Forecasting stock exchange movements using neural networks: Empirical evidence from
Kuwait
Mohamed M. Mostafa a , a New York Institute of Technology, Global Program in Bahrain, Manama, Bahrain
Available online 23 February 2010.
Financial time series are very complex and dynamic as they are characterized by extreme volatility. The major aim of this research is to forecast the Kuwait stock exchange (KSE) closing price movements using data for the period 2001–2003. Two neural network architectures: multi-layer perceptron (MLP) neural networks and generalized regression neural networks are used to predict the KSE closing price movements. The results of this study show that neuro-computational models are useful tools in forecasting stock exchange movements in emerging markets. These results also indicate that the quasi-Newton training algorithm produces less forecasting errors compared to other training algorithms. Due to their robustness and flexibility of modeling algorithms, neuro-computational models are expected to outperform traditional statistical techniques such as regression and ARIMA in forecasting stock exchanges’ price movements.
Keywords: KSE; Neural networks ; Forecasting
1. Introduction and related work
2. Methodology
2.1. Data
2.2. Multi-layer perceptron
2.3. Generalized regression neural network
3. Results
3.1. MLP-based forecasting
3.2. GRNN-based forecasting
4. Implications, limitations and future research
References
The increasing globalization of the financial markets has heightened the interest in emerging markets. In the Middle East there are 11 formal stock markets monitored by Standard and
Poors Emerging Markets Database ( Smith, 2007 ). Stock markets are usually classified as either developed or emerging. Three of the Middle East stock markets are categorized as developed: Kuwait, United Arab Emirates and Qatar. Middle East stock markets are, however, relatively small by world standards as the 11 formal stock markets in the region account for around 0.9% of world stock market capitalization ( Smith, 2007 ).

Kuwait stock exchange (KSE) was officially established in 1984 after the crash of
Almanakh (over-the-counter market). In the post liberation period (1992) KSE witnessed a lot of reforms among which were the government privatization program to sell its holdings of shares in local shareholding companies, the launching of mutual funds, and the emergence of institutional investors as the dominant players in the market ( Al-Loughani & Chappell, 2000 ).
Throughout its history, KSE has been characterized by an irregularity in trades and price formation process. However, the Kuwaiti parliament has recently introduced some measures that permit foreign investors to trade in KSE ( Al-Loughani & Chappell, 2000 ). This study aims to forecast the KSE movements using neural network (NN) models.
NN models have been successfully used in prediction or forecasting studies across many disciplines. One of the first successful applications of MLP is reported by Lapedes and Farber
(1988) . Using two deterministic chaotic time series generated by the logistic map and the
Glass–Mackey equation, they designed an MLP that can accurately mimic and predict such dynamic non-linear systems. Another major application of MLP is in electric load consumption (e.g. [Darbellay and Slama, 2000] and [McMenamin and Monforte, 1998] ).
Many other problems have been solved by MLP. A short list includes air pollution forecasting
(e.g. Videnova, Nedialkova, Dimitrova, & Popova, 2006 ), maritime traffic forecasting
( Mostafa, 2004 ), airline passenger traffic forecasting ( Nam & Yi, 1997 ), railway traffic forecasting ( Zhuo, Li-Min, Yong, & Yan-hui, 2007 ), commodity prices ( Kohzadi, Boyd,
Kemlanshahi, & Kaastra, 1996 ), ozone level ( Ruiz-Suarez, Mayora-Ibarra, Torres -Jimenez,
& Ruiz-Suarez, 1995 ), student grade point averages ( Gorr, Nagin, & Szczypula, 1994 ), forecasting macroeconomic data ( Aminian, Suarez, Aminian, & Walz, 2006 ), financial time series forecasting ( Yu & Lai Wang, 2009 ), advertising ( Poh, Yao, & Jasic, 1998 ), and market trends ( Aiken & Bsat, 1999 ).
Due to its good performance in noisy environments, the generalized regression neural network (GRNN) has been extensively used in various prediction and forecasting tasks in the literature. For example, Gaetz, Weinberg, Rzempoluck, and Jantzen (1998) analyzed EEG activity of the brain using a GRNN. Chtioui, Panigrahi, and Francl (1999) used a GRNN for leaf wetness prediction. Ibric, Jovanovi, Djuri, Paroj, and Solomun (2002) used a GRNN in the design of extended-release aspirin tablets. Cigizoglu (2005) employed a GRNN to forecast monthly water flow in Turkey. In this study, GRNN forecasting performance was found to be superior to the MLP and other statistical and stochastic methods. Kim and Lee (2005) used a
GRNN-based genetic algorithm to predict silicon oxynitride etching. Hanna, Ural, and Saygili
(2007) developed a GRNN model to predict seismic condition in sites susceptible to liquefaction. Shie (2008) used a hybrid method integrating a GRNN and a sequential quadratic programming method to determine an optimal parameter setting of an injectionmolding process.
There is an extensive literature in financial applications of NNs (e.g. [Harvey et al., 2000] and
[Kumar and Bhattacharya, 2006] ). For example, Cao, Leggio, and Schniederjans (2005) used
NNs to predict stock price movements for firms traded on the Shanghai stock exchange.
The authors compared the predictive power using linear models to the predictive power of the univariate and multivariate NN models. Results showed that NN models outperform the linear models. These results were statistically significant across the sample firms and indicated that
NN models are useful for stock price prediction. Kryzanowski, Galler, and Wright (1993) used NN models with historic accounting and macroeconomic data to identify stocks that will outperform the market. McGrath (2002) used market to book and price earnings ratios in

a NN model to rank stocks based on the likelihood estimates. (Ferson and Harvey, 1993) and (Kimoto et al., 1990) used a series of macroeconomic variables to capture predictable variation in stock price returns. McNelis (1996) used the Chilean stock market to predict returns in the Brazilian markets. Yumlu, Gurgen, and Okay (2005) used various NN architectures to model the performance of Istanbul stock exchange over the period 1990–
2002. Leigh, Hightower, and Modani (2005) used NN models and linear regression models to model the New York Stock Exchange Composite Index data for the period from 1981 to
1999. Results were robust and informative as to the role of trading volume in the stock market. Chen, Leung, and Daouk (2003) predicted the direction of return on market index of the Taiwan stock exchange using a probabilistic NN model. The results were then compared to the generalized methods of moments (GMM) with Kalman filter. From this literature survey we find that no previous studies have attempted to predict the movements of
KSE. In this study we aim to fill this research gap through the application of MLP and GRNN models to forecast the movements of the KSE.
2.1. Data
This study covers the time period of June 17, 2001 through November 30, 2003. Thus, the data set contained 612 data points in time series. This data set is quite similar in length to data sets in previous studies of similar nature. We used data for all listed companies traded on the
KSE. The data consists of daily closing prices. The source of the closing price data is the KSE
( www.Kuwaitse.com
). Table 1 shows descriptive statistics of the open and closing prices of the data used in this study.
Table 1.
Descriptive statistics of KSE opening and closing prices.
Open
Mean
Standard Error
Median
Mode
Close
2550.021242 Mean
36.7838047 Standard Error
2196.25 Median
1746.7 Mode
2553.124346
36.81831376
2194.55
1764.2
Standard Deviation 909.9810725 Standard Deviation 910.8347795
Sample Variance 828065.5523 Sample Variance 829619.9955
Kurtosis
Skewness
-0.63889504 Kurtosis
0.891027464 Skewness
-0.64911602
0.88577702
Range
Minimum
2945.2
1578.7
Range
Minimum
2942.8
1579

