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1 This article was downloaded by:[kavussanos, Manolis G.] On: 9 April 2008 Access Details: [subscription number ] Publisher: Routledge Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK The European Journal of Finance Publication details, including instructions for authors and subscription information: Hedging effectiveness of the Athens stock index futures contracts Manolis G. Kavussanos a ; Ilias D. Visvikis b a Athens University of Economics and Business, Athens, Greece b ALBA Graduate Business School, Athens, Greece Online Publication Date: 01 April 2008 To cite this Article: Kavussanos, Manolis G. and Visvikis, Ilias D. (2008) 'Hedging effectiveness of the Athens stock index futures contracts', The European Journal of Finance, 14:3, To link to this article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 The European Journal of Finance, Vol. 14, No. 3, April 2008, Hedging effectiveness of the Athens stock index futures contracts Manolis G. Kavussanos a and Ilias D. Visvikis b a Athens University of Economics and Business, Athens, Greece; b ALBA Graduate Business School, Vouliagmeni, Athens, Greece This paper examines the hedging effectiveness of the FTSE/ATHEX-20 and FTSE/ATHEX Mid-40 stock index futures contracts in the relatively new and fairly unresearched futures market of Greece. Both in-sample and out-of-sample hedging performances using weekly and daily data are examined, considering both constant and time-varying hedge ratios. Results indicate that time-varying hedging strategies provide incremental risk-reduction benefits in-sample, but under-perform simple constant hedging strategies out-of-sample. Moreover, futures contracts serve effectively their risk management role and compare favourably with results in other international stock index futures markets. Estimation of investor utility functions and corresponding optimal utility maximising hedge ratios yields similar results, in terms of model selection. For the FTSE/ATHEX Mid-40 contracts we identify the existence of speculative components, which lead to utility-maximising hedge ratios, that are different to the minimum variance hedge ratio solutions. Keywords: hedging effectiveness; futures markets; constant and time-varying hedge ratios; utility functions; VECM-GARCH-X 1. Introduction One of the most important functions of derivatives futures contracts is the ability they offer to investors to hedge their risks from a long or a short position that they have in an underlying commodity. Thus, a position in a futures contract is taken, which is opposite to that in the cash market. An important issue, in this process of hedging risks, is the calculation of the correct number of futures contracts to use for each cash position held. The solution to this problem depends on a number of parameters, and can make a big difference for investors hedging effectiveness results. These parameters include: the choice of data frequency and thus the investment rebalancing horizon of the investor; the determination of optimal hedged portfolio selection criteria; his level of risk aversion; risks and returns emanating from hedged portfolios; the models that are used to estimate empirically the optimal hedges; and whether in-sample or out-of-sample horizons are considered as relevant for the investor. This paper presents a framework that can be utilised to answer these important issues for investors who wish to engage in derivatives markets. Often the issues come down to selecting a particular policy from a number of alternatives and this paper shows, through an empirical application, that results can be different according to the choice made. The index futures contracts traded in the under-researched derivatives market of Greece are utilised for the investigation. More specifically, the stock index futures FTSE/ATHEX-20 and the less liquid FTSE/ATHEX Mid-40 contracts are examined by estimating hedge ratios, computed Corresponding author. mkavus@aueb.gr ISSN X print/issn online 2008 Taylor & Francis DOI: /

3 244 M.G. Kavussanos and I.D. Visvikis from several model specifications, in an effort to identify the hedging models that generate: (i) the highest price risk reduction and (ii) the maximum utility increase for hedgers involved in these markets. In answering the latter question (i.e. utility maximising hedges), most papers assume that the futures rate follows a martingale i.e. that expected futures prices for next period are equal to today s futures prices; E(F 1 ) = F 0 and resort back to answering the former part of the question that is, they estimate hedge ratios that achieve the highest reduction in portfolio volatility. In this paper, this assumption is removed, and we let the data determine the empirical validity of this question. This is important when there are suspicions that markets may not work perfectly efficiently. The operation of the organised derivatives market in Greece rests with the Athens Derivatives Exchange (ADEX), which was founded in April The first stock index futures contract of ADEX was the FTSE/ATHEX-20, which was introduced in August 1999, with the underlying asset being the FTSE/ATHEX-20 stock index, which consists of the 20 highest capitalisation stocks listed in the Athens Exchange (ATHEX). The FTSE/ATHEX Mid-40 index futures was created, a few months later, in January 2000, and is based on 40 medium capitalisation stocks listed in the ATHEX. 