The effectiveness of different GARCH models estimating the optimal hedging ratio : a static,dynamic and copula-based approach

UNIVERSITY OFAMSTERDAM

The effectiveness of different GARCH models

estimating the optimal hedging ratio: a static,

dynamic and copula-based approach

Job Stekelenburg 11043059

BSc Econometrics & Operations Research Bachelor’s Thesis

Supervisor: Ms D. Güler June 26, 2018

This document is written by Student Job Stekelenburg who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

A

An important task for risk management is hedging against fluctuations in foreign exchange rates. The conventional model, CCC-GARCH, DCC-GARCH and a student’s t-copula GARCH model are used in order to estimate the optimal hedging ratio for the exchange rates EUR/GBP, EUR/USD and EUR/CAD. This study examines which model provides the best variance reduc-tion when the optimal amount of opposite futures posireduc-tions against the spot posireduc-tions are hold. The empirical results support the conclusion that the conventional model and student’s t-copula GARCH model are superior to the CCC-GARCH and DCC-GARCH model. Thus, according to this study the field of risk management should apply either the conventional model or the copula-based model in comparison with CCC-GARCH and DCC-GARCH for minimizing the risk of fluctuations in the foreign currency.

C

2.4 Static and dynamic models . . . 10

3 Data description 14 4 Econometric models and methodology 16 4.1 Estimation of optimal hedging ratio and hedging effectiveness . . . 17

4.2 Conventional method . . . 19

4.3 Constant conditional correlation (CCC) GARCH . . . 19

4.4 Dynamic conditional correlation (DCC) GARCH . . . 20

4.5 Student’s-t copula GARCH . . . 21

5 Empirical Results 23 5.1 Parameter estimation . . . 24 5.2 Hedging performance . . . 27 6 Conclusion 29 References 31 7 Appendix 33

1 I

Due to the increasing financial market liberalization and globalization, cross-border trades and investments are more usual than ever. As a consequence there lies an important task for institutions to deal with currency risks. When an institution possesses foreign assets in its portfolio, it is exposed to foreign exchange rate risk. For example, if a British company possesses S&P500 assets, the returns are remunerated in US Dollars. The British company is therefore exposed not only to volatility of the assets return, but also to the fluctuations in the exchange rate of the GBP/USD. An important task for risk management is to reduce this risk in order to prevent significant losses. Notice that depending on the fluctuations in the exchange rate, it can benefit the net returns as well. However, a portfolio with a substantial amount of foreign assets has a higher volatility due to currency risk. Moreover, an institution wants to minimize the volatility of its portfolio returns, therefore hedging this risk is a significant task for risk management. For example, Eun and Resnick (1988) find empirical evidence that a hedged portfolio provided better gains than an unhedged portfolio.

One way of controlling the exposure to currency risk is currency hedging. In order to hedge the value of your assets against potential losses due to currency risk, hedgers short an amount of futures contracts of the foreign currency. There have been several studies according to whether it is beneficial to hedge at all. Froot (1993) for example concludes, that it is not always rewarding to hedge these risks over the long horizon. Since hedging and thus buying future contracts brings transaction costs, it is a deliberation to hedge at all. However, if it is decided to hedge against currency risk, it is important to hedge in such way the volatility of the returns minimizes. A significant question for risk managers is how many future contracts should be held in order to minimize the volatility of the returns. The hedging ratio is the ratio between the amount of future contracts held for each unit of the underlying asset.

In a number of earlier studies the hedging performance of different models has been demon-strated. The main discussion between the different econometric models is in what way the time dependency of the correlation, tails of the distribution and other factors are present in the data and therefore have to be modelled. Furthermore, the debate regarding this topic

is whether a more complex model is significantly better than a basic model. To summarize, there have been numerous studies comparing model performance for estimating optimal hedging ratios (OHR). However, there has not been found a model yet which outperforms the remaining models in most empirical research. The contribution by this paper to the existing literature is to find empirical evidence of the effectiveness of different GARCH-models regard-ing currency hedgregard-ing. By answerregard-ing the question: which model performs better estimatregard-ing the optimal hedging ratio comparing the effectiveness of different static, dynamic and copula based GARCH models for the exchange rates EUR/USD, EUR/GBP and EUR/CAD. Moreover, the results can be used to carry out further research, especially for using the most effective model for forecasting optimal hedging ratios. Forecasting optimal hedging ratios is useful for practical purposes in the field of risk management and investors. Risk managers and investors can apply the best performing model in order to determine a hedging strategy against currency risk. Besides that, the results and conclusion can be used in other studies in order to invigorate the selection of one or more specific models.

In order to estimate the OHR, early research implied using the slope of the classical linear regression model of the spot prices on future prices, estimated by Ordinary Least Squares (OLS) by Myers and Thompson (1989). Nevertheless, this approach assumes a time-invariant hedging ratio. This is a strong assumption regarding financial returns. Bollerslev (1986) and Engle (1982) stated that conditional variances of most asset returns change overtime. Accordingly a generalized autoregressive conditional heteroskedasticity (GARCH) model allows time-dependent covariances. The first GARCH model that is applied is the Constant Conditional Correlation (CCC) GARCH model introduced by Bollerslev (1990). In this model the conditional variance is time-varying, however the conditional correlation is not. Studies show that the assumption of a constant correlation of the returns on spot and future contracts can be a strong assumption. In order to model this conditional correlation to vary over time, Engle (2002) introduced the Dynamic Conditional Correlation (DCC) GARCH model. The DCC model is in this way an extension of the CCC model. Moreover, the earlier stated models do not account for higher moments of the distributions such as skewness and the kurtosis. However, it is commonly known in finance that returns often follow a skewed and/or fat tailed distribution.

According to Harris and Shen (2003), the CCC and DCC model can therefore attach to much weight to extreme observations, leading to the estimation of the optimal hedging ratio being inefficient. To model these different higher moments, the copula-based GARCH models of Lai (2018) and Hsu et al. (2008) are as well used in this paper.

In order to empirically distinguish the effectiveness of the above four models (OLS, CCC, DCC and copula-based GARCH) data from three different exchange rates is used. As mentioned before, these are the following exchange rates: EUR/USD, EUR/GBP and EUR/CAD. In order to model the optimal hedging ratios, the spot and future prices of the exchange rates on a daily basis are obtained.

The remainder of this paper is organized as follows. Section 2 gives an overview of the existing literature regarding futures contracts, currency hedging and different models for estimating hedging ratios. The description of the data is reported in section 3. In section 4 the estimation of the hedging ratio is described as well as the different econometric models are presented. In section 5 the empirical results of the different econometric models are presented and discussed. Finally, section 6 provides conclusions and limitations of this research as well as suggestions for further research.

2 L

Futures contracts play an important role in financial markets. One of the main reasons the futures market to exist, is the reason that futures contracts are known for their ability to hedge against different sorts of price risk. First of all, the definition of futures contracts is discussed in this chapter. Secondly, the futures market and properties of futures contracts is discussed. The role futures contracts fulfill specifically in hedging against currency risk is extensively explained in chapter 2.2. Econometric models and existing literature based on these models are explained in chapter 2.4.

2.1 FUTURES CONTRACTS

Ar first, the theoretical definition of a futures contract. According to NASDAQ (1999) the definition of a futures contract is as follows:

A legally binding agreement to buy or sell a commodity or financial instrument in a designated future month at a price agreed upon at the initiation of the contract by the buyer and seller. Futures contracts are standardized according to the quality, quantity, and delivery time and location for each commodity. A futures contract differs from an option in that an option gives one of the counterparties a right and the other an obligation to buy or sell, while a futures contract represents an obligation to both counterparties, one to deliver and the other to accept delivery. A future is part of a class of securities called derivatives, so named because such securities derive their value from the worth of an underlying investment.

