Universiteit van Amsterdam
Econometrics and Operations Research
Bachelor Thesis
The Brexit effect: evaluating currency
hedging performance using different
models in UK futures markets
Author:
Brendan Nak
11037482
Supervisor:
Derya Güler
26 June 2018
Abstract
As a result of the Brexit, the GBP largely decreased in value and foreign exchange markets entered a period of increased volatility. Current paper compares the hedging performance of three multivariate models and the static OLS model for four currency pairs in attempt to find the optimal hedging strategy for minimizing investors’ exposure to currency risk during the Brexit. Literary background on modelling time series implies the multivariate models as a better fit for the time-varying nature of exchange rate returns. Empirical research conducted in current paper proves the static model to be the most effective hedging strategy in most cases.
This document is written by Brendan Nak who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it. The faculty of Economics and Business is responsible solely for the supervision and completion of the work, not for the contents.
Contents
1 Introduction 4
2 Theoretical framework 6
2.1 Currency risk: Brexit example . . . 6
2.2 Hedging . . . 7
2.2.1 How to hedge . . . 7
2.2.2 To hedge or not to hedge . . . 7
2.3 Futures . . . 8
2.4 Optimal hedging ratio . . . 9
2.4.1 The historical hedge ratios . . . 9
2.4.2 Objective optimizing hedge ratio . . . 10
2.5 Model specification . . . 10
2.5.1 Static approach . . . 10
2.5.2 Conditional heteroscedasticity modelling . . . 11
2.5.3 Constant conditional correlation . . . 11
2.5.4 Asymmetric modelling . . . 12
2.5.5 Dynamic conditional correlation . . . 13
2.5.6 Past results . . . 13 2.5.7 Hedging effectiveness . . . 14 3 Data 14 4 Methodology 16 4.1 OLS . . . 17 4.2 CCC . . . 17 4.2.1 CCC model specification . . . 17 4.2.2 CCC estimation procedure . . . 19 4.3 VARMA-AGARCH . . . 19
4.3.2 VARMA-AGARCH estimation procedure . . . 20
4.4 DCC . . . 21
4.4.1 DCC model specification . . . 21
4.4.2 DCC estimation procedure . . . 22
4.5 Optimal Hedge Ratio . . . 22
5 Empirical results 24 5.1 Estimation results . . . 24 5.1.1 OLS . . . 24 5.1.2 GARCH(1,1) . . . 24 5.1.3 VARMA-AGARCH . . . 26 5.1.4 DCC . . . 27 5.2 Correlation . . . 28
5.2.1 Constant conditional correlation . . . 28
5.2.2 Dynamic conditional correlation . . . 29
5.3 Hedging performance . . . 29 5.3.1 Australian Dollar . . . 30 5.3.2 Euro . . . 31 5.3.3 Japanese Yen . . . 32 5.3.4 US Dollar . . . 33 6 Conclusion 34
1
Introduction
On the 23rd of July 2016, the United Kingdom (UK) voted in favour of leaving the European Union (EU) after 43 years of membership. The decision to leave the EU, referred to as the Brexit, initiated immense economic and political uncertainty, causing a sharp devaluation of the Great British Pound sterling (GBP). At the end of the first trading day after the announcement, the GBP had lost almost 10% of its value. This led to a severe decrease in the value of assets held in the UK by international investors (Plakandaras, Gupta, & Wohar, 2017). The question as to how investors can deal with such unexpected exchange rate fluctuation arises.
Powered by globalization, investors increasingly diversify their portfolios by including international securities (Aiolfi & Sakoulis, 2017). Because of the relatively low correlation between securities from different geographical areas, investors are able to reduce their portfo-lio risk substantially by including international investments (Heston & Rouwenhorst, 1994). Along with the increased consumer demand for foreign goods and services, these investments led to an average daily trading volume on exchange markets that exceeded $5 trillion in April 2016 (BIS,2016). The enormous volume results in a continuously fluctuating exchange rate, exposing investors to another form of risk, specifically currency risk. Solnik (1974) makes a case for the benefits of international diversification when investors impose a hedging strategy that partly offsets the risk of currency devaluation.
Hedging is a strategic form of investing that makes use of derivatives in order to reduce the risk that investors face from potential future movements of a market variable (Hull, 2012, p.47). Currency hedging therefor has its purpose to reduce part of the risk that comes from exchange rate fluctuations. The general way currency hedging occurs is described by Chang, González-Serrano and Jimenez-Martin (2013) as short selling an amount of futures contracts when holding a long position in the underlying currency. The hedge ratio is defined as the amount of short positions on futures contracts per long position on the spot rate and can be used the argument in optimization processes of investors. Investors minimize their portfolio risk by choosing a so called Optimal Hedge Ratio (OHR).
Several models have been used to estimate the optimal hedge ratio, starting with the ordinary least squares (OLS) regression where the return on spot rates functions as dependent variable, the return on futures rates as independent variable and the OHR is defined as the slope of the regression line. Chang et al. (2013) and Lien, Tse and Tsui (2002) describe this method as the static approach because it assumes spot and futures returns to be time-invariant. This assumption is often not statistically supported and therefor rejected when tested against an alternative hypothesis. The rejection motivates researchers to look further and examine the performance of models that allow for time-varying second moments.
In past research, the Generalized Autoregressive Conditional Heteroscedatic (GARCH) model by Bollerslev (1986) proves to perform well in capturing the time-varying nature of returns and volatility. Based on research conducted by Chang et al. (2013), this paper analyzes three multivariate models; the Constant Conditional Correlation GARCH model (CCC), the Dynamic Conditional Correlation GARCH model (DCC) and the Vector Autore-gressive Moving-Average Asymmetrical GARCH model (VARMA-AGARCH), all of which are an extension of the original GARCH model by Bollerslev (1986). Along with the static OLS approach, the above-mentioned models are used to calculate the optimal hedge ratios. Their performance is thereafter compared on the basis of the Hedging Effectiveness Index (HE). A higher HE score indicates a larger relative risk reduction and a better performing hedge strategy (Ku, Chen, H.C., & Chen, K.H., 2007).
Current research evaluates the effect the Brexit had on hedging performances of four mod-els (OLS, CCC, VARMA-AGARCH and DCC). By comparing the hedging effectiveness over three periods, for four currency pairs (GBP/USD, GBP/EUR, GBP/JPY and GBP/AUD), this paper contributes to existing literature in providing a clear result regarding the ability to minimize investors’ exposure to currency risk when political influences cause large shocks on the foreign exchange market . Additionally, the empirical research on all observations in-dicates a best-performing hedging strategy. The fact that this model outperforms the other models for three out of four currency pairs, implies that for investors with long term holding periods, the simplest model often leads to the largest relative risk reduction.
section presents and shortly analyzes the data used for current research. The fourth section illustrates the research methodology and provides the derivation of the OHR using different econometric model specifications. The fifth section analyzes the hedging performance in UK futures markets of all four models before and after the Brexit. The last section concludes.
2
Theoretical framework
The following subsections discuss the literature regarding currency hedging, investors’ main tool for reducing their exposure to currency risk. At first, the definition of hedging is for-mulated and its importance is discussed. Thereafter, the hedging process and the hedging strategies this paper considers are described. Finally, a literary review on the use of different econometric models relevant to current research is discussed.