1 The study is also of great importance for market participants in the derivatives market of the ATHEX, who need to cover their risk exposure from holding portfolios of stocks in ATHEX. The introduction of a derivatives exchange in a capital market is considered beneficial as derivatives can complete the market, improve efficiency, transfer risk and discover prices, among others (see, for instance, Kavussanos, Visvikis, and Alexakis 2008). The ATHEX is included in the Morgan Stanley International Index (MSCI), as it constitutes an important market for international investors who wish to invest in world markets. ATHEX has been upgraded from a developing market to a mature are, and the Euro has replaced the Greek drachma as the national currency since According to ADEX (as of 2005), the developing Greek derivatives market, despite operating only for a few years, is already in the seventh place among European derivatives markets, in terms of value of daily transactions ( million in 2005), following the established derivatives markets of Germany, UK, Italy, Euronext, Spain and Sweden. The international investor participation has increased from 23.9% in December 2001 to 42.1% in May 2006 and the net capital inflows from international investors in 2005 was 5.2 billion. The strategic planning of ADEX includes collaboration with East Europe (Romania, Bulgaria and Slovakia) and Mediterranean (Israel, Egypt and Cyprus) capital markets in the design and launch of indices and new derivatives products. Some special properties that differentiate the Greek capital market from other well-established markets are the following: (i) the ownership structure in ATHEX is different to that of other more mature markets, such as those of the US and the UK. In Greece, the structure is familyowned, concentrated in block-holders, whereas in other markets the structure is diffused; (ii) the privatisation of the public sector entities that started in early 2000 continues today. The ATHEX is a fully privatised group aiming at value maximisation; (iii) several reforms have taken place, in adopting the EU regulatory framework; (iv) although liquidity has increased lately, for several listed companies, the market is thin; and (v) even though the Greek market is characterised as mature since 2001, some emerging market characteristics may still remain. Thus, according to Bakaert and Harvey (1997), emerging market returns are characterised by low liquidity, thin trading, higher sample averages, low correlations with developed market returns, non-normality, better predictability, higher volatility and short samples. In addition, market imperfections, high transaction and insurance costs, less informed rational traders and investment constraints may also affect the risks and returns involved. Therefore, it is deemed important to empirically examine

4 The European Journal of Finance 245 the effectiveness of risk management strategies, using derivatives contracts for such markets. For the Greek derivatives market, it is often argued that it is characterised by the absence of highly specialised traders, by the absence of a respective large number of foreign derivatives traders (for the period examined in this study) and by the value of trading in derivatives during a normal day in ADEX representing a fraction of that traded in the underlying cash market. These issues, among others, make the Greek market interesting to investigate. This study contributes to the existing literature in a number of ways. First, it is important to know whether hedging effectiveness results coming from newly established derivatives markets are in accordance with corresponding results from well-established derivatives markets. The latter have been examined considerably in the literature, with results indicating that hedging effectiveness in stock index futures ranges from 80% to 99% (Yau 1993; Lee 1994; Park and Switzer 1995, among others). Second, the criteria used for model selection consider both risk-averse investors, who wish to minimise their risks (by considering their variance-return position) through hedging, and investors that aim to maximise expected utility, by taking into account returns, risks and preferences towards risk (the degree of risk aversion). 2 The latter method is practical and relevant for investors who are not entirely risk-averse, but are willing to undertake some risks in order to increase their returns and their utility from their investments, as a consequence. Moreover, taking into account the degree of the investor s risk aversion may result in different optimal model selection in comparison to the mean variance maximisation criterion. Raju (2005) investigates optimal hedging solutions for Ecuador oil and argues that the optimal hedging strategy is not significantly different for high levels of risk aversion. Third, different model specifications are estimated and compared so as to select the model, which takes into account the properties of cash and futures prices. Thus, the hedging effectiveness of dynamic hedge ratios is in contrast with the effectiveness of constant hedge ratios. The assumption by many models of constant hedge ratios is considered restrictive and is in contrast with the empirical evidence in a number of markets (e.g. Kroner and Sultan 1993; Bera, Garcia, and Roh 1997, among others), which indicates that the issue of whether hedge ratios are constant or time-varying (as a consequence of the time-varying distributions of returns and futures prices) is an empirical one. Moreover, the selection of a particular model among a number of alternatives, to estimate optimal hedge ratios, addresses (partly) the concerns raised by Simons (1997), among others, regarding the model risk problem, which arises when results rely on a particular potentially misspecified model. Fourth, economic analysis suggests that the prices of the cash asset and the futures contract are jointly (simultaneously) determined (Stein 1961). Consequently, the estimation of hedge ratios by univariate models may be subject to simultaneity bias; as a consequence, the estimated hedge ratios may not be optimal. Furthermore, hedge ratios estimated by univariate models are potentially misspecified because they ignore the existence of a long-run cointegrating relationship between cash and derivatives prices (Engle and Granger 1987) and fail to capture the short-run dynamics by excluding relevant lagged variables; this results in smaller-than-optimal hedge ratios (Kroner and Sultan 1993; Ghosh 1993). Fifth, the squared lagged disequilibrium error term between cash and futures prices is allowed to enter the specification of the variances of the estimated models, thus allowing disequilibrium effects to influence risk levels in the two markets. Sixth, the distribution of the data used is determined empirically; in line with the evidence in other (emerging or newly matured) markets, the data set for the Greek market follows a Student-t distribution with the degrees of freedom determined empirically during estimation.

