An empirical comparison: autoregressive conditional heteroskedasticity and multi- factor models in asset return predictability by

Theodoros Zigkiris [206568]

M.Sc. Tilburg University 2016

A thesis submitted in partial fulfillment of the requirements of the degree of Master of Science in Econometrics and Mathematical Economics

Tilburg School of Economics and Management Tilburg University

Supervisor

Prof. dr. B. Melenberg

Second reader

Dr. P. Cizek

Date: February 29, 2016 Abstract

In time series analyses of asset returns, the volatility appears to be time-dependent. Several approaches have been proposed in the academic literature in order to predict future stock returns and measure their risk as well. This thesis focuses on the performance of the standard GARCH(1,1) and EGARCH(1,1) models, in comparison with two alternative linear factor models in predicting future stock returns. The linear models are the three-factor model of Fama and French and a macroeconomic four-factor model whose factors are the CPI inflation rate, USD/GBP exchange rates, and the price of crude oil barrel in UK. The final results after the forecast show that the standard GARCH(1,1) as well as EGARCH(1,1) models perform better than the other two multi-factor models in predicting future stock returns.

2

Table of Contents Abstract ...... 2

1. Introduction ...... 4

2. Literature ...... 7

3.Data description ...... 10

3.1 Sample statistics ...... 11

4. Models ...... 20

4.1 Three-factor model ...... 20

4.2 GARCH(1,1) model ...... 23

4.3 Four-factor model ...... 27

4.4 EGARCH(1,1) model ...... 29

5. Comparison of the models ...... 34

5.1 Forecast ...... 34

5.2 Vector Autoregressive Models ...... 41

6. Conclusions ...... 51

References ...... 53

7. Appendix ...... 55

3 1. Introduction

There have been a great variety of researches in order to analyse the relationship between risk factors (market, value, size, momentum, etc.) and the cross-section of stock returns. Are there any arbitrage opportunities that financial investors can exploit? Do investors believe in the efficient market hypothesis? Are investors able to explain and more importantly predict future stock returns based on the public information available? This topic at least goes back to Dow Jones1 around 1900 who tried to exploit predictability in asset returns. Fama (1970) suggested that “a market in which prices always “fully reflect” available information is called "efficient””. Some years later, Black (1972) suggests the Capital Asset Pricing Model CAPM which can be considered as an extension of EMH, in which a market factor is added in the model in order to determine the stocks’ prices. Later on, Fama and French (1992, 1993) propose the three-factor model, so as to understand better the portfolio performance and estimate future returns. Engle (1982) proposed the autoregressive conditional heteroskedasticity (ARCH) models for time series, where volatility is allowed to be time-dependent. A lot of different approaches have been proposed in the already existing literature as far as prediction of future returns is concerned. This thesis is aiming to give answers to the following research questions: Which one of the four models – three-factor model of Fama and French, the standard GARCH(1,1), four-factor model and EGARCH(1,1) - can give the most efficient and accurate forecasts as far as future returns are concerned? Which factors should we include in the multi-factor model so that it describes the data better? The models are chosen due to their structures, which can capture some of the stylized facts (volatility clustering and leverage effect) of stock returns as well as give some insights with respect to the correlations across the assets. Linear regression analysis is used in order to estimate two alternative multi-factor asset pricing models. Multi-factor models in financial econometrics are essential not only to describe the data but also to make the estimation of the correlations feasible. In other words, by taking into consideration the large number of stocks that are available in the

1 https://en.wikipedia.org/wiki/Technical_analysis#History

4 market, greater than 15000 (Rachev, et al. (2007)), we are not able to perform the estimation of the correlations without any simplifications. Indices such as the S&P 500 or FTSE 100 contain many stocks and therefore hundreds of thousands of individual correlations. However, the available samples are inadequate to calculate this large number of correlations due to the difficulty in estimating the inverse covariance matrix. Consequently, factor models assist in order to describe all pairwise correlations in respect of less number of correlations among factors. The two linear multi-factor models used in this thesis are the three-factor model of Fama and French (1992, 1993) and a four-factor model that consists of three macroeconomic factors as well as the market’s factor. In particular, the three new factors are the excess returns-in excess of the risk free return rate-of the CPI inflation rate, the USD/GBP exchange rates, and the price of crude oil barrel in UK. In both multi-factor models, the estimated betas are found to be statistically and economically significant for all stock retuns. In addition, the standard GARCH and EGARCH models are used, so as to address the issue of volatility of stock returns, which is also very important as far as forecasts are concerned. GARCH models can capture the "volatility clustering" phenomenon, which is a well known stylized fact of asset returns. In addition, the EGARCH model takes into account the leverage effect, meaning the negative correlation between the past returns and the future volatility. This thesis focuses on the performance of the standard GARCH(1,1) and EGARCH(1,1) models in asset returns’ predictability. According to the standard GARCH(1,1) model it can be observed that four of the thirteen stocks used as sample present high persistence in volatility. Moreover, the results after using the EGARCH(1,1) model show that a leverage effect can be observed in the returns of six stocks. The final purpose of the thesis is to compare the four different models with each other, so as to find which one gives better predictions about future asset returns. The estimated forecasts based on the estimated root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) for each model show that both standard GARCH(1,1) and EGARCH(1,1) fit better the data while they also give the most accurate forecasts between the four models. This could be expected since GARCH models capture some of the stylized facts of time series data, while they also successfully

5 predict conditional variances. In addition, VAR(1) models are used in order to test whether the future values of the factors used in the four previous models can be predicted from their past values so as to assist in the accuracy of the conclusions concerning the forecasts. The results show that the forecasted values of the factors and indices used in the models are equal to their conditional means which is optimal. The thesis consists of six parts. The first part is the introduction, which consists of the purpose of study and the problem formulation. In the second part, there is a literature review on various approaches that have been suggested in order to predict future returns. In the third part, data sources are discussed. There is a description of the data which consists of descriptive statistics of returns, indices, and factors as well as multiple graphs. In the fourth part, the model assumptions and formulations of the three-factor model, four- factor model, the standard GARCH (1,1) model, and EGARCH (1,1) model are presented. After that, the coefficients of the estimated models are presented and interpreted as well. In the fifth part, model selection criteria are used in order to compare the four different models with each other and to decide which one fits better on the actual stock returns. In addition, VAR models are estimated for the factors and the indices, used as explanatory variables, in order to examine if the future values can be predicted from the past ones. The sixth chapter consists of the conclusions after a small summary, as well as further possible extensions of the models used for this thesis.

6 2. Literature

Stock returns may have risk premiums that compensate financial investors for bearing different kinds of uncertainties in their investment portfolios. The studies from Sharpe (1964) and Lintner (1965) and later Black (1972) give rise to the Capital Asset Pricing Model (CAPM), which have been widely studied and used in order to investigate the existence of additional risk factors besides the market portfolio. Banz (1981) presents evidence that CAPM is misspecified, suggesting that his results might be at least contaminated by the size effect. In addition, Chen, Roll and Ross (1986), propose a general multifactor model in order to test whether innovations in macroeconomic factors are risks that are compensated in the stock market. They find that these sources of risk are significantly priced. Moreover, Fama and French (1992, 1993) introduced three risk factors for returns on stocks: an overall market factor, a factor about the size of firms, and a factor related to book-to-market equity- which started new and enthusiastic discussions and studies on asset pricing. Carhart (1997) suggests a four-factor model for U.S returns so as to capture momentum. Avramov and Chordia (2006) find that Carhart’s four-factor model fails to capture momentum. Also, Novy-Marx (2013), Titman, Wei and Xie (2004) and others, support that the three-factor model of Fama and French is incomplete for expected returns since its three factors fail to spot a lot of the variation in average returns related to profitability and investment. Regarding that, Fama and French (2015) introduce a five- factor asset pricing model, which consists of the three factors of the old model plus two more factors. The first extra factor is profitability (stocks with higher profitability perform better) and the second one is an investment factor (which is the difference between the returns on diversified investment portfolios consisting of stocks of low and high investment firms). However, it is controversial if the new model performs better than the old one since the 5-factor model still ignores momentum and low volatility. Engle (1982) introduced ARCH models which are widely used in econometrics for modelling time series where volatility appears to be time-dependent (volatility clustering). Furthermore, Geweke (1986) and Pantula (1986) proposed the log-ARCH model, while Higgins and Bera (1992) suggested a more general class of models, which is the

7 nonlinear ARCH (NARCH). Bollerslev (1986) introduced linear Generalized ARCH (GARCH) models, which allow for a more flexible lag structure. Maximum likelihood estimators for GARCH are obtained by using the algorithm of Berndt, Hall, Hall and Hausmann (1974). Fiorentini, Calzolari and Panattoni (1996), used first and second derivatives of the log-likelihood in order to estimate the parameters of the GARCH model and they compared different algorithms so as to maximize the likelihood. However, the standard GARCH model presents some drawbacks due to its model’s formulation, for instance, it does not account for a leverage effect. That was the reason why Nelson (1991) proposed the exponential GARCH (EGARCH) model, as a solution to the drawbacks of standard GARCH. Hansen, Lunde and Voev (2014) propose a multivariate GARCH model with realized measures of volatility. The model incorporates realized measures of variances and covariances, which extract information about the degree of volatility and correlations from high frequency data. In general, there is a variety of extensions of GARCH models according to different empirical investigations. Vector autoregressive (VAR) models suggested by Sims (1980) in order to describe the dynamics of the variables. In VAR models each of the variables in the model depend linearly on their own lagged values as well as on the values of the rest variables in the selected vector. Lütkepohl and Poskitt (1991) give a more technical paper in which they use VAR models in time series in order to estimate orthogonal impulse responses. Moreover, VAR models are widely used in forecasting due to their structure since if the variables can be represented as VAR models, then their future values can be predicted from their past ones. For instance, Liu, et al. (1994) test how alternative VAR models are performing in forecasting exchange rates, while Hoque and Latif (1993) are forecasting the exchanging rate of Australian dollar to US dollar using restricted and unrestricted VAR models. Most of the literature chosen is regarded to be pioneering in the fields of asset pricing and autoregressive conditional heteroskedasticity models. Many applications of GARCH models exist as well. Engle (2001) presents an example of risk measurement by using the standard GARCH(1,1) model which can be used in financial decisions. Pagan (1996) shows a set of stylized facts of financial time series, while later he is using GARCH models in order to model those characteristics. Andersen, and Bollerslev (1998) show that

8 volatility models give extremely accurate forecasts for the latent volatility factor. Furthermore, they show that the standard GARCH(1,1) model does perform very well in characterizing volatility clustering. In this thesis there is an attempt to compare and test the efficiency and accuracy of the three-factor model of Fama and French, a four-factor model consisting of microeconomic factors, as well as the standard GARCH(1,1) and an EGARCH(1,1) model in order to forecast the future returns of thirteen stocks. The thesis contributes in the existing literature by providing a review and empirical evidence for the performance of GARCH and multi-factor models in stocks return predictability and providing evidence that GARCH models are able to model the volatility of stock returns and give more accurate forecasts than multi-factor models.