Open
Maximum
Count
4523.9
612
Close
Maximum
Count
4521.8
612
2.2. Multi-layer perceptron
MLP consists of sensory units that make up the input layer, one or more hidden layers of processing units (perceptrons), and one output layer of processing units (perceptrons). The
MLP performs a functional mapping from the input space to the output space. An MLP with a single hidden layer having H hidden units and a single output, y , implements mappings of the form
(1)
(2) where Z h
is the output of the h th hidden unit, W h
is the weight between the h th hidden and the output unit, and W
0
is the output bias. There are N sensory inputs, Xj . The j th input is weighted by an amount
β j
in the h th hidden unit. The output of an MLP is compared to a target output and an error is calculated. This error is back-propagated to the neural network and used to adjust the weights. This process aims at minimizing the mean square error between the network’s prediction output and the target output.
MLP was first developed to mimic the functioning of the brain. It consists of interconnected nodes referred to as processing elements that receive, process, and transmit information. MLP consists of three types of layers: the first layer is known as the input layer and corresponds to the problem input variables with one node for each input variable. The second layer is known as the hidden layer and is useful in capturing non-linear relationships among variables. The final layer is known as the output layer and corresponds to the classification being predicted
( Baranoff, Sager, & Shively, 2000 ). Fig. 1 represents the typical structure of MLP.

Fig. 1. General structure of MLP.
Full-size image (28K)
First of all the network has to be trained to produce the correct output with minimum error.
To achieve the minimum error the network first has to be trained until it produces a tolerable error. This is how the training is done. Input is fed to the input nodes, from here the middle layer nodes take the input value and start to process it. These values are processed based on the randomly allocated initial weight of the links. The input travels from one layer to another and every layer process the value based on the weights of its links. When the value finally reaches the output node, the actual output is compared with the expected output. The difference is calculated and it is propagated backwards, this is when the links adjust their weights. After the error has propagated all the way back to first layer of middle level nodes, the input is again fed to the input nodes. The cycle repeats and the weights are adjusted over and over again until the error is minimized. The key here is the weight of different links. The weights of the links will decide the output value.
The MLP is the most frequently used neural network technique in pattern recognition
( Bishop, 1999 ) and classification problems ( Sharda, 1994 ). However, numerous researchers document the disadvantages of the MLP approach. For example, Calderon and Cheh (2002) argue that the standard MLP network is subject to problems of local minima. Swicegood and Clark (2001) claim that there is no formal method of deriving a MLP network configuration for a given classification task. Thus, there is no direct method of finding the ultimate structure for modeling process. Consequently, the refining process can be lengthy, accomplished by iterative testing of various architectural parameters and keeping only the most successful structures. Wang (1995) argues that standard MLP provides unpredictable solutions in terms of classifying statistical data.
2.3. Generalized regression neural network
GRNN was devised by Specht (1991) , casting a statistical method of function approximation into a neural network form. The GRNN, like the MLP, is able to approximate any functional relationship between inputs and outputs ( Wasserman, 1993 ). Structurally, the
GRNN resembles the MLP. However, unlike the MLP, the GRNN does not require an estimate of the number of hidden units to be made before training can take place.
Furthermore, the GRNN differs from the classical MLP in that every weight is replaced by a distribution of weight which minimizes the chance of ending up in local minima. Therefore, no test and verification sets are required, and in principle all available data can be used for the training of the network ( Parojcic, Ibric, Djuric, Jovanovic, & Corrigan, 2005 ).

The GRNN is a method of estimating the joint probability density function (pdf) of x and y , giving only a training set. The estimated value is the most probable value of y and is defined by
(3)
The density function f ( x , y ) can be estimated from the training set using Parzen’s estimator
( Parzen, 1962 )
(4)
The probability estimate f ( x , y ) assigns a sample probability of width
σ
for each sample x i
and y i , and the probability estimate is the sum of these sample probabilities ( Specht, 1991 ).
Defining the scalar function
(5) and assessing the indicated integration yields the following:
(6)
The resulting regression (6) is directly applicable to problems involving numerical data.
The first hidden layer in the GRNN contains the radial units. A second hidden layer contains units that help to estimate the weighted average. This is a specialized procedure. Each output has a special unit assigned in this layer that forms the weighted sum for the corresponding output. To get the weighted average from the weighted sum, the weighted sum must be divided through by the sum of the weighting factors. A single special unit in the second layer calculates the latter value. The output layer then performs the actual divisions (using special division units). Hence, the second hidden layer always has exactly one more unit than the output layer. In regression problems, typically only a single output is estimated, and so the second hidden layer usually has two units. Fig. 2 shows the general structure of the GRNN.
The GRNN can be modified by assigning radial units that represent clusters rather than each individual training case: this reduces the size of the network and increases execution speed.
Centers can be assigned using any appropriate algorithm (i.e., sub-sampling, K -means or
Kohonen). Fig. 2 shows the General structure of the GRNN.

Full-size image (26K)
Fig. 2. General structure of the generalized regression neural network (GRNN).
3.1. MLP-based forecasting
There are many software packages available for analyzing MLP models. We chose
NeuroIntelligence package ( Alyuda Research Company, 2003 ). This software applies artificial intelligence techniques to automatically find the efficient MLP architecture.
Typically, the application of MLP requires a training data set and a testing data set ( Lek &
Guegan, 1999 ). The training data set is used to train the MLP and must have enough examples of data to be representative for the overall problem. The testing data set should be independent of the training set and is used to assess the classification/prediction accuracy of the MLP after training. Following Lim and Kirikoshi (2005) , an error back-propagation algorithm with weight updates occurring after each epoch was used for MLP training.
Fig. 3 shows the actual versus the fitted closing price values for the whole series using the quick propagation training algorithm. Fig. 4 shows the negative exponential decay in the error rate. As can be seen from Fig. 4 , the best network was obtained after around 500 epochs
(trials). This figure is quite similar to typical convergence of errors in MLP models (similar error graphs were obtained using other training algorithms).
Full-size image (104K)
Fig. 3. Actual versus fit using the quick propagation training algorithm.

Full-size image (47K)
Fig. 4. Error conversion and best network error.
Fig. 5 shows the actual versus the fitted closing price values for the whole series using the conjugate gradient descent training algorithm. Fig. 6 shows the actual versus the fitted closing price values for the whole series using the quasi-Newton training algorithm.
Full-size image (93K)
Fig. 5. Actual versus fit using the conjugate gradient descent training algorithm.
Full-size image (94K)
Fig. 6. Actual versus fit using the quasi-Newton training algorithm.
Based on the descriptive statistics of different training algorithms reported in Table 2 along with the plots of the training algorithms used, it seems that the quasi-Newton training algorithm produces less forecasting errors compared to the other two methods. This is also evident from the target vs. output graph and from the error dependence graph using the quasi-
Newton training algorithm ( Fig. 7 ).
Table 2.

