Value-at-Risk using the variance-covariance approach

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Value-at-Risk using the variance-covariance approach

European evidence

Thesis

Master of Science in Business Administration Specialisation: Finance

University of Groningen

Faculty of Economics and Business

Author: David de Weerd Student number: 1830759 Supervisor: Dr. P.P.M. Smid

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Abstract

This paper tests whether the variance-covariance approach to derive VaR produces accurate estimates of market risk for a bank`s risky assets. This is done using the Moving Average- (MA), the Exponentially Weighted Moving Average- (EWMA), the Generalized Autoregressive Conditional Heteroscedasticity- (GARCH) and the Treshold-GARCH- (TGARCH) model to estimate volatility. These models produce one-day ahead forecasts of VaR using a rolling window procedure and using data of the AEX, CAC40, DAX and FTSE100 indices covering the period January 1985 – February 2011. Generated values of VaR are back-tested and evaluated by the Kupiec test of unconditional coverage, the Christoffersen test of conditional coverage and the Lopez test. This paper finds the EWMA-model to be the only model which is able to produce accurate measures of VaR for all four indices. The GARCH-, TGARCH- and especially the MA-model are not able to accurately derive values of VaR and thus are not appropriate for variance-covariance VaR approach purposes. This paper therefore concludes that the variance-covariance VaR approach does provide accurate measures of market risk but only in case the EWMA-model is used to produce forecasts of volatility.

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1. Introduction

Mainly due to the last credit crunch, more and more focus of banks is transferred to the estimation and reduction of market risk. Banks use Value-at-Risk (VaR) as tool to indicate market risk. This section presents the background of market risk and explains the concept of VaR in general and, more specifically, that of volatility forecasting with the variance-covariance approach. Finally the purpose and outline of the thesis is presented.

1.1 Background market risk

The default of large banks like Lehman Brothers (in 2008) made the banking industry realize banks have to take into account the possibility of large losses and even bankruptcy due to market risk. A result of the credit crunch is the introduction of the new international regulatory framework “Basel III” (Basel Committee on Banking Supervision, 2010) in September 2009 by the Bank for International Settlements (BIS). The importance of measuring risk in portfolios of assets and securities is not only from recent time but goes back to the theory of portfolio selection introduced by Markowitz (1952, 1959).

Market risk can be defined as the risk of changes in the market value of a position or portfolio as a result of changes in market conditions (stock prices, interest rates and exchange rates) which are not expected (Berkowitz and O`Brien, 2002). Market risk, therefore, is included in currencies, bonds, stocks and other financial assets and liabilities a bank trades. The Basel Committee on Banking Supervision of the BIS and the US Securities and Exchange Commission (SEC) have strict rules and require banks to measure their market risk. The Basel Accords (Basel Committee on Banking Supervision, 2004) by the BIS include regulations to estimate and analyze market risk and to stimulate risk management. The update of the Basel Accords aims to “strengthen global capital and liquidity regulations with the goal of promoting a more resilient banking sector” and to “improve the banking sector's ability to absorb shocks arising from financial and economic stress, whatever the source, thus reducing the risk of spillover from the financial sector to the real economy” (Basel Committee on Banking Supervision, 2009B: page 1). However, not only supervisory authorities like the BIS and SEC but also the management of banks take into account measures of market risk. Banks do this in order to make investment decisions and to allocate capital. Market risk gained in importance over the last years due to rapid growth of trading accounts at banks.

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1.2 Value-at-Risk

To manage risk involved in trading, financial institutions developed models to measure the risk of a position or portfolio the institution faced. RiskMetrics™, introduced by US commercial bank J.P. Morgan (1995), was one of the first VaR models made public. VaR quantifies the exposure of a bank (or portfolio) to future market fluctuations. VaR is the maximum loss a bank is exposed to, given the defined time horizon and confidence level and can be defined by the following equation:

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where is the market value of portfolio i, stands for the critical value regarding a required confidence level and is the volatility forecast of the return of portfolio i (Jorion, 2001). The larger VaR, the more capital is charged by central banks to compensate for market risk. According to the capital structure theory of Modigliani and Miller (1958), the value of a firm is unaffected by how it is financed but depends on the return on assets. However, banks in general prefer not to hold too much capital to compensate for market risk since equity is more expensive than debt finance. Therefore, banks in general prefer measures of VaR which cover possible losses to a pre-specified degree but which do not require them to hold to much capital to compensate for market risk.

Value at risk is a popular approach to measure market risk on a daily basis since it presents risk in a single number (Morgan, 1995; Jorion, 1996; Brooks and Persand, 2002, 2003). However, one must consider VaR as a first-order approximation to possible losses, an estimate of risk. Since estimates in most cases are based on historical data, they will be affected by estimation risk and thus cannot be taken at face value. A major benefit of VaR for companies is the structured methodology to critically think about risk (Jorion, 1996). Other advantages of VaR are: management of banks can set overall risk targets, are able to determine capital requirements, and can guide decisions regarding investment/trading (Drowd, 2005). One of the biggest problems of VaR is non-additivity: the value of VaR of a portfolio of assets may be higher than the sum of the value of VaRs of the individual assets composing the portfolio. A second drawback is that VaR cannot describe the size of losses beyond the confidence level.

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purposes, this paper will not consider the method of implied volatility to forecast volatility. The simulation approaches estimate volatility by simulating a large number of possible scenarios with regard to future market developments. Difference between the historical simulation and Monte Carlo simulation approach relate to how the scenarios of risk factors are generated. Monte Carlo simulation generates VaR by calculating the value change of an asset using price scenarios which are randomly generated. First, certain assumptions about the distribution of asset prices need to be made. Historical simulation assumes the distribution of future asset returns equal the distribution of asset returns in the past. In this approach VaR is estimated by starting at the current value of the asset and calculate the change of the asset using actual movements in the past (chew, 1996).

According to Resti and Sironi (2009: page 115) the variance-covariance approach is “undoubtedly the one which is most widespread among financial institutions” for calculating VaR. This paper provides an answer whether the variance-covariance approach for deriving VaR for banks provides accurate measures for market risk. Using four volatility forecasting methods, this paper also recommends which method to use for deriving the most accurate VaR measures.

1.3 Volatility forecasting

When calculating VaR, future volatility is a difficult variable to predict. Volatility is a measure of dispersion of asset returns and is used to quantify the risk of it. Since the estimation of VaR depends critically on volatility, a lot of research is conducted to investigate which model most accurately forecasts volatility. Due to the different methodologies of predicting volatility, different estimates of volatility and thus different values of VaR are calculated. This so called model risk is an important issue in risk management and strengthens the choice of good model selection.

Volatility, in the light of risk management, is used in deriving VaR. In a broader sense, forecasting volatility is key in economic decision making and analysis since volatility is used in pricing options (using e.g. the several versions of Black and Scholes (1973) option pricing formula), in investment decisions and monetary policy making (since these decisions depend on fluctuating macroeconomic variables). Estimating future volatility using a model that most accurately forecasts volatility therefore plays a crucial role.

