Currency hedging in emerging markets : multivariate GARCH and copula modelling

Pavel Bugneac 10864725

Amsterdam Business School MSc Quantitative Finance

Master Thesis

Supervisor: Ms. Derya Güler June 2018

Currency hedging in emerging markets: multivariate GARCH and

copula modelling

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Statement of originality

This document is written by Pavel Bugneac who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is 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 solely responsible for the supervision of completion of the work, not for the contents. (University of Amsterdam, 2018)

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Acknowledgements

First of all, I would like to thank my supervisor Ms Derya Güler for her continuous encouragements and timely feedback. She was a responsible and caring supervisor who deserves all the praise she can get.

I would also like to thank my manager and colleagues at BNP Paribas for being so understanding and allowing me time off work to focus on this thesis. Also to my parents, girlfriend and friends who have continuously supported me.

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Abstract

Emerging markets stocks have been increasingly traded in the recent years for their higher expected returns. However, these returns come at a higher risk, one of which is the currency risk. Currency hedging has been extensively covered in the literature, however, little focus has been paid to emerging markets. These have higher hedging costs and the decision to hedge is still controversial. In this thesis we analyze whether hedging emerging markets stocks using currency futures leads to lower portfolio risk. Since there is not a general consensus on the best method for estimating the optimal hedge ratio, we employ 6 sophisticated methods such as: Normal distribution GARCH DCC, Normal distribution GJR-GARCH DCC, T-student GJR-GARCH DCC, Gaussian copula, T-T-student copula and the Clayton copula. We additionally look into highly diverse portfolios since these also have been overlooked by numerous authors. We use a 12-assets portfolio comprised of 11 emerging markets stock indices and the US stock index. The results of the analysis prove that currency hedging is beneficial for emerging market stocks since, as a consequence, portfolio risk is significantly reduced. Copula methodology generally leads to higher country-wise risk reduction. However, when the diversification effect is taken into account, by using highly diverse portfolios, all of the hedge ratio calculation methods lead to similar results, except for the symmetric normal distribution GARCH DCC method. The general finding is that investors should unequivocally hedge their foreign currency risk exposure for emerging market stocks.

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Contents

1. Introduction ... 5

2. Literature Review ... 8

2.1. Overview ... 8

2.2. Developed vs Emerging markets ... 8

2.3. Hedge ratio estimation ... 10

2.3.1. GARCH methodology ... 11

2.3.2. Dynamic Conditional Correlation model ... 12

2.3.3. Copula methodology ... 14 3. Data ... 17 3.1. Descriptive statistics ... 18 4. Methodology ... 21 4.1. Volatility modelling ... 21 4.2. Portfolio composition ... 23 4.3. Multivariate GARCH DCC ... 25 4.4. Copula functions ... 27 4.4.1. Gaussian Copula ... 27 4.4.2. T-student Copula... 30 4.4.3. Clayton Copula ... 31 4.5. Out-of-sample analysis ... 33 4.6. Hypothesis testing ... 34 5. Empirical results ... 36 5.1. In-sample analysis ... 36 5.1.1. GARCH DCC ... 36 5.1.2. Copula GARCH-DCC ... 43

5.1.3. Methodology performance comparison ... 48

5.2. Out-of-sample analysis ... 49

5.3. Discussion & Comparability ... 53

6. Conclusion ... 56

6.1. Concluding remarks ... 56

6.2. Limitations & Further research ... 58

7. Reference list ... 60

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1. Introduction

Investors are continuously seeking ways of increasing their returns with either a relatively low risk or no risk at all, something known as the “free lunch”. For this reason, in the recent years investors have regularly diversified their portfolios by including international stocks and securities, and in particular emerging market (EM) stocks. The main reason being that EM stock returns have been leading in the recent years, even surpassing U.S. stock returns, and are expected to continue to do so (Constable, 2017; Fisher, 2017). However, EM stock returns are no “free lunch”, for these have an even higher risk attached to them than the developed market stocks, one of the main risks being currency risk. In fact, even if an investor gains in EM stock returns he may still arrive to a loss due to unfavourable foreign currency value movements. Additionally, it has been proven that for EM currency hedging may come at a greater cost, as a consequence of factors such as lower liquidity, underdeveloped derivatives markets, higher volatility, political uncertainty, etc. (Aggarwal & Demaskey, 1997; Crabb, 2004; Atwill, 2015; Kim, 2012). Considering that the existing literature mostly focuses on developed markets, it raises the question whether currency risk should be hedged at all for EM especially when accounting for the diversification effect for large portfolios. Furthermore, there is not currently a widely accepted method for obtaining the hedge ratio, there is still debate on the optimal technique. Therefore, this thesis attempts to bring new insights on whether currency risk should be hedged for EM stocks as well as for the most adequate technique to obtain the optimal hedge ratio. This leads to the aim and research question of this thesis: Is currency hedging beneficial for EM stocks? Which quantitative method leads to the superior hedging performance?

Currency hedging involves the use of currency futures where investors short a specific amount of futures per share, known as the hedge ratio, in order to reduce portfolio variance. The accurate measurement of the hedge ratio is crucial for every investor due to its effect on portfolio returns. It is therefore important to understand the dynamics of currency hedging in EM which are expected to be riskier than developed markets. Conventional hedging strategies imply minimizing the portfolio variance by estimating a constant hedge ratio given by the Ordinary Least Squares method (OLS), an approach also known as the mean-variance optimization. However, throughout the published literature, this method has been proven to be inefficient and inaccurate due to its static nature (Lai, Chen & Gerlach, 2009; Lee, 2009; Chang, González-Serrano & Jimenez-Martin, 2013; Algaba, Boudt & Vanduffel, 2018; Yang

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& Allen, 2005; Park & Switzer, 1995). Instead, we see that returns have a heteroskedastic nature with dynamic variances and correlations, which if unaccounted for, can hinder hedging performance.

For the reasons above, in this thesis we use a more sophisticated methodology to obtain the optimal hedge ratios. This involves the use of Generalized Autoregressive Conditional Heteroskedastic (GARCH) conditional variances which have been proven to be effective in capturing the dynamic volatility of financial asset returns. However, GARCH volatilities are only the first step in obtaining the optimal hedge ratio which is affected by the correlation between stocks and futures. This correlation needs to be estimated through multivariate models that can account for the dynamic and heteroskedastic nature of financial asset returns.

An effective model that can account for this dynamic nature in correlations is the Dynamic Conditional Correlation (DCC) model first proposed by Engle (2002). This model has been adapted and modified throughout literature, yet its performance is still optimal and above more simplistic methods such as OLS (Cheng et al., 2013; Algaba et al., 2018; Lien, Tse & Tsui, 2009). Additionally as noted by Algaba et al. (2018) and Engle (2002), the DCC model is less computationally costly when applied to a portfolio comprised of numerous assets. This is an important factor for investors as these costs might overcome hedging benefits. Moreover, Alexander, Prokopczuk and Sumawong (2013), state that other more complicated models, such as the BEKK model, might suffer from instability in parameter estimates over rolling estimation windows which may lead to unrealistically high hedge ratios. This further supports our choice of the DCC model.

However, the DCC model has its limitations due to the fact that it assumes a bivariate normal distribution which does not hold empirically. One of the main reason for this empirical failure is the existence of tail dependences that remain unaccounted for when assuming a multivariate normal distribution (Y. S. Lai, 2018; Hsu, Tseng & Wang, 2008; J. Y. Lai, 2012; Lee, 2009; Park & Jei, 2010). Simple GARCH-DCC models may therefore be inappropriate and inefficient. For this reason we will also make use of copula functions that will allow us to correctly model the bivariate distribution of stocks and futures.