5 246 M.G. Kavussanos and I.D. Visvikis Seventh, different data frequencies (weekly and daily) are employed, which allow us to examine the question of optimal frequency rebalancing in the presence of transaction costs. In order to answer this question, the benefits from frequent portfolio rebalancing are compared to the higher transaction costs involved. This is more pragmatic than simply taking an ad hoc frequency and estimating hedge ratios. It also answers the important question of optimal rebalancing and investment horizons of hedged portfolios. Eight, in-sample and out-of sample tests are employed to assess the hedging effectiveness of futures contracts. In-sample tests are based mainly on historical information, when it is more relevant for practitioners to examine the out-of-sample forecasting performance of hedging ratios. This paper extends the extant literature by examining the hedging performance both in-sample and out-of-sample, for a number of alternative model specifications, identifying the appropriate model in each case. The empirical models selected turn out to be different between the in-sample and out-of-sample periods and justify our approach. Finally, these contributions are over and above what we have already seen published in the literature regarding the ADEX market (see, for instance, Floros and Vougas 2004; Kavussanos and Visvikis 2005). The remainder of this paper is organised as follows. Section 2 presents the investor s mean variance operating framework and the empirical models that may be used to determine alternative optimal hedge ratios. Section 3 discusses the properties of the data. Section 4 presents the empirical results and evaluates the hedging effectiveness of the proposed strategies. Finally, Section 5 concludes the paper. 2. Optimal hedge ratio models Consider a hedger with a long (short) position in the cash market. He takes a short (long) position in the futures market of magnitude (as a percentage) γ t to offset the cash position. That is, the gains in one market will offset the losses in the other. Equation (1) shows the hedger s cash-futures portfolio return, whereas Equation (2) shows the variance of this portfolio return: R H,t = S t γ t F t (1) σ 2 H,t = Var t ( S t γ t F t ) = Var t ( S t ) + γ 2 t Var t ( F t ) 2γ t Cov t ( S t, F t ) (2) where S t = S t S t 1 is the logarithmic change in the cash price between time periods t 1 and t; F t = F t F t 1 is the logarithmic change in the futures price between t 1 and t; R H,t is the conditional return of the hedged portfolio; σ 2 H,t is the conditional variance of the return of this portfolio; σ 2 S,t Var t ( S t ) and σ 2 F,t Var t ( F t ) are the conditional variances of the returns on cash and futures positions, respectively; σ S,F,t Cov t ( S t, F t ) is the conditional covariance of returns between the cash and futures positions; and γ t is the hedge ratio the futures contracts traded as a percentage of the cash position at time t. This hedge ratio may be constant over the period of the hedge or time-varying as in Equations (1) and (2). When γ t = 0 the cash position remains unhedged, when γ t = 1 the futures position is equal in magnitude (and opposite) to the cash position this is known as naïve hedge, and provides a perfect hedge when cash and futures prices move by the same amount; that is, when they are perfectly correlated and the volatilities in the two markets are equal. Usually, cash and futures prices do not move perfectly together.as a consequence, hedgers select γ t = 1, in order to improve their hedging effectiveness, in what may be called a conditional hedge. Moreover, if the distributions of cash and futures

6 The European Journal of Finance 247 prices are time-varying, hedgers improve further the effectiveness of their hedges by allowing γ t to be time-varying. It is often forgotten that hedging involves an opportunity cost in terms of foregone returns. It is more pragmatic then to compare this cost with the benefit emanating from the reduction in risk. The degree of risk aversion of the hedger plays an important role in resolving this issue. To this effect, assume that the risk-averse investor aims to maximise the expected utility (E t U(R H,t+1 )) from his portfolio, given the information set available at time period t. That is, he aims to maximise the expected return from his hedged portfolio of cash and futures positions subject to the expected risks (variances) that he faces, and a certain level of risk aversion that describes his preferences regarding risks. He derives utility from higher returns, but has disutility from higher variances (risks) and vice versa, where the returns and variances of his portfolio determine the framework under which he operates. Consider, thus, the following mean variance expected utility function: E t U(R H,t+1 ) = E t (R H,t+1 ) kvar t (R H,t+1 ) (3) where k is the risk-aversion coefficient, measuring the degree of risk aversion (k > 0) of the individual investor; where higher (lower) values of k imply higher (lower) levels of risk aversion. The model given here assumes that the hedger has a quadratic utility function or that returns are normally distributed in a Markowitz (1959) framework (see also Kroll, Levy, and Markowitz 1984 for quadratic utility functions). Optimising Equation (3) with respect to γ t yields the following utility maximising hedge ratio (UMHR, γt ): γ t = [ Covt ( S t+1, F t+1 ) = γ t + Var t ( F t+1 ) [ Bias t+1 2kVar t ( F t+1 ) ] [ Et (F t+1 ) F t 2kVar t ( F t+1 ) ] ] = γ t + [ ] (Ft E t (F t+1 )) 2kVar t ( F t+1 ) where γt is the minimum variance hedge ratio (MVHR), that is, the following hedge ratio that minimises, with respect to γ t, the variance σh,t 2 in Equation (2): [ ] γt Covt ( S t, F t ) σ S,t = = ρ SF,t (5) Var t ( F t ) σ F,t where Cov t ( S t, F t ) and ρ SF,t denote the conditional covariance and correlation coefficient, respectively, between cash and futures price returns, while Bias t+1 = E t (F t+1 ) F t denotes the bias in the futures market between periods t and t + 1. γt is the MVHR, and is the optimal hedge ratio for investors who are completely risk averse; that is, who are not concerned about returns, but simply wish to minimise the variance of returns of their hedged portfolios, as described in Equation (2). In other words, in their utility function of Equation (3), the term E t (R H,t+1 ) is not relevant. Thus, the UMHR (γt ) equals the risk-minimising MVHR (γt ), augmented by an element which accounts for the existence of speculative components in the hedge. If futures prices at time period t(f t ) are unbiased predictors of expected futures prices for time period t + 1(E t (F t+1 )), technically if futures returns follow a martingale, and for a finite k, then γt = γt ; that is, in this case, the hedge ratio that generates the minimum portfolio variance is also the hedge ratio that maximises the investor s utility. This would be the case in well-functioning efficient futures markets. There may be markets though where futures prices are biased (F t = E t (F t+1 )); if not (4)

