Until now, four fundamental methods are developed to measure risk of a financial position. They are Notional-amount approach; (2) Factor-sensitivity measures; (3) Risk measure based on scenarios; (4) Risk measures based on the loss distribution.
1.1.1 Notional-amount approaches
It is the most traditional approach to calibrate the risk of a portfolio of risky assets which is described as the weighted aggregate notional values of the securities in this portfolio. The weight is determined by the historical assessment of the riskiness of the same asset class.
Due to its apparent simplicity, it has two main disadvantages. One is the difficulty to reflect the reduced risk as a result of diversification. If it is used to measure a portfolio combining some long and short positions in various assets, the overlapping count for the risk of different positions will occur and the risk of portfolio will be magnified. The other one is the incapability to deal with portfolio containing derivatives. In fact, the nominal value and economic value of the derivative position are not consistent.
1.1.2 Factor-sensitivity approaches
It provides the fund managers useful information about the specific risk-factor related to a certain event. For instance, the duration and convexity for bond portfolio are frequently used. However, these measures cannot be used to calculate the overall riskiness, because the sensitivity with respect to fluctuations in various risk factors cannot be aggregated directly.
1.1.3 Risk measure based on scenarios
It is mostly used when the fund managers need to consider some possible future uncertain changes in risk-factors and thus prepare for the potential crash. In practice, the fund managers can allocate varied weights on them in accordance with different extreme cases and get different losses, the risk is the maximum loss of the portfolio under all scenarios.
1.1.4 Risk measures based on the loss distribution
It is developed fast in recent years with the appearance of many new statistic tools. In fact, they are statistical quantities which describe the loss distribution of the portfolio over some time horizon. There are many reasons for applying these tools to describe the risk. Firstly, the risk management focus on the properties and amount of losses, thus it is precise to measure risk by constructing a distribution. Secondly, the loss distributions are appropriate for either a single financial instrument or a diversified financial portfolio. They can illustrate both the tendency of overall risk and characterization of specific risk together.
In the following sections, the risk measures are based on statistic distributions. Meanwhile, Value-at-Risk (VaR) and Expected shortfall (ES) are used to characterize the tail behaviour of loss distribution.
1.2 Value-at-Risk
1.2.1 Basic Value-at-Risk
VaR is a statistic quantile used in finance to demonstrate how large the loss can occur at a given confidence level. To be more precise, VaR is the smallest number at which the probability that the loss L exceeds is less or equal to at the confidence level .The mathematical form is expressed in the following formula,
Most widely used values for are and ; normally, the time horizon is 1 day.
1.2.2 The VaR for univariate t loss distribution
The univariate t loss distribution is denoted by , consequently, the moments are and when , then the VaR can be denoted by
Where is the cumulative density function of standard univariate t distribution, and it can be written as following,
Where the is the incomplete beta function, and
The inverse cumulative density function of univariate t distribution , can be calculated by MATLAB code as shown in Appendix B
where p is the solution of .
1.3 GARCH theory
1.3.1 Basic GARCH
In practical risk analysis, capturing the critical features of whole time series are more dominant than too deep analysis of arbitrary one time series. Since there are so many risk factors and data to deal with, we cannot analyse every factor's time series. Otherwise, the analysis will be unnecessarily complicated and difficult for managers to handle. Fortunately, the GARCH model families provide a feasible path to ease the challenging computational problems compared with the extreme value theory (EVT) models and Monte Carlo methods.
According to the empirical observations, most daily series of risk factors, including logarithmic returns on stocks, options, exchange rates and commodity prices, have some common characters. For instance, the daily returns of assets do not advocate identical independent assumption. Meanwhile, we non-zero conditional expected returns and time-varying volatilities appear frequently, clustering extreme value and leptokurtic return series are obvious as well.
1.3.2 Multivariate GARCH
1.3.2.1 VEC model
Bollerslev, Engle and Wooldridge firstly introduce the vector error correction model in their article and construct its equation as follow
where is vector and are series of matrices, . In this model, the number of parameters increases fast as n tends to infinity. Therefore, for a large n, the estimation of parameters becomes impossible. At the same time, due to the extremely strict conditions imposed on positive definiteness of , the estimates of parameters are difficult to execute. In fact, this model cannot be widely used.
Since the heavy tail can be modelled perfectly in Pareto distribution, Matteo Bonato (2006) uses the Pareto distribution to capture the feature of heavy tails and fits the daily US stock returns in a multivariate GARCH models and simulates the Gaussian-type covariance matrix to describe the evidence of volatility clustering.
