Hybrid method of using neural networks and ARMA model to forecast value at risk (VAR) in the Chinese stock market

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1 Hybrid method of using neural networks and ARMA model to forecast value at risk (VAR) in the Chinese stock market Hae-Ching Chang * Department of Business Administration National Cheng Kung University No.1, University Road, Tainan City 701 Taiwan (R.O.C.) Jian-Hsin Chou Graduate Institute of Management National Kaohsiung First University of Science and Technology No 2, Jhuoyue Rd., Nanzih District, Kaohsiung City 811 Taiwan (R.O.C.) Cheng-Te Chen Department of Business Administration National Cheng Kung University No.1, University Road, Tainan City 701 Taiwan (R.O.C.) Chin-Shan Hsieh Graduate Institute of Management National Kaohsiung First University of Science and Technology No 2, Jhuoyue Rd., Nanzih District, Kaohsiung City 811 Taiwan (R.O.C.) Abstract Conventional VAR (Value at Risk) estimation includes historical simulation, variance/covariance, and the Monte Carlo simulation method. This study is the first to present a hybrid method of estimating VAR, combining ARMA and Neural Network. Empirical results demonstrate that the hybrid method obtained superior results to the conventional method in estimating VAR. In terms of accuracy, both the conventional and hybrid methods performed well when applied to the Chinese stock market, with the only poorly performing method being the HS method when applied to the Shanghai A share market. In terms of conservativeness, the hybrid method was superior to the conventional method, while in terms of efficiency, the hybrid method outperformed the conventional method when applied to the Shenzhen stock market. Thus, using hybrid Neural Network with the ARMA method to compare with the conventional method in estimating VAR offers certain advantages. Consequently, this study suggests that investors use the hybrid method to estimate VAR. Keywords: Neural Networks, ARMA Model, Value at Risk * Associate Professor, chc5728@ms54.hinet.net Professor, jian@ccms.nkfust.edu.tw Ph.D. candidate at the National Cheng Kung University and the Lecturer of Far East University, nike@cc.feu.edu.tw Ph.D. candidate at the National Kaohsiung First University of Science and Technology and the Lecturer of Kao Yuan University, u @ccms.nkfust.edu.tw

2 2 1. Introduction According to Jorion [22], VAR summarizes the worst expected loss over a target horizon and within a given confidence interval. Estimating VAR requires determining the two parameters of holding period and confidence level. Duffie & Pan [11] developed the idea of applying VAR to compare the risks of stocks or investment portfolios, market price and historic prices, and to compare risks between different markets. Dowd [10] demonstrated that VAR is a simple numeric value for representing the total risks of an investment portfolio in different markets. Most financial asset returns exhibit a fat-tail distribution, but conventional VAR estimation models assume that returns are normal distribution. Fama [13] proposed that the distribution of stock price and stock return usually follows a leptokurtic and fat-tailed distribution. Notably, Hull & White [20] studied the statistical characteristics of asset returns and discovered that normal distribution was associated with systematic errors, particularly in situations involving returns with a fat-tailed distribution. The conventional VAR (Value at Risk) estimation method includes the historical simulation, variance/covariance, and Monte Carlo simulation methods. Recently, Neural Network simulation has been extremely popular in forecasting stock prices, but has almost never been applied to VAR estimation. On the other hand, does stock price data usually exhibits time series correlation, Neural Network simulation forecasting of stock prices cannot be influenced by time series correlation. This study proposes a hybrid model, which combines ARMA and Neural Network, for estimating VAR. To improve the forecasting efficiency of Neural Network simulation, the first step is to adopt the ARMA model to determine significant independent variables. The second step then involves employing these variables as Neural Network input variables. This study has the following objectives: (1) to propose a hybrid model, combining ARMA and Neural Network to estimate VAR; (2) to compare the forecasting performance between the conventional and hybrid models in terms of estimating VAR. 2. Conventional Method 2.1. Historical Simulation Method The historical simulation method directly observes the empirical distribution of investment portfolio return. This method first identifies the assets included in the portfolio and observes the historic data of each asset over a specific period (Stambaugh [29]). Historical data are applied to the weight of the current

