Currency hedging: an evaluation of econometric models for estimating the optimal hedge ratio in European markets

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Faculty Economie en Bedrijfskunde, Amsterdam School of Economics Bachelor thesis Econometrics

Currency hedging: an evaluation of econometric

models for estimating the optimal hedge ratio in

European markets

Yunus Emre Öztürk - 11066989

26/06/2018

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Statement of originality

This document is written by Student Yunus Emre Ozturk who declares to take full respons-ibility 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 comple-tion of the work, not for the contents.

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1 Introduction

With the tranquilizing of the chaos caused by the financial crisis in 2008, many economic agents have again started gaining trust in the global economy. This expansion of trust drives these agents into investing in financial products. Even though the major economies of the globe are slowly recovering from the crisis and further growing in size, these investments are not risk-free. Risk management is essential for companies and investors to reduce the risk. Many agents are eager to know how to minimize the risks attached to investing in a certain commodity. The specific financial products analyzed in this paper are foreign exchange rates. That is the case as foreign exchange markets are arguably the most liquid and one of the largest asset markets there is. Fundamentally, in order to know the potential loss caused by a significant depreciation of a foreign currency, one should understand how much the spot rate fluctuates and should respond accordingly. When minimizing the risk attached to interest rate risk, many international companies and also many investors opt to hedge in the form of futures contracts (Hull, 2011, pp. 71-72). Hedging in futures contracts is a financial instrument which helps reduce the volatility of a portfolio. The main problem with futures contracts is deciding the optimal hedge ratio. This is where making use of financial econometrics is helpful. By applying econometric models one can estimate the optimal hedge ratio. The problem that arises is deciding which model to use and if the choice of models changes in different times of uncertainty. The main research question is: Which model is better to estimate the optimal hedge ratio comparing the effectiveness of different GARCH models and OLS before and after the global financial crisis; based on European futures markets versus the US Dollar?

It is well established in literature that financial time series exhibit volatility clustering and heteroscedasticity (Heij, De Boer, Franses, Kloek, & Van Dijk, 2012, p.621). The Ordinary Least Squares (OLS) method assumes that the second moments are constant. Using the OLS method would therefore result in inaccurate results for the optimal hedge ratio. The Gener-alised Autoregressive Conditional Heteroscedastic (GARCH) model as proposed by Bollerslev (1986), changes this assumption and works with time-dependant second moments. This should theoretically produce precise estimations. However, when Lien, Tse and Tsui (2002) compared OLS with vector GARCH (VGARCH or CCC) they concluded that OLS performed better in estimating the optimal hedge ratio. Yang and Allen (2005) performed the same test, but

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used dynamic hedge ratios (DCC) rather than constant hedge ratios and observed that the multivariate GARCH model was superior. The univariate Glosten-Jagannathan-Runkleand (GJR) model adds to the CCC model by also incorporating asymmetry of positive and negat-ive shocks and should therefore in theory perform better than the CCC model, but the findings of Da Veiga, Chan, and McAleer (2008) suggest the opposite. It is obvious that previous stud-ies have diverging conclusions and further research is required as the best model has yet to be decided upon.

This thesis relies on older studies and expands the research by combining models used in earlier findings and concentrates on estimating the optimal hedge ratio in European markets, compared to the US dollar for different periods. This research compares the effectiveness in estimating optimal hedge ratios of the OLS model and the following 3 GARCH models: GJR, Constant Conditional Correlation (CCC) and Dynamic Conditional Correlation (DCC) models. By applying these models, empirical evidence is presented in whether dynamic hedge ratios perform better than constant hedge ratios and if asymmetry in shocks is empirically crucial. This is done by using the Chang et al. (2013) method to evaluate the effect of different models on conditional correlations forecasts for estimating the returns on futures and spot rates.

The primary focus is on European markets, as the spot and futures prices of the following 3 exchange rates are considered: USD/Euro, USD/British Pound and USD/Swiss Franc. Daily data of spot and historical futures rates is used in the period between 2005-2015. This period is then divided in two separate periods before and after the global financial crisis (GFC), in order to analyze the differences in optimal models for different volatility levels. These periods are respectively Q1 2005 - Q3 2007 and Q3 2007 - Q1 2015. The purpose of this research is first of all to add a contribution to the unsolved debate in literature on which models perform better than others, secondly to analyze what the performance of the models are in the specific case of the European markets and finally to broaden the existing literature by also analyzing the effects of the GFC on the performance of the models.

The thesis is structured as follows: In section (2) the theoretical framework including the main contributions of previous studies are discussed. Section (3) describes the data and variables. Section (4) continues with the explanation of the econometric models used in this thesis. This is followed by the results and the analysis of the research in section (5). Finally,

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some concluding remarks and recommendations are presented in section (6).

2 Theoretical Framework

In this section the contribution of previous studies are discussed. Firstly, the term hedging is defined and its importance in the financial world is emphasized. Secondly, the futures contracts which are used in order to hedge are explained. This is followed by a description of the optimal hedge ratio of these futures contracts. Finally, the econometric models which can be used in order to estimate this optimal hedge ratio are evaluated.

2.1 Hedging and its importance

Every kind of investment has risk attached to it. This risk is mostly due to fluctuations in the price of the invested financial product. If the price of a financial product drops, the investor has to sell this product for a lower price than it was purchased for, resulting in a loss for the investor. Another investment that counteracts this loss and therefore protects the risk attached to the former investment is called a hedge. The Sydney Futures Exchange (SFE) defines hedging as ’the act that reduces the price risk of an existing or anticipated position in the cash market’. In this thesis, the relevant price risk described in the definition of the SFE is the fluctuation of spot prices of European exchange rates in comparison to the United States Dollar.

Eun and Resnick (1988) conclude that fluctuating exchange rates make investments more risky and thereby diminish any potential gains. They analyse an international portfolio and conclude that all hedging strategies outperform unhedged portfolios by increasing the returns of the investments and therefore emphasize the importance of hedging when investing in risky financial products, especially in times of very high volatility such as financial crises. Moreover, Hull (2011, pp. 71-72) states that hedging is especially important for large international com-panies, as many companies do not have the expertise in the field of analyzing and forecasting foreign exchange rates. Therefore, it is in their best interest to hedge against exchange-rate risk in the foreign exchange market caused by unpleasant fluctuations of the spot prices. Fur-thermore, Allayannis and Ofek (2001) support this statement in their analysis of a sample from the S&P 500 index. In their paper, it is investigated why firms use currency derivatives.

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Given the observation that the exchange-rate exposure faced by the international companies is significantly reduced by utilizing currency derivatives, Allayannis and Ofek conclude that these currency derivatives are used in order to hedge, rather than for speculative purposes, affirming the relevance of hedging for large international companies.

2.2 Futures contract

According to the SFE, most organisations use derivatives called futures contracts as their fun-damental method of hedging in order to manage risk. Harvey (2012) defines futures contracts as follows:

"A legally binding agreement to buy or sell a commodity or financial instrument in a des-ignated 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."