A lot of different commodities and financial instruments can form a futures contract of the underlying asset. The commodities include for example gold, sugar and cattle, but there are more commodities that can be named. The futures contracts regarding financial instruments include for example currencies, stocks and treasury bonds as underlying asset. In this study the interest lies on futures contracts of foreign currencies. The different futures contracts are traded on the futures market. Two well-known futures markets are for example the New York Mercantile Exchange and the Chicago Mercantile Exchange.

According to Duffie (1989) the futures market originally suits the needs of three different economic agents. Those who want information about future prices, those who want to speculate and those who want to hedge. This paper focusses on the economic agents who wants to hedge against risk, more specifically currency risk.

In order to fully explain the significance of a future contract, an example regarding a future contract from the underlying exchange rate EUR/USD follows.1If a long position is taken into a futures contract of the above mentioned underlying index. This allows the buyer to buy the index EUR/USD on the expiry date of the futures contract at a fixed price now. By taking a

long position in a futures contract a rise of the price of the index is expected between now and the expiry date. As with stocks it is possible to take short position into a futures contract, expecting the price to fall between now and the expiry date. Getting a position in futures contracts opposite to the position taken in the underlying asset is a way to hedge against price risk. The details of this method regarding currencies is discussed in chapter 2.2.

2.2 CURRENCY HEDGING

Before hedging currency risk with futures contracts is explained in detail. The definition of currency risk and the importance of currency hedging is first of all discussed. Exchange rate risk, commonly referred to as currency risk, is the risk that a depreciation of the underlying currency affects the value of the assets in a negative way. Big firms, institutions and individual investors often hold large amounts of foreign assets. Especially, if the returns of those assets are denominated in foreign currency, these investors increase there risk to gain significant losses due to the fluctuations in the exchange rate. By this the volatility of their portfolio is directly affected by the volatility of the exchange rate market. Most investors wish to hold a portfolio which has a minimized volatility (for the same level of return). Therefore, hedging currency risk is often beneficial for investors.

Futhermore, Eun and Resnick (1988) empirically studied the performance of a hedged portfolio against an unhedged portfolio, they did this according to three different hedging methods (EQW, MVP and BST).2Their findings were that the gains of the hedged portfolio substantially increases in comparison to the unhedged portfolio with respect to all three methods. The MPV hedging method provides the best hedging results in this study.

Nevertheless, there is an ongoing discussion about whether to hedge against currency risk at all. For example, Froot (1993) found that the volatility of foreign stocks displayed greater volatility when hedged than unhedged. His conclusion is that if the investment horizon stretches, the optimal hedging ratio approaches zero. Besides, buying futures contracts brings transaction costs, so hedging exchange rate risk is not just a costless procedure and therefore

2_{EQW is the equally weighted portfolio, MPV is the minimum portfolio variance and BST is the Bayes-Stein}

hedging with futures contracts is an important consideration for investors. The contribution by this paper to the discussion whether to hedge at all is marginal, since this paper only examines the hedging effectiveness of different models, conditionally on the fact the hedge is made. However, empirical results can contribute to arguments that risk reduction is significant by using one or more specific models.

To summarize, empirical studies show that hedging exchange rate risk can be valuable in order to increase gains of the investments or reduce the volatility of the portfolio. But a critical view is needed whether hedging is profitable, especially when investing over the long-horizon. Nonetheless, this paper examines the effectiveness of different GARCH-models regarding currency hedging conditionally the hedge is made.

2.3 HEDGING RATIO

As mentioned previously, hedging with futures contracts enables hedgers to protect the value of their assets against the risk of price fluctuations in the foreign currency. Hull (2012, pp. 48-49) stated that hedgers can execute this hedge by taking a position in the futures market opposite to the position of the physical position they hold to benefit. Hence, hedgers are mainly interested in the optimal amount of futures contracts that should be held for each unit of the underlying asset in order to reduce the exposure to currency risk at its best. According to Hull (2012, p. 56), the ratio between the amount of futures contracts against the amount of the underlying asset is the so-called hedging ratio. Generally, the hedge ratio is greater when the market trend is not stable.

Chen et al. (2013) offers several theoretical approaches in order to find the optimal hedging ratio (OHR). For example, the minimum variance hedge ratio, sharpe hedge ratio, maximum expected utility hedge ratio and five more. A widely-used method in order to find the optimal hedging ratio is the minimum variance hedge ratio, based on minimizing the variance of the portfolio. In this specific study the minimum variance hedge ratio as discussed in Chang et al. (2011), is the method used for determining the optimal hedging ratio. As mentioned previously, Eun and Resnick (1988) use three different methods to determine the optimal hedging ratio and find that the minimum variance hedge ratio provide the best results.

In order to quantify the hedging performance of the different models, the hedging effective-ness (HE) introduced by Ederington (1979) is applied. The hedging effectiveeffective-ness is nothing more than the variance reduction of the hedged portfolio relative to the variance of the un-hedged portfolio. The hedging effectiveness of the different models is therefore later on compared, consequently a conclusion can be formed regarding the performance of the mod-els. The method of constructing a minimum variance portfolio and quantity for the hedging effectiveness is explained in chapter 4.1.

2.4 STATIC AND DYNAMIC MODELS

In the field of hedging with futures contracts, earlier research implied estimating static hedging ratios in order to achieve the highest risk reduction. More recent studies introduced multi-variate conditional volatility models. These models allow the hedging ratio to be time-variant. Most recently, a new class of copula-based GARCH models are introduced. These models allows the joint distribution of the returns to be specified with full flexibility. In this section the four different static, dynamic and copula-based models are discussed. Together with the hedging performance of these models concerning earlier studies.

The first model used to estimate the optimal hedging ratios is ordinary least squares (OLS) regression of the spot returns on the futures returns, discussed in Lien et al. (2002). As a result, the slope of the regression line is the estimated optimal hedging ratio. However, OLS does not account for time-varying joint distributions of the spot and future prices. The OHR is therefore assumed to be static and constant over time. This assumption, however, is not supported by most empirical research. Since most financial returns suffer from heteroscedasticity due to periods with high and low volatility, also known as volatility clustering. Furthermore, Herbst et al. (1989) find evidence that the conventional OLS method is inappropriate due to the suffering of serial correlation and heteroscedasticity. This result implies that the assumption of time invariant variances and covariances is violated and the static hedging strategy is in this case not suitable for futures hedging. Bera et al. (1993) also find heterskedasticity as well as non-normality in the disturbance term of the OLS regression. They find that a model which

takes the ARCH-effects into account outperforms the conventional OLS model. However, note that running the ARCH-model is computationally more demanding than running an OLS regression. Moreover, Baillie and Myers (1991) find strong support that the estimated optimal hedging ratios are time-varying and thus non-stationary. Their findings are find by adopting a bivariate GARCH(1, 1) model with vech parameterization to estimate the optimal hedging ratios for six different commodities. They also compared the conventional model with the bivariate GARCH model with respect to hedging performance and find the GARCH model superior to the conventional model. On the other hand, Tong (1996) compares the conventional OLS model for the spot and futures prices of the Tokyo Stock Index. He finds support that the dynamic hedging reduces risk just slightly more than static hedging.