2.1
Currency risk: Brexit example
In order to get a better understanding of the severity of currency risk, one should consider an US based investor who bought in on the FTSE 100 UK market index on June the 23rd of 2016, with the purpose of holding the index stock for a week. Besides the negative return of 3.1% on the index that week, the GBP/USD exchange rate decreased by 10.5%1. The negative return on the FTSE index is the result of market risk, a form of risk that all investors face, analyze and balance of against the risk premium (Markowitz, 1952). Markowitz (1952) concludes that by extensively diversifying ones portfolio over a wide range of different asset classes, market risk exposure, would be the only risky factor of this portfolio. This conclusion does not hold for the US based investors in the situation mentioned above as the currency risk they were exposed to significantly declined the value of their assets held in the UK.
Currency risk is recognized as a relevant factor that influences portfolio risk, and even in case investors solely hold domestic securities they should be aware of the consequences of currency risk and the tools to deal with it. Early research by Solnik (1974) provides clear empirical evidence that, by including international stocks, investors are able to reduce the
riskiness of their portfolio. Heston and Rouwenhorst (1994) conclude that because industries are unequally represented over countries, the correlations between international stock in-dices are relatively small and significant portfolio improvements are obtained when investors include securities from a wide range of different countries. For this reason, they advise do-mestic investors to be aware of the benefits of international diversification because, in terms of opportunity costs, domestic investors suffer losses by excluding international stocks. Most investors find the benefits of international diversification appealing but fear the added risk that comes from currency fluctuations. For this reason, current research aims at evaluating the best performing econometric models for hedging purposes.
2.2
Hedging
2.2.1 How to hedge
By making use of derivatives, hedging allows investors to reduce losses when assets in their portfolio decrease in value (Yang & Allen, 2004). In the process of constructing a portfolio, hedgers identify the risks they face and decide what risks are acceptable. When the unaccept-able risks are identified, hedgers will implement a strategy that reduces or even eliminates that risk (Hull, 2012, p. 11). A widely used hedging strategy is to take on a short position in the futures contract when holding a long position in the underlying security (Chang, McAleer, & Tansuchat, 2011) . When the market variable unexpectedly decreases in value, the short position generates a positive return and thus functions as a form of insurance against adverse price movements. On the other hand, when the market variable increases in value, the short position will reduce profits as well (Husted, Rogers, & Sun, 2017). This raises the question whether hedging really is an effective investment strategy.
2.2.2 To hedge or not to hedge
Based on the analysis of 132 currency pairs, Kim (2012) concludes that not hedging against currency risk does not show any significant differences with respect to an optimally hedged portfolio when optimizing the Sharpe ratio and that unhedged portfolios outperform fully
hedged portfolios in most cases.
Glen and Jorion (1993) on the other hand, provide strong statistical evidence favouring the use of a hedge strategy over not hedging when considering an international portfolio containing bonds and equities. Their analysis focuses on the relation between monetary policy uncertainty and the excess return on the foreign exchange market. They conclude that due to increasing uncertainty regarding monetary policies, investors’ view on the riskiness of assets held in foreign countries change, and an excess return on exchange rates is demanded. This conclusion is in line with the conclusion by Froot (1993). He argues that a full-hedge strategy is favourable in short term investing horizons but becomes less effective when investors have long term goals. He claims that because the expected returns on exchange rates equal zero in the long run, fully hedging a position would increase costs through transaction costs without reducing the risk of that position. Like Glen and Jorion (1993), he concludes that in short term investing, especially in times of political uncertainty, the expected return on exchange rates can differ from zero, and a full-hedge strategy optimally deals with the exposure to currency risk.
Perold and Schulman (1988) state that a full-hedge strategy should be the norm when investing in foreign stock and bonds at all time. They conclude that a fully hedged portfolio does not decrease returns while significantly reducing risk compared to an unhedged portfolio.
2.3
Futures
The Brexit example, as described in §2.1, makes a case in favour of implementing a hedge strategy in order to offset the riskiness of exchange rate fluctuations. To reduce losses incurred due to currency devaluation, investors use futures to take on a second, opposite, position in a currency.
Chang et al. (2011) describe a futures contract as an agreement between parties to buy and sell products or securities for a given price at a given time. Futures on exchange rates therefor, allow investors to buy or sell currencies at a prespecified price in the future. While the majority of investors use the futures to hedge the value of their assets against potential losses (Yang & Allen, 2004), Chang et al. (2011) recognize speculators as another group
of investors who extensively make use of futures. Speculators, in contrary to hedgers, use futures to increase their risk by taking a long position in futures contracts when holding a long position in the underlying asset. The double bet of speculators increases their exposure to risk but will generate high returns in the best case scenario. For both parties holds, when the prespecified price differs from the price at time of maturity, profits and losses arise due to an arbitrage opportunity (Hull, 2012, p. 13).
2.4
Optimal hedging ratio
According to Yun and Kim (2010), the matter of interest for hedgers is the decision as to what amount of short positions in futures contracts optimally offsets their exposure to currency risk. They define the amount of short positions in a futures contract per long position in the underlying security as the hedge ratio.
2.4.1 The historical hedge ratios
In early research by Perold and Schulman (1988) and Eun and Resnick (1988) the fully hedged portfolio was often considered. A full-hedge, or a fully hedged portfolio, implies that investors take a short position in the futures contract with the same magnitude as they do in the long position on the spot rate. In other words, a hedge ratio of one. Froot (1993) argued that a full-hedge strategy is favourable when investors are not able to forecast future price movements. He concludes that, because returns on exchange rates are expected to equal zero in the long run, a fully hedged portfolio loses effectiveness in long horizon investments. He states that investors with long term goals should implement a hedge ratio considerably smaller than one.
Fischer Black (1990) aimed at estimating an universal optimal hedge ratio, close to but not equal to one, but its performance was widely criticized. Glen and Jorion (1993) compare the hedging performance of Black’s universal hedge ratio with the performance of a conditional OHR and conclude that the latter significantly outperforms the universal hedge ratio.
2.4.2 Objective optimizing hedge ratio
Chen, Lee and Shresda (2003) state that, depending on their objective, investors choose the so-called optimal hedge ratio (OHR) that leads to the optimal level of objective-value. In general, papers on currency hedging assume investors to minimize their portfolio variance (Chang et al., 2011, Chang et al., 2013, Ku et al., 2007 and Lai, 2018). Ku et al. (2007) state that this method simplifies the comparison of hedge performances through the hedg-ing effectiveness index (HE). Chen et al. (2003) argue that the minimum-variance method completely ignores excess returns. They formulate multiple alternative hedge strategies like the mean-variance expected utility maximization, used by Park and Switzer (1995) or the Sharpe ratio maximization, used by Yang and Allen (2004) and Kim (2012). All of the al-ternative objectives proposed by Chen et al. (2003) include the effect of expected returns on the estimation of the optimal hedging ratio. Including this effect is a more realistic objective for fund managers in their overall portfolio selection, but as this paper solely focuses on the ability to reduce the exposure to exchange rate fluctuations and does not consider returns on the exchange rates a priority, the optimal hedge ratios this paper derives are the result of variance minimization, further referred to as the Minimum-Variance (MV) hedge ratio.
