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This paper adds further contribution to the existing literature, by not only testing for the fractal adaptive moving average using a price crossover strategy, but also robust testing its performance over different time intervals, and comparing the results with a naïve buy-and-hold strategy. This paper is the first one to introduce a performance measure which accounts for the number of trades in a strategy due to the fact that different strategy might end up with different number of long and short trades. The aim of this paper is to test the use of an optimized fractal adaptive moving average crossover trading strategy for the Euro/ US dollar currency pair, and whether it outperform the naïve buy-and-hold strategy. The rest of the paper provides some background to the study, lays the framework on the data and research methodology, conducts the analysis, before ending with some conclusive remarks.


Data Assumptions:

All entry and exit prices are on the day of the signal at the close.

The frequency of data is set to minute wise, hourly, daily, weekly and monthly.

GCBF ♦ Vol. 11 ♦ No. 1 ♦ 2016 ♦ ISSN 1941-9589 ONLINE & ISSN 2168-0612 USB Flash Drive 370 Global Conference on Business and Finance Proceedings ♦ Volume 11 ♦ Number 1 The effect of discounts due to bulk trading and taxes are ignored.

All orders occur at market prices. Limit and stop order options are not allowed at this stage.

Annualized return and annualized standard deviation of daily returns would be used as measurements of risk and return. The Sharpe ratio would then be used to compare the performance of the fractal based strategy with the buy-and-hold strategy. Due different strategies yielding different number of long and short trades, and the fact that their Sharpe values are compared for selection purposes, this study proposes the use of a new ratio which will allow for better comparisons among the different trading strategies as


R t −rf

–  –  –

Where, R t = Annualized return rf = Risk free rate σt = Annualized risk γ = Number of long trades δ = Number of short trades The Sharpe to Total Trades ratio adjusts for the number of total trades, and does not discriminate between long and short trades since both would usually have the same trading costs. A higher value of the ratio suggests comparatively a better trading strategy when comparing excess return, risk and number of trades.

In regards to Assumption 3, Wilcox and Crittenden (2005) argue that trend following systems still work well in the equities market after adjusting for taxes. Further, it is expected in this current paper, that any optimized strategy based on moving averages of shorter durations would result in more trading transactions, creating sufficient losses for the investor to benefit from during taxation times. Further, more transactions would result in more discounts from the trading platforms offered by brokers. Assumption 4 is maintained at this stage to prevent any subjectivity from the trader’s beliefs what about are the boundaries of prices that could be entered as limit or stop orders. The time window under analysis is set from 4th January 2000 till the 25th June 2015. FXCM trading station is used to obtain the daily, minute wise, weekly, monthly data.


While the concept of fractured dimensions was emphasized in Mandelbrot (1967) where he initiated the coastline paradox, its applications range from characterizing turbulence (Mandelbrot, 1982), urban growth (Chen, 2011), human physiology (King et al, 2009) medicine (Losa, 2006) and more importantly in our case, market trends (Peters, 1991; Mandelbrot, 2004). Mandelbrot (1967) suggests that the smaller the increment of measurement, the longer the measured length becomes such that if one were to measure a stretch of coastline with a yardstick, one would get a shorter result than if the same stretch were measured with a one-foot ruler. This is due to the fact that one would be laying the ruler along a more curvilinear route than that followed by the yardstick. The empirical evidence suggests a rule which that the measured length increases without limit as the measurement scale decreases towards zero. Essentially, he used a scale G which is raised to the power of 1- , where is the dimension of the coastline, for instance.

Richardson’s law that the measured length of various geographic borders was a function of the measurement Applied in the context of financial market prices, if a stock price series looks smooth, it should have a the fractal dimension is a measure of how completely a fractal appears to fill space as one zooms in, dimension close to 1, and the more volatile the stock prices, the closer the dimension is to 2. In short, GCBF ♦ Vol. 11 ♦ No. 1 ♦ 2016 ♦ ISSN 1941-9589 ONLINE & ISSN 2168-0612 USB Flash Drive 371 Global Conference on Business and Finance Proceedings ♦ Volume 11 ♦ Number 1 where the fractal is a fragmented shape that can be split into parts, which is at least similar to a reduced size of the original. The fractal dimension provides a statistical index of complexity comparing how detail in a pattern changes with the scale at which it is measured. The feature of self-similarity defines a fractal shape and is well existent in financial markets, where one can easily be misguided by the similarity that exists in price movements across different time frames. Similarly, various finance theories such as the stochastic Wiener process which charts Brownian motion - used in the Black Scholes Option Pricing Model, are backed by the concept of fractals.

Using the concept of fractal dimension, Ehlers (2010) developed a method of identifying the D for stocks, simply by averaging the measured fractal dimension over different scales. The fractal dimension is then adopted into the Fractal Adaptive Moving Average (FRAMA) model. Traditionally, the dimension of a shape is uncovered by covering the shape with a number of similar small objects that are various sizes and comparing how many of each fit across the surface. For instance, if a square is broken into 25 similar sub squares then a magnification factor of 5 is needed to reveal the original shape, such that a square can be broken into N2 self-similar copies and to reveal the original shape, a magnification factor of N is needed.