If the assumption of a static hedging ratio is inappropriate, the hedging ratio can be modeled as time dependent. This indicates that a risk manager must adjust his hedging ratio over time in order to reduce risk optimally. According to Hull (2012), buying and selling futures contracts bring transactions costs, due to different kind of fees, for example brokerage fees. Thus, a hedger needs to be aware that a dynamic hedging ratio lead to higher costs.

To take account for volatility clustering and serial correlation in time series, generalized autoregressive conditional heterscedasticity model (GARCH) is proposed by Engle (1982) and Bollerslev (1986). An extension of this model introduced by Bollerslev (1990) is the multivariate model with a constant correlation GARCH(1, 1) structure (CCC-GARCH). The GARCH term allows the second moments to be time-varying, and as a result of this modeling the OHR to be time-varying. Lien et al. (2002) studied the performance of the above CCC-GARCH model in comparison with the conventional OLS model regarding exchange rates and commodities. In both the exchange rate market as the commodity market, the CCC-GARCH model did not outperform the conventional OLS method.

Nevertheless, the assumption of a constant correlation in the CCC-GARCH model can be a strong assumption as well. Engle (2002) suggest the dynamic conditional correlation GARCH model (DCC-GARCH). The DCC-GARCH models allows the correlation between the spot and future prices to be time-varying and is in that way a more flexible model than the CCC-GARCH model. In this study the performance of DCC-GARCH is studied.

This paper closely follows the framework of Chang et al. (2013) with an addition of the study of Hsu et al. (2008), where in the latter they introduced a copula-based GARCH model in order to estimate the optimal hedging ratio. In Chang et al. (2013) the importance of modeling conditional variances and covariances is examined due to model CCC-GARCH, DCC-GARCH, BEKK and VARMA-GARCH. His empirical findings support that modeling the conditional correlation is not crucial for better risk reduction. Hsu et al. (2008) compares the performance of the conventional, CCC-GARCH, DCC-GARCH and copula-based GARCH models. They do this by estimating the optimal hedging ratio for S&P500, FTSE100 and two exchange rates, namely MSCI-SWI and USD/SWF. The empirical results indicate that the proposed copula-based provide substantially better hedging performance.

A significant evolution in modeling dependence structures between two or more random variables, is now known as copulas. As mentioned earlier a copula-based GARCH model permits full flexibility of the joint distribution of the two assets. Among which nonlinearity and asymmetric dependence between the two assets in a cross-hedge portfolio. In most studies the copula-based GARCH models outperformed both static and dynamic GARCH models. In the study of Hsu et al. (2008) the effectiveness of three different copulas is examined, namely the Gaussian, Gumbel and Clayton copula. However, in this study the student’s t-copula defined in Liu et al. (2010) is used in order to capture the dependence structure between spot and futures returns.

The paper of Liu et al. (2010) employs a study on finding the optimal hedging ratio for electricity returns on spot and futures for Nordic power market using a copula-based GARCH model. He compares the copula-based GARCH model with the one-to-one hedge ratio, ECM and DCC model.3The results indicate that the student’s t-copula GARCH model yields greater risk reduction compared with the other models. The DCC-GARCH model provides slightly worse risk reduction than the student’s t-copula GARCH model.

The findings of Lien et al. (2002) support that the OLS model outperforms the CCC-GARCH model at risk reduction considering currency risk. However, Hsu et al. (2008) conclude that

3_{The one-to-one hedging ratio is defined by taking the equal amount of futures positions as spot positions in}

CCC-GARCH outperforms the conventional OLS model. Furthermore, the copula-based model provided the best results in this study and in second the DCC-GARCH model. Also Yang and Allen (2005) find evidence in the Australian futures market that in post-sample analysis the multivariate GARCH models provide superior hedge ratios. On the other hand, Chen et al. (2013) do not find significant evidence that modeling a dynamic correlation (DCC) versus constant correlation (CCC) yields better hedging results. Thus, interesting is whether this study is able to find a model which outperforms other models and consequently is a valuable addition to the field of currency risk management.

To summarize, the main goal of the futures market is hedging. In this paper the risk of fluctuations in foreign exchange rates is hedged, by finding the optimal hedging ratio esti-mated by different models. There are several studies finding optimal hedging ratios in both commodity, stocks and currency markets using different kind of models, which again differ in complexity and time dependency. The conventional OLS model is in most studies less effective than the GARCH models, however there are some exceptions. In order to model the time-varying second moments a CCC-GARCH and DCC-GARCH model is introduced, where the DCC-GARCH model differs in modeling time-varying conditional correlation between the spot and future prices. The most recent studies introduce a more flexible class of models, namely copulas. These copula-based GARCH models are able to model the joint distribution with full flexibility. Most empirical results indicate that copula-based GARCH models out-perform relatively less complex models, such as OLS, CCC-GARCH and DCC-GARCH. In this study the hedging effectiveness based on the conventional model, CCC-GARCH, DCC-GARCH and student’s t-copula GARCH are compared in order to provide significant results for the field of risk management. Especially whether a more complex and therefore computationally more demanding models, outperform relative elementary models.

3 D

In this section, the data used for this study is discussed and analyzed. First of all, the trans-formation of the data and the time series of the returns are discussed. Secondly, summary statistics of the returns are given. At last, it is required to apply different tests on the data in order to select and apply the appropriate model for the data.

In this study daily closing spot and futures prices of the following exchange rates, GBP/EUR, USD/EUR and EUR/CAD are obtained from Datastream. The spot and futures prices are acquired over the period from 1 January 2004 to 29 December 2017, for the exchange rates EU-R/GBP and EUR/USD, which yields a total of 3652 observations. The total of 3699 observations of the exchange rate EUR/CAD are obtained over the period 13 March 2004 to 15 June 2018.4 The original source of the spot prices of the Euro to one Great British Pound (GBP/EUR), the Euro to one United States Dollar (USD/EUR) and the Canadian Dollar to one Euro (EUR/CAD) is WM/Reuters. In order to retrieve the correct spot prices in terms of the exchange rates EUR/GBP as the EUR/USD, the inverse of the original spot prices is taken. For the futures prices, the original source of all three exchange rates is the Chicago Mercantile Exchange (CME). For the closing futures prices of the exchange rate EUR/USD, the price of an E-mini contract is used. However, the closing price of this contract does not differ from the price of a regular contract, the contract size is just $62500 instead of $125000.

In order to retrieve the returns of the spot (RS,t) and futures (RF,t) from the data, the following

transformation on the spot and futures prices is executed

RS,t= log µ _St St −1 ¶ · 100% (3.1) RF,t= log µ _Ft Ft −1 ¶ · 100% (3.2)

where Stand Ftare the ’closing’ prices at time t for the different exchange rates.5In 3.1 and 3.2

log is defined as the natural logarithm. The names and description of the obtained variables

4_{Note that the sample period excludes Saturday and Sunday (weekends) and holidays for EUR/GBP, EUR/USD}

and EUR/CAD

5_{Note that the total observations of the returns reduce by one compared with the closing prices, resulting in a}

are reported in table 7.1. In figure 7.1 the daily returns are plotted against the time. Note that all the returns exhibit larger volatilities for the period around the year 2008. In this period the fall of Lehman Brothers took place and triggered the global economic crisis. As a result of this crisis the volatilities of the different exchange rates increased. In the period after 2016 there is an increase in volatility due to the withdrawal of the United Kingdom from the European Union (Brexit) and the election of the President of the United States D. J. Trump.6The Brexit had probably more influence on the volatility of the EUR/GBP, whereas the election of D.J. Trump had more effect on EUR/USD and EUR/CAD. These periods of high and low volatility present in the series are known as volatility clustering. Moreover, the returns of spot and futures move in the same pattern, therefore the correlation between the returns of spot and futures are relatively high. The correlation are presented in 3.1 and are 0.8642, 0.8126 and 0.8064 for respectively EUR/GBP, EUR/USD and EUR/CAD.