9 3.Data description

The ‘’Data stream’’ of Tilburg University Data Lab was used in order to download the asset and index returns. All asset returns consist of monthly returns of thirteen stocks- HSBA, LLOY, BARC, RBS, STAN, BCB, IMT, BATS, BP, RDSB, VOD, TSCO, RR - (abbreviations are in the table 3.1) from the London Stock Exchange, over the period 16/09/2000 until 16/09/2015. In particular, the stocks’ time series sample consists of returns of five UK banks which are HSBC, Barclays, Lloyds, Royal Bank of Scotland, and Standard Chartered. In addition, there are two tobacco industries, Imperial Tobacco and British American Tobacco, two oil and gas companies, BP and Royal Dutch Shell, one from telecommunications industry, which is Vodafone, one from retailing industry, which is Tesco, and the last one is Rolls-Royce, from the aerospace, defence, and marine industry. Furthermore, the two indices- FTSE100 and S&PCOMP- used are on a monthly frequency and are over the same period. Finally, the currency of the stock returns is UK pounds. As far as the European factors and risk free return rate are concerned, they are taken from the data library of Kenneth R. French and they are monthly as well. The factors used for the second factor model (4-factor model) consist of Crude Oil-Brent which are the monthly prices of crude oil barrels in UK pounds, BOE which is the monthly exchange rate of USD and GBP and UK CPI which is the monthly UK CPI inflation rate for all items (abbreviations are in the table 3.1). Both of the data sources are trustworthy in the sense that there are no missing values which makes the data analysis easier and reliable.

Table 3.1 Abbreviations of stocks and indices Abbreviations  FTSE100 - FTSE 100 _ RETURN INDEX  S&PCOMP - S&P 500 COMPOSITE _ RETURN INDEX  HSBA - HSBC HDG.  LLOY - LLOYDS BANKING GROUP  BARC - BARCLAYS  RBS - ROYAL BANK OF SCTL.GP.

10  STAN - STANDARD CHARTERED  BCB - BCB HOLDINGS  IMT - IMPERIAL TOBACCO GP.  BATS - BRITISH AMERICAN TOBACCO  BP. - BP  RDSB - ROYAL DUTCH SHELL B  VOD - VODAFONE GROUP  TSCO - TESCO  RR. - ROLLS-ROYCE HOLDINGS  CRUDE OIL-BRENT-CRUDE OIL-BRENT M+1 UK Close U$/BBL  BOE- USD TO GDP  UK CPI – CPI INFLATION

3.1 Sample statistics

In this part, descriptive statistics and graphs are presented to describe the data. The following tables present the descriptive statistics for both returns of stocks and indices. The results are estimated in table 3.1.1.

11

Table 3.1.1 Descriptive statistics of monthly returns BARC BATS BCB BP HSBA LLOY IMT RBS RDSB RR TSCO STAN VOD Mean 0.0058 0.0136 -0.0136 -0.0008 -0.0007 -0.0004 0.0126 -0.0053 0.0008 0.0143 0.0012 0.0027 0.0012 Median 0.0061 0.0119 -0.0190 0.0028 -0.0001 0.0067 0.0190 -0.0002 0.0032 0.0199 0.0031 0.0086 0.0046 Maximum 1.3321 0.1756 0.2985 0.2277 0.2670 0.9009 0.1468 0.4106 0.3401 0.3709 0.3082 0.2242 0.2699 Minimum -0.3218 -0.1830 -0.5600 -0.3644 -0.2507 -0.4638 -0.1450 -0.6563 -0.2104 -0.3150 -0.2361 -0.2662 -0.1877 Std. Dev. 0.1425 0.0527 0.1205 0.0724 0.0662 0.1231 0.0543 0.1201 0.0661 0.0941 0.0644 0.0797 0.0700 Skewness 4.3804 -0.1641 -0.3741 -0.5652 -0.0825 1.8035 -0.3555 -0.7583 0.3907 0.0929 0.3556 -0.3899 0.3096 Kurtosis 43.2028 3.9885 5.3078 6.0838 5.8460 19.7171 3.2665 8.8395 6.2712 4.9905 6.6167 3.8588 4.4619

Jarque-Bera 12697.5900 8.1365 44.1434 80.9044 60.9519 2193.5280 4.3234 272.9980 84.8328 29.9757 101.8973 10.0905 18.9046 Probability 0.0000 0.0171 0.0000 0.0000 0.0000 0.0000 0.1151 0.0000 0.0000 0.0000 0.0000 0.0064 0.0001

Observations 180 180 180 180 180 180 180 180 180 180 180 180 180

Table 3.1.1 shows that all the means of stock returns are close to zero, as expected. However, five values of the estimated means turn out to be negative, which shows that almost half of the companies had a financial loss or lackluster performance on some investments throughout the particular time period. Another reason about that fact is that, as can be seen in the histograms of the returns, there are some outliers in the loss direction which pull the value of the means down. In general, the means of the stock returns are between -0.0053 (RBS) and 0.0143 (RR). Moreover, observing the values of the medians, it can be seen that the distribution of both IMT and BATS returns are symmetric, due to the fact that their means 0.0126 and 0.136 are approximately equal to their medians with values (0.0190) and (0.0119), respectively. Concerning BP, BATS, STAN, TSCO and VOD, the distribution of the data returns present an asymmetry (positively skewed distributions), since the mean is greater than the median for each of these returns. On the other hand, the data returns of BARC, BP, HSBA, LLOY, RDSB, RR, and RBS have negatively skewed distributions since the value of median is greater than the mean for each stock, respectively. Moreover, BATS has the lowest standard deviation with a value of 0.0527, while the highest one belongs to BARC with a value of 0.1425. In addition, by checking the value of the Jarque-Bera test, there is evidence that for IMT only, the null hypothesis that its return follows normal distribution cannot be rejected, since the JB-statistic is equal to 4.3234, smaller than the critical 5% level of

12 significance which is equal to 5.99. This can also be seen from the value of the kurtosis (3.26) which is close to 3.

Table 3.1.2 Descriptive statistics of monthly indices and risk-free rate

Risk-free FTSE_100 S_P_500 rate Mean 0.0009 0.0027 0.13 Median 0.0085 0.0095 0.08 Maximum 0.0962 0.1478 0.56 Minimum -0.2317 -0.2201 0.00 Std. Dev. 0.0469 0.0453 0.15 Skewness -11.149 -0.8617 0.97 Kurtosis 59.434 58.056 2.68

Jarque-Bera 1.022.656 813.144 28.93 Probability 0.0000 0.0000 0.00

180 180 180 Observations

From table 3.1.2, it can be seen that the means of both index returns are positive and close to zero, with values 0.0009 and 0.0027 for FTSE 100 and S&P 500, respectively. As far as the distribution of the indices' returns is concerned, it presents an asymmetry (positively skewed distributions), since the mean is greater than the median for each index. FTSE100 and S&P 500 have approximately the same standard deviation which is around 0.046. The mean of the risk free return rate is 0.13, while the its maximum value is equal to 0.56 and minimum value is equal to 0.

13 Graph 3.1.1 Multiple series line graphs

BARCLAYS BARC BRITISH AMERICAN TOBACCO BATS BCB HOLDINGS BCB BP BP. 1.6 .2 .4 .4

1.2 .2 .1 .2

0.8 .0 .0 .0 0.4 -.2

-.1 -.2 0.0 -.4

-0.4 -.2 -.6 -.4 2016 2018 2020 2022 2024 2026 2028 2030 2016 2018 2020 2022 2024 2026 2028 2030 2016 2018 2020 2022 2024 2026 2028 2030 2016 2018 2020 2022 2024 2026 2028 2030

HSBC HDG. (ORD $0.50) HSBA IMPERIAL TOBACCO GP. IMT LLOYDS BANKING GROUP LLOY ROYAL BANK OF SCTL.GP. RBS .3 .2 1.00 .6

.2 0.75 .4 .1 .2 .1 0.50 .0 .0 .0 0.25 -.2 -.1 0.00 -.4 -.1 -.2 -0.25 -.6

-.3 -.2 -0.50 -.8 2016 2018 2020 2022 2024 2026 2028 2030 2016 2018 2020 2022 2024 2026 2028 2030 2016 2018 2020 2022 2024 2026 2028 2030 2016 2018 2020 2022 2024 2026 2028 2030

ROYAL DUTCH SHELL B RDSB ROLLS-ROYCE HOLDINGS RR. .4 .4

.2 .2

.0 .0

-.2 -.2

-.4 -.4 2016 2018 2020 2022 2024 2026 2028 2030 2016 2018 2020 2022 2024 2026 2028 2030

Graph 3.1.1 presents graphs for the returns of each stock. It can be observed that there are periods of high and low volatility. Volatility appears to be time dependent, which indicates a well-known stylized fact, namely volatility clustering. High volatility events tend to be followed by high price variations, whereas low volatility events tend to be followed by low variations in a way that they create ''clusters'' in time of which volatility stays high or low.

14 Graph 3.1.2 Line graph for S&P 500 and FTSE 100

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FTSE 100 - RETURN INDEX FTSE100 S&P 500 COMPOSITE - RETURN INDEX S&PCOMP

Graph 3.1.2 shows that S&P 500 and FTSE 100 tend to move together with approximately the same standard deviation.

Graph 3.1.3 Line graph for the risk- free return rate 풓

Risk- free rate

.006

.005

.004

.003

.002

.001

.000 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15

15 Graph 3.1.3 shows that the risk-free return rate presents a sharp decrease from 2000 till 2004 with its values getting close to zero. The next three years there exists an increase, however after 2007 there is a decline, so that the risk-free rate return is zero the last years of the sample period.

Table 3.1.3 Descriptive statistics of CRUDE OIL, CPI INFL, USD/GBP, HML, and SMB

CRUDE_OIL_BRENT UK_CPI_ USD_TO_GBP HML SMB Mean 41.35 2.18 1.66 0.005 0.002 Median 37.65 2.00 1.60 0.004 0.002 Maximum 77.69 5.20 2.08 0.109 0.048 Minimum 13.04 -0.10 1.39 -0.046 -0.069 Std. Dev. 19.72 1.12 0.17 0.025 0.021 Skewness 0.24 0.50 0.63 0.70 -0.49 Kurtosis 1.69 3.00 2.32 4.53 3.76

Jarque-Bera 14.62 7.70 15.56 32.11 11.62 Probability 0.00 0.02 0.00 0.00 0.00

180 180 180 Observations 180 180

In table 3.1.3, descriptive statistics for CRUDE OIL BRENT, UK CPI, USD TO GBP, HML, and SMB are summarized. The mean of the monthly crude oil price is equal to 41.35 pounds, with a maximum price of 77.69 and a minimum price of 13.04 pounds. Moreover, the distribution of the values of UK CPI and USD TO GBP are symmetric since the values of their means, which are 2.18 and 1.66, respectively, are very close to the values of their means which are 2 and 1.6, respectively. The maximum value for the CPI is 5.2 while the minimum is equal to -0.1. Additionally, the exchange rate has a maximum value of 2.08 and a minimum value of 1.39. Finally, all factors present quite high standard deviations with the highest one belonging to crude oil barrel price. Concerning the two factors of Fama and French model, it can be seen that the means of HML and SMB are close to zero with values 0.005 and 0.002, respectively. Furthermore, the distribution of the values of both factors is symmetric since the value of median is

16 0.004 for HML which is very close to its median and 0.002 for SMB which is equal with its mean. The standard deviation is low in both factor with value approximately 0.023.