Similarly, a line segment can be broken down into N self-similar parts and a magnification factor of N will reveal the original shape. Alternatively stated, the dimension of an object is the exponential of the number of self-similar pieces with a magnification factor of N which a shape can be fragmented into. Due to the fact that prices on a stock chart are homogeneously spaced, Ehlers (2010) used the average slope of the

price curve as an estimation of the number of self-similar pieces, and is calculated as follows:

–  –  –

is the period under analysis and is the dimension value. The lower the dimension value, the closer the price graph is to a straight line, suggesting strong trend presence. Alternatively, higher values of suggest more complex fractals which are usually witnessed in volatility geared markets. As part of adding value to the traditional exponential moving average (EMA) method, Ehlers (2010) made use of the dynamic D to adjust the alpha of the EMA such that there is a quicker reaction in a trending market and a conversely slower reaction time to more volatile markets where traditional moving average techniques results usually

in lots of whipsawed trades. The FRAMA model is provided as follows:

= −1 + ( − −1 ) (3), where = (−1). The FRAMA model is similar to the EMA model, except for the fact that adapts Ehlers (2010) set the value to be -4.6 under the assumptions that the FRAMA moves between a fast to dynamic, by continuously adapting to changes in the fractal dimensions. It is important to note that moving average of 1 day and a slow moving average of 198 days. The value of −1 is set to the closing price initially. The higher the value of , the slower the EMA (higher N value) and the lower the , the faster the EMA (lower N value). The lower the , the closer the stock chart is to a straight line and the stronger the trend. Higher values tend to support charts operating within channels. Alternatively stated, the smoothing factor, , which is used like in the EMA series is lower when D is higher and vice versa, meaning that a higher D results in a lower weighting factor of the last price in the EMA series, such that N tends to be large and the EMA resulting in a slow EMA.

–  –  –

Research Findings The currency index depicts the evolution of a currency relative to the entire forex where the index of the currencies is a simple arithmetic average of the variation in each pair. While a currency pair, say the EUR/USD, is trending when the value of one of the two currencies increases by reference to the other, the most interesting trend is when the Euro (EUR) and the Dollar (USD) are in opposing trends such that the movement of the EUR/USD is the result of the combined variation of the two currencies. An index value of 100 is set at the start of the period under consideration. For consistency purposes, the Japanese yen, British pound and Swiss Francs are used against the US dollar and Euro in the calculation of both the Dollar and Euro Index. Figure 1 shows the performance of the EUR/USD, the Dollar and Euro indices for the period January 2000- June 2015. While the EUR/USD had a closing value of 1.02469 at the beginning of the period and a value 1.12064 at the end, its value trended upwards to reach a maximum of 1.59918 in April 2008, before generally experiencing lower highs to reach a low value of 1.12064 by the 25th June

2015. Noticeably, the EUR/USD has been falling constantly around May 2014, attributable mostly to a stronger Dollar index, accompanied by a Euro currency dropping in value against other major global currencies. Although not reported here, the correlation coefficient between the EUR/USD and the Dollar Index for the last 15 years, using daily data, was -0.94, compared to a value of 0.88 for the EUR/USD and the Euro Index, showing the dollar had a stronger negative relationship on the currency pair. Those relationships became stronger when looking at the correlation between the three variables for the May 2014June 2015 period. While the Dollar Index had a strong correlation value of -0.97 against the EUR/USD, the Euro Index had a strong positive correlation value of 0.95 against the EUR/USD. The correlation between the Dollar Index and Euro Index changed from -0.70 for the last 15 years to -0.85 over the last year, suggesting the EUR/USD is more affected by both currencies than before.

–  –  –

This figure shows the performance of the Euro/USD, the dollar index and Euro index during 2000-2015.

Due to the coverage of various moving average (MA) techniques in existing literature, and with various studies having mixed findings regarding their superiority among each other, it important to start the analysis by carrying out a simple correlation analysis over different time frames, over different moving averages, with different data ranging from opening, high, low to closing values of the EUR/USD. Gurrib (2015) provides a good overview of the importance of changing the number of periods in the models whereby increasing the number of periods results in the latest data added in the calculation in having a subdued effect on the moving average calculation. This would result in greater smoothing of the moving average data series. A 1 day MA would almost faultlessly reflect the more volatile and everyday currency pair value GCBF ♦ Vol. 11 ♦ No. 1 ♦ 2016 ♦ ISSN 1941-9589 ONLINE & ISSN 2168-0612 USB Flash Drive 373 Global Conference on Business and Finance Proceedings ♦ Volume 11 ♦ Number 1 while a longer period MA, say 100, would be flatter and indicate a long-run and more stable state. While the number of periods is important, the frequency in the data interval is also critical in determining the behavior of the MA dataset, whether the interval is daily, weekly, monthly or quarterly. Gurrib (2015) found that a daily MA tracks the currency pair values more closely, compared to a higher data interval like the monthly or quarterly series, where the quarterly interval MA series is less prone to the daily fluctuations of the prices resulting in a smoother MA series.

For the purpose of this study, the simple moving average (SMA) and exponential moving average (EMA), which are the most commonly covered moving averages in existing literature, are analysed together with the fractal adaptive moving average (FRAMA). The Open, High, Low, and Close (OHLC) values of the EUR/USD are used over the minute wise, daily, weekly, and monthly frequencies. Lag periods of 10, 100 and 200 are selected to provide an overview of the change in Pearson product-moment correlation coefficient values over different frequency intervals over a set time period. Results suggest an EMA model with a lag period of 10 is superior to the SMA and FRAMA for all the OHLC values. As the frequency interval is increased to monthly, correlation values drop. Importantly, correlations between EMA and OHLC are nearing zero values at higher frequency intervals, and correlations between SMA and OHLC turned negative at higher intervals. For the monthly data, only SMA is calculated from 2009 due to time frame under analysis. As part of the selection process, minute wise data is discarded for later analysis due to high correlation values among all variables.

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