Forecasting the Exchange Rates of CHF vs USD Using Neural. networks

Transcription

1 Forecasting the Exchange Rates of CHF vs USD Using Neural networks Name Jingtao Yao Yili Li Chew Lim Tan A. PhD Student Master Student Associate Professor Tel Mailing Address: Department of Information Systems and Computer Science National University of Singapore Singapore Abstract In this paper, an experimental research based on a neural network forecasting methodology is discussed. The exchange rates between Swiss Franc and American Dollar are predicted a reasonable out-of-sample prots achieved in the experiment. The results show that a simple backpropagation network with ecient learning and a simple set of technical indicators as inputs serves well as a predictive model for a six month forecasting period. The paper discusses several issues on the frequency of sampling, choice of network architecture, forecasting periods, and measures for evaluating the model's predictive power. After presenting the experimental results, a disscusion on future research concludes the paper. 1

2 Forecasting the Exchange Rates of CHF vs USD Using Neural networks. Abstract In this paper, an experimental research based on a neural network forecasting methodology is discussed. The exchange rates between Swiss Franc and American Dollar are predicted a reasonable out-of-sample prots achieved in the experiment. The results show that a simple backpropagation network with ecient learning and a simple set of technical indicators as inputs serves well as a predictive model for a six month forecasting period. The paper discusses several issues on the frequency of sampling, choice of network architecture, forecasting periods, and measures for evaluating the model's predictive power. After presenting the experimental results, a disscusion on future research concludes the paper. 1 Introduction Currency exchange rates are an important economic index in the international monetary markets. Since 1973, with the abandonment of the xed foreign exchange rates and the implementation of the oating exchange rate system by industrialized countries, researchers have been striving for an explanation of the movement of exchange rates. Foreign exchange rates are aected by many highly correlated factors. These factors could be economic, political and even psychological factors. The interaction of these factors is in a very complex fashion. Therefore, to forecast the changes of foreign exchange rates is generally very dicult. Technical and fundamental analyses are the major forecasting methods which are in popular use in the nancial area.

3 In addition to the classical time series forecasting method such as Box-Jenkins [Box and Jenkins 1976], the neural network method is now widely used for nancial forecasting. Examples using neural networks in currency applications include Green [Green and Pearson 1994], Manger [Manger 1994], Rawani [Rawani 1993], Refenes [Refenes 1992], Weigend [Weigend 1991], and Zhang [Zhang 1994]. Feed-forward backpropagation networks are the most commonly used networks and meant for the widest variety of applications. In theory, a neural network model that ts any kind of functions and data could be built. The main consideration when building a suitable neural network for nancial application is to make a trade-o between convergence and generalization. It is important not to have too many nodes in the hidden layer because this may allow the neural network to learn by example only and not to generalize which is widely known as the over-tting of the network to the learning samples [Baum and Hassler 1989]. Actually, the better a neural network has memorized the training data, the worse it is likely to perform on future data. Computation cost for training is one of the disadvantages for neural network application. An objective of this research is to nd a simple and fast way to train the network which can perform good forecasting later on. In this paper, the research results on using neural networks to forecast the exchange rates between the US dollar (USD) and Swiss Franc (CHF) are presented. Simple technical indicators (moving averages) are applied to a simple backpropagation network (with only one hidden layer) for forecasting. This study shows that without the use of extensive market data or knowledge, useful prediction can be made and signicant paper prot can be achieved with simple technical indicators. The average paper prot achieved for 12 dierent data segments is 28.62% yearly comparing with a 9.88% average annual return of a buy-and-hold strategy was not assisted by any forecast information. A 1% commission was 2

4 deducted for each transaction to simulate a realistic trading. The following section will describe the methodology used in the construction of our neural network model. In Section 3 and 4, we will discuss the data preparation and the network training, based on which the model's predictive power will be evaluated. Section 5 will present a simulation of training with the model to generate paper prots as a measurement of the model's forecasting performance. Section 6 will discuss the experimental results followed by the concluding section which outlines several future research directions. 2 Research Methodology The backpropagation neural network [Rumelhart and McClelland 1986], the most popular network in business applications [Wong 1995], was used to model foreign exchange time series in this study. A hyperbolic tangent function was used as the activation function. Indicators derived from past exchange rate time series were fed to the neural networks and the succeeding week's exchange rates were used to supervise the training. We used one hidden layer for each neural network model. Various architectures were tried, in each of which the hidden layer had about half of the neurons of the input layer. For each architecture, dierent training, validation and testing data segments were also experimented. Hence, a set of the best architectures measured with such parameters as the NMSE was chosen for use in forecasting. The out-of-sample testing data were then used to measure the performance of these models. The input data were rst preprocessed by normalizing them within the scale of?1 to 1. To capture the relationship between the exchange rates of today and the future through a neural network, the moving averages are considered simple and powerful indicators to predict the markets. Following our previous work reported in [Yao at al. 1996], we chose ve moving averages, namely, 3