7 248 M.G. Kavussanos and I.D. Visvikis at all times, for at least some periods of time. Then the UMHR (γt ) contains both hedging and speculative components which investors may wish to utilise to increase their utilities. Moreover, one can distinguish these two components empirically; the first component in Equation (4) represents the MVHR, whereas the second represents the speculative component. Thus, in case futures prices are biased, there is a speculative motivation to trade so as to take advantage of the bias in the futures market (Lien and Tse 2002). The speculative component essentially captures the effect of short hedging (short in futures long in cash) on expected returns. If the expected futures price is less (more) than the current futures price, the hedger benefits from selling (buying) futures contracts. Finally, it should be noted that the UMHR also equals the MVHR for investors who are completely risk averse (κ ). Thus, with infinite risk aversion, the UMHR is independent of the bias in futures prices and is simply equal to MVHR. 2.1 Empirical models for the estimation of optimal hedge ratios The question addressed next is how to estimate the UMHR and the MVHR empirically. Starting with the conventional or constant UMHR (γt ), this can be estimated (where appropriate) from Equation (4), with the Bias t+1 and Var t ( F t+1 ) estimated as constants over the sample period. The conventional (constant) MVHR (γt ) can be estimated as the slope coefficient (γ )inthe following ordinary least-squares (OLS) regression (see, for instance, Ederington 1979): S t = h 0 + γ F t + ε t, ε t iid(0,σ 2 ) (6) Two potential problems exist with this specification. First, if cash and futures prices are cointegrated, an error-correction term (ECT) should be included in Equation (6); if the ECT is not included, then Equation (6) suffers from omitted variables bias, resulting in downward biased values of the coefficient γ (Kroner and Sultan 1993). Second, if the distributions (and their moments) of cash and futures prices are time-varying, so will γ in Equation (6). Once more, the effectiveness of the hedges may be improved upon by examining this issue empirically. To account for the first problem, of the potential omission of the ECT from Equation (6), the bivariate vector-error correction model (VECM) of Equation (7) is also used to estimate γ : p 1 X t = Ɣ i X t i + X t 1 + ε t, ε t 1 distr(0,h) (7) i=1 where X t is a (2 1) vector (S t,f t ) of non-stationary I(1) logarithmic cash and futures prices, respectively; denotes the first difference operator; and ε t isa(2 1) vector of regression equation error terms (ε S,t, ε F,t ), which are serially independent and follow an as-yet-unspecified conditional (on the available information set, t 1 ) bivariate distribution with mean zero and variance covariance matrix H. The VECM specification contains information on both the shortand long-run adjustments to changes in X t, via the estimated parameters in Ɣ i and, respectively. Johansen (1988, 1991) tests are used first to determine whether the series stand in a long-run relationship between them, that is, to test whether they are cointegrated. Kavussanos, Visvikis, and Alexakis (2008) show that this is the case for the pairs ATHEX-20 FTSE/ATHEX-20 and ATHEX Mid-40 FTSE/ATHEX Mid-40 cash and futures prices in the Greek markets. If such a long-run relationship is verified, then the VECM model of Equation (6) should be estimated, and includes the lagged ECT, X t 1 (S t 1 β 1 β 2 F t 1 ), as suggested by Engle and Granger (1987). In this case, constant VECM estimates of γ (and γ ) can be obtained as in the following ratio of the covariance of the error terms of the cash and futures equations (Cov(ε S,t, ε F,t ) = σ SF )

8 The European Journal of Finance 249 over the variance of the error term of the futures equation (Var(ε F,t ) = σf 2 ) from the bivariate VECM of Equation (7): γ = Cov(ε S,t,ε F,t ) Var(ε F,t ) = σ SF σ 2 F Kroner and Sultan (1993) show how to resolve the second issue arising from estimating constant MVHRs (and UMHRs) from Equation (7), when the conditional distributions of cash and futures prices are time-varying. Thus, in the VECM of Equation (7), the variance covariance matrix (H ) of the bivariate regression error term is allowed to become time-varying (H t ) in a generalised autoregressive conditional heteroskedasticity (GARCH) error structure (Bollerslev 1987), such as that of Equation (9). In this bivariate VECM-GARCH model, daily cash and futures prices react to the same information, and hence have non-zero covariances conditional on the available information set ( t 1 ). Moreover, in this paper we introduce a lagged squared ECT [(X t 1 ) 2 ] in the specification of the variance, in what we call a VECM-GARCH-X model. Following Lee (1994), the inclusion of this lagged squared ECT of the cointegrated cash and futures prices can capture the potential relationship between disequilibrium (measured by the ECT) and uncertainty (measured by the conditional variance). This is specified using the BEKK (Baba, Engle, Kraft, Kroner) augmented positive definite parameterisation (for more details, see Baba et al. 1995) 3 : H t = A A + B H t 1 B + C ε t 1 ε t 1 C + G (X t 1 ) 2 G (9) ( ) ( ) σ 2 ( ) ( ) ( )( ) S,t σ SF,t a11 0 a11 0 b11 0 σ 2 σ SF,t σf,t 2 = + S,t 1 σ SF,t 1 b11 0 a 21 a 22 a 21 a 22 0 b 22 σ SF,t 1 σf,t b 22 ( ) ( )( ) ( ) ( ) ( ) c11 0 ε1,t 1 ε1,t 1 c11 0 g (X 0 c 22 ε 2,t 1 ε 2,t 1 0 c 22 g t 1 ) 2 g11 22 g 22 where A isa(2 2) lower triangular matrix of coefficients, B and C are (2 2) diagonal coefficient matrices, with bkk 2 + c2 kk < 1,k = 1, 2 for stationarity and G isa(1 2) vector of coefficients of the lagged squared ECT [(X t 1 ) 2 ]. Matrices B and C are restricted to be diagonal because this results in a more parsimonious representation of the conditional variance. In this representation, the conditional variances are a function of their own lagged values (persistence term, H t 1 ), their own lagged squared error terms (lagged shocks, εt 1 2 ) and a lagged squared ECT parameter, (X t 1 ) 2 = (S t 1 β 1 β 2 F t 1 ) 2, whereas the conditional covariance is a function of lagged covariances and lagged cross-products of the error terms. 