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Application of Dynamic Financial Time-Series Prediction on the interval Artificial Neural Network Approach with Value-at-Risk Model Hsio-Yi Lin An-Pin Chen
ZHANG Wei
Fuzzy-VaR BPN Model This paper propose a hybrid model – Fuzzy-VaR BPN Model to predict financial time series. Fuzzy-VaR BPN Model Back-propagation Neural Network (BPN) Fuzzy Membership Function Value-at-Risk Methodology
Back-Propagation Neural Network BPN uses back-propagation influenced by gradient descent algorithm. Goal: determine a set of weights which minimize the errors between the predictive and the target value: 1X E= (yk ¡ ok )2 2 k • ok : the actual output of the network • yk : the desired output
Back-Propagation Neural Network Learning Process: Firstly, the jth neuron of the hidden layer receives the activation function: X Hj = xi wijh i
• xi : the signal to the input neuron i h • wij : the weight of the connection between the ith input neuron and the jth neuron of the hidden layer.
Back-Propagation Neural Network Learning Process: Firstly, each neuron of the hidden layer receives the activation X function: Hj = xi wijh i Secondly, this activation function produces output by a transfer function f of the hidden layer
hj = fj (Hj ) = fjh
Back-Propagation Neural Network Learning Process: Firstly, each neuron of the hidden layer receives the activation X function: Hj = xi wijh i Secondly, this activation function produces output by a transfer function f of the hidden layer: hj = fj (Hj ) = fjh Thirdly, output neuron receives the hidden layer’s output and produces the final results X o Ok = wjk ¤ hj j
o w • jk is the weight of the connection between hidden neuron j and
output neuron k
Back-Propagation Neural Network Learning Process: Firstly, each neuron of the hidden layer receives the activation X function: Hj = xi wijh i Secondly, this activation function produces output by a transfer function f of the hidden layer: hj = fj (Hj ) = fjh Thirdly, output neuron receivesX the hidden layer’s output and produces the final results: Ok = wjko ¤ hj j Finally, the result is transformed by the transfer function ok = fk (Ok ) = fko
Back-Propagation Neural Network Backward Process The output value may differ from the target value. This error can be adjusted by adjusting the weights of learning epochs
Fuzzy Set In order to improve the single-point shortcoming of BPNs, a fuzzy membership function is added to BPNs. Fuzzy set is completely characterized by its membership function. MF of the fuzzy-interval approach is a normal distribution and specified by two parameters {c, ¾} 1 ¡ 12 ( x¡c )2 ¾ e f(x; c; ¾) = p 2¼¾ • c: the mean of weekly returns • ¾: the standard deviation of weekly returns
The fuzzy-interval MF is centered on c and the extend to which it spreads out around c is added and subtracted 1¾, 2¾,3¾.
Fuzzy BPNs Fuzzy BPN consists of a BPN and a fuzzy membership function maintain the BPNs’ nonlinear features improve the single-point shortcoming of BPNS
Fuzzy BPNs - BPNs BPN is used to learn c and ¾ . c and ¾ are used to find the fuzzy-interval MF
BPNs frame for producing the fuzzy-interval MF
Framework of Fuzzy BPNs
Value-at-Risk Model In order to provides an effective financial risk control, Value-at-Risk model is added to Fuzzy BPN. VaR is the maximal loss of a financial position during a given time period for a given probability. In this paper, all models are constructed and tested with 2.5 confidence level in a weekly (five-days)ptime horizon. • V aRt=t¡5 (2:5%) = 1:96 ¤ W ¤ ¾ ¤ T . – W is the investment – T is the holding period – the weekly loss will exceed this value in 2.5%.
Experimental Data Data set are taken from the Taiwan Top50 Tracker Fund (TTT ) which is composed of daily Net Asset Value (NAV). Data set include 1073 observations.
This study utilizes the past w days to predict the succeeding weekly Net Asset Value. In this paper five different w are considered: 5, 10,15 and 20. A sliding window is proposed with different window width w+5 moving from the first period to the last period of the entire data set labeled Si (i is from 1 to 1073-w-4).
Sliding Window
N (1073) observations are divided into N-w-4 samples. Every sample comprises time-series data containing w+5 NAV observation. 25% of the samples are used in the test and 75% in the training.
Experimental Design This paper takes natural logarithmic transformation to stabilize the time-series of NAV via normalization. The normalizations of w-input: Pk Ik = In( ) P0;w • P0,w is the normalized basic day of input variables • k is from 1 to w
Experimental Design The normalizations of two-output variables of ETF NAVs.
meanSi and SDSi respectively represents the mean and standard deviation for the following week NAVs during period Si .
Empirical Results BPNs Model The BPNs model used in this study is a three-layer feed forward network and is trained to map the next weekly-day mean and standard deviation for the coming w days using a back-propagation algorithm. The BPNs models are tried for w=5,10,15 and 20.
Empirical Results BPNs’ Parameters Mean Squared Error (MSE) is used to assessed the forecasting performance. Pnw 2 (F ¡ O ) i i M SE = i nw ¡ 1 • nw is the number of example sequences, nw =N-w-4 • Oi is the target value • Fi is the predicted value
The final determined parameters of each w-days BPNs are based on the smallest MSE. • The values of MSE between the training set and testing set will be compared with emphasis on the testing set analysis.
