Function of Plant Organs Essay Example for Free

Function of Plant Organs Essay As was noted in the previous chapter, most plant cells are specialized to a greater or lesser degree, and arranged together in tissues. A tissue can be simple or complex depending upon whether it is composed of one or more than one type of cell. Tissues are further arranged or combined into organs that carry out life functions of the organism. Plant organs include the leaf, stem, root, and reproductive structures. The first three are sometimes called the vegetative organs and are the subject of exploration in this chapter. Reproductive organs will be covered in Chapter 5. The relationships of the organs within a plant body to each other remains an unsettled subject within plant morphology. The fundamental question is whether these are truly different structures, or just modifications of one basic structure (Eames, 1936; Esau, 1965). The plant body is an integrated, functional unit, so the division of a plant into organs is largely conceptual, providing a convenient way of approaching plant form and function. A boundary between stem and leaf is particularly difficult to make, so botanists sometimes use the word shoot to refer to the stem and its appendages (Esau, 1965). The Leaf -The plant leaf is an organ whose shape promotes efficient gathering of light for photosynthesis, but the form of the leaf must also be balanced against the fact that most of the loss of water a plant might suffer is going to occur at its leaves. Leaves are extremely variable in details of size, shape, and adornments like hairs. Although the leaves of most plants carry out the same very basic functions, there is nonetheless an amazing variety of leaf sizes, shapes, margin types, forms of attachment, ornamentation (hairs), and even color. Examine the Leaves (forms) page to learn the extensive terminology used to describe this variation. Consider that there are functional reasons for the modifications from a basic type. The Stem The stem arises during development of the embryo as part of the hypocotyl-root axis, at the upper end of which are one or more cotyledons and the shoot primordium. The Root The root is the (typically) underground part of the plant axis specialized for both anchoring the plant and absorbing water and minerals. Root (Follow any links for terms you do not understand and to gain a complete picture of root structural variation) Be sure to read about and understand the meaning of each (at a minimum) of the following terms: adventitious roots, endodermis, epidermis, gravitropism, root cap, root hair, stele, taproot. Most of the material you have read discusses the root organ as found in the angiosperms (flowering plants). However, among the vascular plants, only Psilotales lack such an organ, having i nstead rhizomes that bear hair-like absorbing structures called rhizoids (Eames, 1936 in Esau, 1965).

Speech Males And Females Communication Differences English Language Essay

Speech Males And Females Communication Differences English Language Essay Introduction: The differences in linguistic styles between males and females have exercised linguistic researchers for decades (e.g. Trudgill 1972; Lakoff 1975; Labov 1990; Coates 1998). It has been argued for some time that some consistent differences exist in speech, Holmes (1993). Although the interpretation of such differences remains somewhat elusive. Most previous work has investigated apparent phonological and pragmatic differences between male and female language use in speech (e.g. Trudgill 1972; Key 1975; Holmes 1990; Labov 1990; Eckert 1997). Several statistical phenomena have emerged that appear to be fairly stable across a variety of contexts. For example, females seem to talk more about relationships than do males (Aries Johnson 1983; Tannen 1990) and use more compliments and apologies (Holmes 1988; Holmes 1989) and facilitative tag questions (Holmes 1984). Holmes (1993) has suggested that these and other phenomena might be generalized to a number of universals including that females are more attentive to the affective function of conversation and more prone to use linguistic devices that solidify relationships. However, interpretation of the underlying linguistic phenomena, particularly as regards their specific communicative functions, is the subject of considerable controversy (Bergvall et al 1996). For example, it has been argued (Cameron et al 1988) that the use of facilitative tag questions by women might be more plausibly interpreted as signs of conversational control than as signs of subordination, as had been previously contended (Lakoff 1975). Nevertheless, broadly speaking, the differences between female and male language use appear to be centered about the interaction between the linguistic actor and his or her linguistic context. Are gender issues just women issues? No, but it is understandable that many people think so. This is because in most societies women are subordinated to men. And they are thought to be inferior to men. Many women do not accept this, and therefore they challenge the way their culture and society ascribes them an inferior position and an inferior role. That means, it often tends to be women who raise the issue of gender. But gender refers just as much to the position and role of men in society. Are gender differences in communication patterns related to power? When people are strangers, they expect less competence from women than from men. But if women are known to have prior experience or expertise related to the task, or if women are assigned leadership roles, then women show greatly increased verbal behaviors in mixed-sex groups. Educated professionals who have high social status were less likely to use powerless language, regardless of gender. Thus, differences are linked to power, and are context-specific. Differences are socially created and therefore may be socially altered. Studies have found that talking time is related both to gender (because men spend more time talking than women) and to organizational power (because the more powerful spend more time talking than the less powerful). Who Talks More, Men or Women?    A common cultural stereotype describes women as being talkative, always speaking and expressing their feelings. Well, this is probably true; however, do women do it more than men? No! In fact an experiment designed to measure the amount of speech produced suggested that men are more prone to use up more talking time than women. An experiment b y Marjorie Swacker entailed using three pictures by a fifteenth century Flemish artist, Albrecht Durer which were presented to men and women separately. They were told to take as much time as they wanted to describe the pictures. The average time for males: 13.0 minutes, and the average time for women 3.17 minutes. Why is this?   Ã‚  Ã‚  Ã‚  Ã‚  Ã‚   Sociolinguists try to make the connection between our society and our language in a way that suggests that women talk less because it has not always been as culturally acceptable as it has been for men. Men have tended to take on a more dominant role not only in the household, but in the business world. This ever-changing concept is becoming less applicable in our society, however, the trend is still prominent in some societies across the world. It is more acceptable for a man to be talkative, carry on long conversation, or a give a long wordy speech, however it is less acceptable for a women to do so. It has been more of a historical trend for men have more rights to talk. However , it is common for men to be more silent in situations that require them to express emotion. Since childhood, they have been told to keep their cool and remain calm, be a man. Literature Review The different sources that Ive read and used in the literature review presented different points of view and analysis for the subjects by reliable writers and authorities in this issue. All the sources that were used assume that there is a real difference in communication between males and females, and they agreed that there were many misunderstandings or misinterpretations in communication between genders. However, each article and source presented its own examination of the miscommunication, and they proposed different ways by which to investigate this social issue. According to an article entitled Differences in Gender Communication (2005) there is another form of differences between genders. Communication can be verbal, non verbal, or written because people can communicate also using the mail system, the written way is added to the interpersonal communication in addition of course to the verbal and non verbal ways. From reading this article, it was clear that gender differences in communication existed also in the written way because we can determine the gender of a person just by reading its written words. The differences that exist between genders and the reason why women cant be more like a man play an important role in the creation of misunderstanding in communication. Also differences present the essential causes that lead to a disagreement in communication between the two genders. There are some factors that contribute to the instinctive differences that exist between genders; for instance, there are biological ones, also there is a kind of competitiveness that exists between men and women. In addition, the cultural part enters also into consideration. Another point stated by Hill (2002) is that, there are many styles of communication. These styles are the result of many factors where were from, how and where we were brought up, our educational background, our age, and it also can depend on our gender (p. 87). In communication, generally men and women have special manners and styles of speaking also a specific subject, Coates (1986, p 23). Many studies have been done to clarify the difference in communication between men and women. According to Canaray and Dindia (2006): Researchers typically report that men are more likely to emerge as leaders, to be directive and hierarchical, to dominate in groups by talking more and interrupting more. In contrast, women are found to be more expressive, supportive, facilitative, egalitarian, and cooperative than men, and to focus more on relationships and share more personally with others From this description of the difference between men and women at a level of behaviors, it was clear that women convey their ideas and feelings and cooperate more than men who want to be the leaders and to direct. In addition, the difference between genders in communication causes misunderstanding and leads to conflicts. For example, women might disclose their feelings and the problems that they are facing, but men think that women need help, so they start giving advice and trying to help. However, women behave like that in order to get closer to others not to get solutions (Gray, 1992, p 96). Another important point argued in the research of Tanner (2002) is the reason why difference exists in communication between genders. It is said that these differences should not exist because men and women might belong to the same environment as being neighbors or brothers and sisters, yet the difference is present even if they have a similar background. The dissimilarity rises from the games that boys and girls play since their childhood, and the groups they form. For example, boys form a big group and one is the leader; however, girls tend to form small groups and they disclose their feelings and their opinions (Tanner, 2002). Torppa claims that women and men sometimes perceive the same messages to have different meanings (2002, p112). That is due to the difference in the way of interpreting messages by the two genders. In fact, women are more likely to depend on others. In other words, women want to establish an emotional and passionate interdependence with men. Moreover, women try to satisfy the others as much as possible to make everyone satisfied, merry and happy whereas men more often stick to their independence and try to keep it intact whatever the situation may be. Besides, the spirit of competition inside them let these ones think of themselves more than any other one. Nevertheless, the misunderstandings between these two genders are mostly due to a difference in the way each one expresses oneself. Many examples of the normal life analysed by Torppa revealed that misunderstandings can be caused by a distortion in the manner people want to manifest their emotions and feelings to the other sex: wome n tend more to use words while men prefer generally to show them with acts. The possible way for coping with miscommunication is to try to be aware of the differences that exist with the other sex as well as to figure out what is the point of view or the angle from which the other sees the situation. Lakoff (1975) pioneering work suggested that womens speech typically displayed a range of features, such as tag questions, which marked it as inferior and weak. Thus, she argued that the type of subordinate speech learned by a young girl will later be an excuse others use to keep her in a demeaning position, to refuse to treat her seriously as a human being (1975, p.5). As what was said before, the difference of communication between men and women can trigger some misunderstandings, and in order to overcome this problem men and women should deal with each other as if they are from different cultures. They should not misinterpret words and body language, and try to understand what the other wants to convey. Moreover, they have to clarify and make sure that they understand the other by asking questions and doing perception checking (Lathrop, 2006). Conclusion The differences between genders in communication exist, not only in the verbal way but also in the non-verbal one was proved true. Moreover, these differences in communication are mostly due to culture, education and biologic origin. The consequences of these differences in communication which is misunderstandings that also proved right. However, new ideas also found in such as the fact that a significant number of men are concerned about this issue and have feelings of disappointment when facing misunderstandings with the other sex. No matter what communication style there is both men and women will communicate in different ways. Men will take the approach of instrumental communication style where they want the answer right away and establish their hierarchy. Women, on the other hand, will be more of an expressive style of communication as they will be able to confide to others and are more sensitive to issues than men and they will be able to build, maintain, and strengthen their relationship.