Graph 3.1.4 Multiple series line graphs for CRUDE OIL, CPI INFL, and USD/GBP

Crude Oil-Brent M+1 UK Close U$/BBL (~£ ) OILBLC1(P)~£ 80

60

40

20

0 2002 2004 2006 2008 2010 2012 2014

UK CONTRIBUTIONS TO CPI INFLATION: ALL ITEMS NADJ UKCPANNL(ESA) 6

4

2

0

-2 2002 2004 2006 2008 2010 2012 2014

USD TO GBP (BOE) - EXCHANGE RATE STUSBOE(ER) 2.2

2.0

1.8

1.6

1.4

1.2 2002 2004 2006 2008 2010 2012 2014

Graph 3.1.4 present line graphs of the monthly prices of crude oil barrels in UK pounds, the monthly exchange rates of USD and GBP, and the monthly UK CPI inflation rate for all items. All three graphs show a sharp decrease around 2008, as a result of the 2007 crisis in UK.

17 Graph 3.1.5 Line graph for SMB and HML

.12

.08

.04

.00

-.04

-.08 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15

HML SMB

Graph 3.1.5 shows that both factors tend to move together in most of the time periods with approximately the same standard deviation.

Table 3.1.4 Descriptive statistics of the excess returns of USD/GBP, CPI INFL, and CRUDE OIL

EX_USD_GBP EX_CPI_INFL EX_CRUDE_OIL Mean -0.0006 0.0206 0.0063 Median 0.0001 0.0179 0.0086 Maximum 0.0639 0.0520 0.2162 Minimum -0.0956 -0.0010 -0.2786 Std. Dev. 0.0263 0.0115 0.0907 Skewness -0.2897 0.6252 -0.2661 Kurtosis 3.811 2.933 3.167

Jarque-Bera 7.375 1.163 2.309 Probability 0.025 0.002 0.315

Observations 178 178 178

18 In table 3.1.4, descriptive statistics for the excess returns of the three factors EX_CRUDE_OIL, EX_CPI_INFL and EX_USD_GBP are summarized. The mean of the excess returns of all three factors close to zero. In addition, the standard deviation for EX_USD_GBP and EX_CPI_INFL is quite low with values 0.0263 and 0.0115, respectively, while EX_CRUDE_OIL has standard deviation equal to 0.0907. Moreover, Jarque-Bera test shows that the excess returns of crude oil follow the normal distribution.

Graph 3.1.6 Line graph for the excess returns of the three factors

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EX_USD_GBP EX_CPI_INFL EX_CRUDE_OIL

Graph 3.1.6 shows that excess returns of CPI_INFL and USD_GBP tend to move together with EX_USD_GBP having a slightly higher standard deviation. The excess returns of CRUDE_OIL present a higher and more persistent volatility.

19 4. Models

4.1 Three-factor model

In this part, the three-factor model, Fama and French (1992, 1993), is estimated for each of the assets using as market returns those of FTSE 100 index. The three-factor model is an expansion of the CAPM by adding two extra performance factors to it. A factor about the size of firms (Small Minus Big) and a factor related to book-to-market equity (High Minus Low). The regression model representation is the following:

푖 푖 푖 푖 푖 푖 푅푒,푡 = 훼 + 훽푀(푅푀 − 푟)푡 + 훽푆푀퐵 푆푀퐵푡 + 훽퐻푀퐿 퐻푀퐿푡 + 휀푡,

where 푅푀 indicates the return of the market portfolio, 푟 is the risk free return rate and 푖 푖 푅푒,푡 = 푅푡 − 푟 indicates the calculated excess return of the asset 푖. 푆푀퐵푡 and 퐻푀퐿푡 are the two risk factors. Assumptions for the estimation include:

0 1 0 푖 푅 −푟 1. 퐸 휀푡 ( 푀,푡 ) = (0) 푆푀퐵푡 0 ( 퐻푀퐿푡 ) 2. Stationarity and ergodictity 3. Gordin’s condition. See Hayashi, F. (2000) for a detailed formulation of the required conditions

Since the assumptions cannot meet the standard assumptions of the regression analysis due to the fact that the errors of the stock returns exhibit serial correlation, Newey-West Standard Errors (Newey and West (1987)), are used in order to overcome possible autocorrelations (or correlation) and heteroskedasticity in the error terms of the model. In this way, possible autocorrelations in the returns are taken into account as well. Even though these autocorrelations decrease over time, they still seem to be present on a small time scale.

20 Hence, the covariance matrix used for the regression model estimation is the following:

′ −1 ′ −1 훺 = (퐸(푋푡푋푡)) 푊(퐸(푋푡푋푡)) ,

′ where 퐸(푋푡푋푡) is estimated by using the sample average, and with

푙−1 1 푙 − 푘 1 ̂푊 = ∑ 푒2푥 푥′ + ∑ ( ) ( ∑(푒 푒 푥 푥′ + 푒 푒 푥 푥′)), 푇 푡 푡 푡 푙 푇 푡 푡+푘 푡 푡+푘 푡 푡+푘 푡+푘 푡 푡 푘=1 푡 ′ ̂ where 푒푡 = 푦 + 푡 − 푥푡 (훽), 푙 is the bandwidth which is automatically selected by EViews ,

푘 is the number of the regressors, and 푥푡 are the past returns on time t. The results of the regression are summarized in table 4.1.1.

21 Table 4.1.1 Regression results by using Newey-West Standard Errors Est. t- Est. t- S.D p-value S.D p-value Coeff. Statistic Coeff. Statistic BARC RBS Alpha 0.00 0.009 0.04 0.96 Alpha -0.01 0.007 -1.53 0.12 Beta 1.59*** 0.194 8.19 0.00 Beta 1.59*** 0.221 7.21 0.00 SMB 0.00 0.004 0.31 0.75 SMB 0.003 0.003 0.91 0.36 HML 0.01*** 0.002 2.88 0.00 HML 0.01*** 0.002 3.70 0.00 BATS RDSB Alpha 0.01*** 0.002 3.84 0.00 Alpha 0.00 0.003 0.22 0.82 Beta 0.47*** 0.085 5.43 0.00 Beta 1.05*** 0.057 1.81 0.00 SMB 0.00 0.001 0.56 0.57 SMB 0.00 0.002 0.01 0.98 HML 0.002** 0.001 2.14 0.03 HML 0.00 0.001 1.05 0.29 BCB RR Alpha -0.02* 0.008 -1.84 0.06 Alpha 0.01*** 0.005 2.70 0.00 Beta 0.37** 0.157 2.35 0.02 Beta 1.16*** 0.148 7.78 0.00 SMB 0.00 0.006 0.81 0.42 SMB 0.00 0.004 0.23 0.81 HML 0.00 0.003 0.55 0.58 HML -0.00 0.002 -0.22 0.82 BP STAN Alpha -0.00 0.003 -0.41 0.67 Alpha 0.00 0.004 0.45 0.65 Beta 1.02*** 0.063 1.62 0.00 Beta 1.18*** 0.115 10.26 0.00 SMB -0.00 0.002 -0.10 0.91 SMB 0.00 0.002 0.84 0.39 HML 0.00 0.001 0.55 0.58 HML 0.00 0.001 0.64 0.51 HSBA TSCO Alpha -0.00 0.003 -0.73 0.46 Alpha 0.00 0.004 0.06 0.95 Beta 0.97*** 0.139 6.96 0.00 Beta 0.73*** 0.124 5.91 0.00 SMB -0.00 0.001 -0.86 0.39 SMB 0.00* 0.002 1.76 0.07 HML 0.00 0.001 1.27 0.20 HML -0.00 0.001 -0.14 0.88 IMT VOD Alpha 0.01*** 0.003 3.18 0.00 Alpha 0.00 0.004 0.55 0.58 Beta 0.48*** 0.104 4.67 0.00 Beta 0.78*** 0.074 10.54 0.00 SMB -0.00 0.001 -0.85 0.39 SMB 0.00* 0.002 -1.76 0.07 HML 0.00* 0.001 1.62 0.10 HML 0.00** 0.001 -2.25 0.02 LLOY Alpha -0.00 0.007 -0.76 0.44 Beta 1.47*** 0.154 9.52 0.00 SMB -0.00 0.005 -0.50 0.61 HML 0.01*** 0.002 3.33 0.00 * 10% significance level, ** 5% significance level and *** 1% significance level

From table 4.1.1, it can be seen that the alphas for BATS, IMT, and RR are statistically significant in 5% significance level, while betas are statistically and economically significant for all stocks. As far as the coefficients of SMB and HML are concerned, most

22 of them are insignificant and close to zero in 5% significance level. HML is statistically significant for BARC, BATS, RBS, VOD and LLOY, while SMB is insignificant for all stocks, which means it does not have any power to describe the stock returns. This could be due to the fact that the sample consists of growth stocks. The coefficient 훼푖 indicates the performance of investing in a particular stock after taking into account the other risks included (market, size and value). As it refered above, only three stocks have significant alphas with values close to zero. In other words, the investment has made an adequate return in relation to its risk. Beta measures how risky is the stock in relation to the market, which has a beta equal to 1. High-beta stocks are supposed to be more volatile than low-beta stocks. As mentioned above, all of the stocks have significant and positive betas. However, betas of BATS, BCB, IMT, TSCO, and VOD are found to be lower than one, with values 0.467, 0.370, 0.485, 0.735 and 0.785, respectively, which means that the stocks move less than the market. In addition, betas of BARC, RBS, RR, STAN, and LLOY are found to be higher than one with values 1.594, 1.595, 1.159, 1.183 and 1.471, respectively. All the other stocks’ betas are not statistically significantly different from one, so that the stocks move in line with the market. In other words, the stocks that have a beta of one, they will move the same direction as the index FTSE 100. 푖 In addition, the coefficient 훽푆푀퐵 shows the sensitivity of the stock to movements in small 푖 stocks, while the coefficient 훽퐻푀퐿 shows the sensitivity of the stock to movements in value stocks.

4.2 GARCH(1,1) model

The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model was suggested by Bollerslev (1986) as an extension of the ARCH model. A GARCH(1,1) specification is used in order to predict the future values of the stock returns. The standard GARCH(1,1) is simple and it satisfies the properties of ergodicity and stationarity. The formulation of the model is the following:

23 2 2 2 휎푡 = 푐 + 푎1휀푡−1 + 훽1휎푡−1,

where 휎푡 is the conditional variance of 휀푡 (conditional on the information at time 푡). The general form of the model has the following structure:

푟푡 = 푏0 + 푏1푥푡 + 푏2 푧푡 + 푢푡, where

 푢푡 = 휎푡−1 휀푡 2 2  휎푡−1 = √푐 + 훼1 푢푡−1 + 훽1 휎푡−2, is the conditional variance of 휀푡 at time 푡 − 1

 휀푡 zero mean i.i.d random variables

 훼1 ≥ 0 and 훽1 ≥ 0 should be satisfied in order to ensure that 휎푡−1 > 0

 푥푡 is the FTSE 100 returns

 푧푡 is the S&P 500 returns

In addition, a necessary and sufficient condition so that the moment of 2m,t order exists is that

푚 푚 휇(훼 , 훽 , 푚) = ∑ ( ) 훼 훼푗훽푚−푗 < 1, 1 1 푗 푗 1 1 푗=0

푗 where 훼0 = 1, 훼푗 = ∏푖=1(2푗 − 1), 푗 = 1, … .