1.4 Purpose of study

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shows these models accurately forecast volatility. Integrating outcomes of each model in the calculation of VaR shows whether the variance-covariance VaR approach provides accurate measures of market risk and which method for forecasting volatility results in the most accurate measure of VaR. This paper uses daily data since ARCH type models require a large sample of data to estimate robust values for its coefficients (Figlewski, 1994). Therefore, ARCH type models work better with daily data compared to data of lower frequency (Brooks, 2008). One-day ahead measures of VaR are derived since this paper focuses on short-term market risk of a bank`s risky assets. Since market value of these assets fluctuate and must be managed on a daily basis, one-day ahead measures are calculated. Besides, since most bank`s change it composition of risky assets on a daily basis, a one-day ahead estimate of VaR is most appropriate. Estimates of VaR over a longer period (for example weekly, monthly or quarterly) are more appropriate for mid-term, short-term or reporting purposes of market risk.

Using periods of data including a relatively low and high volatile period, this paper also tests the performance of forecasting models and outcomes of VaR under different conditions.

The above mentioned problems result in the following research question and sub questions:

Does the variance-covariance Value-at-Risk approach provide accurate measures of market risk?

- Does a significant difference exist in the accuracy of the volatility forecasting methods to measure Value-at-Risk using the variance-covariance approach?

- Does the ranking of the volatility forecasting methods in measuring Value-at-Risk differ in case of different levels of volatility?

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Testing the variance-covariance VaR with these four models for volatility forecasting over such a period of European data and under different levels of volatility can be seen as an update of existing evidence.

1.5 Outline of the paper

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2. Literature review

As stated in section 1.2, the variance-covariance approach is the most widespread among financial institutions to derive VaR. This section primarily explains motives for its widespread use and continues with possible drawbacks. Next section explain the Moving Average- (MA), the Exponentially Weighted Moving Average- (EWMA), the Generalized Autoregressive Conditional Heteroscedasticity- (GARCH) and the Treshold-GARCH- (TGARCH) model to forecast volatility, its most important variable in calculating VaR. Section 2.3 conclude with empirical evidence regarding the performance of the four volatility models.

2.1 Variance-covariance VaR

The main reason for the widespread use of the variance-covariance approach in deriving VaR is its simplicity. This is not related to the conceptual framework of the approach but to the time consuming and calculation intensiveness. Moreover, since it is the original version and firstly introduced model of VaR and because of the presence of RiskMetrics™ (the database based upon this approach) which is used by related products, the approach was rapidly spread among banks (Resti and Sironi, 2009). Drawbacks of the approach are essentially linked to the underlying assumptions of the variance-covariance approach. The assumptions are (Jorion, 1996): (i) asset returns are normally distributed with zero mean and stable volatility, (ii) changes between an assets‟ value and risk factor are linear, (iii) changes in the value of assets are obtained using the historical correlation matrix, and (iv) VaR is found by multiplying the value of the assets by its volatility and a scaling factor (dependent on the chosen confidence level).

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structures. By using convexity and gamma for respectively bonds and options, this problem of non-linearity can be overcome (Resti and Sironi, 2009).

Historical simulation and Monte Carlo simulation have the advantage of not relying on a distribution of the returns of assets and to have recourse to full valuation. This means that these models can also deal with derivatives which have non-linear payoffs. Although with historical simulation the distribution of asset returns does not need to be estimated, it assumes that the distribution in the future will equal the distribution in the past. Historical simulation thus assumes that the distribution of asset returns is stable over time. Furthermore, historical simulation assigns equal weight to returns of the whole period and number of possible scenarios.

2.2 Volatility forecasting

Section 2.2 describes the four methods this paper uses to test whether the variance-covariance approach is an accurate method of deriving VaR.

2.2.1 Moving average

Models like the Random Walk-model (RW), the Historical Average-model (HA), the MA- and the EWMA-model are the simplest models in today‟s volatility forecasting. The MA-model forecast volatility by calculating the unweight average of observed squared returns over a fixed length of historical time. The one-day ahead forecast is derived by:

∑ (2)

where is the forecasted variance which is the square of volatility, T is the length of the fixed

period over which volatilities are averaged and is the return at day calculated by:

( - )

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where presents the current return and presents the return of the day before (Brailsford and Faff,

2006). According to Brailsford and Faff (1996) this simple model is often used in traditional time-series analyses.

2.2.2 Exponentially Weighted Moving Average

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( ) (4)

where is the predicted variance which need to be squared to present volatility,

( ) which is the estimate of volatility produced at the end of day , is the most recent

asset return (see equation 3, section 2.2.1) and is the decay factor which must lie between 0 and 1 (Hull, 2009).

In case the decay factor equals 0, only weight is given to most recent actual volatility and the prediction of the next period volatility equals the current volatility. In this case the outcome of EWMA-model will equal the outcome of a RW-model (the volatility forecast of a RW-model equals the most recent observation of actual volatility). If the decay factor is close to 1, forecasts for a small degree are influenced by current actual volatility and largely reflect predicted volatility over previous time periods. The optimal value of the decay factor is determined empirically as the value which minimizes the in-sample sum of squared prediction errors. Brown (1962) recommends using an estimate of the decay factor in the range of 0.05 to 0.30 (in absence of further information). However, more recent studies (Dimson and Marsh, 1990; Brailsford and Faff, 1996) show that higher values of the decay factor are optimal (0.76 and between 0.51 and 0.98 respectively). RiskMetrics™ (which uses EWMA as standard approach in deriving VaR using the variance-covariance method) recommends a decay factor of 0.94.

Advantage of the EWMA-model, in contrast with the MA-model, is that in calculating volatility from historical returns, it attributes non-equal weights to these returns. This leads to quicker responding to market factor changes and avoids an echo effect (Figlewsky, 1994). An echo effect is defined as a change in estimated volatility not only due to a recent market change but also in case this market change leaves the sample because it is replaced by more recent data. This second change in estimated volatility is not justified since it does not present a new piece of data of which estimated volatility needs to take account of.

2.2.3 GARCH

ARCH-family models, introduced by Engle (1982), are popular since they take into account econometrical issues the MA- and EWMA-model cannot deal with. These issues include:

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(ii) Co-movement in volatility. This is the possibility of cross-correlations in financial time series of different financial markets. Black (1976) found volatility of individual assets to change. More specifically, Harris (1986) and Bollerslev (1991) indicate that at the beginning and at the end of the day levels of volatility are highest.

(iii) Long memory of volatility. Bollerslev et al. (1992) found that volatility is highly persistent over time. The idea behind ARCH type models is mean reversion of volatility to a kind of long run average.

ARCH type models hence are able to predict volatility using a regression based upon past volatility and generate an estimate of volatility that changes over time. Especially the GARCH-model developed by Bollerslev (1986) and Taylor (1986) is often used as model to forecast volatility. The most applied GARCH-model is the GARCH (1,1) version where one-day ahead variance is estimated by:

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where = and represents the long-run average variance rate, are weights which must sum to 1, is the most recent asset return (see equation 3, section 2.2.1) and

which is the previous period estimate of variance (Hull, 2009).