VaR Models in Predicting Equity Market Risk

VaR Models in Predicting Equity Market Risk Chapter 3 Research Design This chapter represents how to apply proposed VaR models in predicting equity market risk. Basically, the thesis first outlines the collected empirical data. We next focus on verifying assumptions usually engaged in the VaR models and then identifying whether the data characteristics are in line with these assumptions through examining the observed data. Various VaR models are subsequently discussed, beginning with the non-parametric approach (the historical simulation model) and followed by the parametric approaches under different distributional assumptions of returns and intentionally with the combination of the Cornish-Fisher Expansion technique. Finally, backtesting techniques are employed to value the performance of the suggested VaR models. 3.1. Data The data used in the study are financial time series that reflect the daily historical price changes for two single equity index assets, including the FTSE 100 index of the UK market and the SP 500 of the US market. Mathematically, instead of using the arithmetic return, the paper employs the daily log-returns. The full period, which the calculations are based on, stretches from 05/06/2002 to 22/06/2009 for each single index. More precisely, to implement the empirical test, the period will be divided separately into two sub-periods: the first series of empirical data, which are used to make the parameter estimation, spans from 05/06/2002 to 31/07/2007. The rest of the data, which is between 01/08/2007 and 22/06/2009, is used for predicting VaR figures and backtesting. Do note here is that the latter stage is exactly the current global financial crisis period which began from the August of 2007, dramatically peaked in the ending months of 2008 and signally reduced significantly in the middle of 2009. Consequently, the study will purposely examine the accuracy of the VaR models within the volatile time. 3.1.1. FTSE 100 index The FTSE 100 Index is a share index of the 100 most highly capitalised UK companies listed on the London Stock Exchange, began on 3rd January 1984. FTSE 100 companies represent about 81% of the market capitalisation of the whole London Stock Exchange and become the most widely used UK stock market indicator. In the dissertation, the full data used for the empirical analysis consists of 1782 observations (1782 working days) of the UK FTSE 100 index covering the period from 05/06/2002 to 22/06/2009. 3.1.2. SP 500 index The SP 500 is a value weighted index published since 1957 of the prices of 500 large-cap common stocks actively traded in the United States. The stocks listed on the SP 500 are those of large publicly held companies that trade on either of the two largest American stock market companies, the NYSE Euronext and NASDAQ OMX. After the Dow Jones Industrial Average, the SP 500 is the most widely followed index of large-cap American stocks. The SP 500 refers not only to the index, but also to the 500 companies that have their common stock included in the index and consequently considered as a bellwether for the US economy. Similar to the FTSE 100, the data for the SP 500 is also observed during the same period with 1775 observations (1775 working days). 3.2. Data Analysis For the VaR models, one of the most important aspects is assumptions relating to measuring VaR. This section first discusses several VaR assumptions and then examines the collected empirical data characteristics. 3.2.1. Assumptions 3.2.1.1. Normality assumption Normal distribution As mentioned in the chapter 2, most VaR models assume that return distribution is normally distributed with mean of 0 and standard deviation of 1 (see figure 3.1). Nonetheless, the chapter 2 also shows that the actual return in most of previous empirical investigations does not completely follow the standard distribution. Figure 3.1: Standard Normal Distribution Skewness The skewness is a measure of asymmetry of the distribution of the financial time series around its mean. Normally data is assumed to be symmetrically distributed with skewness of 0. A dataset with either a positive or negative skew deviates from the normal distribution assumptions (see figure 3.2). This can cause parametric approaches, such as the Riskmetrics and the symmetric normal-GARCH(1,1) model under the assumption of standard distributed returns, to be less effective if asset returns are heavily skewed. The result can be an overestimation or underestimation of the VaR value depending on the skew of the underlying asset returns. Figure 3.2: Plot of a positive or negative skew Kurtosis The kurtosis measures the peakedness or flatness of the distribution of a data sample and describes how concentrated the returns are around their mean. A high value of kurtosis means that more of data’s variance comes from extreme deviations. In other words, a high kurtosis means that the assets returns consist of more extreme values than modeled by the normal distribution. This positive excess kurtosis is, according to Lee and Lee (2000) called leptokurtic and a negative excess kurtosis is called platykurtic. The data which is normally distributed has kurtosis of 3. Figure 3.3: General forms of Kurtosis Jarque-Bera Statistic In statistics, Jarque-Bera (JB) is a test statistic for testing whether the series is normally distributed. In other words, the Jarque-Bera test is a goodness-of-fit measure of departure from normality, based on the sample kurtosis and skewness. The test statistic JB is defined as: where n is the number of observations, S is the sample skewness, K is the sample kurtosis. For large sample sizes, the test statistic has a Chi-square distribution with two degrees of freedom. Augmented Dickey–Fuller Statistic Augmented Dickey–Fuller test (ADF) is a test for a unit root in a time series sample. It is an augmented version of the Dickey–Fuller test for a larger and more complicated set of time series models. The ADF statistic used in the test is a negative number. The more negative it is, the stronger the rejection of the hypothesis that there is a unit root at some level of confidence. ADF critical values: (1%) –3.4334, (5%) –2.8627, (10%) –2.5674. 3.2.1.2. Homoscedasticity assumption Homoscedasticity refers to the assumption that the dependent variable exhibits similar amounts of variance across the range of values for an independent variable. Figure 3.4: Plot of Homoscedasticity Unfortunately, the chapter 2, based on the previous empirical studies confirmed that the financial markets usually experience unexpected events, uncertainties in prices (and returns) and exhibit non-constant variance (Heteroskedasticity). Indeed, the volatility of financial asset returns changes over time, with periods when volatility is exceptionally high interspersed with periods when volatility is unusually low, namely volatility clustering. It is one of the widely stylised facts (stylised statistical properties of asset returns) which are common to a common set of financial assets. The volatility clustering reflects that high-volatility events tend to cluster in time. 3.2.1.3. Stationarity assumption According to Cont (2001), the most essential prerequisite of any statistical analysis of market data is the existence of some statistical properties of the data under study which remain constant over time, if not it is meaningless to try to recognize them. One of the hypotheses relating to the invariance of statistical properties of the return process in time is the stationarity. This hypothesis assumes that for any set of time instants ,†¦, and any time interval the joint distribution of the returns ,†¦, is the same as the joint distribution of returns ,†¦,. The Augmented Dickey-Fuller test, in turn, will also be used to test whether time-series models are accurately to examine the stationary of statistical properties of the return. 3.2.1.4. Serial independence assumption There are a large number of tests of randomness of the sample data. Autocorrelation plots are one common method test for randomness. Autocorrelation is the correlation between the returns at the different points in time. It is the same as calculating the correlation between two different time series, except that the same time series is used twice once in its original form and once lagged one or more time periods. The results can range from  +1 to -1. An autocorrelation of  +1 represents perfect positive correlation (i.e. an increase seen in one time series will lead to a proportionate increase in the other time series), while a value of -1 represents perfect negative correlation (i.e. an increase seen in one time series results in a proportionate decrease in the other time series). In terms of econometrics, the autocorrelation plot will be examined based on the Ljung-Box Q statistic test. However, instead of testing randomness at each distinct lag, it tests the overall randomness based on a number of lags. The Ljung-Box test can be defined as: where n is the sample size,is the sample autocorrelation at lag j, and h is the number of lags being tested. The hypothesis of randomness is rejected if whereis the percent point function of the Chi-square distribution and the ÃŽ ± is the quantile of the Chi-square distribution with h degrees of freedom. 3.2.2. Data Characteristics Table 3.1 gives the descriptive statistics for the FTSE 100 and the SP 500 daily stock market prices and returns. Daily returns are computed as logarithmic price relatives: Rt = ln(Pt/pt-1), where Pt is the closing daily price at time t. Figures 3.5a and 3.5b, 3.6a and 3.6b present the plots of returns and price index over time. Besides, Figures 3.7a and 3.7b, 3.8a and 3.8b illustrate the combination between the frequency distribution of the FTSE 100 and the SP 500 daily return data and a normal distribution curve imposed, spanning from 05/06/2002 through 22/06/2009. Table 3.1: Diagnostics table of statistical characteristics on the returns of the FTSE 100 Index and SP 500 index between 05/06/2002 and 22/6/2009. DIAGNOSTICS SP 500 FTSE 100 Number of observations 1774 1781 Largest return 10.96% 9.38% Smallest return -9.47% -9.26% Mean return -0.0001 -0.0001 Variance 0.0002 0.0002 Standard Deviation 0.0144 0.0141 Skewness -0.1267 -0.0978 Excess Kurtosis 9.2431 7.0322 Jarque-Bera 694.485*** 2298.153*** Augmented Dickey-Fuller (ADF) 2 -37.6418 -45.5849 Q(12) 20.0983* Autocorre: 0.04 93.3161*** Autocorre: 0.03 Q2 (12) 1348.2*** Autocorre: 0.28 1536.6*** Autocorre: 0.25 The ratio of SD/mean 144 141 Note: 1. *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. 2. 