Investing with minimum risk : evaluating the futures hedging performance of GARCH models

(1)

Faculty of Economics and Business,

Amsterdam School of Economics

Econometrics

Bachelorthesis Econometrics

Investing with minimum risk

Evaluating the futures hedging performance of GARCH

models

Name: Wesley Beerepoot

Studentnumber: 10191186

Supervisors: Ms D. (Derya) Güler

Date: June 26, 2018

Group 1

2017-2018

Block 5 and 6

(2)

Statement of Originality

This document is written by Wesley Beerepoot who declares to take full responsibil-ity 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

Acknowledgements

First, I would like to thank the University of Amsterdam to provide me with the skills to complete my bachelorthesis. Secondly, I would like to thank my supervisor Derya Güler for helping me to write my bachelorthesis. She gave good advice and she helped me when necessary. Finally, I would like to thank my fellow students, girlfriend and parents who supported me in writing the bachelorthesis.

(3)

Abstract

This research investigates the effectiveness of the minimum variance currency hedge ratio of the European, Australian and Canadian currency futures market (EU-R/USD, AUD/USD and CAD/USD) using four different methods (OLS, CCC, BEKK and DCC). The first expectation was that dynamic models (DCC and BEKK) would have better effectiveness in determining the optimal hedge ratio than the static models (OLS and CCC) on all three markets. Futhermore, there was expected that the dynamic BEKK model was more effective in determining the optimal hedge ratio than the dynamic DCC model. Finally, there was expected that static OLS model was more effective in determining the optimal hedge ratio than the static CCC model. These predictions were tested with futures and spot data from Jan-uari 2003 until April 2018 of the European, Australian and Canadian currency futures markets. The results showed that the dynamic models had more effective-ness than the static models when determining the minimum variance hedge ratio on the three currency futures markets (only DCC performed worse on the Cana-dian currency futures market) than the static models. The BEKK model had the highest effectiveness on all three currency futures markets when determining the minimum variance hedge ratio. There were some mixed results of the effectiveness of the minimum variance hedge ratio based on the OLS and CCC models.

(4)

1 Introduction

Making money with the minimum amount of risk is something every investor is striving for. Hull (2012) indicates that reducing the amount of (systematic) risk on the stock market can be done by diversification. He states that diversification is choosing a large portfolio with different assets. However, on the currency mar-ket the simplest way of managing marmar-ket risk is by hedging with future contracts (Chang, González-Serrano, & Jimenez-Martin, 2013, p. 165). Hedging could be done by taking a long position in a currency and short an amount of future contracts in that currency or vice versa (Chang et al., 2013, p. 165). Here a long position refers to buying the currency and a short position refers to selling the future contracts with the assumption that those future contracts are bought back at some point in the future.

Because foreign exchange rate markets are the largest and most liquid of all asset markets (Chang et al., 2013, p. 165), this research focused its aim on the currency market. When hedging with futures contracts in the currency market, in-vestors want to find out how many futures contracts should be held for each unit of the underlying currency. The most optimal way of choosing the ratio is by calcu-lating the ratio with the minimum amount of variance, also known as the optimal hedge ratio (OHR) (Chang et al., 2013, p. 165). The OHR is constructed in such a way that it is equal to the covariance between the spot and future returns divided by the variance of the future returns (Lien, Tse, & Tsui, 2002, p. 791). The spot re-turn is the rere-turn on an asset today (Hull, 2012, p. 5), in this case the rere-turn on one unit of the underlying currency. There are different methods of finding the OHR.

The conventional approach to calculate the optimal hedge ratio is the Ordinary least squares (OLS) method (Lien et al., 2002, p. 791). In the OLS model the return on one unit of the underlying currency is the dependent variable and the return of the future contracts is the independent variable. The estimate of the independent

(7)

variable in the OLS model corresponds to the estimated OHR (Lien et al., 2002, p. 791). However, Herbst, Kare and Marshall (1989) state that the OLS method is not suitable for estimating the OHR, because time is an important variable in computing the hedge ratio with futures. In other words, a model that takes serial correlation and heteroskedasticity into consideration might be a better model.

In response to Herbst, Kare and Marshall (1989), Bollerslev (1990) proposed a different model that could compute the OHR. His model takes serial correlation and heteroskedasticity into account. The model is called Constant Conditional Cor-related Generalized Autoregressive Conditional Heteroskedastic (CCC) model. In this model the variance of the error terms is conditional on past information. How-ever, Yang and Allen (2004) show that the OLS method could lead to better results than the CCC method when hedging on the Australian currency market. Moreover, other research from Lien et al. (2002) shows that the OLS hedge ratio outperformed the CCC hedge ratio on the European and Japanese currency market.

A more dynamic model is the GARCH model of Baba, Engle, Kraft and Kroner (BEKK) (Engle & Kroner, 1995). The BEKK model is similar to the CCC model of Bollerslev (1990). However, the BEKK model is optimized via a bivariate GARCH process and the CCC model is optimized via two seperate univariate GARCH pro-cesses (Engle & Kroner, 1995). The advantage of the BEKK model is that it ensures the positive-definiteness of the variance-covariance matrix (Su & Huang, 2010). But, Chang, McAleer and Tansuchat (2013) found that the BEKK model performed worse than the CCC model when calculating the optimal hedge ratio.

Perhaps a more preferred model would be the Dynamic Conditional Correlated (DCC) model from Engle (2002). His model is based on the model of Bollerslev (1990) but also estimates additional parameters that capture the effects of previous shocks. Research shows that the DCC model performs better than the CCC model and the OLS model on the British and Japanese markets (Ku, Chen, & Chen, 2007). On the other hand, there is also research that suggests that the CCC model and the DCC

(8)

model provide similar hedging effectiveness (Chang et al., 2013).

It seems that there is no general consensus between the herefor mentioned mod-els and the DCC model and the BEKK model are not even tested on the Canadian and Australian market in the literature up to now. Therefore, the main focus of this paper was to compare the effectiveness of the minimum variance currency hedge ratio of the European, Australian and Canadian currency futures markets (EU-R/USD, AUD/USD and CAD/USD) using four different methods (OLS, CCC, BEKK and DCC).

The paper is structured as followed. In section 2 the theoretical background is discussed. Data is discussed in section 3. Section 4 is about the methodology and the results of the paper are discussed in section 5. Finally, the conclusion is discussed in section 6.

2 Literature review

To formulate hypotheses for the research question of this paper, a literature re-view is presented in this section. The first subsection is about the foreign exchange market. In subsection 2 the futures market and futures contract are discussed. Sub-section 3 is about futures hedging. SubSub-section 4 is about econometric methods to de-termine the optimal hedge ratio. This subsection comprises the four methods (OLS, CCC, BEKK and DCC) used in this research. Finally, the hypotheses of the present paper are discussed in subsection 8.

2.1 Foreign exchange market

Chang et al. (2013) state that the foreign exchange rate markets are the largest and most liquid of all asset markets. Investors and companies are able to buy, sell, speculate and exchange currencies on these markets. Developments in the foreign exchange market influence national trade, monetary policies, the competitiveness

(9)

of nations and the foreign exchange market is also important for companies that engage in cross-border trade and investment (Chang et al., 2013, p. 165). Monetary policy consists of the actions of the central bank to lower or elevate the money in cir-culation to control indicators as interest rate and unemployment. Because so many different financial institutions deal with the foreign exchange market, it is great importance for these institutions to use the most effective methods and financial instruments when investing in these market. One such financial instrument is a futures contract.