4 The most parsimonious specification for each model is estimated by excluding insignificant variables from the mean and the variance of the estimated models. The Broyden, Fletcher, Goldfarb, and Shanno (BFGS) algorithm (Shanno and Phua 1980) is used for estimation. Following Bollerslev (1987), in order to take into account the possibility of non-normality during estimation, the conditional Student-t distribution is used as the density function of the bivariate error term, where the degrees of freedom are allowed to be determined empirically by the data. Following estimation of the VECM-GARCH-X or VECM-GARCH models, time-varying covariances and biases (where appropriate) are used to calculate γt and γt as in Equations (8) and (4), respectively. (8) 2.2 Hedge ratios and hedging effectiveness measures Following estimation of OLS, VECM, VECM-GARCH and VECM-GARCH-X models, corresponding constant and time-varying MVHRs and UMHRs are computed. For each market,

9 250 M.G. Kavussanos and I.D. Visvikis we consider five different hedge ratios: time-varying hedge ratios computed from VECM-GARCH and VECM-GARCH-X specifications; constant hedge ratios generated from VECM with constant variances, estimated as (seemingly unrelated regressions SUR) systems (Zellner 1962); constant OLS hedge ratios from Equation (6); and naïve hedges, by taking futures positions, which are same in size as the cash positions (i.e. setting γ = 1). Comparison between the effectiveness of optimal hedge ratios, computed from different models, is made by constructing portfolios implied by the computed ratios each week (day, for daily data) and then comparing the variances or the expected utilities of the returns of these constructed portfolios. Two measures of hedging effectiveness are considered. The first measure is the variance reduction (VR) statistic of Equation (10), which compares the variance of the returns of the hedged portfolios (Var(R H,t )), to the variance of the unhedged positions, i.e. to Var( S t ): VR = Var( S t) Var(R H,t ) 100 (10) Var( S t ) The greater the reduction in the unhedged variance, the better the hedging effectiveness. That is, the higher the value of VR in Equation (10), the greater is the hedging effectiveness. Notice that for the OLS the degree of VR of the hedged portfolio, achieved through hedging, is provided by the coefficient of determination (R 2 ) of the regression, since this represents the proportion of the variability (risk) in the cash market that is explained (eliminated) through hedging (the variability of the futures position); the higher the R 2, the greater is the effectiveness of the hedge. We introduce a second measure of hedging effectiveness, which considers the economic benefits from hedging, as obtained from the hedger s utility function of Equation (3) and takes hedgers preferences into account. Consider the following utility increasing (UI) statistic: UI = E t U(R H,t ) E t U( S t ) (11) It compares the expected utility increase/decrease for investors, by comparing the expected utility of the hedged with that of the unhedged portfolios. The greater the increase in the utility of a strategy, in relation to the unhedged position, the better the hedging effectiveness. 3. Data The data sets used consist of weekly (250 Wednesday prices) and daily (1186 prices) cash and futures prices of the FTSE/ATHEX-20 market from 1 September 1999 to 7 June 2004, and weekly (228 Wednesday prices) and daily (1082 prices) cash and futures prices of the FTSE/ATHEX Mid-40 market from 1 February 2000 to 7 June When a holiday occurs on Wednesday, Tuesday s observation is used in its place. Futures prices are always those of the nearby contract because it is the most liquid and active contract. To avoid thin markets and expiration effects (when futures contracts approach their settlement days, their trading volume decreases sharply), we rollover to the next nearest contract one week before the nearby contract expires. Cash price data are obtained from the ATHEX, whereas futures price data are from the ADEX. For analysis, all price series are transformed into natural logarithms. Most studies in the economic literature (see, for example, Kroner and Sultan 1993; Gagnon and Lypny 1997, among others) use weekly data to calculate hedge ratios of futures contracts. The choice seems to be justified as it implies that hedgers in the market rebalance their futures positions at no less than a weekly basis, due to excessive transaction costs which would be