2.5
Model specification
Chen et al. (2003) extensively review literary research regarding the estimation of the OHR. Besides multiple different objectives, a wide range of model specifications are considered. In line with the increased research on modelling time series, the demand for a more flexible OHR estimate arises. Where Black (1990) proposed a universal hedge ratio, current research focuses on modelling the OHR in a way it suits individual investors’ objectives, holding periods and the markets of interest.
2.5.1 Static approach
Early research regarding the estimation of the OHR dates back as far as 1960, when Johnson, like many other researchers (Stein, 1961; Ederlington, 1979) at that time, aimed at estimating
the OHR using a OLS-based method which relates changes in cash prices to changes in futures prices (Yang & Allen, 2004). The OLS estimation of a variance minimizing hedge ratio was later referred to by Herbst, Kare and Marshall (1989) as the JSE-approach2. In the OLS framework, all first and second moments, as well as the estimated MV hedge ratio, are assumed to be constant over time. Chang et al. (2013) therefor refer to this method as the static approach. However, for the OLS to be statistically correct, the error terms in each period need to be uncorrelated and homoscedastic (Heij, De Boer, Franses, Van Dijk, & Kloek, 2004, pp. 125-126), which is often not backed up by empirical research on financial time series data. An example is provided by Hsieh (1988). He performs an empirical study on the time-varying nature of exchange rates between 1974 and 1983 and concludes that the volatility of exchange rates is in fact not constant over time.
2.5.2 Conditional heteroscedasticity modelling
Mandelbrot (1963) concludes that large changes tend to be followed by large changes while small changes tend to be followed by small changes. Bollerslev (1987) recognizes the au-toregressive conditional heteroscedasticity (ARCH) model by Engle (1982) and the GARCH model by Bollerslev (1986) as the optimal solutions for modelling such temporal dependence. Evidence favouring the assumption that, with the arrival of new information the riskiness of financial products change, is provided by Bollerslev (1990) and Kroner and Sultan (1993). They state that the time-varying nature of the returns and volatility of exchanges rates call for a time-varying OHR.
2.5.3 Constant conditional correlation
The constant conditional correlation (CCC) model was introduced by Bollerslev in 1990. As most empirical research concluded returns and volatility to be time-variant, Bollerslev aimed at explaining the coherence between short-run nominal exchange rates by implementing a model that allowed for the heteroscedasticity in the error terms. The CCC model he proposed corresponds with the GARCH model as it has time-varying conditional variances
and covariances. According to Bollerslev (1990), the assumption of a constant conditional correlation greatly simplifies the estimation procedure.
Yang and Allen (2004) point out that estimating hedge ratios using multivariate models is far more complex than estimates based on the simple OLS approach. In their analysis they come to an inconsistent result but do find that in maximizing utility in Australian future markets, the OLS-based hedge ratio outperforms a multivariate approach. On the other hand, Kroner and Sultan (1993) and Herbst et al. (1989) conclude that a multivariate approach significantly reduces risk without reducing returns compared to the OLS approach.
2.5.4 Asymmetric modelling
Based on the CCC model by Bollerslev (1990), Ling and McAleer introduced the GARCH model in 2003. The constant conditional correlation assumption of the VARMA-GARCH model is in line with the CCC model of Bollerslev. However, Ling and McAleer (2003) argue that the CCC model does not account for interdependent relations in the mation of the conditional variance-covariance matrix. They propose a matrix GARCH esti-mation for the variance matrix to allow for interdependent volatility across assets (McAleer, Hoti and Chan, 2009). McAleer et al. (2009) attempt to construct a model that treats pos-itive and negative shock asymmetrically by extending the VARMA-GARCH model by Ling and McAleer (2003), which led to the origin of the VARMA-AGARCH model (McAleer et al., 2009). In currency hedging, where risk reduction is the main objective, it seems rather odd to treat negative shocks the same as positive shock. Currency appreciation is not the goal of investors, currency depreciation on the other hand, is seen as a regret and hedging strategies are designed to reduce the risk when this happens.
A strong depreciation of the GBP followed from the Brexit, which implies some great negative shocks in the Pound sterling denominated exchange rates. The VARMA-AGARCH model, which treats the negative shocks not the same as it would treat positive shocks, is therefor expected to be able to adjust the optimal hedge ratio more suitable for the Brexit situation than the CCC model.
2.5.5 Dynamic conditional correlation
Tse (2000) argues that due to the computational simplicity of the constant conditional cor-relation model, most papers regarding the estimation of optimal hedge ratios, rarely perform any empirical research on this assumption. He introduces a Lagrange Multiplier test (LM) and found that depending on the data sets analyzed, there is statistical evidence supporting and rejecting the constant conditional correlation hypothesis. In general, the correlation be-tween two assets is defined by dividing the covariance of these two assets by the product of their standard deviations (Heij et al., 2004, p. 25). When both the conditional covariance and standard deviations are time-varying it seems rational to test the constant conditional correlation hypothesis against a dynamic conditional correlation hypothesis. For this reason, Engle (2002) introduced the dynamical conditional correlation (DCC) model. It attempts to model the correlation matrix in a way it accounts for the time-varying nature of conditional correlations. It seems restrictive to hold on to the assumption of a constant conditional cor-relation when regarding the GBP denominated exchange rates. As the DCC model allows for a more flexible computation of the correlation matrix, the model is expected to be able to adapt the optimal hedge ratio estimates more accurate than the CCC.
2.5.6 Past results
Chang et al. (2013) conclude that the VARMA-AGARCH is more effective than the DCC and the CCC in reducing the variance in the USD/GBP exchange rates, while they can not find any statistical differences in performance between these three models when considering the exchange rates between the US Dollar and the Japanese Yen. Furthermore, the analysis regarding the USD/EUR exchange rate favors the VARMA-AGARCH and DCC model over the CCC model in reducing portfolio variance. Chang et al. (2011) use the same models for the estimation of the OHR in crude oil markets. They conclude that the DCC model is superior in reducing risk compared to the CCC and VARMA-AGARCH model.
2.5.7 Hedging effectiveness
Hedging performance is compared on the basis of the hedging effectiveness index (HE) . The HE is denoted as:
HE = V ar(Ru) − V ar(Rh) V ar(Ru)
(1) where Ru represents the return on the unhedged portfolio and Rh the return on the hedged
portfolio. This index measures the percentage variance reduction as a result of hedging. The model which estimates the OHR in a way that leads to the largest relative risk reduction or largest HE measure, is considered to be superior compared to the other, less effective, models (Ku et al., 2007).
3
Data
This section presents a brief description and analysis of the data used for current research. First the data source and measuring period are discussed, whereafter the returns series are analyzed. Finally, the programs used for the analysis part are motivated.
Daily spot rate prices (Ps) and daily futures prices (Pf) are obtained from the Thomson
Reuters Datastream, acquired from the Excel Tool at the University of Amsterdam. The maturity of the futures contracts this paper analyzes is one month, or near-month futures, as they are referred to by Chang et al., (2013). The currencies this papers examines are; US Dollar (USD), Japanese Yen (JPY), Australian Dollar (AUD) and the Euro (EUR) as these currencies accompanied by the Great Britain Pound (GBP) together amounted for more than 80% of the total daily turnover on exchange markets in April 2016 (BIS, 2016). Exchange rates are considered from perspective of investors who hold assets, or just currency, in the UK. For clarity, let GBP/USD equal 1.50, this indicates that 1 Great British Pound is traded on the foreign exchange market for 1.50 US Dollars.