Generally there is a consensus in the literature that copula methods outperform simpler GARCH-based methods. However, all of the authors that employ copula methods, focus on developed markets with small portfolios of up to 5 assets. Since in our approach we use a very diverse 12-asset portfolio made up of EM stock indices, GARCH DCC models might still bring

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comparable results to the copulas. Consequently, in this thesis we employ a wide spectrum of GARCH-DCC and copula methods for an in-sample analysis which will increase the robustness of our results.

Our choice of models for the in-sample analysis comprises the normal distribution GARCH DCC, the normal distribution GJR-GARCH DCC, the T-student GJR-GARCH DCC and copulas such as the Gaussian, T-student and Clayton copulas. After analysing the in-sample performance, we choose the optimal methodology in order to perform an out-of-sample analysis which will further increase the robustness of our results. The data used in this thesis covers the period between January of 2009 and April of 2018. Our in-sample analysis uses the full sample period to obtain the hedge ratio on each day. Meanwhile the out-of-sample analysis uses one day ahead forecasts to obtain the hedge ratio and then analyses its effectiveness against the realized returns of the last 600 days of the full sample, until April of 2018. The details of this procedure are explained in the methodology section.

All of the published literature that employs similar methodology only focuses on developed markets and only use a few methods. This thesis employs a vast spectrum of sophisticated methods and focuses on emerging markets. This assures the robustness of the results since these might be method-sensitive. The contribution of this thesis is considerable since the approach partaken here has not been applied to the emerging markets so far. Additionally, the authors that use quantitative methods do not analyse the benefits of diversification. By allowing for several assets in the portfolio, we account for the diversification effect which may render hedging ineffective. Therefore our thesis bridges a gap between the existing literatures, as authors that employ sophisticated GARCH-based models do not consider EM stock nor highly diverse portfolios while on the other hand, authors that analyse diverse portfolio assume a constant hedge ratio.

This thesis is structured as follows. In the next section we overview the related literature in order to help us compare and situate our work, as well as explain our empirical results. Section 3 contains data description and section 4 describes our extensive methodology of estimating the optimal hedge ratio as well as testing its effectiveness. Next we present the results of the in-sample and out-of-sample analyses, and compare them with the existing literature. Finally we conclude and summarize our findings, state limitations and propose further research questions.

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2. Literature Review

We start this section by providing an overview of the currency hedging theoretical framework. As explained by Chang et al. (2013), due to market internationalization and liberalization, foreign currency financial assets are increasingly being traded and markets have become more competitive and volatile. This increases the need for hedging portfolios against currency risk exposure, especially in the case EM stocks which are associated with higher risk. Currency hedging is the practice of minimizing the portfolio exposure to foreign currency risk. Practically it implies taking a short position in foreign currency futures or forwards for every foreign stock in the portfolio. The ratio of currency futures per unit of stock is known as the hedge ratio.

For developed markets the general consensus is that hedging is necessary and beneficial. On the other hand for emerging markets there is still debate on the decision of hedging currency risk exposure, especially in the case when the portfolios are diverse which decreases country-specific risk. For this reason it is insightful to analyse the case of emerging markets from a different perspective and by using a more sophisticated methodology.

Finally, obtaining the optimal hedge ratio has in itself been a controversial issue with plentiful of methodology propositions due to the fact that an accurate estimation of the hedge ratio is what determines hedging effectiveness. Moreover, most of these intricate models have only been applied to developed markets stocks and their efficiency for EM stocks is relatively unknown. This is why, as earlier explained, we employ a wide variety of hedge ratio calculation methods in order to obtain robust results that may help with the open debates in the published literature.

2.2. Developed vs Emerging markets

There is extensive literature on currency hedging on international portfolios with multiple ingenious approaches. Most of this literature focuses on the developed economies and finds that on average hedging results in lower risk. One such article, by Glen and Jorion (1993), examines the benefits from currency hedging for speculative and risk minimization motives for international portfolios. They use the mean-variance analysis to obtain the optimal hedge ratio which consists in maximizing the returns per unit or risk of a portfolio and investing into the final risky portfolio depending on an investor’s risk aversion. They impose several other

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restrictions such as no short selling and find statistically significant improvements in the performance of these portfolios. In this thesis we also use the Sharpe ratio, however, we do not use it to obtain the optimal hedge ratio, but rather to measure hedging performance.

Similarly, Campbell et al. (2010) analyse global currency hedging and use data from 7 developed economies to run regressions of portfolio returns on a constant and the vector of currency excess returns. Their conclusion is that the desired exposure to a certain currency depends on the correlation of that currency to the stock market. Their overall finding is that investors prefer to not fully hedge and have a certain amount of currency risk exposure depending on the correlation of currencies to the stock market and each other.

The benefits of hedging are again supported by Perold and Schulman (1988) stating that investors should hedge against currency risk since hedging currency exposure greatly enhances the diversification potential of foreign investors. Meanwhile, Schmittmann (2010) highlights the importance of risk hedging both for short term and long term investors.

Numerous other authors also focus on developed markets when attempting to obtain the optimal hedge ratio using more sophisticated models that stem from the basic GARCH model (Chang et al., 2012; Park & Switzer, 1995; Yang & Allen, 2005; Lien et al., 2002; Y. S. Lai, 2018). They obtain mixed results regarding each of the techniques but the main conclusion is still the same which is that currency hedging is indeed optimal for international investors.

As can be seen, there are numerous papers covering currency hedging for developed markets. The general consensus is that currency hedging is beneficial although the level of the optimal hedge ratio as well as the superior method for obtaining it is still a debate. Meanwhile, as explained by Aggarwal and Demaskey (1997), very few authors focus on solely the emerging markets. Although theirs is a relatively old paper, to this day the literature on EM currency risk hedging is not extensive and still open to debate.

The existing EM currency hedging literature is still open to debate. Mc Nally and Murray (2010) explain that EM have a higher growth and their currencies are more volatile, being subject to sudden devaluations. Using a mean-variance analysis to obtain the optimal hedge ratio, they find that this ratio depends on the expected returns of emerging markets stocks. When these expected returns are zero investors should fully hedge while for positive returns investments should be left unhedged. Similarly, Kim (2012) states that the cost of hedging an EM currency is very high which is why investors are reluctant to hedge EM

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currency risk. He compares the Sharpe ratios of the three portfolios: unhedged, fully hedged and optimally hedged. In order to obtain the hedge ratio for the optimally hedged portfolio he maximizes the ex-post Sharpe ratio, meaning that he uses past information and does not perform forecasts. He concludes that the performance of non-hedged portfolios is not statistically superior to otherwise hedged portfolios. Contrary to Kim, we allow for time-varying hedge ratios which is expected to lead to more accurate results.

A more recent paper by Atwill (2015), examines the implementation issues of currency hedging for emerging markets stocks. This author confirms the difficulties of currency hedging for EM expressed through higher costs or lower accessibility. He adds that the benefits of hedging dissipate in the long run. Lastly, he notes that the decision to hedge is a “market call” on the direction of the emerging markets due to the strong correlation of the hedging benefits with market returns. The conclusion regarding the effect of currency hedging in the long and short runs contradicts the results previously mentioned by Schmittmann (2010).

Lai et al. (2009) employ copula and DCC models, which will be discussed in the next subsection, to Asian emerging markets and find that hedging leads to lower portfolio risk. Contrary to this thesis, they only examine 5 Asian countries and do not look at the full portfolio but rather at each individual country when analysing the hedging performance. Our sample comprises 11 different EM countries from different continents and our in our analysis we look at the performance of the full portfolio.