MLP training algorithms major statistics.
Target Output AE
Quick propagation a
ARE
Mean 2534.744 2537.564 48.450 0.021
SD 908.299 890.698 31.966 0.015
Min 1579.000 1708.253 0.336 0.000
Max 4521.800 4369.447 155.094 0.084
Conjugate gradient descent b
Mean 2534.744 2535.314 26.598 0.010
SD 908.299 902.767 22.605 0.009
Min 1579.000 1653.851 0.334 0.000
Max 4521.800 4433.294 170.075 0.049
Quasi-Newton algorithm c
Mean 2534.744 2535.313 24.956 0.010
SD 908.299 903.801 21.542 0.008
Min 1579.000 1646.779 0.303 0.000
Max 4521.800 4442.282 166.271 0.046 a
Correlation = 0998; R
2
= 0.996. b Correlation = 0999; R 2 = 0.999. c
Correlation = 0999; R
2
= 0.999.
Full-size image (96K)
Fig. 7. Predicted Vs. actual and error dependence graph using the quasi-Newton training algorithm.

3.2. GRNN-based forecasting
There are many computer software packages available for building and analyzing NNs.
Because of its extensive capabilities for building networks based on a variety of training and learning methods, NeuralTools Professional package ( Palisade Corporation, 2005 ) was chosen to conduct GRNN analysis in this study. This software automatically scales all input data. Scaling involves mapping each variable to a range with minimum and maximum values of 0 and 1. NeuralTools Professional software uses a non-linear scaling function known as the
‘tanh’, which scales inputs to a (−1, 1) range. This function tends to squeeze data together at the low and high ends of the original data range. It may thus be helpful in reducing the effects of outliers ( Tam, Tong, Lau, & Chan 2005 ). Table 3 shows the basic properties of the GRNN model used in this study. [Fig. 8] and [Figure 9] show the error distribution and the predicted versus actual results using the GRNN network. These figures indicate the robustness of the
GRNN and its normality of error distributions.
Table 3.
GRNN architecture.
Summary
Net Information
Configuration GRNN Numeric Predictor
Training
Number of Cases
Number of Trials
490
61
Reason Stopped Auto-Stopped
% Bad Predictions (30% Tolerance) 0.0000%
Root Mean Square Error 24.51
Mean Absolute Error
Std. Deviation of Abs. Error
17.13
17.53
Testing
Number of Cases 122
% Bad Predictions (30% Tolerance) 0.0000%
Root Mean Square Error
Mean Absolute Error
32.12
22.30

Summary
Std. Deviation of Abs. Error
Data Set
Number of Rows
23.12
612
Full-size image (22K)
Fig. 8. GRNN histogram of residuals (training/testing).
Full-size image (25K)
Figure 9. GRNN predicted versus actual values (training/testing).
Our results confirm the theoretical work by Hecht-Nielson (1989) who has shown that NNs can learn input–output relationships to the point of making perfect forecasts with the data on which the network is trained. The good performance of the NN models in predicting KSE closing price movements can be traced to its inherent non-linearity. This makes an NN ideal for dealing with non-linear relations that may exist in the data. Thus, neuro-computational models are needed to better understand the inner dynamics of stock markets. Our results are also in line with the findings of other researchers who have investigated the performance of
NN compared to other traditional statistical techniques, such as regression analysis, discriminant analysis, and logistic regression analysis. For example, in a study of creditscoring models used in commercial and consumer lending decisions, Bensic, Sarlija, and
Zekic-Susac (2005) compared the performance of logistic regression, neural networks and decision trees. The PNN model produced the highest hit rate and the lowest type I error.
Similar findings have been reported in a study examining the performance of NN in

predicting bankruptcy ( Anandarajan, Lee, & Anandarajan 2001 ) and diagnosis of acute appendicitis ( Sakai et al., 2007 ).
Despite the significant contributions of this study, it suffers from a number of limitations.
First, despite the satisfactory performance of the NN models in this study, future research might improve the performance of the NN models used in this study by integrating fuzzy discriminant analysis and genetic algorithms (GA) with NN models. Mirmirani and Li (2004) pointed out that traditional algorithms search for optimal weight vectors for a neural network with a given architecture, while GA can yield an efficient exploration of the search space when the modeler has little apriori knowledge of the structure of problem domains.
Second, future research might use other NN architectures such as self-organizing maps
(SOMs) to classify movements in KSE. Due to the unsupervised character of their learning algorithm and the excellent visualization ability, SOMs have been recently used in myriad classification tasks. Examples include classifying cognitive performance in schizophrenic patients and healthy individuals ( Silver & Shmoish, 2008 ), mutual funds classification
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Expert Systems with Applications
Volume 37, Issue 9 , September 2010, Pages 6302-6309
Forecasting model of global stock index by stochastic time effective neural network
Zhe Liao a and Jun Wang , a , a
Institute of Financial Mathematics and Financial Engineering, College of Science, Beijing
Jiaotong University, Beijing 100044, PR China
Available online 8 June 2009.
In this paper, we investigate the statistical properties of the fluctuations of the Chinese
Stock Index, and we study the statistical properties of HSI, DJI, IXIC and SP500 by comparison. According to the theory of artificial neural networks, a stochastic time effective function is introduced in the forecasting model of the indices in the present paper, which gives an improved neural network – the stochastic time effective neural network model. In this model, a promising data mining technique in machine learning has been proposed to uncover the predictive relationships of numerous financial and economic variables. We suppose that the investors decide their investment positions by analyzing the historical data on the stock market, and the historical data are given weights depending on their time, in detail, the nearer the time of the historical data is to the present, the stronger impact the data have on the predictive model, and we also introduce the Brownian motion in order to make the model have the effect of random movement while maintaining the original trend. In the last part of the paper, we test the forecasting performance of the model by using different volatility parameters and we show some results of the analysis for the fluctuations of the global stock indices using the model.
Keywords: Brownian motion; Stochastic time effective function; Data analysis; Neural network ; Returns; Predict

1. Introduction
2. Methodology for stochastic time effective function
2.1. Introduction of stochastic time effective function
2.2. Procedure followed in the stochastic time effective model
3. Experiment analysis
3.1. Selection and preprocessing of data
3.2. Training stochastic time effective neural network
4. Conclusions
Acknowledgements
References
Recently, some progress has been made in the work done on the fluctuations of the Chinese stock market, for example, see (Ji and Wang, 2007) and (Li and Wang, 2006) . In the present paper, we investigate the statistical properties of the fluctuations of the indices of the Chinese
Stock Exchange, and study the statistical properties of HSI, DJI, IXIC and SP500 by comparison. A predictive model of the stock prices is constructed using the theory of artificial neural networks and a stochastic time effective function, further the data from the
Chinese stock markets are analyzed in the model. China has Stock A and Stock B in the stock markets. The indices of Stock A and Stock B play an important role in the
Chinese stock markets, and the database of the indices is from the website www.sse.com.cn
.
Recently, the properties of the fluctuations of the stock markets have been studied in many research fields, for example, see (Azoff, 1994) , (Nakajima, 2000) , (Pino et al., 2008) , (Shtub and Versano, 1999) and (Wang, 2007) . Artificial neural networks are one of the technologies that have made great progress in the study of the stock markets. Usually stock prices can be seen as a random time sequence with noise, artificial neural networks, as large-scale parallel processing nonlinear systems that depend on their own intrinsic link data, provide methods and techniques that can approximate any nonlinear continuous function, without a priori assumptions about the nature of the generating process, see (Pino et al., 2008) and (Shtub and Versano, 1999) . They have good self-learning ability, a strong antijamming capability,and have been widely used in the financial fields such as stock prices, profits, exchange rate and risk analysis and prediction. Although the historical data have a great influence on the investors’ positions, the degree of impact of the data depends on the date at which they occurr (or time), we get a high level effect of the data when they are very near the current state. Furthermore, we also introduce the Brownian motion in the model (see
Wang, 2007 ), in order to make the model have the effect of random movement while maintaining the original trend. We test the forecasting performance of the model by using different volatility parameters, and we show some results of the analysis for the fluctuations of the global stock index using the model. In this work, the forecasting model is developed to estimate the level of returns on Stock A Index of China. In Section 3 , the results show that the forecasting model can predict the index behavior better in a short time interval than in a long time interval, and show the different performances of HSI, DJI, IXIC and SP500 by comparison, see (Abhyankar et al., 1997) , (Austin et al., 1997) and (Balvers et al., 1990) .

