4.1 Back-testing the Bollinger Band indicator ... 10

4.2 The walk-forward optimization of the Bollinger Band indicator ... 12

5. Results... 13

5.1 Results of the back-test ... 13

5.2 Results of the walk-forward optimization ... 15

6. Conclusions ... 17

Bibliography ... 19

Appendix A ... 21

Appendix B ... 22

Appendix C ... 23

Appendix D ... 24

Appendix E ... 25

Appendix F ... 26

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1. Introduction

When an investor is considering buying a security for speculative purposes, he is expecting future price of a stock to be higher so it can be sold for profit. Similar logic applies when the investor is expecting price to fall - the security is being borrowed and sold right now and bought back later for a lower price for positive returns. The expectations regarding future price movements of a security are based on two types of analysis: fundamental and technical.

The fundamental analysis attempts to measure fair value of a stock by examining quantitative and qualitative factors that might affect security's value. These include macroeconomic events such as fluctuations of interest rates, unemployment figures, commodity prices as well as microeconomic news such as financial position of company, management performance, growth possibilities.

The technical analysis focuses on use of past prices and other market data such as volume to guide traders expectation regarding the future. A simple technical trading rule might suggest buying a stock if it is more than 5% above its average price of 5 last days. Technical traders also known as "technicians" study the price patterns therefore exploiting psychological regularities. Technicians argue that security prices move in trends and these are essential for the success of this type of analysis as trends imply predictability and therefore profit opportunities and this is captured by popular saying "the trend is your friend" (Neely & Weller, 2011, p. 2). In addition, history tends to repeat itself and market confronted by similar conditions will react similar during different times. This implies that price patterns would repeat themselves providing opportunities to benefit from such repetitions.

However, the efficient market hypothesis (EMH) first introduced by Maurice Kendall in 1953 questions the ability of such investors to outperform the market. The efficient market hypothesis states that all available information is already represented in security's price and therefore securities are priced correctly. As a result, if EMH indeed held, no single investor would ever be able to outperform other investors or overall market. Specifically, weak form EMH argues that prices already reflect all information that can be derived from the past prices and according to this specific branch of EMH, technical analysis is a fruitless activity (Bodie, Kane & Marcus, 2014).

Despite its appealing reasoning and support by the academics, it is unclear why technical analysis is such a widespread tool amongst professional traders and investors. This is supported by Taylor and Allen (1992) survey on chief foreign exchange dealers in London. According to the survey, almost all traders in the London foreign exchange markets use technical analysis to some degree and it is usually combined with the fundamental analysis. Cheung and Chinn (2001) confirmed this study on the U.S foreign exchange traders and arrived to a conclusion that 30% of all traders could be

5 characterized as technical traders and that an increasing percentage of traders is implementing technical analysis.

The purpose of this paper is to check whether the Bollinger Band technical trading rule can outperform the market and generate positive risk-adjusted returns in the foreign exchange market therefore challenging the weak-form efficient market hypothesis. In this paper, four most traded currency pairs will be used: EUR/USD; GBP/USD; USD/CHF; USD/JPY. The Bollinger Bands indicator will be back tested as a technical rule for all of the currency pairs during the recent period of 1995-2015. Research will also include the walk-forward optimization of the Bollinger Band indicator to determine whether a real-time investor could outperform the market during the specified time period by adjusting parameters of the Bollinger Band as time goes by. Results of the back test and walk-forward optimization will identify whether it was possible to earn positive risk-adjusted returns during this period.

The structure of the remainder of this paper is organized as follows: Section 2 reviews the existing contrary literature on performance of technical analysis. Section 3 elaborates on the theory behind efficient market hypothesis, behavioral finance, technical analysis and the Bollinger Band indicator. Section 4 elaborates on research methodology based on the back-test and walk-forward optimization techniques as well as their parameters in order to check whether the Bollinger Band indicator could deliver positive risk-adjusted returns during the specified period. Section 5 presents the results from the research highlighting inability of this indicator to deliver constant returns while Section 6 summarizes these results and concludes this paper with regards to both efficient market hypothesis and this particular indicator.

2. Literature review

The literature on the performance of technical analysis started in as early as 1970s and has been popular amongst both academics and economists. However, regardless of its significance, research papers do not provide clear conclusions on whether use of technical trading rules can result in abnormal risk-adjusted returns.

One of the earliest and most well known researches was done by Fama and Blume (1966) where researchers applied filter rules for 30 individual stocks of Dow Jones Industrial Average. Filter rules produce a buy signal whenever the exchange rate rises by more than a given percentage from its previous low value and produces a selling signal when exactly opposite condition is met. Filter size is chosen by a technician but typically ranges from 0.5% to 10%. After the commissions, only 4 out of 30

6 securities had positive average returns and were inferior to buy & hold strategy over the period of 1956-1962. These results are contrary to those conducted by Brock, Lakonishok and LeBaron (1992) where previous researches were identified as premature because significant positive returns for DJIA were found for period from 1897 to 1986.

Park and Irwin (2007) conclude that technical analysis is more profitable in the foreign exchange and commodity futures markets than in the stock markets. This is also supported by foreign currency traders after era of floating exchange rates which started in 1970s (Neely & Weller, 2011).