As stated in the definition, futures contracts are important financial instruments which are used to manage the risk of an investment. Managing the risk, so reducing the volatility of the prices, can potentially come with a cost however. A prominent argument used in existing literature against hedging with futures contracts is the fact that the price is set beforehand and that the actual value of the price on the day of transaction, in the case of a positive fluctuation, could have been better without hedging or by using a different hedge ratio. Atwill (2015) concludes that currency hedging in the case of emerging markets only leads to decreased profits and that it should be left unhedged. However, the researh of Atwill is limited as it only focuses on emerging markets, which is indicated in the paper by the statement that emerging markets are in many ways different from developed markets and that the same research applied to developed currency markets could lead to contrasting results. Atwill also solely compares fully hedging to no hedging. A hedge ratio or different hedging strategies are not discussed

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in this paper. For instance, in their analysis on foreign exchange rates, share price indexes and consumer price indexes, Morey and Simpson (2001) describe how using different hedging strategies, such as selectively hedging with the use of a hedge ratio, outperforms complete hedging and no hedging at all. The conclusion of Morey and Simpson is adopted and the hedge ratio is considered in this thesis. Naturally, hedge ratios which do not significantly differ from respectively one or zero imply that it is respectively preferred to fully hedge or leave the investment unhedged.

2.3 Optimal Hedge Ratio

As this thesis is concerned with testing which models outperform each other on estimating the optimal hedge ratio, there needs to be a consensus on what the optimal hedge ratio is. Luckily, there has been extensive research on this topic dating back sixty years. Johnson (1960) and Stein (1961) are the pioneers on this subject. They conclude that an OLS estimation between the spot price as dependent variable and the futures price as independent variable gives the optimal variance-minimizing hedge in which the coefficient of the futures price gives the optimal hedge ratio. If this OLS is performed for data that have already occurred in the past, it gives the exact optimal hedge ratio for that time period. If however, the OLS model is used in order to forecast the spot and futures prices, the coefficient provides an estimate of the optimal hedge ratio for the specific time period in the future. The latter OLS model is also adopted in this thesis in order to estimate the optimal hedge ratio, in which more information is provided in section 3. Later, Ederington (1979) extends the research of Johnson and Stein in terms of returns in hedging against volatilities in foreign exchange markets. Ederington concludes that the optimal hedge ratio in the case of an investment in exchange markets can be calculated by dividing the covariance between the returns on the spot prices and returns on futures prices by the variance of the returns on futures prices. This is equal to the findings of Johnson (1960) and Stein (1961) as per definition the coefficient of the independent variable equals the covariance of the independent and dependent variable divided by the variance of the independent variable (Heij et al., 2004, p.81).

Another way of comparing models, next to the variance-minimizing hedge, is by examining the means of the portfolios that are created by the optimal hedge ratios, rather than ranking the portfolios based on lower variances. One method is not regarded better than the other, it

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is up to the investors’ choice on which hedging strategy to utilize. However as Czekierda and Zhang (2010) and many more researchers mention, choosing a different hedging strategy may result in some models exceeding the performance of others. Both methods will be assessed in this thesis in order to gain a broader view on the performance of the different models.

2.4 Comparison of models

With any sort of hedging, the optimal hedge ratio can only be calculated after the investment is already done, as the covariance and variance of spot and futures prices can only be calculated after the data are already presented. This provides no added value to the hedging process itself. That is why this thesis is interested in estimating the optimal hedge ratio, before the transaction has taken place. This is done by forecasting the spot and futures prices and thereby estimating the optimal hedge ratio for future hedges, which is further discussed in section 3. Forecasting can be done by using multiple econometric models. This is also where there is no real consensus in the empirical studies that have been published and more research is still being performed. Many researchers have diverging findings on the best model to use in order to forecast and estimate optimal hedge ratios. In the paragraphs below the OLS, CCC, DCC and GJR models are respectively discussed in comparison to each other.

In many empirical studies it is well established that financial time series contain hetero-skedasticity (Heij et al., 2012, p.621) and that using the Ordinary Least Squares method is not suited for estimating the optimal hedge ratio as the OLS method assumes that the second moments are constant. GARCH models on the other hand do take into account the hetero-skedasticity, which should in theory lead to better estimates for the optimal hedge ratios, as GARCH models should describe the actual distribution more accurately than the OLS model. However, some empirical studies suggest that the OLS outperforms these GARCH models. Lien et al. (2002) compare the OLS method to the GARCH method in order to forecast the prices and estimate the optimal hedge ratio for different assets including currency futures, commodity futures and stock index futures. They find that the OLS model outperforms the GARCH model as the OLS method provided estimates that were closer to the optimal hedge ratio. Lien (2005) later confirms his findings in favour of the superiority of the OLS hedge ratio in comparison to the error-correction model. Miffre (2004) adds to the research of Lien et al. (2002) as Miffre concludes that the conditional OLS hedge ratio surpasses the hedge

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ratio estimated by the GARCH(1,1) error correction model when six different currencies were analysed, with the British Pound being the only exception to this statement. They suggest that the superiority of the OLS model is caused by the fact that GARCH models tend to produce variance forecasts that are too variable.

As the theory of heteroskedasticity in time series would suggest, there are many publica-tions that are in favour of the GARCH models in comparison to the OLS model. The problem in this case is that there are many different GARCH models that can be used in order to estimate the optimal hedge ratio and that again, there is no real consensus on which GARCH model is best to forecast the spot and futures prices of exchange rates. This paper only focuses on three GARCH models. These are the CCC GARCH model which assumes constant condi-tional correlations, the DCC model which assumes dynamic condicondi-tional correlations and the GJR GARCH model which assumes asymmetric shocks. In the case of the foreign currency market, there are many papers which disfavour the CCC GARCH model when compared to the DCC GARCH model. Bera and Roh (1991) test whether the constant conditional correl-ation (CCC) hypothesis holds for financial time series in financial markets and reject the null hypothesis, implying that in financial time series a dynamic conditional correlation (DCC) is better suited. Chakraborty and Barkoulas (1999) perform similar tests in which the dynamic model is tested against the static model with the same conclusion in favour of the DCC model, this is however only the case for the Canadian dollar and not for the other foreign currencies that were tested upon, which is not further elaborated upon in their research. Furthermore, when Lien et al. (2002) conclude that the OLS model outperforms the GARCH model, the only GARCH model that is considered assumes constant conditional correlations. These pa-pers imply that the CCC GARCH model performs worse than the DCC GARCH model and the OLS model. However, Sim and Zurbruegg (2001) suggest that high volatility negatively impacts the DCC model. In their research on the effect of the Asian financial crisis they suggest that due to this crisis the CCC performs better than the DCC and that if the costs attached to creating time-varying hedge ratios are incorporated, again the CCC outperforms the DCC model.

Research on the comparison between the OLS and DCC GARCH models suggest the superiority of the DCC model. Yang and Allen (2005) evaluated the OLS model and DCC model in their paper by comparing the effectiveness in the futures market of indices using

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two different approaches of calculating the hedging effectiveness. They concluded that both approaches suggest the superiority of the DCC model in comparison to the OLS model. Ku, Chen and Chen (2007) tested the DCC model against the OLS and CCC models in the currency market by comparing the hedging effectiveness of these models when used for hedging the Japanese Yen and the British Pound. They conclude that the DCC model is superior and also confirm the conclusion of the last paragraph, asserting that CCC performs worse than OLS. These empirical studies suggest that in general the DCC model is superior to both the OLS and CCC models. Again, there is no real consensus on the comparison of the DCC and OLS models, as again there are papers that suggest superiority of the OLS model. Cotter and Hanly (2006) describe in the comparison of different indices that the DCC model shows no significant improvement over the OLS model and in their further research in 2012 suggest the superiority of the OLS model and also affirm there to be different results for the best model to use when different performance indices are examined instead of the usual variance minimizing capacity of the models.