95% critical value for the augmented Dickey-Fuller statistic = -3.4158 Figure 3.5a: The FTSE 100 daily returns from 05/06/2002 to 22/06/2009 Figure 3.5b: The SP 500 daily returns from 05/06/2002 to 22/06/2009 Figure 3.6a: The FTSE 100 daily closing prices from 05/06/2002 to 22/06/2009 Figure 3.6b: The SP 500 daily closing prices from 05/06/2002 to 22/06/2009 Figure 3.7a: Histogram showing the FTSE 100 daily returns combined with a normal distribution curve, spanning from 05/06/2002 through 22/06/2009 Figure 3.7b: Histogram showing the SP 500 daily returns combined with a normal distribution curve, spanning from 05/06/2002 through 22/06/2009 Figure 3.8a: Diagram showing the FTSE 100’ frequency distribution combined with a normal distribution curve, spanning from 05/06/2002 through 22/06/2009 Figure 3.8b: Diagram showing the SP 500’ frequency distribution combined with a normal distribution curve, spanning from 05/06/2002 through 22/06/2009 The Table 3.1 shows that the FTSE 100 and the SP 500 average daily return are approximately 0 percent, or at least very small compared to the sample standard deviation (the standard deviation is 141 and 144 times more than the size of the average return for the FTSE 100 and SP 500, respectively). This is why the mean is often set at zero when modelling daily portfolio returns, which reduces the uncertainty and imprecision of the estimates. In addition, large standard deviation compared to the mean supports the evidence that daily changes are dominated by randomness and small mean can be disregarded in risk measure estimates. Moreover, the paper also employes five statistics which often used in analysing data, including Skewness, Kurtosis, Jarque-Bera, Augmented Dickey-Fuller (ADF) and Ljung-Box test to examining the empirical full period, crossing from 05/06/2002 through 22/06/2009. Figure 3.7a and 3.7b demonstrate the histogram of the FTSE 100 and the SP 500 daily return data with the normal distribution imposed. The distribution of both the indexes has longer, fatter tails and higher probabilities for extreme events than for the normal distribution, in particular on the negative side (negative skewness implying that the distribution has a long left tail). Fatter negative tails mean a higher probability of large losses than the normal distribution would suggest. It is more peaked around its mean than the normal distribution, Indeed, the value for kurtosis is very high (10 and 12 for the FTSE 100 and the SP 500, respectively compared to 3 of the normal distribution) (also see Figures 3.8a and 3.8b for more details). In other words, the most prominent deviation from the normal distributional assumption is the kurtosis, which can be seen from the middle bars of the histogram rising above the normal distribution. Moreover, it is obvious that outliers still exist, which indicates that excess kurtosis is still present. The Jarque-Bera test rejects normality of returns at the 1% level of significance for both the indexes. So, the samples have all financial characteristics: volatility clustering and leptokurtosis. Besides that, the daily returns for both the indexes (presented in Figure 3.5a and 3.5b) reveal that volatility occurs in bursts; particularly the returns were very volatile at the beginning of examined period from June 2002 to the middle of June 2003. After remaining stable for about 4 years, the returns of the two well-known stock indexes in the world were highly volatile from July 2007 (when the credit crunch was about to begin) and even dramatically peaked since July 2008 to the end of June 2009. Generally, there are two recognised characteristics of the collected daily data. First, extreme outcomes occur more often and are larger than that predicted by the normal distribution (fat tails). Second, the size of market movements is not constant over time (conditional volatility). In terms of stationary, the Augmented Dickey-Fuller is adopted for the unit root test. The null hypothesis of this test is that there is a unit root (the time series is non-stationary). The alternative hypothesis is that the time series is stationary. If the null hypothesis is rejected, it means that the series is a stationary time series. In this thesis, the paper employs the ADF unit root test including an intercept and a trend term on return. The results from the ADF tests indicate that the test statistis for the FTSE 100 and the SP 500 is -45.5849 and -37.6418, respectively. Such values are significantly less than the 95% critical value for the augmented Dickey-Fuller statistic (-3.4158). Therefore, we can reject the unit root null hypothesis and sum up that the daily return series is robustly stationary. Finally, Table 3.1 shows the Ljung-Box test statistics for serial correlation of the return and squared return series for k = 12 lags, denoted by Q(k) and Q2(k), respectively. The Q(12) statistic is statistically significant implying the present of serial correlation in the FTSE 100 and the SP 500 daily return series (first moment dependencies). In other words, the return series exhibit linear dependence. Figure 3.9a: Autocorrelations of the FTSE 100 daily returns for Lags 1 through 100, covering 05/06/2002 to 22/06/2009. Figure 3.9b: Autocorrelations of the SP 500 daily returns for Lags 1 through 100, covering 05/06/2002 to 22/06/2009. Figures 3.9a and 3.9b and the autocorrelation coefficient (presented in Table 3.1) tell that the FTSE 100 and the SP 500 daily return did not display any systematic pattern and the returns have very little autocorrelations. According to Christoffersen (2003), in this situation we can write: Corr(Rt+1,Rt+1-ÃŽ ») ≈ 0, for ÃŽ » = 1,2,3†¦, 100 Therefore, returns are almost impossible to predict from their own past. One note is that since the mean of daily returns for both the indexes (-0.0001) is not significantly different from zero, and therefore, the variances of the return series are measured by squared returns. The Ljung-Box Q2 test statistic for the squared returns is much higher, indicating the presence of serial correlation in the squared return series. Figures 3.10a and 3.10b) and the autocorrelation coefficient (presented in Table 3.1) also confirm the autocorrelations in squared returns (variances) for the FTSE 100 and the SP 500 data, and more importantly, variance displays positive correlation with its own past, especially with short lags. Corr(R2t+1,R2t+1-ÃŽ ») > 0, for ÃŽ » = 1,2,3†¦, 100 Figure 3.10a: Autocorrelations of the FTSE 100 squared daily returns Figure 3.10b: Autocorrelations of the SP 500 squared daily returns 3.3. Calculation of Value At Risk The section puts much emphasis on how to calculate VaR figures for both single return indexes from proposed models, including the Historical Simulation, the Riskmetrics, the Normal-GARCH(1,1) (or N-GARCH(1,1)) and the Student-t GARCH(1,1) (or t-GARCH(1,1)) model. Except the historical simulation model which does not make any assumptions about the shape of the distribution of the assets returns, the other ones commonly have been studied under the assumption that the returns are normally distributed. Based on the previous section relating to the examining data, this assumption is rejected because observed extreme outcomes of the both single index returns occur more often and are larger than predicted by the normal distribution. Also, the volatility tends to change through time and periods of high and low volatility tend to cluster together. Consequently, the four proposed VaR models under the normal distribution either have particular limitations or unrealistic. Specifically, the historical simulation significantly assumes that the historically simulated returns are independently and identically distributed through time. Unfortunately, this assumption is impractical due to the volatility clustering of the empirical data. Similarly, although the Riskmetrics tries to avoid relying on sample observations and make use of additional information contained in the assumed distribution function, its normally distributional assumption is also unrealistic from the results of examining the collected data. The normal-GARCH(1,1) model and the student-t GARCH(1,1) model, on the other hand, can capture the fat tails and volatility clustering which occur in the observed financial time series data, but their returns standard distributional assumption is also impossible comparing to the empirical data. Despite all these, the thesis still uses the four models under the standard distributional assumption of returns to comparing and evaluating their estimated results with the predicted results based on the student distributional assumption of returns. Besides, since the empirical data experiences fatter tails more than that of the normal distribution, the essay intentionally employs the Cornish-Fisher Expansion technique to correct the z-value from the normal distribution to account for fatter tails, and then compare these results with the two results above. Therefore, in this chapter, we purposely calculate VaR by separating these three procedures into three different sections and final results will be discussed in length in chapter 4. 3.3.1. Components of VaR measures Throughout the analysis, a holding period of one-trading day will be used. For the significance level, various values for the left tail probability level will be considered, ranging from the very conservative level of 1 percent to the mid of 2.5 percent and to the less cautious 5 percent. The various VaR models will be estimated using the historical data of the two single return index samples, stretches from 05/06/2002 through 31/07/2007 (consisting of 1305 and 1298 prices observations for the FTSE 100 and the SP 500, respectively) for making the parameter estimation, and from 01/08/2007 to 22/06/2009 for predicting VaRs and backtesting. One interesting point here is that since there are few previous empirical studies examining the performance of VaR models during periods of financial crisis, the paper deliberately backtest the validity of VaR models within the current global financial crisis from the beginning in August 2007. 