Modelling Financial Market Returns and Volatility - Lab Report Example

Finance and Accounting Modelling Financial Market Returns and Volatility Lecturer’s Executive Summary This paper evaluates the 3 GARCH models using three random distributions to compare the forecast of volatility power of their return for the London stock exchange group, market price index for the 12 years from 2001 to 2012. This study uses the three models GARCH model, DRIFT + GARCH model and EGARCH model against the Normal and General distributions of Error. The next forecast we make is that of the volatility of the stock market for the UK through its stock return using the three models GARCH model, DRIFT + GARCH model and EGARCH model and then performing the comparison of the forecasting of their performances. The outcome demonstrates the fact that the return volatility can change under the influenced of the significant persistence as well as asymmetric effects. We also approximate the variance for the models for in the full sampling period by use of a static forecast. After the comparison of the forecasting performances of the models, the EGARCH model is found to be having the most accurate forecast of the performance compared to other models. Table of Contents 1. INTRODUCTION 2. DISCUSSION TO THE STOCK 3. ECONOMETRIC METHODOLOGY 3.1. ARCH Model 3.2. GARCH Model 3.3. DRIFT + GARCH Model 3.4. Exponential GARCH (EGARCH) Model 4. DATA STATISTICS 4.1. Time Series Analysis 4.2. Descriptive Statistics 4.3. ADF Test 5. CONCLUSION 1. Introduction Features such as volatility clustering and pooling, leptokurtosis and leverage effects commonly characterize the financial data. It is not possible for linear structure models and time series models to demonstrate most of the vital features. The three features are the tendency to develop the returns on the financial assets (Vošvrda & Žıkeš, 2004). The definition of these tendencies is stated as to make the distributions to demonstrate bigger tails and superfluous mean. This makes the volatility of the financial markets to create large returns and to make the volatility increase more because of the more rapid drop in prices than to follow the price increase to the same capacity (Hájek 2007, p. 65). The most common non-linear models of financial forecasts are the ARCH or GARCH models applied in the volatility modeling and forecasting. In this research, we undertake to investigate the volatile nature of the stock market in the UK by studying the dynamics of its return with the GARCH model, DRIFT + GARCH model and the EGARCH model to do comparison of the forecasting of their performances. The next section in this discussion is the exploration of the stock of the London stock Exchange group in relation to the GARCH models as well as the volatility of the stock market (Scheicher 2001, p. 27). The third part of this paper briefly provides information about the GARCH model and the ARCH models, with the presentation of the estimated outcomes. 2. Discussion to the Stock The projected volatility of the return in the stock and the financial market is the major ingredient in the assessment of the asset and portfolio risks. The projection is very essential in the computation and the estimation of the pricing models for the derivatives. The analysis of the movement of short stock rates in the emerging markets in the UK and the entire Central Europe show the region as the prime emergent European stock market (Yu 2002, p. 34). This project estimates the VEC model and performs the volatility modeling using a Multivariate GARCH referred to as the M-GARCH model. The results show that the London Stock Exchange group that was investigated has minimal interaction and its financial return volatility has a specific regional behavior (Wang & Wu 2012, 54). The study of the behavior of the stock return volatility and its distributional features in the region shows the data for the stock markets covering the years between 2009 and 2013. The dataset uses the UK PX-50 index and obtains the statistically significant outcome for the GARCH (1, 1) model. This leads to a conclusion that the financial returns volatility on the PX -50 is persistent. The analysis by Brooks (2008, p. 42) investigates the association linking the emerging stock market in the UK and the other countries in the Central Europe. The investigation uses the multivariate