Estimating pre&post crisis equity market risk in financially stressed european countries with univariate and multivariate analysis.

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University of Amsterdam, Amsterdam Business School

Master in International Finance

Estimating Pre&Post Crisis Equity Market Risk in

Financially Stressed European Countries with

Univariate and Multivariate Analysis

September 2013

Author:

Supervisor:

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1. Introduction and Aim of the Research

According to the Basel II framework, banks are obligated to put apart a minimum amount of regulatory capital to cover losses in their trading book deriving from the sudden changes in the market and their exposure to credit and operational risk. The specified

regulatory capital requirement for the market risk is calculated upon the Value-at-Risk (VaR) metric. It basically describes the maximum likelihood of the loss from the financial assets in a given time interval and confidence level. Basel II requires the use of a VaR estimate for a period of 10 days at a confidence level of 1%. Moreover, it also allows the 10-day VaR estimates to be computed from VaR estimates for shorter periods by using the square-root-of time- rule

However since the financial crisis in 2008, the robustness of the financial institutions due to losses in their portfolios and the adequacy of regulatory capital became highly

debatable. By the effect of turmoil in global financial markets and stressed period, in many banks’ trading books losses have been considerably larger than the VaR-based minimum capital requirement, causing VaR violations. Banks systematically underestimated their VaR and their market risk in this period and authorities looked for stronger criteria for defining minimum capital requirement which will absorb shocks from market and credit risk. This underestimation is due to the fact that VaR estimates typically are measured by using historical data. Following a calm period in financial markets, the VaR estimate and the capital requirement amount can decline to low levels, and then they would probably underestimate the market risk during a period of stress that comes after.

Regulators are also interested in market conditions during a stressed period because they focus on the protection of the financial system against extraordinary events which can be a source of systematic risk. According to regulators, the capital put apart by a bank has to cover the possible losses such that it does not result in bankruptcy or great instability even after a great market shock.

However the recent crisis reflected that stress testing was often not considered as

credible and significant by business units and management. Especially, catastrophic scenarios or significant market shocks were regarded as nearly impossible. One of the main reasons why financial institutions avoid stress testing is that senior management does not know what to do

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4 with stress testing results, without their probability estimates. Even implied stress tests in the recent crisis period was mostly ad hoc. Due to these reasons, the Basel Committee on Banking Supervision (BCBS) suggested supplementing an incremental risk capital charge, which consists of default risk and migration risk for unsecuritized credit products to the existing VaR-based estimation for trading book. Moreover BCBS also proposed to enhance the framework by asking banks to calculate the stressed VaR, in addition to the current VaR, taking into account a one- year observation period which contains significant losses. In order to simplify stress testing and bringing some enlightenment to the financial institutions, The European Banking Authority is required to publish guidelines on Stressed VaR and monitor the practices in this area.

Considering the importance and accuracy of VaR estimates especially in financially stressed times, the key objective of this thesis is estimating the equity market risk of

European countries who experienced financial stress in recent crisis. Both pre and post crisis period in the 2008 crisis are investigated on Spainish, Italian, Portuguese and Greek equity indices with univariate and multivariate analysis. For both analysis, two different models are used and they are compared via backtesting with each other in terms of percentage VaR hit ratio, independence, unconditional coverage and conditional coverage tests. Hence the aim would be to investigate which methodology is better for different periods and of which financial institutions pay more attention to. Implementation is based on both 1% VaR and 5% VaR estimates for pre and post crisis periods.

Univariate analysis is implemented based on an equally weighted portfolio of the above mentioned four countries’ stock indices in which VaR is calculated via two

methodologies: the T-Garch model based on a t-distribution of equity market portfolio log returns and the CAViaR Adaptive model which is indeed a quantile regression model over the same constructed portfolio.

In multivariate analysis, each country’s stock index VaR estimates are computed separately by the CAViaR Adaptive model estimating correlations from the exponential smoothing, portfolio VaR is calculated. On the other hand, multivariate portfolio VaR is estimated by RiskMetrics as well and similarly results are compared with those of Multivariate CAViaR via backtesting as it is done in univariate analysis.

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2. Literature Review

The European Banking Authority Guidelines(2012) provide a common understanding among financial institutions across the EU on Stressed VaR. As a

requirement, historical data is suggested to be used to calibrate the Stressed VaR estimate which has to contain a continuous 12- month period of financial stress. In order to determine this historical period, judgment based or formulaic approaches can be used. Formulaic approach can be risk- factor based, in which institutions identify the risk factors as relevant proxy for the movements in the portfolio, or it can be VaR based, which produces highest resulting measure for the current portfolio. Moreover it is mentioned that Stressed VaR calculations must be at least weekly, but institutions can compute more frequently, like daily. It is also suggested that in principle, changes in institution’s current VaR model should be reflected in Stressed VaR methodology as well. In general, consistency between VaR and Stressed VaR must hold in terms of confidence level and holding period. However it is not mandatory to hold in terms of weighting scheme, backtesting and frequency of comp utation.