1.3.2.2 BEKK model
The estimate equation is constructed as follow,
Where is the conditional covariance matrix at time t, A, B and C are all matrices and A is a triangular matrix, n is the number of options in system.
The structure of BEKK model implies the positive definiteness, whereas, the likelihood surfaces are too flat as a consequence of quickly increasing number of parameters to estimate as the dimension of the system ascend. Thus, the strict restrictions are imposed on the parameters in order to avoid convergence problems.
1.3.2.3 Constraint nonlinear programming
Aslihan Altay-Salih, Mustafa C. Pinar and sven Leyffer (2003) recommend the application of constrained nonlinear programming in modelling bivariate and trivariate GARCH volatility and prove its goodness in practice. This approach can derive a better optimized solution than the complicated process of calculation in BEKK and DVEC models
1.3.3 Generalized orthogonal GARCH
Roy (2002) relaxes the strict orthogonal conditions imposed on orthogonal GARCH to a weaker invertible condition. In this general model, the covariance matrices do not need to be orthogonal, any invertible matrix can be simplified. It is emphasized that the connection hidden in some unobservable independent components has indeed a influence on the estimated correlations in orthogonal GARCH.
2 Literature reviews
2.1 Application of multivariate GARCH models in risk analysis
In the autoregressive conditional heteroscedasticity models including only one risk factor or one asset and its generalized form GARCH models, the varying variance or volatility of financial data over time horizon is precisely simulated. Actually, none of risk managers or investors just holds one financial asset in their portfolios, they construct long and short positions to diversify assets and make them apart from risk and be safer. Thus, extending the univariate form to multivariate models is so attractive that numerous methods are developed.
Wenling Yang, David E. Allen (2004) calculate the hedge ratio of futures in four different models, Ordinary Least Squares (OLS)-based models, Vector Autorregression (VAR) models, Vector Error-Correction models (VEC) and diagonal VEC (DVEC) multivariate GARCH models. By comparing four outcomes, they find that the last type of models is the most efficient method to use. In addition, the decrease of risk contained in portfolio is largest in MGARCH models.
Rombouts and Verbeek (2004) impose a Value-at-Risk constraint on building the optimal portfolios and try to fit the asset returns in a multivariate GARCH models to estimate the VaR of the portfolio. They also demonstrate this estimate of VaR can be generated from the distribution of innovations.
C. Brooks, A.D. Clare and G. Persand (2002) apply the multivariate GARCH models in evaluating the lowest requirement of capital risk given a portfolio. They conclude that the lowest requirement of capital risk can be more precise in multivariate case.
Kroner and Sultan (1991) assume the conditional correlations in currency markets to be horizontal so that the conditional covariance matrices are positive definite. Their assumption becomes popular due to its simple calculation. However, the behaviours on stock markets do not sustain the constant correlation.
Luc Bauwens and Sebastien Laurent (2004) illustrate that evaluating VaR in multiple case requires weaker and less constraints. On the contrary, the distribution of innovations must be determined in univariate case. Hence, the multivariate student t distributions are used in multivariate GARCH model and obtain better simulate the financial data and predicting the VaR of portfolio.
Y. K. Tse and Albert K. C. Tsui (1998) utilise the diagonal VEC model in exchange rate data, stock data and economic industrial data respectively. Some meaningful empirical outcomes are given by generalizing the constant correlation to fluctuating correlation over time. When they compare the BEKK model with DVEC model, they find the constant correlation assumption is not sufficient.
Jelena Minović (2007) analyses the Serbian financial market in multivariate GARCH models, such as BEKK, DVEC and constant conditional covariance models. Only the bivariate and trivariate time series models are taken into account in discussion. She notices an apparent fluctuation in historical equities' conditional covariance over time.
Hence, the log returns of stock and index have an instable correlation. In addition, she verifies the BEKK, DVEC and CCC models after decreasing number of parameters can give exact outcomes.
Kroner and Sultan (1991) construct a hedge position of currency. They propose a bivariate error correction model with a GARCH error structure and focus on minimizing the risk of currency futures hedge ratios. By comparing the within-sample with out-of-sample, the risk contained in new hedge portfolio has been significantly reduced compared with traditional portfolio. In fact, the bivariate model can provide investors more tools to cover the currency exposure.
Tsay (2005) adopts the RiskMetrics developed by J.P.Morgan to calculate the VaR of overall financial positions in multivariate GARCH models. He assumes the loss data follows the student-t distribution and using the IBM data to compare the VaR under different distributions of innovations.
