TRADITIOAL AD IDEX TRACKIG METHODS FOR PORTFOLIO COSTRUCTIO BY MEAS OF EURAL ETWORKS A. Zorin and A. Borisov Keywords: neural networks, portfolio management, back-propagation, time series analysis 1. Introduction eural networks have seen an explosion of interest over the last few years, and are being successfully applied across an extraordinary range of problem domains, in areas as diverse as finance, medicine, engineering, geology and physics. Indeed, wherever that there are problems of prediction, classification or control, neural networks are being introduced. eural networks are very sophisticated modelling techniques, capable of modelling extremely complex functions. In particular, neural networks are non-linear. For many years linear modelling has been the commonly used technique in most modelling domains, since linear models had well-known optimisation strategies. Where the linear approximation was not valid (which was frequently the case) the models suffered accordingly. eural networks also keep in check the curse of dimensionality problem, which bedevils attempts to model non-linear functions with large numbers of variables. Forecasting the behaviour of a given system follows two approaches which are trying to predict future values of time series by extracting knowledge from the past. One common approach is to relate the changes in one time series to other phenomena in the economy. The other approach also adopted in this paper states that stock exchange indices embody all the knowledge that is required to predict future behaviour. The experiments presented here are based on the assumption, that the past values of the indices contain all the knowledge that is needed to predict future behaviour. Portfolio management involves the construction of a portfolio of securities (stocks, bonds etc.) that maximises the investor s utility. The construction of such a portfolio consists of two main stages. In the first stage, the investor evaluates the securities available in the market. There are two main factors, which must be taken into account in the portfolio construction process: the expected return and risk. The biggest problem is to forecast the behaviour of securities (expected returns). Using superior neural network abilities can solve this problem. When all the information is available, one can use simple techniques for portfolio construction. Another approach for portfolio construction is the so called Index Tracking method. It is a kind of passive portfolio management which attempts to match the performance of a theoretical portfolio, such as Dow Jones Riga Stock Exchange index (DJ RSE), as closely as possible. In this particular case we can also use a neural network to predict the index future value. Both of these methods are discussed in this paper. 2. The data set The training set for the first portfolio construction method consists of seven enterprises stock returns that are quoted at the Riga Stock Exchange (Rigas kugu buvetava (RKB), Rigas
transporta flote (RTF), Ventspils afta (VF), Rigas Balzams (RBL), Grindex (GRX), Latvijas Gaze (LGZ), Daugavpils pievadkezu rupnica (DPR)). Our portfolio can therefore be constructed from those seven stocks. The data set covers the period from December 3, 2000 to December 10, 2001 on daily updates for the 253 trading days. The test set consists of the seven above mentioned companies stock returns data for December 10-29, 2001 on daily updates for the 14 trading days. The test sets are only used to evaluate neural network prediction accuracy and are not included in the learning phase. The training and the test sets for the Index Tracking method consist of Dow Jones RSE index data for the same period as in the first case. The results of neural network performance are presented for each of the stock (the results are summed up in table 1). All neural network simulations were completed using STATISTICA eural etworks 4.0 software. The data set pre-processing techniques used in the experiments are not the same for different time series. They will be presented with the network architectures and learning parameters. The learning algorithm is error back-propagation. 