Finally, empirical studies on the GJR model are discussed. This model is an univariate GARCH model developed by Glosten et al. (1993) which is very similar to the multivariate CCC model, however assumes that positive and negative shocks of the error terms are not equal, while the CCC assumes those to have equal impacts on the prices. Hakim and McAleer (2009) tested this model even further and found that indeed there are asymmetric properties of positive and negative shocks in financial time series regarding financial assets such as stocks, bonds and foreign exchange rates. They were cautious however by stating that even though asymmetric models describe the data better than the CCC model, that incorporating these asymmetric properties does not produce better forecasting abilities. Ramasamy and Munisamy (2012) add to the results of Hakim and Mcaleer (2009) and state that the asymmetric effect of the GJR model is negligible and does not improve the results of the ordinary GARCH model when used to hedge currencies, suggesting that one should prefer the ordinary GARCH model over the GJR model. Copeland and Zhu (2011) compare the OLS model to the GJR model with the use of index futures markets and conclude that the OLS model surpasses the GJR model. The GJR and DCC models have not been directly compared to each other. This thesis contributes to previous research by comparing these two models for the first time.

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by the OLS, CCC and GJR models, however results can vary depending on different financial products, the time period and economic factors such as an economic crisis. As explained before there are many more GARCH models which have been researched and published, however as this thesis only focuses on CCC, DCC and the GJR models, the other models are not within the scope of this thesis and therefore they are not discussed.

3 Data

As this research focuses on the hedging of currency markets with the use of futures on the European market, three different exchange rates compared to the Unites Stares Dollar are analyzed. These three exchange rates are the USD/Euro (EUR), USD/British Pound (BP) and USD/Swiss Franc (CHF). From these exchange rates both the spot and futures prices were gathered from the library of the University of Amsterdam using the program called Datastream which contains historical financial data. The value of the spot prices is defined as the ’middle exchange rate’ by Datastream, in which the value is the midpoint between the bid rate and the offered rate. The futures contracts are defined as continuous average settlement prices of the exchange rate. The expiration date of the futures contracts provided by Datastream is a month. The data is analyzed in three periods. The first period reaches from the first of January 2005 until the first of January 2015. Then the period is split in order to account for the effects of the global financial crisis (GFC). The second period, which is before the GFC, spans from the first of January 2005 until the begin of Q3 of 2007. The third period involves Q3 2007 until 1 january 2015. As the exchange rates are only traded with from Monday to Friday, the whole period with all daily frequencies gives 2609 observations on returns to work with. The pre-GFC period consists of 650 observations and the period during and after the GFC consists of 1959 observations. Datastream contains no missing observations on both the spot and futures prices for all three exchange rates.

As explained in the Theoretical Framework, not the spot and futures prices themselves, but the returns of these spot and futures prices are analyzed. The returns are calculated as follows: r_j,t = ln(Pj,t/Pj,t−1), in which Pj,t stands for the spot or futures prices for exchange rate j, on day t. Descriptive statistics on the returns for both the spot and futures prices per exchange rate are given in table 1. The descriptive statistics for the two different periods are

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portrayed in the appendix as tables A1 and A2.

Note: A significance on the 1% level is indicated by *.

Spot return Fut. return

USD/BP USD/CHF USD/EUR USD/BP USD/CHF USD/EUR Mean 7.64e-5 -5.54e-5 4.09e-5 6.82e-5 -6.55e-5 3.09e-5 Maximum 0.03919 0.08475 0.03845 0.05128 0.08731 0.03065 Minimum -0.0447 -0.0545 -0.0462 -0.0346 -0.0552 -0.03086 Std. Dev. 0.00607 0.00678 0.00613 0.00583 0.00675 0.00595 Skewness 0.06821 0.56663 -0.1648 0.45536 0.63681 0.06272 Kurtosis 8.07722 16.2343 6.73238 7.84562 17.9630 5.19964 Jarque-Bera 2803.25* 19172.1* 1525.59* 2641.63* 24505.8* 527.483*

Table 1: Descriptive Statistics Q1 2005 - Q1 2015

As the returns of the first day can not be calculated, the amount of observations drops from 2609 to 2608. All descriptive statistics for the different periods portray similar characteristics. It is clear that the means of the returns are close to zero. The standard deviation is relatively high compared to the mean, which is also observed from the high kurtosis of all variables. Furthermore, the spot returns of the USD/EUR show negative skewness, indicating that the returns result in losses more often than profits for this particular exchange rate, which is not the case for the other spot and futures returns. The USD/EUR exchange rate does have positive skewness in the period before the GFC, which indicates that the GFC might be the cause for the negative skewness.

The high Jarque-Bera values indicate that the errors do not follow a normal distribution. The null hypothesis for the Jarque-Bera normality test assumes a normal distribution. For a large amount of observations this can be tested by the chi-squared distribution with 2 degrees of freedom. The high Jarque-Bera values provide significant evidence to reject the null hypothesis, which is hardly surprising given the fact that these variables are financial time series.

In figure 2 the time series plots of all spot and futures rates are given. It is very clear that the financial crisis caused high returns and also high losses. Also the spot prices of the CHF portray large spikes in 2012, in which the Swiss Franc massively lost value. This was due to

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government interventions which issued an exchange-rate peg in order to create a ceiling for the CHF to counteract the increase in demand by investors that saw the CHF as a safe haven. In order to better visualize the volatility that is caused by the financial crisis the square of the estimated returns are calculated and the time series plot is portrayed in figure A3 in the appendix. It is observed that the volatilites are rather stable, with a sudden increase in the financial crisis. The crisis in 2012 in Switzerland is remarkable as it portrays even higher volatility than during the financial crisis. In these figures one can also observe volatility clustering, suggesting that the use of a GARCH model could result in optimal estimations. Finally, Augmented Dickey-Fuller tests have been performed on the returns in order to test for stationarity. The null hypothesis assumes the series to have an unit root and thus be non-stationary. The results for the Dickey-Fuller tests can be found in tables A4 and A5. All returns gave high critical values which reject the null hypothesis, implying that there is significant evidence to assume that all returns are stationary.

(a) Spot BP (b) Spot CHF (c) Spot EUR

(a) Future BP (b) Future CHF (c) Future EUR

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4 Methodology

This thesis focuses on the Chang et al. (2013) method to evaluate the OLS, CCC, GJR and DCC models. First the econometric models are discussed, which is followed by the explanation on how these models are compared by using a formula for the hedging effectiveness. Finally, the Akaike Information Criterion is discussed in order to compare GARCH models with differing lags.

4.1 OLS model

The equation for the OLS method is given below.

Yt= α + Xtβt+ t with Yt∼ N (Xtβt, σ2) and t∼ N (0, σ2) (1)

In this case Yt is the dependent variable which stands for the spot return rates, Xt portrays the future return rates, _t portrays the disturbance error for period t and α is the constant that is added to the model. The coefficient βt gives the estimate for the optimal hedge ratio. The errors are assumed to be independent and identically normally distributed.