Santos, Nogales, Ruiz and Van Dijk(2012) mentioned the importance of optimal portfolios with minimum capital requirements. They figure out that in calm periods generally banks are overestimating their risks, hence they require much amount of regulatory capital. This result does not only have direct effect on the profitability of the bank but also on the reputation of the bank hence it appears more risk y than it is.

Baptista, Alexander and Shu Yan(2012) worked on the comparison of the original and revised Basel market risk frameworks. The original framework was set in 1996 for trading portfolios and requires VaR and stress testing to prevent substantive tail risk, however the minimum capital requirement set by this framework is based only on the VaR metric.

As trading losses exceeded the minimum capital requirement for many banks in last financial crisis, the revised framework added Stressed VaR to preventing tail risk and minimum capital requirement methodology. Since CVaR(conditional VaR) has advantages over VaR, Baptista, Alexander, Shu Yan(2012) included CVaR and Stressed CVaR in their comparison methodology as well. They used Rockafellar and Uryasev(2002)’s 8th

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6 proposition in defining VaR, Stressed VaR and CVaR. They used January 2008-December 2008 as stressed period as suggested by Basel Committee and specifically also looked at U.S Stock Market Crash(1987) and the attacks in U.S. (September 2001). They implemented their analysis focusing on the revised method’s effectiveness in controlling tail risk and adequacy of minimum capital requirements. As expected, (stressed) CVaR of each risky asset is larger than its (stressed) VaR and stressed VaR(CVaR) of that is larger than its VaR(CVaR). In terms of measuring tail risk, risk management systems based on the Basel framework leads to the selection of the portfolios which have small efficiency losses with respect to mean-CVaR frontier. A portfolio belongs to the efficient frontier if and only if there is no other portfolio with the same expected return and has a smaller CVaR. In this respect, portfolio’s efficiency loss is the difference between its CVaR arising from selecting it instead of another portfolio with the same expected return and has minimum CVaR on this frontier. If the set of constraints allow the selection of portfolios with efficiency losses then that model is not effective in controlling tail risk. On the other hand relative increase in

minimum capital amount with the revised method is calculated by dividing Stressed VaR by the VaR estimate.

Baptista, Alexander, Shu Yan(2012) repeated their analysis both when short selling is allowed and disallowed. In both cases, they used a two-sample Kolmogorov-Smirnov test with the null hypothesis that the cumulative distribution function (cdf) of losses from the model based on the original framework matches with the cdf of losses from the model based on the revised framework. The alternative hypothesis is that they differ. Similarly they also used the Wilcoxon rank sum test with the null hypothesis that the median of the distribution of losses from the model based on the original framework equals the median of the

distribution of losses from the model based on the revised framework. The alternative hypothesis is that the two medians are different. It was found that at the 1% level, both tests were rejected and it shows that efficiency losses in the revised framework are bigger than those in the original one. However when tail risk-to-minimum capital requirement ratios for portfolios with maximum efficiency losses are examined, ratio values in the revised

framework are smaller than those of the original model. These results suggest that the probability of the minimum capital requirements set by the revised framework being wiped out by losses during a period of one day is lower than with those of the original model. All in all, this study reflects that the adaptation of revised framework has some costs and benefits. When it comes to adequacy of minimum capital requirements, it is better than the original

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7 framework, however, it is less effective in protecting banks from taking substantive tail risk.

Sanjay Basu(2011) compared simulation models for market risk stress testing. He based his study on more than six years of daily data on USD-INR and EUR-INR exchange rates. In his study, he required his models to work with both hypothetical and historical data. In addition, those models should be able to capture volatility clustering and correlation breakdowns in stressed markets. Since past data and statistical models are not efficient in future episodes, three hypothetical shocks were considered reflecting the appreciation of INR against EUR and USD by 5%, 10%, 15% respectively. However RMA survey pointed out that most of the large financial institutions use sensitivity analysis as the form of stress testing from the historical data. Banks generally avoid model risk due to large and complex portfolios driven by thousands of risk factors and they don’t want internal capital estimates to differ day to day. Hence they use nonparametric historical simulation(HS) as a flexible method of VaR estimation.

Sanjay Basu(2011) indicates that GARCH models are often used in finance in order to capture the volatility persistence. However for large portfolios with many risk factors estimation and updating the GARCH parameters are very time-consuming. Moreover estimation of several parameters can make a GARCH model very unstable. Hence due to those reasons, Sanjay

Basu(2011) chose EWMA which is special case of GARCH(1,1) in order to model the combined impact of fat tails and volatility spikes under stress. Different from related studies on this subject, Sanjay Basu(2011) gives special importance to correlation break-down in abnormal market conditions, since in stressed times correlation between assets increases and diversification benefit decreases.

Sanjay Basu(2011) investigated six simulation models fo r market stress testing with EUR-INR and USD-INR rates between 1st March 2002 and 22nd April 2008 by applying shocks of INR appreciation of 5%,10% or 15%. Those methods are the simple HS(Historical Simulation) model, Hull and White’s(1998) VWHS model (Volatility Weighted Historical Simulation), Hybrid HS, MCS(Monte Carlo Simulation), MCS-JD(Monte Carlo Simulation with Jump Diffusion) and EXT-HS(Extreme Value Theory-Historical Simulation).