Concerning the estimation of the GARCH model, (quasi) maximum likelihood is used. Table 4.2.1 shows the estimated models for the stock returns by using as explanatory variables the two indices.

24 Table 4.2.1 Regression results of GARCH (1,1) model for each stock Est. p- Est. z- p- S.D z-Stat. S.D Coeff. value Coeff. Stat. value BARC RR

b0 0.01 0.00 1.38 0.17 b0 0.01*** 0.00 2.69 0.01 FTSE 100 1.35*** 0.14 9.42 0.00 FTSE 100 0.96*** 0.15 6.60 0.00 S&P 500 -0.17 0.20 -0.89 0.37 S&P 500 0.21 0.17 1.22 0.22 c 0.00 0.00 0.29 0.78 c 0.00 0.00 1.57 0.11 alpha 0.36*** 0.09 3.79 0.00 alpha 0.12*** 0.04 2.82 0.00 beta 0.76*** 0.06 13.78 0.00 beta 0.84*** 0.05 15.30 0.00 BATS STAN

b0 0.01*** 0.00 4.70 0.00 b0 0.00 0.00 -0.05 0.96 FTSE 100 0.76*** 0.12 6.29 0.00 FTSE 100 1.01*** 0.12 8.41 0.00 S&P 500 -0.28** 0.12 -2.39 0.02 S&P 500 0.24*** 0.10 2.51 0.01 c 0.00*** 0.00 2.61 0.01 c 0.00 0.00 0.41 0.68 alpha -0.07 0.05 -1.51 0.13 alpha 0.03 0.04 0.63 0.52 beta 1.03*** 0.04 23.31 0.00 beta 0.85*** 0.33 2.54 0.01 BCB TSCO

b0 -0.01 0.01 -1.15 0.25 b0 0.00 0.00 0.05 0.61 FTSE 100 0.39* 0.24 1.65 0.10 FTSE 100 0.83*** 0.13 6.51 0.00 S&P 500 -0.20 0.31 -0.65 0.51 S&P 500 -0.23* 0.13 -1.83 0.07 c 0.01*** 0.00 4.58 0.00 c 0.00** 0.00 2.30 0.02 alpha 0.62*** 0.14 4.50 0.00 alpha 0.30** 0.14 2.22 0.03 beta 0.00 0.11 0.01 0.99 beta 0.32 0.24 1.33 0.18 BP VOD

b0 0.00 0.00 0.37 0.71 b0 0.00 0.00 0.72 0.47 FTSE 100 1.28*** 0.12 10.39 0.00 FTSE 100 0.87*** 0.16 5.35 0.00 S&P 500 -0.40*** 0.13 -3.07 0.00 S&P 500 -0.10 0.15 -0.70 0.48 c 0.00*** 0.00 8.15 0.00 c 0.00* 0.00 1.70 0.09 alpha 0.35*** 0.07 5.26 0.00 alpha 0.09* 0.05 1.81 0.07 beta -0.24*** 0.05 -4.89 0.00 beta 0.82*** 0.09 9.50 0.00 HSBA IMT

b0 0.00 0.00 -0.92 0.36 b0 0.01*** 0.00 3.88 0.00 FTSE 100 0.98*** 0.13 7.52 0.00 FTSE 100 0.52*** 0.16 3.30 0.00 S&P 500 -0.02 0.13 -0.14 0.88 S&P 500 -0.14 0.15 -0.94 0.34 c 0.00 0.00 1.39 0.16 c 0.00 0.00 1.21 0.22 alpha 0.19*** 0.08 2.48 0.01 alpha 0.14 0.11 1.34 0.18 beta 0.71*** 0.13 5.46 0.00 beta 0.63*** 0.22 2.89 0.00 * 10% significance level, ** 5% significance level and *** 1% significance level

25 According to this table, as far as the explanatory variables are concerned, FTSE 100 is positive and significant for all the assets except of BCB, while S&P 500 is statistically significant for BATS, BP, STAN and TSCO at the 5% significance level. In addition, alphas and betas are both significant for BARC, BP, HSBA, and RR at the 5% significance level. In order to test the null hypothesis that 푎 = 푏 = 0 for BARC, BP, HSBA, and RR, we perform a Wald test. Table 4.2.2 shows the estimated Wald-statistics of the test.

Table 4.2.2 Wald test results for BARC, BP, HSBA and RR Wald test Wald-statistic P-value BARC 1867.89 0.00 BP 39.533 0.00 HSBA 919.883 0.00 RR 346.86 0.00

Table 4.2.2 shows that the p-values of the Wald-statistic are smaller than 0.05 for BARC, BP, HSBA, and RR, so there is evidence to reject the null hypothesis.

T able 4.2.3 Sum of 휶ퟏ + 휷ퟏ for BARC, BP, STAN and TESCO

휶ퟏ + 휷ퟏ BARC 1.12 BP 0.11 HSBA 0.9 RR 0.96

In the table 4.2.3 it can be seen that the sums are smaller than 1 except of the one for

BARC. The process {푢푡} is stationary if 훼1 + 푏1 < 1. In addition, the sum 훼1 + 푏1 is measuring the persistence in volatility. As can be seen in table 4.2.2, the sums are 1.12, 0.11, 0.9, and 0.96 for BARC, BP, STAN and TESCO, respectively. Hence, high persistence in volatility can be observed in the returns of BARC, HSBA, and RR, while BP's returns have quite lower persistence.

26 4.3 Four-factor model

In the previous part, a three-factor model was estimated for each of the stock returns. This time, a four-factor model is estimated in an attempt to compare the significance of the new factors with the previous ones in the regression model (three-factor model). As it has been proved in the existing literature (Chen, et al. (1986)), macroeconomic factor models use economic time series, such as interest rates and inflation, in order to predict the future values of the stock returns. In this part, the three new factors that will be used in the regression model are the excess returns of the CPI inflation rate, the USD/GBP exchange rates, and the price of crude oil barrel in UK. The following are the calculated returns of the three factors in excess of the risk free return rate 푟:

1. 퐶푃퐼푡

2. 푂퐼퐿푡

3. BOEt

The representation of the three-factor regression model is the following:

푖 푖 푖 푖 푖 푖 푖 푅푒 푡 = 훼 + 훽푀(푅푀 − 푟)푡 + 훽퐶푃퐼 CPIt + 훽푂퐼퐿 푂퐼퐿푡 + 훽퐵푂퐸 BOEt + 휀푡,

푖 where 푅푀 is the market return of FTSE 100 index, 푟 is the risk free return rate and 푅푒,푡 = 푖 푅푡 − 푟 indicates the calculated excess return of the asset 푖. Assumptions for the model estimation are similar to those of the three-factor model. In order to perform the regression, Newey-West Standard Errors are used as in the three- factor model, so as to correct for possible autocorrelations and heteroskedasticity in the error terms of the model. In table 4.3.1 are summarized the regression results for each factor as well the alphas (constant).

27 Table 4.3.1 Regression results by using Newey-West Standard Errors Est. t- p- Est. t- p- S.D S.D coeff. Stat. value coeff. Stat. value BARC RBS Alpha 0.01 0.013 0.92 0.35 Alpha 0.01 0.013 1.28 0.20 Beta 1.58*** 0.232 6.81 0.00 Beta 1.54*** 0.200 7.69 0.00 Ex_CPI -0.35 0.863 -0.41 0.68 Ex_CPI -1.11 0.756 -1.48 0.14 Ex_Oil 0.07 0.091 0.83 0.40 Ex_Oil -0.01 0.086 -0.15 0.88 Ex_USD_GBP 0.77 0.524 1.48 0.14 Ex_USD_GBP 0.21 0.244 0.86 0.38 BATS RDSB Alpha 0.01** 0.007 2.16 0.03 Alpha -0.01 0.006 -1.16 0.24 Beta 0.45*** 0.091 4.97 0.00 Beta 1.04*** 0.071 1.46 0.00 Ex_CPI -0.17 0.291 -0.60 0.54 Ex_CPI 0.35 0.289 1.22 0.22 Ex_Oil -0.02 0.043 -0.57 0.56 Ex_Oil 0.06 0.049 1.31 0.19 Ex_USD_GBP -0.01 0.113 -0.16 0.86 Ex_USD_GBP 0.39*** 0.128 -3.06 0.00 BCB RR Alpha 0.00 0.021 0.35 0.72 Alpha 0.00 0.012 0.55 0.57 Beta 0.31*** 0.171 1.78 0.00 Beta 1.15*** 0.148 7.74 0.00 Ex_CPI -1.06 0.993 -1.07 0.28 Ex_CPI 0.30 0.406 0.75 0.45 Ex_Oil -0.04 0.087 -0.47 0.63 Ex_Oil -0.02 0.072 -0.37 0.70 Ex_USD_GBP -0.52 0.360 -1.45 0.14 Ex_USD_GBP -0.27* 0.162 -1.66 0.09 BP STAN Alpha -0.00 0.006 -0.22 0.82 Alpha 0.01* 0.006 1.73 0.08 Beta 1.01*** 0.062 1.62 0.00 Beta 1.15*** 0.129 8.93 0.00 Ex_CPI -0.01 0.348 -0.04 0.97 Ex_CPI -0.43 0.294 -1.47 0.14 Ex_Oil 0.09* 0.052 1.77 0.07 Ex_Oil 0.13** 0.057 2.30 0.02 Ex_USD_GBP -0.14 0.229 -0.62 0.53 Ex_USD_GBP 0.03 0.141 0.19 0.84 HSBA TSCO Alpha 0.00 0.006 0.23 0.81 Alpha 0.00 0.010 0.85 0.39 Beta 0.98*** 0.149 6.61 0.00 Beta 0.70*** 0.121 5.78 0.00 Ex_CPI -0.14 0.351 -0.39 0.69 Ex_CPI -0.38 0.389 -0.97 0.32 Ex_Oil -0.01 0.042 -0.37 0.71 Ex_Oil -0.03 0.047 -0.62 0.53 Ex_USD_GBP 0.16 0.170 0.94 0.34 Ex_USD_GBP 0.01 0.138 0.09 0.92 IMT VOD Alpha 0.02*** 0.006 3.26 0.00 Alpha -0.01 0.009 -0.65 0.51 Beta 0.48*** 0.104 4.66 0.00 Beta 0.84*** 0.083 10.0 0.00 Ex_CPI -0.50* 0.307 -1.64 0.10 Ex_CPI 0.35 0.317 1.11 0.26 Ex_Oil -0.01 0.043 -0.21 0.87 Ex_Oil -0.10* 0.059 -1.78 0.07 Ex_USD_GBP -0.12 0.131 -0.96 0.33 Ex_USD_GBP 0.24* 0.132 1.81 0.07 LLOY Alpha 0.01 0.011 1.55 0.12 Beta 1.48*** 0.169 8.75 0.00 Ex_CPI -0.89 0.638 -1.39 0.16 Ex_Oil 0.05 0.079 0.69 0.48 Ex_USD_GBP 0.78** 0.374 2.09 0.03 * 10% significance level, ** 5% significance level and *** 1% significance level

28 The results in table 4.3.1 show that all alphas are insignificant except of those of BATS and IMT, which are significant and positive in 5% significance level. Betas are statistically and economically significant for all stocks in 5% significance level. The factor Ex_CPI is insignificant for all stocks. The exchange rate Ex_USD_GBP is significant for RDSB and LLOY, while Ex_Oil is significant and positive for STAN in 5% significance level. As mentioned in chapter 4.1, beta measures how risky is the stock in relation to the market. The values of the significant betas higher than one estimated in table 4.3.1 are 1.586, 1.1486, 1.545, 1.151 and, 1.153 for BARC, LLOY, RBS, RR, and STAN, respectively. The beta of BARC has the highest value, which shows that it is the most volatile stock. Moreover, the values of betas for BATS, BCB, IMT, TSCO, and VOD are found to be equal to 0.454, 0.306, 0.485, 0.700, and 0.839, respectively. These stocks move less than the market since their betas are lower than one. The rest of the stocks’ betas are not statistically significantly different from one, which shows that stocks move in line with the market. In addition, comparing the estimated betas of the tables 4.1.1 and 4.3.1, it can be observed that the low-beta stocks and high-beta stocks estimated with the three-factor model are the same with those estimated with the four-factor model.