3 3 portfolio to simulate possible portfolio changes during the observation period. Portfolio VAR can be obtained given a certain confidence level. This method is easy to understand and explain since it makes no statistical assumptions about the distribution of returns (Badík [3]). The main difficulty in implementing historical simulation is that it requires a time series of the relevant market factors covering on the last N days. This can be problematic in situations were reliable data is not readily available. Another disadvantage is that the N days considered in the calculation may be atypical owing to market events Variance/Covariance Method The simplest and perhaps most widely used method of modeling changes in portfolio value is the variance/covariance method popularized by RiskMetrics (Glasserman, Heidelberger & Shahabuddin [16]). The most notable feature of variance/covariance method is its assumption that future returns on assets are normal distribution, and that investment portfolio gain or loss is also normal distribution in order to simplify VAR estimation. Thus, the VAR over a specific holding period and at a certain confidence level can be directly estimated based on investment portfolio standard deviation. VAR is calculated by multiplying the distributed standard deviation by the standardized normal distribution z-value, so when α is fixed (1-α is confidence coeficient), VAR is affected only by the standard deviation. If we know the variation and correlated coefficient of asset returns, we can estimate the VAR of the investment portfolio. Britten-Jones and Schaefer [7] and Glasserman et al. [15] are good references for the delta-gamma approach. The easy availability of the necessary data makes VAR computation relatively easy, and hence it is the most widely used method (Badík [3]). Despite its ease of calculation, it may be difficult to explain to senior management due to its reliance on the statistical properties of normal distribution when used to calculate. Another significant drawback is the assumption of normality of asset returns, which does not always hold Monte Carlo Simulation method The Monte Carlo simulation method assumes that the volatility of investment portfolio returns follows a certain stochastic process, meaning the paths of stock prices can be computationally simulated hundreds, thousands, or even millions of times to establish portfolio returns distribution and estimate VAR (Stambaugh [29]). Monte Carlo simulation is basically an empirical method based on the law of large numbers, with a larger number of iterations leading to an average close to the theoretical value.

4 4 Monte Carlo simulation is a natural alternative for handling nonlinear portfolios (Jin & Zhang [21]), and is designed to simulate the stochastic process of asset price and risk factors, with each simulation yielding a period-end asset value. Iterated simulations can be used to produce the distribution of period-end portfolio values can be created. The simulated distribution can then be used to derive the VAR with the given confidence level. In terms of precision, it is perhaps the most effective of all methods, particularly in situations involving more complex instruments (Badík [3]). It is also extremely flexible, since it makes no definite assumptions regarding asset returns. The Monte Carlo simulation procedure can be quite complex and time consuming, requiring expensive intellectual and technological skills Empirical studies of VAR Alexander & Leigh [2] used the simple weighted average, exponential weighted moving average, and GARCH models to estimate VAR. Maximum likelihood estimate (MLE), root mean square error (RMSE), back-testing, and forward testing methods were applied for the model testing, and demonstrated that the exponential weighted moving average method tended to underestimate VAR, and GARCH did not differ significantly in terms of statistical verification but yielded a more correct 99% VAR estimate. Hull & White [20] pointed out that under Historical Simulation, better VAR estimates could be obtained when the daily change of market factors estimated using the GARCH or index weighted moving average models was used to adjust historical changes. Papageorgiou & Paskov [28] compared the speed and accuracy of the Monte Carlo and quasi-monte Carlo methods by estimating 34 European stock index options and foreign exchange call options. Moreover, Chang, Chen & Hsieh [9] used the percentile statistics method to estimate value at risk, and demonstrated that the percentile of cluster method was more accurate and conservative than the percentile of statistics method. 3. ARMA model The Autoregressive moving average (ARMA) model is commonly used to estimate VAR (Mapa & Beronilla [27]). The ARMA model was popularized by Box and Jenkins [6], and thus is also known as the Box-Jenkins model, and is typically applied to time series data. Given a time series of data X t, the ARMA model can predict future values for a time series of data. In practice, Berkowitz & Brien [4] employed the ARMA model to forecast VAR at commercial banks. Cabedo and Moya [8] proposed a hybrid approach that combines the historical

