Journal of Derivatives & Hedge Funds

, Volume 15, Issue 3, pp 186–205 | Cite as

The robustness of neural networks for modelling and trading the EUR/USD exchange rate at the ECB fixing

  • Christian L Dunis
  • Jason Laws
  • Georgios Sermpinis
Original Article

Abstract

The objective of this study is to investigate the use, the stability and the robustness of alternative novel neural network (NN) architectures when applied to the task of forecasting and trading the Euro/Dollar (EUR/USD) exchange rate using the European Central Bank (ECB) fixing series with only autoregressive terms as inputs. This is achieved by benchmarking the forecasting performance of three different NN designs representing a Higher Order Neural Network (HONN), a Recurrent Neural Network (RNN) and the classic Multilayer Perceptron (MLP) with some traditional techniques, either statistical, such as an autoregressive moving average model, or technical, such as a moving average convergence/divergence model, plus a naïve strategy. More specifically, the trading performance of all models is investigated in a forecast and trading simulation on the EUR/USD ECB fixing time series over the period January 1999 – August 2008 using the last 8 months for out-of-sample testing. Our results in terms of their robustness and stability are compared with a previous study by the authors, who apply the same models and follow the same methodology forecasting the same series, using as out-of-sample the period from July 2006 to December 2007. As it turns out, the HONN and MLP networks present a robust performance and do remarkably well in outperforming all other models in a simple trading simulation exercise in both studies. Moreover, when transaction costs are considered and leverage is applied, the same networks continue to outperform all other NN and traditional statistical models in terms of annualised return – a robust and stable result as it is identical to that obtained by the authors in their previous study, examining a different period for the series.

Keywords

higher order neural networks recurrent neural networks leverage multilayer perceptron networks quantitative trading strategies 

INTRODUCTION

Neural networks (NNs) are an emergent technology with an increasing number of real-world applications, including finance.1 However, their numerous limitations and the contradicting empirical evident around their forecasting power often create scepticism with regard to their use among practitioners.

The objective of this study is to investigate not only the use of several new NNs techniques that try to overcome these limitations, but also the stability and robustness of their performance. This is achieved by benchmarking three different NN architectures representing a Multilayer Perceptron (MLP), a Higher Order Neural Network (HONN) and a Recurrent Neural Network (RNN). Their trading performance on the Euro/Dollar (EUR/USD) time series is investigated, and is compared with some traditional statistical or technical methods, such as an autoregressive moving average (ARMA) model or a moving average convergence/divergence (MACD) model and a naïve strategy. In terms of the stability and robustness of our findings, we compare them with the conclusions of Dunis et al,2 who apply the same models and follow the same methodology to forecast the same series, but use a different out-of-sample period. With regard to the inputs of our NNs, we use the same selection of inputs and lags as those of Dunis et al.2 Similarly, our MACD model is identical to that of of Dunis et al;2 our ARMA model, however, is different, as we need all coefficients to be significant in the new in-sample period.

Moreover, our conclusions can supplement not only those of Dunis et al,2 but also those of Dunis and Chen3 and Dunis and Williams,4 who conduct similar forecasting comparisons between the EUR/USD and the Dollar/Japanese yen (USD/JPY) foreign exchange rates using similar networks but with multivariate series as inputs.

With regard to our data, the EUR/USD daily fixing is published by the European Central Bank (ECB) and is a tradable quantity, as it is possible to leave orders with a bank and trade on that basis.

As it turns out, the MLP and HONN demonstrate a remarkable performance and outperform the other models in a simple trading-simulation exercise. Moreover, when transaction costs are considered and leverage is applied, the MLP and HONN models continue to outperform all other NN and traditional statistical models in terms of annualised return. As these results are identical to those of Dunis et al,2 who follow the same methodology but for a different period of the EUR/USD ECB fixing series, we can argue that the forecasting superiority of the HONN and MLP is stable and robust over time. In terms of the RNNs, their poor performance in this research may be a result of their inability to provide adequate results when only autoregressive terms are used as inputs.

The remainder of the article is organised as follows. In the next section, we present the literature relevant to the RNN and HONN. The section after that describes the data set used for this research and its characteristics. An overview of the different NN models and statistical techniques is provided in the subsequent section. The penultimate section presents the empirical results of all the models considered and investigates the possibility of improving their performance with the application of leverage. In that section we also test the robustness of our models. The final section provides some concluding remarks.