3. Index Tracking The Index Tracking approach is used by fund managers when they do not feel confident enough of out-performing the market, and are content to follow the average performance [1]. Matching the performance of an index can be performed in different ways. The most common one, also adopted in this paper, is full replication, in which an investment is made in every constituent of the index proportional to its market share. The objective was to forecast, at every time step, the succeeding sales value based only on the information contained in the time series. o external indicators have been included as additional inputs to the neural network. The multilayer perceptron with error backpropagation was chosen due to the positive experience with this type of neural network [4]. The comprehensive description of neural networks and back-propagation algorithm can be found in [5]. The data set pre-processing includes normalising the data. To this end, a linear transformation into the range [0;1] was applied. The data were split into training and test sets. The training data were used for neural network training and the test data for validating the network s generalisation ability. The network architecture at the input and output levels is largely determined by the application. The most common method of identifying regularities within a time series contaminated by noise is windowing (see [3] for further reference). After a number of experiments with different network architectures and learning parameters the final neural network that has the best results was found to be as follows: learning constant and momentum have the same value 0.1 (here it is better to use relatively small values because they give minimal learning error); squashing function is defined by (1): c f x) k 1 e ( (1) network architecture is 45-25-1, which means 45 input layer neurones, 25 hidden layer neurones and 1 output neurone (a less complicated architecture cannot deal with the given kind of time series); weight initialisation range from 0 to 0.01; Tx
training set data has normalised in range [0; 1]. Fig. 1 and Fig. 2 show the neural network performance on the training and the test sets. Dow Jones RSE 165 150 135 120 105 90 75 60 01.04.00 06.28.00 12.15.00 06.07.01 11.23.01 Actual Forecast Fig. 1. eural network performance on the training set (MSE = 8.93; MAPE = 2.15%). Index value 142 141 140 139 138 137 136 10.12.01 17.01.01 23.01.01 Actual Forecast Fig. 2. eural network performance on the test set (MSE = 1.18; MAPE = 0.83%). The results are quite good on the training as well as on the test set. The MAPE on the test set is below 1% and our trained neural network can predict the main changes in the data rather precisely. Using prediction results obtained from neural network we can compute the expected return of the portfolio, which consists of nine enterprises stocks. The expected return will be 0.07%. The computed return of real data is 1.17%. In both cases this return is not big enough to use this kind of portfolio construction technique. We therefore need to use something different. 4. Forecasting expected returns of the stocks This section presents forecasting results for only five enterprises. The results depend very much on the input layer size. The authors try to keep MAPE below 5% and to use less complicated network architectures (below 40 input units).
After a number of the experiments the final parameters of the neural network with the best performance for the RKB stock returns are as follows: learning constant and momentum have the same value 0.1 (here it is better to use relatively small values because they give a minimal learning error); squashing function - sigmoid; network architecture is 32-15-1, which means 32 input layer neurones, 15 hidden layer neurones and 1 output neurone (less complicated architecture cannot deal with the given kind of time series); weight initialisation range from 0 to 0.01; training set data has normalised in range (0; 1). Fig. 3 gives neural network performance on training set (solid line gives actual values and dashed line shows forecast). 0.22 0.2 0.18 0.16 0.14 0.12 0.1 1.12.2000. 5.04.2001 7.08.2001 4.12.200 Fig.3. eural network performance on the RKB training set (MSE=0.00005; MAPE=3.34%). This neural network implementation in the RKB stock price forecasting for two weeks ahead gives MAPE=1.35%. The neural network and data set pre-processing parameters for the RTF stock returns are as follows: learning constant and momentum have the same value 0.1; squashing function - sigmoid; network architecture is 37-17-1; weight initialisation range from 0 to 0.01; training set data has normalised in range (0; 1); training data set has transformed using simple 5 th order moving average. The neural network parameters for the VF are as follows: learning constant and momentum have the same value 0.1; squashing function - sigmoid; momentum - 0.3; network architecture is 38-15-1; weight initialisation range from 0 to 0.01; training set data has normalised in range (0; 1). Figures 4 and 5 give neural network performance on RTF and VF training sets.