4.2 CCC model

The GARCH model, as defined by Bollerslev (1986), is a generalisation of the ARCH model. This Generalised ARCH (GARCH) model differs from the OLS method as it assumes that the variances are not constant in a time series. Furthermore, the CCC and GJR models embrace constant conditional correlations while the DCC model embraces dynamic conditional correl-ations. The CCC multivariate GARCH model is described as follows:

Yt= E[Yt|Wt−1] + t, with t∼ N (0, ht) (2)

hit = αi+Pqj=1αij2i,t−j+

Pp

j=1βijhi,t−j (3)

Here, Y_t denotes an independent and identically distributed vector (y_1t, ..., ynt) of the re-turns of the spots or futures rates. In this vector for Yt, n signifies the amount of exchange rates and futures that are present in the used data. W_t denotes the information set available

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through time t, h_it denotes the conditional variance for each return and again α_i denotes the constant of regression i. The formula for hit portrays that the conditional variance follows a multivariate GARCH process in which βij denotes the GARCH effect, so the extent to which previous variances affect the present variance, and αij denotes the ARCH effect, which in this case portrays the short run persistence of shocks to returns. The combined sum of αij and βij stands for the long run persistence of shocks to returns. The first results in the thesis that are discussed only focus on the GARCH(1,1) model, which implies that p and q in the formula are both equal to one. This assumption is dropped in section 5.4.

4.3 GJR model

The CCC model and GJR model are very similar. The difference between the two is that the CCC model assumes symmetric impacts. Positive and negative shocks of epsilon are assumed to have an equal impact on the conditional variance. Glosten et al. (1993) were the first to change this assumption. The formula for the conditional variance is expanded in the following manner:

hit = αi+Pqj=1αij2i,t−j+

Pp

j=1βijhi,t−j+Pj=1q Ii,t−jzij2i,t−j (4)

Ii,t is an indicator function which equals 1 if it ≤ 0 and 0 if it > 0. In the case that the indicator function or the coefficient z_ij equals 0, the GJR model turns into the CCC model. By adding the indicator function the GJR model gives different magnitudes to positive and negative shocks depending on the size of the coefficients z_ij. This univariate model assumes there to be no covolotility between time series unlike the multivariate CCC model.

4.4 DCC model

The final model that is assessed, the DCC model, drops the assumption of constant conditional correlations. This model was first described by Engle (2002). The conditional correlations now become time dependent and are formulated as follows:

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In which D_t is the matrix below: D_t=           √ h1t 0 . . . 0 0 √h2t 0 ... .. . . .. . .. 0 0 . . . 0 √hnt           (6)

Dtis a diagonal matrix containing the squared roots of the conditional variances htper asset. The conditional variances h_t follow the same formula as the CCC model and W_t again por-trays the information set through time t. Rtis the conditional covariance matrix and Θtis the matrix which contains the dynamic conditional correlations ρ_t. These need to be estimated for the DCC model. This is done in the following manner:

Θt= (diag(Rt)− 1 2)R_t(diag(R_t)− 1 2) (7) Rt= (1 − λ1− λ2)var(t|Wt) + λ1ηt−1ηt−10 + λ2Rt−1 (8)

In the second formula λ₁ and λ₂ are non-negative scalar coefficients which respectively meas-ure the effects of previous shocks and dynamic conditional correlations of one lagged time period. These coeffients are estimated simultaneously with the DCC model itself. If both of these coefficients equal zero at the same time, the DCC model turns into the CCC model. If however, these do not equal 0, there is substantial evidence that the DCC model describes the time series more accurately than the CCC model. This is tested and discussed in section 5. Furthermore, ηtis obtained by dividing the specific return by the square root of its conditional variance.

4.5 Hedging effectiveness

Ku et al. (2007) emphasize the importance of variance reduction of the hedged portfolios acquired by the model estimations compared to the unhedged portfolio. In order to compare the models, an index for the hedging effectiveness is defined. This index, which is defined as HE, gives the decrease in variance of the unhedged portfolio when compared to the hedged portfolio in terms of percentages. The formula for the HE is stated as follows:

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HE = varunhedged−varhedged

varunhedged ∗ 100% (9)

In this formula, varunhedged is the variance of the spot returns. Varhedged is the conditional variance of the rate of returns of the hedged portfolio. The rate of returns and variance of the hedged portfolio are calculated with the use of the formula as defined by Johnson (1960):

RH,t= Rs,t− γ ∗ Rf,t (10)

V ar(RH,t|Wt−1) = var(Rs,t|Wt−1) − 2γt∗ cov(Rs,t, Rf,t|Wt−1) + γt2∗ var(Rf,t|Wt−1) (11)

The optimal hedge ratio as defined in section 2.3 is given by γ in this formula. Var(R_H,t|Wt-1) is the conditional hedged variance, Var(R_s,t|W_t-1) and Var(R_f,t|W_t-1) respectively are the conditional variances of the spot and futures returns. Cov(Rs,t,Rf,t|Wt-1) stands for the condi-tional covariance of the spot and futures returns given past information set W_t-1. By utilizing this formula, the hedged variance is calculated and this hedged variance is then substituted into the HE index. As the HE index gives the decrease in variance, a higher value implies that the hedged portfolio reduces the variance more than a portfolio with a lower HE value. Therefore, when comparing the different models a higher HE index would imply a better model for estimating the optimal hedge ratio.

4.6 Akaike Information Criterion

Finally, the Akaike Information Criterion (AIC) is discussed. This thesis broadens its research by also incorporating an analysis on the performance of the GARCH models when a higher amount of lags is used instead of the usual GARCH(1,1) model. The decision of which amount of lags to use in order to get the best possible performance of the models is done by examining the AIC values. These AIC values are calculated as follows:

AICi = −2log(Li) + 2Ki (12)

Ki denotes the number of estimated parameters while Li denotes the maximum log-likelihood of the specific model that is acquired from the results. The AIC value estimates the

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informa-tion loss by the model, which implies that a lower AIC value results in a lower informainforma-tion loss. Therefore, in the comparison of the model with different amount of lags, the AIC values are examined, and the model with the lowest AIC value is chosen to be the model which should in theory perform better than the original model. As the amount of lags are arbitrary numbers, an infinite number of combinations of lags are possible, which is why only combinations of lags in between the GARCH(1,1) and GARCH(5,5) models are examined.

5 Results

This research compares four models in the estimation of the optimal hedge ratio for three different currency and futures rates in three different periods. The results acquired by the OLS, GJR, CCC and DCC models respectively in the period Q1 2005 - Q1 2015 are provided in tables 3-7. The estimations of the models in the period before the GFC (Q1 2005 - Q3 2007) are provided in tables A6-A9 and in the period post-GFC (Q3 2007 - Q1 2015) are provided in tables A10-A13 in the appendix. Every table contains the Akaike information criterion (AIC). This is added in order to compare the GARCH models when using a different amount of lags in section 5.4.

5.1 Estimation Results

Tables 3, A6 and A10 give the OLS estimates. C stands for the constants and the coefficients of the futures in the linear regression. As to be expected with returns of financial time series, none of the constants are significant. The coefficients of the futures rates are defined as the optimal hedge ratios which are used in later estimations. These ratios differ per currency and per period. For all currencies, the ratios for pre-GFC are higher and the ratios for post-GFC are lower than the ratios for the period including all observations. The OLS therefore suggests higher ratios to be more optimal in times of less volatility, which is in line with the findings of Sim and Zurbruegg (2001). Also, in every period and for every model, the estimated hedge ratio is highest for the British Pound, followed by respectively the Euro and the Swiss Franc. The GJR estimates are provided in tables 4, A7 and A11. Again, the coefficients give the estimates for the optimal hedge ratio. In all GJR estimations a GARCH(1,1) model is estimated, in which the ARCH coefficient is given by α and the GARCH coefficient is given

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by β in the tables. The combined sum of α and β stands for the long run persistence of shocks to returns, which should be smaller than one due to the observed non-stationarity. This is the case in all GJR estimates. In section 5.4 the negative GARCH coefficients in the pre-GFC period as reported in tables A7, A8 and A9 are discussed. The values under ’z’ are the coefficients of the indicator function as explained in equation (4). All currencies in all periods have coefficients z that are not significant on the 10% P-value. This implies that the asymmetric impacts do not have any empirical substance. Positive and negative shocks of epsilon are thus assumed to have an equal impact on the conditional variance, suggesting that using the univariate GJR model does not provide additional benefits over the multivariate CCC model, even in times of high volatility such as the GFC.