GARCH models such as GARCH, EGARCH, DRIFT + GARCH, AGARCH and VGARCH. This study does the investigation and analysis in two phases, the univariate models phase and the multivariate models phase (Thomas & Mitchell 2005, p. 5). On the basis of the univariate model, the analysis concludes that there is a strong GARCH influence appearing for all the European market inclusive of the UK. In the UK, the lagged squared financial returns, which has been non-significant 50 % of the specifications of the conditional stock return volatility (Haroutounıan andPrice 2001, p. 38). In the tests of the hypothesis of having an efficient market in the capital market of the UK between 1995 and 2005, the data shows an existence of efficiency as well as linear dependency on a number of index closing. It shows a similar feature for the Stock Exchange in Prague and comes to conclusion that there is a significant linear relationship between the daily return on stock and the daily index returns. It also concludes that the methodology must be consistent with heteroskedasticity, before it is applied to prevent the biases in relation to statistical significance. According to Mittnik, Paolella & Rachev (2002, p. 65), the application of the GARCH model results show similar results for the UK, Poland and Hungary, with an existence of significant GARCH influences. 3. Econometric Methodology In this analysis, we apply four styles of GARCH models as listed below: ARCH model GARCH model DRIFT + GARCH model EGARCH model 3.1. ARCH Model The analysis with the ARCH models runs on the basis of variance in the error term calculated at the time t. The model depends on the values obtained from the square of the error terms at time t – 1 and before. The model ARCH is expressed as shown in below: yt = u ut ~N (0, ht) (1) ht = α0 + ∑( αjut2 - i) for t from t – 1 to q (2) In the ARCH (q) model, q is the order of the derivative of the lagged squared financial and stock returns. Assuming that the analysis uses ARCH (1) model, the equation becomes: ht = α0 + α1u2t – 1 (3) The values of ht must be positive because ht represents a conditional variance. It doesn’t make any sense to have a conditional variance (α0) with a negative value. In order to obtain a conditional variance approximate value, it is important to make all the coefficients in the conditional variance positive. Therefore, the coefficients will have to satisfy α0 > 0 and α1 >=0. 3.2. GARCH Model The GARCH (p, q) model permits the conditional α0 variance of variable to be dependent on the former lags. The first lag in the square of the residual obtained from the mean equation gives information concerning the previous time’s volatility as shown below: ht = α0 + ∑ αiu2t - i + ∑βiht – 1 (for I from 1 to q, I from 1 to p) (4) In the use of the simple GARCH (1, 1) model, the conditional variance is presented as shown below: ht = α0 + α1u2 t – 1 + β1ht-1 (5) In the hypothesis of having a stationery covariance, the model obtains the unconditional variance using the unconditional value in equation 5. The variance is found as shown below h = α0 + α1h + β1h (6) By solving the equation 5, the result is: h = α0 / (1 – α1 – β1) (7) For the calculation to obtain the unconditional variance then the following conditions must be satisfied: α1 + β1 < 1 To obtain a positive value of the conditional variance, then the following condition has to be true: α0 > 0 3.3. DRIFT + GARCH Model The DRIFT + GARCH model is a taken as a simple extension of the GARCH, having an additional term to represent the possibility of asymmetries. The GARCH model permits the conditional variance with varying responses from the previous innovations, both positive and negative. ht = α0 + ∑ αiu2 t – i + Yiu2t-idt-i + ∑βjht-j (8) In this model, d t - 1 is represents a dummy variable: In the drift + GARCH model, the impact of high return is shown by αi, while low return shows the impact as. Additionally, if the Y is greater than of Less than 0, then the return impact is concluded to be asymmetric. If Y > 0, then there is a leverage effect. For the satisfaction of the non-negative conditional coefficients, the following conditions have to be true α1 > 0 αi > 0 β > 0 αi + Yi >=0 That model is acceptable, provided that Yi > 0. 