0.09 0.08 0.07 0.06 0.05 0.04 1.12.2000. 5.04.2001 7.08.2001 4.12.2001 Fig.4. eural network performance on the RTF training set (MSE=0.000017; MAPE=4.93%). 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 1.12.2000. 5.04.2001 7.08.2001 4.12.2001 Fig.5. eural network performance on the VF training set (MSE=0.00047; MAPE=2.06%). The neural network in Fig.4. forecasts the RTF stock price for two weeks ahead with MAPE=4.41%, whereas the network in the Fig.5. forecasts the VF returns with MAPE=1.66%. Fig.6 shows these forecasts. 0.08 0.075 0.07 0.065 0.06 0.055 0.05 0.045 0.04 10.12.2001 17.12.2001 22.12.2001 0.75 0.7 0.65 0.6 0.55 10.12.2001 17.12.2001 22.12.2001 a) b) Fig. 6. eural network performance on the test sets:
a) RTF test set with MAPE=4.41% and b) VF test set with MAPE=1.66%. The final network parameters and data set pre-processing features for GRX are: learning constant and momentum have the same value 0.1; squashing function - sigmoid; momentum - 0.3; network architecture is 35-15-1; weight initialisation range from 0 to 0.01. Fig.7 presents neural network performances on GRX training sets. 0.55 0.5 0.45 0.4 0.35 1.12.2000. 5.04.2001 7.08.2001 4.12.2001 Fig.7. eural network performance on the GRX training set (MSE=0.0024; MAPE=2.41%). The neural network in the Fig.7 forecasts the test set values of GRX with MAPE=2.41%. There is no need to show all the figures of the companies stocks. The neural network predictions on the test sets will come in useful in the next section, where the portfolio construction process is presented. 5. Portfolio construction The portfolio theory assumes that the investor s choice can be represented by a utility function implicitly used in making investment decisions. The goal is therefore to maximise this utility function. The decision-maker utility maximisation can be formulated as a twoobjective problem: maximisation of the expected return of the portfolio and minimisation of the risk involved. These goals cannot be achieved at the same time, so the investor should find an alternative between the two. There are many ways to measure the expected return and risk. The most common is the mean of the time series (expected return) and the standard deviation of the return (risk). When the investor has both of these measures, he can construct the portfolio. The expected returns are neural network predictions and the risk measures are standard deviations of the stock prices for the period from December 3, 2000 to December 10, 2001. There are many different approaches to solving the portfolio problem. Elton and Gruber [2] present solution techniques for different types of financial situations. We take the first situation, in which short sales are allowed and riskless lending and borrowing is possible. In this case there is only one portfolio of risky assets that is preferred to all other portfolios. The efficient set is determined by finding that portfolio with the greatest ratio of excess return (expected return minus risk-free rate) to standard deviation that satisfies
the condition that the sum of the proportions invested in the assets equals 1. In equation form we need to maximise the objective function given by (2): subject to constraint R P R F X i i1 Stating the expected return and standard deviation of return in the general form yields i1 X i1 2 i 2 i P 1 X ( Ri i R i1 j1 where - X i and X j are fractions of capital invested in stock i and j; - i is standard deviation of stock return i; - R F is risk-free rate; - ij is the covariance between stock returns i and j; - R i is the expected return of stock i. F X ) i X j ij (2) (3) Table 1 gives information required to construct the portfolio. Input data for portfolio construction Table 1 RKB RTF VF RBL DPR GRX LGZ Expected return 0.50-18.83 6.00 2.43 1.66-3.27 2.63 Standard deviation 12.97 11.82 8.63 4.64 9.52 8.73 30.10 Correlation coefficients RKB 1.00 0.11 0.51-0.43 0.24 0.61 0.28 RTF 1.00 0.32 0.03 0.14 0.06 0.07 VF 1.00-0.30 0.37 0.40 0.08 RBL 1.00-0.03-0.01 0.07 DPR 1.00 0.04-0.38 GRX 1.00 0.52 LGZ 1.00 Risk-free rate is 2.3%. As a result we have constructed a portfolio with these proportions invested in stocks: RKB - 0 RTF - 0 VF - 0.615 RBL - 0.366 DPR - 0 GRX - 0 LGZ - 0.019 The expected return on the portfolio is R P = 0.615 * 6 + 0.366 * 2.43 + 0.019 * 2.63 = 4.63%