5 Name Training Period Validation Period Testing Period CHF0 6 Jan July 91 2 Aug July 93 6 Aug 93-3 Nov 95 CHF1 6 Jan Dec 88 6 Jan Dec 89 5 Jan June 90 CHF2 6 July June 89 7 July June 90 6 July Dec 90 CHF3 4 Jan Dec 89 5 Jan Dec 90 4 Jan June 91 CHF4 5 July June 90 6 July June 91 5 July Dec 91 CHF5 3 Jan Dec 90 4 Jan Dec 91 3 Jan June 92 CHF6 4 July June 91 5 July June 92 3 July Dec 92 CHF7 2 Jan Dec 91 3 Jan Dec 92 8 Jan June 93 CHF8 3 July June 92 3 July June 93 2 July Dec 93 CHF 9 8 Jan Dec 92 8 Jan Dec 93 7 Jan June 94 CHF10 1 July June 93 2 July June 94 1 July Dec 94 CHF11 6 Jan Dec 93 7 Jan Dec 94 6 Jan June 95 CHF12 7 July June 94 1 July June 95 1 July 95-3 Nov 95 Table 1: Time periods For Each Data Segments MA5, MA10, MA20, MA60 and MA120 which refer to averages over ve days, 10 days, 20 days, 60 days and 120 days, respectively. The Friday's ve moving averages and its closing price were fed to a neural network as inputs. The supervising targets of the neural network was the succeeding Friday's closing prices. 3 Data Preparation Exchange rates between CHF and USD were used in this study. The data were recorded from March 1, 1983 to November 3, 1995, or 3309 daily data, in Singapore Forex market. 13 dierent data segments were chosen. They were named from CHF0 to CHF12, respectively, and their time periods are show in Table 1. The statistics summary of each segment is show in Table 2. In a real situation, there is no closing price for Forex since its trading takes place 24 hours a day all over the world. However, in view of the diculties in managing the huge data, xed sampled time series are often used to model the behavior of exchange rate changes. Daily data are easily available compared with tick-by-tick data. Trading too frequently will cost much on transaction. Weekly forecasts over 7 daily data were adopted to reduce the amount of transaction. 4

6 Name Mean Median Max Std. Dev. Variance Skew CHF CHF CHF CHF CHF CHF CHF CHF CHF CHF CHF CHF CHF Table 2: Data Statistics of Foreign Exchange Rates of CHF for Dierent Data Segments The weekly closing prices which refer to the Friday's closing prices in the local market were used as the prediction targets in our experiment. In the event of Friday being a public holiday, the preceding available daily closing price for the currency was used. The target data segments in this study consist of 668 weekly data in total. Of the 13 data segments, the rst(i.e. CHF0) spans over a time period of about 12 years. The other twelve (namely CHF1 to CHF12) each covers an overlapping period of years. The twelve year periods are progressively displaced by 1 2 year. Thus, the rst year period (for CHF0) spans from January 1984 to June The next period is slid along the time horizon for half a year, i.e., it spans from July 1984 to December The last time period spans from July 1989 to November 1995 as shown in Table 1. Amongst the 6 1 year data, the rst 6 years' data, or 312 weeks, were used for training and validation 2 while the remaining half a year's data, or 26 weeks, were used to test the performance of the neural network model. Of the 312 data, 260 were used as training data and 52 are used as validation. 5

7 As mentioned earlier, the data were normalized. The normalization formula is: y = 2x? (max + min) max? min (1) where x is the data before normalizing y is the data after normalizing All neural network outputs were scaled back using an inverse formula before any other calculation (trading strategy, prots, etc). 4 Training Results A usual measure to evaluate and compare the predictive power of the model is the Normalized Mean Squared Error (NMSE) [Levin 1994][Yao and Poh 1995]. Additional evaluation measures include the calculation of a correct matching number of the actual and predicted values, x t and ^x t, respectively, in the testing set with respect to the sign and directional change (expressed in percentages). Directional change statistic is the average of a k where a k = 1 if (x t+1? x t )(^x t+1? x t ) 0, and a k = 0 otherwise for Grad1 and a k = 1 if (x t+1? x t )(^x t+1? ^x t ) 0 and a k = 0 otherwise for Grad2. The two statistics, Grad1 and Grad2, are trend correctness when using Strategy 1 and Strategy 2 (to be discussed in Section 5) respectively. These statistics are desirable because the NMSE measures prediction only in terms of levels and tness. For each data segment, a variety of network architectures were experimented. The best architecture, in term of NMSE, for each data segment, is presented in the Table 3. The network architecture is denoted by p-q-r, where p, q, r stand for the number of input nodes, hidden nodes and output nodes, respectively. The maximum iteration time was xed at 10,000. The maximum error for one pass 6