3.3.2. Calculation of VaR 3.3.2.1. Non-parametric approach Historical Simulation As mentioned above, the historical simulation model pretends that the change in market factors from today to tomorrow will be the same as it was some time ago, and therefore, it is computed based on the historical returns distribution. Consequently, we separate this non-parametric approach into a section. The chapter 2 has proved that calculating VaR using the historical simulation model is not mathematically complex since the measure only requires a rational period of historical data. Thus, the first task is to obtain an adequate historical time series for simulating. There are many previous studies presenting that predicted results of the model are relatively reliable once the window length of data used for simulating daily VaRs is not shorter than 1000 observed days. In this sense, the study will be based on a sliding window of the previous 1305 and 1298 prices observations (1304 and 1297 returns observations) for the FTSE 100 and the SP 500, respectively, spanning from 05/06/2002 through 31/07/2007. We have selected this rather than larger windows is since adding more historical data means adding older historical data which could be irrelevant to the future development of the returns indexes. After sorting in ascending order the past returns attributed to equally spaced classes, the predicted VaRs are determined as that log-return lies on the target percentile, say, in the thesis is on three widely percentiles of 1%, 2.5% and 5% lower tail of the return distribution. The result is a frequency distribution of returns, which is displayed as a histogram, and shown in Figure 3.11a and 3.11b below. The vertical axis shows the number of days on which returns are attributed to the various classes. The red vertical lines in the histogram separate the lowest 1%, 2.5% and 5% returns from the remaining (99%, 97.5% and 95%) returns. For FTSE 100, since the histogram is drawn from 1304 daily returns, the 99%, 97.5% and 95% daily VaRs are approximately the 13th, 33rd and 65th lowest return in this dataset which are -3.2%, -2.28% and -1.67%, respectively and are roughly marked in the histogram by the red vertical lines. The interpretation is that the VaR gives a number such that there is, say, a 1% chance of losing more than 3.2% of the single asset value tomorrow (on 01st August 2007). The SP 500 VaR figures, on the other hand, are little bit smaller than that of the UK stock index with -2.74%, -2.03% and -1.53% corresponding to 99%, 97.5% and 95% confidence levels, respectively. Figure 3.11a: Histogram of daily returns of FTSE 100 between 05/06/2002 and 31/07/2007 Figure 3.11b: Histogram of daily returns of SP 500 between 05/06/2002 and 31/07/2007 Following predicted VaRs on the first day of the predicted period, we continuously calculate VaRs for the estimated period, covering from 01/08/2007 to 22/06/2009. The question is whether the proposed non-parametric model is accurately performed in the turbulent period will be discussed in length in the chapter 4. 3.3.2.2. Parametric approaches under the normal distributional assumption of returns This section presents how to calculate the daily VaRs using the parametric approaches, including the RiskMetrics, the normal-GARCH(1,1) and the student-t GARCH(1,1) under the standard distributional assumption of returns. The results and the validity of each model during the turbulent period will deeply be considered in the chapter 4. 3.3.2.2.1. The RiskMetrics Comparing to the historical simulation model, the RiskMetrics as discussed in the chapter 2 does not solely rely on sample observations; instead, they make use of additional information contained in the normal distribution function. All that needs is the current estimate of volatility. In this sense, we first calculate daily RiskMetrics variance for both the indexes, crossing the parameter estimated period from 05/06/2002 to 31/07/2007 based on the well-known RiskMetrics variance formula (2.9). Specifically, we had the fixed decay factor ÃŽ »=0.94 (the RiskMetrics system suggested using ÃŽ »=0.94 to forecast one-day volatility). Besides, the other parameters are easily calculated, for instance, and are the squared log-return and variance of the previous day, correspondingly. After calculating the daily variance, we continuously measure VaRs for the forecasting period from 01/08/2007 to 22/06/2009 under different confidence levels of 99%, 97.5% and 95% based on the normal VaR formula (2.6), where the critical z-value of the normal distribution at each significance level is simply computed using the Excel function NORMSINV. 3.3.2.2.2. The Normal-GARCH(1,1) model For GARCH models, the chapter 2 confirms that the most important point is to estimate the model parameters ,,. These parameters has to be calculated for numerically, using the method of maximum likelihood estimation (MLE). In fact, in order to do the MLE function, many previous studies efficiently use professional econometric softwares rather than handling the mathematical calculations. In the light of evidence, the normal-GARCH(1,1) is executed by using a well-known econometric tool, STATA, to estimate the model parameters (see Table 3.2 below). Table 3.2. The parameters statistics of the Normal-GARCH(1,1) model for the FTSE 100 and the SP 500 Normal-GARCH(1,1)* Parameters FTSE 100 SP 500 0.0955952 0.0555244 0.8907231 0.9289999 0.0000012 0.0000011 + 0.9863183 0.9845243 Number of Observations 1304 1297 Log likelihood 4401.63 4386.964 * Note: In this section, we report the results from the Normal-GARCH(1,1) model using the method of maximum likelihood, under the assumption that the errors conditionally follow the normal distribution with significance level of 5%. According to Table 3.2, the coefficients of the lagged squared returns () for both the indexes are positive, concluding that strong ARCH effects are apparent for both the financial markets. Also, the coefficients of lagged conditional variance () are significantly positive and less than one, indicating that the impact of ‘old’ news on volatility is significant. The magnitude of the coefficient, is especially high (around 0.89 – 0.93), indicating a long memory in the variance. The estimate of was 1.2E-06 for the FTSE 100 and 1.1E-06 for the SP 500 implying a long run standard deviation of daily market return of about 0.94% and 0.84%, respectively. The log-likehood for this model for both the indexes was 4401.63 and 4386.964 for the FTSE 100 and the SP 500, correspondingly. The Log likehood ratios rejected the hypothesis of normality very strongly. After calculating the model parameters, we begin measuring conditional variance (volatility) for the parameter estimated period, covering from 05/06/2002 to 31/07/2007 based on the conditional variance formula (2.11), where and are the squared log-return and conditional variance of the previous day, respectively. We then measure predicted daily VaRs for the forecasting period from 01/08/2007 to 22/06/2009 under confidence levels of 99%, 97.5% and 95% using the normal VaR formula (2.6). Again, the critical z-value of the normal distribution under significance levels of 1%, 2.5% and 5% is purely computed using the Excel function NORMSINV. 3.3.2.2.3. The Student-t GARCH(1,1) model Different from the Normal-GARCH(1,1) approach, the model assumes that the volatility (or the errors of the returns) follows the Student-t distribution. In fact, many previous studies suggested that using the symmetric GARCH(1,1) model with the volatility following the Student-t distribution is more accurate than with that of the Normal distribution when examining financial time series. Accordingly, the paper additionally employs the Student-t GARCH(1,1) approach to measure VaRs. In this section, we use this model under the normal distributional assumption of returns. First is to estimate the model parameters using the method of maximum likelihood estimation and obtained by the STATA (see Table 3.3). Table 3.3. The parameters statistics of the Student-t GARCH(1,1) model for the FTSE 100 and the SP 500 Student-t GARCH(1,1)* Parameters FTSE 100 SP 500 0.0926120 0.0569293 0.8946485 0.9354794 0.0000011 0.0000006 + 0.9872605 0.9924087 Number of Observations 1304 1297 Log likelihood 4406.50 4399.24 * Note: In this section, we report the results from the Student-t GARCH(1,1) model using the method of maximum likelihood, under the assumption that the errors conditionally follow the student distribution with significance level of 5%. The Table 3.3 also identifies the same characteristics of the student-t GARCH(1,1) model parameters comparing to the normal-GARCH(1,1) approach. Specifically, the results of , expose that there were evidently strong ARCH effects occurred on the UK and US financial markets during the parameter estimated period, crossing from 05/06/2002 to 31/07/2007. Moreover, as Floros (2008) mentioned, there was also the considerable impact of ‘old’ news on volatility as well as a long memory in the variance. We at that time follow the similar steps as calculating VaRs using the normal-GARCH(1,1) model. 3.3.2.3. Parametric approaches under the normal distributional assumption of returns modified by the Cornish-Fisher Expansion technique The section 3.3.2.2 measured the VaRs using the parametric approaches under the assumption that the returns are normally distributed. Regardless of their results and performance, it is clearly that this assumption is impractical since the fact that the collected empirical data experiences fatter tails more than that of the normal distribution. Consequently, in this section the study intentionally employs the Cornish-Fisher Expansion (CFE) technique to correct the z-value from the assumption of the normal distribution to significantly account for fatter tails. Again, the question of whether the proposed models achieved powerfully within the recent damage time will be assessed in length in the chapter 4. 3.3.2.3.1. The CFE-modified RiskMetrics Similar VaR Models in Predicting Equity Market Risk VaR Models in Predicting Equity Market Risk Chapter 3 Research Design This chapter represents how to apply proposed VaR models in predicting equity market risk. Basically, the thesis first outlines the collected empirical data. We next focus on verifying assumptions usually engaged in the VaR models and then identifying whether the data characteristics are in line with these assumptions through examining the observed data. Various VaR models are subsequently discussed, beginning with the non-parametric approach (the historical simulation model) and followed by the parametric approaches under different distributional assumptions of returns and intentionally with the combination of the Cornish-Fisher Expansion technique. Finally, backtesting techniques are employed to value the performance of the suggested VaR models. 