As the equation shows, the strategy behind the GARCH-model is a rather intuitive prediction since the conditional variance estimate is derived from the long-run variance, the previous period expected variance and a shock relating to the last period. The weight given to each component determines the degree to which the model reverts to its long run mean. Here the historical variances have declining weights which cannot have the value zero. The optimal factors of these decay factors are determined using the likelihood function of the GARCH-model.

A limitation of the GARCH-model is that it does not take into account asymmetry; it considers the impact of shocks in the market on volatility forecasting to be independent of its direction (Balaban, 2004). This is due to the fact that errors are squared. The TGARCH-model overcomes this problem.

2.2.4 TGARCH

The TGARCH-model introduced by Zakoian (1994) takes into account the possible asymmetry of past forecast errors. The TGARCH-model deals with the observation that negative and positive shocks of equal size have different effect on stock price volatility. Also the negative skewness in series of stock market returns, which may be due to the fact that negative shocks are greater in both absolute size and frequency and happen more quicker as positive shocks (Franses and van Dijk, 1996), can be captured by the TGARCH-model. The TGARCH-model to estimate one-day ahead variance is presented as:

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where represents the long-run average variance rate , weights amd must sum to 1, is the latest period asset return (see equation 3, section 2.2.1), _- _- _- _- which is the previous period estimate of variance and represent a dummy variable with the value 1 if and 0 if (Gonzalez-Rivera et al., 2004). Thus, for example, in case of positive news the impact will be while in case of negative news the impact will be . The impact of positive news on volatility will therefore be greater if and for negative news the impact will be greater if .

Other popular types of volatility forecasting models not described so far and not included in this paper are stochastic volatility (SV) models (Taylor, 1982, 1986). SV-models model the “variance as an unobserved component that follows a particular stochastic process” (Sadorksy, 2005: page 121). Examples of SV-models are quasi-maximum likelihood estimation (QMLE) (Harvey et al., 1994) and generalized method of moments (GMM) (Duffie and Singleton, 1993). Harvey et al. (1994) and Jacquier et al. (1994) popularized SV-models although, according to Poon and Granger (2003) and Sadorksy (2005), they did not forecast volatility better than the MA-, EWMA-, GARCH- or TGARCH-model.

2.3 Empirical literature

Brailsford and Faff (1996), who uses 20 years of daily Australian stock data, investigate the performance of the more difficult ARCH type models, and the more simple volatility models like MA, EWMA, historical average (HA) and exponential smoothing (ES). Brailsford and Faff (1996) conclude that although results are dependent on the error statistics used, in general the more complex ARCH type models estimate more accurate forecasts. Especially the Glosten, Jagannathan and Runkle (1992) GARCH (GJR-GARCH) model, which also takes into account asymmetry, provides good estimates though the authors state no model is superior. If only the more simple volatility models are taken into account a regression model performs best. Used error statistics are Mean Error (ME), Mean Absolute Error (MAE), Root Mean Squared Error (RMSE) and Mean Absolute Percentage Error (MAPE). Earlier results by research of Tse (1991) disagree with Brailsford and Faff (1996) that more complex models provide better forecasts. Tse (1991) uses data from the Tokyo stock exchange in the period 1986-1989 and concludes that the more technical ARCH family models do not provide good forecasts. This paper states that the EWMA-model appears to perform better and makes it attractive in real applications because of its simplicity. Tse and Tung (1992) confirm this with data from the Singapore stock market.

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addition, an RW-model. This paper uses weekly data of five European indices over the period 1986-1994 (the stock market crash of 1987 is excluded in their study). For their evaluation an Aikaike Information Criterion (AIC), a Log Likelihood (LL) and a Box-Pierre test statistic is used. Out of the five used forecast periods, in three periods the QGARCH-model outperforms where in the other two periods the RW-model surpasses the ARCH type models. The evidence for the RW-model is partly shared with McMillan et al. (2000). McMillan et al. (2000) analyze the performance of ten volatility models including the MA-, EWMA-, GARCH- and TGARCH-model at the monthly, weekly and daily frequencies. In this paper data from the United Kingdom is used and tested under both symmetric and asymmetric loss functions. Asymmetry is evaluated using the Mean Mixed Errors statistic for respectively over prediction (MME (O)) and under prediction (MME (U)). Under symmetric loss the GARCH- and MA-model are superior. Under asymmetric loss, when penalizing over (under) predictions more heavily, the RW- (HA-) model performs more accurate.

Sarma et al. (2003) test the performance of an EWMA-, RiskMetrics™-, GARCH- and Historical Simulation-model to decide which model most accurately forecasts volatility for VaR purposes. American and India data over the period 1980-1999 is used to show that the GARCH-model performs most excellent. Using India data and the test of Christoffersen (1998) for evaluating VaR, it is shown that the GARCH-model forecasts volatility correctly at a 99% confidence level. The superiority of the GARCH-model in the paper of Sarma et al. (2003) is in contrast with results of Wong et al. (2003). Wong et al. (2003) compare the performance of time-series models and simulation models to apply them on the VaR back-testing criteria of the Basle Committee. This paper concludes that the ARCH type models are not reliable for forecasting VaR or for risk management and that simulation methods perform equal to time-series models. This is done by using Australian stock data over the period 1983-1999 and using MSE, MAE and relative VaRs as evaluation methods. Ferreira (1983-1999) find the GARCH-model to be best of the ARCH family. The ES-model provides best results if ARCH type models are not taken into account. Walsh and Tsou (1998) find the EWMA-model, after the GARCH-model, to provide best estimates.

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data as a benchmark of true volatility. McMillan and Speight (2004) also state that many failures of the GARCH-model in earlier studies arise from a failure in using the correct benchmark, not from a failure of the GARCH-model itself.

Gonzalez-Rivera et al. (2004) test the EWMA-, MA-, ARCH type- and SV-model in order to find which model can best be used for option-, utility-, VaR- and predictive likelihood- purposes. For VaR purposes this is evaluated using the Q loss function (Koenker and Bassett, 1978). Remarkably, the EWMA-model proposed by RiskMetrics™ to calculate VaR is the worst performing model. Also the MA- GARCH- and TGARCH-model do not derive accurate estimates of VaR. American data over the period 1970-2000 shows that the SV-model performs best. The relative poor performance of the EWMA-model is supported by Ederinton and Guan (2005). The asymmetric EGARCH-model performs equal with the GARCH-model according to Ederinton and Guan (2005). Ederinton and Guan (2005) use data from the S&P 500 index, the yen/dollar exchange rate, the three-month Eurodollar rate, the ten-year treasury bond rate and five equities from the Dow-Jones index to test the forecasting ability of different time series models including the GARCH-, EGARCH- and EWMA-model. Ederinton and Guan (2005) find no clear favorite between the ARCH type models but conclude these models yield better forecasts compared to the simpler EWMA-model. This is in line with Akgiray (1989) which states that the GARCH-model is superior to respectively the ARCH-, EWMA- and HA-model when using US monthly stock data.