2.2 Futures market and future contracts

According to Hull (2012) futures contracts can be described as an agreement be-tween two parties to buy or sell commodities (such as oil or wool) or financial finan-cial instruments (such as bonds or shares) at a certain time in the future at a price agreed upon today. In the Australian currency market this means that you buy or sell Australian dollars at some point in the future, for instance two months from now, at a price agreed upon today. The contract is legally binding, meaning that you have the obligation to comply with the terms of the futures contract (Hull, 2012). This is especially important, because that way investors and companies can use futures contracts to make better decisions when investing in the foreign exchange market.

Futures contracts in the foreign exchange market could be different in price from the price of the underlying currency today. Hull (2012) states that the difference is that the price in the futures contract is a prediction about what the price will be at the end date of the futures contract. He claims that the price of the futures contract will fluctuate and will be equal to the price of the underlying currency at the day of expiration. For example, imagine that the price of one Australian dollar today is 0.75 in American dollars and the price of one Australian dollar in a futures contract expiring two months from is worth 0.73 in American dollars today. In the days

(10)

leading up to the day of expiration both prices fluctuate and will differ from each other. But at the day of expiration both prices will be the same in American dollars. Figure 2.1 shows the course of futures contract and the price of the underlying currency until the expiration date of the futures contract.

Figure 2.1: Futures and spot prices over time.

Note: Relationship between future price and spot price as the expiration date of the futures

contract approaches. The spot price in this case is the price of the currency. (a) Futures price is higher than spot price; (b) Spot price is higher than futures price. The figure is adapted from Options, Futures, And Other Derivatives (Hull, 2012, p. 27).

Hull (2012) indicates that there are a lot of advantages with using futures con-tracts. He states that futures contracts, in many circumstances, are more liquid and easier to trade than the underlying currency. Furthermore, futures contracts entail lower transaction costs than other financial instruments (Hull, 2012, p. 364).

Another advantage of futures contracts is that investors can invest more money with futures contracts than other financial instruments (Hull, 2012, p. 364). Here follows a brief explanation of how that works in the financial market. Investors can set up a margin account by a broker and deposit money into that account (Hull, 2012, p. 27). The key aspect of the margin account is that the investor can use more money than the deposit into the margin account. For instance, an investor wants a futures contract that is worth 10,000$, the broker establishes in the contract that the investor needs to deposit 500$. The amount that the investor must deposit in the margin account is also known as the initial margin (Hull, 2012, p. 27). At the end of

(11)

each trading day, the margin account is adjusted in such a way that it reflects the investor’s gains or losses. This practice is also known as daily settlement or marking to market (Hull, 2012, p.27). If at the end of the day the losses exceed a certain threshold that is established by the broker, the investor is obliged to put additional money into the margin account (Hull, 2012, p. 28). If the investor made some profit the investor could withdraw some money. In other words, with a relatively small amount of money, the investor is able to take a large speculative position (Hull, 2012, p. 13).

The initial margin of a futures contract is lower than other financial instru-ments, so investors can invest more into futures contracts relative to other finan-cial instruments and potentially make more money (Hull, 2012, p. 364). However, the downside is that the investor can lose more money too. That is why it is impor-tant for investors to minimize their risk. One way to minimize the risk on futures contracts is hedging.

2.3 Hedging with futures contracts

Hedging can be described as an investment strategy that tries to reduce the risk of negative price changes in assets (Hull, 2012, p. 11). Hull (2012) states the perfect hedge is a hedge that completely eliminates all risk. Because perfect hedges are rare, the study of hedging with futures contracts is constructing hedges in such a way that they perform as close to perfect as possible (Hull, 2012, p.47).

Chang et al. (2013) state that investors typically hedge with futures contracts on the foreign exchange market, due to the fact that hedging with futures contracts is relatively inexpensive and reliable strategy for hedging on these markets. They in-dicate that it can be done by shorting an amount of futures contracts when holding the long position of the underlying currency or taking a long position in an amount of futures contracts when holding the short position of the underlying currency. For example, a company is trading in Australia and needs Australian dollars to do

(12)

so. The company buys Australian dollars with American dollars at a price of 0.75 American dollars. Let’s assume once again that the price of one Australian dollar in a futures contract two months from now is 0.73 in American dollars. If the company goes short in the futures contract, the company is obliged to sell each Australian dol-lar for 0.73 American doldol-lars two months from now. If the price of one Australian dollar, relative to the American dollar, goes up in these two months, the company profits on the bought Australian dollars and makes losses on the futures contract. However, if the price of one Australian dollar, again relative to the American dollar, goes down in these two months, the company profits on the futures contract and loses on the bought Australian dollars. This way the risk of price fluctuations is as minimal as possible for company that trades in Australian dollars.

But what amount of short positions in futures contract against long positions in the underlying currency would minimize the risk for investors and companies? There has been a lot of research done on this question in the past couple of decades. The hedge ratio that minimizes the amount of risk is better known as the opti-mal hedge ratio (Chang et al., 2013). There are several ways of calculating this optimal hedge ratio. However, the focus in this research will be on the minimum variance hedge ratio, because the minimum variance hedge ratio has some attrac-tive features such as it is quite easy to understand and simple to compute (Chen, Lee, & Shrestha, 2003, p. 437). In this paper, the minimum variance hedge ratio is the hedge ratio that minimizes the variance of the portfolio with short positions in futures contracts and long positions in the underlying currency. In the follow-ing subsections methods of calculatfollow-ing the minimum variance hedge ratio will be discussed.

2.4 Econometrics methods to determine the optimal hedge

This subsection describes four different models (OLS, CCC, BEKK and DCC) that determine the optimal hedge ratio.

(13)

2.4.1 Determining the optimal hedge ratio with OLS

One method to determine the minimum variance hedge ratio is by using ordinary least squares (OLS) method. Johnson (1960) and Stein (1961) laid the ground work for this approach. They argued that a portfolio with futures contracts and the un-derlying commodity or financial instrument does not need to be fully hedged. Ed-erington (1979) extended this work and used the OLS method to construct a ratio between the amount of futures contracts that should be held against one unit of the underlying asset. The objective of this approach was to minimize the variance of the portfolio held by the investor or company (Ederington, 1979, p. 162), now bet-ter known as the minimum variance hedge ratio (Lien et al., 2002, p. 791; Yang & Allen, 2004, p. 302). Ederington (1979) obtained the minimum variance hedge ratio through dividing the covariance of the spot and the futures returns by the variance of the futures returns. He also evaluated the effectiveness of the minimum variance hedge strategy. This was done by comparing the risk on an unhedged portfolio with the minimum variance hedge ratio (Ederington, 1979, p. 164). The risk on the min-imum variance hedge portfolio was lower than the risk on the unhedged portfolio (Ederington, 1979, p. 165).

Nonetheless, Herbst et al. (1989) indicate that optimal hedging with OLS is not appropriate to estimate the minimum variance hedge ratio. They state that time is an important variable in computing the minimum variance hedge ratio. Prob-lems that arise with these kind of time series data are serial correlation and het-eroskedasticity. Myers and Thompson (1989) also point out that OLS uses uncon-ditional sample moments instead of conuncon-ditional sample moments. They state that the covariance and variance in the optimal hedging ratio are clearly conditional moments that depend on the available information at the time of the hedging deci-sion is made (Myers & Thompson, 1989, p. 858). Myers and Thompson (1989) state that a more suitable model would be a model that deals with serial correlation, het-eroskedasticity and has conditional moments that depend on the past. Models that

(14)

deal with these problems will be discussed in the next subsections.