Table 3.1: Descriptive Statistics of the spot and futures returns

EUR/GBP EUR/USD EUR/CAD

Statistic Spot F ut ur es Spot F ut ur es Spot F ut ur es

Mean 0.0063 0.0063 -0.0013 -0.0010 -0.0010 -0.0010 Min -2.7009 -3.5760 -3.8445 -3.0568 -4.4683 -3.8697 Max 6.2224 6.0185 4.6174 3.2520 3.2982 3.1652 Std. dev. 0.5247 0.5209 0.6121 0.6052 0.6083 0.5981 Skewness 0.6215 0.5354 0.1133 -0.0446 0.0010 0.0431 Kurtosis 10.560 10.513 6.0316 4.9591 5.2006 5.0798 Jarque-Bera 8929.2∗ 8760.6∗ 1405.9∗ 585.05∗ 746.2∗ 667.61∗ Q(24) 33.67 55.03∗ _{22.46 26.49 50.47}∗ _28.62 Q2(24) 570.00∗ 762.59∗ 874.68∗ 1164.8∗ 917.51∗ 855.34∗ Corr. 0.8642 - 0.8126 - 0.8064 -ADF −15.779∗ −15.818∗ −14.761∗ −15.028∗ −16.558∗ −16.874∗ Trace 583.31∗ - 385.68∗ - 657.82∗

-***, ** and * denote the significance at a 5%, 2.5% and 1% level, respectively.

Table 3.1 provides descriptive statistics, tests for stationarity, serial correlation and

cointegra-6_{The announcement of the Brexit took place at June 23 2016. The election of D. J. Trump was announced at}

tion. First of all, observe that the means of the returns for all three exchange rates for both spot and futures prices are close to zero. Also note that all returns for all positions are not normally distributed. The null hypothesis of the Jarque-Bera test for normality is rejected in all cases. Moreover, the distribution of the returns show a significant positive kurtosis, which indicates that the distributions are leptokurtic. This is the result of outliers in the returns and a fat-tailed distribution is likely appropriate to model the distribution of the returns. Subsequently, the Ljung-Box test is executed, the null hypotheses indicates that there is no serial correlation in the returns. The Ljung-Box test for serial correlation only finds statistical evidence for serial correlation in the returns of futures for EUR/GBP and the returns of the spot positions of EUR/CAD. In addition, the presence of serial correlation in the squared returns is tested. The Ljung-Box test provides statistical evidence of serial in correlation in all the squared returns. As a result, GARCH-models are an applicable way of modeling the second moments of the returns. In the current financial literature a GARCH(1,1) is mostly used in order to model these second moments. For example, Namugaya et al. (2014) finds that GARCH(1,1) provides the best results in comparison with other GARCH(p, q) models. Moreover, the advantage of an GARCH(1,1) is the simplicity of the model and interpretation of the estimated parameters

In order to test whether the series of the returns are stationary, an Augmented Dickey-Fuller test for stationarity is applied on the data. The results are presented in table 3.1. The null hypotheses of the Augmented Dickey Fuller is that the series has a unit root (non-stationary). Hence, the conclusion based on the ADF is that the series of all returns do not have a unit root and are therefore assumed to be stationary. Finally, the Johansen trace test is executed in order to test for the presence of cointegration. The null hypotheses that there is no cointegration present in the data against the alternative of cointegration is rejected for all returns at a significance level of 1%.

4 E

First of all, the return of the hedged portfolio is defined in this section. Secondly the technical framework of the minimum variance portfolio method to determine the optimal hedging rate

is reported. Followed by the definition of the hedging effectiveness and its importance in order to quantify the hedging performance. Finally, the four different models used in this study are represented in the following order: OLS, CCC-GARCH, DCC-GARCH and finally the student’s

t -copula GARCH.

4.1 ESTIMATION OF OPTIMAL HEDGING RATIO AND HEDGING EFFECTIVENESS

As mentioned previously, hedgers construct a hedged portfolio by retaining a specific amount of futures contracts against the opposite spot position. According to Chang et al. (2013), the return of the hedged portfolio can therefore be obtained by the following expression

RH ,t= RS,t− γtRF,t (4.1)

where RH ,t is the return of the hedged portfolio at time t . RS,t and RF,t are the returns of

respectively the spot positions and futures positions at time t andγtis the hedging ratio at

time t . Note that the portfolio is constructed by selling a specific amount of futures contract opposite to the spot position, in other words the position of the futures contracts in this expression is the short position, this is the reason of the minus sign in (4.1).

In order to determine the optimal hedging ratioγ∗t, the technique of constructing a

mini-mum variance portfolio discussed by Chen et al. (2013) and applied on curreny hedging by Chang et al. (2013) is used in this study. First of all, the conditional variance of the return of the hedged portfolio is given by

V ar¡RH ,t| Ψt −1¢ = V ar ¡RS,t− γtRF,t| Ψt −1 ¢ (4.2) = V ar¡RS,t| Ψt −1¢ + γ2tV ar¡RF,t| Ψt −1¢ − 2γtC ov¡RS,t, RF,t| Ψt −1 ¢ (4.3)

whereΨtis the information set up until time t . The minimum variance portfolio is obtained

by minimizing the variance of the return conditionally on the information set up until time

evaluated inγ∗t and setting it to zero yields ∂ ∂γtV ar¡RH ,t| Ψt −1 ¢¯_¯ ¯ γt =γ∗t= 0 (4.4)

Solving equation (4.4) forγ∗t results in the optimal hedging ratioγ∗t conditionally on the

information set given by

γ∗ t ¯ ¯ ¯ Ψt −1= C ov¡RS,t, RF,t| Ψt −1 ¢ V ar¡RF,t| Ψt −1 ¢ (4.5)

Where C ov¡RS,t, RF,t| Ψt −1¢ is the conditional covariance between the spot and futures returns

and V ar¡RF,t| Ψt −1¢ is the conditional variance of the futures returns. The formula for the OHR,

adapted to the variables used in the multivariate GARCH models, is given by

γ∗ t ¯ ¯ ¯ Ψt −1= ˆ hSF,t ˆ hF,t (4.6)

where ˆhSF,t is the conditional covariance between the spot and futures returns, and ˆhF,tis the conditional variance of futures returns.

To compare whether which model is superior to other models. A quantity that measures the risk reduction between an hedged and hedged portfolio is introduced. In order to distin-guish the hedging effectiveness for the different models. The quantity named the hedging effectiveness index (H E ) used in Chang et al. (2013) is constructed by

H E = ·_{V ar}¡R S,t¢ −V ar ¡RS,t− γ∗tRF,t ¢ V ar¡RS,t ¢ ¸ (4.7)

where the numerator is the variance of the unhedged portfolio minus the hedged portfolio and the denominator is the variance of the unhedged portfolio. Note that a higher H E represents a greater risk reduction as a result of currency hedging. This quantity is therefore in section 5 used to determine which model is more effective in reducing the volatility of the portfolio.