Balaban et al. (2006) test 11 models including the MA-, EWMA-, and GARCH-model to forecast volatility and use data of 15 countries in the period December 1987 to December 1997. In their paper, performance is measured using symmetric and asymmetric error statistics. The authors state that for VaR purposes the ARCH type models perform better. When comparing the basic GARCH-model with the asymmetric GJR-GARCH- and EGARCH-model, no significant difference is found.

Niquez (2008) assess the forecasting performance of the EWMA-model and ARCH type models using the Spanish stock index over the period 1987-2002. Niguez (2008) conclude that the EWMA- and GARCH-model and models which can take account of asymmetry produce significant forecasts. Niquez (2008) uses squared excess returns as benchmark for true volatility and tests significance at a 95% significance level. A sharp result from this paper is that asymmetric models tend to yield a better performance compared to symmetric models.

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MA-model. In order to evaluate this, Sadorsky, besides commonly used error statistics like MSE, Mean Percentage Error (MPE), MAPE, also uses less frequent statistics like Mean Absolute Deviation (MAD), Theil U Statistic, the Diebold and Mariano (1995) test statistic (MD) and the Modified MD (Harvey et al., 1997) test statistic (MDM). Overall, Sadorsky (2005) concludes that, at a 95% confidence level, the more simple volatility models (especially the MA- and ES-model) perform better compared to the SV-model.

Brooks and Persand (2003) state that the gain from using a more complex ARCH type model is minimal compared with the performance of a regular GARCH-model. Brooks and Persand (2003) also find asymmetric models to provide poor estimates for risk management. This is in contrast with Balaban (2004) which favors the asymmetric EGARCH-model over respectively the GARCH- and ARCH-model. Hansen and Lunde (2005) check whether a GARCH-model outperformed 330 other models from the ARCH family. Using the IBM stock and the Danish exchange rate data, Hansen and Lunde conclude this is not the case. No model clearly outperformed. According to Poon and Granger (2003) and Pagan and Schwert (1990) the GARCH-model is the best model of the ARCH family. Dunis et al. (2000) and Blair et al. (2001) also find support for the GARCH-model.

An overview of past research is shown in table 2.1. This table shows that existing literature contains conflicting evidence with regard to the superiority between the simpler MA- and EWMA-models and the more complex ARCH type models. These inconsistencies are disturbing because of the impact of the consequences for risk management.

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Table 2.1: Empirical literature

Article Key question Tested models Data Preferable model Error statistic Significance

Dimson and Marsh (1990)

Best general volatility model

ES, HA, MA, Regression, RW

UK FTA All share index over period 1955-1989

ES, Regression MSE, RMSE No

Brailsford and Faff (1996)

ES, EWMA, HA, MA, Regression, RW, GARCH, GJR-GARCH

Australian Statex-Actuaries Accumulation Index over period 1974-1993

GJR-GARCH ME, MAE,

RMSE, MAPE No

Franses and van Dijk (1996)

RW, GARCH, Q-GARCH, GJR-GARCH

5 European indices over period 1986-1994

Q-GARCH, RW AIC, LL, BPTS No

McMillan et al. (2000)

ES, EWMA, HA, MA, Regression, RW, GARCH, TGARCH, EGARCH, CGARCH

UK FTA All share index over period 1969-1994, UK FTSE100 index over period 1984-1994

MA, GARCH ME, MAE,

RMSE, MME (U), MME (O)

No

Brooks and Persand (2003)

Best volatility model for VaR calculation

ARCH types, Multivariate UK FTSE All share total return

index, 15y UK government bond index, Reuters commodities price index over period 1980-1999

No model MAE , MSE No

Sarma et al. (2003)

Best volatility model for

VaR calculation EWMA, Risk Metrics_{GARCH, Historical} ™,

Simulation (HS)

US S&P500 index, India`s NSE-50 index over period 1990-2000

GARCH Christoffersen for VaR 99% confidence level Wong et al. (2003)

Best volatility model for VaR back-testing

RW, AR, Autoregressive and Moving Average (ARMA), ARCH, GARCH

Australia`s All Ordinary Index (AOI) over period 1983-1999

Variance-covariance approach and Simulation methods perform equally, bad forecasts

MAE, MSE, Relative VaRs

No

Balaban (2004) Best general volatility

model

ARCH, GARCH, EGARCH, GJR-GARCH

US Dollar – Deutsche Mark exchange rate over period 19974-1997

GARCH ME, MAE, MSE,

MAPE, MME (U), MME (O)

No

McMillan and Speight (2004)

ES, MA, GARCH, TGARCH, CGARCH

US Dollar - 17 exchange rate series over period 1990-1996

GARCH ME, MAE,

RMSE

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Table 2.1:

Empi

rical literature (continued)

Article Key question Tested models Data Preferable model Error statistic Significance

Gonzalez-Rivera et al. (2004)

Best volatility model for options, utility, VaR and predictive likelihood

EWMA, MA, ARCH types, SV

US S&P500 index over period 1970-2000 SV Q loss function (Koenker and Bassett, 1978) No Ederington and Guan (2005)

EWMA, Historical Standard Deviation (HSD), GARCH, EGARCH, AGARCH, RLS

US S&P500 index, 5 equities Dow Jones index over period 1962-2002, US Dollar – Euro-, US Dollar – Japanese Yen- exchange rate over period 1971-2003, 10y US treasury bond rate over period 1962-2003

GARCH, EGARCH RMSE No

Hansen and Lunde (2005)

330 ARCH types IBM stock over period 1990-1999,

US Dollar – Danish Mark exchange rate over period 1987-1992

No model MAE, MSE,

QLIKE, R²LOG No

Sadorksy (2005)

ES, HA, MA, RW, Autoregressive (AR5), SV

Future prices on US S&P500 index, 10y US government bond, WTI oil contracts, US Dollar – Canadian Dollar exchange rate over period 1984-2003

MA, ES, AR5 MSE, MAD,

MPE, MAPE, Theil U statistic, DM test statistic, MDM 95% confidence level Balaban et al. (2006)

Best general volatility model and best volatility for VaR calculation

ES, EWMA, HA, MA, Regression, RW, Weighted Moving Average (WMA), ARCH, GARCH, GJR-GARCH, EGARCH

Stock market indices of 15 countries over period 1987-1997

VaR can best use ARCH type models

ME, MAE, RMSE, MAPE, MME (U), MME (O)

No

Niguez (2008) Best volatility model for

VaR calculation

EWMA, ARCH types Spanish IBEX35 index over period

1987-2002

EWMA, GARCH, Asymmetric models

MSPE, MAPE, MME (U), MME (O), AIC, LR

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To test the research question whether the variance-covariance VaR approach provide accurate measures of market risk I state the following hypotheses:

: The variance-covariance Value-at-Risk approach provides accurate measures of market risk : The variance-covariance Value-at-Risk approach does not provide accurate measures of

market risk

In order to test the first sub question, whether a significant difference exist in the accuracy of the volatility forecasting methods to measure Value-at-Risk using the variance-covariance approach, the following hypotheses are included:

: The different methods to measure Value-at-Risk using the variance-covariance approach do not differ significantly with regard to accuracy.