2.4.2 Determining the optimal hedge ratio with CCC

A few decades ago Engle (1982) proposed a model that deals with serial correlation and heteroskedasticity. In his model the variance of the error terms is conditional on past information. He called his model Generelized Autoregressive Conditional Heteroskedastic (GARCH). Bollerslev (1990) extended this model and applied it in the field of currency hedging. The model is better known as Constant Conditional Correlated Generalized Autoregressive Conditional Heteroskedastic (CCC) model. In his model the variance-covariance matrix is conditional on past information. The advantage of this model is that it deals with the problems put forward by Herbst et al. (1993) and Myers and Thompson (1989). Another advantage of this model is that it is easy to compute (Lien et al., 2002, p. 792).

However, a few researchers demonstrate that the CCC method does not outper-form the OLS method (Lien et al., 2002; Yang & Allen, 2004). Yang and Allen (2004) show that the OLS method performed somewhat better than the CCC method while currency hedging on the Australian market. Lien et al. (2002) did research on the European and Japanese market. They show that the OLS method performed even better than the CCC method when determining the minimum variance hedge ratio. Yang and Allen (2004) indicate that dynamic time-varying hedge ratios are supe-rior to static time-varying hedge ratios like the CCC method of Bollerslev. The next subsections describe dynamic time-varying hedge ratios.

2.4.3 Determining the optimal hedge ratio with BEKK

The first dynamic time-varying model is the Baba, Engle, Kraft and Kroner (BEKK) model which was first put forward by Engle and Kroner (1995). The difference be-tween the CCC model of Bollerslev and the BEKK model posed by Engle and Kroner (1995) is that the CCC model optimizes two seperate univariate GARCH processes

(15)

and the BEKK model optimizes the two GARCH processes simultaneously, mak-ing it a bivariate GARCH model. The big advantage of usmak-ing the BEKK model is that it ensures that the variance-covariance matrix is positive-definiteness (Su & Huang, 2010). Chang, Lai and Chuang (2010) show that the BEKK model has bet-ter in-sample hedging effectiveness compared to the CCC model when hedging on the energy market. Futhermore, the BEKK model preformed better than the OLS model when currency hedging on the British, German and Japanese market (Brooks & Chong, 2001).

On the other hand, Chang, McAleer and Tansuchat (2013) found that the CCC model performed better than the BEKK model when determining the optimal hedge ratio for crude oil spots and futures. Besides, Moon, Yu and Hong (2009) state that the CCC and OLS methods had a better hedging performance than the BEKK model on the Korean futures market. Next subsection is about an extended version of the CCC model which could be a better model than the previous models.

2.4.4 Determining the optimal hedge ratio with DCC

The second dynamic time-varying model is an extension of the CCC model from Bollerslev (1990). The model is called the Dynamic Conditional Correlated (DCC) model and was first proposed by Engle (2002). Engle’s (2002) model estimates ad-ditional parameters that capture the effect of previous shocks. His model is easy to compute, because it only has one extra step compared to the CCC model. Ku, Chen and Chen (2007) show that the DCC model outperformed the CCC model and the OLS model on the British and Japanese currency markets. Futhermore, Ham-moudeh, Yuan, McAleer and Thompson (2010) indicate that the DCC model gave better estimates compared to the BEKK model when hedging with metal-exchange rates.

However, Chang et al. (2013) showed that the CCC model, the BEKK model and the DCC model provide similar hedging effectiveness in the European, Japanese

(16)

and British currency markets. Moreover, Park and Jei (2010) indicate that DCC method is not superior to the OLS method at determining the hedge ratio in the soy bean and corn industry. There are contradictory conclusions based on the re-searches of Ku, Chen and Chen (2007) and Chang et al. (2013) for the same markets (Japanese and British currency markets). To able to say which method is better, there needs to be more research for different markets with recent data. Therefor the results of this thesis will contribute a lot to the existing literature for choosing the best method because this thesis will analyze Australian, Canadian and Euro-pean currency markets with the recent data. To compare the performances of OLS, CCC, BEKK and DCC method with each other, there are three hypotheses and these hypotheses will be explained in the next section.

2.5 Hypotheses

Several hypotheses in this paper arise based on previous research. The first hy-pothesis states that the minimum variance hedge ratio is more effective based on the dynamic models (BEKK and DCC model) than the minimum variance hedge ratio based on the static models (OLS and CCC) when hedging on the Australian, Canadian and European currency market (EUR/USD, AUD/USD and CAD/USD). The second hypothesis poses that based on the static OLS model is more effec-tive than the minimum variance hedge ratio based on the static CCC model when hedging on the Australian, Canadian and European currency market (EUR/USD, AUD/USD and CAD/USD). The last hypothesis states that the minimum variance hedge ratio based on the dynamic BEKK model will be more effective than the min-imum variance hedge ratio based on the dynamic DCC model when hedging on the Australian, Canadian and European currency market (EUR/USD, AUD/USD and CAD/USD). The next sections describe the data and the way that the hypothesis are tested.

(17)

3 Data

Daily closing prices of spot and futures for three foreign exchange rate series are used in the present study. These three series are the value of the US dollar to one Australian dollar (AUD/USD), the value of the US dollar to one Canadian dollar (CAD/USD) and the value of the US dollar to one European Euro (EUR/USD).

The 3993 observations from 1 January 2003 to 22 April 2018 are obtained from Thomson Reuters Datastream Financial Database. The spot prices are the values of the three currencies (AUD/USD, CAD/USD and EUR/USD) at the end of each trading day. The futures prices are derived from individual futures contracts. The futures prices are based on the daily continuous average settlement price of a fu-tures contract with an expiration date one month from that particular day. For instance, on the first of May 2018, the futures price is the price of the contract that expires in June 2018.

The returns of currency i at time t are calculated in the following way.

ri,t= log(

p_i,t

p_i,t−1) (3.1)

In (3.1) ri,tis the returns of currency i at time t, pi,tis the closing price of currency i

at time t and p_i,t−1is the closing price of currency i at time t−1. Table 3.1 shows the Augmented Dickey-Fuller test for the price and the returns of the three currencies (AUD/USD, CAD/USD and EUR/USD). The spot and return prices of the currencies are all non stationary and the spot and futures returns of the three currencies are all stationary. Thus, the spot and futures return time series are used for further calculations.

The Box-Pierce portmanteau statistics indicate significant correlation in the squared returns, this suggests the need to model the conditional heteroscedastic-ity (Lien et al., 2002). The estimates are in Appendix table A.1. The Ljung-Box statistics show that there exists serial correlation in some of the spot and futures

(18)

return series. These estimates are in Appendix table A.2.

Descriptive statistics of the return time series for the three currencies (AU-D/USD, CAD/USD and EUR/USD) are given in tables 3.2-3.4. The mean is close to zero for the spot and futures returns of three currencies. The standard deviation of the futures returns is smaller than that of the spot returns in European (EU-R/USD) and Canadian (CAD/USD) market. This indicates that the spot market is more volatile. Spot and futures returns in the Australian (AUD/USD) market show significant skewness to the left, indicating increased presence of losses than gains. Spot returns in the European and Canadian market show a slight skewness to the right, indicating increased presence of gains than losses. The spot and futures re-turns of all currencies show high kurtosis and show that the data is leptokurtic. In other words, the tails of the the spot and futures returns of all currencies are longer and fatter. The Jarque-Bera values suggest that spot and futures returns of the three currencies are not normally distributed.

Figure 3.1 presents plots of the spot and futures daily returns for all the cur-rencies. High positive and negative returns are present somewhere at the end of 2008 and continued in 2009. Thus, a higher volatility was present during the finan-cial crisis. The volatility on the Australian market seems to be higher than on the European and Canadian market during the financial crisis. The spot and futures returns seem to move in the same pattern, this suggest high correlations between the values of the time series data. To give a concrete answer on the research ques-tion, the next section will provide a description of the models that have been used in this paper.