Worth mentioning is the work by Aggarwal and Demaskey (1997), which focuses on cross hedging for Asian emerging markets. Since this article was published, the Asian derivatives market have become more developed which makes it possible for direct hedging with the use of currency derivatives. The results are still interesting since the authors find that hedging is beneficial as measured by the Sharpe ratio. As can be seen, the Sharpe ratio is consistently used throughout our literature to measure hedging performance as it is widely accepted as a risk/return measure. This is one of the reason for our choice of this measure to quantify the hedging performance.

2.3. Hedge ratio estimation

Thus it is clear that there is still some debate whether hedging is beneficial in the case of emerging markets. However, obtaining the optimal hedge ratio has unequivocally been the focus of numerous authors. The vast choice of methodologies have sparked a debate throughout

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the published literature on the most effective hedge ratio calculation method. As explained by authors such as Lai et al. (2009), Chang et al. (2013) and Yang and Allen (2005), the OLS method has been the traditional and widely employed method in the past for obtaining the optimal hedge ratio. This method consists in minimizing the portfolio variance and its functional equation is a linear regression between stock and futures returns whose slope corresponds to the hedge ratio. As a result this hedge ratio is assumed to be constant over time. Another constant ratio was proposed by Black (1989, 1990), a universal hedging formula which does not depend on correlations between currencies and is constant over time.

2.3.1. GARCH methodology

The idea of constant hedge ratio over time has been empirically rejected throughout literature. Park and Switzer (1995) explained that if the joint distribution of stock index and futures prices is non-constant then estimating a constant hedge ratio is inappropriate. Similarly, Lai et al. (2009) indicate that the OLS method contradicts the well-known time-varying nature of the volatility of asset returns. This is again corroborated by Chang et al. (2013) and Yang and Allen (2005) with their argument that the simple OLS method is inadequate as it suffers from the problem of serial correlation and heteroscedasticity encountered in spot and futures price series. Furthermore, Christoffersen (2012) points out that, empirically, correlations between assets increases during financial crisis increasing the need for a dynamic correlation model and subsequently a dynamic hedge ratio.

What is then the solution to this heteroskedastic and time-varying nature in the financial asset volatilities? A widely accepted solution is the generalization of the Autoregressive Conditional Heteroskedastic (ARCH) model first proposed by Engle in 1982. Bollerslev (1986) proposed this generalized form known as GARCH, which models the conditional volatility of asset returns accounting for serial correlation and heteroskedasticity found in financial assets. This model expresses the volatility at each point in time in terms of its own past values and past standardizes shocks to returns, which implies that it is conditional on past information which is why its resulting volatility is known as the conditional volatility.

Due to its highly flexible nature, this model allows for very diverse extensions. A useful extension, employed in our analysis, which accounts for the leverage effect present in financial assets is known as the GJR-GARCH. The leverage effect implies that a negative return increases variance by more than a positive return of the same magnitude, as explained by Christoffersen (2012). This can be explained by the fact that after a negative return the stock

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and firm values drop which makes the firm take on more leverage which subsequently increases stock returns’ risk.

Generally, the GARCH model has proven to be effective in numerous applications and in particular in establishing the optimal hedge ratio. This effectiveness stems from the fact that volatilities of asset returns are time-varying which is correctly captured by this model. For these reasons, multiple authors employ and expand this model in diverse applications. For instance, Park and Switzer (1995) employ a bivariate GARCH model to obtain the hedge ratio and find its performance to excel over its counterpart, the OLS method, even after accounting for transaction costs. Nevertheless, both methods, bivariate GARCH and OLS, lead to variance reduction although in different magnitudes. Similarly, Tse and Tsui (2002) adapt the GARCH model to the multivariate case and compare its performance with a constant correlation multivariate GARCH (MGARCH) model and with the BEKK model. They find that their extension leads to similar results as the BEKK model and surpasses the constant correlation MGARCH model. This again highlights the importance of a dynamic correlation.

Another important analysis is the one performed by Yang and Allen (2005), who employ several multivariate GARCH models, such as the bivariate vector autoregression (VAC), vector error-correction (VECM) and diagonal VEC (DVEC) models. These extensions to GARCH are employed to allow for dynamic correlations, due to the fact that a common finding in financial markets is that the constant correlation hypothesis does not hold. By looking at the variance reduction ratio (VRR) they find that the most complex strategies perform best with a VRR as high as 85.29% which concludes that GARCH models are superior to their simpler counterparts. Their out-of-sample analysis delivers even more support to their GARCH methods for the risk is even lower. However, their results are slightly inconclusive due to the fact that the OLS method brings slightly higher returns and the GARCH method brings lower risk. For this reason, in this thesis we also look at the Sharpe ratio as a measure of hedging performance which takes into account the returns per unit of risk.

2.3.2. Dynamic Conditional Correlation model

Among the numerous applications, the GARCH variance model is often used in conjunction with other models that allow for a dynamic correlation. One such model was proposed again by Engle (2002) known as the Dynamic Conditional Correlation (DCC) model which is merely an extension of the Constant Conditional Correlation (CCC) model by Bollerslev. The DCC model is similar to the GARCH formulation but in terms of correlations,

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which means that at each point in time the correlation depends on its own past values and past standardized shocks to returns. Similarly to GARCH, the correlation is based on all the available past information which results in a conditional correlation. Moreover, it has the advantage that it allows for correlations to vary over time which is crucial for a dynamic hedge ratio as we have seen to be backed by numerous academics.

An important argument by Engle (2002) is that most of the literature that uses the GARCH model never uses more than 5 assets in their analysis, which is currently still the case. Realistic portfolios, however, are comprised of a multitude of assets that allow for diversification which is why we also consider a portfolio containing 12 assets. Most models would require the estimation of a vast number of parameters which is computationally costly. With the DCC model, only two coefficients need to be estimated regardless of how complex the portfolio is, explains Engle, which clearly lessens the computational burden. Additionally, Alexander et al. (2013) note that other complex GARCH models like the bivariate GARCH may suffer from instability in parameter estimates over rolling estimation windows which in turn may lead to upwards biased hedge ratios. The use of such rolling windows in our methodology coupled with our diverse portfolio justifies the usage of the DCC model.

Several authors find the DCC model to outperform the OLS and simple GARCH methods. Chang et al. (2013) provide examples in the available literature regarding the effectiveness of GARCH and DCC hedge ratios over the standard OLS method. In their analysis, they compare several methods such as CCC, vectorial autoregressive moving average asymmetric GARCH (VARMA-AGARCH), DCC and BEKK. They find that dynamic asymmetry does not play an important role in hedge determination as a result of comparing the CCC and VARMA-AGARCH models, with the latter accounting for asymmetries in correlations. This finding is in line with other published works on this topic suggesting a symmetric DCC model might be appropriate for hedging purposes. Furthermore, similarly to our approach of using daily portfolio rebalancing they confirm that models such as DCC or BEKK perform better and lead to more optimal hedging than the rest of the models they have considered.

A very recent paper, yet to be published, by Algaba et al. (2018) similarly compares the DCC model together with a new proposed one, Variance Implied Conditional Correlation, against the traditional OLS model. Their model leads to similar results as the DCC model, the latter outperforming the OLS model and leading to a Variance Reduction Ratio (VRR) between

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6.67% and 38.36%. One important finding from their analysis is that an OLS hedge ratio on an expanding estimation window may lead to worse results than no hedging at all. This highlights the importance of dynamic correlations, especially for our approach in which we make use of expanding estimation windows.