: The different methods to measure Value-at-Risk using the variance-covariance approach do differ significantly with regard to accuracy.

In order to test the second sub question, whether the ranking of the volatility forecasting methods to measure Value-at-Risk differ in case of different levels of volatility, the following hypotheses are included:

: The ranking of the accuracy of volatility forecasting methods does not differ in case of different levels of volatility

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3. Methodology

This section describes the methodology to answer the research questions. First, section 3.1 explains the rolling window procedure applied to produce forecasts of volatility. Section 3.2 presents the back-testing approaches to evaluate the performance of the different methods to derive at measures of VaR: the Kupiec test of unconditional coverage, the Christoffersen test of conditional coverage and the Lopez test.

3.1 Rolling window procedure

An out-of-sample one-day ahead rolling window procedure as is applied by Brooks and Persand (2003) and Koopman et al. (2005) is followed to produce one-day ahead volatility forecasts. This procedure rolls the data sample forward by one trading day while the sample of historical data stays 1250 trading days, as is recommended by the Basel Committee (Basel Committee on Banking Supervision, 2009A). For example, the AEX index including the period of the 4th of July 1990 to the 28th of February 2011 contains 5252 trading days where volatility models estimated 4002 samples out of 5252 observations. Following the rolling window procedure, the sample rolls forward 4001 times beginning from the 4th of July 1990 while the number of 1250 observations for each forecast stays equal. The first sample generates a forecast for the 1251th day from the 4th of July 1990. Generating forecasts using a rolling window procedure with 1250 observations should ensure that estimated volatilities are produced using parameter estimation over a sufficient amount of data. Meanwhile, the rolling window procedure does not incorporate such a long period of data that it may no longer be relevant in a changing financial market.

The Basel Committee recommends using 1250 trading days of historical data while it states banks should at least use 250 trading days of historical data. To test whether a sample of historical observations of 1250 trading days indeed produce most accurate volatility forecasts, this paper also uses samples of 125, 250 and 500 historical observations.

After deriving volatility, measures of VaR are produced using equation 1, see section 1.2. An asset value of €1000 and a confidence level of 99% (as is recommend by the Basel Committee) is used. For the EWMA-model, a decay factor with regard to the degree to which most recent returns and previous estimated volatility are taken into account, needs to be determined. Since RiskMetrics™ (Morgan, 1995) finds a decay factor of 0.94 provides the most accurate results, this paper uses a decay factor of 0.94 for the EWMA-model.

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coefficients. Therefore, and since previous research conclude the TGARCH-model in many cases shows accurate measures of volatility (McMillan and Speight, 2004), this paper includes the TGARCH-model as model to account for asymmetry.

3.2 Back-testing methods for VaR

The accuracy of the forecasting models to produce measures of VaR in this paper are measured using three back-testing methods: the unconditional coverage test by Kupiec (1995), the conditional coverage test by Christoffersen (1998) and the Lopez test based upon a loss function (Lopez, 1999).

3.2.1 Kupiec test of unconditional coverage

This test is one of the first tests for back-testing VaR and is also called „proportion of failures test‟. It is based upon the frequency with which the losses of a portfolio exceed VaR. The test presents a Likelihood Ratio of unconditional coverage ( ) and can be presented by:

* ( )

( ) + (7)

where is the significance level, presents the number of observations in the sample, is the number of exceptions that losses exceed VaR and is the exception rate: (Kupiec, 1995).

follows a chi-square distribution with one degree of freedom. The VaR model is regarded as a good model in case the value of does not exceed the chi-square critical value. The test`s p-value

presents the probability (in case the null hypothesis is correct) that the value of the unconditional coverage test is higher compared to the observed one. In practice, a lower p-value indicates a less reliable model.

The value of increases both in case the number of exceptions exceeds the significance level and

in case the number of exceptions is lower compared to the significance level; it presents the likelihood ratio of deriving the significance level. Since banks are mainly interested in the degree to which exceptions exceed the significance level, exception rates for each model are also presented in the results.

The test of unconditional coverage only takes into account the coverage of VaR estimates and does not test whether these estimates are independently distributed. In order to not only test the probability of unconditional coverage but also the dependency of VaR estimates, the Christoffersen test of conditional coverage, as recommended by Campbell (2005), is used.

3.2.2 Christoffersen test of conditional coverage

Since not considers the distribution of time with regard to the exceptions of losses over VaR, a

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the case when in a sub period VaR is underestimated since there is a high number of exceptions while in another sub period VaR is overestimated since the number of exceptions is low. Because does

not take account of possible clustering of exceptions and the ability to respond to new market conditions, this test is unconditional. Major advantage of the Christoffersen test of conditional coverage (Christoffersen, 1998) is that it also tests whether exceptions are serially independent. It does so by providing a Likelihood Ratio of unconditional coverage which is similar to the test of

Kupiec (1995), a Likelihood Ratio of serial independence and a Likelihood Ratio of conditional

coverage where,

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In this test, takes account for possible clustering of exceptions and can be presented by

(Christoffersen, 1998):

- [ ( - )

( - ) ( - ) )] (9)

In this equation the numerator is defined as the constrained likelihood function which represents the probability of finding the number of exceptions equal to . In equation (9) the denominator represents the maximum probability of obtaining the number of exceptions equal to . and are defined as follows: (10) (11) (12)

where are defined as:

- : number of no exceptions followed by no exceptions

- : number of exceptions followed by no exceptions

- : number of no exceptions followed by exceptions

- : number of exceptions followed by exceptions.

follows a chi-square distribution with one degree of freedom. follows a chi-square

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rejected when exceeds the corresponding significance level. As with and , also values

of need to be as low as possible to evaluate the corresponding model as adequate. 3.2.3 Lopez test

The tests for unconditional and conditional coverage presented above are solely based on the number of times losses exceed VaR. The test proposed by Lopez (1999) not only takes into account the number of exceptions but also the size of exceeded losses. It is based on minimizing a loss function where the loss function increases with the exceeded losses over VaR. The loss of the upcoming day ( ) can be presented by:

{ ( )

} (13)

where is the assets return at and is VaR produced at , related to and calculated by equation 1 (section 1.2) and divided by the value of the portfolio (Resti and Sironi, 2009). The average loss of each exception ( ) in a sample of N days starting at day is:

∑ (14)

Purpose of each model is to minimize the average loss of each exception.

Although the advantage of this method is that also the size of excess losses are taken into account, one must acknowledge VaR is only constructed for indicating the frequency of excess losses. The size of losses therefore is not a proper way for evaluating a VaR model since it is not related to its logic. The reason this back-testing model is included in this paper is because it, nevertheless, is interesting for banks to know how much they relatively can lose in excess of VaR using the four models.

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4. Data description

This data description section first explains the data used and continues with the identification of the sub periods to use for testing the performance of the volatility forecasting models under different levels of volatility. This section ends with descriptive statistics of both the total sample period as well as the sub sample periods.