In HS, possible changes in the market value are estimated from actual pa st returns. Sanjay Basu(2011) multiplied all past returns with current market value. Results show that VaR

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8 estimates for long positions in his portfolios are not sensitive to stress. For long and short

positions, stress VaR only falls when a 5% shock is applied and then remained the same. In the VWHS method, as the volatility of each currency increases with the density of the shocks, the VaR and ES(expected shortfall) loss measures rise sharply. It is observed that an increase in market risk proportional to rise in stress volatility which is captured by VWHS model reflected very well. For Hybrid HS, recent returns have more weight than earlier ones. By means of that, Sanjay Basu(2011) observed that temporary recent shocks can distort VaR severely but if market conditions become moderate again, still the effect of shocks can be observed as a large VaR estimate. When the stressed loss period’s weight decreases below 1%, VaR will fall

instantaneously without any change in market conditions. Hence in the Hybrid HS model VaR does not respond smoothly to market changes.

In the Monte Carlo Simulation method, Sanjay Basu(2011) assumed that future shocks will follow the same distribution as past returns and he fit distributions to historical returns. Since stress scenarios change the parameter estimates of fitted distributions, they have an effect on simulated returns. Due to this fact, he observed a response to stress but its strength is less compared to the VWHS model. The reason is that concentration of central

observations reduces the effect of stress events and stress volatility is higher than historical volatility. In the next model, MCS with jump-diffusion, there is a sharp rise in VaR and ES at both tails after a shock. However loss estimates are lower than VWHS due to the fact that tail returns in VWHS are leptokurtic whereas simulated returns in MCS-JD are Gaussian. On the other hand, as expected, Sanjay Basu(2011) observed the sharp decline in

diversification benefits through the effect of the shocks, reflecting the stress period behavior. During this time, there is a steady increase in correlations and this is captured very well with the MCS-JD model compared to other ones.

The last model that he analyzed was EVT-HS, combining the Generalized Pareto Distribution(GDP) fitted to the extreme left tail and historical simulation. Historical data is sorted in ascending order and the worst 10% returns are chosen to fit the GDP. Results indicate that stress loss estimates show a very slow response to shocks. This is due to the trade-off between scale and thickness parameters. In fat tailed distributions where the thickness parameters are high, large losses tend to be glued to each other. When the tail is very thick compared to the scale parameter, VaR and ES estimates do not respond to stress shocks sensitively.

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9 With the EWMA model for stress test volatility updating, VWHS followed by MCS-JD respond sensitively to stress shocks compared to other models. Sanjay Basu(2011)

implemented some changes to his volatility model afterwards and used a GARCH(1,1) model instead of EWMA. This methodology change has an effect only on VWHS and MCS-JD models. The results are similar to those obtained from the EWMA model and the rankings between the models are not changed.

Simone Manganelli and Robert F. Engle use a different approach in estimating VaR in their paper which was published in October 2004. They offer a conditional autoregressive model for VaR which is called CAViaR (Conditional Autoregressive Value at Risk) The model and its unknown parameters are estimated by Koenker and Bassett’s regression quantile framework which includes the Least Absolute Deviation(LAD) model which is more robust than OLS when the errors have fat-tailed distribution. According to general understanding, VaR is estimated by financial institutions to cover the market risk caused by their operations and changes in capital requirements occur as a result of them. If the risk is not properly estimated, financial institutions may experience excessive capital allocation and inefficient usage of its resources or

underestimate their market risk and allocate a lower capital amount than market risk requires, leading to vulnerability to possible crisis periods.

In their paper, they classify current VaR calculation models in two groups, factor models such as RiskMetrics and portfolio models like historical quantiles. In the former, all portfolio assets are reflected by a limited number of factors by estimating their volatility and correlations. Hence in such models volatility and correlation of the factors go hand in hand with the risk of the portfolio itself. In portfolio models, current portfolio is assumed to follow the past performance of itself. In this methodology, VaR is calculated from historical returns based on a statistical model. Considering how historical returns are constructed and how quantiles are forecasted, different estimates can be observed from one portfolio model to another. Some of the quantile forecasting mechanisms are estimating the volatility of the portfolio just like in a GARCH model, using exponential smoothing and extreme value theory. According to Manganelli and Engle(2004), all those models have drawbacks. Estimating a volatility model assumes that negative extreme returns follow a similar pattern as the rest of the returns. A rolling historical quantile takes a certain period of window into consideration and there might be dramatic changes in VaR estimation due to very bad days

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10 dropping out from the window. Similarly, extreme value theory works only for very low probability quantiles and it might yield poor estimations for some market standard probability levels (5%). In addition to that, the extreme value model is restricted to a

framework of independent and identically distributed variables which is not realistic with the most financial datasets.

Manganelli and Engle(2004) have a different approach to quantile estimation. Either than modeling the distribution itself, they model the quantile. A generic CAViaR

specification formula is:

where x_t be a vector of time t observable variables, and βθ be a p-vector of unknown parameters, p = q + r + 1 the dimension of β, and l is function of a finite number of lagged values of observables.

Four different type of CAViaR model estimates are described in the paper: Adaptive, Symmetric absolute value, Asymmetric slope and Indirect GARCH(1,1).