LITERATURE REVIEW

The objective of this study is to apply some of the most promising new NNs architectures that have been developed recently with the purposes of overcoming the numerous limitations of the more classic neural architectures and assessing whether they can achieve a higher performance in a trading simulation using only autoregressive series as inputs.

RNNs have an activation feedback that embodies short-term memory, allowing them to learn extremely complex temporal patterns. Their superiority against feedfoward networks when performing nonlinear time series prediction is well documented in studies by Connor and Atlas5 and Adam et al.6 In financial applications, Kamijo and Tanigawa7 apply them successfully to the recognition of stock patterns of the Tokyo Stock Exchange, while Tenti8 achieve good results using RNNs to forecast the exchange rate of the Deutsche Mark. Tino et al9 use them to successfully trade the volatility of the DAX and FTSE 100 using straddles, while Dunis and Huang,10 using continuous implied volatility data from the currency options market, obtain remarkably good results from their GBP/USD and USD/JPY exchange rate volatility trading simulation.

HONNs were first introduced by Giles and Maxwell11 as a fast learning network with increased learning capabilities. Although their function approximation superiority over the more traditional architectures is well documented in the literature (see, among others, studies by Redding et al,12 Kosmatopoulos et al13 and Psaltis et al14), their use in finance thus far has been limited. This changed when scientists started to investigate not only the benefits of NNs against the more traditional statistical techniques, but also the differences between the different NN model architectures. Practical applications have now verified the theoretical advantages of HONNs by demonstrating their superior forecasting ability, and have put them at the front line of applied research in financial forecasting. For example, Dunis et al15 use them to successfully forecast the gasoline crack spread, while Fulcher et al16 apply HONNs to forecast the AUD/USD exchange rate, achieving 90 per cent accuracy. However, Dunis et al17 show that, in the case of the futures spreads and for the period under review, the MLPs performed better compared with HONNs and RNNs. Moreover, Dunis et al,18 who also studied the EUR/USD series for a period of 10 years, demonstrate that when multivariate series are used as inputs, the HONNs, RNNs and MLP networks have a similar forecasting power. Finally, Dunis et al,2 in a study with a methodology identical to that used in this study, demonstrate that the HONNs and MLP network are superior in forecasting the EUR/USD ECB fixing until the end of 2007, compared to the RNNs, an ARMA model, an MACD model and a naïve strategy.

THE EUR/USD EXCHANGE RATE AND RELATED FINANCIAL DATA

The ECB publishes a daily fixing for selected EUR exchange rates: these reference mid-rates are based on a daily concentration procedure between central banks within and outside the European System of Central Banks, which normally takes place at 14:15 hours ECB time. The reference exchange rates are published both by electronic market information providers and on the ECB's website shortly after the concentration procedure has ended. Although only a reference rate, many financial institutions are ready to trade at the EUR fixing, and it is therefore possible to leave orders with a bank for business to be transacted at this level.

The ECB daily fixing of the EUR/USD is therefore a tradable level, which makes our application more realistic.19 In Table 1 we present how we divide our data set.
Table 1

The EUR/USD data set

Name of period

Trading days

Beginning

End

Total data set

2474

4 January 1999

29 August 2008

In-sample data set

2304

4 January 1999

31 December 2007

Out-of-sample data set (Validation set)

170

2 January 2008

29 August 2008

Figure 1 shows the total data set for the EUR/USD and its upward trend since early 2006.
Figure 1

EUR/USD Frankfurt ECB fixing prices (total data set).

The observed EUR/USD time series is non-normal (Jarque-Bera statistics confirm this at the 99 per cent confidence interval), containing slight skewness and high kurtosis. It is also non-stationary, and hence we decided to transform the EUR/USD series into a stationary daily series of rates of return,20 using the following formula:

where Rt is the rate of return and Pt is the price level at time t.
The summary statistics of the EUR/USD returns series are presented in Figure 2, and reveal slight skewness and high kurtosis. The Jarque-Bera statistic confirms again that the EUR/USD series is non-normal at the 99 per cent confidence interval.
Figure 2

EUR/USD returns summary statistics (total data set).