(19)

Table 3.1: Augmented Dickey-Fuller test of prices and returns of the three currencies (AUD/USD, EUR/USD and CAD/USD).

ADF test

Price time series Returns time series

Currency (Currency cross code (CSS) ) p_s p_f r_s r_f

Australian Dollar (AUD/USD) 0.13446 0.20025 -44.953** -45.704**

European Euro (EUR/USD) 0.13445 0.32900 -44.788** -44.681**

Canadian Dollar (CAD/USD) 0.15256 0.22072 -45.583** -44.054**

Note: ps is the price of the spot (currency), pf is the price of the futures, rs is the return

of the spot (currency) and rf is the return of the futures. In the Augmented Dickey-Fuller

(ADF) test the null hypothesis states that there is a unit root present in the price or return time series (non stationary). Meaning that if the test statistic is in the critical region the time series is stationary. * indicates p-value < 0.05 and ** indicates p-value < 0.01 (The lower tail critical values are -2.58 and -1.95 at the 1% and 5% significance levels).

Table 3.2: Descriptive statistics of Australian spot and futures returns (AUD/USD) in percentages.

Returns AUD/USD Spot returns Futures returns

Mean 0.00776 0.00875 Maximum 6.70063 6.09168 Minimum -8.82827 -10.11862 Std. Dev. 0.82767 0.82855 Skewness -0.87716** -0.83511** Kurtosis 15.965** 14.256** Jarque-Bera 28473** 21538**

Note: * indicates p-value < 0.05 and ** indicates p-value <

(20)

Table 3.3: Descriptive statistics of European spot and futures returns (EUR/USD) in percentages.

Returns EUR/USD Spot returns Futures returns

Mean 0.00393 0.00582 Maximum 4.61715 3.08562 Minimum -3.84406 -3.06532 Std. Dev. 0.60735 0.59125 Skewness 0.09305* -0.04844 Kurtosis 5.8528** 4.8196** Jarque-Bera 1359.5** 552.26**

0.01.

Table 3.4: Descriptive statistics of Canadian spot and futures returns (CAD/USD) in percentages.

Returns CAD/USD Spot returns Futures returns

Mean 0.00542 0.00590 Maximum 5.04622 5.11602 Minimum -4.33752 -3.72207 Std. Dev. 0.61237 0.58684 Skewness 0.08959* -0.04242 Kurtosis 7.6046** 7.1438** Jarque-Bera 3532** 2857.4**

(21)

Figure 3.1: Spot and futures return

Note: Plots of daily spot and futures returns of the Australian, Canadian and European

currency are presented.

4 Methodology

The first subsection is about the econometric models. The econometric models con-sists of the OLS model, the CCC model, the BEKK model and the DCC model. The last subsection is about obtaining the minimum variance hedge ratio and the eval-uation of the performance of the four different models presented in the econometric models subsection.

(22)

4.1 Econometric models

4.1.1 The OLS model

The OLS model used in this research had the same structure as Lien et al. (2002) used in their study. The equation is constructed as follows.

r_s,t_{= α + βr}_{f ,t}_{+ ε}_t (4.1)

In (4.1) rf ,t is the return on the futures contract at period t (futures return) and

r_s,tis the return on the underlying currency at period t (spot return).εt is the error

term at time t and α is the constant included in the model and β is defined as

follows.

β =Cov(rs,t, rf ,t)

V ar(rf ,t)

(4.2) In (4.2) Cov(rs,t, rf ,t) is defined as the covariance between the spot returns and the

futures returns and V ar(rf ,t) is the variance of the futures returns. Next subsection

describes the CCC model of Bollerslev (1990).

4.1.2 The CCC model

This subsection gives a description about the CCC model first presented by Boller-slev (1990). His CCC model is based on the following equations.

y_t_{= E(y}_t_|Ψ_t−1_{) + ε}_t,

εt= Dtηt

(4.3)

In (4.3) yt is the spot rate returns of the currencies at time t,Ψt−1 is the information

set up until time t − 1, E(yt|Ψt−1) is the expectation of yt giving the information set

Ψt−1 and εt is the innovation at time t (Bollerslev, 1970). Also in (4.3) ηt is the

sequence of independently and identically distributed (i.i.d.) random values defined as_ηt= D−1/2_t εtand Dt= diag(σ2_s,t1/2,σ2_{f ,t}1/2) = diag(σs,t,σf ,t) (McAleer, 2005).σ2_s,t=

(23)

E(ε2_s,t|Ψt−1) is the conditional variance of the spot returns andσ2f ,t= E(ε 2

f ,t|Ψt−1) is

the conditional variance of the futures returns. Bollerslev (1990) indicates that the conditional variance for each return in the equation follows an univariate GARCH process. In this paper these equations are defined as follows.

σ2 s,t= ωs+ αsε2_s,t−1+ βsσ2_s,t−1 σ2 f ,t= ωf+ αfε2_{f ,t−1}+ βfσ2_{f ,t−1} αs+ βs≤ 1 αf+ βf ≤ 1 (4.4)

In equation (4.4)ωs andωf are the constants of the spots and futures return

uni-variate GARCH process, αs and αf represent the ARCH effects, better known as

the short-run persistence of shocks to the spot and futures returns and βs andβf

represent the GARCH effects.αs+ βs and αf + βf are also known as the long-run

persistence of shocks of the spot and futures returns and should be smaller than or equal to one to induce stationarity and avoid negative variances (McAleer, 2005; Bollerslev, 1986).σ2_s,t−1andσ2

f ,t−1are the variances of the spot and futures returns

at time t−1 and ε2_s,t−1andε2_{f ,t−1}are the squared innovations of the spot and futures returns at time t − 1 (Bollerslev, 1990). This paper uses the GARCH(1,1) model, be-cause Choo (1999), Longmore and Robinson (2004) and Engle (1993) studied the performance of GARCH models in forecasting the stock market and other asset prices volatility. They concluded that the GARCH(1,1) model is a parsimonious and effective model for modeling returns volatility.

The conditional correlation matrix of CCC is Γ= E(ηtη0_t|Ψt−1) = E(ηtη0_t), where

Γ= {ρi, j} for i, j = 1,2 (McAleer, 2005). From equation (4.3) the following can be

derived,εtε0_t= Dtηtη0_tDt= (diagQt)1/2 and E(εtε0_t|Ψt−1) = Qt= DtΓDt, where Qt is

the conditional covariance matrix (McAleer, 2005). Thus, the conditional correla-tion matrix is defined asΓ= D−1_t QtD−1_t and each conditional correlation coefficient

(24)

is estimated from the standardized residuals in equations (4.3) and (4.4) (McAleer, 2005). The next subsection describes the BEKK model developed by Engle and Kro-ner (1995).

4.1.3 The BEKK model

This subsection outlines the BEKK model first presented by Engle and Kroner (1995). The starting point of the model is the same as the CCC model of Bollerslev (1990).

y_t_{= E(y}_t_|Ψ_t−1_{) + ε}_t,

εt= Dtηt

(4.5)

The equation presented in (4.5) is the same as the equation in (4.3), so yt is the

spot rate returns of the currencies at time t, Ψ_t−1 is the information set up un-til time t − 1, E(yt|Ψt−1) is the expectation of yt giving the information set Ψt−1

and εt is the innovation at time t (Bollerslev, 1970). And again in (4.5) ηt is a

sequence of independently and identically distributed (i.i.d.) random values with

ηt= D−1/2_t εt and Dt= diag(σ2s,t1/2,σ2f ,t

1/2_{) = diag(σ}

s,t,σf ,t) (McAleer, 2005). Here is

σ2

s,t= E(ε2s,t|Ψt−1) the conditional variance of the spot returns andσ2f ,t= E(ε 2

f ,t|Ψt−1)

the conditional variance of the futures returns.