3.1. Data The data used in the study are financial time series that reflect the daily historical price changes for two single equity index assets, including the FTSE 100 index of the UK market and the SP 500 of the US market. Mathematically, instead of using the arithmetic return, the paper employs the daily log-returns. The full period, which the calculations are based on, stretches from 05/06/2002 to 22/06/2009 for each single index. More precisely, to implement the empirical test, the period will be divided separately into two sub-periods: the first series of empirical data, which are used to make the parameter estimation, spans from 05/06/2002 to 31/07/2007. The rest of the data, which is between 01/08/2007 and 22/06/2009, is used for predicting VaR figures and backtesting. Do note here is that the latter stage is exactly the current global financial crisis period which began from the August of 2007, dramatically peaked in the ending months of 2008 and signally reduced significantly in the middle of 2009. Consequently, the study will purposely examine the accuracy of the VaR models within the volatile time. 3.1.1. FTSE 100 index The FTSE 100 Index is a share index of the 100 most highly capitalised UK companies listed on the London Stock Exchange, began on 3rd January 1984. FTSE 100 companies represent about 81% of the market capitalisation of the whole London Stock Exchange and become the most widely used UK stock market indicator. In the dissertation, the full data used for the empirical analysis consists of 1782 observations (1782 working days) of the UK FTSE 100 index covering the period from 05/06/2002 to 22/06/2009. 3.1.2. SP 500 index The SP 500 is a value weighted index published since 1957 of the prices of 500 large-cap common stocks actively traded in the United States. The stocks listed on the SP 500 are those of large publicly held companies that trade on either of the two largest American stock market companies, the NYSE Euronext and NASDAQ OMX. After the Dow Jones Industrial Average, the SP 500 is the most widely followed index of large-cap American stocks. The SP 500 refers not only to the index, but also to the 500 companies that have their common stock included in the index and consequently considered as a bellwether for the US economy. Similar to the FTSE 100, the data for the SP 500 is also observed during the same period with 1775 observations (1775 working days). 3.2. Data Analysis For the VaR models, one of the most important aspects is assumptions relating to measuring VaR. This section first discusses several VaR assumptions and then examines the collected empirical data characteristics. 3.2.1. Assumptions 3.2.1.1. Normality assumption Normal distribution As mentioned in the chapter 2, most VaR models assume that return distribution is normally distributed with mean of 0 and standard deviation of 1 (see figure 3.1). Nonetheless, the chapter 2 also shows that the actual return in most of previous empirical investigations does not completely follow the standard distribution. Figure 3.1: Standard Normal Distribution Skewness The skewness is a measure of asymmetry of the distribution of the financial time series around its mean. Normally data is assumed to be symmetrically distributed with skewness of 0. A dataset with either a positive or negative skew deviates from the normal distribution assumptions (see figure 3.2). This can cause parametric approaches, such as the Riskmetrics and the symmetric normal-GARCH(1,1) model under the assumption of standard distributed returns, to be less effective if asset returns are heavily skewed. The result can be an overestimation or underestimation of the VaR value depending on the skew of the underlying asset returns. Figure 3.2: Plot of a positive or negative skew Kurtosis The kurtosis measures the peakedness or flatness of the distribution of a data sample and describes how concentrated the returns are around their mean. A high value of kurtosis means that more of data’s variance comes from extreme deviations. In other words, a high kurtosis means that the assets returns consist of more extreme values than modeled by the normal distribution. This positive excess kurtosis is, according to Lee and Lee (2000) called leptokurtic and a negative excess kurtosis is called platykurtic. The data which is normally distributed has kurtosis of 3. Figure 3.3: General forms of Kurtosis Jarque-Bera Statistic In statistics, Jarque-Bera (JB) is a test statistic for testing whether the series is normally distributed. In other words, the Jarque-Bera test is a goodness-of-fit measure of departure from normality, based on the sample kurtosis and skewness. The test statistic JB is defined as: where n is the number of observations, S is the sample skewness, K is the sample kurtosis. For large sample sizes, the test statistic has a Chi-square distribution with two degrees of freedom. Augmented Dickey–Fuller Statistic Augmented Dickey–Fuller test (ADF) is a test for a unit root in a time series sample. It is an augmented version of the Dickey–Fuller test for a larger and more complicated set of time series models. The ADF statistic used in the test is a negative number. The more negative it is, the stronger the rejection of the hypothesis that there is a unit root at some level of confidence. ADF critical values: (1%) –3.4334, (5%) –2.8627, (10%) –2.5674. 3.2.1.2. Homoscedasticity assumption Homoscedasticity refers to the assumption that the dependent variable exhibits similar amounts of variance across the range of values for an independent variable. Figure 3.4: Plot of Homoscedasticity Unfortunately, the chapter 2, based on the previous empirical studies confirmed that the financial markets usually experience unexpected events, uncertainties in prices (and returns) and exhibit non-constant variance (Heteroskedasticity). Indeed, the volatility of financial asset returns changes over time, with periods when volatility is exceptionally high interspersed with periods when volatility is unusually low, namely volatility clustering. It is one of the widely stylised facts (stylised statistical properties of asset returns) which are common to a common set of financial assets. The volatility clustering reflects that high-volatility events tend to cluster in time. 3.2.1.3. Stationarity assumption According to Cont (2001), the most essential prerequisite of any statistical analysis of market data is the existence of some statistical properties of the data under study which remain constant over time, if not it is meaningless to try to recognize them. One of the hypotheses relating to the invariance of statistical properties of the return process in time is the stationarity. This hypothesis assumes that for any set of time instants ,†¦, and any time interval the joint distribution of the returns ,†¦, is the same as the joint distribution of returns ,†¦,. The Augmented Dickey-Fuller test, in turn, will also be used to test whether time-series models are accurately to examine the stationary of statistical properties of the return. 3.2.1.4. Serial independence assumption There are a large number of tests of randomness of the sample data. Autocorrelation plots are one common method test for randomness. Autocorrelation is the correlation between the returns at the different points in time. It is the same as calculating the correlation between two different time series, except that the same time series is used twice once in its original form and once lagged one or more time periods. The results can range from  +1 to -1. An autocorrelation of  +1 represents perfect positive correlation (i.e. an increase seen in one time series will lead to a proportionate increase in the other time series), while a value of -1 represents perfect negative correlation (i.e. an increase seen in one time series results in a proportionate decrease in the other time series). In terms of econometrics, the autocorrelation plot will be examined based on the Ljung-Box Q statistic test. However, instead of testing randomness at each distinct lag, it tests the overall randomness based on a number of lags. The Ljung-Box test can be defined as: where n is the sample size,is the sample autocorrelation at lag j, and h is the number of lags being tested. The hypothesis of randomness is rejected if whereis the percent point function of the Chi-square distribution and the ÃŽ ± is the quantile of the Chi-square distribution with h degrees of freedom. 3.2.2. Data Characteristics Table 3.1 gives the descriptive statistics for the FTSE 100 and the SP 500 daily stock market prices and returns. Daily returns are computed as logarithmic price relatives: Rt = ln(Pt/pt-1), where Pt is the closing daily price at time t. Figures 3.5a and 3.5b, 3.6a and 3.6b present the plots of returns and price index over time. Besides, Figures 3.7a and 3.7b, 3.8a and 3.8b illustrate the combination between the frequency distribution of the FTSE 100 and the SP 500 daily return data and a normal distribution curve imposed, spanning from 05/06/2002 through 22/06/2009. Table 3.1: Diagnostics table of statistical characteristics on the returns of the FTSE 100 Index and SP 500 index between 05/06/2002 and 22/6/2009. DIAGNOSTICS SP 500 FTSE 100 Number of observations 1774 1781 Largest return 10.96% 9.38% Smallest return -9.47% -9.26% Mean return -0.0001 -0.0001 Variance 0.0002 0.0002 Standard Deviation 0.0144 0.0141 Skewness -0.1267 -0.0978 Excess Kurtosis 9.2431 7.0322 Jarque-Bera 694.485*** 2298.153*** Augmented Dickey-Fuller (ADF) 2 -37.6418 -45.5849 Q(12) 20.0983* Autocorre: 0.04 93.3161*** Autocorre: 0.03 Q2 (12) 1348.2*** Autocorre: 0.28 1536.6*** Autocorre: 0.25 The ratio of SD/mean 144 141 Note: 1. *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. 2. 