As inputs to our networks and based on the autocorrelation function and some ARMA experiments, we selected a set of autoregressive and moving average terms of the EUR/USD exchange rate returns and the 1-day Riskmetrics volatility series. Our set of NN inputs is presented in Table 2.
Table 2

Explanatory variables

Number

Variable

Lag

 1

EUR/USD exchange rate return

1

 2

EUR/USD exchange rate return

2

 3

EUR/USD exchange rate return

3

 4

EUR/USD exchange rate return

7

 5

EUR/USD exchange rate return

11

 6

EUR/USD exchange rate return

12

 7

EUR/USD exchange rate return

14

 8

EUR/USD exchange rate return

15

 9

EUR/USD exchange rate return

16

10

Moving average of the EUR/USD exchange rate return

15

11

Moving average of the EUR/USD exchange rate return

20

12

1-day Riskmetrics volatility

1

In order to train our NNs, we further divided our data set, as shown in Table 3.
Table 3

The neural networks data sets

Name of period

Trading days

Beginning

End

Total data set

2474

4 January 1999

29 August 2008

Training data set

1794

4 January 1999

31 December 2005

Test data set

510

2 January 2006

31 December 2007

Out-of-sample data set (Validation set)

170

2 January 2008

29 August 2008

FORECASTING MODELS

Benchmark models

In this study, we benchmark our NN models with three traditional strategies, namely, an ARMA model, an MACD model and a naïve strategy.

Naïve strategy

The naïve strategy simply takes the most recent period change as the best prediction of the future change, that is, a simple random walk. The model is defined by:

where Yt is the actual rate of return at period t and Ŷt+1 is the forecast rate of return for the next period.

The performance of the strategy is evaluated in terms of trading performance via a simulated trading strategy.

Moving average

The moving average model is defined as:

where Mt is the moving average at time t; n is the number of terms in the moving average; and Yt is the actual rate of return at period t.

The MACD strategy used is quite simple. Two moving average series are created with different moving average lengths. The decision rule for taking positions in the market is straightforward. Positions are taken if the moving averages intersect. If the short-term moving average intersects, the long-term moving average from below a ‘long’ position is taken. Conversely, if the long-term moving average is intersected from above, a ‘short’ position is taken.21

The forecaster must use judgement when determining the number of periods n on which to base the moving averages. The combination that performed best over the in-sample subperiod was retained for out-of-sample evaluation. The model selected was a combination of the EUR/USD and its 24-day moving average, namely, n=1 and 24, respectively, or a (1, 24) combination. The performance of this strategy is evaluated solely in terms of trading performance.

ARMA model

ARMA models assume that the value of a time series depends on its previous values (the autoregressive component) and on previous residual values (the moving average component). For a full discussion of the procedure, refer to the studies by Box et al22 or Pindyck and Rubinfield.23

The ARMA model takes the following form:

where Yt is the dependent variable at time t; Yt−1, Yt−2 and Ytp are the lagged dependent variables; φ0, φ1, φ2 and φp are regression coefficients; ɛt is the residual term; ɛt−1, ɛt−2 and ɛtp are previous values of the residual; and w1, w2 and wq are weights.
Using as a guide the correlogram in the training and test subperiods, we have chosen a restricted ARMA (11, 11) model. All of its coefficients are significant at the 95 per cent confidence interval. The null hypothesis that all coefficients (except the constant) are not significantly different from zero is rejected at the 95 per cent confidence interval (see Table A1).
Table A1

The output of the autoregressive moving average model used in this study is presented below

Variable

Coefficient

SE

t-Statistic

Prob.

C

0.000128

0.000128

1.002067

0.3164

AR(1)

−1.216599

0.042416

−28.68222

0.0000

AR(2)

−0.475940

0.081259

−5.857069

0.0000

AR(7)

−0.139565

0.046558

−2.997679

0.0027

AR(11)

0.197421

0.055779

3.539363

0.0004

MA(1)

1.213517

0.039315

30.86659

0.0000

MA(2)

0.473609

0.077725

6.093399

0.0000

MA(7)

0.152391

0.044815

3.400464

0.0007

MA(11)

−0.217830

0.054607

−3.989048

0.0001

R2

0.008541

Mean dependent var

0.000122

Adjusted R2

0.005067

SD dependent var

0.006169

SE of regression

0.006153

Akaike info criterion

−7.339687

Sum squared resid

0.086446

Schwarz criterion

−7.317159

Log likelihood

8420.282

F-statistic

2.458312

Durbin–Watson stat

1.997623

Prob(F-statistic)

0.011925

Inverted AR roots

0.75

0.65+0.46i

0.65−0.46i

0.23+0.77i

 

0.23−0.77i

−0.21−0.82i

−0.21+0.82i

−0.73−0.67i

 

−0.73+0.67i

−0.94−0.20i

−0.94+0.20i

Inverted MA roots

0.76

0.66+0.46i

0.66−0.46i

0.24+0.78i

 

0.24−0.78i

−0.21+0.83i

−0.21−0.83i

−0.73−0.68i

 

−0.73+0.68i

−0.94+0.20i

−0.94−0.20i

Dependent variable: RETURNS.