In this paper, forecasting based on neural network involves two major steps, data preprocessing and structure design. In the pretreatment stage, the collected data should be normalized and properly adjusted, in order to reduce the impact of noise in the stock markets. At the design stage, different data training sets, validations and data processing will cause the different results, see (Breen et al., 1990) and (Campbell, 1987) . In stock markets, the environment and behavior of the markets may change greatly, for example see the Chinese stock markets in 2007. As a result, the data in the data training set should be time-variant, reflecting the different behavior patterns of the markets at different times. If all the data are used to train the network equivalently, the network system may not be consistent with the development of the stock markets, see (Chenoweth and Obradovic, 1996) , (Chenoweth and
Obradovic, 1996) , (Cybenko, 1989) and (Demuth and Beale, 1998) . Especially in the current
Chinese stock markets, stock market trading rules and management systems are changing rapidly, for example, the daily price limit (now 10%), shareholding reformation, the direct investment of Hong Kong stock markets, the reorganization of A share, B share and H share, and the establishment of financial derivatives such as futures and options. Therefore, using the historical data of the past it is difficult to reflect the current Chinese stock markets’ development. However, if only the recent data are selected, a lot of useful information will be lost which the early data hold. In the financial model of the present paper, a promising data mining technique in machine learning is proposed to uncover the predictive relationships of numerous financial and economic variables. Considering the abovementioned financial situation, this paper presents an improved neural network model, the stochastic time effective series neural network model: each historical datum is given a weight depending on the time it occurs in the model, and we also use the probability density functions to classify the various variables from the training samples, see (Desai and Bharati,
1998) , (Duda et al., 2001) and (Elton and Gruber, 1991) .
2.1. Introduction of stochastic time effective function
In this section, first we describe a three-layer BP neural network model (see Azoff, 1994 ), which is shown in Fig. 1 . The neural network model includes three layers: input layer, hidden layer and output layer. In the model, the proper number of the hidden layer nodes requires validation techniques to avoid under-fitting (too few neurons) and over-fitting (too many neurons). Generally, too many neurons in the hidden layers, and, hence, too many connections, produce a neural network that memorizes the data and lacks the ability to generalize.
Full-size image (27K)
Fig. 1. Three-layer neural network.

Suppose that a three-layer neural network has neurons, and for any fixed neuron n ( n =1,2,…, N ), the model has the following structure: let { x i
( n ): i =1,2,…, p } denote the set of input of neurons, { y j
( n ): j
=1,2,…, m } denote the set of output of the hidden layer neurons; V i
is the weight that connects the node i in the input layer neurons to the node j in the hidden layer,
W j
is the weight that connects the node j in the hidden layer neurons to the node k in the output layer; and { o k
( n ): k
=1,2,…, q } denote the set of output of neurons. Then the output value for a unit is given by the following function where are the neural thresholds, is the sigmoid activation function.
Let T k
( n ) be the actual value of the data sets, then the error of the corresponding neuron k to the output is defined as
ε k
= T k
o k
. In this paper, the error of the output is defined as then the error of the sample n ( n
=1,2,…,
N ) is defined as
, where ( t ) is the stochastic time effective function. Now we define ( t ) as follows: where
τ
(>0) is the time strength coefficient, t
1
is the current time or the time of the newest data in the data set, and t n
is an arbitrary time point in the data set.
μ
( t ) is the drift function (or the trend term), σ ( t ) is the volatility function, and B ( t ) is the standard Brownian motion
( Wang, 2007 ). The stochastic time effective function implies that the recent information has a stronger effect for the investors than the old information. In detail, the nearer the events happened, the greater the investors and markets affected. And the impact of data follows the time exponential decay, see (Nakajima, 2000) and (Pino et al., 2008) . Then the total error of all the data training sets in the set output layer with the stochastic time effective function is defined as
2.2. Procedure followed in the stochastic time effective model
Note that the training objective of stochastic time effective neural network is to modify the weights so as to minimize the error between the network’s prediction and the actual target.

When all the training data are data (that is t
1
= t n
), the stochastic time effective neural network is the general neural network model. In Fig. 2 , the training algorithms procedures of stochastic time effective neural network are shown, which are as follows:
Step 1: Train a stochastic time effective neural network by choosing five kinds of stock prices in the input layer: daily opening price, daily closing price, daily highest price, daily lowest price and daily trade volume, and one price of the stock prices in the output layer: the closing price of the next trading day. Then set the connective weights, and input the training data sets.
Step 2: At the beginning of data processing, connective weights V i
and W j
follow the uniform distribution on (-1,1), and let the neural threshold
θ k
,
θ j
be 0.
Step 3: Introduce the stochastic time effective function t
in the error function e ( n , t ). Choose different volatility parameters. Give the transfer function from the input layer to the hidden layer and the transfer function from the hidden layer to the output layer.
Step 4: Establish an error acceptable model and set pre-set minimum error. If the output error is below pre-set minimum error, go to Step 6, otherwise go to Step 5.
Step 5: Modify the connective weights: Calculate backward for the node in the output layer:
Calculate
δ
backward for the node in the hidden layer: where o ( n ) is the output of the neuron n , T ( n ) is the actual value of the neuron n in the data sets, o ( n )[1o ( n )] is the derivative of the sigmoid activation function and h
′
is each of the nodes which connect with the node h and in the next hidden layer after node h . Modify the weights from the layer to the previous layer: where
η
is the learning step, which usually takes constants between 0 and 1.
Step 6: Output the predictive value.

Full-size image (66K)
Fig. 2. Procedure followed in the stochastic time effective model.
3.1. Selection and preprocessing of data
In this paper, we select the data of Stock A Index (SAI) and Stock B Index (SBI) of
China for each trading day in a 18-year period, that is from December 19, 1990 to June 7,
2008, which are from the Shanghai and Shenzhen Stock Exchange. And we also choose the data of HSI, DJI, IXIC and SP500 by contrast. First, we study the statistical properties of the returns of the index using the stochastic time effective neural network model, and then study the relativity between the Chinese stock indices and foreign country stock indices.
Fig. 3 presents the related coefficients between SAI, SBI, HSI, DJI, IXIC and SP500. In Fig.
3 , we can see that the related coefficients between SAI and SBI, HSI, DJI, IXIC and SP500 are 0.885, 0.629, 0.463, 0.108, 0.329; and the related coefficients between SBI and SAI, HSI,
DJI, IXIC and SP500 are 0.885, 0.817, 0.688, 0.458, 0.623.
Full-size image (60K)
Fig. 3. Related coefficients.
In the model of this paper, we suppose that the network inputs include five kinds of data, daily opening price, daily closing price, daily highest price, daily lowest price and daily trade volume, and the network outputs include the closing price of the next trading day.