According to the Adaptive Model, when tomorrow’s VaR estimate is exceeded by tomorrow’s return, the VaR estimate is immediately increased and if it is not exceeded then it is slightly decreased. Symmetric and Indirect GARCH models both react

symmetrically to previous returns however the asymmetric slope model reacts differently to positive and negative returns. The Indirect GARCH model is correctly specified if the data are GARCH(1,1) and having independent and identically error distribution. The Symmetric absolute value and asymmetric slope models are implemented by GARCH process in which the standard deviation is estimated having a symmetrically and asymmetrically independent and identically error distribution.

In their paper, they mentioned an implementation of their methodology on real data based on GM, IBM, S&P500 daily prices. They estimated 1-day VaR for both the 1% and 5% probability levels with all four different CAViaR specifications, took a sample of 3392 daily observations, 2892 of which are used for the parameter estimation, and the last 500

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11 observations are used for out-of-sample tests. They published the parameters estimated, standard errors of those parameters, and one-sided p-values of them, the regression quantile objective function value, the VaR exceedance percentage, and the p-value of the Dynamic Quantile Test for both in-sample and out-of-sample periods. Dynamic quantile is a goodness-of- fit test for the estimated CAViaR specifications and their parameters.

According to the empirical results from the above- mentioned data, they observed that indirect GARCH and symmetric absolute value models’ past returns have a symmetric impact on VaR. Moreover the coefficient of the autoregressive parameter is significant in all estimatio ns. It indicates that there is volatility clustering in the tails. In addition, it is also observed that all the models give close to nominal results in terms of percentage of in-sample hits.

In their implementation, they used the Nelder-Mead simplex algorithm and the quasi-Newton method. All computations are implemented in Matlab via fminsearch and fminunc for optimization. They programmed the recursive quantile function calls in the C language and call them from Matlab.

3. Methodology and Data

3.1 Methodology

In this thesis, Value-at-Risk estimate will be calculated for the indices of four financially stressed European countries (Spain, Italy, Greece, Portugal) both through

univariate and multivariate analysis. In the univariate analysis, the Adaptive CAViaR and T-Garch methodologies are used separately on a portfolio which is constructed from the equally weighted country indices log returns, 25% of each, and results are obtained via backtesting in terms of the percentage VaR hit ratio, and the hit sequence will be analyzed considering independence, unconditional and conditional coverage tests. For multivariate CAViaR, portfolio VaR is calculated from the individual country indeces Adaptive CAViaR results together with a correlation matrix which is estimated by exponential smoothing. Another multivariate analysis has been implemented based on the RiskMetrics methodology, in which a portfolio VaR estimate is calculated from the standard deviation of the portfolio, which is calculated from the covariance matrix and the country weights.

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12 Environment and output results are uploaded in excel files. For CAViaR, input data is

provided by .txt files to the programs while for the T-Garch Univariate analysis, it is provided by .mat files. Backtesting VaR Hit ratios are calculated over the excel file results and indepencence, conditional and unconditional coverage regression tests are

implemented in Eviews 7.

For the CAViaR model implementation, matlab code of Robert F. Engle and Simone Manganelli(2004) is used and some improvements have been added on them and configured according to the thesis structure. Similarly for the UnivariateT-Garch model

implementation, the Econometrics Toolbox by James P. LeSage(2010) is used and configured according to the thesis scope.

In all models and analyses, the first 1300 observations are used in order to estimate the starting parameters and values. Afterwards, for all of them a one day moving window is implemented and all the parameters are re-estimated based on the most recent 1300

observations. Hence, accuracy and up-to-date structure of the models are aimed to be increased.

3.2.1. Univariate T-Garch Analysis

In this model, log returns of equally weighted country indices portfolio are taken as input and a StudentsT GARCH(1,1) model is fitted to them. After estimating the standard deviation of the initial 1300 observations, starting from the 1301th day, the Garch

parameters and degrees of freedom are calculated from the most recent 1300 observations for the remaining 2132 days of the sample period. By means of those parameters,the previous day’s standard deviation and the previous log return, the following day’s standard deviation is estimated as:

where ω,α,β are the Garch parameters,Rt is return and σt is thestandard deviation.

After estimating the next day’s standard deviation, the implied VaR is calculated from the standardized t distribution as:

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13 where d is degrees of freedom, t−1_p (d) is the 100p% percentile of the t-distribution.

For t−1_p (d) Matlab’s built- in tinv function is used separately both for the 1% and 5% VaR estimates. The VaR estimates are compared with the following day’s actual log return and a hit sequence vector, where 1s represents VaR exceedance, and estimated VaR ratios are saved in excel files as an output both for 1% and 5% VaR probability levels. In addition, the degrees of freedom from the fitted Garch model are also saved in excel file among other outputs

3.2.2. Adaptive CAViaR Univariate Analysis

The CAViaR univariate model is implemented as having two input arguments, model number and theta. Model number represents several CAViaR models described in Robert F. Engle and Simone Manganelli(2004)’s paper, and theta represents VaR

probability levels. In Adaptive CAViaR univariate model, equally weighted country portfolio data which in percent is read by the program from a txt file as input. For the implementation, the Adaptive CAViaR model is used among others since the VaR estimate is very sensitive to previous day’s exceedance and adjusted immediately. Hence it is

expected that VaR coverage would remain close to nominal levels. Below is Adaptive CAViaR process representation:

Other then the Adaptive CAViaR model, Symmetric Absolute Value model is also described in F. Engle and Simone Manganelli(2004)’s paper. This model responds

symmetrically to previous returns and it is mean-reverting considering the coefficient on the lagged VaR is not restricted to 1. Its quantile specifications would be truly implemented by a GARCH process where standard deviation is modeled symmetrically with independent and identically distributed(iid) errors. Below is Symmetric Absolute Value CAViaR process representation:

Another model offered in the paper is Asymmetric Slope which responds differently to positive and negative returns unlike Indirect Garch(1,1) and Symmetric Absolute Value models. It is also mean-reverting considering the coefficient on the lagged VaR is not

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14 restricted to 1. Its quantile specifications would be truly implemented by a GARCH process where standard deviation is modeled asymmetrically with iid errors. Below is Asymmetric Slope CAViaR process representation:

The last model which is also explained in F. Engle and Simone Manganelli(2004)’s paper is Indirect GARCH(1,1) model. It responds symmetrically to previous returns and it is mean-reverting considering the coefficient on the lagged VaR is not restricted to 1. It is truly implemented if the underlying data are a GARCH(1,1) with an iid error distribution. Below is Indirect GARCH(1,1) CAViaR process representation:

Over the first 300 observations, the starting empirical quantile is calculated, which will be used in the optimization routine afterwards. In that routine, optimal initial conditions and parameters are estimated via the fminunc and fminsearch functions of Matlab on the ‘RQobjectiveFunction’ method of Robert F. Engle and Simone Manganelli(2004). Starting from estimating empirical quantile, initial conditions, parameters and starting VaR estimate are calculated over the following 1000 observations. As in the Univariate T-Garch model, a moving window is implemented starting from the 1301th observation. Over the most recent 1300 days of observations, the initial conditions and parameters are re-estimated. A new function named ‘RQobjectiveFunctionThesis’ is developed from the ‘RQobjectiveFunction’ method of Robert F. Engle and Simone Manganelli(2004) in order to make VaR exceed check over the calculated VaR estimate. Estimated parameters, previous day’s log return, its VaR estimate, the VaR probability level and the following day’s actual log return is given to the function as input parameters. From the previous day’s log return, the previous day’s VaR estimate, the parameters and the specified VaR probability level, the following day’s VaR estimate is calculated by the Adaptive method call. Adaptive and other 3 models’ functions are implemented in C program by Robert F. Engle and Simone Manganelli(2004) and called from the Matlab environment through .dlls. Since those .dlls and C codes were developed in a 32 bit environment, they were not compatible with the 64 bit Windows 7 environment where the Matlab codes for the thesis are generated, so that those C codes were converted to Matlab functions and directly called from the Matlab environment in order to run the

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15 CAViaR programs. The calculated VaR estimate is set as starting VaR at the end of the loop and it is used in the following day’s (next loop iteration) VaR calculation. As an output, the calculated 5% and 1% VaR estimates, corresponding hit sequence which represents whether the loss exceeded the day’s VaR figure or not are generated and saved in Excel files.

3.2.3. RiskMetrics MultiVariate Analysis

In RiskMetrics Multivariate analysis, log returns of each country’s index in percent, daily closing index prices, and the degrees of freedom vector are read by the program from .txt files as input. In the program itself, the first 1300 daily observations are used to estimate starting values for the covariance matrix from the log returns and their squares. As in the univariate analysis, starting from the 1301th day, a moving window is applied. In each of the following day calculations, a portfolio is constructed as including one share of each index and the asset weight vector is generated from the closing asset price of that day. Hence in terms weights of indices in the portfolio, the asset weight vector changes over time in each iteration. The following day’s individual index variances are calculated from the previous day’s variance and the squared log return of that asset according to the EWMA/RiskMetrics equation:

where 0.94 is used for λ as RiskMetrics recommends for daily returns.

Similarly, the covariances between individual indices are calculated from the same EWMA/RiskMetrics Formula, where following day’s covariance is estimated from the previous day’s covariance together with the multiplication of the related asset log returns.

Here is the corresponding formula used to calculate the next day covariance:

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16 For each iteration starting from the 1301th day, the covariance matrix is generated from the following day’s variance and covariance as estimated by above formulas. The following day’s portfolio variance is estimated by the formula below from the covariance matrix and asset weight vector:

where w represents the asset weight vector and ∑_t+1 represents the covariance matrix. After estimating the next day portfolio variance, the following day 5% and 1% VaR estimates are calculated from the next day’s portfolio standard deviation and a quantile from the

standardized t-distribution with the same degrees of freedom numbers estimated from T-Garch univariate analysis. The actual portfolio return is calculated from the portfolio value changing ratio between previous and following day and this return in multiplied by 100 to Express it in percent.The real portfolio return, 1% and 5% VaR estimates, and the

corresponding hit sequence vectors are saved in excel files as output.