Method: Least squares.

Date: 9 March 2008; Time: 1728 hours.

Sample (adjusted): 12 2303.

Included observations: 2292 after adjustments.

Convergence achieved after 59 iterations.

Backcast: ? 0.

The selected ARMA model takes the following form:

The model selected was retained for out-of-sample estimation. The performance of the strategy is evaluated in terms of trading performance.

Empirical results for the benchmark models

A summary of the empirical results of the three benchmark models on the validation subset is presented in Table 4. The empirical results of the models in the training subperiod are presented in Table A3, while Table A2 shows the performance measures.
Table 4

Trading performance of the benchmark models out-of-sample

 

NAIVE

MACD

ARMA

Sharpe ratio (excluding costs)

1.28

−0.02

−0.45

Annualised volatility (excluding costs)

9.51%

9.54%

9.54%

Annualised return (excluding costs)

12.14%

−0.19%

−4.32%

Maximum drawdown (excluding costs)

−3.69%

−9.85%

−11.39%

Positions taken (annualised)

108

28

185

Table A3

Training subperiod trading performance

 

NAIVE

MACD

ARMA

MLP

RNN

HONN

Sharpe ratio (excluding costs)

−0.10

0.51

1.17

0.46

0.16

0.48

Annualised volatility (excluding costs)

10.36%

10.44%

10.42%

9.97%

10.38%

10.40%

Annualised return (excluding costs)

−1.07%

5.31%

12.19%

4.62%

1.67%

4.95%

Maximum drawdown (excluding costs)

−29.39%

−15.35%

−12.91%

−17.63%

−23.82%

−18.20%

Positions taken (annualised)

79

11

113

75

63

106

Table A2

Trading simulation performance measures

Performance measure

Description

 

Annualised return

RA=252(1/Nt=1NRt

(A1)

 

With Rt being the daily return

 

Cumulative return

RCt=1NRt

(A2)

Annualised volatility

Open image in new window

(A3)

Sharpe Ratio

SR=(RA)/(σA)

(A4)

 

Maximum negative value of Σ(Rt) over the period

 

Maximum drawdown

Open image in new window

(A5)

As can be seen, the naïve strategy outperforms all other models by far. These results are surprising not only because of the simplicity of the naïve model, but also based on the training subperiod results (see Table A3). There the naïve strategy presents an annualised return of −1.07 per cent and a Sharpe ratio of −0.10. Moreover, with a closer look at the returns in our out-of-sample period, we observe that the positive and negative returns are clustered. In order to verify that the sequence of signs of the returns in the validation period is not random, we conduct the Wald–Wolfowitz or runs test for randomness.24 The test confirms that the sequence of signs is not random at 99 per cent and 95 per cent confidence intervals. Therefore, as the naïve strategy is only using today's return as a forecast for tomorrow, it is able to exploit this phenomenon and present a remarkable performance. This anomaly was not present in previous years, and this is why the performance of the naïve strategy in-sample is so much worse. In the circumstances, this phenomenon is accidental, and we have no reason to believe that it should continue in the future. We thus discard the results of the naïve strategy from our conclusions.

Neural networks

Neural networks exist in several forms in the literature. The most popular architecture is the MLP.

A standard NN has at least three layers. The first layer is called the input layer (the number of its nodes corresponds to the number of explanatory variables). The last layer is called the output layer (the number of its nodes corresponds to the number of response variables). An intermediary layer of nodes, the hidden layer, separates the input from the output layer. Its number of nodes defines the amount of complexity the model is capable of fitting. In addition, the input and hidden layer contain an extra node, called the bias node. This node has a fixed value of 1 and has the same function as the intercept in traditional regression models. Normally, each node of one layer has connections to all the other nodes of the next layer.