95% critical value for the augmented Dickey-Fuller statistic = -3.4158 Figure 3.5a: The FTSE 100 daily returns from 05/06/2002 to 22/06/2009 Figure 3.5b: The SP 500 daily returns from 05/06/2002 to 22/06/2009 Figure 3.6a: The FTSE 100 daily closing prices from 05/06/2002 to 22/06/2009 Figure 3.6b: The SP 500 daily closing prices from 05/06/2002 to 22/06/2009 Figure 3.7a: Histogram showing the FTSE 100 daily returns combined with a normal distribution curve, spanning from 05/06/2002 through 22/06/2009 Figure 3.7b: Histogram showing the SP 500 daily returns combined with a normal distribution curve, spanning from 05/06/2002 through 22/06/2009 Figure 3.8a: Diagram showing the FTSE 100’ frequency distribution combined with a normal distribution curve, spanning from 05/06/2002 through 22/06/2009 Figure 3.8b: Diagram showing the SP 500’ frequency distribution combined with a normal distribution curve, spanning from 05/06/2002 through 22/06/2009 The Table 3.1 shows that the FTSE 100 and the SP 500 average daily return are approximately 0 percent, or at least very small compared to the sample standard deviation (the standard deviation is 141 and 144 times more than the size of the average return for the FTSE 100 and SP 500, respectively). This is why the mean is often set at zero when modelling daily portfolio returns, which reduces the uncertainty and imprecision of the estimates. In addition, large standard deviation compared to the mean supports the evidence that daily changes are dominated by randomness and small mean can be disregarded in risk measure estimates. Moreover, the paper also employes five statistics which often used in analysing data, including Skewness, Kurtosis, Jarque-Bera, Augmented Dickey-Fuller (ADF) and Ljung-Box test to examining the empirical full period, crossing from 05/06/2002 through 22/06/2009. Figure 3.7a and 3.7b demonstrate the histogram of the FTSE 100 and the SP 500 daily return data with the normal distribution imposed. The distribution of both the indexes has longer, fatter tails and higher probabilities for extreme events than for the normal distribution, in particular on the negative side (negative skewness implying that the distribution has a long left tail). Fatter negative tails mean a higher probability of large losses than the normal distribution would suggest. It is more peaked around its mean than the normal distribution, Indeed, the value for kurtosis is very high (10 and 12 for the FTSE 100 and the SP 500, respectively compared to 3 of the normal distribution) (also see Figures 3.8a and 3.8b for more details). In other words, the most prominent deviation from the normal distributional assumption is the kurtosis, which can be seen from the middle bars of the histogram rising above the normal distribution. Moreover, it is obvious that outliers still exist, which indicates that excess kurtosis is still present. The Jarque-Bera test rejects normality of returns at the 1% level of significance for both the indexes. So, the samples have all financial characteristics: volatility clustering and leptokurtosis. Besides that, the daily returns for both the indexes (presented in Figure 3.5a and 3.5b) reveal that volatility occurs in bursts; particularly the returns were very volatile at the beginning of examined period from June 2002 to the middle of June 2003. After remaining stable for about 4 years, the returns of the two well-known stock indexes in the world were highly volatile from July 2007 (when the credit crunch was about to begin) and even dramatically peaked since July 2008 to the end of June 2009. Generally, there are two recognised characteristics of the collected daily data. First, extreme outcomes occur more often and are larger than that predicted by the normal distribution (fat tails). Second, the size of market movements is not constant over time (conditional volatility). In terms of stationary, the Augmented Dickey-Fuller is adopted for the unit root test. The null hypothesis of this test is that there is a unit root (the time series is non-stationary). The alternative hypothesis is that the time series is stationary. If the null hypothesis is rejected, it means that the series is a stationary time series. In this thesis, the paper employs the ADF unit root test including an intercept and a trend term on return. The results from the ADF tests indicate that the test statistis for the FTSE 100 and the SP 500 is -45.5849 and -37.6418, respectively. Such values are significantly less than the 95% critical value for the augmented Dickey-Fuller statistic (-3.4158). Therefore, we can reject the unit root null hypothesis and sum up that the daily return series is robustly stationary. Finally, Table 3.1 shows the Ljung-Box test statistics for serial correlation of the return and squared return series for k = 12 lags, denoted by Q(k) and Q2(k), respectively. The Q(12) statistic is statistically significant implying the present of serial correlation in the FTSE 100 and the SP 500 daily return series (first moment dependencies). In other words, the return series exhibit linear dependence. Figure 3.9a: Autocorrelations of the FTSE 100 daily returns for Lags 1 through 100, covering 05/06/2002 to 22/06/2009. Figure 3.9b: Autocorrelations of the SP 500 daily returns for Lags 1 through 100, covering 05/06/2002 to 22/06/2009. Figures 3.9a and 3.9b and the autocorrelation coefficient (presented in Table 3.1) tell that the FTSE 100 and the SP 500 daily return did not display any systematic pattern and the returns have very little autocorrelations. According to Christoffersen (2003), in this situation we can write: Corr(Rt+1,Rt+1-ÃŽ ») ≈ 0, for ÃŽ » = 1,2,3†¦, 100 Therefore, returns are almost impossible to predict from their own past. One note is that since the mean of daily returns for both the indexes (-0.0001) is not significantly different from zero, and therefore, the variances of the return series are measured by squared returns. The Ljung-Box Q2 test statistic for the squared returns is much higher, indicating the presence of serial correlation in the squared return series. Figures 3.10a and 3.10b) and the autocorrelation coefficient (presented in Table 3.1) also confirm the autocorrelations in squared returns (variances) for the FTSE 100 and the SP 500 data, and more importantly, variance displays positive correlation with its own past, especially with short lags. Corr(R2t+1,R2t+1-ÃŽ ») > 0, for ÃŽ » = 1,2,3†¦, 100 Figure 3.10a: Autocorrelations of the FTSE 100 squared daily returns Figure 3.10b: Autocorrelations of the SP 500 squared daily returns 3.3. Calculation of Value At Risk The section puts much emphasis on how to calculate VaR figures for both single return indexes from proposed models, including the Historical Simulation, the Riskmetrics, the Normal-GARCH(1,1) (or N-GARCH(1,1)) and the Student-t GARCH(1,1) (or t-GARCH(1,1)) model. Except the historical simulation model which does not make any assumptions about the shape of the distribution of the assets returns, the other ones commonly have been studied under the assumption that the returns are normally distributed. Based on the previous section relating to the examining data, this assumption is rejected because observed extreme outcomes of the both single index returns occur more often and are larger than predicted by the normal distribution. Also, the volatility tends to change through time and periods of high and low volatility tend to cluster together. Consequently, the four proposed VaR models under the normal distribution either have particular limitations or unrealistic. Specifically, the historical simulation significantly assumes that the historically simulated returns are independently and identically distributed through time. Unfortunately, this assumption is impractical due to the volatility clustering of the empirical data. Similarly, although the Riskmetrics tries to avoid relying on sample observations and make use of additional information contained in the assumed distribution function, its normally distributional assumption is also unrealistic from the results of examining the collected data. The normal-GARCH(1,1) model and the student-t GARCH(1,1) model, on the other hand, can capture the fat tails and volatility clustering which occur in the observed financial time series data, but their returns standard distributional assumption is also impossible comparing to the empirical data. Despite all these, the thesis still uses the four models under the standard distributional assumption of returns to comparing and evaluating their estimated results with the predicted results based on the student distributional assumption of returns. Besides, since the empirical data experiences fatter tails more than that of the normal distribution, the essay intentionally employs the Cornish-Fisher Expansion technique to correct the z-value from the normal distribution to account for fatter tails, and then compare these results with the two results above. Therefore, in this chapter, we purposely calculate VaR by separating these three procedures into three different sections and final results will be discussed in length in chapter 4. 3.3.1. Components of VaR measures Throughout the analysis, a holding period of one-trading day will be used. For the significance level, various values for the left tail probability level will be considered, ranging from the very conservative level of 1 percent to the mid of 2.5 percent and to the less cautious 5 percent. The various VaR models will be estimated using the historical data of the two single return index samples, stretches from 05/06/2002 through 31/07/2007 (consisting of 1305 and 1298 prices observations for the FTSE 100 and the SP 500, respectively) for making the parameter estimation, and from 01/08/2007 to 22/06/2009 for predicting VaRs and backtesting. One interesting point here is that since there are few previous empirical studies examining the performance of VaR models during periods of financial crisis, the paper deliberately backtest the validity of VaR models within the current global financial crisis from the beginning in August 2007. 3.3.2. Calculation of VaR 3.3.2.1. Non-parametric approach Historical Simulation As mentioned above, the historical simulation model pretends that the change in market factors from today to tomorrow will be the same as it was some time ago, and therefore, it is computed based on the historical returns distribution. Consequently, we separate this non-parametric approach into a section. The chapter 2 has proved that calculating VaR using the historical simulation model is not mathematically complex since the measure only requires a rational period of historical data. Thus, the first task is to obtain an adequate historical time series for simulating. There are many previous studies presenting that predicted results of the model are relatively reliable once the window length of data used for simulating daily VaRs is not shorter than 1000 observed days. In this sense, the study will be based on a sliding window of the previous 1305 and 1298 prices observations (1304 and 1297 returns observations) for the FTSE 100 and the SP 500, respectively, spanning from 05/06/2002 through 31/07/2007. We have selected this rather than larger windows is since adding more historical data means adding older historical data which could be irrelevant to the future development of the returns indexes. After sorting in ascending order the past returns attributed to equally spaced classes, the predicted VaRs are determined as that log-return lies on the target percentile, say, in the thesis is on three widely percentiles of 1%, 2.5% and 5% lower tail of the return distribution. The result is a frequency distribution of returns, which is displayed as a histogram, and shown in Figure 3.11a and 3.11b below. The vertical axis shows the number of days on which returns are attributed to the various classes. The red vertical lines in the histogram separate the lowest 1%, 2.5% and 5% returns from the remaining (99%, 97.5% and 95%) returns. For FTSE 100, since the histogram is drawn from 1304 daily returns, the 99%, 97.5% and 95% daily VaRs are approximately the 13th, 33rd and 65th lowest return in this dataset which are -3.2%, -2.28% and -1.67%, respectively and are roughly marked in the histogram by the red vertical lines. The interpretation is that the VaR gives a number such that there is, say, a 1% chance of losing more than 3.2% of the single asset value tomorrow (on 01st August 2007). The SP 500 VaR figures, on the other hand, are little bit smaller than that of the UK stock index with -2.74%, -2.03% and -1.53% corresponding to 99%, 97.5% and 95% confidence levels, respectively. Figure 3.11a: Histogram of daily returns of FTSE 100 between 05/06/2002 and 31/07/2007 Figure 3.11b: Histogram of daily returns of SP 500 between 05/06/2002 and 31/07/2007 Following predicted VaRs on the first day of the predicted period, we continuously calculate VaRs for the estimated period, covering from 01/08/2007 to 22/06/2009. The question is whether the proposed non-parametric model is accurately performed in the turbulent period will be discussed in length in the chapter 4. 