However, the difference is that the conditional variance is calculated in a dif-ferent way in this model. In the CCC model the conditional variance of the spot returns and the futures returns where obtained with a univariate GARCH estima-tion. After that the conditional covariance was obtained with Qt = DtΓDt. In the

BEKK model the covariance matrix Qt is obtained via a bivariate GARCH model

with the following formula (Engle & Kroner, 1995).

(25)

In (4.6) A and B are square coefficient matrices, C is a triangular coefficient matrix of constants,ε_t−1 are the residuals at time t − 1 and Qt−1 is the conditional

covari-ance matrix at t − 1 (Caporin & McAleer, 2005). Also the conditional correlations matrix is now dynamic instead of static. In the CCC model the conditional correla-tion matrix is a time-invariant 2-by-2 matrix. In the BEKK model the condicorrela-tional correlation matrix is still a 2-by-2 matrix, but now for every time period t. Thus, the conditional correlation matrix is now time-varying. The next subsection describes the DCC model first presented by Engle (2002).

4.1.4 The DCC model

In this subsection the DCC model of Engle (2002) will be discussed. Just like the BEKK model, the starting point of the model is in line with the CCC model of Boller-slev (1990).

y_t_{= E(y}_t_|Ψ_t−1_{) + ε}_t,

εt= Dtηt

(4.7)

Just as before the variables presented in (4.7) are specified in the same as in equa-tion (4.3). The first step is to calculate the univariate condiequa-tional variances with the same formula as Bollorslev (1990) used in the CCC model.

σ2 s,t= ωs+ αsε2_s,t−1+ βsσ2_s,t−1 σ2 f ,t= ωf+ αfε 2 f ,t−1+ βfσ 2 f ,t−1 αs+ βs≤ 1 αf+ βf ≤ 1 (4.8)

(26)

The variables presented in (4.8) are specified in the same way as in (4.4) . The sec-ond step in the DCC model is estimated using the GARCH(1,1) standardized resid-ualsηs,tandηf ,t(Caporin & McAleer, 2005). But, in order to interpret the dynamic

components as valid conditional correlations, an appropriate standardization of the conditional correlations is required. The standardized dynamic conditional correla-tions (for each timeperiod t) are estimated with the following equation.

Γt= (1 − θ1− θ2)Γ+ θ1ηt−1η0_t−1+ θ2Γt−1 (4.9)

In (4.9) θ1 and θ2 are scalar parameters obtained via loglikelihood optimization,

Γis the static 2-by-2 constant conditional correlation matrix, _η_t−1 is the matrix of standardized residuals at time t − 1 andΓt−1 is the dynamic 2-by-2 conditional

cor-relation matrix at time t − 1 (Caporin & McAleer, 2005). The conditional covariance matrix Qt in the DCC model is estimated in the same way as in CCC, Qt= DtΓtDt.

The only difference is that the 2-by-2 dynamic conditional correlationΓtis now

cal-culated for each timeperiod t. In other words, the dynamic conditional correlation is time-varying. However, if _θ1= θ2= 0 the DCC model of Engle (2002) will have

exactly the same results as CCC model of Bollerslev (1990). The next subsection is about obtaining the minimum variance hedge ratio with the four different models.

4.2 Obtaining the minimum variance hedge ratio

In this subsection the minimum variance hedge ratio for the four models is pre-sented. Chang et al. (2013) put forward the following equation for a hedged portfo-lio.

r_h,t_{= r}_s,t_{− γr}_{f ,t} (4.10)

In equation (4.10) rh,t refers to the return of the hedged portfolio between

timepe-riod t − 1 and t, rs,t is the spot returns of the underlying currency between

(27)

and t, and γ is the hedge ratio (Chang et al., 2013). Johnson (1960) indicates that the variance of a hedged portfolio is equal to the following equation.

V ar(rh,t) = V ar(rs,t) − 2γCov(rs,t, rf ,t) + γ2V ar(rf ,t) (4.11)

V ar(rh,t) in (4.11) is the variance of the hedged portfolio. V ar(rs,t), V ar(rf ,t) and

Cov(rs,t, rf ,t) are the variance of the futures and spot returns and the covariance

between the spot and futures returns. Taking the derivative of this equation with respect to γ minimizes the variance of the hedged portfolio (Johnson, 1960). This results in the following equation.

γ =Cov(rs,t, rf ,t)

V ar(rf ,t)

(4.12)

For the OLS model this means that the estimate ofβ is the time-invariant minimum variance hedge ratio. Because, as stated earlier,β is defined as:

β =Cov(rs,t, rf ,t)

V ar(rf ,t)

(4.13)

The other three (GARCH) models have a minimum variance hedge ratio that is time-varying and thus a vector of values. Also in these three models the variance and covariance are conditional on the previous t − 1 values. According to Johnson (1960), the following equation of the conditional variance of the hedged portfolio arises.

V ar(rh,t|Ψt−1) = V ar(rs,t|Ψt−1) − 2γtCov(rs,t, rf ,t|Ψt−1) + γ2tV ar(rf ,t|Ψt−1) (4.14)

V ar(rh,t|Ψt−1) in (4.14) is the time-varying variance of the hedged portfolio

condi-tional on the information set prior to time period t. V ar(rs,t|Ψt−1), V ar(rf ,t|Ψt−1)

and Cov(rs,t, rf ,t|Ψt−1) are the time-varying variance of the futures and spot

(28)

information set prior to timeperiod t.γtis the time-varying hedge ratio. Again,

min-imum variance hedge ratio is obtained by taking the derivative of the time-varying hedge ratioγt. The following equation is the result.

γt|Ψt−1=

Cov(rs,t, rf ,t|Ψt−1)

V ar(rf ,t|Ψt−1)

(4.15)

Thus, to calculate the time-varying minimum variance hedge ratio in the three GARCH models, the conditional covariance between the spot and futures returns and the conditional variance of the futures returns need to be obtained.

Ku et al. (2007) constructed a way to evaluate the performance of the minimum variance hedge ratios computed with the four different models. They did this with hedging effectiveness. Hedging effectiveness measures the variance reduction for any hedged portfolio compared with the unhedged portfolio (Ku et al. 2007). The hedging effectiveness index (HE) is defined as.

HE =var(u) − var(h)

var(u) (4.16)

In (4.16) var(u) is the variance of the unhedged portfolio, in this case the variance of the spot returns, and var(h) is the variance of the hedged portfolio, in this case the variance of the in equation (4.10) defined rh,t. A higher HE indicates more variance

reduction, so for instance if the DCC model has a higher HE than the CCC model than the DCC model is regarded as a better hedging strategy than the CCC model. In the next section gives a overview of the results obtained in this research.

5 Empirical results

The first subsection describes the estimates of the four models used in this research (OLS, CCC, BEKK and DCC) and comparison with existing literature. The second subsection shows the minimum variance hedge ratios (optimal hedge ratios) of the

(29)

four models (OLS, CCC, BEKK and DCC). It also states the performance of calcu-lated hedge ratios of the four models (OLS, CCC, BEKK and DCC) and compari-son with existing literature. Finally, the last subsection presents some robustness checks. These checks involve optimal hedge ratios of the four models (OLS, CCC, BEKK and DCC) for different time periods.

5.1 Estimates of the models

In this paper four models (OLS, CCC, BEKK and DCC) were used to calculate the minimum variance hedge ratio. The estimated parameters are reported in tables A.3-A.6 of the Appendix.