4.4 EGARCH(1,1) model

Standard GARCH models are used in econometrics in order to model time series where volatility appears to be time-dependent (volatility clustering). However, they present some drawbacks due to their model formulation. In particular, GARCH models assume that the 2 future values of 휎푡 depend only on the value of 휀푡, but they do not depend on the sign of 2 휀푡 (positive or negative). In other words, the conditional variance of 휎푡 is considered to be symmetric towards the past shocks 휀푡−1. According to that, there could exist significant problems as far as the estimation of the model is concerned. For instance, GARCH models do not take the leverage effect into account which is related to the good news (variance of returns decreases) and bad news (variance of returns increases) of the stock market.

29 Due to that reason, Nelson (1991) proposed the Exponential GARCH (EGARCH) model, which gives a solution to the drawbacks of GARCH since the Generalized Error Distribution (Box and Tiao 1973) is used in order to account for the fat tails of the returns. The probability density function of GED with mean zero and variance equal to one is the following:

1 푧 푣 푣∗푒푥푝[− | | ] 2 휆 푓(푧) = 푣 , −∞ < 푧 < +∞, 푣 > 0, 1+ 1 휆2 2 Γ( ) 푣

1 2 − 1 2 2 푣Γ( ) 푣 where 휆 = [ 3 ] , Γ(. ) is Gamma distribution and 푣 is a parameter which Γ( ) 푣 parameterizes how fat the tails of the return distribution is. EGARH(1,1) will be used in order to predict the future values of the stock returns. The model with explanatory variables has the following formulation:

푟푡 = 푏0 + 푏1푥푡 + 푏2 푐푡 + 휀푡 2 2 ln(휎푡 ) = 푎0 + 훽1 ln(휎푡−1) + 휃1푧푡−1 + 훾1(|푧푡−1| − 퐸|푧푡−1|), where 휀푡 = 푧푡휎푡, 푟푡 is the stock return at time 푡, 푥푡 is the FTSE 100 returns, 푐푡 is the S&P 2 500 returns, 푧푡 is iid and follows the GED with mean zero and variance 1, 휎푡 is the conditional variance of 휀푡 at time 푡, while 휎푡 = 푧푡 = 0 푓표푟 푡 < 0. Under the assumption of

GED for 푧푡, it holds that

2 훤 (푣) 퐸|푧푡−1| = 1 , 푧푡~퐺퐸퐷 1 3 2 [훤 (푣) 훤 (푣)]

If the coefficient 훽1 is equal to zero then the model is the exponential ARCH (EARCH), since the equation of the conditional variance consists only of the values of 퐸|푧푡−1|.

30 The relation between the stock returns and their future volatility can be “captured” with a function which depends on the magnitude and the sign of 푧푡. This function is a linear combination of 푧푡−1 and |푧푡−1|, and it has the following formulation:

g(zt−1) = θ1zt−1 + γ1(|zt−1| − E|zt−1|)

2 Let for instance θ1 be equal to zero and γ1 > 0. Then the innovation of ln(휎푡 ) is positive

(negative) when |zt−1| is greater (smaller) than the value of E|zt−1|. This term represents the influence of the magnitude. The term θ1zt−1 indicates the influence of the sign. 2 The way that ln(휎푡 ) is estimated by the EGARCH(1,1) model allows the conditional variance to be positive in every time period, so that the parameters of the model can be used without any restrictions. In table 4.4.1 the results of an EGARCH(1,1) model are illustrated for each stock.

31 Table 4.4.1 Regression results of EGARCH(1,1) Est. Est. S.D z-Stat. p-value S.D z-Stat. p-value Coeff. Coeff. BARC RR

b0 -0.00 0.004 -0.99 0.31 b0 0.01*** 0.004 2.45 0.01 FTSE100 1.38*** 0.138 10.00 0.00 FTSE100 0.74*** 0.158 4.74 0.00 S&P 500 -0.10 0.162 -0.63 0.52 S&P 500 0.40** 0.173 2.32 0.02 α0 -0.30 0.195 -1.57 0.11 α0 -0.29 0.192 -1.55 0.12 β1 0.20* 0.117 1.78 0.07 β1 0.13 0.084 1.59 0.11 θ1 -0.22*** 0.073 -3.15 0.00 θ1 -0.17*** 0.055 -3.14 0.00

γ1 0.96*** 0.029 33.5 0.00 γ1 0.96*** 0.029 32.8 0.00 BATS STAN

b0 0.01*** 0.004 2.78 0.00 b0 0.00 0.002 0.07 0.93 FTSE100 0.76*** 0.127 5.99 0.00 FTSE100 1.27*** 0.085 14.9 0.00 S&P 500 -0.30*** 0.111 -2.69 0.00 S&P 500 -0.06 0.088 -0.76 0.44

α0 -0.04*** 0.000 -11.4 0.00 α0 12.01*** 0.475 -25.3 0.00

β1 -0.05*** 0.000 -18.1 0.00 β1 0.07 0.065 1.19 0.23 θ1 -0.01 0.025 -0.76 0.44 θ1 0.03 0.060 0.59 0.55

γ1 0.98*** 0.000 21.1 0.00 γ1 -1.00*** 0.033 -30.3 0.00 BCB TSCO b0 -0.01*** 0.006 -2.38 0.01 b0 0.00 0.003 0.15 0.88 FTSE100 0.41 0.250 1.64 0.09 FTSE100 0.78*** 0.114 6.86 0.00 S&P 500 -0.18 0.282 -0.65 0.51 S&P 500 -0.15 0.120 -1.31 0.19 α0 -3.37*** 1.169 -2.88 0.00 α0 -0.53*** 0.229 -2.35 0.01 β1 0.78*** 0.237 3.30 0.00 β1 -0.04 0.076 -0.58 0.56 θ1 -0.31** 0.164 -1.92 0.05 θ1 -0.20*** 0.059 -3.38 0.00

γ1 0.36 0.246 1.46 0.14 γ1 0.90*** 0.038 24.1 0.00 BP VOD b0 0.00 0.003 -0.03 0.97 b0 0.00 0.004 0.07 0.94 FTSE100 1.41*** 0.103 13.70 0.00 FTSE100 0.89*** 0.152 5.90 0.00 S&P 500 -0.49*** 0.107 -4.61 0.00 S&P 500 -0.12 0.149 -0.80 0.42 α0 -8.02*** 2.234 -3.59 0.00 α0 -0.44 0.300 -1.47 0.13 β1 0.46*** 0.189 2.41 0.01 β1 0.11 0.099 1.14 0.25 θ1 -0.33 0.144 -2.27 0.02 θ1 -0.16** 0.070 -2.28 0.02

γ1 -0.24 0.360 -0.65 0.51 γ1 0.94*** 0.044 21.5 0.00 HSBA IMT b0 -0.01*** 0.002 -2.59 0.01 b0 0.01*** 0.004 3.40 0.00 FTSE100 0.75*** 0.057 13.2 0.00 FTSE100 0.64*** 0.170 3.74 0.00 S&P 500 0.17*** 0.043 3.89 0.00 S&P 500 -0.18 0.147 -1.21 0.22 α0 12.86*** 0.320 -40.1 0.00 α0 -1.93 1.667 -1.16 0.24 β1 0.19*** 0.084 2.35 0.01 β1 0.17 0.180 0.97 0.33 θ1 0.02 0.028 0.59 0.55 θ1 0.16 0.134 1.18 0.23

γ1 -0.96*** 0.040 -24.2 0.00 γ1 0.70*** 0.267 2.64 0.00

* 10% significance level, ** 5% significance level and *** 1% significance level

32 Table 4.4.1 shows that the majority of the coefficients for both FTSE 100 and S&P 500 are significant. However, a main drawback of the standard GARCH (1,1) which was estimated in chapter 4.2, is that it does not take into account the leverage effect as mentioned in the beginning of this chapter. A leverage effect is found when the coefficient

휃1 is significant and negative. In the above table it can been seen that 휃1 is significant and negative for BARC,BCB, BP, TSCO, VOD and RR in 5% significance level. That means that according to the estimates there is a negative correlation between the past returns and future volatility of those assets. The higher the leverage occurs due to a negative return, results in lower equity prices. As a result, a positive shock (good news) 2 has less effect on the conditional variance [ln(휎푡 )] compared to a negative shock (bad news).

33 5. Comparison of the models

5.1 Forecast

In this section the performance of the previous estimated models is evaluated. In order to do that, the following methodology is used. Initially, the models are estimated by using a sample of the period 2000-2010 according to the four previous methods. Since there are thirteen stocks, for each method two stocks are chosen in order to predict their future returns. Concerning the multifactor models, two stocks are chosen randomly according to their betas since all estimated betas are found to be significant in both models, so that there is a high-beta stock and a low-beta stock. For the standard GARCH(1,1) two stocks are chosen randomly from the stocks that found to have persistence in volatility, while for the EGARCH(1,1) model two stocks are chosen randomly from the stocks that found to have a leverage effect. The second step is to forecast the future returns for the period 2010-2015, which is chosen as forecast sample. The method used by EViews in order to perform the forecasting is Dynamic forecasting for the standard GARCH(1,1) as well as for EGARCH(1,1) and Static forecasting for the 3-factor and 4 -factor model. Dynamic forecasting uses the forecasted value of the lagged dependent variable, while static forecasting uses the actual value of the lagged dependent variable. Both methods always give the same results in the first period of a multi-period forecast. The forecasts are created from the regression equations used in the previous four models. According to that, given the vector of explanatory variables, the resulting forecast of the returns of the stock is estimated by applying the estimated regression coefficients to the explanatory variables. Furthermore, forecasts are estimated with error which mainly has two kinds of sources. The first one is due to the residuals in the equation since they are unknown for the forecasted period. In particular, in dynamic forecasting the residual uncertainty comes from the fact that lagged dependent variables depend on lagged residuals. EViews sets these innovations equal to their expected values, which differ from their actual ones. Generally, the forecast standard errors estimated by EViews account for both residual and coefficient uncertainty. After the forecasting, both models (forecast and sample model) are opened as a group so as to compare the performance of the estimated model to the forecasted model. Forecasts are

34 evaluated by estimating the forecasts errors which show the quality of the forecasting model.