3.2.4. CAViaR MultiVariate Analysis

In this model, initially Adaptive CAViaR 1% and 5% VaR ratios are calculated for each country index separately and they are used as an input in the model. In addition to VaR figures, log returns of individual indices and closing index prices are provided as input to the program. Initially, the covariance between individual indices, variances, and square returns are calculated from the first 1300 observations as starting values. Just like in other models, a moving window is applied to the CAViaR Multivariate model as well, starting from the 1301th observation. In each iteration, initially the Euro VaR is calculated from each index percentage VaR ratio, which is given as input. As in the RiskMetrics Multivariate Analysis, the covariance matrix is obtained from the EWMA formula, where the following day’s covariance is calculated from the previous covariance and multiplication of the related asset log returns. Similarly, the following day’s individual asset variances are calculated with the EWMA formula just like in the RiskMetrics Multivariate analysis by taking into account the previous variance and square of the log returns. In addition to that, correlations between the country indices are calculated by dividing the covariance of the assets by their standard deviations:

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17 With correlations between country indices and individual euro VaR estimates, the Euro VaR of the portfolio is calculated from below RiskMetrics Formula:

The following day’s return is calculated from the difference between the previous and next day’s index closing prices constituting the portfolio value. It is compared with the VaR ratio which is calculated by dividing the Euro VaR estimate by the previous day’s portfolio value. Just like for the other models, the 1% and 5% VaR hit sequence, the corresponding VaR estimates, and the portfolio values are saved in excel files as output.

3.2. Data

In this thesis, daily stock index closing prices of financially stressed European countries are used, which are: Spain (IBEX:IND) , Italy (FTSEMIB:INDEX), Greece (ASE:IND) and Portugal (PSI20:IND)

The period of analysis starts with the closing price of 04/01/2000 and ends with 31/05/2013. Tables 1-4 show the index movements of specified country stock exchanges during these periods:

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Table 2 Italy FTSEMIB Index

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19 Source: DataStream

Omitting all the common holiday days for all indices in that period, the data set consists of 3432 observations in total. In both the univariate and multivariate analysis and for each methodology, the first 1300 days of observations are used to estimate the starting parameters and values for the models. The remained 2132 daily observations is used for out-of-sample testing. It starts from 03/02/2005 and continues till 31/05/2013. This out-out-of-sample period is also divided into two sections as pre-crisis period and post-crisis. The Pre-crisis period is between 03/02/2005 and 29/08/2008 which is 914 daily observations in total. The remaining 1218 daily observations which are between 01/09/2008 and 31/05/2013, constitute the post-crisis period. In both pre-crisis and post-crisis periods, one day rolling window is used to re-estimate the required parameters from recent 1300 days in each iteration.

All individual index log returns are regressed on a constant separately and ARCH Heteroskedasticity tests and Jarque–Bera tests for Normality are applied. For all indices, ARCH tests are rejected with p-value 0, indicating that there is heteroskedasticity as can be seen from test results in Appendix A

Similarly Jarque–Bera tests on the log returns of all indices indicate that normality does not hold since p-values for all tests are 0 as can be seen from results here below: Table 4 Portugal PSI20 Index

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Table 5 Spain Histogram Normality

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Table 7 Greece Histogram Normality

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22 Those results show that we should not use normal distribution in our analysis so that t-distribution would be an appropriate choice. Moreover existence of

heteroskedasticity gives rise the thought of volatility clustering which is consistent with the models applied in this thesis.

4. Results

After running the programs for the univariate and multivariate analysis, VaR hit sequences, estimated VaR ratios for 1% and 5% probability levels, and portfolio returns are saved in excel files in order to use them in backtesting afterwards. In total, the last 2132 days constitute the out-of-sample period where 914 days correspond to the pre-crisis period and 1218 days to the post-crisis term. For constructing the hit sequence, the following day’s VaR estimate is compared with the following day’s portfolio return. If following day’s return is less than minus the VaR estimate, then that day’s value of the sequence is set to 1, else 0. Afterwards, the hit ratio is compared with the VaR probability level. Here below is the specified formula for that:

where p represents the VaR probability level

Backtesting consists of tests for independence, unconditional coverage, and conditional coverage for the VaR hit sequence, which is called the Engle-Manganelli test in the literature. The independence test checks whether the VaR violations are independent over time. Under the null hypothesis, the hit sequence consists of independent Bernoulli random variables where:

where It represents hit sequence vector

The unconditional coverage test checks whether the number of VaR violations consistent with the VaR probability level:

Independence and uncoditional coverage test together constitute the conditional coverage test, which tests whether the number of VaR violations is

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23 correct, and they are independent over time, represented as:

In order to test these hypothesis, regression tests were applied on below linear regression model in Eviews:

For the independence test, a t-test is applied with the null hypothesis :

For the unconditional coverage test, a t-test is applied with the null hypothesis:

For the conditional coverage test, a Wald test (F test) is applied with the null hypothesis:

For all univariate and multivariate analysis, both pre-crisis and post-crisis periods are investigated seperately. The pre-crisis period covers the daily

observations between 03/02/2005 and 31/08/2008 and consists of 914 days. The post-crisis period includes 1218 days which are between 01/09/2008 and