5.1.1 Estimates of the OLS model

Table A.3 shows the estimates of the OLS model. Constants in the three currency

markets (AUD/USD, EUR/USD and CAD/USD) are not significant. The β

coeffi-cient in the three currencies markets are all significant, ranging from 0.79051 for the Australian Dollar and 0.82604 for the European Euro. The coefficient of deter-mination (R2) is in the range of 0.5986 for the Canadian market and 0.6467 for the European market.

5.1.2 Estimates of the CCC model

Table A.4 displays the estimated paramaters of the CCC model. The second moment and log moment condition are satisfied in spot and futures returns of all markets. This is a sufficient condition for Quasi-Maximum Likelihood Estimation (QLME) and asymptotically normal distributed (McAleer, Hoti, & Chan, 2009). The long run persistence of shocks,α + β, is quite high in all markets with the lowest value the Canadian futures returns (0.99215) and highest value the European spot returns (0.99748). All values ofα+β satisfy the condition stated in the methodology section.

(30)

ef-fects are small (highest value 0.04700) and the GARCH efef-fects are high (smallest value 0.94899). The ARCH effects are relatively higher on the Australian market (0.04700 and 0.04423 against 0.03007 and 0.02775 on the European market and 0.03436 and 0.03411 on the Canadian market). Also the GARCH effects are rela-tively lower on the Australian market (0.94899 and 0.95204 against 0.96741 and 0.96942 on the European market and 0.96013 and 0.95804 on the Canadian mar-ket). All constant conditional correlations (ρ) are significant.

The conditional correlation on the Canadian market is relatively lower (0.76590 against 0.80489 on the European market and 0.80203 on the Australian market. The conditional covariance of the three currency markets calculated with the CCC model are given in Figure A.1 of the Appendix. High spikes in the conditional co-variance are perceived at the time of the financial crisis. The log-likelihood is the highest (lowest) on the European market, -4587.44, (Australian market -6437.79) indicating that the log-likelihood is best (worst) optimized on the European market (Australian market).

5.1.3 Estimates of the BEKK model

In Table A.5 are the estimated parameters of the BEKK model displayed. Almost all constants in the triangular constants matrix C are significant. Furthermore, almost all coefficients in the A and B matrices are significant. These results indicate that there is a time-varying market risk, strong evidence for presence of GARCH effects (the B matrix) and weak evidence for presence of ARCH effects (the A matrix). Also ARCH and GARCH effects of the spot returns in almost all currency markets had an significant effect on the futures returns.

Conditional correlations and covariances are presented in Figures A.2 and A.3. Figure A.2 shows time-varying conditional correlations with a lot of downward spikes in the Australian and European market. Figure A.3 shows time-varying co-variances with high spikes at the time of the financial crisis. The log-likelihood

(31)

is the highest (lowest) on the European market, 4527.35, (Australian market -6437.79) indicating that the log-likelihood is best (worst) optimized on the European market (Australian market).

5.1.4 Estimates of the DCC model

The estimates of the DCC model are presented in Table A.6. The long run per-sistence of shocks, α + β, is quite high in all markets with the lowest value the Canadian futures returns (0.99272) and highest value the European futures re-turns (0.99764). All values ofα + β satisfy the condition stated in the methodology section. Just like CCC, ARCH effects () and GARCH effects () are all significant. Again, the ARCH effects are low (all lower than 0.055) and the GARCH effects are high (all higher than 0.940).θ1 is significant in all markets, but θ2 is only

signifi-cant in the Canadian market. Table A.7 shows thatθ1andθ2are jointly significant,

indicating that the DCC is a better optimized model compared to the CCC.

The conditional correlation and covariances are given in Figure A.4 of the Ap-pendix. There seem to be downward spikes in the conditional correlations, espe-cially after the financial crisis in 2008 and 2009. Conditional covariances have ex-tremely high spikes during the financial crisis. The log-likelihood is the highest (lowest) on the European market, -4478.79, (Australian market -6343.61) indicat-ing that the log-likelihood is best (worst) optimized on the European market (Aus-tralian market).

5.1.5 Comparison existing literature

The OLS estimates are comparable with the estimates in the research of Lien et al. (2002). The estimates based on the CCC and DCC model are for the most part in line with the estimates that Chang et al. (2013) found in their study. An important difference was thatθ2 was significant in the research of Chang et al. (2013), but it

(32)

are roughly the same as the estimates Chang et al. (2013) found. The only differ-ence is that some off diagonal values in the A and B matrices are negative in this research. The next subsections shows results of the optimal hedge ratio (minimum variance hedge ratio).

5.2 Hedge ratios and hedging effectiveness

5.2.1 Hedge ratios

The hedge ratio obtained in the OLS is the calculatedβ in the model. For the three GARCH models the (conditional) variances and covariances of the futures and spot returns of all markets need to be obtained with the underlying parameters of each model. Afterwards, the optimal hedge ratio and the hedging effectiveness are calcu-lated with equations (4.15) and (4.16).

Tables 5.1-5.3 report the optimal hedge ratio, variance of the hedged portfolio, the hedging effectiveness and the variance of the unhedged portfolio for the three currencies using spot and futures returns. Tables 5.1-5.3 show that hedging is ef-fective in reducing the variance of the portfolio with all models for every currency. The highest average optimal hedge ratio (0.8291) is on the European market using the CCC model. This means that in order to minimize the risk on European mar-ket, a long (buy) position of one euro should be hedged with a short (sell) position of 0.8291 euros in futures contracts. The lowest optimal hedge ratio (0.7867) is on the Australian market using the BEKK model. The optimal hedge ratios calculated with the four models on the Australian market are close together. However, there seem to be some differences in the optimal hedge ratio between the different mod-els on the European and Canadian markets. On the European market the highest optimal hedge ratio value is obtained with the CCC model (0.8291) and the lowest optimal hedge ratio is obtained with the BEKK model (0.7894). The highest optimal hedge ratio on the Canadian market is obtained with the BEKK model (0.8127) and the lowest optimal hedge ratio is obtained with the DCC model (0.7936).

(33)

5.2.2 Hedging effectiveness

The hedging effectiveness of all models for the three currency markets is at least 58%, meaning that all models ensure at least 58% variance reduction. The hedging effectiveness is relatively smaller on the Canadian market (ranging from 58.98% until 62.10%) than on the European (ranging from 64.62% until 68.73%) and Aus-tralian (ranging from 62.62% until 64.96%) markets. Dynamic time-varying optimal hedge ratios (BEKK and DCC models) seem to have better hedging effectiveness than static optimal hedge ratios (OLS and CCC) on the Australian and European markets. The optimal hedge ratio of the DCC model (58.98%) has the worst hedging effectiveness on the Canadian market. However, the hedging effectiveness of the BEKK model (62.10%) is by far the best on the Canadian market. The hedging ef-fectiveness of the BEKK model is the best on all currency markets. The OLS model has by far the worst hedging effectiveness on the Australian market (62.62%). The CCC model has the worst hedging effectiveness on the European (64.62%) market.

Figures A.6-A.8 show the optimal hedge ratios of the four different models on each currency market over time. The optimal hedge ratio based on the dynamic models (DCC and BEKK model) fluctuate more than the static models (OLS and CCC model) on all currency markets. Figure A.6 shows the optimal hedge ratio on the Australian market. The optimal hedge ratios of the dynamic models (DCC and BEKK) fluctuate more during and after the financial crisis than before that finan-cial crisis on the Australian market. Figure A.7 shows the optimal hedge ratio on the European market. The optimal hedge ratio based on the BEKK model fluctu-ates more than the DCC model on the European market. Furthermore, the optimal hedge ratio based on the CCC, BEKK and DCC model all show an upward spike at the beginning of 2017 on the European market. Figure A.8 shows the optimal hedge ratio on the Canadian market. The optimal hedge ratio based on the BEKK model fluctuates more than the DCC model on the Canadian market. Also, during the financial crisis and few years after the financial crisis the optimal hedge ratios

(34)

based on the CCC, BEKK and DCC model seem to fluctuate more than in other time periods on the Canadian market.