10 The European Journal of Finance 251 incurred if rebalancing takes place more frequently than once a week. Time-varying hedging strategies have higher implementation costs than constant strategies, as they require frequent updating and rebalancing of hedged portfolios. Thin trading, relatively low liquidity, high bid-ask spreads and high transaction costs can make daily rebalancing expensive. Therefore, the choice of rebalancing frequency, in the presence of transactions costs, is of high importance to the decisionmaker. Weekly data provide adequate number of observations (N = 250 in FTSE/ATHEX-20 and N = 228 in FTSE/ATHEX Mid-40) to allow investigation of the in- and out-of-sample performance of GARCH-based hedge ratios. Daily data are also used and give qualitatively the same results in terms of model specification and optimal hedge ratio selection. Summary statistics of logarithmic first-differences of weekly and daily cash and futures prices in the two markets are presented in Table 1. The results indicate excess kurtosis in all cases. There is also evidence of excess skewness in all series for daily but not for weekly data. In turn, Jarque and Bera (1980) tests indicate departures from normality for cash and futures prices. The Ljung- Box Q(i) statistics (for i = 4, 12 and 24) (Ljung and Box 1978) on the first 4, 12 and 24 lags of the sample autocorrelation function of the series, the Q 2 (i) statistics (for i = 4, 12 and 24) of the squared series and the ARCH(i) statistics (for i = 4, 12 and 24) (Engle 1982) indicate existence of serial correlation, heteroskedasticity and time-varying heteroskedasticity in the returns series of both markets. Given the time-series nature of the data, the first step in the analysis is to determine the order of integration of each price series using augmented Dickey Fuller (ADF 1981) and Phillips Perron (PP 1988) tests. Application of the ADF and PP unit root tests on the log-levels and log-first differences of the daily and weekly cash and futures price series indicate that all variables are log-first difference stationary, all having a unit root on the log-level representation (not shown due to lack of space). 4. Empirical results Having identified that cash and futures prices are I(1) variables (integrated of order 1), Johansen (1988) cointegration tests are used next to examine the existence of long-run relationships between cash and futures prices in the FTSE/ATHEX-20 and FTSE/ATHEX Mid-40 markets. The results indicate that at both frequencies (daily and weekly) the corresponding cash and futures prices in both markets are cointegrated, and thus, stand in a long-run relationship between them. 5 In order to examine whether the exact lagged basis or an unrestricted spread should be included as an ECT in the mean and variance equations of the VECM and VECM-GARCH-X models, the following cointegrating vector, β X t = (β 0 S t β 1 β 2 F t ) is examined, testing whether β = (1, 0, 1). This would imply that the ECT is the lagged basis; that is, that X t 1 = S t 1 F t 1. For both weekly and daily data the results, which are not shown here due to lack of space, indicate that the restrictions are not accepted in both markets. Thus, in the ensuing analysis, the unrestricted spread is used in the estimation of the VECM, VECM-GARCH and VECM-GARCH-X models. 4.1 Estimated models Having determined that cash and futures prices are cointegrated and that the basis is unrestricted, maximum-likelihood estimates of the most parsimonious VECM-GARCH and VECM-GARCH- X models, based on weekly data, selected on the basis of LR tests, for each market, are presented in Table 2. 6 The estimates of the coefficients of the mean and variance equations are shown in panels A and B, respectively. In the FTSE/ATHEX-20 market, a VECM-GARCH(1,1) specification is

11 Table 1. Descriptive statistics of weekly and daily logarithmic first-differences of cash and futures prices. ARCH ARCH ARCH ADF PP ADF PP Mean Skew Kurt Q(4) Q(12) Q(24) Q 2 (4) Q 2 (12) Q 2 (24) (4) (12) (24) J-B Lev Lev first differences Panel A: FTSE/ATHEX-20 weekly cash and futures price series (09/99 to 06/04) Cash (249) [0.1892] [0.103] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] Q 2 (12) [0.000] [0.000] [0.000] [0.000] (1) (5) (1) (10) Futures (249) [0.1969] [0.220] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] (1) (7) (1) (7) Panel B: FTSE/ATHEX Mid-40 Weekly Cash and Futures Price Series (02/00 to 06/04) Cash (249) [0.0321] [0.964] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] (0) (10) (0) (9) Future (249) [0.0418] [0.729] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] (0) (7) (0) (6) Panel C: FTSE/ATHEX-20 Daily Cash and Futures Price Series (09/99 to 06/04) Cash ,867 27, ,961 23,951 33,947 11, (1081) [0.1552] [0.010] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] (0) (1) (0) (5) Futures ,874 27, ,017 24,089 31,138 10, (1081) [0.1824] [0.019] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] (1) (3) (0) (1) Panel D: FTSE/ATHEX Mid-40 Daily Cash and Futures Price Series (02/00 to 06/04) Cash ,171 22, ,582 17,738 41,183 13, (1081) [0.0183] [0.006] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] (1) (9) (0) (8) Future ,131 22, ,431 17,306 29, (1081) [0.0412] [0.090] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] (0) (6) (0) (7) 252 M.G. Kavussanos and I.D. Visvikis Data series are measured in logarithmic first differences. Numbers in parentheses below the headers Cash and Futures are number of observations. Figures in square brackets [.] indicate exact significance levels. Mean is the sample mean of the series. Skew and Kurt are the estimated centralised third and fourth moments of the data; their asymptotic distributions under the null are T â 3 N(0, 6) and T(â 4 3) N(0, 24), respectively. Q(i) and Q 2 (i) are the Ljung-Box (1978) Q statistics on the first i (for i = 4, 12 and 24) lags of the sample autocorrelation function of the raw series and of the squared series. ARCH(i) is the Engle (1982) test for ARCH effects (for i = 4, 12 and 24). J-B is the Jarque and Bera (1980) test for normality. ADF is the Augmented Dickey Fuller (1981) test. The ADF regressions include an intercept term; the lag-length of the ADF test (in parentheses) is determined by minimising the SBIC (1978). PP is the Phillips and Perron (1988) test; the truncation lag for the test is in parentheses. Lev and first differences correspond to price series in log-levels and log-first differences, respectively. The 95% critical value for the ADF and PP tests is The results indicate excess kurtosis and departures from normality for the cash and futures prices in both markets. The Q(i) statistic indicate significant serial correlation, while the Q 2 (i) statistic and the ARCH(i) statistic indicate existence of heteroskedasticity in the price series in both markets. The ADF and PP tests indicate that all variables are log-first difference stationary, all having a unit root on the log-levels representation.