2.2 The choice of the GARCH models
2.2.1 The comparison between GARCH models and exponentially weighted moving average models
Due to the perfect convergence of short-term volatility and correlation forecasts to their long-term average levels, however the short-term and long-term forecasts lie in the constant term structure produced in EWMA model.
Engle (1982) and Bollerslev (1986) pose the univariate GARCH models and succeed in estimating and forecasting volatility in financial markets. Engle and Kroner (1993) try to extend the univariate GARCH models to multivariate GARCH models, however it is still extraordinarily tough in practical application of these models due to the large number of frustrating parameters produced in mean and covariance matrix to be estimated.
Engle, Ng and Rothschild (1990) employ the Capital Asset Pricing Model to produce the volatilities and correlations between equities. But there exists a problem that the variance generated by univariate GARCH model of the market risk factor cannot be directly extended to multivariate GARCH model when the risk factors' covariance matrix are not orthogonal, equivalently, there exists weak dependence among the different unobservable components. However, some multivariate GARCH models are still be developed.
2.2.2 The comparison between orthogonal GARCH models and multivariate GARCH models
Although these multivariate GARCH models describe some important features by adopting multi-distribution and vector auto-regression, the complex computational problems have not solved yet, there still are many large matrices to be solved.
In order to relieve the high-dimension calculations, reducing the dimension of risk-factors by principal component analysis can be an appropriate method. C Alexander (2000) fits daily data in the orthogonal factor models to narrow the size of relatively large covariance matrix and combines the principal components with some widely used volatility models, such as GARCH and EWMA. He also demonstrates the dimension reduction methods.
2.2.3 The dimension reduction in MGARCH
Alexander (2001) applies the Principal Component Analysis (PCA) to simplify the calculation by reducing the dimensionality of the risk-factor data. Using multivariate GARCH models to simulate a multivariate risk-factor return series. In practice, the risk managers or fund managers always control risk by constructing a diversified portfolio containing many different corporations' stocks. Thus, the dimensionality of portfolio can be really high, for instance, the original dimensionality of portfolio is 20 if the portfolio consists of 20 stocks. In this case study, we will use the PCA method to simplify the MGARCH models.
If the diagonal matrix is estimated in a GARCH model, this equation denotes an orthogonal GARCH. The orthogonal GARCH models just replace the dependent variables (scalar variances) in classical GARCH models by covariance matrix, and it deduces the volatilities and correlations of these assets from the principal component volatilities.
It is unnecessary to impose constraints on dimensionality of the original matrices in orthogonal GARCH. Nevertheless, the multivariate GARCH models can just solve single figure dimensions. More importantly, there must be some illiquid assets in financial market and the risk-factors associated with them cannot be distinguished obviously if every factor needs to be estimated in ordinary multivariate GARCH models. However, given the orthogonal condition, the orthogonal GARCH models can avoid evaluating every factor in original matrix and hence can be employed in a wider range.
2.2.4 The comparison between orthogonal GARCH models and generalize the orthogonal GARCH models
Although the transformation of original covariance matrix into orthogonal matrix can obtain some excellent and ideal statistical properties and outcomes, the inherent assumptions of linearly conversion seem too restrictive to be guaranteed in reality. It is impossible that no connection exists among economic factors since the economy is an integrated entity. Therefore, the potential linkage in these matrices should be taken into account.
Furthermore, the matrix cannot be estimated in orthogonal matrix unless the information is unconditional rather than conditional however, using the conditional information can bring more precise estimation. Thus, a more general GARCH model should be introduced to allow for both conditional information and natural relationship contained in data.
Roy (2002) relaxes the strict orthogonal conditions imposed on orthogonal GARCH to a weaker invertible condition. In this general model, the covariance matrices do not need to be orthogonal, any invertible matrix can be simplified. It is emphasized that the connection hidden in some unobservable independent components has indeed a influence on the estimated correlations in orthogonal GARCH.
2.3 The choice of distribution of innovation
2.3.1 The defects of using multivariate normal distributions
2.3.1.1 The thin tail
Univariate marginal distributions have too thin tail which is contradict with real case. In fact, the most of true risk-factor return data have heavy tails which stems from ignoring to assign adequate weight to extreme events. Although the probability of occurring extreme events is very low, the risk of them should be considered. Furthermore, the joint tails of the multivariate distributions are also thin as a result of not allocating enough weight to joint extreme outcomes.