31/05/2013. Based on the above mentioned tests, below is the summary of the results for univariate analysis:

Table 9 Univariate Analysis 5% VaR Results

11

Pre Crisis VaR 5% Post Crisis VaR 5% Unconditional Coverage Adaptive CAViaR t-test p-value 91.97% 66.38% Unconditional Coverage T-Garch t-test p-value 6.25% 1.70% Independence Hypothesis Adaptive CAViaR t-test p-value 82.63% 20.09% Independence Hypothesis T-Garch t-test p-value 14.83% 87.85% Conditional Coverage CAViaR Wald Test p-value 97.48% 43.63% Conditional Coverage T-Garch Wald Test p-value 10.58% 4.20% Adaptive CAViaR VaR Violation Ratio 5.03% 5.09%

T-Garch VaR Violation Ratio 6.24% 6.81%

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24

Table 10 Univariate Analysis 1% VaR Results

Pre Crisis VaR 1% Post Crisis VaR 1% Unconditional Coverage Adaptive CAViaR t-test p-value 38.25% 43.65%

Unconditional Coverage T-Garch t-test p-value 3.00% 15.10%

Independence Hypothesis Adaptive CAViaR t-test p-value 68.78% 66.36%

Independence Hypothesis T-Garch t-test p-value 54.39% 60.08%

Conditional Coverage CAViaR Wald Test p-value 65.20% 69.46%

Conditional Coverage T-Garch Wald Test p-value 9.04% 33.52%

Adaptive CAViaR VaR Violation Ratio 1.31% 1.23%

T-Garch VaR Violation Ratio 1.97% 1.48%

Portfolio log returns and univariate VaR estimates for the 1% and 5%

probability levels for both the pre and post crisis periods are shown in Appendix B. A summary of results from backtesting for the multivariate analysis is shown below:

Table 11 Multivariate Analysis 5% VaR Results

Pre Crisis VaR 5% Post Crisis VaR 5%

Unconditional Coverage Adaptive CAViaR t-test p-value 79.40% 97.33%

Unconditional Coverage RiskMetrics t-test p-value 1.36% 3.06%

Independence Hypothesis RiskMetrics t-test p-value 91.02% 77.90%

Conditional Coverage CAViaR Wald Test F-statistics 94.11% 87.33%

Conditional Coverage RiskMetrics Wald Test F-statistics 3.44% 6.63%

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25

Table 12 Multivariate Analysis 1% VaR Results

Portfolio log returns and multivariate VaR estimates fort he 1% and 5% probability levels for both pre and post crisis periods are shown in Appendix C

5. Conclusion

As a result of the Engle-Manganelli test, for the univariate analysis and at the 5% probability level, all p-values for both the T-Garch and CAViaR models are bigger than 5% in the pre-crisis period. In the post-crisis period, it is observed that the univariate T-Garch model’s unconditional and conditional coverage p-values are below 5% for the probability level of 5%. For the CAViaR model, all post crisis p-values are greater than 5% for the same probability level. In terms of VaR violation ratios, the Adaptive CAViaR model has 5.03% and 5.09% for pre and post crisis. T-Garch model’s VaR violation ratios are 6.24% and 6.81% respectively. For the pre-crisis period, the CAViaR model performs better than T-Garch in terms of

independence, conditional and unconditional coverage tests for the 5% probability level. For the post-crisis period, the T-Garch performs better than CAViaR only for independence tests, while for the other coverage tests the CAViaR model performs better again. When pre and post crisis models are compared internally for each model, it is observed that pre-crisis p- values for the Engle-Manganelli tests are higher than

Pre Crisis VaR 1% Post Crisis VaR 1% Unconditional Coverage Adaptive CAViaR t-test p-value 38.25% 59.47%

Unconditional Coverage RiskMetrics t-test p-value 5.33% 43.65%

Independence Hypothesis RiskMetrics t-test p-value 26.98% 66.36%

Conditional Coverage CAViaR Wald Test F-statistics 65.20% 81.62%

Conditional Coverage RiskMetrics Wald Test F-statistics 5.91% 69.46%

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26 those post-crisis except for the T-Garch model independence test results. Moreover unconditional coverage is more closer to the 5% probability level in the pre-crisis period as well compared to the post-crisis period for both models. All those results can be observed from Table 9.

In univariate analysis for the 1% probability level, all Adaptive CAViaR model p-values are higher than 5% for both the pre-crisis and post-crisis period. For the T-Garch model, the unconditional coverage test pre-crisis p-value is less than 5%. The rest of the tests do not reject and p- values are higher than 5% before and after the crisis. CAViaR VaR unconditional coverage is 1.31% and 1.23% and those of the T-Garch model are 1.97% and 1.48% for pre and post crisis respectively. For all tests, the Adaptive CAViaR model performs better than the T-Garch model in both the pre and post crisis period at the 1% probability level. When the pre and post crisis results are compared internally for each model at the 1% probability level, post-crisis p-values are higher and unconditional coverate is lower in the post-crisis period than in the pre-crisis period for all models, except for the independence test for the Adaptive CAViaR model. All those results can be observed from Table 10.