Fig. 4 presents the plot of the time sequence log return of SAI, SBI, IXIC and SP500. We denote the price sequence of SAI, SBI, IXIC and SP500 of time then R ( t ) denotes the logarithm of return rate, respectively, by t by ,
In Fig. 4 , we can see that the prices of the index fluctuate wildly, and this indicates that there is a big noise in the data that causes difficulty in forecasting. Thus, we should go through data preprocessing before forecasting, so the data are normalized as follows:
Similarly to (5), the normalized values of the above-mentioned five kinds of data can also be given.
Full-size image (107K)
Fig. 4. The plot of log returns of SAI (a), SBI (b), IXIC (c), and SP500 (d).
3.2. Training stochastic time effective neural network
Data sets are divided into two parts, data training set and data testing set. We collect the data of SAI in 1990–2006 as the training set and the data of SAI in 2007–2008 as the testing set.
According to the procedures of the three-layer network introduced in Section 2 , the number of neural nodes in the input layer is 5, the number of neural nodes in the hidden layer is
20 and the number of neural nodes in the output layer is 1, and the threshold of the maximum training cycles is 100 and the threshold of the minimum error is 0.0001. We take the μ ( t ) (the drift parameter) and σ ( t ) (the volatility parameter) ( μ ( t ), σ ( t )) to be (1, 0), (1, 1) and (1, 2). While (
μ
( t ),
σ
( t )) is (1, 0), the model has the effect of only time effective function; while (
μ
( t ),
σ
( t )) is (1, 1), the model has the effect of both time effective function and normal randomization; and while ( μ ( t ), σ ( t )) is (1, 2), the model has the effect of intensive randomization.
In Table 1 the parts of training errors of different trading dates for (a) SAI, (b) SBI, (c) HSI,
(d) DJI, (e) IXIC and (f) SP500 are given. It can be clearly seen that during the years 1991 and 1992, the relative error is larger than that in the year 2007, this clearly shows the effect of time effective function. Furthermore, the gap between the relative error of SAI and SBI is much more greater than the relative error of the foreign stock markets. So we can conclude that the value of the historical data in the foreign stock markets is greater than that in the

Chinese stock markets, this means that the Chinese stock markets fluctuate more sharply than the foreign markets.
Table 1.
Comparison of errors of different dates for SAI, SBI, HSI, DJI, IXIC and SP500 (( μ ( t ), σ ( t )) is
(1, 1)).
Time
(a) SAI
Actual Predictive Error
90/12/20 104.39 144.57
91/05/10 108.53 149.06
06/09/20 1820.8 1830.3
−0.384
−0.373
−0.005
−0.0001
07/02/13 2973.4 2974.0
(b) SBI
92/02/24 124.65 128.88 −0.034
93/12/24 93.14 87.64 0.059
05/07/11 59.88 60.8308 −0.02
06/11/15 108.27 106.37 0.017
(c) HSI
01/07/31 12316.7 12123.8 0.016
02/01/29 11014.2 10712.2 0.027
06/10/11 17862.8 17849.5 0.001
07/09/26 26430.2 26230.8 0.008
(d) DJI
91/12/23 3022.6 2948.1 0.025
−0.014
92/07/07 3295.2 3340.8
06/07/26 11102.5 11125.5
−0.002
07/12/28 13365.9 134129 −0.004

Time
(e) IXIC
Actual Predictive Error
91/11/08 548.08
92/03/05 621.97
540.3
635.6
06/03/07 2268.38 2290.0
0.014
−0.022
−0.010
0.005 07/05/29 2572.06 2559.6
(f) SP500
91/01/03 321.91
92/02/04 413.85
339.06
408.38
−0.053
0.013
06/10/31 1377.94 1376.05 0.001
07/02/05 1446.99 1448.47 −0.001
Fig. 5 shows the fluctuations of the time sequence of relative errors during the years 1991 to
2007 for the prices of SAI, SBI, HSI, DJI, IXIC and SP500. In these plots, 0 represents the farthest data to the current date, and the larger t (date) represents the date that is nearer to the current date. Fig. 5 also clearly indicates that the stochastic time effective neural network model can be realized by assigning different weights to the data of different times. Time sequence of relative errors of (b) SBI and (d) DJI in Fig. 5 clearly reflects the model of randomization by the effect of the Brownian motion. And we also conclude that SBI is similar to DJI, and that SAI is similar to HIS. By coincidence, the conclusion is supported by the relativity analysis shown in Fig. 3 .
Full-size image (137K)
Fig. 5. Relative errors of SAI, SBI, HSI, DJI, IXIC,and SP500 ((
μ
( t ),
σ
( t ))) is (1, 1).

In order to test the validity of the volatility parameter
σ
( t ), we take
σ
( t ) to be 0 or 2. If
σ
( t ) is
0, then the model has no effect of randomization but has only the effects of the time effective function. If σ ( t ) is 2, then the model has intensive effect of the wild fluctuation.
In Table 2 , the predictive values and relative errors of SAI by the different values
σ
( t ) are given. This shows that the relative error is the smallest when
σ
( t )=1 (almost below 1%) and the relative error is the largest when σ ( t )=0 (almost over 10%). So we can conclude that adding in the Brownian motion in this financial model is propitious for increasing the precision of prediction. However, the volatility parameter should not be too large, otherwise it will increase the error of the prediction.
Table 2.
Predictive values and errors of time effective neural network model of SAI by different
σ
( t ).
Time
(a) σ ( t )=1
Actual Predictive Error
08/06/03 3605.859 3627.168
−0.0059
08/06/04 3535.809 3604.064 −0.0193
08/06/05 3516.219 3530.171
−0.0039
08/06/06 3493.189 3507.505 −0.0040
(b)
σ
( t )=0
08/06/03 3605.859 3655.273 −0.014
08/06/04 3535.809 3689.849
−0.044
08/06/05 3516.219 3559.069
−0.012
08/06/06 3493.189 3502.137
−0.003
(c) σ ( t )=2
08/06/03 3605.859 4060.890
−0.126
08/06/04 3535.809 4034.350 −0.141
08/06/05 3516.219 4001.172
−0.138
08/06/06 3493.189 3968.767
−0.136