2.3.1.2 Elliptically symmetric forms
The multivariate normal distributions have elliptically symmetric forms which contradict the notable observation for stock returns. They present heavier tails in losses than gains.
Thus, the following model called multivariate normal mean-variance mixtures are introduced to endeavour to add asymmetric factors into the multivariate normal distributions by combining normal distributions with various means and variances. In this model, a random vector has a distribution:
where ; is a non-negative, scalar-valued random variable which is independent of Z; is a matrix; is a measurable function. Where d denotes the number of risk factors, k represents the number of the stocks in portfolio, t is used to count time.
Furthermore, if we assume , where denotes a Gaussian Inverse Gamma distribution, and use the mean-variance mixture construction above, then we obtain a generalized hyperbolic (GH) distribution.
2.3.2 The advantages of selecting t distribution
Alexander J.Mcneil, Rudiger Frey and Paul Embrechts (2005) fit this multivariate GH distributions to real stock return data and conclude that for the daily data, the best simulation distribution is skewed t distribution, it produces a maximized objective function value (lnL) which cannot be increased by any other general distributions, meanwhile, the elliptically symmetric t distribution cannot be rejected and hence give a simple parsimonious model for these data.
In comparison with the multivariate normal distribution, the multivariate t distribution has heavier marginal tails and is more likely to generate extreme values. Therefore, t distribution is closer to the reality.
2.3.2.1 Multivariate t loss distributions
Suppose W in formula () to be an raxzndom variable with an inverse gamma distribution , then has a multivariate t distribution with υ degrees of freedom, and moments are , the density of multivariate t distribution is
2.3.2.2 The tail of univariate student t distribution
Using the Hill method, the tail has a form when . The slowly varying function can be taken out of the integral according to Karamata's theorem, and the degree of freedom is replaced by tail index. Finally, the tail function of t distribution is
2.3.2.3 Building the likelihood based on t distribution
For the multivariate GARCH(1,1) models, given T+1 linearized losses of option data values given and for the conditional covariance matrix. .
The conditional joint density is
Let the multivariate innovation follow a spherical distribution where the density function is h(u), then
Hence, the conditional likelihood can be shown as
where is the same as formula (2.2).
The orthogonal factorisation can simplify the process of generating large covariance matrices based on daily data by principal component analysis (PCA) of the risk factors included in original covariance matrices. Carol Alexander (2000) embeds the principal components into standard volatility estimation methods to produce large positive semi-definite covariance matrices.
3. Methodology
3.1 Methods for measuring Market risks
3.1.1 Variance-Covariance Method
Conditional case of this method assumes that the realization of multivariate normal time series is known, thus,, where represent the conditional mean and covariance matrix given information until current time t.
Although the variance-covariance method provides a convenient analytical solution under some unreasonable assumptions, we can overcome the defects of this method by utilising some other more accurate approaches and techniques.
If the multivariate normal distribution is replaced by multivariate t distribution, then the heavy tail which is a vital characterization of financial data is described in the model. At the same time, the estimates will be more reliable.
If the risk-factor changes are approximated by delta-gamma approximation rather than simple linear delta approximation, then the non-linear relationship between the value of portfolio and risk factors can be explained when the assets in portfolio are derivatives, such as European call options.
3.1.2 Historical Simulation Method
The loss distribution is estimated by empirical distribution of past and current risk data . In addition, .Meanwhile, the historical simulated loss data are used to infer the loss distribution and risk measures. Under some strict conditions, the empirical density function of the data is a consistent estimator of the density function of loss operator . when the size of sample is large enough, .
3.1.2.1 The advantages of historical simulation method
Firstly, the implementation and calculation of risk-measure estimation problem are simplified to a one-dimensional problem. Secondly, the statistical estimation of the multivariate distribution of risk-factor changes' vector X is neglected. Finally, it is unnecessary to take the dependence structure of X into account.
3.1.2.2 The disadvantages of historical simulation method
The strengths of historical simulation method are apparent and meaningful only if the sufficient empirical data for all risk factors can be collected. For example, the gaps in existing risk-factors' history and emergence of new risk factor will make estimates of VaR and ES inaccurate. In addition, this unconditional method is less relevant for daily market risk in reality than other conditional approaches.
3.1.3 Monte Carlo method
This method simulates risk-factor changes to an explicit parametric model. There are two main stages, the first step is choosing the model and fitting this model to historical data, the second step is generating N independent realizations of risk-factor changes for the future.