Tables 5, A8 and A12 give the CCC estimates. The optimal hedge ratio in this case and also for the DCC model is calculated by dividing the estimated standard deviation of the spot returns by the estimated standard deviation of the futures returns, and multiplying this by the correlation. The GARCH coefficients of the BP and EUR in tables 5 and A12 show estimates in which the sum of α and β are greater than one. This suggest that the returns are non-stationary, which is in contradiction with the results of the ADF tests given in tables A4 and A5. The GARCH coefficients under table A8, so pre-GFC, again give negative coefficients. Both of these problems are tackled in section 5.4.

Finally, the DCC estimates are provided in tables 7, A9 and A13. Due to the large amount of essential coefficients to be displayed, these tables are flipped 90 degrees in order to be able to fit the table and still be readable. Again, the same problems of large sums of α and β, and some negative coefficients in the period pre-GFC arise as in the CCC model, which are again discussed in section 5.4. Two coefficients, λ1 and λ2 are added in the tables of the DCC model. Because some of these estimates are not significant on the 5% P-value level, a test is done for all DCC estimations to check whether both of these coefficients are equal to zero at the same time, so λ₁ = λ₂ = 0. In all models, this hypothesis is rejected on the 1% P-value level, implying that there is substantial evidence that none of these hypotheses hold true. This suggests that these coefficients certainly do have an effect on the estimations, that the DCC model should not turn into the CCC model and that the DCC model should more accurately describe the data set than the CCC model does. Now that this preliminary analysis is done, the hedging effectiveness estimated by the models are compared in section 5.2.

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Table 3: OLS estimates Q1 2005 - Q1 2015

Returns C Std. Err. R2 AIC

BP Constant .0000180 (0.79) .0000677 0.6761 -22160.01 BP Future .8566815 (0.00) .0116165 CHF Constant -2.11e-06 (0.98) .0000781 0.6544 -21410.79 CHF Future .8136575 (0.00) .0115836 Eur Constant .0000152 (0.83) .0000709 0.6508 -21913.2 Eur Future .8307429 (0.00) .0119208

Note: The P-values are given in brackets.

Table 4: GJR estimates Q1 2005 - Q1 2015

Returns C α β z AIC

BP Spot/Fut. .8990903 (0.00) .4824685 (0.00) .3881448 (0.00) .0310619 (0.58) -22776.15 CHF Spot/Fut. .8353568 (0.00) .4759351 (0.00) .0267011 (0.41) .0193003 (0.77) -21864.64 Eur Spot/Fut. .8559925 (0.00) .5020616 (0.00) .2608554 (0.00) .0208642 (0.74) -22444.34

Table 5: CCC estimates Q1 2005 - Q1 2015

Returns C α β Correlation AIC

BP Spot* -3.12e-06 (0.98) .0757893 (0.00) 1.268795 (0.00) .8250095 (0.00) -41880.85 BP Fut.* .0000128 (0.90) .1026236 (0.00) 1.043515 (0.00)

CHF Spot -1.25e-06 (0.99) .1304605 (0.00) .7173711 (0.00) .8110972 (0.00) -40294.16 CHF Fut. -.0000145 (0.91) .0779191 (0.00) .8239989 (0.00)

Eur Spot* 8.50e-06 (0.94) .0667126 (0.00) 1.349342 (0.00) .8113657 (0.00) -41386.03 Eur Fut.* -4.89e-06 (0.97) .0474860 (0.00) 1.554242 (0.00)

Note: The P-values are given in brackets. The star next to the exchange rates indicate the sum of α and β to be greater than 1.

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Table 6: DCC estimates Q1 2005 - Q1 2015

Returns C α β Correlation λ1 λ2 AIC

BP Spot -3.90e-06 (0.97) .2039156 (0.00) .7524752 (0.00) .8709984 (0.00) .1417626 (0.00) .4030028 (0.00) -42131.77 BP Fut. 2.57e-06 (0.98) .2286745 (0.00) .6972661 (0.00) CHF Spot -.0000656 (0.60) .202823 (0.00) .5361644 (0.00) .846357 (0.00) .1198937 (0.00) .2116329 (0.07) -40433.67 CHF Fut. -.0000526 (0.67) .1430928 (0.00) .6467533 (0.00) Eur Spot* -.000042 (0.71) .1281999 (0.00) 1.055061 (0.00) .8518713 (0.00) .1555559 (0.00) .1378229 (0.07) -41578.67 Eur Fut.* -.0000524 (0.64) .1122827 (0.00) 1.049929 (0.00)

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5.2 Minimum variance comparison

The hedging effectiveness index estimated by the models are given in tables 7, 8 and 9 for respectively the BP, CHF and EUR exchange rates. This hedging effectiveness evaluates the variance reducing capabilities of the models, in which a higher HE indicates a better estimation, as this lowers the variance of the optimal portfolio more than a model with a lower HE. In every table, the optimal hedge ratio as estimated by the models, the unhedged variance, the hedged variance and the HE is given. First of all, in every table is observed that the HE drops for times of high volatility in the returns. This is to be expected, as it is more difficult for the models to accurately forecast the values of the returns. In all results, the OLS method scores very similarly to the CCC model and slightly outperforms the CCC model which is in line with the findings of Lien et al. (2002) and Miffre (2004). There is no case in which the CCC model is superior to the OLS model.

The DCC model estimates provide surprising results. This model, together with the GJR model are outperformed by the OLS and CCC models in most cases. The DCC model only outperforms the CCC model in case of the pre-GFC BP exchange rate and the GJR model only outperforms the CCC model in case of the pre-GFC CHF exchange rate. Both models never surpass the OLS model. In the comparison of the GJR and DCC models themselves, the DCC only exceeds the GJR model in the pre-GFC EUR exchange rate and the pre-GFC BP and post-GFC BP exchange rates, but not in the period containing all observations. This one case suggests that the GJR could provide better estimations for longer periods and the DCC for shorter periods, however this is not reinforced by the other currencies.

In summary, the OLS scores best in all estimations, followed by the CCC, GJR and DCC models. The economic crisis does not seem to create a significant impact in the comparison of the models, wtih the only exception being that the GJR and DCC models only surpass the CCC model in the pre-GFC period suggesting that these models could only perform better in low volatility exchange rates. The findings for the superiority of the OLS model on the GJR and CCC, and the superiority of the CCC model on the GJR model are in line with the findings at the end of the Theoretical Framework, which suggested the ranking to be in the following order: DCC, OLS, CCC and GJR. However, the low performance of the variance reduction by the DCC model is surprising. Czekierda and Zhang (2010) find similar results and suggest

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that not only a minimum variance comparison should be performed, but also a comparison of the mean returns, as some models can decrease the variance of the hedged portfolio, but at the cost of lowering the mean returns. This comparison is performed in section 5.3.