Researches which focused specifically on foreign exchange markets reach conclusions which are in favor of profitability of technical analysis however some exceptions in later studies still exist. Poole (1967) applied filter rules on 9 exchange rates for period 1919-1929 and 1950-1962. His findings were significantly supportive towards technical analysis. 7 out of 9 exchange rates resulted in average annual gross returns of more than 25% and these filter rules beat the buy & hold strategy by large difference. Sweeney (1986) focused on a shorter period of 1975-1980 on 10 different exchange rates. His results show that smaller filter rules (0.5% to 5%) beat simple buy & hold strategy, even when transaction costs are accounted for. Similar researches by Levich and Thomas (1991), Gencay (1999) and Martin (2001) found evidence that technical trading rules can generate persistent profits. On the other hand, Dooley & Shafer (1984) examined 9 foreign currencies in the New York market also using filter rules for period of 1973-1981. While results were different for each currency, small filter rules generally produced positive returns while larger ones resulted in a loss.

Most recent research using Bollinger Bands as an indicator was carried on by Nikola Gradojevic and Camillo Lento in their article "Investment information content in Bollinger Bands?" (2007). This specific indicator involves two bands plotted around a moving average line and indicates a buying signal when current price touches lower band and selling signal when current prices touches upper band therefore implying that prices have a tendency to reverse towards the mean. In this research, Bollinger Bands were used for TSX, DJIA, NASDAQ and CDN/US$ currency pair to measure results for different markets and focused on the period of 1995-2004. Also, different values for Bollinger Bands were considered (moving average for 20 and 30 days; standard deviation for values 1 and 2). Conclusions indicated that Bollinger Bands were not profitable and did not outperform relative to buy & hold strategy as the indicator turned out to be profitable for only 2 of 12 tests. Opposite results were observed when contrarian approach was utilized. It turned out that even though rationale behind the Bollinger Bands is sound, profitability was greatly improved when opposite positions were being entered

7 meaning buying a stock when an indicator is suggesting to sell it and vice versa (Lento, Gradojevic & Wright, 2007).

Weller and Neely (2011) summarize literature on technical analysis stating that it has established that simple technical trading rules on dollar exchange rates provided 15 years of positive, risk-adjusted returns during the 1970s and 80s and these were extinguished during later years.

This conclusion was puzzling as it was challenging weak-form efficient market hypothesis which states that past prices should not produce positive risk-adjusted returns. Weller and Neely (2011) put out several possible explanations for this apparent success of technical analysis: data snooping, publication bias and data mining1. All of these related but distinct problems tend to result in positive bias on conclusion of technical trading rules.

3. Theoretical Framework

3.1 Efficient and Adaptive Market Hypothesis

The notion that stocks' prices already reflect all available information is referred to as the efficient market hypothesis (EMH). It highlights that any information that could be used to predict security performance should already be reflected in its price since if something could be predicted then the prediction would be part of today's information. Thus, changes in security prices in response to previously unpredicted information also must move unpredictably. This is the essence of the argument behind EMH - prices should follow a random walk meaning that they should be random and unpredictable.

An alternative to EMH was put forward by Andrew Lo (2004) who introduced a concept of adaptive market hypothesis (AMH). It is based on assumption that markets gradually arbitrage away patterns as they become known and only more complex ones survive. The AMH modifies the efficient markets view of the world to assert that learning, competition and evolutionary selection pressures are forces that drive prices toward their efficient levels. Investors are no longer 'hyper-rational' beings of the standard paradigm but rather rational "satisficers" - terminology used by Herbet Simon (1955). The AMH predicts that profit opportunities will exist in the financial markets but it is learning and competitions that erode these opportunities as they become known. While some strategies decline as they become less profitable, learning will result in more complex strategies which adjust to the changing market environment. However, it is still unclear to what extent can learning eliminate cognitive biases as behavioral finance argues that such biases are persistent rather than disappearing.

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3.2 Behavioral Finance

In contrast to conventional theories such as EMH which assume that investors are rational, a relatively new field of study, behavioral finance, starts with the assumption that they are not (Bodie, Kane & Marcus, 2014). It argues that the conventional financial theory ignores how real people make decisions and provide proof for irrationalities that characterize individuals' decision making. These irrationalities fall into two categories: first, investors do not absorb information correctly and therefore their probability distributions about future rates of return are incorrect; and second that even given a correct probability distribution of returns, investors make inconsistent or suboptimal decisions.

Weller and Dunham (2009) show that confirmation bias alone can generate price momentum and reversal. It refers to a phenomenon that is characterized by the search for evidence in ways that favor existing beliefs or expectations. It has been described as "perhaps the best known and most widely accepted notion of inferential error to come out of the literature on human reasoning" (Evans (1989) quoted in Nickerson (1998 p. 41)).

Even though behavioral explanations of efficient market anomalies do not give guidance as to how to exploit such irrationalities, an important implication is that if prices are distorted, capital markets will give misleading signals and incentives as to where an investor's money should be allocated. Investors who are aware of the potential biases in information processing and decision making that characterize other investors should have an edge and be better able to avoid such errors. Therefore, behavioral critique of the efficient market hypothesis is significant, regardless of any implication for investment strategies.