The network processes information as follows: the input nodes contain the value of the explanatory variables. As each node connection represents a weight factor, the information reaches a single hidden layer node as the weighted sum of its inputs. Each node of the hidden layer passes the information through a nonlinear activation function and passes it on to the output layer if the calculated value is above a threshold.

The training of the network (which is the adjustment of its weights in the way that the network maps the input value of the training data to the corresponding output value) starts with randomly chosen weights and proceeds by applying a learning algorithm called backpropagation of errors (for a more detailed discussion see the study by Shapiro25). Backpropagation networks are the most common multilayer networks and are the most commonly used type in financial time series forecasting (see Kaastra and Boyd26). The learning algorithm simply tries to find those weights that minimise an error function (normally the sum of all squared differences between target and actual values). As networks with sufficient hidden nodes are able to learn the training data (as well as their outliers and their noise) by heart, it is crucial to stop the training procedure at the right time to prevent overfitting (this is called ‘early stopping’). This can be achieved by dividing the data set into three subsets, respectively called the training and test sets, used for simulating the data currently available to fit and tune the model, and the validation set, used for simulating future values. The network parameters are then estimated by fitting the training data using the above-mentioned iterative procedure (backpropagation of errors). The iteration length is optimised by maximising the forecasting accuracy for the test data set. Our networks, which are specially designed for financial purposes, will stop training when the profit of our forecasts in the test subperiod is maximised. Then the predictive value of the model is evaluated applying it to the validation data set (out-of-sample data set).

The MLP model

The MLP network architecture
The network architecture of a ‘standard’ MLP looks as presented in Figure 327 where xt[n] (n=1, 2, …, k+1) are the model inputs (including the input bias node) at time t; ht[m] (m=1, 2, …, j+1) are the hidden nodes outputs (including the hidden bias node); t is the MLP model output; and ujk and wj are the network weights.
Figure 3

A single-output, fully connected MLP model.

The error function to be minimised is:
Empirical results of the MLP model
The trading performance of the MLP on the validation subset is presented in Table 5. We chose the network with the highest profit in the training subperiod. The trading strategy applied is simple: go or stay long when the forecast return is above zero and go or stay short when the forecast return is below zero. Table A3 provides the performance of the MLP in the training subperiod, while Table A4 provides the characteristics of our network. The results of other models are included for comparison.
Table 5

Out-of-sample trading performance of the Multilayer Perceptron (MLP)

 

MACD

ARMA

MLP

Sharpe ratio (excluding costs)

−0.02

−0.45

0.74

Annualised volatility (excluding costs)

9.54%

9.54%

9.18%

Annualised return (excluding costs)

−0.19%

−4.32%

6.82%

Maximum drawdown (excluding costs)

−9.85%

−11.39%

−5.78%

Positions taken (annualised)

28

185

107

Table A4

Network characteristics

Parameters

MLP

RNN

HONN

Learning algorithm

Gradient descent

Gradient descent

Gradient descent

Learning rate

0.001

0.001

0.001

Momentum

0.003

0.003

0.003

Iteration steps

20 000

20 000

10 000

Initialisation of weights

N(0,1)

N(0, 1)

N(0, 1)

Input nodes

12

12

12

Hidden nodes (one layer)

7

5

0

Output node

1

1

1

As can be seen, the MLP outperforms our benchmark statistical models.

The RNN

Our next model is the RNN. While a complete explanation of RNN models is beyond the scope of this study, we present below a brief explanation of the significant differences between RNN and MLP architectures. For an exact specification of the RNN, see the study by Elman.28

A simple RNN has activation feedback, which embodies short-term memory. The advantages of using RNNs over feedforward networks, for modelling nonlinear time series, have been well documented in the past. However, as described in by Tenti,8 ‘the main disadvantage of RNNs is that they require substantially more connections, and more memory in simulation, than standard backpropagation networks’ (p. 569), thus resulting in a substantial increase in computational time. However, having said this, RNNs can yield better results in comparison to simple MLPs owing to the additional memory inputs.

The RNN architecture
A simple illustration of the architecture of an Elman RNN is presented in Figure 4 where xt[n] (n=1, 2, …, k+1), ut,[1]ut[2] are the model inputs (including the input bias node) at time t; t is the recurrent model output; dt [f] (f=1, 2) and wt[n] (n=1, 2, …, k+1) are the network weights; and Ut [f] (f=1, 2) is the output of the hidden nodes at time t.
Figure 4

Elman RNN architecture with two nodes on the hidden layer.