Daily data is collected from 28 July 2006 till 1 January 2018 as GBP/AUD one month futures’ data collection started on July the 28th 2006. The total sample contains 2892 observations. The measuring period is divided into two periods. The first period, 28 July
2006 till 31 December 2015, is referred to as the Pre-Brexit and contains 2460 observations. The second period, 1 January 2016 till 1 January 2018, is referred to by the Brexit period and contains 522 observations. The British government announced the referendum regarding the Brexit on February the 20th of 2016, the referendum took place on the 23rd of June 2016. The Brexit period therefor contains all the observations effected by the Brexit procedure.
Table 1 contains the descriptive statistics of the spot and futures returns of the four currency pairs over the entire measuring period. The returns on spot and futures rates at time t are calculated by Log(Pi,t/Pi,t−1) × 100% , and referred to by Ri , where i = {s, f }
representing spot and futures respectively.
The mean returns are all extremely close to zero. For all currencies, the standard deviation of spot returns is greater than the standard deviation of futures returns, indicating that the returns on spot rates is more volatile than the market for futures. The GBP/AUD exchange rate shows positive skewness measures for spot and futures returns, while the other three currency pairs show negative skewness measures. A negative skewness measure indicates longer left tails, which means that there were more extreme losses than extreme gains (Chang et al., 2013). Also, the return series display high kurtosis. This indicates that there is a relatively high probability for extreme returns, either positive or negative.
Mean Stdev Skewness Kurtosis JBstat P-val ADFstat P-val
AUDRS -0.011449 0.75987 0.27478 13.933 14884 0.001 -56.22 0.001 AUDRF -0.011358 0.75771 0.59597 16.51 22845 0.001 -56.972 0.001 EURRS -0.0087313 0.5558 -0.61815 10.123 6491.9 0.001 -53.29 0.001 EURRF -0.0087191 0.55012 -0.54667 10.111 6428.4 0.001 -51.799 0.001 JPYRS -0.011387 0.90571 -1.1825 18.099 29013 0.001 -52.043 0.001 JPYRF -0.01125 0.88437 -1.356 19.109 33147 0.001 -52.447 0.001 USDRS -0.010782 0.63567 -0.78739 16.706 23642 0.001 -52.18 0.001 USDRF -0.010784 0.61633 -1.1913 16.628 23773 0.001 -52.137 0.001
Table 1 contains the Bera (JB) statistics and its p-values as well. The Jarque-Bera measure follows from testing whether there is enough statistical evidence to reject the Null-hypothesis, which states that the series follow a normal distribution. For all returns series, the table shows p-values smaller than 0.05 and the Null-hypothesis is rejected for all series.
The second test statistics are the result of an Augmented Dickey-Fuller (ADF) test. The hypothesis of the ADF test states that the series contain a unit root. This Null-hypothesis is rejected for all returns series, which indicates that the return series are station-ary. For stationary series, the conditional expectations are equal for every period.
In figure 1, the return series of spot and futures rates of all four currency pairs are plotted. In all eight plots the day of the Brexit is clearly displayed by a large negative shock after the 2500th observation. Furthermore, in all plots there is a significant increase in volatility after the initiation of the subprime mortgage crisis in September 2008.
The analysis is performed by making use of three different programs; Excel, Eviews9 and Matlab. For most computations regarding GARCH and maximum likelihood (ML) estimations, Eviews9 is preferred because it contains a set of build-in functions that allow for quick and straightforward estimation procedures. Because of its flexibility in modelling large data matrices and vectors, Matlab is preferred for computing conditional variances, correlations and plotting.
4
Methodology
The following subsections present a discussion on model specifications, as well as the deriva-tion of the optimal hedge ratio estimates using the four different models. Literary background concerning the performances, shortcomings and upsides of all four models are discussed in the theoretical framework.
4.1
OLS
When considering the simple regression model of the returns on spot rates (Rs,t) between t
- 1 and t on a constant (α) and the returns on futures rates (Rf,t) between t - 1 and t, as
shown in equation two, the MV hedge ratio is easily estimated.
Rs,t = α + βRf,t+ t (2)
The OLS estimate of β equals the Minimum Variance (MV) hedge ratio. The OLS estimate of β, or ˆbOLS, follows after minimizing the squared sum of the residuals. Where
t represents
the normal distributed error term at time t.
For the OLS to be statistically correct, the error terms need to be independent identical distributed (Heij et al., 2004, pp. 125-126). The static OLS approach assumes returns to be time-invariant which leads to a constant OHR. Because most empirical research implies the returns to vary over time, it seems rational that the optimal hedge ratio differs over time as well . The research by Lien et al. (2003) however concluded that the multivariate CCC model does not outperform the OLS method in terms of hedging effectiveness. If their result holds, the simplicity of the OLS method would make it more attractive, as a more complex computation of the OHR is more costly and harder to understand.
4.2
CCC
4.2.1 CCC model specification
The CCC model is presented by Tim Bollerslev in 1990 and formulated by:
Yt = E[Yt|ψt−1] + t, t= Dtηt (3)
Where Yt is a 2 x 1 vector containing the spot and futures returns at time t and ψt−1
contains all the past information of these time series. Where ηt represents a 2 x 1 vector
with independent identical distributed error terms. This specification shows that the error terms i do not need to be identical and independently distributed but instead can be
The Dt matrix is best explained when considering the conditional variance.
V ar(Yt|ψt−1) = V ar(t|ψt−1) = Ht= Dtηtηt0D 0
t (4)
The Ht matrix represents the conditional variance-covariance matrix of spot and futures
re-turns. Bollerslev (1990) describes hij,t as the {i,j}’th element of the conditional variance
matrix (H) and computes the conditional correlation of assets i and j as:
ρi,j =
hij,t
phii,thjj,t
(5) Note that in this case, i and j represent either 1 for spot returns or 2 for futures returns. The following matrix notation for the conditional variance matrix follows after rewriting equa-tions (4) and (5): Ht= h11,t h12,t h21,t h22,t = ph11,t 0 0 ph22,t 1 ρs,f ρs,f 1 ph11,t 0 0 ph22,t (6)
Where the diagonal matrix represents the Dt matrix and the conditional correlation matrix
in the center is represented by Γ .
he CCC model assumes that the conditional variances follow an univariate GARCH pro-cess (Chang et al., 2011):
hi,t = ωi+ r X j=1 αij2i,t−j + s X j=1 βijhi,t−j (7)
In equation (7), r represents the amount of lagged squared error terms that have significant effect on the conditional variance, αij represent the magnitude the j’th lagged squared error
term effects the conditional variance. The GARCH effect is represented by βij and s
repre-sents the amount of lagged conditional variance terms that have a significant effect on the conditional variance in some period. Where ωi represents the constant term of the
condi-tional variance of either spot or futures returns. Because variances are non-negative at all time, all the parameters ωi, αij and βij must be non-negative as well.
4.2.2 CCC estimation procedure
In order to estimate the OHR using a CCC model, or any of the multivariate models, the first parameter of interest is the conditional expectation. The results of the ADF tests, as described §3, lead to the conclusion that all return series are stationary. This means that the conditional expectation is equal for all periods. After subtracting the unconditional mean from the return series, i,t is obtained. This series is GARCH(1,1) estimated with no
additional regressors in Eviews9. The parameters are used in the process of computing the conditional variance vectors.