5 5 simulation approach with the ARMA model to forecast VAR, and showed that this combined approach significantly outperforms the HS approach alone. The power of ARMA models lies in their ability to incorporate both autoregressive terms and moving average terms. Which is usually then referred to as the ARMA(p,q) model where p is the order of the autoregressive part and q is the order of the moving average part. The notation AR(p) indicates the autoregressive model of order p. The AR(p) model is written X t c p i 1 X (1) i t i t where i are the parameters of the model, c is a constant and t is an error term. The notation MA(q) indicates the moving average model of order q: X t q (2) t i 1 i t i where the i are the parameters of the model and the t are the error terms. The notation ARMA(p, q) indicates the model with p autoregressive terms and q moving average terms. This model contains the AR(p) and MA(q) models, p q i t i i 1 i 1 X t X (3) t i t i 4. Neural Network Simulation Method Neural Network (NN) is an artificial intelligence (AI) methods Kai & Wenhua [24] Neural Network is a highly non-linear, large scale, continuous, time-based and dynamic system. The knowledge of NN is stored in the relationship among numerous nodes in the form of a weigh matrix. This method has become extremely important in making stock market predictions. Backpropagation neural networks comprise input, hidden and output layers (Fig. 1). Input layer hidden layer output layer Fig. 1. the structure of Neural Network A common advantage shared by NN applications is their ability to deal with uncertain and robust data. Therefore, NN can be efficiently used in stock markets to predict either stock prices or returns. NN is particularly flexible and efficient

6 6 in situations when certain data are unavailable. Previous studies have demonstrated that NN outperforms classical forecasting and statistical methods. The combination of several NN can obtain extremely accurate value predictions, because the combined methods can focus on different data set characteristics that are important for calculating output (Zekic [30]; Afolabi & Olude [1]). Kaastra & Boyd [23] used backpropagation neural networks and ARIMA to forecast futures trading volume, and found that neural network forecasting ability is also benchmarked relative to the ARIMA model. Furthermore, Franses & Griensven [14] investigated the performance of artificial neural networks (ANNs) for technical trading rules for forecasting daily exchange rates, showed that ANNs perform well, and frequently outperform linear models. 5. Data and Methodology The research subject comprised daily data from the Chinese stock market, including the Shenzhen and Shanghai A and B share index, with the data period spanning 2000/01/04~2005/12/30, and the estimation period spanning 2006/01/04~2006/12/29, employing the rolling-window method, and with data being source from TEJ (Taiwan Economic Journal Co. Ltd.) data on the Chinese security market. Previous studies of VAR all used conventional methods (history simulation, variance/covariance, and Monte Carlo simulation method) to estimate VAR. This study instead uses the hybrid Neural Network and ARMA models for VAR estimation to compare the performance between the conventional and hybrid methods Hybrid Method This study proposes a hybrid model for estimating VAR that combines ARMA and Neural Network. Adopting the ARMA model requires determining which independent variables are significant then employing these significant variables as Neural Network input variables. A hybrid model incorporating the variance/covariance method (NN_VCV), or the Monte Carlo simulation method (NN_MCS), according to a hybrid model for forecast result, which combine variance/covariance (VCV) or Monte Carlo simulation (MCS) to estimate VAR. ARMA model dependent variable: close price, independent variables: 5-day Bias, 5-day MA (Moving Average), 6-day RSI (Relative Strength Index), RSV (Raw Stochastic Value), volume. The Neural Network simulation method is based on Back-propagation Network (BPN), which is used to identify the relationship between input and output variables. The input layer comprises five neurons, including 5-day Bias, 5-

7 7 day MA, 6-day RSI, RSV and volume. The output layer is close price. The training data comprise 1,430 records, while the testing data comprise 241 records Assessment methods for VAR models The hybrid and convention methods were compared in terms of conservativeness, accuracy, and efficiency (Engel & Gizyck [12]), using a confidence level of 95%. Kupiec testing method was also used in Goorbergh & Vlaar [17], Billio & Pelizzon [5], Guermat & Harris [18], Lin, Chang Chien & Chen [26], and Chang, Chen & Hsieh [9]. 6. Empirical Results Table 1 lists the results of parameter estimation for the ARMA model, and reveals that all variables were significant at P-Value <0.1, SSR= 55,535.58, AIC=6.5098, Schwarz s SBC=6.5393, R 2 =0.9971, Adjusted R 2 =0.9971, F- statistic=70,864.35, Durbin-Watson stat= Table 1. Estimate parameter result of ARMA model Variable Coefficient Std. Error t-statistic P-Value. C day Bias *** 5-day MA(Moving Average) *** 6-day RSI (Relative Strength Index) * RSV (Raw Stochastic Value) * VOLUME ** AR(1) *** MA(1) * *: P-Value <0.1, **: P-Value <0.05, ***: P-Value <0.001 The simplest generalized autoregressive conditional heteroskedasticity (GARCH) model of dynamic variance can be written as t 1 R t t, with 1 (4) Table 2 lists the result of parameter estimation for the GARCH(1,1) model in different stock markets. These parameters (α, β, ω) are used to estimate the daily variance for VAR forecasting.