8 Model Archit. NMSE Grad1 Grad2 CHF % 62% CHF % 84% CHF % 72% CHF % 76% CHF % 36% CHF % 84% CHF % 76% CHF % 68% CHF % 84% CHF % 72% CHF % 92% CHF % 68% CHF % 82% Table 3: The Technical Details of Chosen Models and Their Forecasting Potentials (: learning rate; : momentum rate; NMSE: Normalized Mean Squared Error; Grad1, Grad2: correctness of gradients for testing data.) through the whole training set was Figure 1 to Figure 6 are the testing (i.e. forecasting) results for 6 data segments, namely CHF0, CHF1, CHF5, CHF8, CHF9, and CHF10, respectively. Other testing data were not shown here in the interest of space. The actual target(solid line) can be compared with the neural network output (dotted line) in each instance. The vertical axis represents the exchange rate between CHF and USD while the horizontal axis represents the week number of the testing period. From Figure 1, we found that the rst 40 weeks forecasts t with the actual values quite well. As found in a similar work conducted by us earlier [Yao at al. 1996], there was indication from Figure 1 that the network should be retrained every half a year. The half a year prediction was adopted for other data segments (from CHF0 to CHF12). It can be seen from Figure 2 to 6 that the neural network outputs closely follow the actual target values. 7

9 chf.tar chf.out Figure 1: Prediction of the Weekly CHF/USD Aug Nov. 1995(CHF0) 1.8 chf1.tar chf1.out Figure 2: Prediction of the Weekly CHF/USD Jan June 1990(CHF1) chf5.tar chf5.out Figure 3: Prediction of the Weekly CHF/USD Jan June 92(CHF5) 8

10 1.5 chf8.tar chf8.out Figure 4: Prediction of the Weekly CHF/USD July Dec 93(CHF8) chf9.tar chf9.out Figure 5: Prediction of the Weekly CHF/USD Jan June 94(CHF9) chf10.tar chf10.out Figure 6: Prediction of the Weekly CHF/USD July Dec 94(CHF10) 9

11 5 Testing with Trading Strategies A trader's real concern is the prots or nancial gains. It does not matter whether the forecasts are accurate or not in terms of NMSE or Grad. After the forecast results were obtained from the neural network models, a program simulating the real trading was developed to test the possible monetary gains. Since this is not the real trading, we name it as paper prots. Assume that a certain amount of seed money is used in this program. The seed money is used to buy a certain amount of one currency on Friday when the prediction shows a rise in that currency in succeeding Friday. At the end of the testing period, the currency should be converted to the original currency of the seed money using the exact direct or cross rate of that day. The calculation of paper prot is: M oneyobatined Return = SeedM oney 52 n? 1 (2) where MoneyObtained = The amount of the money obtain on the last day of testing SeedMoney = Amount of money used for trading on the the rst day of testing n= No. of weeks in testing period There are two kinds of trading strategies used in this study. One uses the dierence between predictions, and another uses the dierence between the predicted and the actual levels to trade as the prediction standard. Strategy 1: if(^x t+1? ^x t ) > 0 then buy else sell (3) Strategy 2: if(^x t+1? x t ) > 0 then buy else sell (4) 10

12 Name Ret1 Ret1 U SD Ret2 Ret2 U SD Bench. I Bench. II CHF CHF CHF CHF CHF CHF CHF CHF CHF CHF CHF CHF CHF Table 4: Benchmark results for dierent time period For example, assume that x t = 10; x t+1 = 11; ^x t = 8; ^x t+1 = 9, Strategy 1 says that the market will go up and Strategy 2 says that the market will be going down. In actual trading, practitioners may choose one of the strategies. A conservative trading strategy would require a trader to act only when both strategies recommend the same actions. As a transaction cost is incurred in real trading, a 1% transaction cost was included in the calculation. The paper prots can be compared with two benchmarks. Benchmark 1 uses a `buy-and-hold' strategy while Benchmark 2 uses `Trend-follow' strategy. The Benchmark 1's strategy is to buy the USD at the beginning of the testing period and then sell it at the end of the testing period. The Benchmark 2's strategy is to buy when the market is continually up for two weeks and sell when it is down for one week. Table 4 shows the dierences between paper prots and their benchmark prots. All trade on Fridays. All the prots are based on the same calculation using Equation 2. Prots under Ret1 and Ret2 denote the returns obtained in CHF (seed money is CHF), while Ret1 U SD and Ret2 U SD in USD (seed money is USD). The dierences between two currencies is due to the general changes of Forex. For benchmarks, USD are used as seed money. 11