3.1.3.1 The disadvantages of Monte Carlo method
In the real market, the loss distribution has a heavy tail. However, the simulated distribution generated by Monte Carlo cannot guarantee this characterization. Actually, there is an obviously rising tendency of computational cost of the Monte Carlo method when the portfolio has numerous derivatives.
3.2 The approximation of loss distribution
There are three existing and prevail methods for approximating the loss distribution
of some portfolios. The first one is the non-parametric historical simulation method mentioned above; the second one is using some variation of GARCH models where the volatility is dynamic and conditionally normal distributed; the last one is the models obtained in accordance with extreme value theory.
In the historical simulation method, the empirical distribution describing the past losses is the estimated loss distribution. Nevertheless, in some extreme cases, the quantiles are too difficult to estimate since the estimators have high variances.
In the econometric models of dynamic volatilities with conditionally normal innovation, such as GARCH models, VaR estimates reveal the volatility background. However, the major drawback is the reliability of conditional normality. In fact, Danielsson and de Vries (1997) prove that the collected data are not consistent with this strict assumption, especially when estimating large quantiles of loss distribution.
In the extreme value theory, the main concentration is estimating the unconditional distribution. Two sound reasons for choosing EVT-based methods to estimate tail estimation exist. The first one is that these methods are constructed on the good statistical theory. The second one is that the parametric form of tail distribution results in feasibility of extrapolation beyond the size of data even though the conditions at the endpoints are strict. However, the VaR estimates generated by extreme value theory cannot illustrate the current volatility.
McNeil, A.J. and R. Frey (2000) suggest that an approach which combines three methods above together can overcome the drawbacks. The main idea of their approach has three steps. Firstly, the conditional volatility can be estimated by GARCH modelling and pseudo-maximum-likelihood. Statistical tests and exploration prove that the residuals follow approximately identically independent distributions and have a heavy tail. At the next step, the central part of the residuals distribution can be estimated by historical simulation and the upper tail part of the residuals distribution can be estimates by threshold exceedances methods. Finally, after the conditional mean and volatility and distribution of residuals are estimated, the conditional return distribution can be constructed.
Their innovation stands in employing the extreme value theory to model residuals and inferring the conditional return distribution from the estimated residual distribution. At the end, they conclude that the new estimates of Value-at-Risk and expected shortfall are more precise and practical than previous studies using unconditional extreme value theory or modelling return series in GARCH where the error terms are normally distributed provide poor estimates of Value-at-Risk.
3.3 Selecting the appropriate risk factors in multivariate GARCH models
Given a fixed portfolio of n stocks, the volume of stock i in the portfolio at time t is denoted by , the time series of price of stock i is denoted as . Using the logarithmic prices as risk factors, we get , the change in risk-factors
The value of the portfolio is presented by
hence, the loss is defined as
and the linearized loss is denoted by
Where the weight gives the proportion of the portfolio value invested in stock i at time t. Consequently, the corresponding unconditional mean and variance of are and .
3.4 Estimation of VaR for each asset in portfolio
For the excess loss data , and assume , the formula for tail probabilities is given by
where theis the standard t distribution. Thus, the Value-at-Risk can be obtained by transforming this formula. If , the VaR is shown below,
Jon Danielsson and Casper G. (1997) use data drawn from Olsen Corporation to explain how to estimate the tail shape of forex returns by Monte Carlo simulation. They attempt to find the feature of tail shape in second order expansion and make the estimators more efficient. They conclude that the tail shape of sum of variables which are subject to identical independent student t distribution with a degree of freedom 3 is not close to normal distribution and its tail index α do not change.
According to the study of Smith (1987), estimation of tail probabilities is proposed as
Smith states this formula can be obtained by extrapolation based on empirical method for large extreme tail probabilities.
Using this formula, the is replaced by simple average, the, are replaced by , , which are estimated by maximizing the . Because it is convenient to gain the parameters required, we use this estimator in the following case study.
3.5 Using RiskMetrics to forecast the VaR of portfolio
The daily return data is denoted by and the information set at time is presented by . The return data is assumed to follow a conditional normalized distribution , and the conditional mean and variance of return data are decided by
Where the is usually located in (0.9,1).
In fact, the conditional variance of is a certain percentage of time horizon k. The daily VaR of the portfolio calculated in RiskMetrics ith 95% confidence level is denoted by
and for k days, the VaR is
In the multivariate case, a random walk IGARCH (1,1) model is used in RiskMetrics. Assume VaR for two positions are and denote the correlation coefficient of return data by . Then the integrated VaR is
Jelena Minovic and Ivana Simeunovic (2008) describe the movement of multivariate conditional covariance matrix over time. They apply the covariance as inputs in calculating the optimized portfolio, approximating the Value-at-Risk and disposition of asset in portfolio, pricing the financial assets.