3.3 Technical analysis

The technical analysis attempts to exploit recurring and predictable patterns in security prices to generate abnormal profits. One of the best-known behavioral tendencies is the disposition effect, which refers to the investors behavior of holding on to losing investments (Bodie, Kane & Marcus, 2014). Such investors seem reluctant to realize losses. This effect can lead to momentum in stock prices even if prices indeed follow a random walk (Grinblatt & Haan, 2005). Behavioral biases are also consistent with technical analysts using volume data. It refers to overconfidence bias, a systematic tendency to overestimates one's abilities. As traders become increasingly confident, they trade more therefore creating a pattern between trading volume and market returns. Daniel, Hirshleifer and Subrahmanyam (1998) developed a model in which investors are overconfident and it manifests itself as overestimation of the precision of private information. They argue that individuals are used to thinking that favorable

9 outcomes are due to skill and unfavorable ones are due to bad luck and these biases result in investors placing excessive weight on their private information. This generates a price overreaction that is not instantly reversed as new public information is revealed. In this way, price momentum and reversal are generated. Another example of behavioral importance in technical analysis is significance of prices crossing round levels also known as barriers. Creswell (1995) reported that "The 100 yen level for the dollar is still a very big psychological barrier and it will take a few tests before it breaks. But once you break 100 yen, it's not going to remain there for long. You'll probably see it trade between 102 and 106 for a while. Also, Osler (2003) shows that when an exchange rate approaches a round number, such as 100 yen to the dollar, it tends to reverse its path. But when it crosses such a level, it tends to move rapidly past it. Such observations highlight importance of psychology when making trading decision therefore contradicting rational investors assumption by EMH and providing reason for existence of technical analysis.

Two broad categories of technical trading rules involve momentum and mean-reversal strategies (Serban, 2010). Momentum strategies are also referred to "trend following" and evolve around anticipating a future trend and buying a stock when an uptrend is expected and selling a stock when a downtrend is expected. On the other hand, mean reversion refers to a security being too far from what is considered its fair value or its mean price. These strategies involve betting against the trend anticipating that security's price will reverse back towards its particular number of periods mean value. The Bollinger Bands indicator used in this paper is an example of a mean-reversion indicator.

3.4 The Bollinger Bands

John Bollinger, a long time technician of the markets, introduced Bollinger Bands technical indicator in the 1980s (Bollinger, 1992). In his paper "Using Bollinger Bands" John Bollinger highlights that this indicator does not give absolute buy and sell signals. Instead, it provides information on whether prices are high or low on a relative basis. Using this, investor can make buy or sell decision by using indicator to confirm price action. Any indicator involving trading bands consists of lines plotted in and around the price structure to form a band. And it is the prices that are near the outer lines of the band that traders are particularly interested in. The idea of trading bands became popular in mid 1970s and refers to a concept of moving average up and down by a certain number of points or fixed percentage to obtain a channel. In order to create such a chart it is required to calculate and plot the desired average and then add upper band by adding any fixed amount or percentage to plot the upper band and subtract any fixed amount or percentage to form a lower band. For the DJIA, the two most popular averages are 20

10 and 21-day averages while most popular percentage shifts are between 3.5 and 4 (Bollinger, 1992). John Bollinger focused on volatility as the key variable and after testing multiple volatility measures settled with standard deviation as the method by which to set band width as opposed to fixed subjective number. Reasoning behind this is that standard deviation is sensitive to extreme deviations and as a result Bollinger Bands react extremely quick to large moves in the market. Bollinger Bands are plotted two standard deviations above and below a simple moving average and data used to calculate the standard deviation is the same as those for the simple moving average.

𝜎 = (𝑋𝑖− 𝑋 ) 2 𝑁 𝑖=1 𝑁 𝑋 = 𝑋𝑖 𝑁 𝑖=1 𝑁

𝜎 denotes standard deviation of a period while 𝑋 is a formula for the moving average of the same period. Therefore, middle band represents the n-day mean. The upper band is the n-day mean plus 2 standard deviations whereas the lower band is the n-day mean minus 2 standard deviations where n is chosen such that it describes the intermediate-term trend (Bollinger, 1992).

The intuitive idea behind Bollinger Bands when trading is straightforward. When the current price touches the upper Bollinger Bands the prices are considered to be overbought and should be expected to reverse towards its mean, therefore indicating a selling signal. Contrary, if the lower band is touched, prices are thought to be oversold and also likely to reverse to its mean indicating a buying signal.

4. Research Methodology

4.1 Back-testing the Bollinger Band indicator

This paper attempts to check whether Bollinger Bands indicator generated positive risk-adjusted returns in the foreign exchange market. For this purpose 4 most commonly currency pairs will be used: USD/EUR; USD/GBP; USD/CHF; USD/JPY. For each pair, performance during the period of 1995 - 2015 will be presented. Such period contains significant amount of daily data therefore resulting in more reliable results.

For Bollinger Bands, moving average value and standard deviation values have to be set. In his paper "Using Bollinger Bands" John Bollinger highlights the use of 20 periods for moving average and 2 standard deviations. This research paper focuses on daily historic prices and therefore price fluctuations

11 within the same day are unknown. As a result, John Bollinger suggests using the weighted close price as an approximation for average price which is calculated as follows:

𝑊𝑒𝑖𝑔𝑕𝑡𝑒𝑑𝑃 =𝑕𝑖𝑔𝑕 + 𝑙𝑜𝑤 + 𝑐𝑙𝑜𝑠𝑒 + 𝑐𝑙𝑜𝑠𝑒

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In this formula, high represents the maximum price that currency has reached during a day, low represents the minimum price that has been reached during the same day and close is the closing price of that day.

These values will be set as default parameters for Bollinger Band indicator. Therefore, overall trading strategy when to enter the trade can be constructed as follows:

𝑠 = 1, 𝐵𝑜𝑙𝑙𝑖𝑛𝑔𝑒𝑟 2,20 , 𝐿𝑜𝑤𝑒𝑟 > 𝑊𝑒𝑖𝑔𝑕𝑡𝑒𝑑𝑃

−1, 𝐵𝑜𝑙𝑙𝑖𝑛𝑔𝑒𝑟 2,20 , 𝑈𝑝𝑝𝑒𝑟 < 𝑊𝑒𝑖𝑔𝑕𝑡𝑒𝑑𝑃

This algorithm can be interpreted as buying a currency pair (1) when current weighted price is below the lower boundary of the Bollinger Bands as selling a currency pair (-1) when current weighted price is above the upper boundary. Values of moving average and standard deviation will be kept constant at 20 and 2 respectively as proposed by John Bollinger. Even though it is possible to optimize these parameters by changing their values, this will be subject to data mining problem and will not be done in this paper.