3.3.2.2. Parametric approaches under the normal distributional assumption of returns This section presents how to calculate the daily VaRs using the parametric approaches, including the RiskMetrics, the normal-GARCH(1,1) and the student-t GARCH(1,1) under the standard distributional assumption of returns. The results and the validity of each model during the turbulent period will deeply be considered in the chapter 4. 3.3.2.2.1. The RiskMetrics Comparing to the historical simulation model, the RiskMetrics as discussed in the chapter 2 does not solely rely on sample observations; instead, they make use of additional information contained in the normal distribution function. All that needs is the current estimate of volatility. In this sense, we first calculate daily RiskMetrics variance for both the indexes, crossing the parameter estimated period from 05/06/2002 to 31/07/2007 based on the well-known RiskMetrics variance formula (2.9). Specifically, we had the fixed decay factor ÃŽ »=0.94 (the RiskMetrics system suggested using ÃŽ »=0.94 to forecast one-day volatility). Besides, the other parameters are easily calculated, for instance, and are the squared log-return and variance of the previous day, correspondingly. After calculating the daily variance, we continuously measure VaRs for the forecasting period from 01/08/2007 to 22/06/2009 under different confidence levels of 99%, 97.5% and 95% based on the normal VaR formula (2.6), where the critical z-value of the normal distribution at each significance level is simply computed using the Excel function NORMSINV. 3.3.2.2.2. The Normal-GARCH(1,1) model For GARCH models, the chapter 2 confirms that the most important point is to estimate the model parameters ,,. These parameters has to be calculated for numerically, using the method of maximum likelihood estimation (MLE). In fact, in order to do the MLE function, many previous studies efficiently use professional econometric softwares rather than handling the mathematical calculations. In the light of evidence, the normal-GARCH(1,1) is executed by using a well-known econometric tool, STATA, to estimate the model parameters (see Table 3.2 below). Table 3.2. The parameters statistics of the Normal-GARCH(1,1) model for the FTSE 100 and the SP 500 Normal-GARCH(1,1)* Parameters FTSE 100 SP 500 0.0955952 0.0555244 0.8907231 0.9289999 0.0000012 0.0000011 + 0.9863183 0.9845243 Number of Observations 1304 1297 Log likelihood 4401.63 4386.964 * Note: In this section, we report the results from the Normal-GARCH(1,1) model using the method of maximum likelihood, under the assumption that the errors conditionally follow the normal distribution with significance level of 5%. According to Table 3.2, the coefficients of the lagged squared returns () for both the indexes are positive, concluding that strong ARCH effects are apparent for both the financial markets. Also, the coefficients of lagged conditional variance () are significantly positive and less than one, indicating that the impact of ‘old’ news on volatility is significant. The magnitude of the coefficient, is especially high (around 0.89 – 0.93), indicating a long memory in the variance. The estimate of was 1.2E-06 for the FTSE 100 and 1.1E-06 for the SP 500 implying a long run standard deviation of daily market return of about 0.94% and 0.84%, respectively. The log-likehood for this model for both the indexes was 4401.63 and 4386.964 for the FTSE 100 and the SP 500, correspondingly. The Log likehood ratios rejected the hypothesis of normality very strongly. After calculating the model parameters, we begin measuring conditional variance (volatility) for the parameter estimated period, covering from 05/06/2002 to 31/07/2007 based on the conditional variance formula (2.11), where and are the squared log-return and conditional variance of the previous day, respectively. We then measure predicted daily VaRs for the forecasting period from 01/08/2007 to 22/06/2009 under confidence levels of 99%, 97.5% and 95% using the normal VaR formula (2.6). Again, the critical z-value of the normal distribution under significance levels of 1%, 2.5% and 5% is purely computed using the Excel function NORMSINV. 3.3.2.2.3. The Student-t GARCH(1,1) model Different from the Normal-GARCH(1,1) approach, the model assumes that the volatility (or the errors of the returns) follows the Student-t distribution. In fact, many previous studies suggested that using the symmetric GARCH(1,1) model with the volatility following the Student-t distribution is more accurate than with that of the Normal distribution when examining financial time series. Accordingly, the paper additionally employs the Student-t GARCH(1,1) approach to measure VaRs. In this section, we use this model under the normal distributional assumption of returns. First is to estimate the model parameters using the method of maximum likelihood estimation and obtained by the STATA (see Table 3.3). Table 3.3. The parameters statistics of the Student-t GARCH(1,1) model for the FTSE 100 and the SP 500 Student-t GARCH(1,1)* Parameters FTSE 100 SP 500 0.0926120 0.0569293 0.8946485 0.9354794 0.0000011 0.0000006 + 0.9872605 0.9924087 Number of Observations 1304 1297 Log likelihood 4406.50 4399.24 * Note: In this section, we report the results from the Student-t GARCH(1,1) model using the method of maximum likelihood, under the assumption that the errors conditionally follow the student distribution with significance level of 5%. The Table 3.3 also identifies the same characteristics of the student-t GARCH(1,1) model parameters comparing to the normal-GARCH(1,1) approach. Specifically, the results of , expose that there were evidently strong ARCH effects occurred on the UK and US financial markets during the parameter estimated period, crossing from 05/06/2002 to 31/07/2007. Moreover, as Floros (2008) mentioned, there was also the considerable impact of ‘old’ news on volatility as well as a long memory in the variance. We at that time follow the similar steps as calculating VaRs using the normal-GARCH(1,1) model. 3.3.2.3. Parametric approaches under the normal distributional assumption of returns modified by the Cornish-Fisher Expansion technique The section 3.3.2.2 measured the VaRs using the parametric approaches under the assumption that the returns are normally distributed. Regardless of their results and performance, it is clearly that this assumption is impractical since the fact that the collected empirical data experiences fatter tails more than that of the normal distribution. Consequently, in this section the study intentionally employs the Cornish-Fisher Expansion (CFE) technique to correct the z-value from the assumption of the normal distribution to significantly account for fatter tails. Again, the question of whether the proposed models achieved powerfully within the recent damage time will be assessed in length in the chapter 4. 3.3.2.3.1. The CFE-modified RiskMetrics Similar

Zambezi Valley :: essays research papers

Zambezi Valley If the average person was asked about the Zambezi Valley, how many would actually have anything to say? From all the places I have been in the world, the Zambezi Valley stands out most in my mind. The mighty Zambezi River forms the border between Zimbabwe and Zambia as they lie on the maps in our libraries. Few people have been graced the opportunity to be in the presence of this majestic silver python as it carves away at the crust of our earth. There is no better way to experience this natural wonder than by organizing an expedition and venturing into the unknown wilderness of the "Dark Continent" for memories that will last you a lifetime. Unfortunately these days you have to do it through a Safari company that will charge you an arm and a leg for a week long tour, only skimming the surface and not taking you into the darkest of Africa of which you have read in so many adventure novels. Traveling is a very stimulating hobby, but Africa is part of me. Darkness overcame all as Mother Earth turned her back on the center of our solar system. In the heart of Africa everything is sleeping, or so you are meant to think. The ruler of that kingdom is patrolling his territory in absolute silence. His bushy black mane casts a shadow in the pale moonlight. Eyes like those of an eagle penetrate the darkest shadows of the bush. The soft gray pads of his paws tread along the game path barely leaving any evidence of his presence. The great beast strides graciously along before disappearing into black night. He will soon find either a dense thicket or some tall Buffalo grass swaying back and forth on the rhythm of the early morning breeze where he can lay his giant body down and get some rest. Stars begin to fade as a mysterious yellow glow takes their place in the East. The bush is coming to life. Birds are singing their songs of joy and hippos are snorting out of pure pleasure for a new day has come. This will be a day where the fight for survival takes over like an uncontrollable urge, nevertheless, little is known as to who should be feared. Should it be the predators lurking around, wanting to fill their own stomachs, or will it be the natives searching for food in the land on which they have lived for thousands of years. Remember that this is done in an effort to rise above the ever present poverty.