Table 7: Results USD/BP

Optim. Hedge Ratio Var. Unhedged Var. Hedged Hedge Effectiveness

All obs.

OLS 85.67 3.68e-5 1.194e-5 67.6057

GJR 89.91 3.68e-5 1.200e-5 67.4400

CCC 85.54 3.68e-5 1.194e-5 67.6055

DCC 89.95 3.68e-5 1.200e-5 67.4368

Pre-GFC

OLS 89.19 2.44e-5 6.525e-6 73.2025

GJR 90.59 2.44e-5 6.530e-6 73.1842

CCC 88.89 2.44e-5 6.526e-6 73.2018

DCC 89.30 2.44e-5 6.525e-6 73.2024

Post-GFC

OLS 84.97 4.1e-5 1.372e-5 66.5237

GJR 89.38 4.1e-5 1.379e-5 66.3450

CCC 84.09 4.1e-5 1.372e-5 66.5165

DCC 89.27 4.1e-5 1.379e-5 66.3537

Table 8: Results USD/CHF

All obs.

OLS 81.37 4.60e-5 1.591e-5 65.4374

GJR 83.54 4.60e-5 1.593e-5 65.3908

CCC 81.10 4.60e-5 1.591e-5 65.4367

DCC 84.92 4.60e-5 1.597e-5 65.3124

Pre-GFC

OLS 83.64 3.03e-4 9.900e-6 67.3321

GJR 84.18 3.03e-4 9.901e-6 67.3293

CCC 84.63 3.03e-4 9.903e-6 67.3226

DCC 81.92 3.03e-4 9.909e-6 67.3035

Post-GFC

OLS 80.93 5.12e-5 1.790e-5 65.0715

GJR 83.56 5.12e-5 1.793e-5 65.0026

CCC 80.59 5.12e-5 1.790e-5 65.0704

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Table 9: Results USD/EUR

All obs.

OLS 83.07 3.76e-5 1.312e-5 65.0786

GJR 85.60 3.76e-5 1.314e-5 65.0185

CCC 83.13 3.76e-5 1.312e-5 65.0785

DCC 86.24 3.76e-5 1.316e-5 64.9838

Pre-GFC

OLS 83.70 2.34e-5 7.510e-6 67.9200

GJR 86.25 2.34e-5 7.525e-6 67.8571

CCC 83.68 2.34e-5 7.510e-6 67.9200

DCC 84.87 2.34e-5 7.513e-6 67.9069

Post-GFC

OLS 82.95 4.23e-5 1.498e-5 64.5581

GJR 85.44 4.23e-5 1.501e-5 64.5004

CCC 83.16 4.23e-5 1.498e-5 64.5578

DCC 86.30 4.23e-5 1.503e-5 64.4531

5.3 Mean return of hedged portfolio comparison

The mean returns of the hedged portfolios for all currencies per period and model are given in table 10. The mean returns of the unhedged portfolios are given in brackets. It is observed that hedging indeed minimizes the risks attached to investing, as the means of the hedged portfolios are much closer to zero than the unhedged portfolios. This entails that the profits could have been higher without hedging, however it also indicates potential losses to be lower in comparison to no hedging.

On the performance of the models, table 10 suggests that the OLS still outperforms CCC, with the only exceptions being the BP exchange rate for the whole observation set and post-GFC, suggesting a slight edge to the CCC in higher volatility investments. This is however only the case for the BP exchange rate and is not reinforced by the other exchange rates. The GJR model shows a single exceptional performance in the BP exchange rate for the pre-GFC period in which it surpasses every model. This could be a coincidence however as every other result suggest there to be no particular improvement of the GJR model over the other models. In contrary in some cases the GJR model even gets outperformed by the DCC model. The DCC model shows superiority over the other models in all pre-GFC currencies. The post-GFC

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shows the opposite however, as the DCC is the worst in post-GFC periods, with the exception of the GJR performing worse for the BP exchange rate. The low performance of the DCC model in times of economic crisis is in line with the findings of Sim and Zurbruegg (2001). In the whole observation set, there is not much difference in the ranking, with the exception of the CHF exchange rate where the DCC model exceeds other models by having a positive return while other models estimate optimal portfolios with negative returns.

In general it can clearly be stated that examining the mean returns favours the DCC more often than examining the minimum variance, especially in the pre-GFC period. The DCC outperforms every other model in times of low volatility in terms of mean return. All of these results assume GARCH(1,1) models, which can result in problematic estimates as observed in section 5.1. A different amount of lags could result in different results, which is discussed in section 5.4.

Table 10: Mean Returns on all currencies of hedged portfolios

USD/BP USD/CHF USD/EUR

All obs.

OLS 1.80e-5 (7.64e-5) -2.11e-6 (-5.54e-5) 1.52e-5 (4.09e-5) GJR 1.51e-5 (7.64e-5) -6.90e-7 (-5.54e-5) 1.44e-5 (4.09e-5) CCC 1.81e-5 (7.64e-5) -2.28e-6 (-5.54e-5) 1.52e-5 (4.09e-5) DCC 1.51e-5 (7.64e-5) 2.18e-7 (-5.54e-5) 1.42e-5 (4.09e-5) Pre-GFC

OLS 4.44e-6 (-8.13e-5) 2.14e-5 (1.0e-4) 1.26e-6 (-5.09e-6) GJR 5.79e-6 (-8.13e-5) 2.09e-5 (1.0e-4) 1.45e-6 (-5.09e-6) CCC 4.16e-6 (-8.13e-5) 2.05e-5 (1.0e-4) 1.25e-6 (-5.09e-6) DCC 4.55e-6 (-8.13e-5) 2.31e-5 (1.0e-4) 1.34e-6 (-5.09e-6) Post-GFC

OLS 2.45e-5 (1.3e-4) -1.11e-5 (-1.0e-4) 1.99e-5 (5.61e-5) GJR 1.91e-5 (1.3e-4) -8.02e-6 (-1.0e-4) 1.88e-5 (5.61e-5) CCC 2.55e-5 (1.3e-4) -1.15e-5 (-1.0e-4) 1.98e-5 (5.61e-5) DCC 1.92e-5 (1.3e-4) -6.26e-6 (-1.0e-4) 1.84e-5 (5.61e-5)

Note: The mean returns of the unhedged portfolios are given in brackets.

5.4 Different GARCH lags comparison

In section 5.1 is observed that some estimates show a sum of the ARCH and GARCH coef-ficients larger than one, which contradicts the ADF tests. These coefcoef-ficients can be altered by adding more lags to the GARCH models. The decision of the amount of lags to be added to the model is done by examining the AIC values of the models. If a model with a certain

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amount of lags has a lower AIC value than the same model with different amount of lags, it is assumed to describe the data set better. There are seven results which have sums greater than one in the post-GFC period and the period containing all observations. Surprisingly this is not the case for the pre-GFC period. These results are indicated in tables 5, 6, A12 and A13 with a star. All of these results are re-estimated by using higher lags and the best model is decided upon by examining the AIC values. The best models with the optimal amount of lags had sums which did not exceed one. The new models with optimal amount of lags are used to calculate the new HE and the new mean return of the optimal portfolio. These results can be observed in table 11. In this table also a column is added in which the improvements in performance in comparison to the other models is described. That is, if the new model performed better than the old model. If the new model performed worse than this is indicated in the table by ’N/A’. If the new model with different lags does indeed perform better, but does not exceed any of the other OLS or GARCH models, this is indicated by ’none’.

