Hedging with Stock Index Futures Contracts in the Athens

HEDGING WITH STOCK INDEX FUTURES CONTRACTS IN THE ATHENS DERIVATIVES EXCHANGE MANOLIS G. KAVUSSANOS 1 and ILIAS D. VISVIKIS 2 1

Athens University of Economics and Business, 76 Patission St., 10434, Athens, Greece. Email: [email protected] 2

ALBA Graduate Business School, Athinas Ave. & 2A Areos St., 16671, Vouliagmeni, Athens, Greece. Email: [email protected]

ABSTRACT 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 minimal 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, which are lower than minimum variance hedge ratios.

Keywords: Hedging Effectiveness, Futures Markets, Constant and Time-Varying Hedge Ratios, Utility Functions, VECM-GARCH-X. JEL Classification: G13, G14, C32. 1

Corresponding Author:

Professor Manolis G. Kavussanos, Athens University of Economics and Business, 76 Patission St., 10434, Athens, Greece, Tel: 0030 210 8203167, Fax: 0030 210 8203196, Email: [email protected] Acknowledgement: The authors would like to thank Dr. Nikos Porfiris from the Athens Derivatives Exchange for providing the data. Thanks are also due for their comments to the editor and two anonymous referees, as well as participants of the following seminars and conferences, where earlier versions of this study were presented: International Conference on Advances in Applied Financial Economics (AFE), Samos, Greece, May 2004; 8th Annual European Conference of the Financial Management Association International (FMA), Zurich, Switzerland, June 2004 and University of Piraeus, May 2006.

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 which can be utilized to answer these important issues for investors wishing 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 investigated in this paper. 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 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 (that is, that expected futures prices next period are equal to today’s futures prices; E(F1= F0) and resort back to answering the former part of the

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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 answer to 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), founded in April 1998. The first stock index futures contract of ADEX was the FTSE/ATHEX-20 futures contract, 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 ATHEX1.

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, amongst others – see for instance Kavussanos et al. (2007). The ATHEX is included in the Morgan Stanley International Index (MSCI), as it constitutes an important market for international investors wishing to invest in world markets. ATHEX has been upgraded from a developing to a mature market, and the Euro has replaced the Greek drachma as the national currency since 2001. According to ADEX (as of 2005), the developing Greek derivatives market, despite operating only for a few years, is already in the 7th place amongst European derivatives markets, in terms of the value of daily transactions (50-100 million euro in 2005), following the established derivatives markets of Germany, UK, Italy, Euronext, Spain and Sweden. The

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international investor participation has increased from 23.9% in December 2001 to 42.1% in May 2006 and the net capital inflow from international investors in 2005 was 5.2 billion euros. 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 the 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 privatization of the public sector entities that started in the early 2000, continues today. The ATHEX is a fully privatized group aiming at value maximization; (iii) several reforms have taken place, in adopting the E.U. regulatory framework; (iv) although liquidity has increased lately, for several listed companies, the market is thin; and (v) even though the Greek market is characterized 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 transactions 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 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, the absence of a respective large number of foreign derivatives traders (for the period examined in this study) and a value of trading in derivatives during a normal day in ADEX representing a fraction of that traded in the

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underlying cash market. These issues, amongst 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 if 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% (see Yau, 1993; Lee, 1994 and Park and Switzer, 1995, amongst 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.

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 contrasted with the effectiveness of constant hedge ratios. The assumption by many models of constant hedge ratios is considered restrictive and contrasts with the empirical evidence in a number of markets (e.g. Kroner and Sultan, 1993; Bera et al., 1997, amongst others), which indicates that the issue of whether hedge ratios are static or

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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 amongst a number of alternatives, to estimate optimal hedge ratios, addresses (partly) the concerns raised by Simons (1997), amongst 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 (see 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 (see Kroner and Sultan, 1993 and 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.

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

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transactions costs involved, in order to answer this question. 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 insample and out-of-sample, for a number of alternative model specifications, identifying the appropriate model in each case; the empirical models selected may be different between the insample 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) and Kavussanos and Visvikis (2005).