The error function to be minimised is:

In short, the RNN architecture can provide more accurate outputs because the inputs are potentially taken from all previous values (see inputs Uj−1 [1] and Uj−1 [2] in Figure 4).

Empirical results of the RNN model

The RNNs are trained with gradient descent, as are the MLPs. However, the increase in the number of weights, as mentioned before, makes the training process extremely slow, taking 10 times as long as the MLP.

We follow the same methodology as we did for the MLPs for the selection of our optimal network. The characteristics of the network that we use are shown in Table A4, while a summary of the performance of the network in the training subperiod is presented in Table A3.

The trading strategy is the same as that followed for the MLP. As shown in Table 6, the RNN has a worse performance than the MLP model when measured by the Sharpe ratio and annualised return. The results of other models are included for comparison. Table 6 presents the out-of-sample trading performance results of the RNN.
Table 6

Out-of-sample trading performance results of the Recurrent Neural Network (RNN)

 

MACD

ARMA

MLP

RNN

Sharpe ratio (excluding costs)

−0.02

−0.45

0.74

0.40

Annualised volatility (excluding costs)

9.54%

9.54%

9.18%

9.54%

Annualised return (excluding costs)

−0.19%

−4.32%

6.82%

3.83%

Maximum drawdown (excluding costs)

−9.85%

−11.39%

−5.78%

−5.91%

Positions taken (annualised)

28

185

107

93

The HONN

HONNs were first introduced by Giles and Maxwell,11 and were called ‘Tensor Networks’. Although the extent of their use in finance has so far been limited, Knowles et al29 show that, with shorter computational times and limited input variables, ‘the best HONN models show a profit increase over the MLP of around 8 per cent’ on the EUR/USD time series (p. 7). For Zhang et al,30 a significant advantage of HONNs is that ‘HONN models are able to provide some rationale for the simulations they produce and thus can be regarded as “open box” rather then “black box”. Moreover, HONNs are able to simulate higher frequency, higher order non-linear data, and consequently provide superior simulations compared to those produced by ANNs (Artificial Neural Networks)’ (p. 188).

The HONN architecture
Although they have already experienced some success in the field of pattern recognition and associative recall (see Karayiannis et al31), HONNs have not yet been widely used in finance. The architecture of a three-input second-order HONN is shown in Figure 5 where xt[n] (n=1, 2, …, k+1) are the model inputs (including the input bias node) at time t; t is the HONNs model output; and ujk are the network weights.
Figure 5

Left, MLP with three inputs and two hidden nodes; right, second-order HONN with three inputs.

The error function to be minimised is:

HONNs use joint activation functions; this technique reduces the need to establish the relationships among inputs when training. Furthermore, this reduces the number of free weights and means that HONNs are faster to train than even MLPs. However, because the number of inputs can be very large for higher-order architectures, orders of four and over are rarely used.

Another advantage of the reduction of free weights is that the problems of overfitting and local optima affecting the results of NNs can be largely avoided. For a complete description of HONNs, see the study by Knowles et al29.

Empirical results of the HONN model
We follow the same methodology as we did with the RNNs and MLPs for the selection of our optimal HONN. The trading strategy is that followed for the MLP. A summary of our findings is presented in Table 7, while Table A3 provides the performance of the network in the training subperiod and Table A4 provides its characteristics. The results of other models are included for comparison. Table 7 presents the out-of-sample trading performance results of the HONN.
Table 7

Out-of-sample trading performance results of the Higher Order Neural Network (HONN)

 

MACD

ARMA

MLP

RNN

HONN

Sharpe ratio (excluding costs)

−0.02

−0.45

0.74

0.40

0.86

Annualised volatility (excluding costs)

9.54%

9.54%

9.18%

9.54%

9.49%

Annualised return (excluding costs)

−0.19%

−4.32%

6.82%

3.83%

8.16%

Maximum drawdown (excluding costs)

−9.85%

−11.39%

−5.78%

−5.91%

−7.21%

Positions taken (annualised)

28

185

107

93

170

We can see that the HONN performs significantly better than the RNN and the traditional MLP models.