Let 푠휏 be the actual and 푠̂휏 the forecast volatility at time 휏, while the forecast period is between 휏 + 1 to 휏 + 푛. Then the forecast error statistics are the following:

1 2  푅푀푆퐸 = √[ ∑푡+푛 ( 푠̂ − 푠 ) ] 푛 휏=푡+1 휏 휏 1  푀퐴퐸 = ∑푡+푛 |푠̂ − 푠 | 푛 휏=푡+1 휏 휏

1 푡+푛 푠̂휏 − 푠휏  푀퐴푃퐸 = ∑휏=푡+1 | | ∗ 100 푛 푠휏

Table 5.1 has the estimated root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) for each model.

Table 5.1.1 Evaluation statistics for the performance of the models Evaluation statistics Model: 3-factor Forecast RMSE MAE MAPE RBS 0.088 0.069 289.105 BCB 0.122 0.083 93.033 Model: GARCH(1,1) Forecast RMSE MAE MAPE HSBA 0.034 0.027 250.439 RR 0.058 0.041 183.361 Model: 4-factor Forecast RMSE MAE MAPE BARC 0.087 0.068 154.700 BCB 0.120 0.080 85.622 Model: EGARCH(1,1) Forecast RMSE MAE MAPE BARC 0.081 0.062 142.890 BP 0.065 0.038 164.642

As can be seen from table 5.1.1, forecasted returns for HSBA after using standard GARCH(1,1) are the closest to the eventual returns since they have the smallest error values, both RMSE and MAE are low. Concerning the EGARCH(1,1) model BP

35 forecasted returns have small errors as well. In addition, RR and BARC forecasts present small RMSE and MAE as well. Both factor models have higher evaluation statistics than the GARCH models. Concerning MAPE, it is noteworthy that it gets quite high values in all four models, particularly for RBS and HSBA. A reason for that would be that 푠휏 is much smaller than 푠̂휏 − 푠휏 (in absolute values), so the deviation of them, gives too high numbers. Hence, it can be concluded that both standard GARCH(1,1) and EGARCH(1,1) fit better the data while they give the most accurate forecasts between the four models. The graphs 5.1.1, 5.1.2, 5.1.3 and 5.1.4 depict the forecasted returns in comparison with the actual returns in the out-of-sample forecasting interval (2010-2015).

3-factor model

Graph 5.1.1: Predicted versus actual returns for RBS and BCB. RBSF and BCBF are the forecasted values.

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EX_RBSF EX_RBS

36 .4

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EX_BCB EX_BCBF

Graph 5.1.1 shows that the forecasted returns for RBS follow the ups and downs of the actual returns in most of the of the time period, however in some periods there is a higher deviation from the actual values which can be attributed to the factors included in the model. Concerning the forecasted returns of BCB, they present a high deviation from the actual returns in periods of high volatility, as the evaluation statistics prove.

37 GARCH(1,1) model

Graph 5.1.2: Predicted versus actual returns for RR and HSBA. RRF and HSBAF are the forecasted values.

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ROLLS-ROYCE HOLDINGS RR. RRF

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HSBC HDG. (ORD $0.50) HSBA HSBAF

Graph 5.1.2 shows that both forecasted returns of RR and HSBA fit well on their actual returns, as it was expected and the evaluation statistics show the deviation from the eventual returns is low.

38 4-factor model

Graph 5.1.3: Predicted versus actual returns for BARC and BCB. BARCF and BCBF are the forecasted values.

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EX_BARCF EX_BARC

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EX_BCBF EX_BCB

Graph 5.1.3 shows that the forecasted returns of BARC tend to move in line with the actual returns, while in many time periods they have a really good fit on the actual returns.

39 In addition, forecasted returns of BCB present a high deviation from the actuals ones in time periods of high volatility, while they tend to fit well in periods of low volatility.

EGARCH(1,1) model

Graph 5.1.4: Predicted versus actual returns for BARC and BP. BARCF and BPF are the forecasted values.

40 Graph 5.1.4 shows that both forecasted returns of BARC and BP fit very well on their actual returns and particularly those of BP which have the smallest RMSE, MAE and MAPE.

5.2 Vector Autoregressive Models

In this chapter, vector autoregressive models are estimated for the factors of both 3-factor and 4-factor models and the indices used as explanatory variables in the standard GARCH(1,1) and EGARCH(1,1) models. If the factors and indices can be represented as VAR models, then future values can be predicted from the past values. ′ Let 푌푡 = (푦1푡, 푦2푡, … , 푦푁푡) be a 푁 × 1 vector of multivariate stochastic time series, then a vector autoregressive model of order p [VAR(p)] has the following form:

푌푡 = 푐 + 훤1 푌푡−1 + 훤푡−푝 푌푡−푝 + 휀푡,

where 훤푖 are 푁 × 푁 coefficient matrices and 휀푡 ~ 푊ℎ푖푡푒 푁표푖푠푒 with mean zero. The process to get the VAR models is the following: 1. Select the number of lags by using Schwarz-Bayesian criteria Hannan-Quinn information criterion. 2. Estimate the VAR model. 3. Evaluate the significance of the coefficients.

Many different consistent criteria have been proposed in order to determine the correct order of the VAR models, such as FPE (Final Prediction Error), AIC (Akaike information criterion) and HQ (Hannan-Quinn information criterion). Bayesian information criterion (BIC) which proposed by Schwartz (1978) is quite popular and it selects the model that minimizes the following equation:

푙표푔푇 퐵퐼퐶(푝) = 푙표푔|훴̂(푝)| + 2푝푁2, 푇

41 where 훴̂(푝) is the estimate of the residual covariance matrix of a model of order 푝, 푡 = 1, … , 푇 observations and 푁 is the number of variables. The formulas of the other estimators are the following:

푇 + 푁푝 + 1 푁 퐹푃퐸(푝) = [ ] det (훴̂(푝)) 푇 − 푁푝 + 1 2푝푁2 퐴퐼퐶(푝) = 푙표푔|훴̂(푝)| + 푇 2log (푙표푔푇) 퐻푄(푝) = 푙표푔|훴̂(푝)| + 푝 푁2 푇

FPE and AIC are inconsistent estimators in the sense that they determine the optimal model order in the limit of an infinite sample ((Rachev, et al. (2007)). HQ is close to BIC, however it gives an extra penalty for extra coefficients. In order to select the number of lags in the VAR models the Schwarz-Bayesian criteria and Hannan-Quinn information criterion will be used. FPE and AIC will be estimated in order to compare their results to the other two criteria.

 3-factor model

Table 5.2.1 VAR lag order selection criteria Lag FPE AIC SC HQ 0 0.044143 5.393.314 5.448651* 5.415769 1 0.041015 5.319777 5.541128 5.409599* 2 0.039757 5.288513 5.675876 5.445700 3 0.039002* 5.269072* 5.822448 5.493625 4 0.039638 5.284767 6.004156 5.576686 5 0.042826 5.361355 6.246757 5.720641 6 0.043777 5.382172 6.433587 5.808824 7 0.042734 5.356494 6.573922 5.850512 8 0.043745 5.377813 6.761253 5.939197

* indicates lag order selected by the criterion

42 Table 5.2.1 shows that based on Bayesian information criterion the optimal lag order is zero, while HQ show that the optimal lag order is 1. Both FPE and AIC show that the optimal lag order is three.

Table 5.2.2 Estimated means for MRF, SMB, and HML MRF SMB HML C 0.000128 0.001639 0.005433 (0.00353) (0.00155) (0.00190) [ 0.03629] [ 1.05610] [ 2.86405]

* Standard errors in ( ) & t-statistics in [ ]

Table 5.2.2 shows the means of MRF, SMB, and HML, which are all very close to zero.

Table 5.2.3 Estimated coefficients for MRF (FTSE 100 - Risk free), and HML with one Lag* MRF SMB HML MRF(-1) -0.087636 0.077709 0.43604 (0.07684) (0.03321) (0.03912) [-1.14057] [ 2.33979] [ 1.11457]

SMB(-1) 0.010485 0.027215 0.047272 (0.00175) (0.07559) (0.08904) [ 0.05995] [ 0.36001] [ 0.53088]

HML(-1) -0.165593 0.114074 0.331413 (0.00140) (0.06060) (0.07138) [-1.18118] [ 1.88248] [ 4.64289]

C 0.001166 0.000869 0.003419 (0.00364) (0.00157) (0.00185) [ 0.32037] [ 0.55272] [ 1.84542]

* Standard errors in ( ) & t-statistics in [ ]

43 Table 5.2.3 has the estimated coefficients of MRF, SMB, and HML after using VAR(1) model in order to test if future values can be predicted from the past values. It can be observed that for MRF none of the estimated coefficients is significant. Concerning SMB, the estimated coefficients of MRF(-1) is statistically and economically significant. Finally, the coefficient of HML(-1) is statistically and economically significant for HML.

Table 5.2.4 Evaluation statistics for the performance of the factors using VAR(1)

Evaluation statistics Model: 3-factor

Forecast RMSE MAE MAPE

MRF 0.047 0.034 355.153 HML 0.021 0.016 734.913 SMB 0.025 0.018 610.737

From table 5.2.4 it can be seen that the estimated values of RMSE, MAE, and MAPE are low, with lowest those of HML and SMB. As a result, the future values of MRF, HML, and SMB estimated by VAR(1) models present a quite low deviation from the actual ones.

Graph 5.2.1: Predicted vs actual values of MRF, HML, and SMB

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-.25 -.08 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 MRF MRF (FORECASTED) HML (FORECATSED) HML

44 .06

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SMB SMB (FORECATSED)

Graph 5.2.1 shows that the forecasted values of MRF, HML, and SMB are close to zero and one, while they move according to their conditional means, so the VAR(1) model is optimal.

 4-factor model

Table 5.2.5 VAR lag order selection criteria Lag FPE AIC SC HQ 0 1.57e-12 -1.582644 -1.575266 -1.579650 1 1.13e-13 -1.845695 -18.08803* -18.30725* 2 1.07e-13* -18.51273* -1.784868 -1.824.327 3 1.16e-13 -1.843823 -1.747904 -1.804900 4 1.30e-13 -1.832272 -1.706840 -1.781373 5 1.41e-13 -1.824033 -1.669087 -1.761158 6 1.52e-13 -1.817.125 -1.632666 -1.742274 7 1.60e-13 -1.812678 -1.598706 -1.72850 8 1.66e-13 -1.809795 -1.566310 -1.710.992

Table 5.2.5 shows that based on Bayesian information criterion the optimal lag order is one. The same lag gives the HQ information criterion, while FPE and AIC show that the optimal lag order is two.