12 The European Journal of Finance 253 Table 2. Maximum likelihood estimates of VECM-GARCH-X models. (FTSE/ATHEX-20: 1999: :06, FTSE/ATHEX Mid-40:2000: :06) VECM-GARCH(1,1) VECM-GARCH(1,1)-X Cash Futures Cash Futures Coefficients FTSE/ATHEX-20 FTSE/ATHEX-20 FTSE/ATHEX-40 FTSE/ATHEX-40 Panel A: Conditional Mean Parameters a j,j = S,F (0.909) (2.401) (1.898) (3.130) [0.363] [0.016] [0.058] [0.002] a j,1,j = S,F ( 2.386) ( 1.731) (1.801) (1.769) [0.017] [0.083] [0.072] [0.069] b j,1,j = S,F (2.453) (1.875) 0.38 ( 1.851) ( 1.879) [0.014] [0.061] [0.063] [0.075] Panel B: Conditional Variance Parameters a (8.151) [0.000] (0.663) [0.507] a (6.166) [0.000] (1.243) [0.214] a (6.168) [0.000] (2.339) [0.019] b kk, k = 1, (1.359) (2.514) ( ) ( ) [0.174] [0.012] [0.000] [0.000] c kk, k = 1, (3.403) (3.566) (2.457) (2.459) [0.000] [0.001] [0.000] [0.014] [0.013] g kk, k = 1, (5.166) (4.559) [0.000] [0.000] v (3.941) [0.000] (3.504) [0.000] Panel C: Diagnostic tests on standardised residuals Log-Likelihood Skewness [0.449] [0.405] [0.325] [0.240] Kurtosis [0.000] [0.000] [0.000] [0.000] Q(12) [0.672] [0.725] [0.151] [0.305] Q 2 (12) [0.638] [0.756] [0.832] [0.874] ARCH(12) [0.790] [0.907] [0.751] [0.839] AIC SBIC The maximum-likelihood estimates of the preferredvecm-garch andvecm-garch-x models, selected on the basis of LR tests, for each market are given in panels A and B for the mean and variance equations, respectively. Diagnostic tests for the standardised residuals are in panel C. All variables are in natural logarithms. The GARCH models in both markets are estimated using the Student-t distribution; v is the estimated degrees of freedom of the Student-t distribution. The BFGS algorithm is used to estimate the models. Figures in parentheses (.) and in squared brackets [.] indicate t-statistics and exact significance levels (P -values), respectively. Q(12) and Q 2 (12) are the Ljung-Box (1978) tests for 12th order serial correlation and heteroskedasticity in the standardised residuals and in the squared standardised residuals, respectively. ARCH(12) is Engle s (1982) F test for Autoregressive Conditional Heteroskedasticity. SBIC is the Schwartz Bayesian Information Criterion (Schwartz 1978). p 1 p 1 S t = a S,i S t i + b S,i F t i + a S X t 1 + ε S,t i=1 i=1 p 1 p 1 F t = a F,i S t i + b F,i F t i + a F X t 1 + ε F,t ; ε t = i=1 i=1 ( ) εs, t εf, t t 1 t dist(0,h t,v) (7) H t = A A + B H t 1 B + C ε t 1 ε t 1 C + G (X t 1 ) 2 G (9)

13 254 M.G. Kavussanos and I.D. Visvikis selected. In the FTSE/ATHEX Mid-40 market, a VECM-GARCH(1,1)-X model is appropriate, where the lagged squared unrestricted ECT in the variance is found significant in both the cash and in the futures equations. The lagged ECT in the mean of the cash equation is insignificant in the FTSE/ATHEX-20 market, whereas it is significant only at the 10% level for the FTSE/ATHEX Mid-40 market. Normally, one would expect negative coefficients of the ECT terms in the cash market equations; in this case they are statistically zero or marginally positively significant. The lagged ECT term in the mean of the futures equations is significant and positive in both markets, which is in line with a priori expectations. The above is an indication that last week s disequilibrium effect between the cash and futures price series has an impact on the futures market, but not on the cash market. Thus, in response to a positive forecast error, only the futures price series increases in value to restore the long-run equilibrium. This may indicate a relative weakness in the cash markets, in comparison to their futures price series counterparts, to respond to a long-run disequilibrium in the cash futures markets relationship (see also Kavussanos, Visvikis, and Alexakis 2008 for further evidence on this). In relation to the short-run dynamics, it is observed that the lagged cash price change ( S t i ) coefficients are negative, whereas the lagged futures price change ( F t i ) coefficients are positive in the FTSE/ATHEX-20 cash and futures markets, but the situation is the opposite in the FTSE/ATHEX Mid-40 markets. It seems that in the relatively more liquid FTSE/ATHEX-20 market, last week s information from the futures market influences positively the current cash and futures returns, whereas in the illiquid FTSE/ATHEX Mid-40 market last week s information has the opposite effect. In contrast, previous (last) week s information from the cash markets influences negatively the current returns in the cash and futures FTSE/ATHEX-20 markets, and negatively the current returns in the cash and futures FTSE/ATHEX Mid-40 markets. In the FTSE/ATHEX-20 cash market equation, the persistence of lagged cash and futures price changes in the mean is zero as the coefficients and , effectively add up to 0. This persistence in the corresponding futures equation is also minimal, standing at In the FTSE/ATHEX Mid-40 cash market equation the persistence of lagged cash and futures price changes in the mean is 0.17, whereas this persistence in the corresponding futures equation stands at +0.19, both being non-zero. These results for the FTSE/ATHEX Mid-40 market make sense, because this market displays much lower liquidity in comparison to the FTSE/ATHEX-20 market, with illiquidity manifesting itself in higher persistence effects. 