The standard deviation of the return on the portfolio is P = [0.615 2 * 8.63 2 + 0.366 2 * 4.64 2 + 0.019 2 * 30.1 2 +2 * 0.615 * 0.366 * 8.63 * 4.64 * (- 0.3) + + 2 * 0.165 * 0.019 * 8.63 * 30.1 * 0.08 + 2 * 0.366 * 0.019 * 4.64 * 30.1 * 0.07] 0.5 = 5.16% The portfolio constructed has a respectable expected return of 4.63% with 5.16% standard deviation. In real market conditions our portfolio would give a 5.76% profit. 6. Conclusions We have presented neural network implementation possibility in a portfolio management task. The index tracking approach is not good enough as compared with to the second and most commonly used portfolio construction methods. The neural network prediction is the key factor in our research. However, there are many problems, which have not been solved yet. eural networks are black boxes and it is often hard to explain their behaviour. The next step will be to try to implement the neural network technology in the problem domain using another prediction approach, based on the assumption that changes in one timeseries are related to other phenomena in the economy. There is a possibility to use neural networks not only in the prediction but also in the portfolio construction procedure. That is the subject for our future work. References 1. Baestaens D. E., Van den Bergh W. M. Tracking the Amsterdam Stock Index Using eural etworks // In: eural etworks in Capital Markets, 1995- Vol. 5. P. 149-161. 2. Elton E. J., Gruber M. J. Modern Portfolio Theory and Investment Analysis // In: ew York, John Whiley and Sons, 1995-716 p. 3. Refenes A.., Azema-Barac M., Chen L., Karoussos S. A. Currency Exchange Rate Prediction and eural etwork Design Strategies // In: Springer-Verlag, London Limited, 1993 - p. 46 58. 4. Refenes A.., Zapranis A., Francis G. Stock Performance Modelling Using eural etworks // In: eural etworks, 1994 - Vol. 7. o 2. P. 357 388. 5. Zurada J. M. Introduction to Artificial eural Systems // In: St. Paul: West Publishing Company, 1992-684 p. Alexey Zorin PhD Student, Decision Support Systems Group, Institute of Information Technology, Riga Technical University, Kalku Str.1, Riga LV-1658, Latvia. E-mail: alex@ru.lv Arkady Borisov Prof., Dr. hab. sc. comp., Riga Technical University, Head of Department, Kalku Str.1, Riga LV-1658, Latvia. E-mail: aborisov@egle.cs.rtu.lv Zorins A., Borisovs A. TRADICIOL U IDEKSA SEKOŠAAS METODE PORTFEA VEIDOŠAAI AR EIROU TKLIEM. Šaj rakst ir pardts neironu tklu pielietojums akciju portfea veidošanas proces, kas sastv no diviem soiem. Pirmaj sol investors novrt un izvlas tirg pieejamus vrtspaprus. Otraj sol tiek pieemts lmums
par to, kdu kapitla dau ieguldt attiecg akcij, tdjdi izveidojot portfeu. Rakst ir apskatta neironu tklu izmantošana akciju portfea veidošanas pirmaj sol. Otrais solis ir realizts saska ar klasisko finansu teoriju. Papildus ir apskatts ar indeksa sekošanas metode portfea veidošanai. Zorin A. and Borisov A. TRADITIOAL AD IDEX TRACKIG METHODS FOR PORTFOLIO COSTRUCTIO BY MEAS OF EURAL ETWORK. This paper presents neural network application in portfolio construction, which consists of two major steps. In the first step the decision-maker (investor) evaluates and selects the securities that are available in the market. In the second step the investor decides on the amount of the capital available that should be invested in each security, thus constructing a portfolio. The neural network implementation in the first step of portfolio management process is presented in this paper. The second stage of the portfolio construction procedure is as in the financial theory. The paper also mentions an alternative index tracking portfolio construction method..,.....,,....