4.3
VARMA-AGARCH
4.3.1 VARMA-AGARCH model specification
A restrictive assumption of the GARCH specification of the conditional variance of Boller-slev’s CCC model is the fact that he assumes that the conditional variance of spot and futures returns follow two independent GARCH processes. Ling and McAleer (2003) present the VARMA-GARCH model which builds on the CCC model of Bollerlsev (1990). The as-sumption of a constant conditional correlation is kept in place but the GARCH process for estimating the conditional variance is altered. Instead of two independent GARCH processes, the VARMA-GARCH model estimates the conditional variances of spot and futures returns in a joint GARCH process, allowing for interdependent volatility across returns (Ling and McAleer, 2003). Consider the VARMA-GARCH model:
Yt = E[Yt|ψt−1] + t, t= Dtηt (8)
This part is specified the same way as the CCC model. As mentioned above, the difference lays in the computation of the conditional variances.
Ht= W + r X l=1 Al2t−l~ + s X l=1 BlHt−l (9)
In equation (9), W represents a 2 x 1 vector with its elements representing constant terms in the estimating process of conditional variances. Al and Bl represent a 2 x 2 matrix. The
elements of this A matrix represent the coefficients that measure the effect of the l’th lagged squared error term vector on the conditional variance and ’r’ represents the amount of lagged ARCH terms that show a significant effect on the estimation of the conditional variance. In equation (9), ~t2 represents the 2 x 1 squared error terms vector at time t. The elements of
the B matrix represent the coefficients that measure the effect of the l’th lagged conditional variance terms on the conditional variance of the current period.
The VARMA-AGARCH model by McAleer et al. (2009) extends the VARMA-GARCH model by considering an additional factor that effects the conditional variances. This factor is premultiplied with an indicator function, with the purpose of separating negative and pos-itive shocks. I(ηi,t) = 0, if it > 0 1, if it ≤ 0 (10)
Consider the conditional variance in the VARMA-AGARCH model:
Ht = W + r X l=1 Al~2t−l+ s X l=1 BlHt−l+ r X l=1 ClIt−l~2t−l (11)
In equation (11), Cl represents a 2 x 2 matrix. Its elements represent the coefficients that
measure the additional effect a negative shock has on the conditional variance. The elements of the C matrix are denoted by γi. By including the C matrix, multiplied with the indicator
function as specified in equation (10), the VARMA-AGARCH model by McAleer et al. (2009) treats positive and negative shock asymmetrically.
4.3.2 VARMA-AGARCH estimation procedure
In order to estimate the OHR using the VARMA-AGARCH model, the indicator function has to be defined first. This paper identifies all ’negative’ shocks as a return that falls short of their expectation in some period. After identifying the negative shocks, a quasi maximum
likelihood estimation (QMLE) procedure is performed. A total of 12 parameters, as specified in equation (12), are estimated and thereafter used for computing the conditional variance vector. h11,t h22,t = ω1+ α121,t−1+ α222,t−1+ γ1(I11,t−121,t−1) + β1h11,t−1+ β2h22,t−1 ω2+ α321,t−1+ α422,t−1+ γ2(I22,t−122,t−1) + β3h11,t−1+ β4h22,t−1 (12)
4.4
DCC
4.4.1 DCC model specificationIn previous model specifications, the conditional correlation is assumed to be constant over time, which seems to be somewhat unrealistic. In general, the correlation between two assets is defined by dividing the covariance of these two assets by the product of their standard deviations (Heij et al., 2004, p. 23). When both are conditionally time-varying it seems ra-tional to test the constant condira-tional correlation hypothesis against a dynamic condira-tional correlation hypothesis. For this reason, Engle (2002) proposed the DCC model. The DCC model allows for a time-varying computation of the conditional correlation. The DCC model is defined as:
Yt|ψt−1 ∼ N (0, Qt) (13)
Qt = DtΓtDt0 (14)
The DCC model assumes the returns on spot and futures rates at time t to follow a normal distribution with mean zero and variance matrix Q, conditionally on the past information set. The computation of the conditional variance matrix, as shown in equation (14), is in line with the computation in equation (4), with the exception of the time-varying correlation.
This model follows the CCC model in the specification of the conditional variance:
hi,t = wi+ r X k=1 αi,k2i,t−k+ s X k=1 βi,khi,t−k (15)
All parameter definitions are in line with those in equation (7). As mentioned above, the difference lays in the computation of the conditional correlation matrix, which is defined as:
Γt= diag(Qt)−1/2 Qt diag(Qt)−1/2 (16)
It is well known that the correlation matrix has all ones on the diagonal as any asset is perfectly correlated with itself. The Qtmatrix, the conditional variance matrix, is computed
as follows:
Qt = (1 − θ1− θ2) ¯Q + θ1ηt−1η0t−1+ θ2Qt−1 (17)
Where θ1 and Θ2are non-negative scalars that, respectively, represent the effects of previous
shocks and dynamic conditional correlations on present dynamic conditional correlation. In equation (17), ¯Q represents a 2 x 2 unconditional variance matrix of error terms ηt .
4.4.2 DCC estimation procedure
In order to estimate the OHR using a DCC model, the returns series, are estimated by a GARCH(1,1) model. The standardized residuals of these processes form ηi,t. The theta
parameters are thereafter estimated by maximum likelihood and used for the computation of the exact conditional variances and covariances as well as the dynamic conditional correlation.
4.5
Optimal Hedge Ratio
Investors who intend to hedge their international holding against currency risk consider the following portfolio:
Rh,t = Rs,t− γRf,t (18)
Where the return on holding the hedged portfolio (Rh,t) between t - 1 and t, is computed
by subtracting the hedge ratio γ times the return on the futures (Rf,t) positions between t
- 1 and t from the return on the spot positions (Rs,t). As mentioned in §2.4.2, this paper
considers variance minimization as investors’ main objective. The variance of the hedged portfolio is denoted as:
V ar(Rh,t) = V ar(Rs,t) + γ2V ar(Rf,t) − 2γCov(Rs,t, Rf,t) (19)
The minimum variance (MV) hedge ratio follows after taking the derivative with respect to γ , setting the derivative equal to zero and solving for the hedge ratio.
∂V ar(Rh,t)
∂γ = 2γV ar(Rf,t) − 2Cov(Rs,t, Rf,t) = 0 (20) γ∗ = Cov(Rs,t, Rf,t)
V ar(Rf,t)
(21) With γ∗ equals the MV hedge ratio.
The MV hedge ratio is thus obtained by dividing the covariance of spot and futures returns by the variance of futures returns. The differences in estimating the optimal hedge ratios arise from the different computations of conditional variances and covariances across the four models. The CCC model estimates conditional variances by two independent GARCH pro-cesses, conditional covariances are estimated restricted with the assumption of a constant conditional correlation. The VARMA-AGARCH model estimates conditional variances by a joint GARCH process along with extra factors to allow for an asymmetrical treatment of positive and negative shocks. The DCC model differs from the CCC model in the specifica-tion of the condispecifica-tional correlaspecifica-tion matrix. The MV hedge ratio, as shown in equaspecifica-tion (21), can also be denoted as follows:
γ∗ = ρs,f
pV ar(Rs,t) pV ar(Rf,t)
(22) Equation (22) clearly shows the effect the computation of the correlation has on the estimation of the MV hedge ratio. The assumption of a constant conditional correlation may limit the ability of models to adjust the OHR estimates during the Brexit.