4. Case study
4.1 Identifying the distribution
4.1.1 Testing the normality of return series
4.1.1.1 The Jarque-Bera statistic
It can be used to examine the distribution of return series (see Appendix MATLAB codes). The value h is defined as jbtest(x) where the x is the vector of returns' time series of selected stocks. The Jarque-Bera test has the null hypothesis and alternative:
The sample in vector x follows a normal distribution.
The sample are not normally distributed.
This test returns the value h = 1 if the null hypothesis is rejected at the 5% significance level, and h = 0 implies that the sample are normally distributed.
The Jarque-Bera test is substantially useful in examining the bilateral goodness of fit if the null distribution cannot be fully-specified and its parameters are not able to be estimated. The test statistic is
where N is the sample size, s represents third movement (skewness) of sample, and k denotes the fourth moment (kurtosis) of sample. In accordance with the central limiting theorem, as N becomes sufficiently large, the test statistic follows chi-square distribution in which the degrees of freedom are two.
Figure 1 shows that all of return series cannot pass the Jarque-Bera test and hence they do not follow normal distributions.
4.1.1.2 QQ plot
QQ plot (X) displays a Quantile-Quantile plot of the sample quantiles of X versus theoretical quantiles from a normal distribution. If the distribution of X is normal, the plot will be close to linear. Figure 2.1 illustrates that a linear relationship between the distribution of return series of Barclays and normal distribution only locates in [-2, 2]. Beyond this interval, equivalently, when the quantile of normal distribution is closer to -4 and 4, more apparent dispersion is observed and the heavy tail is verified in distribution of return series of Barclays. Similarly, the QQ plot of BP's, HSBC's, RBS's, WHSmith's and Tesco's return data all violate the normal assumptions. The QQ plots are shown in figure 2.2, 2.3, 2.4, 2.5, 2.6 respectively.
4.1.2 Testing the identical distribution of return series
The Quantile-Quantile plot is chosen to present the relationship between two return series. QQ plot(X,Y) displays a Quantile-Quantile plot of two samples. If the samples are from the same distribution, the plot will be linear. The results are shown in Appendix C. The figure 3.1 shows the relationship between the distribution of Barclays' return data and that of BP's data. The same phenomenon is presented as the distance between red line and blue line becomes larger at the edges of dotted blue line. However, most data are gathered and not far away from the straight red line which demonstrates the distribution of Barclays' return data and that of BP's data can be thought as the same one. Similarly, the distributions of other combination of arbitrary two assets' returns can be assumed to be identical corresponding to the figure 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 3.10, 3.11, 3.12, 3.13, 3.14, 3.15 respectively.
4.1.3 Stationary test of return series
Stationary test (ADF) and ML method in t loss distribution are used to estimate the mean of time series (μ) and hence get the time series of . Then the Ljung-Box test is used to test randomness and test iid property of to tabulate the p-value. In statistics and econometrics, the existence of an unit root in a time series sample can be tested by augmented Dickey-Fuller test (ADF). It is the generalized form of the Dickey-Fuller test applied in a larger and more complex time series models. The augmented Dickey-Fuller (ADF) statistic, used in the test, is a negative number. As it is more negative than the critical value, it is more likely to reject the null hypothesis that there is a unit root at some level of confidence.
4.1.3.1 ADF test
The testing equation for the ADF test is
where α is a constant, β is the coefficient on a time trend and p is the lag order of the autoregressive process. The constraints α=0 and β=0 are imposed in accordance with a random walk and the constraint β=0 is imposed to model a random walk with a drift. By including lags of the order p, the ADF formulation takes higher-order autoregressive processes into account. This means that the lag length p has to be determined before test. One possible approach is to test from high orders to low orders and examine the t-values on coefficients. The other approach is to examine information criteria such as the Akaike information criterion, Bayesian information criterion or the Hannan-Quinn information criterion.
The null hypothesis is , the alternative hypothesis is . The test statistic is
4.1.3.2 Q-statistic
For daily risk factor return series, it is ideal to reject the strict white noise hypothesis of , although the original data reveals trivial evidence of serial correlation, there is also possible that the raw data follow a white noise but not the strict white noise, in this case, the ARMA models are replaced by GARCH models.