5.2.3 Comparison existing literature

The optimal hedge ratios calculated in this paper are comparable to the optimal hedge ratios reported in Chang et al. (2013) and Yang and Allen (2004). Although, the optimal hedge ratios on the European market seem to be a little bit higher (except the optimal hedge ratio based on the BEKK model) in this research than the optimal hedge ratios reported in the research of Chang et al. (2013). The hedging effectiveness in the present paper and the hedging effectiveness reported in the paper of Chang et al. (2013) are very similar. However, the hedging effectiveness is somewhat higher in the present paper than the hedging effectiveness in the article of Chang et al. (2013) when hedging on the European market. Robustness checks are discussed in the next subsection.

Table 5.1: Hedging performance on the Australian market (AU-D/USD).

Model OHR Var. HP HE Var. UP

OLS 0.7905 0.2560 62.62% 0.6850

CCC 0.7964 0.2453 64.19% 0.6850

BEKK 0.7867 0.2400 64.96% 0.6850

DCC 0.7951 0.2450 64.23% 0.6850

Note: Hedging performance on the Australian market for the four models

(OLS, CCC, BEKK and DCC). OHR is the optimal hedge ratio, also known as the minimum variance hedge ratio (for the CCC, BEKK and DCC the mean of the OHR vector is taken), Var. HP is the variance of the hedged portfolio, HE is the hedging effectiveness in percentage and Var. UP is the variance of the unhedged portfolio. In bold text the most effective hedging strategy.

(35)

Table 5.2: Hedging performance on the European market (EU-R/USD).

OLS 0.8260 0.1303 64.67% 0.3689

CCC 0.8291 0.1305 64.62% 0.3689

BEKK 0.7894 0.1153 68.73% 0.3689

DCC 0.8288 0.1302 64.70% 0.3689

Note: Hedging performance on the European market for the four models

Table 5.3: Hedging performance on the Canadian market (CAD/USD).

OLS 0.8074 0.1505 59.86% 0.3750

CCC 0.7963 0.1532 59.14% 0.3750

BEKK 0.8127 0.1421 62.10% 0.3750

DCC 0.7936 0.1538 58.98% 0.3750

Note: Hedging performance on the Canadian market for the four models

5.3 Robustness checks

The models were tested in three different software programs (RStudio, Eviews and Stata). All software programs led to the same results. Also, optimal hedge ratios and hedging performance for different time periods were calculated. The results

(36)

are presented in the next subsections.

5.3.1 Hedging ratios and performance from 2003 until 2010

Results based on the the time period from 2003 until 2010 are presented in tables A.8-A.10. The optimal hedge ratio based on the DCC model is the most effective on the Australian (66.36%) and on the European market (66.56%). The optimal hedge ratio based on the BEKK model is the most effective on the Canadian market (65.67%). Again, it seems that the optimal hedge ratios of the dynamic models (DCC and BEKK) perform better than the static models (OLS and CCC). This is in line with the results Chang et al. (2013) found. However, the optimal hedge ratio based on the BEKK model is the least effective on the Australian market (60.97%) and the European market (64.91%). The optimal hedge ratio based on the OLS model has the worst effectiveness on the Canadian market (63.91%).

5.3.2 Hedging ratios and performance from 2010 until 2018

Results based on the the time period from 2010 until 2018 are presented in tables A.11-A.13. The optimal hedge ratio based on the BEKK model is the most effective on the Australian (64.76%), European (64.74%) and Canadian market (56.69%). These findings are not in line with the results that Chang et al. (2013) found. It still seems that the optimal hedge ratios based on the dynamic models (BEKK and DCC) are more effective than the optimal hedge ratios based on the static models (OLS and CCC). On the other hand, the optimal hedge ratio based on the OLS model has the second best effectiveness on all currency markets. The optimal hedge ratio based on the CCC model has the worst effectiveness on the Australian (62.11%), European (62.90%) and Canadian market (53.29%). The next section states the conclusion of this research.

(37)

6 Conclusion

This research compared the effectiveness of the minimum variance currency hedge ratio of the European, Australian and Canadian currency futures markets (EU-R/USD, AUD/USD and CAD/USD) using four different methods (OLS, CCC, BEKK and DCC). The findings of this research showed that the BEKK model is the most effective method to calculate the minimum variance currency hedge ratio on all currency futures markets. The other three models (OLS, CCC, DCC) showed some mixed results when calculating the minimum variance hedge ratio on the Euro-pean, Australian and Canadian currency futures market.

The dynamic time-varying models (BEKK and DCC) were more effective than the static models (OLS and CCC) when hedging on the Australian and Canadian currency futures markets. This is in line with the first hypothesis. When determin-ing the effectiveness of minimum variance hedge ratio on the Canadian currency futures market the dynamic BEKK model performed better than the static mod-els (OLS and CCC), but the dynamic DCC model performed worse than the static models (OLS and CCC). This indicates that not all results are in line with the first hypothesis. The effectiveness of minimum variance hedge ratio based on the OLS model was better than the effectiveness of minimum variance hedge ratio based on the CCC model on the European and Canadian currency futures markets. This is in line with the second hypothesis. However, the CCC model performed better than the OLS model when comparing the effectiveness of the minimum variance hedge ratio on the Australian currency futures market. This result shows that not all results correspond with the second hypothesis. Finally, the effectiveness of min-imum variance hedge ratio based on the BEKK model was better the effectiveness of minimum variance hedge ratio based on the DCC model on all currency futures markets. This outcome corresponds with the last hypothesis.

(38)

mod-els (OLS and CCC) when determining the minimum variance hedge ratio on the European and Australian currency futures market and the DCC model was least effective on the Canadian market than the static models (OLS and CCC). It could be that the time period analyzed in this research does not have a strong dynamic time-varying component in it. On the other hand, The BEKK model showed way better effectiveness than the static models (CCC and OLS) when determining the minimum variance hedge ratio on all currency futures markets. Thus, it seemed that there was a dynamic time-varying component present in data. In short, this research suggests that researchers should take into account the dynamic time-varying component when determining the minimum variance hedge ratio on the currency futures markets.

In this research optimization of a bivariate GARCH model (BEKK model) showed way better effectiveness than optimizing two seperate univariate GARCH models (CCC and DCC model) when determining the minimum variance hedge ratio on the Australian, Canadian and European currency futures markets. Based on this re-search, investigators should take into consideration the ARCH and GARCH effects of spot returns on futures returns when determining the minimum variance hedge ratio and vice versa.