In Table 3 , I stands for the average relative error, II stands for the average relative error of the first 1000 days in the data sets, and III stands for the latest 100 days in the data sets. In Table
3 , the time effective function in the model is clearly expressed. Take the relative error in SAI at
σ
( t )=1 for example, the average relative error is 2.6%; the average relative error of the first
1000 days is 6.88% and the average relative error of the latest 100 days is 1.3%. So the latest data are more valuable than the historical data of the past in the stock market.
Table 3.
Average relative errors for SAI, SBI, DJI, IXIC, HSI and SP500 of different
σ
( t ).
SAI SBI DJI IXIC HSI SP500
(a) σ ( t )=1
I 0.026 0.0212 0.0388 0.0152 0.0104 0.0074
II 0.0688 0.0262 0.0697 0.0366 0.0112 0.0077
III 0.013 0.0141 0.0058 0.0078 0.0109 0.0078
(b)
σ
( t )=0
I 0.0766 0.0284 0.0128 0.2143 0.0322 0.0345
II 0.1637 0.0391 0.201 0.7247 0.0379 0.045
III 0.0514 0.0148 0.0103 0.0328 0.0247 0.0366
(c)
σ
( t )=2
I 0.2458 0.1041 0.043 0.1443 0.0636 0.0897
II 0.8686 0.1276 0.083 0.3466 0.0875 0.2256
III 0.0353 0.0985 0.034 0.1041 0.0347 0.0265
In Table 3 , the global stock indices errors of the different values
σ
( t ) are also given. Take the relative error in SAI for example, while
σ
( t )=1, the average relative error is 2.6%; while
σ
( t )=0, the average relative error is 7.66%; while
σ
( t )=2, the average relative error is 24.58%.
So this implies that the appropriate volatility parameter is beneficial for the prediction of the stochastic time effective neural network, and that it is widely applicable to the global stock markets.

In Fig. 6 , the comparison between predictive values and the actual values in SAI, SBI, HSI and IXIC by the stochastic time effective neural network model is shown.
Full-size image (71K)
Fig. 6. Comparison of the predictive values and actual values.
In Fig. 7 , by using the linear regression method, we compare the predictive values of stochastic time effective neural network model with the actual values in SAI (a), SBI (b),
NSDK (c) and SP500 (d). Through the regression analysis, different linear equations in SAI
(a), SBI (b), NSDK (c) and SP500 (d) are obtained. Take the linear equation in SAI (a) for example, the linear equation is
And the correlation coefficient R =0.9984. The linear equation for SBI (b) is
And the correlation coefficient R =0.9978; the linear equation for NSDK(c) is
And the correlation coefficient R =0.9988 and the linear equation for SP500 (d) is
And the correlation coefficient R =0.9991. So we test the accuracy of the results of the forecasting from another angle.

Full-size image (72K)
Fig. 7. Regression of the predictive values and actual values.
This paper introduces a new stochastic time effective function to model a stochastic time effective neural network model. The effectiveness of the model has been analyzed by performing a numerical experiment on the data of SAI, SBI, HSI, DJI, IXIC and SP500, and the validity of the volatility parameters of the Brownian motion is tested. Further, the present paper shows some predictive results on the global stock indices using the stochastic time effective neural network model.
The authors were supported in part by National Natural Science Foundation of China Grant
No. 70771006 and by BJTU Foundation No. 2006XM044.
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Wang, 2007 J. Wang, Stochastic process and its application in finance, Tsinghua University
Press and Beijing Jiaotong University Press (2007).
Corresponding author. Tel./fax: +86 10 51682867.
Expert Systems with Applications
Volume 37, Issue 1 , January 2010, Pages 834-841
Improving returns on stock investment through neural network selection
Tong-Seng Quah , , a and Bobby Srinivasan b a
School of Electrical and Electronic Engineering, Nanyang Technological University,
Nanyang Avenue, Singapore 639798, Singapore b School of Accountancy and Business, Nanyang Technological University, Nanyang Avenue,
Singapore 639798, Singapore

Available online 1 November 1999.
The Artificial Neural Network (ANN) is a technique that is heavily researched and used in applications for engineering and scientific fields for various purposes ranging from control systems to artificial intelligence. Its generalization powers have not only received admiration from the engineering and scientific fields, but in recent years, the finance researchers and practitioners are taking an interest in the application of ANN. Bankruptcy prediction, debtrisk assessment and security market applications are the three areas that are heavily researched in the finance arena. The results, this far, have been encouraging as ANN displays better generalization power as compared to conventional statistical tools or benchmark.
With such intensive research and proven ability of the ANN in the area of security market application and the growing importance of the role of equity securities in Singapore, it has motivated the conceptual development of this project in using the ANN in stock selection.
With its proven generalization ability, the ANN is able to infer from historical patterns the characteristics of performing stocks. The performance of stocks is reflective of their profitability and the quality of management of the underlying company. Such information is reflected in financial and technical variables. As such, the ANN is used as a tool to uncover the intricate relationships between the performance of stocks and the related financial and technical variables. Historical data such as financial variables (inputs) and performance of the stock (output) are used in this ANN application. Experimental results obtained this far have been very encouraging.
Author Keywords: Technical analysis; Fundamental analysis; Neural network ; Economic factors; Political factors; Firm specific factors
1. Introduction
2. Application of neural network in financial and commercial domains
3. Neural architecture
3.1. Select the Appropriate Algorithm
3.2. Architecture of ANN
3.3. Selection of the Learning Rule
3.4. Selection of the Appropriate Learning Rates and Momentum
4. Variables selection
5. Experiment
5.1. Research design
5.2. Design 1 (Basic System)
5.3. Design 2 (Moving Window System)
5.4. Results
6. Conclusion and future works
References

With the growing importance in the role of equities to both the international and local investors, the selection of attractive stocks is of utmost importance to ensure a good return.
Therefore, a reliable tool in the selection process can be of great assistance to these investors.
An effective and efficient tool/system gives the investor the competitive edge over others as he/she can identify the performing stocks with minimum effort.
Trading strategies, rules and concepts based on fundamental and technical analysis, have been devised by both academics and practitioners in assisting the investors in their decision making process. Innovative investors opt to employ information technology to improve the efficiency in the process. This is done through transforming trading strategies into computer known languages so as to exploit the logical processing power of the computer. This greatly reduces the time and effort in short-listing the list of attractive stocks.
In this age where information technology is dominant, such computerized rule based expert systems have great limitations that will affect its effectiveness and efficiency. However, with the significant advancement in the field of Artificial Neural Network (ANN), these limitations have found a solution. In this research, the generalization ability of the ANN is being harnessed in creating an effective and efficient tool for stock selection. Results of the researches in this field have so far been very encouraging.
Research developments in Bankruptcy prediction have showed that the ANN performs better than the conventional statistical methods such as Discriminant Analysis and Logistic
Regression. Alici (1996) , using UK data, shows that ANN, with the architecture consisting of
3 multi layer Preceptron (28 financial inputs, seven hidden neurons and two output neurons) has an average of 71.38%(76.07%) for the failed firms (healthy firms) using neural network. On the other hand, the Discriminant Analysis and Logistic Regression have both achieved 60.12%(71.43%) and 65.29%(71.07%) for the failed firms (Healthy Firms). The feasibility of applying ANN into bankruptcy was also studied by Raghupathi, Schkade and
Raju, 1991 and Odom , and many others.
Salchenberger, Cinar and Lash (1992) use the ANN for classifying failure pertaining to the
Savings and Loans (S&Ls) organizations in US. In the Debt Risk Assessment applications,
Dutta and Shekhar (1993) use neural network to classify bonds.
The generalization ability of the ANN is also extended to commodity trading (copper). Robles and Naylor (1996) are able to show that the ANN outperforms the traditional Weighted
Moving Average rule and a “buy and hold” strategy. In equity,
Gencay and Stengos (1996) show that the ANN outperforms the (linear model) in the identical research methodology. The ratios of the average MSPE of the testing model and benchmark are 0.961 and 1 for the ANN and the , respectively.
Burgess and Refenes (1996) prove that, the ANN has better generalization ability than
Ordinary Least Square in predicting daily return of the applied in predicting the trend of Italian stock market ( index. Neural network is also
). Chinetti and Rossignoli
(1993) attempted to use technical variables to build an accurate prediction system to forecast the timing of buying and selling for the index. Westheider (1994) studied the predictive power of the ANN on stock index returns. He uses economic and fundamental variables to predict monthly, quarterly and yearly returns.