Yet another similar approach to our own is the analysis performed by Park and Jei (2010) who similarly to us fit a T-student GJR-GARCH DCC model. They argue their choice of the GJR-GARCH specification by the fact that the leverage effect is consistently present in financial asset returns. They confirm that asymmetric specifications reduce portfolio risk especially for short horizons such as daily rebalancing or one-day ahead forecasts. The benefits from this specification might decrease, however, as the horizons increases. For this reason their DCC specification is asymmetric. By evaluating the VRR, they again find that the DCC model outperforms the traditional OLS model, especially out-of-sample. The asymmetry effect in the DCC model, however, is not significant which is also why it is omitted in our analysis.

2.3.3. Copula methodology

One important caveat of the DCC model is that it assumes a bivariate normal distribution. This restrictive assumption does not hold empirically which is corroborated by numerous authors (Lai et al., 2009; J. Y. Lai, 2012; Y. S. Lai, 2018; Lee, 2009; Hsu et al., 2008, Park & Jei, 2010). This comes as a result of the failure of the bivariate normal distribution to account for the distribution’s higher moments and for the tail dependence between stocks and futures. The tail dependence is defined as the correlation between assets in their tails, meaning that it quantifies the correlation between assets when these reach extreme values. If left unaccounted for, when both stock and futures returns are large negative or large positive their correlation will be higher which means that hedge ratios will not be accurate. Other GARCH-based methods similarly omit these crucial features. Moreover, Y. S. Lai (2018) explains that standard GARCH models would attach too much weight to extreme observations, which makes hedge ratios inaccurate. This caveat is especially notable when daily or weekly data is used as in our case.

A possible solution to the issues mentioned above are copulas, first proposed by Sklar, which model conditional tail dependence. This methodology decomposes a multivariate distribution into univariate distributions by means of a copula function which in turn is able to describe their multivariate dependence. As explained by Christoffersen (2012) and Lai et al. (2009), one of the biggest advantages of copulas is their flexibility, since we can change the

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multivariate distribution of assets without altering their marginal distributions. In fact we need not make any assumptions regarding the returns’ marginal distributions. For these reasons, copula models are the method of choice for numerous authors in our literature review.

This thesis closely follows certain aspects of the work of Hsu et al. (2008). These academics employ a GJR skewed-t-student GARCH model together with three copulas, Gaussian, Clayton and Gumbel. Their findings indicate that the Gaussian copula has the highest log-likelihood for direct hedging while the Gumbel copulas for cross hedging. All of the copula models considered in their analysis outperform the simpler GARCH-DCC model. More particularly, for direct hedging, the Gaussian copula brings about the highest VRR while for cross hedging the Gumbel copula is the highest. This results does not persist when increasing the holding period, as the Gumbel and Clayton copulas perform even worse than OLS, while the Gaussian copula still performs satisfactorily. Their out-of-sample results, which are also based on one-day-ahead forecast, lead to the same conclusion. An important remark made by these authors is that continuous rebalancing is highly required in a cross hedge which is another reason why we assume daily rebalancing.

Similarly, Lai et al. (2009) use the Gaussian copula plus a mix of the Gumbel and Clayton copulas on EM stocks and find that their copulas outperform traditional OLS hedges both in terms of higher hedged returns and lower risk. On the other hand, Lee (2009) uses a regime-switching asymmetric copula on developed market stocks and reaches similar results as his method outperforms the traditional OLS and CCC methods. He does state that allowing for asymmetries does not necessarily lead to better hedging performance, but it does in conjunction with regime-switching. Even after accounting for transaction costs, Lee’s results remain significant.

Finally a more recent paper by Y. S. Lai (2018) follows closely the work of Hsu et al. (2008) but instead uses high-frequency data. For regular data their conclusions are the same as Hsu et al but for high-frequency data they find that risk reduction is the highest for Clayton copulas. The risk reduction obtained by his methodology ranges from 8% to 11%. His out-of-sample analysis also consists of one-day ahead forecasts and an expanding window and it reaches the same conclusion as his in-sample analysis. In this thesis we likewise closely follow some aspects of the methodology of Hsu et al., such as the univariate T-student GJR-GARCH model. Moreover we also consider a copula that has been given less attention, the T-student copula which assumes symmetrical tail dependence.

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As previously explained, all of the available literature focuses only on developed markets and only consider up to 5 assets. In our approach we use a hypothetical portfolio formed of 12 emerging markets stock indices. For this reason we use the DCC functional form both for the GARCH and copula approaches. Therefore there is one important difference between the existing literature on copulas and our approach. Contrary to other authors who directly estimate the conditional correlations from the copulas themselves, we combine copulas with the DCC model leading to a 3-step estimation process, an approach suggested by Christoffersen (2012). We therefore first fit copulas to the standardized residuals and obtain the copula shocks to which we fit the DCC model. The detailed explanation can be found in the methodology section, the main idea being that this greatly eases computational burden and allows for a simpler interpretation of correlation coefficients. Additionally, we use the Sharpe ratio to analyze the hedging effectiveness due to the possible inconclusiveness of the final state of returns and risk. The optimal weights that compose our portfolio will also depend on the model chosen for the conditional variances and correlations. Therefore this will allow us to fully assess the effectiveness of hedging currency risk, especially when the diversification effect will also play a role in such a diverse portfolio as ours. Therefore, our analysis is very extensive and novel in some aspects which will bring even more insights into the existing currency hedging literature.

In the next section, we describe our sample with the help of several descriptive statistics which will help us gain a general overview of the nature of our data.

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3. Data

In our analysis we make the simplifying assumption that investors allocate their wealth into each countries stock index, in our case provided by the MSCI index. Such an assumption is widely used and realistic since the MSCI index is seen as trustworthy by international investors and it is sometimes preferred to the national stock indexes of each country. As for the countries of choice, it is common practice to choose among the countries that compose the MSCI Emerging Markets index when deciding which EM countries to invest into. In our case the choice of countries, however, is limited by the availability of the currency futures data since for certain EM countries this data is unavailable. Ultimately we focus on the U.S. MSCI index and 11 EM countries such as: Brazil, China, the Czech Republic, Hungary, Israel, Mexico, New Zealand, Poland, Russia, South Africa and South Korea. The spectrum of the countries is chosen so that the sample contains countries from each continent. Some scholars focus on a continent in particular, however, since we want to analyze a highly diverse portfolio we choose countries from different continents. This will lead to a lower correlation between each country thus leading to higher diversification benefits.

Moreover, as explained by Kim (2012), data on emerging-markets stocks is limited which is why he focused on the period between 2001 and 2010. Even if the data is available for earlier periods, it is not completely reliable. For this reason we likewise select a more recent time period, starting from January 2009 until April 2018. Thus our sample contains 2412 observations which ensure the elimination of the small sample bias. Moreover, as pointed out by Tse and Tsui (2002), samples above 500 observations ensure that the bias arising from Maximum Likelihood Estimation (MLE), used throughout our methodology, remains small. Our sample size, both in-sample and out-of-sample satisfies this minimum size condition. For the in-sample-analysis, we use the full sample of 2412 observations while for the out-of-sample analysis we use the last 600 observations to test hedging effectiveness of the forecasted hedge ratios. This choice of size for the out-of-sample analysis is a result of a trade-off between estimation bias and computational burden. A sample smaller than 500 would suffer from bias from the MLE, while a too high sample size would increase computational costs through unrealistically long lasting estimation procedures.

This historical data on MSCI prices is retrieved from DataStream. Currency futures data is also retrieved from the futures continuous series from DataStream, whereas this data is comprised of perpetual series of futures data as predefined in DataStream. Similarly we obtain

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each countries historical foreign exchange rate from DataStream. Our data transformation are straightforward, using the historical foreign exchange rates we convert foreign denominated stock and futures prices into dollar-denominated prices since we take the position of an US investor.