In total there are four CCC models with more lags than the original CCC GARCH (1,1) model. As performance is measured in two different ways, this results in eight new performance indices. In five of these cases the new model performs worse than the old model. In two other cases there is only little improvement, in which the CCC model does not surpass any of the OLS, GJR or DCC GARCH(1,1) models. Only in one of the eight cases does the new CCC model exceed another model. These numbers indicate that using a different amount of lags makes the CCC model estimate worse than it already did, even if the Akaike information criterion indicates that the new CCC model should describe the data set better than the old CCC model. The GARCH(1,1) model seems to be optimal for CCC estimates and should not be altered, which is in line with the findings of Hansen and Lunde (2005), who compared GARCH(1,1) to many other GARCH models and argued that the GARCH(1,1) exceeds all other models for estimation purposes.

The new DCC model on the other hand does show significant benefits over the old DCC model. In all cases the DCC model improves over the GARCH(1,1) model. By using the new models, the DCC model is superior to the GJR model in every case. In the post-GFC estimations for the EUR exchange rate, the DCC model even surpasses the CCC and OLS models. It is clear that a different amount of lags positively affects the DCC model, however there are not many cases in which the DCC model surpasses the OLS and CCC models,

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indicating that the OLS model still performs best in most cases, with exceptions as discussed in earlier sections. Also remarkable is that in every case the amount of GARCH lags equals three, while the ARCH lags differ per currency and period. The amount of GARCH lags could be a pattern, but more research is needed on this topic.

The table does not include any CHF exchange rates as this exchange rate already had sums smaller than one. It also does not include any of the pre-GFC GJR, CCC or DCC improvements, even though negative coefficients were observed for the GARCH coefficients. This is the case, because when new lags were added to this model, the coefficients never became positive and the AIC values suggested that the added lags only made the model worse. In some cases using a specific amount of lags did not solve as the loglikelihood function was not concave and was not able to find an optimum. Nelson and Cao (1992) on their research on negative GARCH coefficients suggest that the negativity could be due to misspecification or sampling errors, but do not necessarily have to. To be sure a different programming code was used to estimate the models with negative GARCH coefficients. This did not create a difference in the negativity of the coefficients. Another explabation is the low amount of observations. The amount of observations for the pre-GFC is 650. Ng and Lam (2006) in their paper titled "How does sample size affect GARCH models" strongly argue that a minimum of 1000 observations is required to acquire reliable estimations. To be completely certain that the amount of observations caused the problem of negative coefficients, a larger period spanning from 1 January 1999 until 12 June 2018 was used and the models were recalculated. Indeed, all of the negative coefficients turned positive as suggested by Ng and Lam (2006). This is a point which can be used in further research on this subject by for example increasing the amount of observations for the pre-GFC period, in order to get more reliable results.

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Table 11: Improvements lags

Old HE New HE Improves on Old Mean New Mean Improves on All Obs.

BP CCC (2,3) 67.6055 67.6025 N/A 1.81e-5 1.84e-5 None

EUR CCC (3,3) 65.0785 65.0754 N/A 1.52e-5 1.50e-5 N/A

EUR DCC (1,3) 64.9838 65.0775 GJR 1.42e-5 1.51e-5 GJR

Post-GFC

BP CCC (2,3) 66.5165 66.5226 None 2.55e-5 2.46e-5 N/A

BP DCC (3,3) 66.3537 66.4991 None 1.92e-5 2.25e-5 None

EUR CCC (2,3) 64.5578 64.5358 N/A 1.98e-5 2.05e-5 OLS

EUR DCC (2,3) 64.4531 64.5578 GJR and CCC 1.84e-5 2.00e-5 OLS, GJR and CCC

Note: In brackets the amount of lags of the new model are given. ’N/A’ implies that the new model performs worse than the old model. ’None’ implies that the new model does perform better, but does not surpass other (OLS, GJR or CCC/DCC) models as the improvement was not significant enough, or it was already superior to the other models.

6 Conclusion

This thesis sheds light on the best model to estimate the optimal hedge ratio when hedging against daily volatility of European currencies relative to the US Dollar with the help of futures. The empirical findings are of importance to hedgers that take future positions for currencies in both low risk and high risk periods. In this research, the OLS model is compared to three GARCH models, namely the GJR, CCC and DCC models. The models are compared by their variance reduction capabilities and the means of the optimal portfolios that are created by the estimated optimal hedge ratio, in which the methods both portray some similarities and differences in the ranking of the best models. For the CCC and DCC models, higher lags are also taken into consideration for a broader research on the superiority of the models.

The empirical results show that there are some differences with the findings in the Theor-etical Framework. The DCC model was suggested to be the best model, however the results show otherwise, as the DCC model shows the worst performance. This is especially the case

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when comparing the variance reduction capabilities of the models. The OLS ranks the best, followed by the CCC, GJR and DCC models. When observing the mean returns of the port-folios the performance of the DCC model increases, as it turns into the best model to use in times of low volatility, such as the pre-GFC period. In the global crisis it still turns out to be the worst performing model. Again, the ranking of the other models does not change as the OLS still outperforms the CCC model, and the CCC model still outperforms the GJR model. Furthermore, when different amount of lags are used in the assessment of the CCC and DCC models, one can observe that this differentiation does not benefit the CCC model, but does significantly improve the performance of the DCC model. All of the new DCC models suggest a GARCH lag of three to be optimal, which is a topic for further research as this could be a pattern. Finally, the results of the GJR model suggest that asymmetric shocks are not empirically crucial, as the coefficient of the asymmetric element of the GJR model are not significant.

To conclude, it seems plausible to assume that the ranking of the OLS, CCC and GJR models is fixed in that chronological order, as these rankings are confirmed over most currencies and periods that are analyzed in this research. Therefore, the CCC and GJR models are not advised to use for estimation purposes, as there are only a negligible amount of occasions in which these models outperform the OLS model. The ranking of the DCC model differs greatly. If the investor is interested in reducing the variance, the OLS model is advised. If however the investor is interested in the highest mean return of his or her portfolio, than the DCC model is advised in times of low volatility and the OLS model is advised in times of high volatility, such as an economic crisis. If the DCC model is being utilized, than it is convenient to use higher lags as higher lags result in greater returns of the portfolios. The amount of lags per period and currency does differ though, which again is a topic for further research. Furthermore, in future research more in-depth analysis of the effect of the global financial crisis can be performed by using a larger amount of observations for the period before the global financial crisis.

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Appendix

Table A1: Descriptive Statistics Q1 2005 - Q3 2007

USD/BP USD/CHF USD/EUR USD/BP USD/CHF USD/EUR Mean -8.13e-5 0.00010 -5.09e-6 -9.62e-5 9.49e-5 -7.58e-5 Maximum 0.01484 0.01659 0.01608 0.01327 0.01981 0.02259 Minimum -0.0201 -0.0188 -0.0196 -0.0169 -0.0211 -0.0188 Std. Dev. 0.00494 0.00551 0.00484 0.00473 0.00540 0.00476 Skewness -0.1016 -0.2852 -0.1905 -0.2011 -0.4081 0.0955 Kurtosis 3.42337 3.50206 3.76031 3.19632 3.93204 4.33302 Jarque-Bera 5.96431*** 15.6173* 19.55772* 5.41864*** 41.5080* 49.0382*

Note: A significance on the 1% level is indicated by * and on the 10% level is indicated by ***.