4.2 CONVENTIONAL METHOD

In order to use the conventional method to estimate the optimal hedging ratio the following linear regression model discussed in Lien et al. (2002) model is applied:

RS,t= α + βRF,t+ εt (4.8)

εt∼ N (0, σ2), for all t (4.9)

whereεtis the disturbance error at time t . In the conventional model, the regression coefficient β provides the optimal hedging ratio. Note that β is time-invariant, the optimal hedging is

therefore assumed to be constant over time in the conventional model. Also note that OLS assumes the second moments to be constant over time, which can be a strong assumption regarding time series of financial returns.

4.3 CONSTANT CONDITIONAL CORRELATION(CCC) GARCH

In order to model a dynamic hedging ratio, the conditional variances and covariances need to be specified. In order to model these second moments, consider the CCC multivariate GARCH model introduced by Bollerslev (1990) and applied in Chang et al. (2011) and Chang et al. (2013) yt= E³yt¯¯ ¯ Ψt −1 ´ + εt , εt= Dtηt (4.10) V ar³εt ¯ ¯ ¯ Ψt −1 ´ = DtΓDt (4.11)

where ytis a vector of the two returns for spot and futures given by (RS,t, RF,t)0andηtrepresents a

2x1 vector with independent identically distributed error terms given by (ηS,t,ηF,t),Ψtis the in-formation set available at time t , Dt= diag(h1/2S , h

1/2

F ) and t = 1,...,n. where n is the number of

observations. AsΓ = E¡ηtη0t

±

Ψt −1¢ = E¡ηtη0_{¢, whereΓ = ¡ρi j¢ for i , j = S,F represents the}

con-stant conditional correlation matrix of the conditional shocks.εtε0t= Dtηtη0tDt= (diagQt)1/2,

and E¡εtε0t/Ψt −1¢ = Qt= DtΓDt, where Qtis the conditional covariance matrix. For the matrix

definite.

According to Bollerslev (1990), the CCC-GARCH model states that the conditional variance

hi ,t, i = S,F , follows a univariate GARCH process. In this study the conditional variance will be

modelled by a GARCH(1,1) model, that is defined as

hS,t= ωS+ αSε 2 S,t −1+ βShS,t −1 (4.12) hF,.t= ωF+ αFε 2 F,t −1+ βFhF,t −1 (4.13)

where the long run persistence is represented byαi+ βifor i = S,F and αi+ βi≤ 1 must hold

for all i = S,F . The short run persistence of the shocks to the returns is given by αifor i = S,F ,

also known as the ARCH effects. The parameterβirepresents the GARCH effect. The condition ofωi,αi,βi> 0 for all i = S, F must hold as well.

4.4 DYNAMIC CONDITIONAL CORRELATION(DCC) GARCH

As mentioned previously, the assumption that the conditional correlations are constant over time can be an unrealistic. In order to model the conditional correlationΓ as a function of time, Engle (2002) proposed a dynamic correlation (DCC) GARCH model and defined by Chang et al. (2013), is defined as follows:

yt¯¯

¯ Ψt −1∼ (0,Qt) , t = 1,2,...,n (4.14)

Qt= DtΓtDt, (4.15)

where Dt is the diagonal matrix of conditional variances given by Dt= diag(h1/2S , h

1/2

F ). Ψt

is the information set available at time t . The conditional variances hi t are modelled as a

univariate GARCH(1,1) model as follows:

hS,t= ωS+ αSε 2 S,t −1+ βShS,t −1 (4.16) hF,t= ωF+ αFε 2 F,t −1+ βFhF,t −1 (4.17)

distributed random variables, with zero mean and variance of one, andηi t is standardized

byηi t= yi t±phi t. After the standardizationηi tfor i = S,F is used to estimate the dynamic

conditional correlations, as follows:

Γt=

n

d i ag (Qt)−1/2oQtnd i ag (Qt)−1/2

o

(4.18)

where the symmetric positive definite matrix Qtis given by

Qt= (1 − θ1− θ2)Q + θ1ηt −1η0t −1+ θ2Qt −1 (4.19)

in whichθ1andθ2 are scalar parameters to estimate the effects of previous shocks and

previous dynamic conditional correlations on the current dynamic conditional correlation. The restrictionsθ1,θ2≥ 0 and θ1+ θ2< 1 must hold, these restrictions imply that Qt> 0. When θ1= θ2= 0, Qtis equivalent to the matrix of the conditional variances in CCC GARCH model.

When Qtis conditional on the vector standardized residuals is a conditional covariance matrix,

and Qtis the unconditional variance matrix ofηt.

For the estimation of the DCC-GARCH model a two-step maximum likelihood estimation is suggested. In the first stage the parameters for the univariate GARCH models are estimated again for i = S,F . In the second stage the parameters θ1andθ2are estimated using a

log-likelihood with the given parameters from the estimated univariate GARCH.

4.5 STUDENT’S-T COPULAGARCH

Previously mentioned models are all estimated on the assumption of multivariate normality. A copula-based approach allows the marginal distributions of the returns and the dependence structure between the returns of spot and futures be modeled with more flexibility, which is a more realistic approach. Sklar’s theorem, introduced by Sklar (1959), provides a theoretical foundation that every multivariate cumulative distribution function of random variabels can be expressed of its marginals and a copula. Sklar’s theorem state that there is a copula function

C (·) for every joint distribution F (n dimensional), with marginal distributions Fi, such that:

spot and futures returns of the exchange rates at period t . Let FS,t(RS,t|Ψt −1) and FF,t(RF,t|Ψt −1)

be their CDF, whereΨt is the information set at time t . The conditional copula function

Ct(ut, vt|Ψt −1) is defined by the time-varying CDF of spot and futures returns of the exchange

rates, where ut= FS,t(RS,t|Ψt −1) and vt= FF,t(RF,t|Ψt −1) for (ut, vt) ∈£0,1¤. From Sklar’s theorem,

the bivariate conditional CDF of RS,tand RF,tcan be written as F (RS,t, RF,t|Ψt −1) = Ct(ut, vt|Ψt −1).

The copula-GARCH model in this study closely follows the copula-GARCH model estimated by Liu et al. (2010).

In this study the marginal distributions of the spot returns as wel as the futures returns are captured by a univariate GARCH(1,1) model defined by

RS,t= µS+ εS,t εS,t ¯ ¯ ¯ Ψt −1∼ t (0, hS,t; vS) (4.20) hS,t= ωS+ αSε 2 t −1+ βShS,t −1 and RF,t= µF+ εF,t εF,t¯¯ ¯ Ψt −1∼ t (0, hF,t; vF) (4.21) hF,t= ωF+ αFε 2 t −1+ βFhF,t −1

whereεi ,tfor i = S,F are the student’s t-distributed error terms with corresponding degrees of freedom vSand vF. Moreover, the conditional variances for the spot and futures returns are

given by hi ,tfor i = S,F .

Next, a copula function that captures the dependence structure between the spot and futures returns is defined. In this study a copula from the elliptical copula class, namely the Student’s

t-copula (T ) is used and defined by CtT(zS,t; zF,t;ρ,vc) = (1 − ρ2)(1/2) Γ(vc+ 2)/2 Γ(vc/2) · Γ(v c/2) Γ(vc+ 2)/2 ¸2 · 1 +z 2 S,t+ z 2 F,t− 2ρzS,tzF,t vc(1 − ρ2₎ ¸−(vc+2)/2 (4.22) ·_³ 1 +z 2 S,t vc ´³ 1 +z 2 F,t vc ´¸(vc+1)/2

where zS,t= tvc−1(ut) and zF,t= tvc−1(vt) and tvc(·) is the CDF of the Student’s-t distribution with

degrees of freedom vc. The linear correlation coefficient is defined asρ and Γ(·) is the Gamma

function.