3.1. Descriptive statistics

We provide extensive descriptive statistic in this section to analyze certain implications of the data characteristics. As can be seen from table 1, both the index and the futures returns have means close to 0 and a much higher standard deviation. This is known as the mean blur and it is one of the stylized factors of assets returns. The only exception is the US, where the mean is non-zero at a 5 % significance level. This indicates that we could assume the mean of the returns to be 0, however in this thesis we allow for a non-zero mean for more accuracy. Additionally, we can see that currency futures returns are generally less volatile, with a standard deviation generally below 1%, than index returns for all the countries. This could indicate toward the currency futures being appropriate hedging instruments for our portfolio.

Furthermore, other stylized factors of financial asset returns are also present. The JB statistic is high for all returns which rejects the null that the distribution of the assets is normal. This is further confirmed by the presence of the excess kurtosis and skewness. This non- normality is particularly present in the futures of China and Russia. We also fit a t-distribution to individual returns whose consequent degrees of freedom again prove that these returns are non-normal. As a rule-of-thumb, near 20 degrees of freedom would indicate normality and as can be seen all of our assets’ degrees of freedom are significantly lower. For this reason we additionally use a T-student GARCH-based model.

We can also see that all returns, stock and futures, have a high negative ADF statistic indicating that we reject the null hypothesis of a unit root which indicates that returns are stationary. As a consequence, the GARCH coefficients will sum to less than one, 𝛼 + 𝛽 < 1, which is necessary condition for our models to function properly as explained by Bollerslev (1986). This also implies that we can expect mean-reversion in the returns conditional volatility.

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Table 1. In-sample data descriptive statistic.

This table present several descriptive statistic of the in-sample data used in this thesis. The presented statics are sample means, standard deviation (SD), Jarque-Bera statistic (JB), excess kurtosis (EK), skewness (SK) and Augmented Dickey Fuller statistic with lags of each country’s MSCI index and currency futures returns. The mean and standard deviations of the returns are expressed in percentages. For the US only the MSCI index returns are the presented since the investors are assumed to be from the US.

Mean (%) SD (%) JB EK SK Degrees of _freedom

ADF Stat. Lags Brazil Index Futures 0,012 -0,015 1,923 0,935 1592,666 1685,572 3,959 3,980 -0,203 -0,482 6,4243 6,2754 -47,144 -24,339 0 4 China Index Futures 0,036 0,004 1,409 0,185 561,623 874344,099 2,356 92,824 -0,093 -4,572 8,0296 3,4214 -10,591 -38,085 19 1 Czech Republic Index Futures -0,013 -0,003 1,480 0,764 1128,016 1307,368 3,349 3,524 0,016 -0,385 8,3095 7,2729 -15,778 -19,671 8 5 Hungary Index Futures 0,025 -0,011 2,139 0,966 1346,651 816,401 3,657 2,735 0,059 -0,401 8,2495 8,4765 -35,742 -26,730 1 3 Israel Index Futures 0,001 0,003 1,169 0,452 2000,863 743,507 4,343 2,679 -0,508 -0,236 3,8563 5,978 -21,453 -46,161 4 0 Mexico Index Futures 0,021 -0,011 1,441 0,761 1926,983 4324,227 4,333 6,443 -0,308 -0,616 7,6514 9,0476 -17,079 -17,208 9 7 New Zealand Index Futures 0,030 0,009 1,227 0,827 292,988 557,313 1,670 2,278 -0,177 -0,299 10,4088 10,4353 -47,158 -50,194 0 0 Poland Index Futures 0,005 -0,006 1,889 0,942 1708,818 1755,065 4,099 4,075 -0,215 -0,463 6,2815 8,1702 -23,843 -21,012 4 5 Russia Index Futures 0,012 -0,024 2,192 1,189 3461,034 498693,971 5,863 70,329 -0,094 1,998 7,5337 5,3798 -13,088 -7,207 12 26 South Africa Index Futures 0,026 -0,010 1,732 1,016 473,204 286,819 2,166 1,629 -0,057 -0,224 9,8222 10,2862 -22,272 -23,752 5 4 South Korea Index Futures 0,043 0,007 1,518 0,655 1405,685 6242,114 3,689 7,794 -0,304 -0,583 7,7542 6,1875 -18,117 -11,391 6 24 US Index 0,045** 1,034 2910,394 5,333 -0,351 4,716 -18,558 7

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Table 2. Sample unconditional correlations between stock and currency futures returns

In this table the sample time-invariant correlation (ρ) between a country’s stock index returns and its currency futures returns are presented. Due to fact that it is assumed that investors are from the US, this country is omitted.

The sample correlation between stock indices and currency futures are moderate as seen in Table 2, only slightly higher than 0.5 for most countries. The lowest value is for China, possibly due to the tight monetary policy of the government with numerous unexpected adjustments to the value of the currency. As we will see in the next section, this will have an effect on the hedge ratio and consequently on hedging effectiveness. All the sample correlations are significantly higher than 0 at the 1% significance level.

As for the sample period for the out-of-sample analysis, more specifically the last 600 days of our full sample, it has very similar features as the full sample, as can be seen in table A1. For some countries the standard deviation of stock indices or futures returns are lower, however, generally speaking the out-of-sample data closely matches the full sample data. Having provided an overview of the sample used, we proceed to carefully describe our methodology step by step. Due to the complexity of some of our models, it is necessary to be clear and concise to avoid misunderstandings.

Brazil China Czech

Rep. Hungary Israel Mexico

New

Zealand Poland Russia

South Africa

South Korea ρ 0,386 0,167 0,534 0,614 0,363 0,651 0,548 0,629 0,557 0,623 0,546

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4. Methodology

The first step after obtaining all the necessary data is to calculate the returns of each index. We first exchange the index prices to US dollars using the historical exchange rates of each country’s currency. Next we obtain the returns by:

𝑟𝑖,𝑡 = ln(𝑆𝑖,𝑡) − ln⁡(𝑆𝑖,𝑡−1) (1) By calculating the returns on the dollar indexes, we obtain a sum of the local price index returns and the exchange rate returns.

In this thesis we assume that investors can only hedge their currency risk using currency futures. As such their hedged portfolio of each pair of country index and their respective currency futures becomes a sum of its positions in the assets. Therefore, the investors goes long in the country indexes, with returns Ri, and short on currency futures, with returns Rfi, leading

to a country portfolio return, Rc, of:

𝑅_𝑐,𝑡 = 𝑅_𝑖,𝑡− 𝛾_𝑡𝑅_{𝑓𝑖,𝑡} (2) Taking the variance of equation 2 we obtain:

𝑉𝑎𝑟(𝑅𝑐,𝑡) = 𝑉𝑎𝑟(𝑅𝑖,𝑡) + 𝛾𝑡2𝑉𝑎𝑟(𝑅𝑓𝑖,𝑡) − ⁡2𝛾𝑡𝐶𝑜𝑣(𝑅𝑖,𝑡, 𝑅𝑓𝑖,𝑡) (3) Since we want to minimize the portfolio volatility, we take the derivative of equation 3 with respect to γt, set it equal to 0 and solve for γt, which gives us the hedge ratio:

𝛾

=

=

= 𝜌

Equation (4) is known as the minimum-variance hedge ratio, since we minimize the portfolio variance, and as earlier explained it was traditionally estimated through an OLS regression with the functional form of Equation 2. We saw that the assumptions of an OLS regression are too restrictive which is why we instead use GARCH-based models to estimate the optimal hedge ratio.