Regarding the multivariate analysis at the 5% probability level, all Adaptive CAViaR p-values are higher than 5% for both the pre-crisis and post-crisis period. However unconditional coverage for pre&post crisis and conditional coverage for pre-crisis period p-values for the Multivariate Riskmetrics model are less than 5%. Unconditional coverate for Multivariate CAViaR are 5.14% and 5.09% respectively for the before and after crisis periods. Those for Riskmetrics are 7.22% and 6.65%. Regarding the independence hypothesis, Riskmetrics p-values are higher than those of Multivariate CAViaR for both pre and post crisis. In the conditional and

unconditional coverage tests, the Multivariate Adaptive CAViaR model outperforms the Riskmetrics model in the pre and post crisis period. In addition to that,

unconditional coverage for the Adaptive CAViaR Multivariate model is very close to 5% in all periods whereas for Riskmetrics, unconditional coverage is not very close to 5% for both periods. When pre and post crisis periods are compared internally for all models, the pre-crisis results are better than those for the post-crisis period with respect to the independence hypothesis for both models and with respect to the conditional coverage test in the Adaptive CAViaR Multivariate model. All those

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27 results can be observed from Table 11.

Finally, for the multivariate analysis at the 1% probability level, all p-values corresponding to the Adaptive Multivariate CAViaR and Riskmetrics models are higher than 5% for all periods. Unconditional coverage for the Multivariate Adaptive CAViaR are 1.31% and 1.15% before and after the crisis respectively. Those for the Multivariate RiskMetrics model are 1.97% and 1.23% respectively. At the 1% probability level, the p-values of the independence and unconditional and conditional coverage tests are higher for the Multivariate Adaptive CAViaR model than for the RiskMetrics Multivariate model for both the pre and post crisis periods. Moreover, the unconditional coverage of the Multivariate Adaptive CAViaR model is closer to 1% when they are compared with those of Multivariate RiskMetrics. Hence at the 1% probability level, the Multivariate Adaptive CAViaR model outperforms the Multivariate RiskMetrics model in all tests in both periods. All those results can be observed from Table 12.

According to those results, both in the univariate and multivariate analysis, it is observed that the CAViaR model performs better than the T-Garch and RiskMetrics models in most of the cases

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28

APPENDIX A

Figure 1 Spain Arch Heteroskedasticity Test

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29

Figure 3 Greece Arch Heteroskedasticity Test

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30

APPENDIX B

CAVIAR UNIVARIATE

.06 .04 .02 .00 -.02 -.04 -.06 -.08

I II III IV I II III IV I II III IV I II III

2005 2006 2007 2008

PreCrisis 1% VaR PreCrisis Portfolio Return .12 .08 .04 .00 -.04 -.08 -.12

III IV I II III IV I II III IV I II III IV I II III IV I II 2008 2009 2010 2011 2012 2013

PostCrisis 1% VaR PostCrisis Portfolio Return

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31 .06 .04 .02 .00 -.02 -.04 -.06 -.08

2005 2006 2007 2008

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32

T-GARCH UNIVARIATE

.06 .04 .02 .00 -.02 -.04 -.06 -.08

2005 2006 2007 2008

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33 .06 .04 .02 .00 -.02 -.04 -.06 -.08

2005 2006 2007 2008

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34

APPENDIX C

CAVIAR MULTIVARIATE

.06 .04 .02 .00 -.02 -.04 -.06

2005 2006 2007 2008

PreCrisis 1% VaR PreCrisis Portfolio Return

.16 .12 .08 .04 .00 -.04 -.08 -.12

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35 .06 .04 .02 .00 -.02 -.04 -.06

2005 2006 2007 2008

PreCrisis 5% VaR PreCrisis Portfolio Return .16 .12 .08 .04 .00 -.04 -.08

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36

RISKMETRICS MULTIVARIATE

.06 .04 .02 .00 -.02 -.04 -.06

2005 2006 2007 2008

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37 .06 .04 .02 .00 -.02 -.04 -.06

2005 2006 2007 2008

PreCrisis 5% VaR PreCrisis Portfolio Return .16 .12 .08 .04 .00 -.04 -.08

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38

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39 Rachev, S.T., Menn, C., Fabozzi, F.J., 2005. Fat-tailed and Skewed Asset Return Distributions: Implications for Risk Management, Portfolio Selection, and Option Pricing. John Wiley & Sons, Hoboken, New Jersey.

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http://www.simonemanganelli.org/Simone/Research.html

Estimating pre&post crisis equity market risk in financially stressed european countries with univariate and multivariate analysis.

University of Amsterdam, Amsterdam Business School

Master in International Finance

Estimating Pre&Post Crisis Equity Market Risk in

Financially Stressed European Countries with

Univariate and Multivariate Analysis

September 2013

Author:

Supervisor:

Table Of Contents

1.

Introduction and Aim of the Research

2. Literature Review

3. Methodology and Data

3.1 Methodology

3.2. Data

4. Results

5. Conclusion

APPENDIX A

APPENDIX B

CAVIAR UNIVARIATE

T-GARCH UNIVARIATE

APPENDIX C

CAVIAR MULTIVARIATE

RISKMETRICS MULTIVARIATE

Bibliography