3.4. Exponential GARCH (EGARCH) Model Exponential GARCH uses a leverage effects the expression. In the EGARCH model, we present the conditional covariance as: Log (ht) = α0 + ∑βjlog(ht-j) + ∑ αi|(ut-i) / √(ht-i)| + ∑Yk ((ut-k) / √(ht-k)) (9) In the EGARCH equation Yk represents the leverage factor, which explains the asymmetric nature of the model. As the basic GARCH model demands restrictions, the EGARCH model permits unrestricted approximation of the unconditional variance. If Yk < 0, then the leverage effect is present. If Yk is greater than or less than 0, then there is an asymmetric. The implication of the existence of leverage effect is a reduction in stock return volatility. 4. Data Statistics In this research, we use weekly data for the UK FTSE100 (London Stock Exchanges Group) covering the period between 2009 and 2013. The data was obtained from www.finance.yahoo.com. We make use of the return, which is defined as: r = log (xt / x t-1) (10) where xt represents the capital index 4.1. Time Series Analysis Figure 1 below shows the plotted graph of stock prices PX. In figure 2, the graph shows the stock return defined as RPX. Figure 1: Stock Price Variation Figure 2: Stock Price Return 4.2. Descriptive Statistics The results for descriptive statistics are presented in table 1 below: Table 1: Descriptive Statistics Statistical Measure RPX  Mean  0.000211  Median  0.000722  Max  0.123640  Jarque-Bera  18821.75  Min -0.1618550  Std Deviation  0.015433  Skewness -0.524062  Kurtosis  15.43874  Probability  0.0000  Sum  0.609243  Sum Sq. Dev.  0.690061  Observations  200 Table 1: Descriptive Statistics Table 1 above gives a summary of the descriptive statistics for prices. The return on the prices gives a negative skewness as well as a highly positive kurtosis. These results signify the series distributions, and the fact that they have a long left tail and leptokurtic. The Jarque-Bera statistics finds the p - value less than 0.5 at the 1 % confidence level and thus rejects the normal distribution’s null hypothesis for the RPX variable. 4.3. ADF Test Table 2 below presents the ADF test results (Augmented Dickey-Fuller test). It concludes that the return is stationary. Table 2: ADF Test Results The test for mean of the ARCH model effect is done in this section using the ARCH-LM Test. The results for the test are presented in Table 3 below. If the test in the statistics generates greater values than the normal distribution’s critical value, then the test rejects the null hypothesis. ARCH (1) Test Dependent Variable of Model ARCH(1)LM Stat P RPX 429,7907*** 0.0000 Note: *** denotes significant at the 1% level. The hull hypothesis, there is no ARCH effect is rejected. Table 3: Statistical Results for the ARCH (1) Test We apply the GARCH, DRIFT + GARCH and the EGARCH model with both the Student- t and the Genera Error distribution added to the Normal distribution. R The results are presented in the appendix section. There appears to be a strong GARCH and DRIFT + GARCH effects for the stock returns. The addition of the coefficient α and β is less than that of the GARCH and DRIFT + GARCH models for all the distribution. In the DRIFT + GARCH model, Yk is found to be greater than 0, meaning that the resultant impact of the prices is asymmetric, hence, volatility increases. The E-GARCH model, there is a significant negative leverage effect in the returns. In all the models, r < 2, but is statistically significant, showing that the RPX is leptokurtic. The evidence gives a historic trend of model performance. An estimate is done for the variance in the whole sample time frame with the static forecast. The next phase is a comparison of the forecasting model performance, in consideration of the 4 statistical forecasting models to be able to derive the forecasting error. The four methods are applicable in evaluating the accuracy of the forecast. The four methods are: MSE (Mean Square Error) MAE (Mean Absolute Error) MSE1 = 1/ n∑ (σt^2 – σt’^2) ^2 for t from 1 to n (11) MSE2 = 1/ n∑ (σt – σt’) ^2 for t from 1 to n (12) MAE1 = 1/ n∑ | (σt^2 – σt’^2)| for t from 1 to n (13) MAE2 = 1/ n∑ | (σt – σt’)| for t from 1 to n (14) In all the methods, n represents the number of forecasts. Coefficient σt^2 represents the actual volatility while σ’t^2 represents the day t forecast of volatility. Table 4 below represents the forecasting results of the GARCH, DRIFT + GARCH and the EGARCH models. The EGARCH model has the largest accuracy of the forecast from