To measure returns accurately, stop loss and take profit values will be assigned. These values will make position automatically exit the trade when one of them is reached. Since relatively short time periods are less likely to be affected by fundamental shocks, position should not be held for a long time. An average basis point movement for a daily trading session is measured to be around 100 basis points where 1 basis point represents 0.0001 of a price change, therefore 100 basis points corresponding to 0.01. For instance, if a currency is bought at a price of 1.0000, it will either exit the trade at 1.0100 when profit target is reached or 0.9900 when stop loss target is reached. Research will also include values of stop loss and take profit for 300 and 500 basis point values. In this way, magnitude of returns of each trade is kept constant and percentage of winning and losing trades is directly related to returns. If mean reversion indeed holds one would expect number of winning trades to be higher and therefore Bollinger Bands to be profitable over any period of time.

It is crucial to account for investors bearing the risk when calculating these returns. If this is not done, reported excess returns might not be an indication of market inefficiency but simply appropriate

12 compensation for the risk incurred. Neely and Weller (2011) identify Sharpe ratio as among the most-practically useful risk-adjustment tools as they permit direct comparison of risk among trading rules. Sharpe ratio measures excess returns per unit of risk and is calculated as follows:

𝑆𝑕𝑎𝑟𝑝𝑒 𝑟𝑎𝑡𝑖𝑜 =𝑟𝑝− 𝑟𝑓 𝜎𝑝

where 𝑟𝑝 measures return of a portfolio, in this case a currency pair; 𝑟𝑓is a risk-free rate and 𝜎𝑝 is the

standard deviation of portfolio. The risk-free rate in this equation is considered to be the opportunity cost of investing into a security as money invested could be stored and earn a risk-free interest. However, this research focuses on currency markets and trading these does not involve opportunity cost as currencies traded can indeed earn risk-free. Therefore, risk-free rate is assumed to be 0% throughout this paper.

4.2 The walk-forward optimization of the Bollinger Band indicator

Another way to measure performance of this trading indicator is to apply an out of sample test for best fitting parameters of the trading rule and then apply it in the sample test. This procedure is defined as walk-forward optimization and can replicate the investor's decisions in real-time more realistically rather than just back-testing the rule with the constant values of the parameters over the historic data.

In order to do this, test will optimize parameters of the Bollinger Band over the period of 2 years so that maximum cumulative profit is reached. The Bollinger Band indicator with these parameter values will then be tested on following period of 1 year so that out of sample test is longer than in the sample.

The parameters and their respective values to be optimized for each 2 year periods are presented below.

Table 1. Parameters and their respective values to optimize

Parameter Minimum value Maximum value Minimum increment

Moving Average 10 50 10

Standard deviation 2 3 1

Take profit/Stop loss 100 500 100

The moving average is optimized within the range most commonly used in other technical indicators. It controls the number of trades as very low value would result in excess amount of trades while a large number would significantly decrease likelihood of entering the trade. Standard deviation used in the other papers is 2 and 3 for the same reason. Standard deviation value of 1 would make the

13 trading strategy to enter trade too frequently whereas parameter of 4 or 5 would do the opposite. Finally, boundaries for take profit and stop loss are set so currency pair stays in the market for at least a day and does not stay too long for it not to be affected by fundamental shocks.

Therefore strategy works as follows: years 1995 and 1996 are optimized for parameters with which maximum cumulative profit would be achieved during this period. This strategy is ran during 1997 and profitability of this rule is recorded. After this, 1996 and 1997 are optimized for best fitting values and applied for year 1998 and continues. Therefore, trading starts at 1997 and finishes at the end of 2014.

5. Results

5.1 Results of the back-test

Profitability of the back-test of each currency pair for the selected take profit and stop loss levels are presented in Table 2.

Table 2. Profitability of the Bollinger Band trading rule back-test Take profit/stop loss Currency pair Cumulative Profit Average Annual Profit Standard deviation Sharpe ratio Number of trades Avg. (days) 100 basis points EURUSD -49.22% -2.46% 5.03% -0.49 433 3.52 GBPUSD -34.86% -1.74% 2.83% -0.62 456 2.14 USDCHF -23.12% -1.16% 4.92% -0.23 409 2.9 USDJPY -30.88% -1.54% 4.86% -0.32 457 3.56 300 basis points EURUSD -66.03% -3.30% 18.41% -0.18 433 17.2 GBPUSD -30.18% -1.51% 11.03% -0.14 456 19.41 USDCHF -35.45% -1.77% 19.57% -0.09 409 20.01 USDJPY -31.93% -1.60% 20.94% -0.08 457 22.42 500 basis points EURUSD -88.59% -4.43% 27.14% -0.16 433 45.16 GBPUSD 72.70% 3.64% 23.96% 0.15 456 38.73 USDCHF -42.07% -2.10% 29.24% -0.07 409 39.56 USDJPY -90.21% -4.51% 41.33% -0.11 457 42.07

Table 2 presents profitability of the back-test of each currency pair for selected take profit and stop loss levels. It can be observed that no Bollinger Bands variant was possible to be consistently

14 profitable and provided positive results only in GBP/USD market with take profit and stop loss profits of 500 basis points. Sharpe ratios range between -0.16 and 0.03 therefore indicating limited amount of risk incurred if trading with stop loss and take profit limits. The number of trades for each currency pair is significant and ranges between 409 to 457 therefore resulting in an average of 22 trades per year. As limits of exiting the trade become wider, average holding period of a currency pair increases which is illustrated by Avg. (days) column.