4.1 Data used

This paper uses European data from four stock indices, namely the Dutch AEX index, the France CAC40 index, the German DAX index and the FTSE100 index from the United Kingdom. These indices represent a weighted average of the 25, 40, 30 and 100 largest companies of respectively The Netherlands, France, Germany and the United Kingdom measured by their market capitalization. These indices are used since they represent well diversified portfolios which actually could represent a bank‟s risky assets. By selecting four European indices this paper attempts to present diversified and robust European evidence. Daily opening, closing and intraday high and low prices are derived from Thomson Reuters DataStream (2010) in order to calculate returns for the forecasting of volatility. Log-returns are calculated using equation 3 (section 2.2.1)

Data from the four indices are derived from the period in which all four daily price variables (opening, closing, high and low) were available. The data sample for all four indices last till the end of February 2011. For the AEX index first observations are identified at the 4th of July 1990 including a total of 5388 observations. Removing the 136 non-trading days during this period induce a total of 5252 observations. The CAC40 index starts at the 4th of January 1988 and shows 6040 observations including 194 non-trading days. The CAC40 index, then, includes a sample of 5846 observations. At the 1st of November 1988 the DAX index shows the first observations. With 5824 observations and 202 non-trading days, this index causes a total of 5622 trading days. The FTSE100 index, starting at the 7th of January 1985, counts 6820 observations of which 218 are non-trading days leaving a total sample of 6602 observations. Non-trading days of the indices are due to national holidays.

4.2 Sub samples

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Figure 4.1: Actual daily volatility of AEX, CAC40, DAX and FTSE100 indices

Note: actual daily volatility is calculated using the extreme-value estimator (Poon, 2005): ( ) (

)

where presents the actual volatility measure, is highest intraday price of the asset, is the lowest intraday price of the

asset, and are respectively the opening and closing prices of the relating day.

AEX

CAC40

DAX

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Figure 4.1 shows high levels of volatility during the period 1997 – 2004 as a result of the so-called dot-com bubble. Another high volatile period which can be identified is presented by the period 2007 – 2010 and is explained by the recent credit crunch. Furthermore, the FTSE100 index shows an outlier at the 19th of October 1987 which is explained by the crash known as Black Monday. This paper uses the period January 2007 – December 2010 as sub sample of relative high volatility. Between the periods of relative high volatility (the dot-com bubble and recent credit crunch) the level of volatility is relatively low. This paper uses this period which contains the period January 1993 – December 1996 as sub sample of relative low volatility.

4.3 Descriptive statistics

Descriptive statistics of the returns of the AEX, CAC40, DAX and FTSE100 indices of the entire sample period are presented in table 4.1.

Table 4.1: Descriptive statistics of the daily returns during the total sample period

AEX CAC40 DAX FTSE100

Mean 0.000 0.000 0.000 0.000 Median 0.001 0.000 0.001 0.001 Maximum 0.100 0.106 0.108 0.094 Minimum -0.096 -0.095 -0.137 -0.130 Std. Dev. 0.014 0.014 0.015 0.011 Skewness -0.146 -0.035 -0.262 -0.517 Kurtosis 9.667 7.918 9.342 13.150 Jarque-Bera 9746.581 5891.698 9486.343 28632.600 Probability 0.000 0.000 0.000 0.000 Observations 5252 5846 5622 6602

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Tables 4.2 and 4.3 presents descriptive statistics of the returns of the AEX, CAC40, DAX and FTSE100 indices during the low and high sub sample period. The low volatile sub sample period shows that the returns of the four indices have means and medians of around zero. As could be expected, the maximum en minimum returns are smaller and vary in a lesser extend compared to the returns of the total sample period. Also the standard deviations of all indices are lower using this period of data compared to the standard deviations of the total sample period. While the sample still is

Table 4.2: Descriptive statistics of the returns during the low volatile period: January 1993 –

December 1996

AEX CAC40 DAX FTSE100

Mean 0.001 0.000 0.001 0.000 Median 0.001 0.000 0.001 0.000 Maximum 0.025 0.032 0.037 0.023 Minimum -0.025 -0.035 -0.041 -0.023 Std. Dev. 0.007 0.010 0.009 0.007 Skewness -0.285 -0.001 -0.324 -0.172 Kurtosis 3.665 3.119 4.459 3.308 Jarque-Bera 32.183 0.589 106.657 8.984 Probability 0.000 0.745 0.000 0.011 Observations 1008 999 1004 1011

negatively skewed, almost no excess kurtosis is identified. Especially the CAC40 index shows to be almost normally distributed. The DAX index, however, still has excess kurtosis of 1.459 which rejects the hypothesis of normally distributed returns.

Table 4.3: Descriptive statistics of the returns during the high volatile period: January 2007 –

December 2010

AEX CAC40 DAX FTSE100

Mean 0.000 0.000 0.000 0.000 Median 0.000 0.000 0.001 0.000 Maximum 0.100 0.106 0.108 0.094 Minimum -0.096 -0.095 -0.074 -0.093 Std. Dev. 0.018 0.018 0.017 0.016 Skewness -0.098 0.200 0.226 -0.065 Kurtosis 9.483 9.025 9.776 8.938 Jarque-Bera 1796.815 1557.212 1952.498 1489.119 Probability 0.000 0.000 0.000 0.000 Observations 1025 1025 1016 1013

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5. Results

This section describes the results this paper provides. It starts with an analysis of the performance of the four different methods, the Moving Average- (MA), Exponentially Weighted Moving Average (EWMA), Generalized Autoregressive Conditional Heteroscedasticity- (GARCH) and the Treshold-GARCH- (TGARCH) model. VaRs are calculated using a historical sample of 1250 days. Section 5.2 presents a check of robustness whether the same results are shown using samples of 500, 250 and 125 days. This section also shows which lengths of historical sample can be best used for each method. Finally, section 5.3 answers the question whether the preference for a method change in case VaR is calculated for periods with different levels of volatility.

5.1 Total sample period

Values of VaR are calculated using equation 1 (section 1.2) with a portfolio value of €1000 and a 99% confidence level. The MA-, EWMA-, GARCH- and TGARCH-model are used to calculate one-day ahead VaR forecasts. The results are presented in table 5.1. Table 5.1 shows that most methods derive an average VaR of around 30 for all four indices. Only the FTSE100 index shows lower VaRs of around 24, implying the average volatility of the selected period is lower compared to the other indices. For every index, the MA-model presents the highest maximum VaR and the lowest minimum VaR. The TGARCH-model approximately shows the opposite implying this model, for almost every index, forecasts the broadest range of VaRs. However, when analyzing the standard deviation of each model and of each index, the EWMA-model shows the highest level of dispersion. Except for the MA-model, each model calculates VaRs which are negatively skewed and have excess kurtosis. This results in a high Jarque-Bera value of which none is statically significant, indicating that none of the models calculate normally distributed VaRs.

In Table 5.2 one can find the outcomes of the Kupiec test which produce the Likelihood Ratio of unconditional coverage ( ), the Christoffersen test which produce the Likelihood Ratio of

independence ( ) and the Likelihood Ratio of conditional coverage ( ) and the Lopez test.