45 Table 5.2.6 Estimated coefficients for MRF (FTSE 100 - Risk free), CRUDE_OIL_BRENT, UK_CPI and USD_TO_GBP with one Lag*

MRF EX_CPI_INFL EX_CRUDE_OIL EX_USD_GBP

MRF(-1) -0.082376 0.001553 0.399800 0.050356 (0.07590) (0.00465) (0.14402) (0.04181) [-1.08536] [ 0.33415] [ 2.77591] [ 1.20429]

EX_CPI_INFL(-1) -0.017932 0.971899 -0.414490 -0.314166 (0.31568) (0.01934) (0.59905) (0.17392) [-0.05680] [ 50.2616] [-0.69191] [-1.80637]

EX_CRUDE_OIL(-1) -0.020896 0.008711 -0.034864 0.034111 (0.03983) (0.00244) (0.07558) (0.02194) [-0.52465] [ 3.57044] [-0.46128] [ 1.55456]

EX_USD_GBP(-1) 0.127366 -0.003386 0.072189 0.016642 (0.13870) (0.00850) (0.26320) (0.07641) [ 0.91831] [-0.39850] [ 0.27428] [ 0.21779]

C 0.000889 0.000500 0.015125 0.005777 (0.00747) (0.00046) (0.01418) (0.00412) [ 0.11898] [ 1.09301] [ 1.06633] [ 1.40292]

* Standard errors in ( ) & t-statistics in [ ]

Table 5.2.6 has the estimated coefficients of MRF, EX_CPI_INFL, EX_CRUDE_OIL, and EX_USD_GBP after using VAR(1) model in order to test if future values can be predicted from the past values. It can be observed that the estimated coefficients for MRF are insignificant for the four endogenous variables. Concerning EX_CRUDE_OIL, the coefficient of MRF(-1) is statistically and economically significant. Moreover, the coefficients of EX_CPI_INFL(-1) and EX_CRUDE_OIL(-1) are significant for EX_CPI_INFL, while none of the coefficients of the four endogenous variables is significant for EX_USD_GBP.

46 Table 5.2.7 Evaluation statistics for the performance of the factors

Evaluation statistics Model: 4-factor

Forecast RMSE MAE MAPE MRF 0.043 0.033 131.820 EX_CPI INFL 0.037 0.033 758.243 EX_CRUDE OIL 0.071 0.054 621.991 EX_USD_GBP 0.024 0.020 364.284

From table 5.2.7 it can be observed that both RMSE and MAE are significantly lower than those estimated for MRF in the 3-factor model approach. In addition, the evaluation statistics for EX_CPI INFL. and EX_USD_GBP are very low as well. EX_CRUDE OIL has the highest values of RMSE and MAE, so its future values show a higher deviation from the actual ones.

47 Graph 5.2.2: Predicted vs actual values for MRF, CPI Inflation, Crude Oil, and USD to GBP

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EX_USD_GBP EX_CRUDE_OIL EX_USD_GBP (FORECASTED) EX_CRUDE_OIL (FORECASTED)

Graph 5.2.2 shows that although the evaluation statistics for EX_CRUDE_OIL are quite higher than the other three factors, its forecasted values move according to its conditional mean. Moreover, forecasted values of MRF, EX_CPI_INFL, and EX_USD_CBP tend to move according to their conditional means as well, so that information about their future values can be obtained.

48  Standard GARCH(1,1) and EGARCH(1,1)

Table 5.2.8 VAR lag order selection criteria Lag FPE AIC SC HQ 0 1.40e-06 -7.804.669 -7.767.777 -7.789.699 1 1.24e-06 -7.920.824 -7.810149* -7.875913* 2 1.24e-06 -7.922.094 -7.737.635 -7.847.243 3 1.29e-06 -7.886.485 -7.628.243 -7.781.694 4 1.30e-06 -7.874.916 -7.542.890 -7.740.183 5 1.23e-06 -7.935.584 -7.529.775 -7.770.912 6 1.21e-06* -7.953693* -7.474.100 -7.759.080 7 1.26e-06 -7.911.977 -7.358.601 -7.687.424 8 1.29e-06 -7.882.868 -7.255.709 -7.628.374

Table 5.2.8 shows that based on Bayesian information criterion the optimal lag order is one. The same lag gives the HQ information criterion, while there is a big difference of the estimated optimal lag order of FPE and AIC, which is equal to six.

Table 5.2.9 Estimated coefficients for FTSE 100 and S&P500 with one Lag* FTSE_100 S_P_500 FTSE_100(-1) 0.006518 0.270012 (0.13324) (0.12719) [ 0.04892] [ 2.12288]

S_P_500(-1) -0.124459 -0.193129 (0.13788) (0.13162) [-0.90268] [-1.46729]

C 0.001961 0.003726 (0.00354) (0.00338) [ 0.55339] [ 1.10152]

* Standard errors in ( ) & t-statistics in [ ]

The table 5.2.9 shows the estimated coefficients of FTSE 100 and S&P 500 after using VAR(1) model in order to test if future values can be predicted from the past values. It can be observed that only the coefficient of FTSE_100(-1) is significant for S_P_500.

49 Table 5.2.10 Evaluation statistics for the performance of the factors

Evaluation statistics Model:

GARCH(1,1)/EGARCH(1,1) Forecast RMSE MAE MAPE FTSE_100 0.038 0.031 1155.19 S&P 500 0.037 0.030 454.198

From table 5.2.10 it can be observed that both RMSE and MAE are low for FTSE_100 and S&P 500, which shows that forecasted values give information about the eventual ones, since they are close to their actual ones. However, MAPE is too high for FTSE_100 and quite high for S&P 500.

Graph 5.2.3: Predicted vs actual values for FTSE 100 and S&P 500

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RETURN INDEX S&PCOMP RETURN INDEX FTSE100 RETURN INDEX S&PCOMP (FORECASTED) RETURN INDEX FTSE100 (FORECASTED)

Graph 5.2.3 shows that the forecasted values of FTSE_100 and S&P 500 are too close to zero as their conditional means which is optimal forecast.

50 6. Conclusions

The four different models proposed is this thesis aim to predict the future returns of thirteen assets. The first model is the three-factor model of Fama and French, the second one is a multi-factor model which includes the following macroeconomic factors: CPI inflation rate, USD/GBP exchange rates and the price of crude oil barrel in UK. In addition, two different GARCH specifications are used. The standard GARCH(1,1) so as to capture the “volatility clustering” and the EGARCH(1,1) in order to take into account the leverage effect. In order to compare the estimated models, root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) for each model are used. In the three-factor model it is found that all stocks have positive and significant betas. Moreover, five stocks are found to have betas less than one, which is an indicator that the stocks move less than the market and five stocks are found to have betas higher than one, which indicates that they are more volatile. The other stocks (BP, HSBA, and RDSB), have betas that are not statistically significantly different from one, so they move in line with the market. Compared to the four-factor model, it can be observed from the estimated betas that the number of stocks which are more volatile, are equal to the number of stocks found in the three-factor model. In addition, five stocks found to have betas lower than one, which is equal to the number of stocks found in the three-factor model, while only three stocks found to move in line with the market. Furthermore, the standard GARCH(1,1) shows that high a persistence in volatility can be observed in the returns of BARC, HSBA and RR, while BP's returns have quite lower persistence according to the sum of coefficients 훼1 and 훽1. Additionally, the EGARCH(1,1) shows that a leverage effect can be found in the data returns of BARC, BCB BP, TSCO, VOD, and RR. The higher leverage occurs due to a negative return which results in lower equity prices. VOD has the highest coefficient 휃1 which accounts for the leverage effect. After forecasting, it can be concluded that the standard GARCH(1,1) and EGARCH(1,1) fit better than the other two multi-factor models in predicting the future returns of HSBA and BP, respectively since they have the smallest error values in the evaluation statistics, both RMSE and MAE are low. In addition, RBS and BARC forecasts with the three-factor

51 model and the four-factor model, respectively, present small RMSE and MAE as well. In final, VAR(1) models are estimated for the factors used in the multi-factor models as well as the market indices used as explanatory variables in the standard GARCH(1,1) and EGARCH(1,1). The results show that according to the factors SMB and HML there is limited predictability in the three-factor model since the evaluation statistics are quite high. However, the factor EX_USD_GBP in the four-factor model have the lower estimated RMSE and MAE, so that information can be obtained from its forecasted values. In the GARCH models there is predictability as far as the two explanatory variables are concerned, since their evaluation statistics are quite low, except the value of MAPE which is quite high in all estimated forecasts. In general, all graphs in the part 5.2 show that the forecasted values of both factors and indices are equal to their conditional means which is optimal. Further extensions of the models used in this thesis would be to choose different factors for the four-factor model, as well as alternative GARCH specifications. In particular, IGARCH (Integrated GARCH) could be used instead of the standard GARCH due to the fact that the sum of the coefficients 훼푖 and 푏푖 is too close to one, which is evidence to use GARCH model according to several empirical studies.

52 References

Andersen, T. G., & Bollerslev, T. (1998). Answering the Skeptics: Yes, Standard Volatility Models do Provide Accurate Forecasts. International Economic Review, 39(4), 885–905. Athreya, K. B., & Pantula, S. G. (1986). A note on strong mixing of ARMA processes. Statistics & Probability Letters, 4(4), 187-190. Avramov, D., & Chordia, T. (2006). Asset pricing models and financial market anomalies. Review of Financial Studies, 19(3), 1001-1040. Banz, R. W. (1981). The relationship between return and market value of common stocks. Journal of Financial Economics, 9(1), 3-18. Berndt, E. R., Hall, B. H., Hall, R. E., & Hausman, J. A. (1974). Estimation and inference in nonlinear structural models. In Annals of Economic and Social Measurement, Volume 3, number 4 (pp. 653-665). Black, F.. (1972). Capital Market Equilibrium with Restricted Borrowing. The Journal of Business,45(3), 444–455. Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), 307-327. Carhart, M. M. (1997). On persistence in mutual fund performance. The Journal of Finance, 52(1), 57-82. Chen, N. F., Roll, R., & Ross, S. A. (1986). Economic forces and the stock market. Journal of Business, 383-403. Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica: Journal of the Econometric Society, 987-1007. Engle, R.. (2001). GARCH 101: The Use of ARCH/GARCH Models in Applied Econometrics. The Journal of Economic Perspectives, 15(4), 157–168. Fama, E. F., Fisher, L., Jensen, M. C., & Roll, R. (1969). The adjustment of stock prices to new information. International Economic Review, 10(1), 1-21. Fama, E. F., & French, K. R. (1992). The cross‐section of expected stock returns. The Journal of Finance, 47(2), 427-465. Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33(1), 3-56. Fama, E. F., & French, K. R. (2015). A five-factor asset pricing model. Journal of Financial Economics, 116(1), 1-22. Fiorentini, G., Calzolari, G., & Panattoni, L. (1996). Analytic derivatives and the computation of GARCH estimates. Journal of Applied Econometrics, 11(4), 399-417. Geweke, J. (1986). Modeling the persistence of conditional variances: A comment. Econometric Review, 5, 57–61. Hamilton, J. D. (1994). Time series analysis (Vol. 2). Princeton: Princeton University Press.