The remainder of this paper is organised as follows. Section two presents the investor’s meanvariance operating framework, and the empirical models that may be used to determine alternative optimal hedge ratios. Section three discusses the properties of the data. Section four presents the empirical results and evaluates the hedging effectiveness of the proposed strategies. Finally, section five 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) γt to offset the cash position. That

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is, the gains in one market will offset the losses in the other. Equation (1) shows the hedger’s cash-futures portfolio return, while Equation (2) shows the variance of this portfolio return:

RH,t = ∆St - γ t ∆Ft

(1)

σ 2H,t = Vart(∆St – γt∆Ft) = Vart(∆St ) + γt2Vart(∆Ft ) – 2γtCovt(∆St, ∆Ft )

(2)

where, ∆St = St – St-1 is the logarithmic change in the cash price between time periods t-1 and t; ∆Ft = Ft – Ft-1 is the logarithmic change in the futures price between t-1 and t; RH,t is the conditional return of the hedged portfolio; σ 2H,t is the conditional variance of the return of this 2 portfolio; σ S,t ≡ Vart(∆St ) and σ 2F,t ≡ Vart(∆Ft ) are the conditional variances of the returns on

cash and futures positions, respectively; σ S, F ,t ≡ Covt(∆St, ∆Ft ) 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 (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 together and 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 prices are time-varying, hedgers improve further the effectiveness of their hedges by allowing γ to be timevarying.

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Assume further that the risk-averse investor aims to maximise the expected utility (EtU(RH,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 visa 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: EtU(RH,t+1) = Et(RH,t+1) – k Vart (RH,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 et al. (1984) for quadratic utility functions.

Optimising Equation (3) with respect to γt yields the Utility Maximizing Hedge Ratio (UMHR,γt**): UMHR:  Cov t (∆S t +1 , ∆Ft +1 )   E t (Ft +1 ) − Ft  * γt**=  −  =γt + Vart (∆Ft +1 )   2k Vart (∆Ft +1 )  

 - (Ft - E t (Ft +1 ))  *  - Bias t +1    =γt +    2k Vart (∆Ft +1 )   2 k Vart (∆Ft +1 ) 

(4)

where, γt* is the Minimum Variance hedge Ratio (MVHR); that is, the hedge ratio that minimizes with respect to γt the variance σ 2H,t in Equation (2). and

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MVHR:

σ S ,t  Cov t (∆S t , ∆Ft )  γ t* =   = ρ SF ,t σ F ,t  Vart (∆Ft ) 

(5)

where, Covt(∆St, ∆Ft) and ρSF,t denotes the conditional covariance and correlation coefficient, respectively, between cash and futures price returns, while Biast+1 = E(Ft+1) – Ft denotes the bias in the futures market between the periods t and t+1. γ* 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 minimize 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 Et(RH,t+1) is not relevant.

Thus, the UMHR (γt**) equals the MVHR (γt*) – the latter minimises the risk of the hedger, augmented by an element which accounts for the existence of speculative components in the hedge. If futures prices at time period t (Ft) are unbiased predictors of expected futures prices for time period t+1 (EtFt+1), technically if futures returns follow a martingale, and for a finite k, then γt** = γt*; that is, 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 (Ft ≠ Et(Ft+1)); if not 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 utilize to increase their utilities. Moreover, one can distinguish these two components empirically; the first component in Equation (4) represents the MVHR, while the second represents the speculative component. In this case, there is a speculative motivation to trade so as to take advantage of the bias in the futures market (see also 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

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(buying) futures contracts. The UMHR equals the MVHR also for investors who are completely risk-averse (κ →∞). Thus, with infinite risk aversion, the UMHR is independent of the bias in futures prices.