TRADING COSTS, LEVERAGE AND ROBUSTNESS

Thus far, we have presented the trading results of all our models without considering transaction costs. As some of our models trade quite often, taking transaction costs into account might change the whole picture.

We therefore introduce transaction costs as well as a leverage for each of our models. Moreover, we examine the robustness of our models by examining and comparing not only the results of our current research, but also the conclusions of a previous study in which we examined the same series with the same models but over a different period. For the purposes of comparability, we use the same selection of inputs for our NNs in both studies.32 The only difference is the out-of-sample period: here it is 2 January 2008 – 29 August 2008, while in the previous study it was 3 July 2006 – 31 December 2007. The purposes of this test are to validate the robustness of our models over time and to provide concrete empirical evidence of the forecasting power of our models.

Transaction costs

The transaction costs for a tradable amount, say USD 5–10 million, are approximately 1 pip (0.0001 EUR/USD) per trade (one way) between market makers. But as the EUR/USD time series considered here is a series of middle rates, the transaction cost is one spread per round trip.

With an average exchange rate of EUR/USD 1.532 for the out-of-sample period, a cost of 1 pip is equivalent to an average cost of 0.007 per cent per position.

In Table 8 we present the performance of our models after transaction costs are considered.
Table 8

Out-of-sample trading performance results with transaction costs (2 January 2008 – 29 August 2008)

 

MACD

ARMA

MLP

RNN

HONN

Sharpe ratio (excluding costs)

−0.02

−0.45

0.74

0.40

0.86

Annualised volatility (excluding costs)

9.54%

9.54%

9.18%

9.54%

9.49%

Annualised return (excluding costs)

−0.19%

−4.32%

6.82%

3.83%

8.16%

Maximum drawdown (excluding costs)

−9.85%

−11.39%

−5.78%

−5.91%

−7.21%

Positions taken (annualised)

28

185

107

93

170

Transaction costs

0.20%

1.30%

0.75%

0.65%

1.19%

Annualised return (including costs)

−0.39%

−5.62%

6.07%

3.18%

6.97%

We observe that although the HONN model presents higher transaction costs, it continues to outperform the other models in terms of annualised return. The MLP comes second, while the RNN demonstrates the third best performance. In contrast, the ARMA and MACD models have rather disappointing performances, as they both present negative annualised returns.

Examining a different out-of-sample period, 3 July 2006 – 31 December 2007, but following the same methodology and using the same selection of inputs for our NNs, the results presented in the table are similar, only the MLP model outperformed the HONN in that out-of-sample period. Table 9 presents the out-of-sample trading performance results with transaction costs.
Table 9

Out-of-sample trading performance results with transaction costs (3 July 2006 – 31 December 2007)

 

MACD

ARMA

MLP

RNN

HONNs

Sharpe ratio (excluding costs)

0.70

−0.40

1.88

0.60

0.99

Annualised volatility (excluding costs)

6.38%

6.38%

6.34%

6.34%

6.37%

Annualised return (excluding costs)

4.44%

−2.53%

11.91%

3.82%

6.29%

Maximum drawdown (excluding costs)

−4.73%

−10.12%

−5.05%

−4.64%

−4.37%

Positions taken (annualised)

20

189

109

167

80

Transaction costs

0.16%

1.51%

0.87%

1.33%

0.64%

Annualised return (including costs)

4.28%

−4.04%

11.04%

2.48%

5.65%

We observe that in both periods the MLP and HONN models clearly outperform the other strategies. In the latest period, the HONN model has a better performance, while 1½ year before the MLP presented better results. In contrast, the ARMA model in both periods presents a rather disappointing trading performance. This empirical evidence allows us to argue that HONNs and MLPs have a consistent and better performance than the RNN, MACD and ARMA models in forecasting the ECB daily fixing of the EUR/USD.

Leverage to exploit low volatility

In order to further improve the trading performance of our models, we introduce a ‘level of confidence’ to our forecasts, that is, a leverage based on the test subperiod that takes into account the low volatility of the trading performance of our models. For the ARMA and MACD models, which show a negative return, we do not apply leverage. The leverage factors applied are calculated in such a way that each model has a common volatility of 10 per cent33 on the test data set.