As mentioned in §2.5.7, the hedging performance is measured on the basis of relative risk reduction, by making use of the Hedging Effectiveness Index (HE). Its derivation is described in equation (1).
5
Empirical results
This section presents the results of analyzing the hedging performance in UK futures markets. At first, the estimation results are discussed, followed by a brief discussion regarding the constant and dynamic conditional correlation. Finally, the hedging performances of the static OLS model and the three multivariate models (CCC,VARMA-AGARCH and DCC) are discussed for three different periods.
5.1
Estimation results
This subsection presents a total of 252 parameter estimates. Estimations printed in bold text are insignificant at the 5% level. Table 2 presents the OLS estimates, table 3 presents the DCC theta estimates and tables 12-15 present the GARCH(1,1) and VARMA-AGARCH parameter estimates.
5.1.1 OLS
The Brexit period shows larger OLS estimates than the Pre-Brexit sample for all currency pairs. The Euro exchange rates shows the largest OLS estimate in the Pre-Brexit period whereas it resulted in the smallest estimate during the Brexit period.
Total Sample Pre-Brexit Brexit AUD 0.840 0.827 0.909 EUR 0.876 0.868 0.904 JPY 0.873 0.859 0.927 USD 0.863 0.841 0.936
Table 2: OLS estimates
5.1.2 GARCH(1,1)
All of the GARCH(1,1) parameter estimates measured over the total sample and the Pre-Brexit sample, are positive and differ significantly from zero. The results for each of the currency pairs share a common trend that, in general, βi estimates are rather large, ranging
are rather small, ranging from 0.0045696 (EUR spot) and 0.097593 (JPY futures) for the total period. These results are in line with empirical research by Chang et al. (2013) and Lien et al. (2002). When considering the fact that, αi is multiplied by 2i, which can take on
some extreme values during the Brexit period, it is somewhat logical that αi takes on a value
close to zero. The condition, α1 + βi < 1, is held for all currency pairs and periods. Chang
et al. (2013) state that when α1+ βi approaches 1, the volatility persistence increases. For
all currency pairs we see that the volatility persistence decreases during the Brexit.
The Australian Dollar shows some remarkable results during the Brexit period, which are not in line with the majority of existing literature. First, for the spot returns the GARCH(1,1) estimates only the autoregressive factor has a significant effect on the conditional variance, when testing at a 5% level. By excluding the effect of a lagged squared epsilon, the conditional variance depends only on previous variances. As the previous variance is multiplied by βs, which is estimated to equal 0.78929, the conditional variance converges to zero rather
quickly.This problem is solved by including all parameters in the estimation procedure even though there does not exists enough statistical evidence to reject the hypothesis that αs and
Cs have no effect. This decision is based on the knowledge that the alternative leads to a
conditional variance of zero, a horizontal line for spot returns during the Brexit period, which is rejected by the returns plot as displayed in figure 1.
Also, αf is estimated to have a significant negative effect and βf is greater than one. A
negative factor in the conditional variance equation could lead to a negative variance. The βf coefficient on the other hand can, if the series contain enough observations, diverge the
conditional variance series to infinity. These results suggest that a GARCH(1,1) method does not fit the Brexit data of the GBP/AUD exchange rate. In order to compare the pre and post-Brexit periods, the coefficients are transformed. To adjust for any negative variance, the absolute value of αf is considered in computing the conditional variance. This transformation
is based on the conditional variance plot in Eviews9.
For the other currency pairs one finds that during the Brexit period, the αi estimates
decline significantly when compared to the Pre-Brexit period, while the βi estimates increase.
Brexit period, the increased conditional variances is partly explained by the larger αi values.
As βi is multiplied by the lagged conditional variance, the larger variances do not persist as
long as they would in the Pre-Brexit period.
5.1.3 VARMA-AGARCH
Over the course of this project the estimation procedure for the VARMA-AGARCH param-eters resulted in some biased results. As the QMLE method requires the initial parameter values to be carefully chosen, the end results highly depend on the initial values. The pa-rameters have been estimated using an extremely wide range of different initial values and the resulting OHRs and hedging effectiveness measures were stored and compared. This led to the remarkable result that, for some initial values3, the QMLE procedure did not, or barely, changed the initial values. However, the OHR and its hedging effectiveness resulting from these initial values were by far the most realistic based on the literary research on the VARMA-AGARCH. This leads to the second point; the literary background is quite scarce. As the VARMA-AGARCH model was introduced in 2009, there does not exist a lot of rel-evant research regarding the estimation procedure and its results. As stated by McAleer et al. (2009), the VARMA-AGARCH require the estimation of 9 parameters, when considering two assets, but in current research we feel this is not sufficient and a total twelve parameters are estimated by the QML method.
Furthermore, in the analysis of the VARMA-AGARCH method, widely ranging parameter values eventually resulted in not so different hedging effectiveness measures. Even though it is expected, when optimizing some objective, that the parameter values that lead to this optimal value will result from the QML method.
Again, the parameters presented in tables 12-15 are achieved after an extensive trial-and-error procedure using different initial values. The results of the parameter values are in no way in line with those of the research by Chang et al. (2013). It might be due to the fact that Chang et al. (2013) do not show any interdependent parameter estimates, so the total number of estimated parameters is not clear, but they only print the results of 8 parameter
estimates.
The research paper by Allen, Amram and McAleer (2013) regarding the volatility spillovers of the Chinese stock market to surrounding nations’ stock markets provides a description of the VARMA-AGARCH method in more detail, which led to the conditional variance equa-tion as described in equaequa-tion (12). As their research is not correlated to current research, a comparison on parameter values is not relevant.
For all currency pairs except the GBP/AUD exchange rate, the QMLE returns the initial values after a few iterations. All of the parameters are significant at the 5% level for the Pre-Brexit sample. For these currencies, during the Pre-Brexit, some estimates show no significant effect even though the values do not change. For the GBP/AUD exchange rate, the QMLE does change the initial values for most parameters. The factors of the A matrix show some remarkable large values while the coefficients of the B matrix stay relatively low. During the Brexit period however, the coefficients from the B matrix are the only factors that prove to have a significant effect on the conditional variances. There is large difference between the Australian Dollar and the other currencies in iterative steps that lead to the parameter estimates.
5.1.4 DCC
In addition to the GARCH(1,1) parameter estimates, the DCC model requires estimates for Θ1, the parameter that measures the effect of standardized lagged shocks, and Θ2, the
pa-rameter that measures the effect of lagged conditional variance, in computing the conditional variance matrix Q.
The first parameter has a significant positive effect for all instances. Its value is substan-tially smaller during the Pre-Brexit compared to the Brexit sample. The relatively small value of Θ1 is in line with the estimated values by Hsu and Yang (2010).
The second parameter on the other hand, does not show any significant effects for the Euro and US Dollar exchange rates. For the Australian Dollar, this parameter is insignificant for the Brexit period only. A comparison between the two periods is thus based on the GBP/JPY only. For this exchange rate the parameter declines during the Brexit.