Assume that the time series of follow the ARMA (p,q) model and is expressed as
Define the estimated correlations of residuals as
Furthermore, the hypothesises are
There is no correlation among the lag variables, the sampling process is random. Equivalently,
There are correlations among the lag variables, the data are dependently distributed. Equivalently, there exist some non-zero autocorrelations .
where n is the sample size, is the sample autocorrelation at lag i. The null hypothesis is rejected only if , is the -quantile of the chi-squared distribution with k degrees of freedom. Notably, the number of lags in the statistic is the same as the degrees of freedom. If the null hypothesis is rejected, then the dependent structure of residual series should be determined.
Form the figure 4.1 to 4.12 as shown in Appendix B, all of six assets have more negative value of t statistic than critical value. Hence, they all have ARCH effect and stationary. Figure 4.1 gives the plot of return series of Barclays, and the table in figure 4.2 implies it is stationary. Since DF value is -69.39194<-1.95. It implies that the null hypothesis should be rejected. Hence, the return series of Barclays does not have a unit root. . Figure 4.3 gives the plot of return series of BP, and the table in figure 4.4 implies it is stationary. Since DF value is -45.792<-1.95. It implies that the null hypothesis should be rejected. Hence, the return series of Barclays does not have a unit root. . Figure 4.5 gives the plot of return series of HSBC, and the table in figure 4.6 implies it is stationary. Since DF value is -70.55907<-1.95. It implies that the null hypothesis should be rejected. Hence, the return series of Barclays does not have a unit root. . Similarly, the return series of RBS, WHSmith and Tesco are all stationary.
4.2 Modelling the volatility
Bollerslev (1986) argues that the future volatility is determined not only by the previous realizations but also the errors of the forecasted volatility. Consequently, the GARCH models are utilized to make prediction of volatility more accurate. Specifically, the GARCH(p,q) process has following objective equations,
where is the squared volatility error realization, is the past prediction of volatility and is the strict white noise with zero mean and unit variance. , and . Stability test of volatility process using the stationary criterion for GARCH (1,1). Using the Matlab toolbox, the GARCH model for Barclays' and other five assets' return series are calculated as
4.2.1 LBQ test for ARCH effect
In the Ljung-Box test, h represents the Vector of Boolean decisions for the tests, if h=1, reject the null of no autocorrelation, and admit that the residual series is auto-correlated, if h=0, there is no autocorrelation in residual series. The outcomes of LBQ test shown in figure 5 imply that all six assets have ARCH effects and they should be modelled in GARCH models.
4.2.2 Selecting GARCH models
The detailed processes are shown in Figure 6.
4.2.2.1 The estimated equation of volatility of residuals of Barclays
It can be described as GARCH (2,1) model
where the and are both defined in formula (4.7)
By using the same Matlab codes, the estimate equation of volatility of residuals of return series of BP can be described as GARCH (1,1) model
By using the same Matlab codes as shown in figure (), the estimate equation of volatility of residuals of return series of HSBC can be described as GARCH(1,1)
By using the same Matlab codes as shown in figure (), the estimate equation of volatility of residuals of return series of RBS can be described as GARCH (2,2)
By using the same Matlab codes as shown in figure (), the estimate equation of volatility of residuals of return series of WHSmith can be described as GARCH (2,1)
By using the same Matlab codes as shown in figure (), the estimate equation of volatility of residuals of return series of Tesco can be described as GARCH (2,2)
4.3 Predicting the volatility
Recently, there are two main paths to predict loss time series which are Box-Jenkins approach and exponential smoothing respectively. Due to the increasing application of VaR on risk control for banks and other financial institutions, the estimates of VaR given by
It is becoming more critical and hence more attentions are paid on forecasting the short-run volatility in this expression. In our case study, the forecasting horizon is 200 days. Construct the forecast horizon by using 5019 returns from 28/08/1992 to 23/11/2011 to predict the variance of BP return series, and compare the realized variance in prediction horizon containing 200 returns from 24/11/2011 to 29/08/2012.
4.3.1 GARCH prediction in MATLAB for return series
By using the MATLAB code as follow,
[SigmaForecast, MeanForecast]=garchpred(garchfit(barclaysreturn),barclaysreturn).
The estimated standard deviations for Barclays, BP, HSBC, RBS, WHSmith, Tesco's return series in forecasting horizon are 3.90%, 1.70%, 2.64%, 4.76%, 1.98%, 1.68%. The estimated means are 0.0008148, 0.0006635, 0.0004883, 0.0006453, 0.0007987, 0.0006415.In fact, the realized standard deviations are 3.05%, 1.39%, 1.41%, 3.06%, 1.43%, 1.60%. And the error percentages are 27.87%, 18.23%, 87.23%, 55.56%, 38.46%, 5.00%.