The estimation of the parameters of the copula function and thereafter the conditional correlations and variances is as follows. At first the parameters of the univariate GARCH-t from equation 4.20 and 4.21 are estimated by maximum likelihood.7Thereafter, the shocks from this estimation are transformed by the probability integral transform in order to retrieve (ut, vt) on the interval£0,1¤. Subsequently, (ut, vt) are used to estimate the parametersρ and vcby applying maximum likelihood on equation 4.22. The estimated parameter vcis used to

define the degrees of freedom in transforming the uniformly distributed values (ut, vt) back to

shocks z∗i ,tfor i = S,F as follows: zS,t∗ = tvc−1(ut) andzF,t∗ = tvc−1(vt), where tvc(·) is the CDF of the

Student’s-t distribution with degrees of freedom vc. The conditional correlations and variances

are obtained by applying the DCC-GARCH model on z∗S,tand zF,t∗ discussed in 4.4.

5 E

R

In this section the results are reported and discussed. First of all, the estimation of the pa-rameters in the different models is discussed. Secondly, the papa-rameters of OLS, CCC-GARCH, DCC-GARCH and the copula-GARCH model. Finally, the hedging performance of the different models is discussed.

5.1 PARAMETER ESTIMATION

The parameter estimates of the OLS regression are reported in table 7.2. For all three currencies the regression coëfficientsβ are all significant at 1% significance level. As discussed earlier these coëfficients represent the optimal hedging ratio for this model and are therefore further discussed in section 5.2. In figure 7.3 there is again strong evidence for the presence of heteroskedasticity. The residuals show high volatility in the period of the global financial crisis in 2008 as well as in the period after 2015. A Breusch-Pagan test for heteroskedasticity is executed, the alternative hypothesis of this test is that there is statistical evidence for the presence of heteroskedasticity in the residuals. The results of the Breusch-Pagan are reported in 7.3 The Breusch-Pagan test rejects the null hypothesis of homoskedasticity for the exchange rates EUR/GBP and EUR/USD. For the exchange rate EUR/CAD there is no statistical evidence for the presence of heteroskedasticity. Consequently, OLS suffers from heteroskedasticity for EUR/GBP and EUR/USD and is therefore no longer the best linear unbiased estimator (BLUE).

The estimation of the univariate generalized autoregressive conditional heteroskedasticity model (GARCH) are presented in table 7.4. Note that the parameter of interestα and β are all significant at a significance level of 1%. The ARCH effects (α) are between 0.02857 and 0.04378, with the lowest ARCH effects for the returns on the exchange rate EUR/USD and the highest on the returns of the exchange rate of EUR/GBP. The GARCH effects are just above 0.95 for EUR/GBP and EUR/CAD , with the exception for EUR/USD which is just above 0.96. After fitting the univariate GARCH models the estimated conditional volatilities of the returns are presented in figure 7.5. First of all, note that for all returns, the period of the global financial crisis presents a higher conditional volatility, which indicates that the univariate GARCH model is modeling this period well. Regarding EUR/GBP, there is an abrupt increase in conditional volatility in 2016, the cause of this increase is probably the announcement of the withdrawal of the United Kingdom from the European Union (Brexit). In the same period there is a lesser increase in conditional volatility for EUR/USD, the cause is probably likewise the Brexit. However, the cause of the increase in volatility in November of 2016 for EUR/USD and EUR/CAD is probably the election of D.J. Trump. The estimated parameters of the univariate GARCH models are used to estimate the CCC-GARCH(1,1) model. The estimated parameters

of CCC-GARCH are reported in table 7.5. First of all, note that the parametersα and β are again all significant at a 1% level and that the conditionα+β ≤ 1 is satisfied for all returns. The parametersα and β do not differ that much from the univariate GARCH estimation. However note thatα + β are all very close to one, α + β is known as the long run persistence. The long run persistence in the results indicates that the volatility persistence is high for these returns. In other words the rate at which the volatility of previous periods dies over time is quite low. The relatively low values ofα points out that the volatility of the today that feeds through in to the next period is relatively low, also known as the short run persistence of the shocks. The CCC-GARCH lies a restriction on a constant conditional correlation and the constant conditional correlations are 0.863, 0.814 and 0.810 for respectively EUR/GBP, EUR/USD and EUR/CAD. The conditional correlation between the spot and futures returns for EUR/USD and EUR/CAD is significantly lower than the conditional correlation of EUR/GBP.

In order to model a dynamic conditional correlation between the spot and futures positions, a DCC-GARCH(1,1) model is fitted on the returns. Table 7.7 reports the estimations of the parameters of this dynamic model. All the parametersα and β are significant at a significance level of 1% and the condition ofα + β ≤ 1 is satisfied for all returns. Just as with the estimation of the CCC model the short run persistence is the lowest for the return on the exchange EUR/USD, namely around 0.029 for both the spot and futures returns. The highest estimated short run persistence is for the exchange rate EUR/GBP, which is around 0.042. The long run persistence is again for all the returns very close to one for all the returns. The parametersθ1

andθ2estimated in the DCC model are all significant with the exception ofθ2for the exchange

rate EUR/CAD. This suggest that the conditional correlation is not constant over time for all the returns, since DCC is equivalent to CCC whenθ1= θ2= 0 holds. However, θ1andθ2

are relativity low for the returns of EUR/GBP and EUR/USD. The conditional correlations estimated by DCC-GARCH are presented in figure 7.7. These figures do not directly indicate that there is a distinct difference in conditional correlation over time, however the parameters

θ1andθ2represent that there is statistical evidence of a time-variant conditional correlation.

But as mentioned previouslyθ1andθ2are relativity close to zero for EUR/GBP, EUR/USD and

The above models are restrictive in a sense of the joint distributions of the returns. For that reason the copula-GARCH model is fitted to model the joint distribution with more flexibility. Before the student’s t-copula is fitted, the marginal distribution is estimated as a univariate GARCH model with student’s t-distributed shocks. The estimated parameters of this univariate GARCH model are reported in table 7.9. The estimates of the parameterα and β of the univari-ate GARCH model with normally distributed shocks compared with student’s t-distributed shocks for EUR/USD, EUR/GBP and EUR/CAD are, apart from some marginal differences, the same. The estimated degrees of freedom in the GARCH(1,1)-t model are relatively low, this indicates that a student’s t-distribution is more appropriate than a normal distribution.8 The reason for the low degrees of freedom can be found in figure 7.1 were the returns of all three exchange rates exhibit extreme values often, especially in periods of high volatility. As a result of the relatively low estimated degrees of freedom, the distributions of the returns are therefore leptokurtic and better approached and estimated by a student’s t-distribution. After estimating the marginal processes, the shocks of this process are transformed to a uniformly distributed variable on the interval [0, 1] by applying the probability integral transform on the shocks. Subsequently, the transformed shocks are used to estimate the parametersρ and

vcof the copula function given in equation 4.22. The estimated parameters of the student’s t-copula are reported in table 7.9. The parameterρ the linear correlation coefficient which captures the correlation between the spot and futures prices is equal to 0.86, 0.82 and 0.81 for respectively EUR/GBP, EUR/USD and EUR/CAD. The estimated degrees of freedom vc

of the copula function given in equation 4.22 are 5.8, 4.4 and 6.7 for respectively EUR/GBP, EUR/USD and EUR/CAD. Subsequently, the uniformly distributed shocks are again trans-formed into shocks by taking the inverse of the estimated student’s t-distribution. These shock are used to estimate the time-varying conditional correlations and variances, estimated by a DCC-GARCH(1,1) model. The resulting estimated parameters are given in 7.9.