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Table 5.1: Results of VaR using 1250 days of historical data

Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque-Bera Observations

AEX MA -31.59 -33.63 -16.83 -43.34 7.86 0.54 -0.97 2814.98 4003 EWMA -30.79 -25.93 -10.84 -119.88 17.66 -1.83 3.96 2378.84 4003 GARCH -30.52 -25.67 -12.21 -143.62 16.93 -2.18 6.24 4919.80 4003 TGARCH -29.87 -25.38 -10.77 -151.10 16.76 -2.25 7.50 6762.69 4003 CAC40 MA -31.29 -30.81 -23.24 -40.95 5.12 -0.17 -1.34 3624.69 4597 EWMA -30.29 -26.84 -12.38 -112.60 14.18 -2.10 5.96 5048.59 4597 GARCH -30.54 -27.48 -13.26 -131.45 12.92 -2.59 10.13 14872.75 4597 TGARCH -29.82 -26.74 -5.15 -138.79 12.81 -2.53 11.05 17335.72 4597 DAX MA -33.14 -33.46 -20.48 -46.20 7.13 0.03 -1.08 3037.49 4373 EWMA -31.61 -27.36 -11.53 -107.44 15.90 -1.67 2.96 2031.95 4373 GARCH -31.70 -27.38 -13.27 -120.53 14.69 -2.02 4.99 3695.10 4373 TGARCH -31.17 -27.28 -12.93 -113.52 14.41 -1.89 4.32 2928.97 4373 FTSE100 MA -24.83 -25.18 -16.35 -34.42 5.07 -0.18 -1.10 3783.73 5353 EWMA -23.69 -20.39 -9.77 -106.68 12.21 -2.42 9.00 13245.36 5353 GARCH -24.18 -21.20 -10.57 -122.59 11.40 -3.13 15.63 44314.78 5353 TGARCH -23.86 -20.87 -10.34 -132.93 11.20 -3.19 17.66 56963.01 5353

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indices and produced by the four models are graphically presented in figure 5.1. In appendix A, one can find a graphical presentation of the portfolio fluctuations and corresponding VaR levels for each model during the selected period.

When including the exception rates into the formula of Kupiec`s unconditional coverage test, the EWMA-model is the only model which does not exceed the chi-square critical value and thus is statistically significant at a 99% confidence level. The TGARCH-model and especially the MA-model

Table 5.2: Test results of forecasted VaRs using 1250 days of historical data

Note: Values of VaR are calculated using an observation period of 1250 days, a portfolio value of €1000 and a confidence

level of 99%. Values with * are statistically significant at a 99% confidence level.

produce high values of . When focusing on the serial independence of exceptions, at all indices,

the MA-model shows values of which are high and insignificant. The EWMA-, GARCH- and

TGARCH-values are significant. This indicates that the exceptions produced by these models are not clustered. Combining these two likelihood ratio`s results in values of . Likelihood ratios of this

test indicate that only the EWMA-model is a proper model to use for calculating VaR. As table 5.2 shows, the MA-model is far from appropriate. The GARCH- and TGARCH-model both do not show statistical significance for any of the indices and thus are not suitable for VaR purposes. Although

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both models produce approximately equal results, the GARCH-model is preferred according to the tests of Kupiec and Christoffersen. When taking Lopez test values into account no clear ranking of performance between the models can be made. Since the EWMA-model at the AEX, CAC40 and FTSE100 indices show to have the highest average excess VaR values, the EWMA-model proves not to be the preferable model at all tests. The higher Lopez test values for the EWMA model indicate that, in case losses exceed the level of VaR, one would face the largest losses over the level of VaR using the EWMA-model. However, since the EWMA-model face the least number of exceptions, a bank would still face the least loss when using the EWMA-model.

Figure 5.1: Comparison of excess VaRs using 1250 days of historical data

Note: Exceptions for each model of each index are presented. The blue line presents the maximum allowed number of

exceptions on a 99% confidence level. As can be seen, for all four indices, EWMA is the only model which proves to be a good model.

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performs better than the ARCH type models. Besides, although the TGARCH-model has the advantage to take into account possible asymmetry, the GARCH-model is more appropriate for VaR purposes.

Table 5.3: Ranking performance of the different models using 1250 days of historical data

Exception rate LR unconditional LR independent LR conditional Lopez value

AEX MA 4 4 4 4 3 EWMA 1 1 2 1 4 GARCH 2 2 1 2 2 TGARCH 3 3 3 3 1 CAC40 MA 4 4 4 4 4 EWMA 1 1 1 1 3 GARCH 2 2 2 2 2 TGARCH 3 3 3 3 1 DAX MA 4 4 4 4 4 EWMA 1 1 3 1 3 GARCH 2 2 2 2 1 TGARCH 3 3 1 3 2 FTSE100 MA 4 4 4 4 2 EWMA 1 1 1 1 4 GARCH 2 2 2 2 3 TGARCH 3 3 3 3 1

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5.2 Robustness check

This section presents whether the results outlined in section 5.1 are the same using historical samples of 500, 250 and 125 days. Tables with outcomes of the back-testing models can be found in appendix B-D.

Appendix B shows results that are almost equal for the models based on a 500 day historical sample compared to the models based on a 1250 day historical sample. The EWMA-model is the only model with exception rates lower than the significance level. Except for the GARCH-model which produces a statistical significant unconditional coverage likelihood ratio for the FTSE100 index, EWMA is the only model of which the unconditional coverage values are significant. When taking into account the test for serial independence, approximately the same models show significance compared to the models based on a 1250 day historical sample. Also with a 500 day historical sample the EWMA-model shows significance for all indices. Besides, also the GARCH- and TGARCH-EWMA-model for all indices score significant serial independence likelihood ratio`s. Christoffersen`s test of conditional coverage shows the same outcomes as the models with a 1250 days historical sample (except for an insignificant test value of the EWMA-model for the CAC40 index). Also test values of the Lopez test are comparable. The only difference is the relative large average loss for each exception for the FTSE100 index. This is due to the 1987 market crash which is only included in the FTSE100 index sample.

The results for the models based on a historical sample of 250 days can be found in appendix C. Main difference with regard to the exception rates is that the EWMA-model for the AEX index is above the significance level. The EWMA-model still produces exception rates which are acceptable for the CAC40, DAX and FTSE100 indices. Results of for the EMWA-model with regard to each index

are significant. Besides the EWMA-model, also the GARCH-model for the FTSE100 index shows statistical significant values of . Test results of the serial independence test of the models based

on a 250 day historical sample equal those of the models based on a 500 day historical sample. Besides the EWMA-model (except for the CAC40 index), also the GARCH- and TGARCH-model show significance. When analyzing values of , only the EWMA-model shows to be a good model

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although both the GARCH- and TGARCH-model show, in general, to have the highest average loss at each exception.

Appendix D shows results for the models based on a historical sample of 125 days. Also these models show the same degree of exceptions as compared to the models based on a historical sample of 250 and 500 days. Again, the EWMA-model is the only model able to produce exception rates no higher than 1% (except a 1.03% exception rate for the AEX index). All values of for the EWMA-model

are significant. Based on the test of serial independence, the TGARCH-model outperforms the EWMA-model. The TGARCH-model shows significance for all indices although the EWMA-model shows significance for three out of the four indices. However, when looking at the conditional coverage likelihood ratio`s, the EWMA-model is the only model which shows significance (only not the case for the CAC40 index). Again the Lopez test values do not show uniformity. For each index another model is producing the lowest and highest average loss for each exception. As with the historical sample of 250 days, Lopez values of the DAX, CAC40 and especially the FTSE100 index are high for the historical sample of 125 days.