Table 3 presents annual performance of each currency pair for every stop loss and take profit limits.

Table 3 again represents inability of the Bollinger Bands to consistently deliver positive results as vast majority of the results have negative returns. However, years 1998, 2003, 2005, 2007, 2009 have performed better than others. It is also visible that the higher stop loss and take profit limits are, more positive returns foreign exchange market deliver. This might indicate that prices require a longer time period, more than a month, to reverse back towards mean. Appendices B, C, D, E present cumulative profit graphs for different limit returns of 100, 300 and 500 respectively which to some extent confirm that better results were achieved for higher stop loss and take profit limits. However, no clear tendency can be observed from these results.

Table 3. Annual profitability of the Bollinger Band trading rule

100 basis points 300 basis points 500 basis points

Period EURUSDGBPUSDUSDCHF USDJPY EURUSDGBPUSDUSDCHFUSDJPY EURUSDGBPUSDUSDCHF USDJPY

1/1/1995 -9.24% -2.05% -3.45% -9.77% -0.73% 6.97% -1.50% -25.01% 13.21% 27.82% 8.90% -56.22% 1/1/1996 -9.46% -1.45% -3.13% -1.46% -12.45% -11.16% -29.33% -0.59% -0.96% -9.23% -22.57% -3.30% 1/1/1997 -6.48% -1.32% 7.02% -8.26% -28.46% -9.21% -7.88% -26.75% -42.08% 4.60% -12.39% -67.90% 1/1/1998 3.05% 0.53% 4.26% -2.00% 23.34% 14.71% -0.96% 18.62% 16.86% 32.77% -3.33% 42.90% 1/1/1999 -9.04% 1.78% -3.90% -4.00% -24.45% 3.34% -14.19% -15.94% -40.88% 19.56% -18.11% -30.56% 1/1/2000 -10.81% -5.98% -4.18% -6.31% -1.41% -31.01% 3.50% 4.78% 4.94% -45.00% 4.57% 9.17% 1/1/2001 0.83% -3.48% 2.99% -3.34% 18.96% -3.16% 1.75% -21.40% 5.59% 7.22% 12.60% -19.01% 1/1/2002 -5.07% -1.49% -4.52% -4.07% -38.34% 6.35% -10.41% -12.56% -53.74% -5.16% -18.71% -7.95% 1/1/2003 0.78% -8.28% -6.37% 1.56% 1.96% 6.88% 6.09% 15.76% -24.39% -7.36% 21.58% 32.72% 1/1/2004 0.14% -3.30% -2.63% -0.41% -11.56% -1.76% 1.07% 16.64% -5.64% 2.74% -4.90% 38.50% 1/1/2005 -1.15% -6.49% -0.79% 0.42% 27.53% 6.28% 18.99% 9.93% 64.43% 51.15% 54.85% 31.97% 1/1/2006 0.18% 0.55% -3.38% -1.90% -8.19% -0.68% -12.06% 8.31% -10.47% 7.98% -5.78% -11.30% 1/1/2007 -6.25% -0.96% 3.88% -3.34% 19.99% 6.27% 16.10% 13.69% 17.92% 33.33% 49.35% 40.01% 1/1/2008 -7.56% -5.30% -9.84% 2.14% -8.13% -13.17% -6.71% -0.02% 5.15% -17.54% -28.17% -40.93% 1/1/2009 -1.12% 1.86% -1.39% 7.34% 2.02% 18.25% 7.14% 55.16% 7.49% 43.52% 27.34% 91.19% 1/1/2010 -0.03% -0.81% 0.93% -5.97% -10.80% -6.04% -19.68% 6.82% -36.85% -18.17% -17.05% -23.85% 1/1/2011 3.84% 0.63% 10.16% 10.51% -0.74% -2.48% -19.64% 18.10% 5.05% 0.90% -48.95% 38.22% 1/1/2012 -6.93% -1.32% -5.21% -5.37% -10.38% 1.05% -0.91% -29.45% -9.49% 4.96% -6.59% -31.97% 1/1/2013 6.12% -4.48% 1.22% -1.29% 20.98% -0.33% 65.80% -14.93% 2.12% 9.53% 62.95% -45.29% 1/1/2014 -5.96% -0.27% -5.50% 1.16% -24.81% -12.58% -4.74% -19.14% -32.13% -25.74% -22.01% -33.08%

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5.2 Results of the walk-forward optimization

The results of the Bollinger Band walk-forward process are summarized in Table 4.

Table 4. Profitability of the Bollinger Band walk-forward optimization

Currency pair Cumulative Profit Average Annual Profit Standard deviation Sharpe ratio Number of trades EURUSD -97.20% -4.86% 25.73% -0.19 360 GBPUSD 530.71% 26.54% 35.33% 0.75 447 USDCHF -85.43% -4.27% 16.37% -0.26 376 USDJPY 97.71% 4.89% 27.52% 0.18 320

It can be observed that 2 out of 4 currency pairs produced positive returns. Especially, GBPUSD performed significantly well with over 530% cumulative profit of the 20 year period time averaging 26.54% annual returns. However, standard deviation of these returns is relatively high resulting in Sharpe ratio of 0.75. It can be observed that standard deviations of all returns are high indicating that this indicator produces volatile returns and might not be attractive to risk-averse investors who are in interest of minimizing their risk. Number of trades did not change significantly and average around 375 trades during whole period or 19 trades per year.

