Market Returns and Volatility (Term Paper Sample)

Finance and Accounting
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Market Returns and Volatility
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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
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 (VoÅ¡vrda & ŽıkeÅ¡, 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.
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.
Econometric Methodology
In this analysis, we apply four styles of GARCH models as listed below:
ARCH model
GARCH model
DRIFT + GARCH model
EGARCH model
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.
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 EMBED Equation.3
 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
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.
Exponential GARCH (EGARCH) Model
Exponential GARCH uses a leverage effects the expression. In the EGARCH model, we present the conditional covariance as:
Log (ht) = Î&plus...