the analysis of MSE2, MAE1 and MAE2. The only method that shows higher accuracy in the DRIFT + GARCH model is MSE1. Table 4:ComparisonForecasting Performance of GARCH Models   Normal Distribution Student t Distribution GED   GARCH DRIFT + GARCH EGARCH GARCH DRIFT + GARCH EGARCH GARCH DRIFT + GARCH EGARCH MSE1 6.31E-07 6.06E-07 6.26E-07 6.33E-07 6.04E-07 6.25E-07 6.36E-07 6.07E-07 6.28E-07 MSE2 0.000481 0.000479 0.000464 0.000479 0.000478 0.000458 0.000479 0.000477 0.000457 MAE1 0.000248 0.000234 0.000235 0.000247 0.000236 0.000235 0.000241 0.000238 0.000236 MAE2 0.015477 0.015384 0.015345 0.015446 0.015397 0.015324 0.015436 0.015386 0.015316 Table 4: Measure of Forecasting Accuracy 5. Conclusion This paper evaluates three major GARCH models, the GARCH, DRIFT + GARCH and EGARCH model. It uses three distributions, the Normal distribution, Student - t distribution and the General Error distribution. The evaluation tests the power (accuracy) of forecasting of volatility in the return of the London Stock Exchange Group, using the PX index. All the models were applied, followed by the interpretation of their coefficients. The results demonstrate the presence of significant effects of ARCH and GARCH in the data. This indicates that the return volatility has a feature of the significant asymmetric influences and persistence. With the GARCH type models applied in the London Stock Exchange Group, we perform comparison between the sample performance forecast models. The results provide reliable evidence that the EGARCH model has the greatest forecasting power and performance under the various criteria for measuring losses. REFERENCES Brooks C 2008, “IntroductoryEconometricsFor Finance: Second Edıtıon ” CambrıdgeUnıversıtyPress Hájek J 2007, “CzechCapital Market Weak-Form Efficiency, SelectedIssues” , PragueEconomicPapers, 4, 303-318 Haroutounıan MK & Price S 2001, “Volatility İn TheTransitionMarkets of Central Europe”, Applied Financial Economics, 11, 93-105 Mittnik S, Paolella MS & Rachev ST 2002, “Stationarity Of StablePower-GARCH Processes”, Journal Of Econometrics, 106, 97–107 Scheicher M 2001, “TheComovements Of StockMarketsInHungary, Poland AndTheCzechRepublıc” InternatıonalJournal Of FınanceAndEconomıcsInt. J. Fin. Econ. 6: 27–39 Thomas S & Mitchell H 2005, “GARCH Modeling of High-FrequencyVolatility in Australia’sNationalElectricity Market”, DiscussionPaper. Melbourne Centrefor Financial Studies. Vošvrda M & Žıkeš F 2004, “An Applıcatıon Of TheGarch-t Model On Central EuropeanStockReturns” PragueEconomıcPapers, 1, 26-39 Wang Y & Wu C 2012, “ForecastingEnergy Market Volatility Using GARCH Models: Can MultivariateModelsBeatUnivariateModels?”, EnergyEconomics. Yu J 2002, “ForecastingVolatility in the New ZealandStock Market”, Applied Financial Economics, 2002, 12, 193-202. APPENDIX Normal Distribution GARCH (1, 1) DRIFT + GARCH E GARCH Value p Value p Value p Meanequation α0 0.00090 0.00000 0.00060 0.00230 0.00050 0.00550 Variationequation α0 4.61E-06 0.00000 6.08E-06 0.00003 -0.51475 0.00000 α0 0.13183 0.00000 0.07273 0.00000 0.25313 0.00000 Y - - 0.10362 0.00000 -0.0685 0.00000 β 0.84961 0.00000 0.84412 0.00000 0.96364 0.00000 AIC -5.91921 -5.9293 -5.92821 SIC -5.91101 -5.91882 -5.91793 DW-stat 1.88631 1.88882 1.88902 ARCH-LM (1) Test P - value of ChiSq 0.306628 0.08466 0.14936 Obs. 200 Student’s T Distribution GARCH(1, 1) DRIFT + GARCH E GARCH Value p Value p Value p Meanequation α0 0.00091 0.00000 0.00081 0.00000 0.00082 0.00013 Variationequation α0 4.34E-06 0.00000 5.36E-06 0.0000 -0.4700 0.0000 α0 0.1192 0.00000 0.06960 0.0000 0.2354 0.0000 Y - - 0.0895 0.0000 -0.613 0.0000 β 0.86164 0.00000 0.8563 0.0000 0.9673 0.0000 AIC -5.9526 -5.9583 -5.9577 SIC -5.9423 -5.9459 -5.9453 DW-stat 1.8855 1.8872 1.8873 ARCH-LM(1) Test p-value of ChiSq. 0.3879 0.1387 0.2596 Obs. 200 Generalized-Error Distribution (GED) GARCH(1, 1) DRIFT + GARCH E GARCH Value p Value p Value p Meanequation α0 0.0009 0.0000 0.0006 0.0001 0.0005 0.0001 Variationequation α0 4.56E-06 0.0000 5.72E-06 0.0000 -0.4947 0.0000 α0 0.1256 0.0000 0.0712 0.0000 0.2446 0.0000 Y - - 0.0960 0.0000 -0.0646 0.0000 β 0.8556 0.00005 0.8495 0.0000 0.9653 0.0000 r 1.5073 0.00005 1.535154 0.0535 1.5323 0.0497 AIC -5.9427 -5.9493 -5.9483 SIC -5.9320 -5.93678 -5.9365 DW-stat 1.8858 1.8876 1.8872 ARCH-LM (1) Test p-value of ChiSq. 0.2896 0.07456 0.19745 Obs. 200 Read More