When taking into account the models with different historical samples, still the EWMA-model is the best performing model for each historical sample. The EWMA-model, for almost all indices, has exception rates not exceeding the 1% significance level and which show significant values of

and . This means that also based on historical samples of 500, 250 and 125 days, of the

research question cannot be rejected implying the variance-covariance VaR approach provides accurate measures of market risk. Again, also using historical samples of different size, of the first sub question is rejected since only the EWMA-model is able to provide accurate measures of VaR. The MA-, GARCH- and TGARCH-model are not able to do so which means that the different methods to measure VaR using the variance-covariance approach do differ significantly with regard to accuracy. As with the models based on a 1250 day historical sample, also for models based on a historical sample of 500, 250 or 125 days, the MA-model is the worst performing model. Furthermore, the GARCH- and TGARCH-model, also with other historical samples, prove not to be appropriate for variance-covariance VaR approach purposes.

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Figure 5.2: Comparison of different days of historical data using the relating exception rates

Note: The darkest blue line indicates the significance level relating to the number of exceptions.

Furthermore one can observe that the MA-model results in exception rates which are high but which are most optimal using a shorter historical sample. In this paper the MA-model performs most optimal using a historical sample of 125 days. This paper therefore recommends using a short historical sample in case the MA-model is used for variance-covariance VaR purposes. For all indices the GARCH- and TGARCH-model produce more accurate estimates for VaR when including more information in the historical sample. Therefore this paper concludes that a 1250 day historical sample can be best used in case of indicating market risk with the variance-covariance approach using the GARCH- or TGARCH-model.

5.3 Low and high volatile sample period

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base VaR on as is recommended by the Basel Committee (Basel Committee on Banking Supervision, 2009A). Appendix C shows results for the total period sample based on a historical sample of 250 days.

Table 5.4 shows the number and rate of exceptions and the outcomes of the three other back-testing models for the period with low volatility. As can be derived from the exception rates, no model is able to produce values of VaR for the AEX index which do not exceed the significance level. However, for the other indices, the EWMA-model is able to produce exception rates which are 1% or less. An unexpected significant exception rate of the MA-model for the CAC40 index is also noted. As could be expected, for most models lower exception rates are calculated for the low volatile period

Table 5.4: Test results of forecasted VaRs during the low volatile period: January 1993 – December

1996 using 250 days of historical data

Obser-vations Excep-tions Exception rate P value Lopez value AEX MA 1008 17 1.69% 3.98* 4.61%* 0.58* 4.56* 1.001 EWMA 1008 12 1.19% 0.35* 55.51%* 1.79* 3.03* 1.002 GARCH 1008 12 1.19% 0.35* 55.51%* 0.29* 0.64* 1.002 TGARCH 1008 25 2.48% 15.80 0.01% 1.27* 17.07 1.001 CAC40 MA 999 9 0.90%* 0.10* 74.88%* 0.16* 0.27* 1.003 EWMA 999 7 0.70%* 1.01* 31.50%* 0.10* 0.20* 1.003 GARCH 999 14 1.40% 1.45* 22.92%* 0.40* 1.84* 1.002 TGARCH 999 14 1.40% 1.45* 22.92%* 0.40* 1.84* 1.002 DAX MA 1004 13 1.29% 0.81* 36.92%* 0.34* 1.15* 1.002 EWMA 1004 10 1.00%* 0.00* 98.99%* 0.20* 0.20* 1.003 GARCH 1004 19 1.89% 6.40* 1.14%* 0.73* 7.13* 1.001 TGARCH 1004 21 2.09% 9.20 0.24% 0.90* 10.09 1.002 FTSE100 MA 1011 12 1.19% 0.34* 56.17%* 0.29* 0.63* 1.001 EWMA 1011 10 0.99%* 0.00* 97.22%* 0.20* 0.20* 1.001 GARCH 1011 12 1.19% 0.34* 56.17%* 0.29* 0.63* 1.001 TGARCH 1011 11 1.09% 0.08* 78.15%* 0.24* 0.32* 1.001

Note: Values of VaR are calculated using an observation period of 250 days, a portfolio value of €1000 and a confidence level

of 99%. Values with * are statistically significant at a 99% confidence level.

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indices. However, this applies to almost every model in each index (except for the AEX and DAX indices with the TGARCH-model). This indicates that, except for the TGARCH-model, the MA-, EWMA- and GARCH-model prove to be able to calculate values of VaR in a relative low volatile period which are not exceeded by market movements at a 99% confidence level according to the test of unconditional coverage. When including the test of serial independence in this analysis, all methods produce exceptions which are not clustered. Thus, all methods in each index prove to be able to deal with this problem. When combining the test of unconditional coverage and serial independence, the likelihood ratios of conditional coverage are derived. For , results comparable with results of the

test of unconditional coverage are presented. The MA-, EWMA- and GARCH-model show significant values of . This implies that these models, for all indices, are able to account for acceptable

coverage and serial independence of exceptions. Furthermore, for three out of the four indices, EWMA shows to have the lowest values for the tests of Kupiec and Christoffersen. This indicates that also in the low volatile period, the EWMA-model proves to be the model which is most accurate in estimating VaR at a 99% confidence level. The TGARCH-model is the only model which fails in two cases (the AEX and DAX indices). When taking into account the values of the Lopez test, a comparable analysis as with the previous section can be made. No clear model outperforms. However, again, the EWMA-model shows to have the highest loss on average in case of an exception. Especially the GARCH- and TGARCH-model show low average losses in case of exceptions. Overall, losses for all models (according to the Lopez values) are low which means that in case losses of a portfolio exceed VaR, this is on a relative small level. This is in line with expectations when using data with a relative low level of volatility.

In table 5.5 the results for the high volatile period are given. Again, as could be expected, almost all exception rates are higher compared to the low volatile period sample or the total period sample. Interesting to see is that, using this period, almost none of the methods for any index is able to calculate values of VaR which do not have more than 1% exceptions. The EWMA-model is the only model which computes less than 1% exceptions for the CAC40 and DAX indices. All other models fail and prove to have difficulty with time series consisting of high volatile trading days. When including the number of exceptions in Kupiec`s test of unconditional coverage, again the EWMA-model proves to be the most accurate EWMA-model. Three out of the four indices show significant likelihood ratio`s for the test of unconditional coverage. All other models seem to have difficulty with estimating VaR for the FTSE100 index when focusing on the high unconditional coverage test values. Values for the test of independence for all models show significance indicating that, also using this sample period, none of the models show difficulty with possible clustering of exceptions. Only the value of for the MA-model in case of the AEX index exceeds the critical value. Christoffersen`s test of

Value-at-Risk using the variance-covariance approach