The computer software selected for the training and testing the network is Neural Planner version 3.71. This software was programmed by Stephen Wolstenholme. It is an ANN simulator strictly designed for only one Back Propagation learning algorithm.
There are four major issues in the selection of the appropriate network:
1. Select the Appropriate Algorithm.
2. Architecture of the ANN.
3. Selection of the Learning Rule.
4. Selection of the Appropriate Learning Rates and Momentum.
3.1. Select the Appropriate Algorithm
Since the sole purpose of this project is to identify the top performing stocks and the historical data that are used for the training process will have a known outcome (whether it is considered Top performer or otherwise), algorithms designed for supervised learning are ideal. Among the available algorithms, the Back Propagation algorithm designed by
Rumelhart, Hinton and Williams (1986) is the most suitable as it is being intensively tested in
Finance. Moreover, it is recognized as a good algorithm for generalization purposes.
3.2. Architecture of ANN
Architecture, in this context, refers to the entire structural design of the ANN (Input Layer,
Hidden Layer and Output Layer). It involves determining the appropriate number of neurons required for each layer and also the appropriate number of layers within the Hidden Layer.
The logic of the Back Propagation method is the Hidden Layer. The Hidden Layer can be considered as the crux of the Back Propagation method. This is because hidden layer can extract higher level features and facilitate generalization, if the input vectors have low level features of a problem domain or if the output/input relationship is complex. The fewer the hidden units, the better is the ANN able to generalize. It is important not to over-fit the ANN with large number of hidden units than required until it can memorize the data. This is because the nature of the hidden units is like a storage device. It learns noise present in the training set, as well as the key structures. No generalization ability can be expected in these.
This is undesirable as it does not have much explanatory power in a different situation/environment.
3.3. Selection of the Learning Rule
The learning rule is the rule that the network will follow in its error reducing process. This is to facilitate the derivation of the relationships between the input(s) and output(s). The generalized delta rule developed by Rumelhart et al. (1986) is used in the calculations of weights. This particular rule is selected because it is heavily used and proven effective in the
Finance researches.
3.4. Selection of the Appropriate Learning Rates and Momentum

The Learning Rates and Momentum are parameters in the learning rule that aid the convergence of error, so as to arrive at the appropriate weights that are representative of the existing relationships between the input(s) and the output(s).
As for the appropriate learning rate and momentum to use, the Software has a feature that can determine appropriate learning rate and momentum for the network to start training with. This function is known as “Smart Start”. Once this function is activated, the network will be tested using different values of learning rates and momentum to find a combination that yields the lowest average error after a single learning cycle. These are the optimum starting values as using these rates improve error converging process, thus require less processing time.
Another attractive feature is that the software comes with an “Auto Decay” function that can be enabled or disabled. This function automatically adjusts the learning rates and momentum to enable a faster and more accurate convergence. In this function, the software will sample the average error periodically, and if it is higher than the previous sample then the learning rate is reduced by 1%. The momentum is “decayed” using the same method but the sampling rate is half of that used for the learning rate. If both the learning rate and momentum decay are enabled then the momentum will decay slower than the learning rate.
In general cases, where these features are not available, a high learning rate and momentum
(e.g. 0.9 for both the Learning Rates and Momentum) are recommended as the network will converge at a faster rate than when lower figures are used. However, too high a Learning Rate and Momentum will cause the error to oscillate and thus prevent the converging process.
Therefore, the choice of learning rate and momentum are dependent on the structure of the data and the objective of using the ANN.
In general, financial variables chosen are constrained by data availability. They are chosen first on the significant influences over stock returns based on past literature searches and practitioners’ opinions and then on the availability of such data. Most of the data used in this research is provided by Credit Lyonnais Securities (Singapore) Pte Ltd. Stock prices are extracted from a financial database called .
Broadly, factors that can affect stock prices can be classified into three categories: economic factors, political factors and firm/ stock specific factors. Economic factors have significant influence on the returns of individual stock as well as stock index in general as they possess significant impact on the growth and earnings’ prospects of the underlying companies thus affecting the valuation and returns. Moreover, economic variables also have significant influence on the liquidity of the stock market. Some of the economic variables used are: inflation rates, employment figures and producers’ price index.
Many researchers have found that it is difficult to account for more than one third of the monthly variations in individual stock returns on the basis of systematic economic influences, and shown that political factors could help to explain some of the missing variations. Political stability is vital to the existence of business activities and the main driving force in building a strong and stable economy. Therefore, it is only natural that political factors such as fiscal policies, budget surplus/deficit etc do have effects on stock price movements.

Firm specific factors affect only individual stock return. For example, financial ratios and some technical information that affects the return structure of specific stocks, such as yield factors, growth factors, momentum factors, risk factors and liquidity factors. As far as stock selection is concerned, firm specific factors constitute to important considerations as it is these factors that determine whether a firm is a bright start or a dim light in the industry. Such firm specific factors can be classified into five major categories:
1. Yield factors: these include “historical P/E ratio” and “Prospective P/E ratio”. The former is computed by price/earning per share. The latter is derived by price/consensus earnings per share estimate. Another variable is the “cashflow yield”, which is basically price/operating cashflow of the latest 12 months.
2. Liquidity factors: the most important variable is the “market capitalization”, which is determined by “price of share×number of shares outstanding”.
3. Risk factors: the representative variable is the “earning per share uncertainty”, which is defined as “percentage deviation about the median EPS estimates”.
4. Growth factors: basically, this means the “return on equity (ROE)”, and is computed by
“net profit after tax before extraordinary items/shareholders equity”.
5. Momentum factors: a proxy is derived by “average of the price appreciation over the quarter with half of its weights on the last month and remaining weights being distributed equally in the remaining two months”.
The inputs of the neural network stock selection system are the above seven inputs and the output is the return differences between the stock and the market return (excess returns).
This is to enable the neural network to establish the relationships between inputs and the output (excess returns).
The training data set will include all data available until the quarter before the testing quarter.
This is to ensure that the latest changes in the relationship of the inputs and the output are being captured in the training process.
The quarterly data required by the project are generally stock prices and financial variables
(inputs to the ANN stock selection system) from 1/1/93 to 31/12/96.
Stock Prices, which are used to calculate stock returns, are extracted from the Financial
Database called . These stock returns, adjusted for dividends, stock splits and bonus issues, will be used as output in the ANN training process.
One unique feature of this research is that Prospective P/E ratio, measured as
(Price/Consensus Earnings Per Share Estimate), is being used as a forecasting variable. This variable has not received much attention in Financial Research. Prospective P/E ratio is used among practitioners as it can reflect the perceived value of stock with respect to EPS
(Earnings per share) expectations. It is used as a value indicator, which has similar implications as that of the Historical P/E ratio. As such, a low Prospective P/E suggests that the stock is undervalued with respect to its future earnings and vice versa. With its