4.1. Volatility modelling

In order to obtain the portfolio volatility we need the volatility of the returns of each country. For this reason we fit several GARCH models to obtain each stocks volatility, starting with the simple GARCH (1,1) model as portrayed by Christoffersen (2012):

~ 22 ~ 𝜎_𝑡+12 _{= 𝜔 + 𝛼𝑅}

𝑡2+ 𝛽𝜎𝑡2 (5) This is a symmetric model and it overlooks certain stylized facts of asset returns such as non-normality of returns and the leverage effect. More complete GARCH models should account for these facts. For this reason we also fit a GJR-GARCH model, also known as GARCH (1,1,1), which accounts for the leverage effect which we earlier defined. This can be accounted for by adding an additional element in the GARCH specification:

𝜎_𝑡+12 _{= 𝜔 + 𝛼𝑅}

𝑡2+ 𝛾𝐼𝑡𝑅𝑡2+ 𝛽𝜎𝑡2, (6) where It is a hit series that takes the value of 1 if Rt < 0 and a value of 0 otherwise.

One of the main drawbacks of the two previous GARCH models, GARCH (1,1) and GARCH (1,1,1) is their normality assumption. One of the stylized facts of assets returns is that these are non-normal, with higher kurtosis and skewness, which was proven in the last section. The filtered errors or standardized returns from a regular GARCH model may still be non-normal and asymmetric. To account for this non-non-normal distribution we can fit a GJR-GARCH model together with a skewed standardized t-distribution. The skewed t-distribution allows for higher kurtosis as well negative skewness. This is an approach seen throughout literature (Lai, 2018; Hsu et al., 2008) and it is expected to bring more accurate results.

The asymmetric T-student distribution density function is defined by:

𝑓_{𝑠𝑘𝑒𝑤−𝑡}= {𝐵𝐶[1 + (𝐵𝑧+𝐴)2 (1−𝑑)2_(𝜈−2)]−(1+𝜈)/2,⁡⁡⁡𝑖𝑓⁡𝑧 < −𝐴/𝐵 𝐵𝐶[1 +_(1+𝑑)(𝐵𝑧+𝐴)_2(𝜈−2)2 ]−(1+𝜈)/2_{,⁡⁡⁡𝑖𝑓⁡𝑧 ≥ −𝐴/𝐵} (7) Where 𝐴 = 4𝑑𝐶𝜈 − 2 𝜈 − 1,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝐵 = √1 + 3𝑑2− 𝐴2,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝐶 = 𝛤(𝜈 + 1_{2 )} 𝛤(𝜈_{2)√𝜋(𝜈 − 2)}

with 𝜈 measuring the degrees of freedom and d the skewness. Applying this distribution to returns is relatively straightforward as there are adequate statistical packages and commands readily available within Matlab that estimate the optimal fit by maximizing the likelihood of the distribution.

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Having an estimate of each return’s conditional volatility, we need to obtain its shocks which we will use

We define the shocks as:

𝑧̌_𝑖𝑡 =𝑅𝑖𝑡

𝜎𝑖𝑡 (8)

To ensure that our residuals or shocks have a mean of 0 and standard deviation of approximately 1, we standardize them by subtracting their mean and dividing by their standard deviation:

𝑧_𝑖𝑡 =𝑧̌𝑖𝑡−⁡𝑧̅𝑖𝑡

𝜎𝑧𝑖 (9)

4.2. Portfolio composition

Our initial strategy is to analyze the sensitivity of the international portfolio value with and without hedging for currency risk. We first obtain the optimal portfolio composition in terms of developed and emerging markets stocks using a mean variance analysis, which has been used throughout literature (Glen & Jorion, 1993; Mc Nally & Murray, 2010; Campbell et al., 2010). Portfolio composition at each point in time is therefore obtained maximizing its Sharpe ratio, Sp, as:

max 𝑆_𝑝,𝑡 = 𝐸(𝑟𝑝,𝑡)

𝜎𝑝,𝑡 , (10)

where 𝜎_𝑝,𝑡 is the portfolio standard deviation and 𝐸(𝑟_𝑝,𝑡) is the expected portfolio return. The full portfolio expected return is defined as follows:

𝐸(𝑟𝑝,𝑡) = ⁡ 𝑤1,𝑡𝐸(𝑟1,𝑡) + 𝑤2,𝑡𝐸(𝑟2,𝑡) + ⋯ + 𝑤12,𝑡𝐸(𝑟12,𝑡), (11) where each expected return corresponds to the 100-day mean US dollar return of the MSCI indexes of each country, including the US. We use a rolling window mean, thus having a different mean each day. Similarly, the portfolio standard deviation is:

𝜎𝑝,𝑡 = ⁡ 𝑤1,𝑡𝜎1,𝑡+ 𝑤2,𝑡𝜎2,𝑡+ ⋯ + 𝑤12,𝑡𝜎12,𝑡+ 𝑤1,𝑡𝑤2,𝑡𝜌1,2,𝑡𝜎1,𝑡𝜎2,𝑡+ ⋯ = 𝜎𝑝,𝑡 = ⁡ ∑ 𝑤𝑖,𝑡𝜎𝑖,𝑡 12 𝑖=1 + ∑ ∑ 𝑤𝑖,𝑡 12 𝑗=1 12 𝑖=1 𝑤𝑗,𝑡𝜎𝑖,𝑡𝜎𝑗,𝑡𝜌𝑖𝑗,𝑡

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𝜎𝑝,𝑡2 = ⁡ ∑12𝑖=1∑12𝑗=1𝑤𝑖,𝑡𝑤𝑗,𝑡𝜎𝑖,𝑡𝜎𝑗,𝑡𝜌𝑖𝑗,𝑡⁡⁡⁡⁡𝑤ℎ𝑒𝑟𝑒⁡𝑖 ≠ 𝑗 (12) which can be expressed in form of vectors and matrixes as:

𝜎𝑝,𝑡2 = ⁡ 𝑤𝑡′Ʃ𝑡𝑤𝑡⁡,

where wt corresponds to the weights vector and Ʃt corresponds to the covariance matrix at each

point in time. Therefore, at each point in time we maximize the portfolio Sharpe ratio with respect to the portfolio weights given the returns and the calculated conditional volatilities and correlations. This returns us daily dynamics optimal portfolio weights since we assumed daily portfolio rebalancing.

Having obtained these optimal weights by maximizing the Sharpe ratio under equation 10, we calculate the full portfolio returns (RP) by multiplying each country’s individual stock

returns by its corresponding optimal weights as:

𝑅_𝑃 = ∑12 𝑅_𝑖𝑤_𝑖

𝑖=1 (13)

For the hedged full portfolio (RHP) we simply replace the individual country returns by

the hedged returns from equation 2, except for the US returns since we assume it is our home country:

𝑅_𝐻𝑃 = ∑11 𝑅_𝐶𝑖𝑤_ℎ𝑖

𝑖=1 +⁡𝑅𝑈𝑆𝑤𝑈𝑆⁡ (14) where whi are the optimal weights of the hedged returns which are obtained the same way as

we previously explained except that we replace the returns in Equation 11 by the hedged returns defined in Equation 2. Similarly, we substitute the volatilities and correlations of unhedged returns by the volatilities and correlations of hedged returns in equation 12 and maximize the Sharpe ratio with respect to the weights. The volatilities of hedged returns are simply obtained by fitting one of the GARCH volatility specifications to equation 2 instead of the individual stock index returns.

Thus we see that for the unhedged portfolio we already make use of the GARCH volatility estimates. Additionally, we need to calculate the correlations between returns in equation 12 and ensure that these correlations are dynamic. Here is where we introduce the DCC model which we use both for optimal hedge ratio calculation as well as for optimal portfolio composition. As can be seen, the optimal portfolio composition, hedged or unhedged, depends on the volatility and correlation methodology used.