Table 5 represents the performance of each currency pair over each year along with the parameters used.

Table 5. Annual profitability of the walk-forward optimization

EURUSD GBPUSD

Year Parameters Cumulative Profit Parameters Cumulative Profit 1997 50/2/300 -29.28% 10/2/500 -3.09% 1998 50/2/400 0.13% 50/2/500 19.14% 1999 20/2/400 -29.52% 40/2/500 45.71% 2000 10/2/500 4.25% 50/2/500 -42.48% 2001 50/2/300 0.30% 50/2/400 37.47% 2002 50/2/500 -75.71% 10/2/300 6.29% 2003 10/2/500 -4.13% 10/2/500 -2.87% 2004 50/3/400 -3.12% 10/2/300 -3.34% 2005 10/2/400 9.27% 30/2/500 62.64% 2006 20/2/500 0.38% 30/2/500 -9.41% 2007 50/2/400 23.59% 20/2/500 22.23% 2008 30/2/500 -49.67% 40/2/500 9.94% 2009 20/2/500 1.84% 40/2/500 27.07% 2010 20/2/500 -32.26% 40/2/500 4.25% 2011 40/3/400 -5.77% 50/2/400 17.66% 2012 20/2/200 -8.05% 50/2/500 116.32% 2013 40/2/500 13.12% 50/2/500 -10.13% 2014 30/2/500 -48.37% 50/2/500 -20.81%

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Table 5 (continued)

USDCHF USDJPY

Year Parameters Cumulative

Profit Parameters Cumulative Profit 1997 10/2/500 -5.97% 50/3/100 0.89% 1998 50/2/500 -2.19% 10/2/300 -0.43% 1999 40/2/500 -2.61% 50/3/500 0.00% 2000 30/2/500 8.52% 50/3/500 0.00% 2001 30/2/500 -3.98% 20/3/500 0.00% 2002 50/2/500 -35.24% 10/2/500 3.62% 2003 50/3/500 0.00% 50/2/500 -37.00% 2004 50/2/300 -13.31% 40/2/500 48.27% 2005 40/2/500 -23.19% 20/2/500 36.37% 2006 20/2/500 9.57% 30/2/500 22.09% 2007 50/2/500 -17.91% 20/2/500 27.86% 2008 20/2/500 -32.41% 50/2/500 59.09% 2009 20/2/400 6.54% 50/2/300 35.92% 2010 20/3/500 0.00% 20/2/300 4.68% 2011 30/2/100 6.90% 50/2/300 13.54% 2012 20/2/100 -5.64% 20/2/300 -37.74% 2013 10/2/300 3.83% 50/2/500 -31.07% 2014 40/2/500 -46.66% 50/3/500 -13.38%

By looking at the yearly performance of the Bollinger Band indicator few important observations can be made. While most profitable moving average values range from 10 to 50 without any clearly observable pattern, standard deviation of 2 seems to be most profitable during optimization periods just as proposed by John Bollinger. Most importantly, it can be observed that strategies with higher stop loss and take profit limits were most profitable during the optimization period. This indicates that researched currency pairs might require more time to reverse back towards the mean. Years 2005, 2006, 2007 and 2009 were most profitable amongst all currencies and are in correlation with the back-test results.

Appendix F presents parameter values and their respective returns during 2 year optimization periods. It can be noticed that even though 74 out of 76 optimization periods resulted in positive returns, this does not continue during the test period as observed in the Table 4 where 35/72 tests resulted in positive results. This is due to data mining problem as parameters can be optimized differently and might return positive returns purely because of chance which does not hold during longer period.

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6. Conclusions

The results presented in the previous section show that the Bollinger Band technical trading rule could not consistently deliver positive risk-adjusted returns in the foreign exchange market represented by four currency pairs during the period of 1995-2015.

When comparing back-test and walk-forward optimization results, several similarities and differences can be observed. First of all, walk-forward optimization performed better as 2 out of 4 currency pairs delivered positive returns with GBP/USD pair averaging 26.54% annual return. This might indicate that optimizing time period and then running an indicator on different time periods is better rather than just performing a test with set parameter values over a long period of time. On the other hand, it can be observed that GBP/USD pair performed particularly well as it was the only observation in the back-test which delivered positive returns in one of the tests. However, there is no possible explanation for this pair to be a better performer than others and no conclusion about that can be drawn. Moreover, wider take profit and stop loss limits performed better in both of the tests indicating that mean reversion is more likely the longer position is held and is not profitable in the short-term trading. Even though profitable years for both tests can be observed and they are correlated, no clear reason can be found for why these particular years would perform better than others. Most importantly, even though optimization periods resulted in positive returns in outstanding 74 out of 76 runs, these results should not be surprising or interpreted as significant. These findings are subject to data mining problem as observing a profitable strategy over historic data is possible in any scenario but it is whether it holds in the future is what technicians are interested in. This particular paper shows that despite its appealing numbers, less than half of the tests delivered positive results on the following out of sample year. While the indicator did not deliver constant positive risk-adjusted returns, issue might be behind this particular indicator and not efficiency of the market.