It seems that different time periods show mixed results when determining the minimum variance hedge ratio. The time-varying minimum variance hedge ratios based on the CCC, BEKK and DCC models perform better than the time-invariant minimum variance hedge ratio based on the OLS model in times with high volatil-ity (the period from 2003 until 2010). There is more fluctuation in the minimum variance hedge ratio during the financial crisis of 2008 and 2009. This suggests that the minimum variance hedge ratio is time-varying during times of high un-certainty. However, the time-invariant minimum variance hedge ratio based on the OLS model has a higher effectiveness than the time-varying minimum variance hedge ratios based on the CCC and DCC model in times with relatively low

(39)

volatil-ity (the period from 2010 until 2018). This suggests that a time-invariant minimum variance hedge ratio is better than time-variant minimum variance hedge ratios in times with relatively low volatility. But, the time-varying minimum variance hedge ratio based on the BEKK model shows better effectiveness than the time-invariant minimum variance hedge ratio based on the OLS model on all currency futures markets. It could be that the minimum variance hedge ratio based on the BEKK model is most effective in times of low volatility en the DCC model is most effective in times of high volatility. That is why it could be of interest to investigate with dif-ferent models that capture time-varying aspects when determining the minimum variance hedge ratio during financial crises. Also, it would be interesting to examine different models that determine the minimum variance hedge ratio in times with relatively low volatility.

This research only used futures contracts that had a expiration date one month from the agreement date. It could be of interest to investigate futures contracts with longer maturities, for instance two or three months until the expiration date. Maybe these futures contracts or a mixture of futures contracts with different maturities could lead to more effectiveness in the minimum variance hedging ratio in currency futures markets.

(40)

Bibliography

Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of econometrics, 31(3), 307-327.

Bollerslev, T. (1990). Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH model. The review of economics and statistics, 72(3), 498-505.

Brooks, C., & Chong, J. (2001). The cross-currency hedging performance of implied versus statistical forecasting models. Journal of Futures Markets, 21(11), 1043-1069.

Caporin, M., & McAleer, M. (2008). Scalar BEKK and indirect DCC. Journal of Fore-casting, 27(6), 537-549.

Chang, C. L., González-Serrano, L., & Jimenez-Martin, J. A. (2013). Currency hedg-ing strategies ushedg-ing dynamic multivariate GARCH. Mathematics and Comput-ers in Simulation, 94(1), 164-182.

Chang, C. Y., Lai, J. Y., & Chuang, I. Y. (2010). Futures hedging effectiveness under the segmentation of bear/bull energy markets. Energy Economics, 32(2), 442-449.

Chang, C. L., McAleer, M., & Tansuchat, R. (2011). Crude oil hedging strategies us-ing dynamic multivariate GARCH. Energy Economics, 33(5), 912-923.

Chong, C. W., Ahmad, M. I., & Abdullah, M. Y. (1999). Performance of GARCH models in forecasting stock market volatility. Journal of forecasting, 18(5), 333-343.

Ederington, L. H. (1979). The hedging performance of the new futures markets. The Journal of Finance, 34(1), 157-170.

Engle, R. (2002). Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. Journal of Business & Economic Statistics, 20(3), 339-350.

Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica: Journal of the

Econometric Society, 50(4), 987-1007.

Engle, R. F. (1993). Statistical models for financial volatility. Financial Analysts Journal, 49(1), 72-78.

Engle, R. F., & Kroner, K. F. (1995). Multivariate simultaneous generalized ARCH. Econometric theory, 11(1), 122-150.

Hammoudeh, S. M., Yuan, Y., McAleer, M., & Thompson, M. A. (2010). Precious met-als–exchange rate volatility transmissions and hedging strategies. International Review of Economics & Finance, 19(4), 633-647.

Herbst, A. F., Kare, D. D., & Marshall, J. F. (1989). A time-varying, convergence adjusted hedge ratio model. Advances in Futures and Options Research, 6(1), 137-155.

(41)

Johnson, L. L. (1960). The theory of hedging and speculation in commodity futures. The Review of Economic Studies, 27(3), 139-151.

Ku, Y. H. H., Chen, H. C., & Chen, K. H. (2007). On the application of the dynamic conditional correlation model in estimating optimal time-varying hedge ratios. Applied Economics Letters, 14(7), 503-509.

Lien, D., Tse, Y. K., & Tsui, A. K. (2002). Evaluating the hedging performance of the constant-correlation GARCH model. Applied Financial Economics, 12(11), 791-798.

Longmore, R., & Robinson, W. (2004). Modelling and forecasting exchange rate dy-namics: an application of asymmetric volatility models. Bank of Jamaica, Work ing Paper, WP2004, 3, 191-217.

McAleer, M. (2005). Automated inference and learning in modeling financial volatil-ity. Econometric Theory, 21(1), 232-261.

McAleer, M., Hoti, S., & Chan, F. (2009). Structure and asymptotic theory for mul-tivariate asymmetric conditional volatility. Econometric Reviews, 28(5), 422-440.

Moon, G. H., Yu, W. C., & Hong, C. H. (2009). Dynamic hedging performance with the evaluation of multivariate GARCH models: evidence from KOSTAR index futures. Applied Economics Letters, 16(9), 913-919.

Myers, R. J., & Thompson, S. R. (1989). Generalized optimal hedge ratio estimation. American Journal of Agricultural Economics, 71(4), 858–868.

Park, S. Y., & Jei, S. Y. (2010). Estimation and hedging effectiveness of time-varying hedge ratio: Flexible bivariate garch approaches. Journal of Futures Markets, 30(1), 71-99.

Su, W., & Huang, Y. (2010). Comparison of multivariate GARCH models with appli-cation to zero-coupon bond volatility.

Yang, W., & Allen, D. E. (2005). Multivariate GARCH hedge ratios and hedging effectiveness in Australian futures markets. Accounting & Finance, 45(2), 301-321.

(42)

A

Appendix

A.1 Statistics of the return time series

Table A.1: Box and Pierce portmanteau test on the squared returns.

Returns Lags Statistic

AUD/USD Spots 5 1533.34 (0.000) 10 2295.93 (0.000) 15 3428.14 (0.000) Futures 5 1127.87 (0.000) 10 1460.99 (0.000) 15 2226.28 (0.000) EUR/USD Spots 5 246.12 (0.000) 10 419.93 (0.000) 15 621.54 (0.000) Futures 5 268.06 (0.000) 10 504.47 (0.000) 15 743.88 (0.000) CAD/USD Spots 5 478.88 (0.000) 10 995.43 (0.000) 15 1624.30 (0.000) Futures 5 459.19 (0.000) 10 975.73 (0.000) 15 1553.07 (0.000)

Note: (AUD/USD) are the returns based on the Australian market, (EUR/USD)

are the returns based on the European market and (CAD/USD) are the returns based on the Canadian market. Lags refers to the amount of delayed time peri-ods of the time series. The statistic is evaluated on the χ2(5) distribution with 5 lags, χ2(10) distribution with 10 lags andχ2(15) distribution with 15 lags. In the Box-Pierce portmanteau test the null hypothesis states that there is no serial correlation (data independent distributed) and the alternative hypothesis states that there is serial correlation. P-values are given next to the statistic in paren-theses.

Investing with minimum risk : evaluating the futures hedging performance of GARCH models

Faculty of Economics and Business,

Amsterdam School of Economics

Econometrics

Bachelorthesis Econometrics

Investing with minimum risk

Evaluating the futures hedging performance of GARCH

models

Name: Wesley Beerepoot

Studentnumber: 10191186

Supervisors: Ms D. (Derya) Güler

Date: June 26, 2018

Group 1

2017-2018

Block 5 and 6

Statement of Originality

Acknowledgements

Abstract

Contents

1

Introduction

2

Literature review

2.1

Foreign exchange market

2.2

Futures market and future contracts

2.3

Hedging with futures contracts

2.4

Econometrics methods to determine the optimal hedge

2.5

Hypotheses

3

Data

4

Methodology

4.1

Econometric models

4.2

Obtaining the minimum variance hedge ratio

5

Empirical results

5.1

Estimates of the models

5.2

Hedge ratios and hedging effectiveness

5.3

Robustness checks

6

Conclusion

Bibliography

A

Appendix

A.1

Statistics of the return time series