13 Name Ret1 Ret1 U SD Ret2 Ret2 U SD Bench. I Bench. II Average Median Portfolio Average Portfolio Acceptability Table 5: Analysis of Results for dierent Models. Average1: average prot from 13 models; Median: median of 13 models; Average2: average except CHF0; Portfolio1: Prot of the sum of 20% of Max, 20% of Min and 60% average of others(inclusive CHF0); Portfolio2: Same as Portfolio1 exclusive CHF0; Acceptability: Percentage of prot greater than 5% annually 6 Discussion Only half a year's forecasts were performed on data segments CHF1 to CHF12, while a two year's forecast was conducted on CHF0. All prediction lines tted quite well with the target lines except for CHF0. The performance on CHF0 shows a degradation in forecasting in about the third quarter after training. This indicates the presence of `recency' problem of the network, namely, the network did retain some memory of the history. A half a year forecast period is thus recommended based on the present study. The study of the 13 data segments shows that the neural network models could be applied to future forecasting. Compared with the two benchmarks, the neural network model is better. As shown in Table 5, for the neural network, the worst percentage of acceptable prot, say > 5%, is among the four strategies (two strategies based on two currencies). While the benchmarks can only achieve and respectively. Referring to Table 5 again, another comparison was made for the forecasting results. Assume that we have a portfolio prot based on 20% of maximum of prot, 20% of minimum prot, and 60% of average prots from other data segments. CHF0 is included in Portfolio 1 and excluded in Portfolio 2. This will indicate the general performance of each strategy. Figure 7 and Figure 8 are the graphical presentation of the prot gained for each strategy. The 12

14 Profit 40 Ret1 Ret1$ Bench1 Bench CHF0 CHF2 CHF4 CHF6 CHF8 CHF10 Models Figure 7: Each Models Prots on Strategy 1 Compare with Benchmarks Profit 60 Ret2 Ret2$ Bench1 Bench CHF0 CHF2 CHF4 CHF6 CHF8 CHF10 Models Figure 8: Each Models Prots on Strategy 2 Compare with Benchmarks strategy 1 is consistently better than other strategies and the benchmarks across time. Whether this will hold over other currencies will be studied later. 7 Conclusion Simple technical indicators can be fed to a neural network to build a neural network based forecasting model. Average paper prots of 11.36%, 15.87%, 24.17% and 27.59% can be achieved for dierent trading strategies over dierent time horizons. This could be considered a useful model for actual forecasting. Long time learning is not ideal for time series forecasting. Half a year's forecasting horizon is an acceptable time according to the results of this research. A backpropagation network used in the present study has proved to be adequate for forecasting. The use of recurrent network will 13

15 be considered in future and compared with the present backpropagation network. More statistical analyses are needed in order to nd out which kind of data segments will best capture the underlying behavior of the market changes. To solve the `recency' and delay problem, a randomly sampling data method instead of time order segregating approach is currently under study. As Zhou [Zhou 1995] stated, increasing observation frequency does not always help to improve the accuracy of forecasting. In this research, the weekly data are used assuming that they have enough information to capture the \rules". Due to the volatility of the currency movement [Muller and Dacoragna 1993], a dierent frequency of data maybe needed than the weekly data. The data could be sampled according to the market character, e.g. bullish, bearish, or trading, etc. In other words, when the market is volatile, we sample more data for training, and vice versa. Nonlinear or volatile time scale will be taken into consideration in our further research. The forecasting target on the percentage of change of actual rate will also be examined. Acknowledgments This work was conducted with Dr. Hean-Lee Poh initially. Special thanks to his invaluable advice, constructive discussions and helpful comments. We thank Mary Aviani, Swee-Yuan Tay and Kok-Sun Lee for their useful discussions. References Baum, E.B. and Hassler, D.(1989), \What Size Net Gives Valid Generalization? " Neural Computation, 1, Box, G.E.P., and Jenkins, G.M.(1976), Time Series Analysis: Forecasting and Control Holden-day, San Francisco 14

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