Empirical Results MSE of BPNs’ Best Parameter Setting Models
Empirical Results GARCH Model Various goodness-of-fit statistics are used to compare the estimated GARCH model.
Forecast-Performance Comparison This paper applies three evaluation criteria to compare the forecasting performance. Mean Absolute Error (MAE) Pnw i jOi ¡ Fi j M AE = nw ¡ 1 Mean Absolute Percentage Error (MAPE) Pnw jOi ¡Fi j MAP E =
Correct Rate
i
Oi
nw ¡ 1
# of correct examples Correct Rate = #of total training=testing examples
Forecast-Performance Comparison Mean Absolute Error (MAE)
• For each w, the MAE value of BPN models are less than that of AR-GARCH models.
Forecast-Performance Comparison Mean Absolute Percentage Error (MAPE)
• For each w, the MAPE value of BPN models are less than that of AR-GARCH models.
Forecast-Performance Comparison Correct Rate
• The correct rates of the traditional BPN models are 0. • For each w, the correct rates of Fuzzy-VaR BPN are higher than AR-GARCH models. • It is rational that the correct rate of the testing data should be lower than those of the training data.
VaR Performance Comparision In this paper, the performance of VaR models are tested based on failure rate using Likelihood-Ratio Test. The failure rate is the proportion of times the actual returns are below the forecasted value at risk Vart/t-5(®). If the VaR model is correctly specified, the failure rate should be equal to the pre-designed ® (2.5%).
The Likelihood-Ratio Statistic is defined as In(1 ¡ q)S¡S1 ¤ q S1 LR = 2 ¤ In(1 ¡ ®)S¡S1 ¤ ®S1
For S observations, S1 is the number of failures q = S1/S, represents the proportion of failures
VaR Performance Comparision
LR threshold is 5.23903. The model will be rejected if its LR> 5.23903. Only 20-day Fuzzy-VaR BPN models are suitable for forecasting the VaR values. It seems that Fuzzy-VaR BPN models are more superior than the GARCH models.
Conclusion Fuzzy BPNs consisted of a fuzzy-interval membership function not only possess artificial neural networks nonlinear capabilities but also improve the shortcoming of single-point estimations in conventional ANN. In terms of interval evaluation, the forecast performance of Fuzzy-VaR BPNs is better than traditional BPNs and VaR-GARCH models. Fuzzy-VaR BPNs provide a loss-alarm effect when the returns are lower than or equal to the computing value of VaR models.
ZHANG Wei
Fuzzy-VaR BPN Model This paper propose a hybrid model – Fuzzy-VaR BPN Model to predict financial time series. Fuzzy-VaR BPN Model Back-propagation Neural Network (BPN) Fuzzy Membership Function Value-at-Risk Methodology
Back-Propagation Neural Network BPN uses back-propagation influenced by gradient descent algorithm. Goal: determine a set of weights which minimize the errors between the predictive and the target value: 1X E= (yk ¡ ok )2 2 k • ok : the actual output of the network • yk : the desired output
Back-Propagation Neural Network Learning Process: Firstly, the jth neuron of the hidden layer receives the activation function: X Hj = xi wijh i
• xi : the signal to the input neuron i h • wij : the weight of the connection between the ith input neuron and the jth neuron of the hidden layer.
Back-Propagation Neural Network Learning Process: Firstly, each neuron of the hidden layer receives the activation X function: Hj = xi wijh i Secondly, this activation function produces output by a transfer function f of the hidden layer
hj = fj (Hj ) = fjh
Back-Propagation Neural Network Learning Process: Firstly, each neuron of the hidden layer receives the activation X function: Hj = xi wijh i Secondly, this activation function produces output by a transfer function f of the hidden layer: hj = fj (Hj ) = fjh Thirdly, output neuron receives the hidden layer’s output and produces the final results X o Ok = wjk ¤ hj j
o w • jk is the weight of the connection between hidden neuron j and
output neuron k
Back-Propagation Neural Network Learning Process: Firstly, each neuron of the hidden layer receives the activation X function: Hj = xi wijh i Secondly, this activation function produces output by a transfer function f of the hidden layer: hj = fj (Hj ) = fjh Thirdly, output neuron receivesX the hidden layer’s output and produces the final results: Ok = wjko ¤ hj j Finally, the result is transformed by the transfer function ok = fk (Ok ) = fko
Back-Propagation Neural Network Backward Process The output value may differ from the target value. This error can be adjusted by adjusting the weights of learning epochs
Fuzzy Set In order to improve the single-point shortcoming of BPNs, a fuzzy membership function is added to BPNs. Fuzzy set is completely characterized by its membership function. MF of the fuzzy-interval approach is a normal distribution and specified by two parameters {c, ¾} 1 ¡ 12 ( x¡c )2 ¾ e f(x; c; ¾) = p 2¼¾ • c: the mean of weekly returns • ¾: the standard deviation of weekly returns
The fuzzy-interval MF is centered on c and the extend to which it spreads out around c is added and subtracted 1¾, 2¾,3¾.