7 In the variance equations, the coefficients of the lagged squared error terms (c kk ) are positive in both the FTSE/ATHEX-20 and in the FTSE/ATHEX Mid-40 markets. However, the coefficients of the lagged variance terms (b kk ) are positive in the FTSE/ATHEX-20 market but negative in the FTSE/ATHEX Mid-40 market. The variance term coefficient (b kk ) can be thought of as reflecting the impact of old news. It is picking up the impact of volatility changes relating to the period prior to the last one, and thus, to volatility news which arrived the period before last period. The negative sign (and the high t-statistics) indicates that the volatility of the FTSE/ATHEX Mid-40 market is more susceptible to negative old news, and vice versa for the positive coefficient observed in the FTSE/ATHEX-20 market; it has to be noted that for the latter market it is only in the futures and not in the cash market that this coefficient is significant. From the above it can be noticed that the selected models for the two markets (avecm-garch for the FTSE/ATHEX-20 and a VECM-GARCH-X for the FTSE/ATHEX Mid-40 market) yield qualitatively different results. Their differences may be due to: the different model specifications (e.g. inclusion of a lagged squared ECT in the variance in the VECM-GARCH-X model), the different compositions of the underlying stock indices of the futures contracts and the different

14 The European Journal of Finance 255 estimation periods. Moreover, the significantly lower level of trading volume and liquidity in the FTSE/ATHEX Mid-40 futures market, when compared with the more liquid FTSE/ATHEX-20 futures market, may be another reason for the observed differences between the two markets in both the mean and variance equations results. Returns follow conditional t-distributions with and degrees of freedom, respectively, for the FTSE/ATHEX-20 and FTSE/ATHEX Mid-40 markets. Panel C of Table 2 reports the diagnostic tests for the standardised residuals (ε t / ĥt), including Ljung-Box (1978) statistics for 12th-order serial correlation in levels and squares of the standardised residuals. They indicate the absence of significant serial correlation, heteroskedasticity and ARCH effects. Moreover, the test statistics for asymmetry (sign bias, negative size bias, positive size bias, joint sign and size bias) developed by Engle and Ng (1993), but not shown here due to lack of space, indicate the absence of any size or sign biases in the standardised residuals of the equations. Thus, the estimated models are well specified and fit the data very well. 4.2 Hedging effectiveness results In-sample hedge ratios Having estimated OLS, VECM, VECM-GARCH and VECM-GARCH-X models, the corresponding optimal constant and time-varying MVHRs (Equation (5)) and UMHRs (Equation (4)), optimal hedged portfolio returns (R H,t ) (Equation (1)), variances (σh,t 2 ) (Equation (2)), and expected utilities (Equation (3)) as well as VR (Equation (10)) and utility increases (UI) (Equation (11)) are presented in Table 3, assuming a risk-aversion parameter of k = 3. The latter is in line with most empirical studies in the literature. 8 However, this assumption is relaxed later and the risk-aversion coefficient is allowed to take several different values. Second, if futures prices are unbiased, that is, if they follow a pure martingale process (i.e. if E t (F t+1 ) = F t ), the MVHR equals the UMHR, and there is no speculative demand component. The existence of such a speculative component in a hedge can only be verified empirically. Panels A and B of the table show the in-sample estimates for daily and weekly frequencies, respectively, whereas Panels C and D present the corresponding out-of-sample estimates. Results for the FTSE/ATHEX-20 market, based on both weekly and daily data, indicate that time-varying hedge ratios, estimated from the VECM-GARCH model, outperform the constant hedge ratios, based on the VR criterion of Equation (10). For the FTSE/ATHEX Mid-40 market the results are almost similar, with time-varying hedge ratios, computed from the VECM-GARCH-X model, outperforming all other hedging strategies for weekly data. For daily data, the conventional (OLS) model produces the highest VR. Not surprisingly, as can be observed in the table, staying unhedged is the worst option in both markets. Finally, the best hedging strategy out of all constant hedge ratio models (naïve, OLS, VECM) for the FTSE/ATHEX-20 market seems to be emanating from the VECM model, whereas for the FTSE/ATHEX Mid-40 market the conventional (OLS) model seems to be more appropriate. The above results are in line with those of Floros and Vougas (2004) for the FTSE/ATHEX- 20 and FTSE/ATHEX Mid-40 markets. However, their analysis is constrained in an in-sample framework with daily data, and also assume, rather than test, that the MVHR is equal to the UMHR. This paper uses both daily and weekly data in-sample and out-of-sample frameworks and examines empirically whether the speculative components of hedges are indeed zero; that is, whether the MVHRs equals the UMHRs. The mean value of the bias (E t (F t+1 ) F t ) for the FTSE/ATHEX-20 market is for weekly data and for daily data, with corresponding t-statistics 1.27 and 1.34,