The transaction costs are calculated by taking 0.007 per cent per position into account, while the cost of leverage (interest payments for the additional capital) is calculated at 4 per cent p.a. (that is, 0.016 per cent per trading day34). Our results are presented in Table 10.
Table 10

Trading performance – Final results (2 January 2008 – 29 August 2008)35

 

MACD

ARMA

MLP

RNN

HONN

Sharpe ratio (excluding costs)

−0.02

−0.45

0.74

0.40

0.86

Annualised volatility (excluding costs)

9.54%

9.54%

10.01%

10.02%

9.96%

Annualised return (excluding costs)

−0.19%

−4.32%

7.43 %

4.02%

8.56%

Maximum drawdown (excluding costs)

−9.85%

−11.39%

−6.30%

−6.21%

−7.57%

Positions taken (annualised)

28

185

107

93

170

Leverage

1.09

1.05

1.05

Transaction costs

0.20%

1.30%

0.99%

0.79%

1.33%

Annualised return (including costs)

−0.39%

−5.62%

6.44%

3.23%

7.23%

As can be seen from the last row in Table 10, the HONN model continues to demonstrate a superior trading performance. Similarly, the MLP and RNN continue to perform well and present the second- and the third-highest annualised return, respectively. In general, we observe that all models in which leverage was applied were able to exploit it and increase their trading performance despite the higher transaction costs.

The performance of our models for a different out-of-sample period, 3 July 2006 – 31 December 2007, is shown in Table 11.
Table 11

Trading performance – Final results (3 July 2006 – 31 December 2007)35

 

MACD

ARMA

MLP

RNN

HONN

Sharpe ratio (excluding costs)

0.70

−0.40

1.88

0.60

0.99

Annualised volatility (excluding costs)

7.27%

6.38%

7.16%

7.23%

7.26%

Annualised return (excluding costs)

5.06%

−2.53%

13.45%

4.35%

7.17%

Maximum drawdown (excluding costs)

−5.39%

−10.12%

−5.70%

−5.30%

−4.98%

Positions taken (annualised)

20

189

109

167

80

Leverage

1.14

1.13

1.14

1.14

Transaction costs

1.02%

1.51%

1.67%

2.19%

1.50%

Annualised return (including costs)

4.04%

−4.04%

11.78%

2.16%

5.67%

We note that in both periods, the MLP and HONN models continue to outperform the other models, as they are able to exploit the leverage and present an increased trading performance in both out-of-sample periods. So even if the ranking of these two models is different in the two out-of-sample forecasting periods retained, they clearly outperform all other models in all cases, a fact that allows us to argue with confidence their forecasting superiority and their stability and robustness over time. In contrast, the RNN model was not able to exploit the extra memory inputs in its architecture and presents rather disappointing results. Moreover, the time spent to derive the RNN results is 10 times longer than the time required with the HONN and MLP models. Similarly, the MACD and ARMA models present a very weak forecasting power, even though their training subperiod performance was promising (see Table A3).

CONCLUDING REMARKS

In this study, we apply MLP, RNNs and HONNs to a 1-day-ahead forecasting and trading task of the EUR/USD exchange rate using the ECB fixing series with only autoregressive terms as inputs. We use a naïve, an MACD and an ARMA model as benchmarks. Our aim is not only to examine the forecasting and trading performance of our models, but also to see whether this performance is stable and robust over time. In order to do so, we develop these different prediction models over the period January 1999 – December 2007, and validate their out-of-sample trading efficiency over the following period from January 2008 to August 2008. To examine the robustness and stability of our models, we compare our results with those from a previous study using the same models and the same selection of autoregressive terms as inputs to the NNs, but with an out-of-sample period between July 2006 and December 2007.

As it turns out, the MLP and HONN models clearly outperform the other models in both out-of-sample periods in terms of annualised return. Our conclusions are the same even after we introduced transaction costs and a leverage to exploit the low volatility of the trading performance of those models. This enables us to argue with confidence their forecasting superiority and their stability and robustness over time. In contrast, the RNN model seems to have difficulty in providing good forecasts when only autoregressive series are used as inputs. Similarly, the ARMA and MACD models present low or even negative annualised returns in this application, despite their satisfactory training subperiod performance.

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Copyright information

© Palgrave Macmillan 2009

Authors and Affiliations

  • Christian L Dunis
    • 1
  • Jason Laws
  • Georgios Sermpinis
  1. 1.Liverpool Business School, CIBEF – Centre for International Banking, Economics and Finance, JMU, John Foster Building, 98 Mount PleasantLiverpoolUK

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