2.1. Empirical Models for the Estimation of Optimal Hedge Ratios The next question to answer is how to estimate the UMHR and the MVHR empirically. The conventional or constant UMHR (γt**) can be estimated (where appropriate) as in Equation (4), with the Biast+1 and Vart(∆Ft+1) estimated as constants over the sample period. The conventional (constant) MVHR can be estimated as the slope coefficient (γ*) in the following Ordinary Least Squares (OLS) regression - see for instance Ederington (1979): ∆St = h0 + γ*∆Ft + εt ; εt ~ iid(0,σ2)

(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 of the coefficient γ*, resulting in downward biased values of γ* – see 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

∆Xt =

∑ Γi∆Xt-i + ΠXt-1 + εt i=1

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;

εt | Ωt-1 ~ distr(0, H)

(7)

where, Xt is a (2x1) vector (St, Ft)' of non-stationary I(1) logarithmic cash and futures prices, respectively, ∆ denotes the first difference operator, and εt is a (2x1) vector of regression equation error-terms (εS,t, εF,t)', which are serially independent and follow an as-yet-unspecified conditional (on the available information set, Ωt-1) bivariate distribution with mean zero and variance-covariance matrix H. The VECM specification contains information on both the shortand long-run adjustment to changes in Xt, via the estimated parameters in Γi and Π, 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 et al. (2007) show that this is the case for the pairs ATHEX-20 - FTSE/ATHEX-20 and ATHEX Mid40 - FTSE/ATHEX Mid-40 cash and futures prices in the Greek markets. If such a long-run relationship is verified, then a VECM model, such as that of Equation (6), can be estimated. This is the correct specification of Equation (6) in this case, and includes the lagged ECT, Xt-1 ≡ (S t-1 – β 1 – β 2 F t-1 ), as suggested by Engle and Granger (1987). Constant VECM estimates of γ* (and γ**) can be estimated as the ratio of the covariance of the error-terms of the cash and futures equations (Cov(εS,t, εF,t) = σSF) from the bivariate VECM of Equation (7) over the variance of the error-terms of the futures equation (Var(εF,t) = σ 2F ) of the VECM:

*

γ =

Cov(ε S, t , ε F,t ) Var(ε F,t )

(8)

Kroner and Sultan (1993) show how to resolve the second issue arising from estimating constant MVHRs (and UMHRs) from Equations (6) or (7). Essentially, they allow the conditional distributions of cash and futures prices to be 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 (Ht) in a Generalised Autoregressive Conditional Heteroskedasticity (GARCH) error structure (see Bollerslev, 1987), as in Equation (8). In this bivariate VECM-GARCH

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model, daily cash and futures prices react to the same information, and hence, have non-zero covariances conditional on the available information set (Ωt-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: Ht = A'A + B' Ht-1 B + C' εt-1εt-1' C + G' (X t −1 )2 G

 σ S2,t σ SF ,t   a 11 0  '   =   σ σ 2   a 21 a 22   SF, t F ,t  '

(9)

' 2  a 11 0   b 11 0   σ S ,t −1 σ SF ,t −1    +   2  a 21 a 22   0 b 22   σ SF, t -1 σ F ,t −1 

 c 11 0   ε 1, t − 1     +   0 c 22   ε 2 , t − 1 

 ε 1, t − 1     ε 2, t − 1 

'

'

 b 11 0     0 b 22 

 c 11 0   g 11    +   (X t −1 )2  0 c 22   g 22 

 g 11     g 22 

where, A is a (2x2) lower triangular matrix of coefficients, B and C are (2x2) diagonal coefficient matrices, with b 2kk + c 2kk < 1, k = 1,2 for stationarity and G is a (1x2) 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, Ht-1), their own lagged squared error-terms (lagged shocks, ε t2−1 ), and a lagged squared ECT parameter, (X t −1 )2 = (St-1 – β1 – β2Ft-1)2, while the conditional covariance is a function of lagged covariances and lagged cross products of the error-terms4.

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The most parsimonious specification for each model is estimated by excluding insignificant variables. The Broyden, Fletcher, Goldfarb, and Shanno (BFGS) algorithm (see 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 VECMGARCH models, time-varying covariances and biases (where appropriate) are used to calculate γ *t and γ *t * as in Equations (5) and (4), respectively.