Table A2: Descriptive Statistics Q3 2007 - Q1 2015

USD/BP USD/CHF USD/EUR USD/BP USD/CHF USD/EUR Mean 0.00013 -0.0001 5.61e-5 0.00012 -0.0001 4.37e-5 Maximum 0.039187 0.08475 0.03845 0.05128 0.08731 0.03065 Minimum -0.0447 -0.0545 -0.0462 -0.0346 -0.0552 -0.0309 Std. Dev. 0.00640 0.00716 0.00650 0.00615 0.00714 0.00630 Skewness 0.08286 0.69408 -0.1610 0.53628 0.78440 0.08159 Kurtosis 8.28665 17.0594 6.70009 8.04436 18.7028 5.05509 Jarque-Bera 2283.55* 16291.7* 1125.96* 2170.89* 20327.9* 346.907*

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Figure A3: Time series of volatility spot and futures prices

(a) Volatilty Spot BP (b) Volatility Spot CHF (c) Volatilty Spot EUR

(a) Volatilty Future BP (b) Volatilty Future CHF (c) Volatilty Future EUR

Table A4: Dickey-Fuller tests on stationarity of spot rates

Test Statistic all obs. -45.431 (0.00) -43.035 (0.00) -43.946 (0.00) Test Statistic Pre-GFC -22.790 (0.00) -23.011 (0.00) -22.719 (0.00) Test Statistic Post-GFC -39.355 (0.00) -36.795 (0.00) -37.832 (0.00)

5% Critical value -2.860 -2.860 -2.860

Table A5: Dickey-Fuller tests on stationarity of futures rates

Test Statistic all obs. -43.082 (0.00) -42.659 (0.00) -42.592 (0.00) Test Statistic Pre-GFC -21.989 (0.00) -21.742 (0.00) -21.505 (0.00) Test Statistic Post-GFC -37.918 (0.00) -36.813 (0.00) -36.819 (0.00)

5% Critical value -2.860 -2.860 -2.860

Note: The ADF tests have been performed without the inclusion of a trend term, as the returns are fluctuating around zero. The P-values are given next to the statistics. It can be observed that all statistics reject non-stationarity on the 1% P-value level.

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Table A6: OLS estimates Q1 2005 - Q3 2007

BP Constant 4.44e-06 (0.97) .0001004 0.7320 -5904.149 BP Future .8918233 (0.00) .0212134

CHF Constant .0000214 (0.86) .0001236 0.6733 -5633.601 CHF Future .8364368 (0.00) .0229050

Eur Constant 1.26e-06 (0.99) .0001077 0.6792 -5812.957 Eur Future .8370217 (0.00) .0226153

Table A7: GJR estimates Q1 2005 - Q3 2007

BP Spot/Fut. .9059283 (0.00) .3572679 (0.00) .339188 (0.00) -.0557294 (0.66) -5943.755 CHF Spot/Fut. .8418083 (0.00) .2629617 (0.01) -.1594549 (0.31) .0120201 (0.92) -5661.034 Eur Spot/Fut. .8624892 (0.00) .4380701 (0.00) .0905676 (0.36) -.0663972 (0.62) -5877.698

Table A8: CCC estimates Q1 2005 - Q3 2007

BP Spot -.0000753 (0.70) .0128378 (0.64) .0832553 (0.92) .8506808 (0.00) -11001.23 BP Fut. -.0000829 (0.65) .0185745 (0.26) -1.060255 (0.00)

CHF Spot .0000861 (0.69) -.0338178 (0.20) -.7726368 (0.05) .8288006 (0.00) -10559.48 CHF Fut. .0000855 (0.69) -.0142121 (0.50) .4655477 (0.59)

Eur Spot -3.54e-06 (0.99) -.0451855 (0.02) .4831349 (0.34) .8269864 (0.00) -10902.46 Eur Fut. -2.49e-06 (0.90) -.0183551 (0.14) .1574568 (0.88)

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Table A9: DCC estimates Q1 2005 - Q3 2007

BP Spot -.000027 (0.87) .0991101 (0.02) .2975404 (0.33) .8711221 (0.00) .1282201 (0.00) .4178256 (0.00) -11026.26 BP Fut. -.0000348 (0.85) .0861838 (0.03) .1790096 (0.54) CHF Spot .0001282 (0.55) .0030598 (0.89) .3236225 (0.52) .821615 (0.00) .1315847 (0.00) .0591358 (0.72) -10582.61 CHF Fut. .0001766 (0.41) .0123899 (0.37) -.948924 (0.00) Eur Spot .0000387 (0.83) .0043312 (0.85) -.7499275 (0.18) .8597399 (0.00) .2044167 (0.00) .2083718 (0.09) -10961.89 Eur Fut. .0000528 (0.78) .0242447 (0.27) .2789877 (0.63)

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Table A10: OLS estimates Q3 2007 - Q1 2015

BP Constant .0000245 (0.77) .0000837 0.6652 -16371.65 BP Future .8497463 (0.00) .0136262 CHF Constant -.0000111 (0.91) .0000956 0.6507 -15851.06 CHF Future .8093027 (0.00) .0134032 Eur Constant .0000199 (0.82) .0000875 0.6456 -16199.24 Eur Future .8295478 (0.00) .013894

Table A11: GJR estimates Q3 2007 - Q1 2015

BP Spot/Fut. .8937832 (0.00) .4939376 (0.00) .3831093 (0.00) .059788 (0.37) -16861.55 CHF Spot/Fut. .8356464 (0.00) .5181986 (0.00) .024154 (0.46) .023581 (0.77) -16225.84 Eur Spot/Fut. .8543661 (0.00) .4857209 (0.00) .2706423 (0.00) .0672032 (0.36) -16602.2

Table A12: CCC estimates Q3 2007 - Q1 2015

BP Spot* .0000389 (0.76) .0791628 (0.00) 1.376525 (0.00) .8188966 (0.00) -31010.73 BP Fut.* .0000638 (0.61) .1082231 (0.00) 1.132149 (0.00) CHF Spot -.0000344 (0.82) .1508205 (0.00) .7180041 (0.00) .8100595 (0.00) -29839.55 CHF Fut. -.0000511 (0.74) .0849963 (0.00) .8484016 (0.00) Eur Spot* .0000292 (0.83) .0723971 (0.00) 1.360009 (0.00) .808895 (0.00) -30612.89 Eur Fut.* .0000167 (0.90) .047607 (0.00) 1.680523 (0.00)

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Table A13: DCC estimates Q3 2007 - Q1 2015

BP Spot* .0000276 (0.83) .2123043 (0.00) .8138555 (0.00) .8666437 (0.00) .1438858 (0.00) .3722993 (0.00) -31196.45 BP Fut.* .0000402 (0.74) .2412901 (0.00) .7641495 (0.00) CHF Spot -.0001113 (0.46) .2269927 (0.00) .5504239 (0.00) .8479699 (0.00) .1212649 (0.00) .2066724 (0.13) -29947.71 CHF Fut. -.0001044 (0.49) .1533700 (0.00) .6728664 (0.00) Eur Spot* -.0000349 (0.80) .1315334 (0.00) 1.079185 (0.00) .8487523 (0.00) .1493071 (0.00) .1169795 (0.18) -30747.92 Eur Fut.* -.0000453 (0.74) .1105051 (0.00) 1.138547 (0.00)

Currency hedging: an evaluation of econometric models for estimating the optimal hedge ratio in European markets