Fuzzy BPNs Fuzzy BPN consists of a BPN and a fuzzy membership function maintain the BPNs’ nonlinear features improve the single-point shortcoming of BPNS
Fuzzy BPNs - BPNs BPN is used to learn c and ¾ . c and ¾ are used to find the fuzzy-interval MF
BPNs frame for producing the fuzzy-interval MF
Framework of Fuzzy BPNs
Value-at-Risk Model In order to provides an effective financial risk control, Value-at-Risk model is added to Fuzzy BPN. VaR is the maximal loss of a financial position during a given time period for a given probability. In this paper, all models are constructed and tested with 2.5 confidence level in a weekly (five-days)ptime horizon. • V aRt=t¡5 (2:5%) = 1:96 ¤ W ¤ ¾ ¤ T . – W is the investment – T is the holding period – the weekly loss will exceed this value in 2.5%.
Experimental Data Data set are taken from the Taiwan Top50 Tracker Fund (TTT ) which is composed of daily Net Asset Value (NAV). Data set include 1073 observations.
This study utilizes the past w days to predict the succeeding weekly Net Asset Value. In this paper five different w are considered: 5, 10,15 and 20. A sliding window is proposed with different window width w+5 moving from the first period to the last period of the entire data set labeled Si (i is from 1 to 1073-w-4).
Sliding Window
N (1073) observations are divided into N-w-4 samples. Every sample comprises time-series data containing w+5 NAV observation. 25% of the samples are used in the test and 75% in the training.
Experimental Design This paper takes natural logarithmic transformation to stabilize the time-series of NAV via normalization. The normalizations of w-input: Pk Ik = In( ) P0;w • P0,w is the normalized basic day of input variables • k is from 1 to w
Experimental Design The normalizations of two-output variables of ETF NAVs.
meanSi and SDSi respectively represents the mean and standard deviation for the following week NAVs during period Si .
Empirical Results BPNs Model The BPNs model used in this study is a three-layer feed forward network and is trained to map the next weekly-day mean and standard deviation for the coming w days using a back-propagation algorithm. The BPNs models are tried for w=5,10,15 and 20.
Empirical Results BPNs’ Parameters Mean Squared Error (MSE) is used to assessed the forecasting performance. Pnw 2 (F ¡ O ) i i M SE = i nw ¡ 1 • nw is the number of example sequences, nw =N-w-4 • Oi is the target value • Fi is the predicted value
The final determined parameters of each w-days BPNs are based on the smallest MSE. • The values of MSE between the training set and testing set will be compared with emphasis on the testing set analysis.
Empirical Results MSE of BPNs’ Best Parameter Setting Models
Empirical Results GARCH Model Various goodness-of-fit statistics are used to compare the estimated GARCH model.
Forecast-Performance Comparison This paper applies three evaluation criteria to compare the forecasting performance. Mean Absolute Error (MAE) Pnw i jOi ¡ Fi j M AE = nw ¡ 1 Mean Absolute Percentage Error (MAPE) Pnw jOi ¡Fi j MAP E =
Correct Rate
i
Oi
nw ¡ 1
# of correct examples Correct Rate = #of total training=testing examples
Forecast-Performance Comparison Mean Absolute Error (MAE)
• For each w, the MAE value of BPN models are less than that of AR-GARCH models.
Forecast-Performance Comparison Mean Absolute Percentage Error (MAPE)
• For each w, the MAPE value of BPN models are less than that of AR-GARCH models.
Forecast-Performance Comparison Correct Rate
• The correct rates of the traditional BPN models are 0. • For each w, the correct rates of Fuzzy-VaR BPN are higher than AR-GARCH models. • It is rational that the correct rate of the testing data should be lower than those of the training data.
VaR Performance Comparision In this paper, the performance of VaR models are tested based on failure rate using Likelihood-Ratio Test. The failure rate is the proportion of times the actual returns are below the forecasted value at risk Vart/t-5(®). If the VaR model is correctly specified, the failure rate should be equal to the pre-designed ® (2.5%).
The Likelihood-Ratio Statistic is defined as In(1 ¡ q)S¡S1 ¤ q S1 LR = 2 ¤ In(1 ¡ ®)S¡S1 ¤ ®S1
For S observations, S1 is the number of failures q = S1/S, represents the proportion of failures
VaR Performance Comparision
LR threshold is 5.23903. The model will be rejected if its LR> 5.23903. Only 20-day Fuzzy-VaR BPN models are suitable for forecasting the VaR values. It seems that Fuzzy-VaR BPN models are more superior than the GARCH models.
Conclusion Fuzzy BPNs consisted of a fuzzy-interval membership function not only possess artificial neural networks nonlinear capabilities but also improve the shortcoming of single-point estimations in conventional ANN. In terms of interval evaluation, the forecast performance of Fuzzy-VaR BPNs is better than traditional BPNs and VaR-GARCH models. Fuzzy-VaR BPNs provide a loss-alarm effect when the returns are lower than or equal to the computing value of VaR models.