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4.3. Multivariate GARCH DCC

In order to obtain the portfolio variance, we need each stock’s volatility and its correlations with the other stock indexes in the portfolio. The DCC model is similar to the GARCH specification in Equation 5 but in terms of correlations. In the DCC model, the correlation between assets is determined by the matrix⁡𝒬𝑡+1:

𝒬_𝑡+1 = 𝐸[𝑧_𝑡𝑧_𝑡′_{](1 − 𝛼 − 𝛽) + 𝛼(𝑧}

𝑡𝑧𝑡′) + ⁡𝛽𝒬𝑡 (15) In the two-asset case we have:

𝒬𝑡+1= [ 𝑞11,𝑡+1 𝑞12,𝑡+1 𝑞12,𝑡+1 𝑞22,𝑡+1] ⁡𝒬_𝑡+1= [_𝜌1 𝜌12 12 1 ](1 − 𝛼 − 𝛽) + 𝛼 [ 𝑧_1,𝑡2 _𝑧 1,𝑡𝑧2,𝑡 𝑧_1,𝑡𝑧_2,𝑡 𝑧_2,𝑡2 ] + 𝛽 [ 𝑞_11,𝑡 𝑞_12,𝑡 𝑞_12,𝑡 𝑞_22,𝑡] (16) Here the variables 𝑞_{𝑖𝑗,𝑡+1} simply represent the update to the correlation and drive the correlation dynamics while not having direct economic interpretation. Equation 16 also makes use of the standardized residuals defined in equation 9. Finally, 𝜌₁₂ is the unconditional correlation between the two assets, which is estimated as:

𝜌̅₁₂=_𝑇1∑𝑇 𝑧_1,𝑡𝑧_2,𝑡

𝑡=1 (17)

This leads us to the correlation between two assets being equal to: 𝜌𝑖𝑗,𝑡+1 =_√𝑞 𝑞𝑖𝑗,𝑡+1

𝑖𝑖,𝑡+1𝑞𝑗𝑗,𝑡+1⁡ (18)

One of the necessary conditions for the functioning of this model is that the matrix 𝒬𝑡+1 has to be positive semidefinite. This means that it can only take positive values or 0. This condition is necessary to ensure that the subsequent correlation and covariance matrices are also positive semidefinite. Otherwise, we could obtain a negative variance which according to our model, does not exist. One way to ensure this condition is to keep the correlation coefficients from Equation 16, α and β, constant. This does not mean that the correlation in itself is constant since this is determined by ρij, which is time varying. It does not assume that

the univariate GARCH coefficients are the same for each asset neither. In fact the volatilities coefficient can vary from asset to asset. Therefore we can have different persistence in volatility for each asset while having a steady persistence in correlation.

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The DCC model can be estimated using a two-step estimation which drastically lessens the computational burden. We first fit the individual GARCH models by maximizing the likelihood of equations 5 or 6. Next we estimate the DCC parameters by quasi maximum likelihood (LC) estimation for two assets and composite likelihood estimation (CLC) for more

than two assets. The composite likelihood equation is equal to:

ln(𝐶𝐿𝐶) = −1₂∑ ∑ ∑ (ln(1 − 𝜌𝑖𝑗,𝑡2 ) + (𝑧_𝑖,𝑡2+𝑧_𝑗,𝑡2 −2𝜌𝑖𝑗,𝑡𝑧𝑖,𝑡𝑧𝑗,𝑡) (1−𝜌_{𝑖𝑗,𝑡}2 ) ) 𝑗>1 𝑛 𝑖=1 𝑇 𝑡=1 (19)

Which in the two asset case simplifies to the quasi maximum likelihood formulation:

ln(𝐿𝐶) = −1₂∑ (ln(1 − 𝜌𝑖𝑗,𝑡2 ) +

(𝑧_𝑖,𝑡2+𝑧_𝑗,𝑡2−2𝜌𝑖𝑗,𝑡𝑧𝑖,𝑡𝑧𝑗,𝑡)

(1−𝜌_{𝑖𝑗,𝑡}2 ) ) 𝑇

𝑡=1 (20)

In equations 19 and 20, ρij, is defined by equation 18 and thus depends on parameters α

and β. Therefore we maximize function 19 or 20 (depending on the number of assets) with respect to coefficients correlation persistence coefficients, α and β. This can be achieved through Matlab with a relatively low computational cost due to the fact that we only estimate two coefficients for any number of assets, a powerful advantage of this model. Another important aspect of our computational implementation, is that the conditional correlation between returns is equal to the conditional covariance between standardized returns (Engle, 2002). In other words, the returns correlation given by Equation 18 is equivalent to the covariance between the standardized shocks, from Equation 9, which we also used in Equation 16. This is important due to the fact that the Matlab toolbox we employed, returns the covariance between standardized returns which is in the fact the correlation between returns. This saves us a considerable amount of time as we need not run separate estimation to obtain the dynamic correlation.

Thus having the formulas for the portfolio returns and volatility, we use optimization techniques through Python to obtain the weights that maximize the Sharpe ratio of the unhedged portfolio at each point in time. Thus we run loops, calculating on each day the 100-day mean return and using these together with the GARCH and DCC estimates to maximize the Sharpe ratio at daily. We impose restrictions on the weights, such as that their sum must be equal to 1 and that short positions are restricted, which implies that weights are always equal or greater than 0. These restrictions assume that investors do not borrow but only invest a

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proportion of their current wealth. The short selling restriction is imposed for simplicity and due to the fact that in some country short selling is not allowed or it is restricted, such as China.

Next we obtain the time varying hedge ratio from equation 4, using the conditional volatility from the GARCH models and the conditional correlation from equation 18. We do this for every foreign country from a US investor perspective, thus obtaining 11 hedge ratio time series for every GARCH DCC combination1_{. Due to the fact that we also evaluate the} impact on the portfolio Sharpe ratio, we establish the individual hedged country returns as in equation 2 and its volatility as in equation 3. Next we maximize Equation 19 to obtain the full portfolio correlation parameters. Next follow the same procedure as explained earlier in the case of the unhedged portfolio Sharpe ratio, by applying GARCH models to the hedged returns and using these together with their correlations to obtain the optimal weights. This consistency will allow us to correctly compare Sharpe ratios and draw valid conclusions.

4.4. Copula functions 4.4.1. Gaussian Copula

In order to allow for more flexibility, we consider GARCH-based copulas for estimating the assets joint distribution. As we saw earlier, numerous authors highlight the importance of the tail dependence present in financial assets. This tail dependence suggests that assets have a higher correlation when they both reach extreme values. This is overlooked by the bivariate normal distribution that we used in our DCC model.

This joint normal distribution does not account for tail dependences while the joint T-student distribution is also too restrictive since it assumes that all assets have the same degrees of freedom. For this reason here we use the copula functions which allow for different univariate distributions. We make use of Sklar’s Theorem, as defined by Christoffersen (2012, p. 203), “for a class of multivariate density functions, F(z1,…, zn), with marginal cumulative distributions functions F1(z1), …, Fn(zn), there exists a unique copula function, G(•) linking the marginal to form the joint distribution” such that:

𝐹(𝑧1, … , 𝑧𝑛) = 𝐺(𝐹1(𝑧1), … , 𝐹𝑛(𝑧𝑛)) (21)

1 _{Normal distribution GARCH DCC; Normal distribution GARCH DCC and T-student distribution} GJR-GARCH DCC

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= 𝐺(𝑢1, … , 𝑢𝑛)

We first start by considering the Gaussian copula to obtain the correlation between the stock indexes and their respective currency futures:

𝐺(𝑢_𝑖, 𝑢_𝑓𝑖; 𝜌∗_{) = ⁡ 𝜙}

𝜌∗(𝜙−1(𝐹𝑖(𝑧𝑖)), 𝜙−1(𝐹𝑓𝑖(𝑧𝑓𝑖))) = 𝜙𝜌∗(𝜙−1(𝑢𝑖), 𝜙−1(𝑢𝑓𝑖)) (22) where ρ* is the correlation between the inverse Gaussian cumulative distribution functions (CDF), 𝜙−1_(𝑢

𝑖) and 𝜙−1(𝑢𝑓𝑖), which is also known as the copula correlation. Note that even though the copula is Gaussian, it does not assume that marginal distributions are normal. If marginal distribution are non-normal, their multivariate distribution will also be non-normal. This is the advantage of copula functions, they allow for greater flexibility.