Arthur Millers Presentation Of John Proctors Moral Journey Essay

Examine Arthur Miller's Presentation Of John Proctor's Moral Journey - The Crucible by Arthur Miller "The Crucible" by Arthur Miller is a play based upon an American settlement during the late 1600's. It is centred around actual events from history to try to portray the way of life in this era. Miller has chosen the confusion of the witch trials of this time, to provide a base for the struggles of his main character, John Proctor. At the beginning of the play the focus is laid mainly on introducing the main characters and storyline, but as the script unfolds, it becomes clear that John Proctor is the main character, something not immediately obvious from the beginning. It is how Miller presents and demonstrates Proctor's moral journey throughout the play, and the different channels he uses to do this that I will focus on. Act One really only sets the scene for the play by portraying the different characters in the Salem and how their ways of life revolve mostly around the 'church' and their religion. The inhabitants can for the most part be sectioned off into three groups; the established figures, eg. Rev. Parris; the citizens, and people who have in theory 'earned' their status, eg. Francis and Rebecca Nurse; and the 'outsiders', eg. Rev. Hale. This set-up seems to work well until the events of the play, when people become separated by their views, and everyone begins blaming others for their shortcomings in order to maintain their authority and status. The main power in the village being the church, naturally the Rev. Parris will do anything to keep his position, especially as Miller informs us that his character feels that for some reason everyone in the world is against him, and his life is jus... ... which he was innocent. At the very end, when he knew what would happen to him, Proctor refused to publish a lie about himself, or admit to a sin he did not commit. This shows at least some remnant of pride was left, even after everything he had been through - and this is what saves him in the end. It makes him realise that John Proctor wasn't as evil as he had thought, that, like everyone else, he was a mixture, and now with absolutely nothing to hide. Possibly, this act may cancel out John's adultery, especially as there is a lot of confusion around what are actually 'evil acts', and what are just natural flaws and instincts. Elizabeth says right at the end of the play, "He have his goodness now. God forbid I take it from him!.." This suggests goodness and reconciliation in Proctor's act, as the once shallow and indecisive John, is finally decisive.

Nature Nurture Debate Essay

?NATURE VERSUS NURTURE IS THE ARGUMENT OF WHETHER IT IS the characteristics that are inherited, or those that are learnt through environmental influences, which effect how we develop. ?WE ARE GOING TO LOOK INTO THE DIFFERENT PSYCHOLOGICAL approaches in relation to whether it is nature or nurture that determines gender: ?Psychodynamic ?Biological ?Social Learning ?Cognitive Sex and Gender Sex and gender are often referred to as one and the same, so it is important to distinguish between certain words and phrases: ?Sex: The biological state of a person; whether they are male or female depending on their genetic makeup. ?Gender: This is the social interpretation of sex. Is an individuals classification on whether a person is male or female. ?Gender Identity/ Role: Through socialisation we learn what is acceptable behaviour from females and males. We learn that each have different expected characteristics. ?Gender Constancy: Is the realisation that gender is fixed. This happens at 4 years of age. Psychodynamic Approach ?Freud backs up both the nature and nurture, with his psychodynamic approach: ?His idea of the personality being in three parts: the Id, Ego, and Superego. ?The Id is what we are said to have acquired naturally in birth. It is the primitive self, who strives to survive. Psychodynamic Approach ?The ego develops a few months in, and continues to be learnt from the outside world, it is when our consciousness comes into play. A child would learn the di$erence between male and female. ?The superego is the internalisation of moral values. We strive to do the right thing. So a child may have it instilled to act a certain way because that is what is expected, so any other feelings may be pushed into the unconscious. Psychosexual Stages ?In Freuds Psychosexual stages, it is the Phallic stage at age 3 to 6 years old that children become gender aware. ?A child is aware of what sex they are biologically, but their gender is e$ected by interactions between mother and father. Psychodynamic Theory ?However the personality model Freud talks about cannot be proved. ?This approach is also deterministic in that there is no free will when it comes to the psychosexual stages Little Hans Case Study Biological Approach All that is psychological is psysiological ? rst ‘Aron’. Social Learning Approach Cognitive Approach Conclusion ? There is no doubt that sex is based solely upon nature, whereas determining gender brings together both nurture and nature as it is the how society views the sexes. All that is psychological is psysiological ? rst ‘Aron’. Thought’s, feeling’s, ect, reside in the mind and are ultimately of biological cause. Biological psychology is the study of the physiological basis of behaviour and experience, it is highly scienti+c in approach, the key areas of study include:- †¢The nervous system and behaviour †¢States of consciousness †¢Biological rythms †¢Motivation †¢Anxiety and stress †¢pain Biological Psychology Cognitive is mainly a consequence of maturation stages of innate structures Biological Biological psychology states:- †¢Human Behaviour can be explained through hormones, genetics, evolution and the nervous system †¢If in theory, behaviour can be explained biologically, causes for unwanted behaviours can be modi+ed or removed using biological treatment such as medication for mental illnesses. †¢Biological psychology believes in experimental treatment conducted using animals, this is due to our biological similarities. Biological Charles Darwin †¢Was an 18th century English naturalist and geologist. †¢Best known for his contributions to evolutionary theory. †¢Whilst on a 5 year expedition, Darwin concluded species of life have descended over time from common ancestors. †¢This created the scienti+c theory to a branching pattern of evolution resulted from a process he called natural selection. †¢For 20 years Darwin worked on his theory †¢1859 The origin of species was published †¢The book was extremely controversial, mainly due to the theory of homo sapiens being another form of animal, leading to a theory of our evolution from apes†¦Ã¢â‚¬ ¦. The church was not happy. Biological Studying psychology within an evolutionary framework has revolutionized the +eld, allowing di$erent approaches to be uni+ed under one banner. Darwin also pioneered one of evolutionary psychology’s most important tools, the comparative method. The results coming from this new +eld continue to change how we view our behavior and mental abilities, as well as those of other animals. Darwin’s impact in biological Psychology Biological Strengths and weaknesses biological psychology and the theories within support nature over nuture. However, it can be argued, that by limiting explanations for behaviour in terms of either nature or nurture, the complexity of human biengs is underestimated. It could be argued, that the interaction of both nature (biology) and nurture (environment) both play vital roles in our behaviour. A strength to the biological approach is its use of scienti+c methods, producing clear evidence, such as neurotransmitters. The biological approach is able to produce clear evidence, scienti+cally, for explanations A weakness to biological psychology is the reductionist explanations provided, which do not fully encompass the full scope of human behaviour. Individuals may posses a predisposition, to particular behavioural traits, however, environmental factors can also be the cause. This is called ‘Diathesis stress model’ of human behaviour. Biological Biological explanation of gender. Through evolution, men in their role as ‘hunter gatherer’, may have developed a stronger ‘+ght or ;ight response than women, who had the role of caring for the children. Due to this males and females may have developed a di$erent physiological response to stress. Taylor et al (2000) suggested that women produce a calmer response to stress due to a hormone. Oxytocin is realised in response to stress and has been shown to lead to maternal behaviour. Taylor called this the ‘tend and berfriend response’ instead of the ‘+ght or ;ight’ response. Leading to the idea women are more likely to seek social support to cope with stress.

Philosophy and Its Branches Essay

The study of the fundamental nature of knowledge, reality, and existence, especially when considered as an academic discipline. Philosophy is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language. It is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational argument. Origin: Middle English: from Old French philosophie, via Latin from Greek philosophia ‘love of wisdom’. Philosophy comes from the Greek for â€Å"love of wisdom,† giving us two important starting points: love (or passion) and wisdom (knowledge, understanding). Philosophy sometimes seems to be pursued without passion as if it were a technical subject. Although there is a role for dispassionate research, philosophy must derive from some passion for the ultimate goal: a reliable, accurate understanding ourselves and our world. Branches of philosophy: The following branches are the main areas of study: †¢Metaphysics is the study of the nature of being and the world. Traditional branches are cosmology and ontology. †¢Epistemology is concerned with the nature and scope of knowledge, and whether knowledge is possible. Among its central concerns has been the challenge posed by skepticism and the relationships between truth, belief, and justification. †¢Ethics, or â€Å"moral philosophy†, is concerned with questions of how persons ought to act or if such questions are answerable. The main branches of ethics are meta-ethics, normative ethics, and applied ethics. †¢Political philosophy is the study of government and the relationship of individuals and communities to the state.