8 8 Table 2. Estimate parameter result of GARCH model Shenzhen stock index Shanghai A share index Shanghai B share index α β ω α+β MLE Figure 2 displays the Neural Network convergent graphs in three stock markets, and reveals that they are convergent. Figure 3 illustrates the expected and forecast values for the Neural Network scatter plot. The Shenzhen and Shanghai B share index have better fitness. Table 3 shows that the correlation coefficient and mean square of error (MSE) for the Neural Network estimate have correlation coefficient exceeding 99% in all markets. (a) Shenzhen stock index (b) Shanghai A share index (c) Shanghai B share index Fig. 2. Neural Network convergent graphs for different stock markets Expected value Forecasting value of NN Expected value Forecasting value of NN Expected value Forecasting value of NN (a) Shenzhen stock index (b) Shanghai A share index (c) Shanghai B share index Fig. 3. Neural Network scatter plot for different stock markets

9 9 Table 3. Correlation coefficient and MSE of Neural Network estimate value Market Correlation coefficient MSE Shenzhen stock index Shanghai A share index Shanghai B share index Figure 4 shows the VAR derived by various methods given maximum tolerable loss of 5% for the Shenzhen stock market. Methods used to forecast VAR included Neural Network with variance/covariance methodology (NN_VCV), Neural Network with Monte Carlo simulation method (NN_MCS), Monte Carlo simulation method (MCS), and variance-covariance methodology (VCV). The historical simulation method (HS) yielded excessively volatile VAR. 4.0% 3.0% 2.0% 1.0% 0.0% Return of Loss -1.0% -2.0% -3.0% -4.0% -5.0% -6.0% Minus Return HS VCV MCS NN_VCV NN_MCS -7.0% -8.0% 1/06 1/06 2/06 3/06 4/06 5/06 6/06 7/06 8/06 9/06 10/06 11/06 12/06 Date (2006) Fig. 4. VAR estimated by each model for the Shenzhen stock index Figure 5 shows that VAR for the Shanghai A share market is analogous to that for the Shenzhen stock market, because the NN_VCV, NN_MCS, MCS, and VCV models yield similar VAR forecasts. The HS method produced lower VAR than the above methods.

10 10 4.0% 3.0% 2.0% 1.0% Return of Loss 0.0% -1.0% -2.0% -3.0% -4.0% -5.0% -6.0% Minus Return HS VCV MCS NN_VCV NN_MCS -7.0% -8.0% 1/06 1/06 2/06 3/06 4/06 5/06 6/06 7/06 8/06 9/06 10/06 11/06 12/06 Date (2006) Fig. 5. VAR estimated by each model for the Shanghai A share index Figure 6 shows the VAR for the Shanghai B share market, and reveals that the NN_VCV and NN_MCS methods are less volatile than the HS, MCS, and VCV methods. 8.0% 7.0% 6.0% 5.0% 4.0% 3.0% Return of Loss 2.0% 1.0% 0.0% -1.0% -2.0% -3.0% -4.0% Minus Return HS VCV MCS NN_VCV NN_MCS -5.0% -6.0% -7.0% 1/06 1/06 2/06 3/06 4/06 5/06 6/06 7/06 8/06 9/06 10/06 11/06 12/06 Date (2006) Fig. 6. VAR estimated by each model for the Shanghai B share index The Kupiec [25] testing method was used to assess the accuracy of the VAR models. Table 4 lists the accumulated failures of each VAR model. Table 5 lists

11 11 the Kupiec [25] likelihood ratio (LR PF ) calculated using the accumulated failures. The result was further compared using χ 2 from Table 6 to verify the model accuracy. Table 4. Accumulated failures of each model Conventional method Hybrid Method Market HS VCV MCS NN_VCV NN_MCS Shenzhen stock index Shanghai A share index Shanghai B share index Table 5. Kupiec (1995) Likelihood Ratio of each model (p=0.05) Method Conventional method Hybrid Method T Shenzhen Shanghai Shanghai B stock index A share index share index HS * VCV MCS NN_VCV NN_MCS Given failure rate p=0.05, regardless of whether the conventional or hybrid method was used, for all three stock markets the LR PF were lower than the Chisquare value, which display the accuracy, just only HS method in Shanghai A share index had a LR PF (5.864) higher than χ 2 (1,α=0.05). Table 6. Test of χ2 distribution χ 2 (1,α=0.01) χ 2 (1,α=0.05) χ 2 (1,α=0.1) The Root Mean Squared Relative Bias (RMSRB) proposed by Hendricks [19] was adopted to assess VAR model conservativeness. RMSRB is a negative indicator, with smaller RMSRB indicating greater conservativeness.