Recapitulatory, both spot and futures returns are suffering from volatility clustering. After estimating the parameters of the different GARCH models in order to model this volatility

clustering, the conclusion is that there is strong evidence for ARCH and GARCH effects in the data. With the estimation of the DCC-GARCH, the assumption of a constant conditional correlation in the CCC-GARCH is violated. To capture the dependence structure between the spot and futures returns a student’s t-copula-GARCH model is fitted on the data. The empirical result regarding the hedging effectiveness for the different models are reported in section 5.2.

5.2 HEDGING PERFORMANCE

The goal of this study is to determine which model provides the best hedging effectiveness. With the estimation of the above parameters, the conditional variances and conditional covariances can be calculated in order to determine the optimal hedging ratio by equation 4.6. With these ratios the variance of the hedged portfolio is calculated and subsequently the hedging effectiveness by equation 4.7. The optimal hedging ratio, variance of the hedged and unhedged portfolio and the hedging effectiveness are reported in table 5.1.

Table 5.1: Overview hedging performance different models

Return OHR Var. UHP. Var. HPF HE

EUR/GBP OLS 0.8706 0.2753 0.0696 74.68% CCC-GARCH 0.8768 0.2753 0.0710 74.21% DCC-GARCH 0.8756 0.2753 0.0710 74.20% Copula-GARCH 0.8628 0.2753 0.0697 74.69% EUR/USD OLS 0.8218 0.3746 0.1273 66.03% CCC-GARCH 0.8230 0.3746 0.1290 65.54% DCC-GARCH 0.8231 0.3746 0.1298 65.34% Copula-GARCH 0.8097 0.3746 0.1289 65.59% EUR/CAD OLS 0.8203 0.3701 0.1294 65.04% CCC-GARCH 0.8272 0.3701 0.1310 64.61% DCC-GARCH 0.8257 0.3701 0.1308 64.66% Copula-GARCH 0.8099 0.3701 0.1296 64.97%

OHR is the optimal hedging ratio. Var. UHP. is the variance of the unhedged portfolio. Var. HPF. is the variance of the hedged portfolio. HE is the hedging effectiveness.

First of all, note that the optimal hedging ratio (OHR) of the copula-GARCH model is the lowest for all exchange rates, followed by OLS. The OHR estimated by DCC and CCC differ marginally from each other. For the exchange rates EUR/GBP and EUR/CAD, DCC results in a lower OHR than the OHR of CCC. The OHR estimated by OLS, in contrast to the OHR of CCC, DCC and Copula-GARCH model, is constant over time. In other words the OHR for EUR/GBP is 0.8706 at all time, regardless there is a period of high or low volatility. The optimal hedging ratios plotted against time for the models CCC-GARCH, DCC-GARCH and copula-GARCH are presented in figures 7.11, 7.13 and 7.15. The variance of the unhedged portfolio (Var. UHP) is obviously equal for all models, since it is defined as the variance of the spot returns. More interesting is the variance reduction of the hedged portfolio after hedging with the optimal amount of futures contracts determined by the OHR. The variance reduction is captured by the quantity, the hedging effectiveness. The OLS model provides the highest hedging effectiveness for EUR/USD and EUR/CAD. For EUR/GBP the copula-GARCH model yields the highest effectiveness, although there is a marginal difference between OLS and copula-GARCH. For EUR/USD and EUR/CAD the copula-GARCH provides the second best hedging effectiveness. The DCC-GARCH model provides the worst variance reduction for the exchange rates EUR/GBP and EUR/USD and have therefore the lowest hedging effectiveness. The CCC-GARCH model provides the worst reduction of variance for the exchange rate EUR/CAD.

The study of Chen et al. (2013) could not find evidence for differences in performance between the CCC and DCC-GARCH models. This corresponds with the findings of this study as well. Moreover, Lien et al. (2002) examined the performance of the conventional method (OLS) and the CCC-GARCH. He finds that the conventional methods outperforms the CCC-GARCH model in all cases. Based on the empirical results of this paper, the same conclusion can be drawn. Furthermore, Chang et al. (2013) find results that the DCC-GARCH model does not outperform the CCC-GARCH model, which again is also present in the results of this study. The results of Hsu et al. (2008) provide evidence that the copula-based GARCH model provides the best hedging performance in comparison with DCC in second and CCC thereafter. However, in this study the Gaussian, Gumbel and Clayton copula are used instead of the Student’s t-copula. Moreover, Liu et al. (2010) find evidence in the electricity market that the Student’s

t-copula GARCH model yields the best hedging performance in comparison with the DCC and conventional model. In this study the copula does only outperform the conventional model for the exchange rate EUR/GBP. However, the copula-GARCH model does outperform the DCC-GARCH and CCC-GARCH model for all exchange rates.

6 C

In this study the risk-minimizing currency hedging ratios are estimated by four different models. The conventional model and three multivariate GARCH models, namely CCC-GARCH, DCC-GARCH and a copula-based GARCH model. Regarding the copula-based GARCH model a student’s t-copula is used. The empirical results are obtained by fitting the models on three different exchange rates, namely EUR/GBP, EUR/USD and EUR/CAD. The findings of this study are important for the field of risk management, whereas there is not yet been found one specific model which outperforms all other models, this study contributes to this research. Therefore, the goal of this study was to determine which of these models performs better estimating the optimal hedging ratio in order to reduce currency risk the most.

First, the CCC-GARCH and DCC-GARCH model provide the worst risk reduction for all three currency pairs. Besides that, the DCC-GARCH only exceeded reduction of risk for the exchange rate EUR/USD. Thus, there is no empirical evidence that modeling a time-variant conditional correlation results in a significant greater risk reduction in comparison with a time-invariant conditional correlation. The conventional model does provide the best hedging effectiveness for the exchange rates EUR/USD and EUR/CAD. Whereas, the copula-based GARCH results in the best hedging effectiveness for EUR/GBP. Based on the empirical results, the conventional model and copula-based GARCH are significantly superior to the CCC-GARCH and DCC-GARCH models. However, there is no distinct difference in the hedging performance between the conventional and the copula-based GARCH model as well as between the CCC-GARCH and DCC-GARCH model. Recapitulatory, the conventional en copula-based GARCH are superior to the CCC-GARCH and DCC-GARCH model.

constant hedging ratio in comparison with a dynamic hedging ratio. This is an important remark for risk managers. Buying and selling futures contracts brings transaction costs, there-fore it is most likely not profitable to change their portfolio with a high frequency. However, a disadvantage for a constant hedging is ratio is during more or less volatile periods the hedging ratio does not change as a result of the volatility changes. Further studies should contribute to build a hedging strategy whether a constant or dynamic hedging ratio is beneficial.

There are several possibilities for improvement of this research. First of all, the above problem can be further examined since this is especially for the practice of risk management an important . Secondly, in this study the models are merely fitted to the data. An improvement would be to include a in-sample and out-of-sample forecast to check whether the models can be used as a forecasting tool in the field of risk management. Furthermore, only a student’s t-copula is used in order to capture the dependence structure between the two derivatives. Further research can include other copulas to distinguish which copula strategy performs best in a multivariate copula-GARCH model. Finally, a robustness check can be executed on the results of this study to verify if they are robust or not. Thus, there are still limitations regarding this research as well as enough possibilities for further research in order to determine the best model and currency hedging strategy.