Research by Hsu and Yang (2010) regarding the hedge ratio changes around the subprime mortgage crisis based on the DCC-GARCH model concludes that the Θ2 parameter does not
show any significant effects on the conditional correlation for the Japanese Yen, US Dollar and Pound sterling. A relevant comparison cannot be made between current research and the research regarding the subprime mortgage crisis. Though, their result proves that Θ2 is
known to have either a relatively large significant effect or no significant effect at all while the Θ1 parameter shows, small significant effects at all time.
Total Sample Pre-Brexit Brexit AUD: Θ1 0.044264 0.041199 0.049828 Θ2 0.919219 0.92827 0.039256 EUR: Θ1 0.12966 0.12283 0.17243 Θ2 0.027227 -0.052898 0.25839 JPY: Θ1 0.080165 0.068893 0.18948 Θ2 0.56474 0.48159 0.30555 USD: Θ1 0.12241 0.11018 0.18611 Θ2 0.14194 0.091101 0.20979 Table 3: DCC estimates
5.2
Correlation
Table 4 presents the OLS correlation estimates and the constant conditional correlation estimated by the CCC and VARMA-AGARCH methods. The plots in figure 2 present the dynamical conditional correlation.
5.2.1 Constant conditional correlation
From table 4 one is able to conclude that for all correlation coefficients, but the JPY CCC estimate, the Brexit sample-based correlations are greater than correlation coefficients based on Pre-Brexit sample. As the conditional correlation is estimated by taking the unconditional expectation of the standardized shocks, a larger correlation coefficient is expected in periods with larger shocks, such as the Brexit. A remarkable result though is the fact that the OLS method results in the largest correlation coefficients for all currencies but the Australian
Dollar during the Brexit period while its correlation estimates during the Pre-Brexit period resulted in the smallest values.
5.2.2 Dynamic conditional correlation
Figure 2 shows the dynamical conditional correlation plots for all four currency pairs, mea-sured over the total sample. All currency pairs show large negative shocks in their conditional correlation plots. The effect of Θ2 is clearly noticed when comparing the AUD and JPY with
the EUR and USD plots. The latter two currencies seem to be centered more around their mean while the plots on the left, where Θ2 has a significant effect on the conditional
corre-lation, the correlation looks more time-varying. Table 5 presents the descriptive stats of the dynamical conditional correlations. It shows that, measured over the total sample, the Aus-tralian Dollar dynamic conditional correlations are most volatile and the Euro conditional correlations are the most stable.
Total Pre-Brexit Brexit AUD: OLS 0.838 0.828 0.892 CCC 0.854 0.843 0.872 VARMA-AGARCH 0.858 0.848 0.898 EUR: OLS 0.868 0.861 0.891 CCC 0.870 0.866 0.870 VARMA-AGARCH 0.870 0.866 0.887 JPY: OLS 0.852 0.840 0.897 CCC 0.856 0.856 0.842 VARMA-AGARCH 0.862 0.860 0.870 USD: OLS 0.836 0.822 0.883 CCC 0.835 0.830 0.853 VARMA-AGARCH 0.836 0.827 0.864
Table 4: Correlation table
Mean Min Max Std AUD 0.845 0.338 0.979 0.048 EUR 0.866 0.529 0.989 0.032 JPY 0.852 0.379 0.981 0.033 USD 0.835 0.460 0.987 0.035
Table 5: DCC correlation statistics
5.3
Hedging performance
This subsection discusses and compares hedging performance over the course of the Brexit process. For the three multivariate models, the mean OHR is displayed in the tables, but the hedged portfolios, based on the multivariate models, take account for the specific OHR
estimates for each period. The std(OHR) column represents the standard deviation of the OHR series. VARu represents the variance of a portfolio, solely consisting of the returns on spots, and VARh represents the variance of the hedged portfolio. RiskReduction is easily computed by subtracting VARh from VARu. Figures 3-6 present the OHR series per currency. In the figures containing OHR plots, the multivariate models are always displayed with the OLS estimates for comparison.The CCC-based OHRs are plotted at the top, the VARMA-AGARCH-based OHRs are plotted in the center and DDC at the bottom. The most effective hedging model is underlined for clarity.
5.3.1 Australian Dollar
Table 6 presents the OHR estimates and hedging performance for each of the four models for the GBP/AUD exchange rate, figure 3 shows the OHR plots for all periods. When considering the total sample, OHR estimates lay relatively close together, ranging from 0.856 (CCC) to 0.805 (VARMA-AGARCH). The OLS results in the largest HE measure, which indicates that a hedging strategy based on OLS estimates is the most effective strategy, closely followed by the VARMA-AGARCH model specification.
The OHR estimates in the Brexit period are all greater than the estimates in the Pre-Brexit sample, while the spot returns were less volatile during the Pre-Brexit sample. The fact that spot returns were less volatile during the Brexit period is unique in current research. During the Brexit period, DCC, VARMA-AGARCH and OLS score almost equally on the hedging effectiveness index.
Some notes on the results of the Brexit sample estimates have to be made. The CCC OHR plot results after altering the GARCH(1,1) coefficients. For the conditional variance of futures returns, the absolute value of the negative alpha is considered, as noted in §5.1.2., for the conditional variance of spot returns the insignificant factors are included. It lead to a significant smaller hedging effectiveness than other models but a correct conclusion cannot be made because a CCC-GARCH model clearly does not fit the AUD data as well as others.
GBP/AUD OHR std(OHR) VARu VARh RiskReduction HE Total Sample: OLS 0.840 0 0.577 0.172 0.405 0.702 CCC 0.856 0.07 0.577 0.180 0.397 0.688 VARMA-AGARCH 0.805 0.076 0.577 0.174 0.404 0.700 DCC 0.844 0.138 0.577 0.176 0.401 0.695 Pre-Brexit: OLS 0.827 0 0.583 0.184 0.399 0.685 CCC 0.843 0.035 0.583 0.194 0.389 0.668 VARMA-AGARCH 0.795 0.035 0.583 0.185 0.398 0.682 DCC 0.836 0.028 0.583 0.189 0.395 0.677 Brexit: OLS 0.909 0 0.550 0.113 0.437 0.795 CCC 0.977 0.061 0.550 0.133 0.417 0.759 VARMA-AGARCH 0.894 0.065 0.550 0.113 0.437 0.795 DCC 0.890 0.026 0.550 0.113 0.437 0.794
Table 6: GBP/AUD: Hedging performance
5.3.2 Euro
Table 7 presents the OHR estimates and hedging performance for each model for the GBP/EUR exchange rate, figure 4 shows the OHR plots for all periods.
The CCC-based mean OHR estimates are larger in the Pre-Brexit period than during the Brexit period, while the returns are significantly more volatile during the last period. As the CCC correlation estimates are almost equal in both periods, the decreased OHR indicates that the futures returns’ volatility increased more than the volatility of spot returns.
The OLS method is the most effective hedging strategy for the total sample and the Pre-Brexit sample, while the OLS, VARMA-AGARCH and the DCC methods show the exact same hedging effectiveness during the Brexit period.