According to the performance of prediction, the return series of Barclays, BP, WHSmith, Tesco stock can be forecasted somewhat accurate due to the small disperse of predicted and realized standard deviation (percentage is 27.87%, 18.23%, 38.46%, 5.00% respectively). Thus, we can choose these four assets in our portfolio.
4.4 Calculating the VaR of assets
Using the estimated mean in the prediction horizon, , , , and estimator of standard deviation , , , . Hence, the Value-at-Risk of these assets at 95% are calculated by the following formula where the mean and standard deviation is replaced by specific value of each asset
Where label the Barclays, BP, WHSmith and Tesco stocks.
Using the Matlab code x=tinv(P,V) in which x is the solution of the cumulative density function integral with parameter ν, and the desired probability p should be supplied. Set the DOF . Then the result in Matlab is , hence the , , , .
4.7 Using RiskMetrics to calculate the Value-at-Risk of the portfolio selected
The integrated VaR can be expressed as
After the calculation,, , , , , . Thus, substitute four VaR and correlation coefficients obtained above into this formula, we can get the VaR of the portfolio is 20.92%. That means that the loss of portfolio has a probability of 95% to locate below 20.92%.
5. Conclusion
According to the analysis in modelling volatility, each corresponding standard deviation of return series is obtained by using MATLAB in GARCH model, and their orders are also determined. However, not all of these initial six assets are suitable to be components in optimized portfolio. It can be seen from the error or distance between realized and forecasted volatility that the return or loss series of HSBC and RBS's stocks cannot be forecasted correctly. In fact, other four assets perform well in simulation in MATLAB, hence they should be taken out of the portfolio. Consequently, only the remaining four assets can construct the final portfolio and their weight can be determined by minimizing the Value-at-Risk of portfolio and keeping at least a predetermined return level. Given the 95% confidence level, the overall VaR of this portfolio is 20.92%. In other words, the probability of losing more than 20.92% of total value of portfolio is less than 5%. Therefore, the constructed portfolio is suitable for risk-averse managers to hold. The remaining question is to determine their weights in portfolio such that the overall volatility is minimized and the predetermined return level can be kept at the same time.
Appendix
Figure 1 JB test outcomes
Figure 2 QQ plots of return series
Figure 2.1 Figure 2.2
Figure 2.3 Figure 2.4
Figure 2.5 Figure 2.6
Figure 3 QQ plots to test identical independent distribution
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Figure 3.1 Figure 3.2
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Figure 3.3 Figure 3.4
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Figure 3.5 Figure 3.6
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Figure 3.7 Figure 3.8
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Figure 3.9 Figure 3.10
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Figure 3.11 Figure 3.12
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Figure 3.13 Figure 3.14
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Figure 3.15
Figure 4 stationary test in eviews
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Figure 4.1 Figure 4.2
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Figure 4.3 Figure 4.4
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Figure 4.5 Figure 4.6
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Figure 4.7 Figure 4.8
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Figure 4.9 Figure 4.10
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Figure 4.11 Figure 4.12
Figure 5. LBQ test outcomes
(1) For Barclays
residuals = barclays-mean(barclays);
h1 = lbqtest(residuals)
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(2) For BP
residualbp=bp-mean(bp)
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(3) For HSBC
residualhsbc=hsbc-mean(hsbc)
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(4) For RBS
residualrbs=rbs-mean(rbs)
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(5) For WHSmith
residualsmith=smith-mean(smith)
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(6) For TESCO
residualtesco=tesco-mean(tesco)
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Figure 6. Determining the order of GARCH models and modelling the volatility
(1) Barclays's volatility model
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(2) BP's volatility model
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(3) HSBC's volatility model
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(4) RBS's volatility model
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(5) WHSmith's volatility model
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(6) Tesco's volatility model
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Figure 7 Predicting the volatility
(1) Prediction of Barclays return series
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(2) Prediction of BP's return series
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(3) Prediction of HSBC's return series
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(4) Prediction of RBS's return series
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(5) Prediction of WHSmith's return series
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(6) Prediction of Tesco's return series
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Figure 8 Calculating the VaR
(1) Barclays's VaR
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(2) BP's VaR
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(3) WHSmith's VaR
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(4) Tesco's VaR
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Figure 9 Calculating the weights of assets
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