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, we consider five different hedge ratios: time-varying hedge ratios computed from VECMGARCH and VECM-GARCH-X specifications; constant hedge ratios generated from VECM with constant variances, estimated as SUR (Seemingly Unrelated Regressions) systems (see Zellner, 1962); constant OLS hedge ratios from Equation (6); and naïve hedges, by taking futures positions, which are the 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 statistic of Equation (10), which compares the variance of the returns of the hedged portfolios (Var(RH,t)), to the variance of the unhedged positions, i.e. to Var(∆St):

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VR =

Var (∆St ) − Var (R H,t ) Var (∆St )

x 100

(10)

The greater the reduction in the unhedged variance, the better the hedging effectiveness. That is, the higher the value of Variance Reduction (VR) in Equation (10), the greater is the hedging effectiveness.

Notice that for the OLS and VECM, the degree of variance reduction of the hedged portfolio, achieved through hedging, is provided by the coefficient of determination (R2) 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 R2 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(∆St )

(11)

It compares the 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 01 September 1999 to 07 June 2004, and

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weekly (228 Wednesday prices) and daily (1082 prices) cash and futures prices of the FTSE/ATHEX Mid-40 market from 01 February 2000 to 07 June 2004. 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, while 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 and Gagnon and Lypny, 1997, amongst 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 transactions costs which would be incurred if rebalancing takes place more frequently than once a week. Time-varying hedging strategies have higher implementation costs than constant strategies, since they require frequent updating and rebalancing of hedged portfolios. Thin trading, relatively low liquidity, high bidask spreads and high transactions costs can make daily rebalancing expensive. Therefore, the choice of rebalancing frequency, in the presence of transactions costs, is of high importance to the decision-maker. 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.

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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, JarqueBera (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 Q2(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, indicates that all variables are log-first difference stationary, all having a unit root on the loglevels 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 one), 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 Mid40 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 them5. 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'Xt = (β0St β1 β2Ft)' is examined, testing whether β' = (1, 0, –1), which would imply that the ECT is the lagged basis, Xt-1 =

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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 VECMGARCH(1,1) specification is selected. In the FTSE/ATHEX Mid-40 market, a VECMGARCH(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, while 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-

17

run disequilibrium in the cash-futures markets relationship – see also Kavussanos et al. (2007) for further evidence on this.

In relation to the short-run dynamics, it is observed that, the lagged cash price change (∆St-i) coefficients are negative, while the lagged futures price change (∆Ft-i) coefficients are positive in the FTSE/ATHEX-20 cash and futures markets, while 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, while in the illiquid FTSE/ATHEX Mid-40 market last week’s information has the opposite effect. On the other hand, 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 -0.707 and +0.712, effectively add up to 0. This persistence in the corresponding futures equation is also minimal, standing at +0.036. In the FTSE/ATHEX Mid40 cash market equation the persistence of lagged cash and futures price changes in the mean is -0.17, while 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, since this market displays much lower liquidity in comparison to the FTSE/ATHEX-20 market, with illiquidity manifesting itself in higher persistence effects7.

In the variance equations, the coefficients of the lagged squared error-terms (ckk) are positive in both the FTSE/ATHEX-20 and in the FTSE/ATHEX Mid-40 markets. However, the coefficients of the lagged variance terms (bkk) are positive in the FTSE/ATHEX-20 market but negative in the FTSE/ATHEX Mid-40 market. The variance term coefficient (bkk) can be thought of as

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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 visa 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 (a VECM-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 (for example, 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 to the different 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 of the observed differences between the two markets in both the mean and variance equations results.

Returns follow conditional t-distributions with 5.773 and 6.410 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 (εt / hˆ t ), including Ljung-Box (1978) statistics for 12th-order serial correlation in levels and squares of the standardised residuals. They indicate 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 absence of

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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 4.2.1. 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 (RH,t) (Equation 1), variances (σ 2H ,t ) (Equation 2), and expected utilities (Equation 3) as well as variance reductions (VR) (Equation 10) and utility increases (UI) (Equations 11) are presented in Table 3. Panels A and B of the table show the insample 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 VECMGARCH-X model, outperforming all other hedging strategies for weekly data. For daily data, the conventional (OLS) model produces the highest variance reduction. 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, while for the FTSE/ATHEX Mid-40 market the conventional (OLS) model seems to be more appropriate.