explanatory power, Prospective P/E ratio qualifies as an input in the stock selection system.
Data on Earnings per Share estimates, which is used for the calculation of EPS Uncertainty and Prospective P/E ratio, is available in . This is a compilation on EPS estimates and recommendations are put forward by financial analysts. The coverage has estimates from countries over the Asia Pacific Rim as from January 1993.
5.1. Research design
The purpose of this ANN stock selection system is to select stocks that are top performers from the market ( Stock that outperformed the market by 5%) and to avoid selecting under performers ( Stocks that underperformed the market by 5%). More importantly, the aim is to beat the market benchmark (Quarterly return on the index) on a portfolio basis.
This ANN stock selection system is a quarterly portfolio re-balancing strategy whereby it will select stocks in the beginning of the quarter and performance (the return of the portfolio) will be assessed at the end of the quarter.
5.2. Design 1 (Basic System)
In this research design, the sample used for training consists of stocks that outperformed and underperformed the market quarterly by 5% from 1/1/93 to 30/6/95.
The inputs of the ANN stock selection system are the seven inputs chosen in Section 4 and the output will be the return differences between the stock and the market return (excess returns). This is to enable the ANN to establish the relationships between inputs and the output (excess returns).
The training data set will include all data available until the quarter before the testing quarter . This is to ensure that the latest changes in the relationship of the inputs and the output are being captured in the training process.
The generalization ability of the ANN in selecting top performing stocks and whether the system can perform is tested across time. The data used for the selection process are from the third quarter of 1995(1/7/95–30/9/95), the fourth quarter of 1995 (1/10/95–31/12/95), the first quarter of 1996 (1/1/1996–31/3/1996), the second quarter of 1996 (1/4/1996–30/6/1996), the third quarter of 1996 (1/7/1996–30/9/1996), the fourth quarter of 1996 (1/10/1996–
31/12/1996). The limited test duration is constrained by the data availability.
The testing inputs are being injected into the system and the predicted output will be calculated using the established weights. After which, the top 25 stocks with the highest output value will be selected to form a portfolio of stocks. These 25 stocks are the top 25 stocks recommended for purchase at the beginning of the quarter. Generalization ability of the ANN will be determined by the performance of the portfolio, measured by excess returns over the market as well as the % of top performers in the portfolio as compared to the benchmark portfolio (Testing Portfolio) at the end of the month.
5.3. Design 2 (Moving Window System)
The Basic System is constrained by meeting the minimum sample size required for training process. However, this second design is going to forgo the recommended minimum sample

size and introduce a Moving Window concept. This is to analyze the ANN ability to perform under a restricted sample size environment.
The inputs and output variables are identical with that of the Basic System, but the training and testing samples are different. The Moving Window System uses three quarters as training sample and the subsequent quarter as the testing sample. The selection criterion is also identical with that of the Basic System in research Design 1.
5.4. Results
The ANN is made to train with 10 000 and 15 000 cycles. The reason for using these numbers of cycles for training is because the error converging is generally slow after 10 000 thus suggesting adequate training. Moreover, it does not converge beyond 15 000. This is an indicator that the network is over-trained.
The training of four hidden neurons for 10 000 cycles takes approximately 1.5 h, eight hidden neurons takes about 3 h and the most complex (14 hidden neurons) took about 6 h on a
Pentium 100 MHz PC. As for those architectures that require 15 000 cycles, it usually takes about 1.5 times the time it takes to train the network for 10 000 cycles.
The results of the Basic Stock Selection System based on the training and testing schedules mentioned are presented in two forms: (1) the excess return format and (2) the percentage of the top performers in the selected portfolio. These two techniques will be used to assess the performance and generalization ability of ANN.
Testing results show that the ANN is able to “beat” the market overtime, as shown by positive compounded excess returns achieved consistently throughout all architectures and training cycles. This implies that the ANN can generalize relationships overtime. Even at individual quarters’ level, the relationships between the inputs and the output established by the training process is proven successful by “beating” the market in 6 out of 8 possible quarters which is a reasonable 75%. ( Fig. 1 .)
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Fig. 1. Graphical presentation of excess returns for 15 000 cycles.
The Basic Stock System has consistently performed better than the testing portfolio overtime. This is evident by the fact that the selected portfolios have higher percentage of top performing stocks (above 5%) than the testing portfolio overtime. This ability has also enabled the network to better the performance of the market ( index) presented earlier. ( Fig. 2 and Fig. 3 .)

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Fig. 2. Performance of portfolios with % of stocks with actual return of 5% above market for 10 000 cycles.
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Fig. 3. Graphical presentation of excess returns of portfolio.
The Moving Window Selection System is designed to test the generalization power of the
ANN in an environment with limited data.
The generalization ability of the ANN is again evident in the Moving Window Stock
Selection System as it outperformed the Testing portfolio in 9 out of 13 testing quarters
(69.23%). This can be seen in the graphical presentation that the line representing the selected portfolio is above the line representing testing portfolio most of the time. ( Fig. 4 .) Moreover, the compounded excess returns and the annualized compounded excess returns are better than that of the testing portfolio by two times over. The selected portfolios have outperformed the market 10 out of 13 (76.92%) testing quarters and excess returns (127.48% for the 13 quarters and 36.5% for the Annualized compounded return), which proved its consistent performance over the market ( index) overtime. ( Fig. 5 .)
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Fig. 4. % of Top performers in the portfolio - moving window stock selection system.
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Fig. 5. Actual returns - moving window stock selection system.

The Selected Portfolios have outperformed the Testing Portfolio in nine (69.23%) and equal the performance in one occasion. This further proves the generalization ability of the ANN.
Moreover, the ability to avoid selecting undesirable stocks is also evident by the fact that the selected portfolios have less of this kind of stocks than the testing portfolio in 10 out of
13 occasions (76.92%).
From the experimental results, the portfolio of the Selected portfolios outperformed the
Testing and Market portfolios in terms of compounded actual returns overtime. The reason is because the Selected portfolios outperform the two categories of portfolios in most of the testing quarters thus achieving better overall position at the end of the testing period.
The ANN has displayed its generalization ability in this particular application. This is evident through the ability to single out performing stock counters and having excess returns in the
Basic Stock Selection System overtime. Moreover, neural network has also showed its ability in deriving relationships in a constrained environment in the Moving Window Stock
Selection System thus making it even more attractive for applications in the field of Finance.
This paper is largely constrained by the availability of data. Therefore, when more data is available, performance of the neural networks can be better assessed in the various kinds of market conditions, such as bull, bear, high inflation, low inflation or even political conditions in which all have different impact on stocks.
Also, as more powerful neural architectures are being discovered by researchers on a fast pace, it is good to repeat the experiments using several architectures and compare the results.
The best performance structure may than be employed.
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Corresponding author; email: itsquah@ntu.edu.sg
Expert Systems with Applications
Volume 17, Issue 4 , November 1999, Pages 295-301