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7 A

Table 7.1: Description of variables

Variable Description

St Price of a spot position at time t

Ft Price of a futures position at time t

RS,t Return of a spot position at time t

RF,t Return of a futures position at time t

Table 7.2: Estimates of OLS

Return α β R2

EUR/GBP 8.49e-4 0.8706∗ 0.7469 EUR/USD 5.10e-4 0.8218∗ 0.6603 EUR/CAD 2.18e-4 0.8203∗ 0.6504

***, ** and * denote the significance at a 5%, 2.5% and 1% level, respectively.

Table 7.3: Breusch-Pagan test for heteroskedasticity Exchange rate Test statistic

EUR/GBP 11.846 (0.0005) EUR/USD 8.3181 (0.0039) EUR/CAD 1.2067 (0.2720)

The test statistic of the Breusch-Pagan test is LM = nR2, distributed asχ2_p−1.

Table 7.4: Estimates of univariate GARCH(1,1)

Return µ ω α β Log-likelihood AIC

EUR/GBP 0.00040 0.00084 0.04137∗ 0.95665∗ -2457.36 1.3483 EUR/GBP_FUT -0.00103 0.00119∗ 0.04378∗ 0.95253∗ -2407.65 1.3211 EUR/USD 0.00490 0.00093 0.02977∗ _0.96775∗ _{-3111.95 1.7069} EUR/USD_FUT 0.00592 0.00105∗ 0.02857∗ 0.96839∗ -3077.03 1.6878 EUR/CAD -0.00382 0.00320∗ 0.03244∗ 0.95865∗ -3233.12 1.7507 EUR/CAD_FUT -0.00255 0.00304∗ 0.03194∗ 0.95921∗ -3166.20 1.7146

***, ** and * denote the significance at a 5%, 2.5% and 1% level, respectively.

Table 7.5: Estimates of parameters CCC-GARCH(1,1)

Return ω α β α + β CCC9 EUR/GBP 0.00114∗ 0.04201∗ 0.95450∗ 0.99652 0.86304∗ EUR/GBP_FUT 0.00136 0.04118∗ 0.95390∗ 0.99508 EUR/USD 0.00082∗ 0.03070∗ 0.96715∗ 0.96887 0.81381∗ EUR/USD_FUT 0.00092 0.02818∗ 0.96917∗ 0.99735 EUR/CAD 0.00255∗ 0.02910∗ 0.96381∗ 0.99292 0.81045∗ EUR/CAD_FUT 0.00263 0.02833∗ 0.96392∗ 0.99225

***, ** and * denote the significance at a 5%, 2.5% and 1% level, respectively.

8_{CCC denotes the Constant Conditional Correlation.}

Table 7.6: Log-likelihood and AIC of CCC-GARCH(1,1)

Return Log-likelihood AIC

EUR/GBP -2372.78 1.3003

EUR/USD -4194.50 2.3016

Table 7.7: Estimates of parameters DCC-GARCH(1,1) Return µ ω α β α + β θ1 θ2 EUR/GBP 0.0004 0.0008∗∗∗ 0.0414∗ 0.9567∗ 0.9980 0.1256∗ 0.0908∗∗∗ EUR/GBP_FUT -0.0010 0.0012∗∗ 0.0438∗ 0.9525∗ 0.9963 EUR/USD 0.0049 0.0009 0.0298∗ 0.9677∗ 0.9975 0.1535∗ 0.0874∗∗∗ EUR/USD_FUT 0.0059 0.0011 0.0286∗ 0.9684∗ 0.9970 EUR/CAD -0.0038 0.0032∗ 0.0324∗ 0.9586∗ 0.9911 0.1018∗ 0.1637 EUR/CAD_FUT -0.0026 0.0030∗ 0.0319∗ 0.9592∗ 0.9911

***, ** and * denote the significance at a 5%, 2.5% and 1% level, respectively.

Table 7.8: Log-likelihood and AIC of DCC-GARCH(1,1)

Return Log-likelihood AIC

EUR/GBP -2288.53 1.2591

EUR/USD -4083.26 2.2423

Table 7.9: Estimates of Copula-based GARCH

EUR/GBP EUR/USD EUR/CAD

Parameter Spot F ut ur es Spot F ut ur es Spot F ut ur es

Marginal process µ -0.0037 -0.0036 0.0055 0.0060 -0.0038 -0.0028 ω 0.0011∗ 0.0010∗ 0.0008∗∗∗ 0.0006 0.0027∗ 0.0022∗ α 0.0416∗ 0.0398∗ 0.0314∗ 0.0320∗ 0.0333∗ 0.0285∗ β 0.9546∗ 0.9570∗ 0.9667∗ 0.9668∗ 0.9592∗ 0.9651∗ α + β 0.9962 0.9968 0.9982 0.9988 0.9925 0.9936 v10 8.2179∗ 7.6703∗ 7.7707∗ 8.1271 9.0841∗ 9.0695∗ Log-likelihood -2393.1 -2342.5 -3066.8 -3031.1 -3197.0 -3128.1 Copula estimates ρ 0.8643∗ 0.8186∗ 0.8110∗ vc11 5.8446∗ 4.3791∗ 6.6948∗ Log-likelihood -2542.4 -2075.4 2013.1 DCC estimates µ 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 ω 0.0015 0.0015 0.0034 0.0052 0.0033 1.2217∗ α 0.0000 0.0000 0.0000 0.0000 0.0000 0.0623∗ β 0.9990∗ 0.9990∗ 0.9981∗ 0.9971∗ 0.9977∗ 0.0743 α + β 0.9990 0.9990∗ 0.9981 0.9971 0.9990 0.1366 θ1 0.1138∗ 0.1720∗ 0.1112∗ θ2 0.1069 0.0204 0.2430 Log-likelihood -9283.6 -10387 -9713.7

***, ** and * denote the significance at a 5%, 2.5% and 1% level, respectively.

9_{v are the estimated degrees of freedom of the univariate GARCH model.} 10_v

Figure 7.1: Daily returns of spot and futures prices

(a) Spot returns EUR/GBP (b) Futures returns EUR/GBP

(c) Spot returns EUR/USD (d) Futures returns EUR/USD

Figure 7.3: Residuals of the OLS estimation

(a) EUR/GBP (b) EUR/USD

Figure 7.5: Estimated conditional volatilities

(a) Conditional volatilities for EUR/GBP (spot) (b) Conditional volatilities for EUR/GBP (futures)

(c) Conditional volatilities for EUR/USD (spot) (d) Conditional volatilities for EUR/USD (futures)

Figure 7.7: Estimated conditional correlations by DCC-GARCH model

(a) EUR/GBP (b) EUR/USD (c) EUR/CAD

Figure 7.9: Estimated conditional correlations of the copula-GARCH model

Figure 7.11: Estimated optimal hedging ratios by CCC-GARCH model

(a) OHR EUR/GBP (b) OHR EUR/USD (c) OHR EUR/CAD

Figure 7.13: Estimated optimal hedging ratios by DCC-GARCH model

(a) OHR EUR/GBP (b) OHR EUR/USD (c) OHR EUR/CAD

Figure 7.15: Estimated optimal hedging ratios by copula-GARCH model