GBP/EUR OHR std(OHR) VARu VARh RiskReduction HE Total Sample: OLS 0.876 0 0.309 0.076 0.232 0.753 CCC 0.891 0.065 0.309 0.078 0.231 0.747 VARMA-AGARCH 0.883 0.064 0.309 0.077 0.232 0.751 DCC 0.867 0.128 0.309 0.078 0.231 0.747 Pre-Brexit: OLS 0.868 0 0.288 0.075 0.213 0.740 CCC 0.886 0.016 0.288 0.076 0.212 0.736 VARMA-AGARCH 0.877 0.015 0.288 0.075 0.212 0.738 DCC 0.861 0.017 0.288 0.077 0.211 0.733 Brexit: OLS 0.904 0 0.409 0.084 0.324 0.794 CCC 0.871 0.057 0.409 0.091 0.317 0.777 VARMA-AGARCH 0.901 0.055 0.409 0.085 0.324 0.794 DCC 0.888 0.074 0.409 0.084 0.324 0.794
Table 7: GBP/EUR: Hedging performance
5.3.3 Japanese Yen
Table 8 presents the OHR estimates and hedging performance for each model specification for the GBP/JPY exchange rate, figure 5 shows the OHR plots for all periods.
The VARMA-AGARCH method is the most effective hedging strategy for the total sample and the Brexit sample, while the DCC method is the most effective strategy during the Pre-Brexit period.
All models are able to reduce the hedged variance during the Brexit to a point that it is even less volatile than during the Pre-Brexit sample. The VARMA-AGARCH approach obtains a hedging effectiveness of 80.6%, which is, according to current research, the most effective hedging performance.
GBP/JPY OHR std(OHR) VARu VARh RiskReduction HE Total Sample: OLS 0.873 0 0.820 0.225 0.596 0.726 CCC 0.872 0.094 0.820 0.224 0.596 0.727 VARMA-AGARCH 0.879 0.086 0.820 0.222 0.598 0.729 DCC 0.854 0.132 0.820 0.223 0.597 0.728 Pre-Brexit: OLS 0.859 0 0.787 0.231 0.556 0.706 CCC 0.868 0.022 0.787 0.230 0.558 0.708 VARMA-AGARCH 0.876 0.022 0.787 0.229 0.558 0.709 DCC 0.843 0.023 0.787 0.228 0.559 0.710 Brexit: OLS 0.927 0 0.978 0.191 0.787 0.805 CCC 0.761 0.051 0.978 0.212 0.765 0.783 VARMA-AGARCH 0.889 0.044 0.978 0.190 0.788 0.806 DCC 0.888 0.091 0.978 0.214 0.764 0.781
Table 8: GBP/JPY: Hedging performance
5.3.4 US Dollar
Table 9 presents the OHR estimates and hedging performance for each model specification for the GBP/USD exchange rate, figure 6 shows the OHR plots for all periods.
The OLS method is the most effective hedging strategy for all periods. The DCC OHR plot seems more centered around a mean value than the CCC OHRs, yet the CCC strategy leads to larger relative risk reduction than the DCC during the Brexit period.
GBP/USD OHR std(OHR) VARu VARh RiskReduction HE Total Sample: OLS 0.863 0 0.404 0.121 0.283 0.699 CCC 0.845 0.078 0.404 0.123 0.281 0.695 VARMA-AGARCH 0.849 0.065 0.404 0.122 0.282 0.697 DCC 0.837 0.109 0.404 0.124 0.280 0.694 Pre-Brexit: OLS 0.841 0 0.371 0.120 0.250 0.676 CCC 0.835 0.017 0.371 0.122 0.249 0.671 VARMA-AGARCH 0.840 0.016 0.371 0.121 0.249 0.672 DCC 0.824 0.022 0.371 0.122 0.249 0.671 Brexit: OLS 0.936 0 0.563 0.124 0.439 0.780 CCC 0.899 0.059 0.563 0.125 0.438 0.778 VARMA-AGARCH 0.881 0.055 0.563 0.125 0.439 0.779 DCC 0.880 0.080 0.563 0.130 0.433 0.769
6
Conclusion
This paper analyses the hedging performance based on the OLS, CCC, DCC and VARMA-AGARCH models and evaluates the best hedging strategy in UK futures markets, during and before the Brexit.
Table 10 summarizes for each currency, the most effective hedging strategy. In this table, 1 indicates a best performing model for the total sample, 2 represents the Pre-Brexit sample and the Brexit sample is represented by 3. For each currency, except the Japanese Yen, the OLS proves to be the most effective in reducing exposure to currency risk. For the currency pairs GBP/AUD and GBP/EUR, the static approach is more effective during the Pre-Brexit period but during the Brexit period the VARMA-AGARCH and DCC method are able to perform as well as the OLS method. For the Japanese Yen, the most volatile currency, the optimal hedging effectiveness is obtained by a multivariate model for all periods.
For all currencies, the models are more effective in minimizing portfolio variance during the Brexit period as opposed to the Pre-Brexit and the mean Optimal Hedge Ratios are larger in the Brexit period as well. In line with Froot’s (1993) statements regarding hedging effectiveness, current paper concludes that large hedge ratios become less effective when investment horizons are relatively large.
Table 11 presents the percentage increase in hedging effectiveness between the Pre-Brexit period and the Brexit period, along with relative volatility increase of the unhedged portfolios. The results in this table are in line with the expectation that the models that allow for time-varying second moments, in most cases, increase their performance more than the static OLS method during the Brexit. We find no correlation between the relative increase in volatility of an unhedged portfolio and the relative increase in hedging effectiveness.
In line with research conducted by Chang et al. (2013), current research finds that the VARMA-AGARCH method is more effective than the CCC method in all periods and for all currency pairs. As the VARMA-AGARCH is easily converted to the CCC-GARCH by excluding some factors, the VARMA-AGARCH model should never under perform compared to the CCC-GARCH.
Current research concludes that for the Euro, US Dollar and Japanese Yen, the DCC method results in the most volatile estimation of the OHR but this flexibility does not ensure a larger hedging effectiveness. The DCC method is even outperformed by the CCC method during the Brexit period for the GBP/JPY and GBP/USD exchange rates. This result is remarkable as the DCC model can be converted to the CCC-GARCH by simply excluding the Θ1 and Θ2 parameters. For this reason the DCC method is expected to perform at least
as good as the CCC model.
Although the hedging effectiveness of multivariate models improve more than the OLS-based hedging effectiveness during the Brexit, the empirical research contradicts common believes that multivariate models will outperform the static OLS approach in minimizing the exposure to currency risk. However, as multivariate models are preferred for the most volatile currency (JPY), further research regarding the correlation between volatility levels and hedging effectiveness of multivariate models compared to the OLS will provide better insight in selecting the optimal hedging strategy that fits each investor’s individual situa-tion. Thereby, current research recognizes there are opportunities for enhancement regarding the estimation of the VARMA-AGARCH method. After extensive research we found that the RATS software, which we were unable to obtain access to, contains a build-in function which allows for a quick and precise estimate of the VARMA-AGARCH parameters. Fur-ther research should include this software in order to perform a robustness check on the VARMA-AGARCH estimation results.
AUD EUR JPY USD OLS 1/2/3 1/2/3 - 1/2/3 CCC - - - -VARMA-AGARCH 3 3 1/3 -DCC - 3 2
-Table 10: Summary hedging performance
∆ VARu ∆ HE: OLS CCC VARMA-AGARCH DCC AUD -0.057 0.162 0.137 0.171 0.174 EUR 0.420 0.072 0.055 0.076 0.083 JPY 0.242 0.140 0.105 0.137 0.099 USD 0.520 0.154 0.159 0.159 0.146
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Appendix
Figure 3: AUD OHR plots
Figure 5: JPY OHR plots