12 12 Table 7. RMSRB of each model Conventional method Hybrid Method Market HS VCV MCS NN_VCV NN_MCS Shenzhen stock index Shanghai A share index Shanghai B share index Table 7 lists the RMSRB of each VAR model. The hybrid method is superior to any of the conventional methods, because of in all three stock markets had a higher level of conservativeness, so the hybrid method was a more conservative method. Mean Relative Scaled Bias (MRSB) is used to assess the efficiency of VAR models. MRSB can be used to identify the VAR model with the smallest VAR given some theoretical failure rate. MRSB is a negative indicator, while smaller MRSB indicates higher efficiency. Table 8. MRSB of each model Conventional method Hybrid Method Market HS VCV MCS NN_VCV NN_MCS Shenzhen stock index Shanghai A share index Shanghai B share index Table 8 shows the MRSB for each VAR model, and reveals that models based on the hybrid method were more efficient than those based on the conventional method when applied to Shenzhen stock market. In other markets, there was no significant efficiency difference. 7. Conclusion Recently Neural Network simulation has been extremely popular for forecasting stock prices, but has rarely been used for VAR estimation. On the other hand, stock price data usually exhibits time series correlation, Neural Network simulation forecasting of stock prices cannot be influenced by time series correlation. This study proposes a hybrid model, which combines ARMA and Neural Network, for estimating VAR. Empirical results demonstrated that the hybrid method outperformed the conventional method in estimating VAR. In terms of accuracy, both the conventional and hybrid methods performed well when applied to all markets, with the only exception being the HS method

13 13 when applied to the Shanghai A share market. In terms of conservativeness, the hybrid method was superior to the conventional method. In terms of efficiency, the hybrid method outperformed the conventional method in the Shenzhen stock market, but in all other markets the two methods performed similarly. To summarize, using hybrid Neural Network with ARMA method to compare with the conventional method in the estimation of VAR has certain advantages. Consequently, investors are suggested to use the hybrid method to estimate VAR when estimating the VAR of asset returns. References [1] Afolabi, M. O., and O. Olude, Predicting Stock Prices Using a Hybrid Kohonen Self Organizing Map (SOM), Proceedings of the 40th Hawaii International Conference on System Sciences 2007, pp. 1-8 (2007). [2] Alexander, CO and CT Leigh, On the covariance matrices used in value at risk models, Journal of Derivatives, spring, pp (1997). [3] Badík, P., Use Of The Var Method For Measuring Market Risks And Calculating Capital Adequacy, Biatec, 13, pp (2005). [4] Berkowitz, T. and J. O, Brien, How Accurate Are Value at risk Models at Commercial Banks? Journal of Finance, 57(3), pp (2002). [5] Billio, M. and L. Pelizzon, Value-at-Risk: A Multivariate Switching Regime Approach, Journal of Empirical Finance, 7, pp (2000). [6] Box,G. and Gwilym M. Jenkins, Time Series Analysis: Forecasting and Control, second edition, Oakland, CA: Holden-Day (1976). [7] Britten-Jones, M. and S. M. Schaefer, Non-linear Value-at-Risk, European Finance Review, 2, pp (1999). [8] Cabedo, J.D. and I. Moya,Estimating oil price value at risk using the historical simulation approach, Energy Economics, 25, pp (2003). [9] Chang, H. C., C. T. Chen and C. S. Hsieh, Forecasting Of Value at Risk By Using Percentile Of Cluster Method, in Proc. 10th Joint Conference on Information Sciences. Salt Lake City: World Scientific Publishing Co., pp (2007). [10] Dowd, K., Beyond Value at Risk, Wiley, New York (1998). [11] Duffie, D. and J. Pan, An overview of value at risk, Journal of Derivatives, 4, pp (1997). [12] Engel, J. and M. Gizycki, Conservatism, Accuracy and Efficiency: Comparing Value-at-Risk Models, Working Paper 2, March (1999). [13] Fama, E.F., The Behavior of Stock Market Prices, Journal of Business 38, pp (1965).

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