There are several limitations to this research which might have an impact on the overall results. First of all, the sample of the most commonly traded currency pairs might not be good representation of overall foreign exchange market as all sample representatives involve U.S. dollar which creates correlation between the currencies used. In addition, walk-forward optimization had arbitrary values of 2 years optimization and 1 year test period. These values can be changed and optimized as well indicating that on might obtain more significant results by changing parameters of the strategy more frequently. What is more, performance measurement of exiting trades with stop losses and take profits functions is questionable. Even though this measure helps to maintain the magnitude of successful and unsuccessful trade constant and makes it easy to interpret results, a portion of positions might be exited

18 before or after reversing towards mean. This causes inconsistent performance measurement for each trade and impacts the end results. Furthermore, constant values of these limits might not be a valid assumption. For example, taking lower profit target than stop loss would be reasonable if probability to reverse is actually increasing as price of the currency is moving further away from the mean. The tests have also shown that currencies deliver better results for longer time periods. 500 basis point stop loss and take profit limits might not be high enough and even though higher limits would result in trades staying in the market for a relative long time this might make returns less volatile. Finally, while the period 1995-2015 is significantly long it is also relatively recent. This might support adaptive market hypothesis theory as such an old and well-known indicator should not be expected to perform well in the recent times. According to AMH, it is more advanced and complex trading rules that are yet to be shared by profitable investors which generate positive results while old techniques become obsolete as no investor would take an opposite side of a trade of a profitable strategy.

Future researches on this topic should focus on several possible improvements. If only efficiency of foreign exchange market is being considered one should include several other pairs which do not correlate amongst themselves. Also, wider variety of exit strategies should be used. If stop loss and take profit limits are used, wider ranges of these values should be tested as well as how these value vary when compared to each other. Most importantly, further researches should use a pool of major technical trading rules to check efficiency of this market. While this induces data mining and scooping biases, results have higher significance compared to using only one indicator with possibly flawed theory. Finally, even if foreign exchange markets are efficient and no positive returns can be achieved by using technical trading rules, researches should include different markets in order to reach a clear conclusion with regards to what extent efficient market hypothesis holds. A possible way would be to analyze portfolios of hedge funds which use technical analysis widely and include wide range of securities in their portfolios.

Researches regarding efficient market hypothesis and effectiveness of technical analysis have not reached a conclusive answer of the use of technical trading rules and it is clear why. Measuring excess returns is open to biases and subjectivity and can be easily interpreted as data mining at any point as one will always be able to find a pattern within the market which returned abnormal profits. However, it is to what extent these abnormal profits are explained by the behavioral analysis or the use of statistics is yet to answer.

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Cheung, Y.-W., & Chinn, M. D. (2001). Currency trades and exchange rate dynamics: a survey of the US market. Journal of International Money and Finance , 439-471.

Creswell, J. (1995). Currency market expects rate cut by Bank of Japan. Wall Street Journal .

Daniel, K., Hirshleifer, D., & Subrahmanyam, A. (1998). Investor Psychology and Security Market Under- and Overreactiions. The Journal of Finance , 1839-1880.

De Bondt, W. F., & Thaler, R. H. (1987). Further Evidence On Investor Overreaction and Stock Market Seasonality. The Journal of Finance , 557-581.

Dooley, M. P., & Shafer, J. R. (1984). Analysis of Short-Run exchange Rate Behavior: March 1973 to November 1981. In D. Bigman, & T. Taya, Floating Exchange Rates and the State of World Trade Payments. Cambridge: Ballinger Publishing Company.

Gencay, R. (1999). Linear, non-linear and essential foreign exchange rate prediction with simple technical trading rules. Journal of International Economics , 91-107.

Levich, R. M., & Thomas, L. R. (1991). The Significance of Technical Trading-Rule Profits in the Foreign Exchange Market: A Bootstrap Approach. Journal of International Money and Finance , 451-474. Martin, A. D. (2001). Technical trading rules in the spot foreign exchange markets of developing countries. Journal of Multinational Financial Management , 59-68.

Neely, C. J., & Weller, P. A. (2011). Technical Analysis in the Foreign Exhange Market. Federal Reserve Bank of St. Louis, Working Paper Series .

Osler, C. L. (2003). Currency Orders and Exchange Rate Dynamics: An Explanation for the Predictive Success of Technical Analysis. The Journal of Finance , 1791-1819.

Park, C.-H., & Irwin, S. H. (2007). hat do we know about the profitability of technical analysis? Journal of Economic Surveys , 786-826.

Poole, W. (1967). Speculative Prices as Random Walks: An Analysis of Ten Time Series of Flexible Exchange Rates. Southern Economic Journal , 468-478.

Serban, A. F. (2010). Combining mean reversion and momentum trading strategies in foreign exchange markets. Journal of Banking & Finance , 2720-2727.

20 Sweeney, R. J. (1986). Beating the Foreign Exchange Market. The Journal of Finance , 163-182.

Taylor, M. P., & Allen, H. (1992). The use of technical analysis in the foreign exchange market. Journal of International Money and Finance , 304-314.

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Appendix A: Data snooping, data mining and publication bias resulting in

positive biases in the performance results

Data snooping refers to researchers choosing technical trading rules which have been already proven profitable on previous data. It is impossible to avoid any data snooping as amount of data sets and technical trading rules is limited however it is argued that researches unconsciously replicate previous researches to arrive to similar results. Publication bias refers to economic journals being more willing to accept submission with positive rather than negative results. This therefore gives incentive for researches to only submit articles with significant results and disregard the rest. Finally, data mining is the problem combining both publication bias and data snooping. It refers to researches testing multiple trading rules and focusing their overall results on the most successful ones. In this way, negative results are ignored while positive are submitted as an indication of profitability of technical analysis.