53 Hansen, P. R., Lunde, A., & Voev, V. (2014). Realized beta GARCH: a multivariate GARCH model with realized measures of volatility. Journal of Applied Econometrics, 29(5), 774-799. Hayashi, F. (2000). Econometrics. 2000. Princeton University Press. Higgins, M. L., & Bera, A. K. (1992). A class of nonlinear ARCH models. International Economic Review, 137-158. Hoque, A., & Latif, A. (1993). Forecasting exchange rate for the Australian dollar via-à- vis the US dollar using multivariate time-series models. Applied Economics, 25(3), 403- 407. Jensen, M. C., Black, F., & Scholes, M. S. (1972). The capital asset pricing model: Some empirical tests. Lintner, J. (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. The Review of Economics and Statistics, 13-37. Liu, T. R., Gerlow, M. E., & Irwin, S. H. (1994). The performance of alternative VAR models in forecasting exchange rates. International Journal of Forecasting, 10(3), 419- 433. Lütkepohl, H., & Poskitt, D. S. (1991). Estimating orthogonal impulse responses via vector autoregressive models. Econometric Theory, 7(04), 487-496. Malkiel, B. G., & Fama, E. F. (1970). Efficient capital markets: A review of theory and empirical work. The journal of Finance, 25(2), 383-417. Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica: Journal of the Econometric Society, 347-370. Newey, W.K., and K.D. West, (1987), A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix, Econometrica: Journal of the Econometric Society, 55, 703–708. Novy-Marx, R. (2013). The other side of value: The gross profitability premium. Journal of Financial Economics, 108(1), 1-28. Rachev, S. T., Mittnik, S., Fabozzi, F. J., Focardi, S. M., & Jašić, T. (2007). Financial econometrics: from basics to advanced modeling techniques (Vol. 150). John Wiley & Sons. Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics, 6(2), 461-464. Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk*. The Journal of Finance, 19(3), 425-442. Sims, C. A.. (1980). Macroeconomics and Reality. Econometrica, 48(1), 1–48. Titman, S., Wei, K. C., & Xie, F. (2004). Capital investments and stock returns. Journal of Financial and Quantitative Analysis, 39(04), 677-700.

54 7.Appendix

7.1 Data description

Graph 7.1.1 Multiple series graphs of Kernel density distributions

BARCLAYS BARC BRITISH AMERICAN TOBACCO BATS BCB HOLDINGS BCB

5 8 5

4 4 6

y 3 y y 3

t t t

i i i

s s s

n n 4 n

e e e

D 2 D D 2

2 1 1

0 0 0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -.25 -.20 -.15 -.10 -.05 .00 .05 .10 .15 .20 .25 -.8 -.6 -.4 -.2 .0 .2 .4 .6

BP BP. FTSE 100 - RETURN INDEX FTSE100 HSBC HDG. (ORD $0.50) HSBA

8 12 8

10 6 6 8

y y y

t t t

i i i

s s s

n 4 n 6 n 4

e e e

D D D 4 2 2 2

0 0 0 -.5 -.4 -.3 -.2 -.1 .0 .1 .2 .3 .4 .5 -.3 -.2 -.1 .0 .1 .2 .3 -.3 -.2 -.1 .0 .1 .2 .3 .4

IMPERIAL TOBACCO GP. IMT LLOYDS BANKING GROUP LLOY ROYAL BANK OF SCTL.GP. RBS

8 6 6

5 5 6 4 4

y y y

t t t

i i i

s s s

n 4 n 3 n 3

e e e

D D D 2 2 2 1 1

0 0 0 -.20 -.15 -.10 -.05 .00 .05 .10 .15 .20 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 -.8 -.6 -.4 -.2 .0 .2 .4 .6 .8

Kernel Normal

55 ROYAL DUTCH SHELL B RDSB ROLLS-ROYCE HOLDINGS RR.

7 6

6 5

5 4

y 4 y

t t

i i

s s 3

n n e 3 e

D D 2 2

1 1

0 0 -.4 -.3 -.2 -.1 .0 .1 .2 .3 .4 -.4 -.3 -.2 -.1 .0 .1 .2 .3 .4 .5

S&P 500 COMPOSITE - RETURN INDEX S&PCOMP STANDARD CHARTERED STAN

12 6

10 5

8 4

y y

t t

i i

s 6 s 3

n n

e e

D D 4 2

2 1

0 0 -.3 -.2 -.1 .0 .1 .2 .3 -.4 -.3 -.2 -.1 .0 .1 .2 .3 .4

VODAFONE GROUP VOD TESCO TSCO

7 8

6 6 5

y 4 y

t t

i i

s s 4

n n e 3 e

D D

2 2 1

0 0 -.3 -.2 -.1 .0 .1 .2 .3 .4 -.4 -.3 -.2 -.1 .0 .1 .2 .3 .4

Kernel Normal

Graph 7.1.1 presents another stylized fact which is “Heavy (Fat) Tails”. Rama Cont's paper suggests that, '' the unconditional distribution of returns seems to display a power- law or Pareto-like tail, with a tail index which is finite, higher than two and less than five for most data sets studied''. In fact, the distribution of the returns, is usual leptokurtic. In other words, the distribution has more heavy (fat) tails than the normal distribution. In order to investigate this stylized fact, first is needed to know what indicates a heavy tail in a distribution. A positive kurtosis indicates a fat tail. For asset returns, it is found that the kurtosis for each of them is positive. All returns have kurtosis more than 3 that is the one for normal distribution.

56

Graph 7.1.2 Multiple series QQ-Plots

BARCLAYS BARC BRITISH AMERICA N TOBACCO BATS BCB HOLDINGS BCB

.6 .2 .4

.4

l l l

a a .1 a .2

m m m

r r r

o o o

N .2 N N

f f f

o o o

.0 .0

s s s

e e e

l l l i .0 i i

t t t

n n n

a a a u u -.1 u -.2 Q -.2 Q Q

-.4 -.2 -.4 -0.4 0.0 0.4 0.8 1.2 1.6 -.2 -.1 .0 .1 .2 -.6 -.4 -.2 .0 .2 .4

Quantiles of BARC Quantiles of BATS Quantiles of BCB

BP BP. FTSE 100 - RETURN INDEX FTSE100 HSBC HDG. (ORD $0.50) HSBA

.3 .15 .2

.2 .10

l l l

a a a .1

m m m r .1 r .05 r

o o o

N N N

f f f

o o o

.0 .00 .0

s s s

e e e

l l l

i i i

t t t n -.1 n -.05 n

a a a u u u -.1

Q Q Q -.2 -.10

-.3 -.15 -.2 -.4 -.2 .0 .2 .4 -.3 -.2 -.1 .0 .1 -.3 -.2 -.1 .0 .1 .2 .3

Quantiles of BP Quantiles of FTSE_100 Quantiles of HSBA

LLOYDS BANKING GROUP LLOY IMPERIA L TOBA CCO GP. IMT ROYAL BANK OF SCTL.GP. RBS .4 .2 .4

l l l

a .2 a .1 a .2

m m m

r r r

o o o

N N N

f f f

o o o

.0 .0 .0

s s s

e e e

l l l

i i i

t t t

n n n

a a a u -.2 u -.1 u -.2

Q Q Q

-.4 -.2 -.4 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 -.2 -.1 .0 .1 .2 -.8 -.6 -.4 -.2 .0 .2 .4 .6

Quantiles of LLOY Quantiles of IMT Quantiles of RBS

57 ROYAL DUTCH SHELL B RDSB ROLLS-ROYCE HOLDINGS RR. .2 .3

.2

l .1 l

a a

m m

r r .1

o o

N N

f f

o o

.0 .0

s s

e e

l l

i i

t t

n n

a a -.1

u u

Q -.1 Q -.2

-.2 -.3 -.3 -.2 -.1 .0 .1 .2 .3 .4 -.4 -.2 .0 .2 .4

Quantiles of RDSB Quantiles of RR

STANDARD CHARTERED STAN S&P 500 COMPOSITE - RETURN INDEX S&PCOMP .3 .15

.2 .10

l l

a a

m m

r .1 r .05

o o

N N

f f

o o

.0 .00

s s

e e

l l

i i

t t

n n

a -.1 a -.05

u u

Q Q -.2 -.10

-.3 -.15 -.3 -.2 -.1 .0 .1 .2 .3 -.3 -.2 -.1 .0 .1 .2

Quantiles of STAN Quantiles of S_P_500

TESCO TSCO VODAFONE GROUP VOD .2 .2

l .1 l .1

a a

m m

r r

o o

N N

f f

o o

.0 .0

s s

e e

l l

i i

t t

n n

a a

u u

Q -.1 Q -.1

-.2 -.2 -.3 -.2 -.1 .0 .1 .2 .3 .4 -.2 -.1 .0 .1 .2 .3 Quantiles of TSCO Quantiles of VOD

Graph 7.1.2 depicts that empirical quantiles do not match the normal quantiles. It can be seen that there are some outliers which change the distribution of the returns. As a result, monthly returns are highly non-normal apart from IMT that its returns are close to normal distribution.

58 Graph 7.1.3 Multiple series Histograms

BARCLAYS BARC BCB HOLDINGS BCB BRITISH AMERICAN TOBACCO BATS

5 8 12

10 4 6 8

y 3 y y

t t t

i i i

s s s

n n 4 n 6

e e e

D 2 D D 4 2 1 2

0 0 0 -.6 -.4 -.2 .0 .2 .4 .6 -.6 -.4 -.2 .0 .2 .4 .6 -.6 -.4 -.2 .0 .2 .4 .6

BP BP. FTSE 100 - RETURN INDEX FTSE100 HSBC HDG. (ORD $0.50) HSBA

8 16 10

8 6 12

y y y 6

t t t

i i i

s s s

n 4 n 8 n

e e e

D D D 4

2 4 2

0 0 0 -.6 -.4 -.2 .0 .2 .4 .6 -.6 -.4 -.2 .0 .2 .4 .6 -.6 -.4 -.2 .0 .2 .4 .6

IMPERIAL TOBACCO GP. IMT LLOYDS BANKING GROUP LLOY ROYAL BANK OF SCTL.GP. RBS

12 6 6

10 5 5

8 4 4

y y y

t t t

i i i

s s s n 6 n 3 n 3

e e e

D D D 4 2 2

2 1 1

0 0 0 -.6 -.4 -.2 .0 .2 .4 .6 -.6 -.4 -.2 .0 .2 .4 .6 -.6 -.4 -.2 .0 .2 .4 .6

Histogram Kernel

59

ROYAL DUTCH SHELL B RDSB ROLLS-ROYCE HOLDINGS RR.

8 7

6 6 5

y y 4

t t

i i

s 4 s

n n e e 3

D D

2 2 1

0 0 -.6 -.5 -.4 -.3 -.2 -.1 .0 .1 .2 .3 .4 .5 .6 -.6 -.5 -.4 -.3 -.2 -.1 .0 .1 .2 .3 .4 .5 .6

S&P 500 COMPOSITE - RETURN INDEX S&PCOMP STANDARD CHARTERED STAN

12 7

10 6

5 8

y y 4

t t

i i

s 6 s

n n e e 3

D D 4 2

2 1

0 0 -.6 -.5 -.4 -.3 -.2 -.1 .0 .1 .2 .3 .4 .5 .6 -.6 -.5 -.4 -.3 -.2 -.1 .0 .1 .2 .3 .4 .5 .6

TESCO TSCO VODAFONE GROUP VOD

8 8

6 6

y y

t t

i i

s 4 s 4

n n

e e

D D

2 2

0 0 -.6 -.5 -.4 -.3 -.2 -.1 .0 .1 .2 .3 .4 .5 .6 -.6 -.5 -.4 -.3 -.2 -.1 .0 .1 .2 .3 .4 .5 .6

Histogram Kernel

Graph 7.1.3 shows that the majority of the returns are concentrated around zero.

60