In order to estimate the Gaussian copula together with its correlation we will make use of its copula probability density function (PDF), given by:

𝑔(𝑢_𝑖, 𝑢_𝑓𝑖; 𝜌∗_{) =} 1 √1−𝜌∗2𝑒𝑥𝑝 {− 𝜙−1(𝑢𝑖)2+𝜙−1(𝑢𝑓𝑖)2−2𝜌∗𝜙−1(𝑢𝑖)𝜙−1(𝑢𝑓𝑖) 2(1−𝜌∗2₎ + ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡+𝜙−1(𝑢𝑖)2+𝜙−1(𝑢𝑓𝑖) 2 2 } (23)

In order to obtain the normal copula and its respective correlation for a bivariate portfolio we maximize the log likelihood (Lg) based on equation 23:

ln𝐿_𝑔 = ∑𝑇 ln𝑔(𝑢_𝑖,𝑡 𝑡=1 , 𝑢𝑓𝑖,𝑡) (24) ln𝐿𝑔 = −𝑇₂ln(1 − 𝜌∗2) − ∑ 𝜙 −1_{(𝑢𝑖,𝑡)}2_+𝜙−1_{(𝑢𝑓𝑖,𝑡)}2_−2𝜌∗_𝜙−1_{(𝑢𝑖,𝑡)𝜙}−1_{(𝑢𝑓𝑖,𝑡)} 2(1−𝜌∗2₎ 𝑇 𝑡=1 + ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡+𝜙−1(𝑢𝑖,𝑡) 2 +𝜙−1_{(𝑢𝑓𝑖,𝑡)}2 2 (25)

The inputs from equation 22, 𝑢_𝑖,𝑡, are the univariate CDFs of the standardized residuals, and in order to obtain them we need to fit the proper distribution to each asset. We do not know however, the exact distribution of each asset and we cannot assume normality as we saw in the descriptive statistics. A possible solution to the unknown univariate distribution issue can be solved by the kernel estimate of the cumulative distribution. Using Parzen’s (1962) definition, “the kernel density estimation is a non-parametric way to estimate the probability density

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function of a random variable”. He further explains that inferences about the population are made based on a limited data sample. The functional form of this method is as follows:

𝑓̂_ℎ(𝑧) = ⁡_𝑛ℎ1 ∑ 𝐾 (𝑧−𝑧𝑖

ℎ )

𝑛

𝑖=1 (26)

where K is the kernel and h the bandwidth. This estimation method is readily available and automated in computational packages. The main advantage of this method is that we do not need to make incorrect assumptions regarding the assets’ marginal distributions. We simply approximate the assets real distribution given the limited empirical sample.

Important noting that the copula correlations we obtain from these models are static, meaning they do not change over time. In our case we want a dynamic correlation for higher accuracy which is why we adopt the solution suggested by Christoffersen (2012). He proposes applying the DCC model to the copula shocks themselves which, in the case of a Gaussian copula, are given by:

𝑧_𝑖,𝑡∗ _{= 𝜙}−1_(𝑢

𝑖,𝑡) = 𝜙−1(𝐹𝑖(𝑧𝑖,𝑡)) (27) Next we apply our DCC model explained earlier to these shocks and instead of using the sample average, 𝐸[𝑧𝑡𝑧𝑡′] as the constant correlation term as suggested by Christoffersen, we make use of the copula correlation which brings more accurate results. Therefore we have the following formula for the correlation between each stock and its respective currency future:

𝒬_𝑡+1= [_𝜌1_∗ 𝜌_{1 ]}∗ (1 − 𝛼 − 𝛽) + 𝛼 [ 𝑧𝑖,𝑡 ∗2 _𝑧 𝑖,𝑡∗ 𝑧𝑓𝑖,𝑡∗ 𝑧_𝑖,𝑡∗ _𝑧 𝑓𝑖,𝑡∗ 𝑧𝑓𝑖,𝑡∗2 ] + 𝛽 [𝑞_𝑞11,𝑡 𝑞12,𝑡 12,𝑡 𝑞22,𝑡] (28) In the n-assets case this takes the shape of equation 13:

𝒬𝑡+1= 𝛶∗(1 − 𝛼 − 𝛽) + 𝛼(𝑧𝑡∗𝑧𝑡∗′) + ⁡𝛽𝒬𝑡 (29) As can be seen the main difference with the standard DCC model we used earlier is that we use the copula correlation, 𝜌∗_{, as the unconditional or long term correlation as well as the} copula shocks, 𝑧_𝑖,𝑡∗ , rather than the GARCH shocks,⁡𝑧_𝑖,𝑡. We follow the same estimation procedures as in the earlier section, namely by maximizing equation 20 using the copula shocks from equation 27.

The advantage of combining copulas with the DCC model is that we can apply it to a portfolio composed of numerous assets without the need to estimate numerous coefficients.

~ 30 ~

Repeating Engle’s (2002) statement, regardless of the number of assets in the portfolio we only need to estimate two coefficients for the DCC model which clearly saves us a substantial amount of time. Moreover, it is simpler to interpret the correlation coefficient obtained by the DCC model.

In order to obtain the correlation between all hedged returns which is needed for the hedged portfolio Sharpe ratio, we need to generalize equation 20 to more than two assets. The general form of this equation is:

𝑔(𝑢_1𝑖, … , 𝑢_𝑛; 𝛶∗_{) = |𝛶}∗_|−1_{2𝑒𝑥𝑝 {−}1 2𝜙

−1_(𝑢)′_(𝛶∗−1_{− 𝐼}

𝑛)𝜙−1(𝑢)} (30) with u being a vector of univariate CDFs, In an n-dimensional identity matrix and finally the

correlation matrix Υ* is obtained by maximizing the following likelihood:

𝑙𝑛𝐿𝑔 = −1₂∑𝑇𝑡=1ln|𝛶∗| −₂1∑𝑇𝑡=1𝜙−1(𝑢𝑡)′(𝛶∗−1− 𝐼𝑛)𝜙−1(𝑢𝑡) (31) Form the above maximization we similarly obtain the copula shocks and use them to maximize equation 19. We then use the output for equation 29 to obtain a dynamic correlation between all assets in the portfolio. We then obtain 66 intercountry correlations which we use in equation 12 and follow the same procedures as earlier to obtain the portfolio Sharpe ratio and subsequently the optimal portfolio composition.

4.4.2. T-student Copula

Despite the fact that the Gaussian copula allows us more flexibility by applying different marginal distributions to our returns it still does not account for the tail dependence present in financial assets. As previously mentioned, tail dependence is an important feature of financial assets which can affect the accuracy of hedging. For this reason the next model we consider is the T-student copula which allows for symmetrical tail dependence and whose CDF is given by:

𝐺(𝑢𝑖, 𝑢𝑓𝑖; 𝜌∗, 𝑑) = ⁡ 𝑡(𝑑,𝜌∗)(𝑡−1(𝐹𝑖(𝑧𝑖); 𝑑), 𝑡−1(𝐹𝑓𝑖(𝑧𝑓𝑖); 𝑑)) (32) where d corresponds to the degrees of freedom of this distribution.