Technicians and researches regarding performance of technical analysis contain a substantial amount of subjective element and avoiding these biases is difficult (Neely & Weller, 2011). While focusing on number of technical trading rules as performance measure would decrease data mining problem, this would increase data snooping bias as these selected rules that might be profitable during some time period purely by chance. "Perhaps the most certain solution to data snooping, data mining and publication bias is to analyze the performance of rules in true out-of-sample tests that occur long after an important study. That is, once can test a group of rules that were examined in studies conducted sufficiently long time ago that one has enough new data to carry out a true out-of-sample test. Of course, this technique has a substantial cost: One must wait years to employ it on past studies" (Christopher J. Neely; Paul A. Weller (2011)).

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Appendix F: USDCHF cumulative profit graphs

USDCHF USDJPY

Parameters From To Cumulative

Profit Parameters From To

Cumulative Profit 10/2/500 1/1/1995 12/30/1996 45.37% 50/3/100 1/1/1995 12/30/1996 -0.17% 50/2/500 1/1/1996 12/30/1997 76.56% 10/2/300 1/1/1996 12/30/1997 5.50% 40/2/500 12/31/1996 12/30/1998 108.49% 50/3/500 12/31/1996 12/30/1998 13.52% 30/2/500 12/31/1997 12/30/1999 22.39% 50/3/500 12/31/1997 12/30/1999 8.70% 30/2/500 12/31/1998 12/29/2000 24.85% 20/3/500 12/31/1998 12/29/2000 4.88% 50/2/500 12/31/1999 12/29/2001 35.45% 10/2/500 12/31/1999 12/29/2001 12.91% 50/3/500 12/30/2000 12/29/2002 2.93% 50/2/500 12/30/2000 12/29/2002 17.93% 50/2/300 12/30/2001 12/29/2003 13.64% 40/2/500 12/30/2001 12/29/2003 47.62% 40/2/500 12/30/2002 12/28/2004 119.77% 20/2/500 12/30/2002 12/28/2004 88.18% 20/2/500 12/30/2003 12/28/2005 51.25% 30/2/500 12/30/2003 12/28/2005 109.59% 50/2/500 12/29/2004 12/28/2006 71.85% 20/2/500 12/29/2004 12/28/2006 25.71% 20/2/500 12/29/2005 12/28/2007 62.44% 50/2/500 12/29/2005 12/28/2007 130.10% 20/2/400 12/29/2006 12/27/2008 4.73% 50/2/300 12/29/2006 12/27/2008 24.95% 20/3/500 12/29/2007 12/27/2009 0.00% 20/2/300 12/29/2007 12/27/2009 64.05% 30/2/100 12/28/2008 12/27/2010 3.80% 50/2/300 12/28/2008 12/27/2010 71.68% 20/2/100 12/28/2009 12/27/2011 10.10% 20/2/300 12/28/2009 12/27/2011 27.42% 10/2/300 12/28/2010 12/26/2012 8.90% 50/2/500 12/28/2010 12/26/2012 58.28% 40/2/500 12/28/2011 12/26/2013 142.99% 50/3/200 12/28/2011 12/26/2013 -0.55% 20/2/300 12/27/2012 12/26/2014 38.24% 20/2/200 12/27/2012 12/26/2014 5.73%

27

EURUSD GBPUSD

Parameters From To Cumulative

Profit Parameters From To

Cumulative Profit 50/2/300 1/1/1995 12/30/1996 58.70% 10/2/500 1/1/1995 12/30/1996 16.09% 50/2/400 1/1/1996 12/30/1997 12.35% 50/2/500 1/1/1996 12/30/1997 32.51% 20/2/400 12/31/1996 12/30/1998 32.45% 40/2/500 12/31/1996 12/30/1998 174.39% 10/2/500 12/31/1997 12/30/1999 30.69% 50/2/500 12/31/1997 12/30/1999 117.12% 50/2/300 12/31/1998 12/29/2000 34.63% 50/2/400 12/31/1998 12/29/2000 14.74% 50/2/500 12/31/1999 12/29/2001 34.65% 10/2/300 12/31/1999 12/29/2001 3.73% 10/2/500 12/30/2000 12/29/2002 2.24% 10/2/500 12/30/2000 12/29/2002 19.05% 50/3/500 12/30/2001 12/29/2003 0.00% 10/2/300 12/30/2001 12/29/2003 8.26% 10/2/400 12/30/2002 12/28/2004 10.22% 30/2/500 12/30/2002 12/28/2004 9.21% 20/2/500 12/30/2003 12/28/2005 48.51% 30/2/500 12/30/2003 12/28/2005 124.26% 50/2/400 12/29/2004 12/28/2006 48.69% 20/2/500 12/29/2004 12/28/2006 44.71% 30/2/500 12/29/2005 12/28/2007 43.48% 40/2/500 12/29/2005 12/28/2007 94.89% 20/2/500 12/29/2006 12/27/2008 12.23% 40/2/500 12/29/2006 12/27/2008 95.53% 20/2/500 12/29/2007 12/27/2009 3.53% 40/2/500 12/29/2007 12/27/2009 39.70% 40/3/400 12/28/2008 12/27/2010 6.19% 50/2/400 12/28/2008 12/27/2010 29.53% 20/2/200 12/28/2009 12/27/2011 20.63% 50/2/500 12/28/2009 12/27/2011 64.95% 40/2/500 12/28/2010 12/26/2012 27.49% 50/2/500 12/28/2010 12/26/2012 212.09% 30/2/500 12/28/2011 12/26/2013 53.48% 50/2/500 12/28/2011 12/26/2013 102.17% 20/2/200 12/27/2012 12/26/2014 9.54% 10/2/500 12/27/2012 12/26/2014 16.01%