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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.

Thus, for estimation purposes, in the expected utility model of Equation (3) initially it is assumed that the risk aversion coefficient is k = 3, implying high levels of risk-aversion. This is in line with most empirical studies in the literature8. 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 Et(Ft+1) = Ft), 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.

The mean value of the bias (Et(Ft+1) – Ft) for the FTSE/ATHEX-20 market is -0.00324 for weekly data and -0.00069 for daily data, with corresponding t-statistics -1.27 and -1.34, respectively; that is, the bias is statistically zero, and thus, in the FTSE/ATHEX-20 market, the MVHR equals the UMHR; that is, investors use futures for hedging purposes and cannot benefit for speculative/investment reasons. Thus, the MVHRs produced by the different models (that is the VECM-GARCH, VECM, OLS, etc.) are used in Equation (3), with k = 3, to estimate the expected utility from the hedges, in the third and fifth columns of the table.

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In the FTSE/ATHEX Mid-40 market, however, the results indicate the existence of a bias in futures prices, with mean values (t-statistics) of -0.00661 (-2.04) for weekly data and -0.00137 (-2.03) for daily data. Therefore, for the FTSE/ATHEX Mid-40 market, assuming that investor’s expectations are formed rationally, the non-zero return of the hedged portfolio ((RH,t)), of Equation (1) and the expected utility function of Equation (3) is derived for each of the alternative hedging models, by utilizing the estimated UMHR of Equation (4) and assuming k = 3. The latter is the MVHR of Equation (5) plus a speculative demand component. The

estimated expected utilities from each of these models are also shown in Table 3.

In the same table, the utility increase with respect to the unhedged position achieved by each model is estimated in each market for both daily and weekly data. As can be observed, the UI statistic selects the time-varying VECM-GARCH and VECM-GARCH-X models in all cases. The results then, in terms of model selection, are similar to those obtained when the MVHR is used as the model selection criterion. The only exception is for the FTSE/ATHEX Mid-40 market with daily data, for which the OLS constant MVHR produces the best results.

However, in the choice between constant or time-varying hedge ratios, an important element is the transaction costs, incurred during the portfolio rebalancing process implied in the timevarying case, in comparison to the constant ratio situation. Take for instance the in-sample average expected weekly portfolio variances of returns from the hedged positions in the FTSE/ATHEX-20 market from the OLS and the VECM-GARCH hedge ratios, reported in Panel B (fourth column) of Table 3; they are 0.000102 and 0.000099, respectively. Then, on average, the investor obtains a weekly expected utility of EtU(RH,t+1) = Et(RH,t+1) – k Vart (RH,t+1) = -0.000148 – 3 x (0.000102) = - 0 . 0 0 0 4 5 4 and EtU(RH,t+1) = -0.000058 – 3 x (0.000099) – φ = -0.000239 – φ, when the OLS and the VECM-GARCH hedge ratios are used, respectively; where, φ

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represents the reduced returns caused by the transactions costs (as a rate of return) incurred due to portfolio rebalancing. Thus, by using the VECM-GARCH model, over the OLS model, hedgers in the market can benefit from an increase in their average weekly expected utility of (0.000215 – φ). The VECM-GARCH hedge strategy will be preferred over the OLS strategy as long as φ < 0.000215, otherwise due to excessive costs it is not economically wise to employ the VECM-GARCH hedging strategy. One daily round trip (one buy and one sell) for established investors in the FTSE/ATHEX-20 market costs on average 16 euros (= 8€ buy + 8€ sell) and in the FTSE/ATHEX Mid-40 market it costs approximately 20 euros (= 10€ buy + 10€ sell)9. Assuming a margined contract size of 1,144€ (= 2,288 index value x 5 multiplier x 10% margin) for the FTSE/ATHEX-20 and 2,435.5€ (= 4,871 index value x 5 multiplier x 10% margin) for the FTSE/ATHEX Mid-40, transactions costs in percentage terms amount to around 0.01399% (= 16/1,144) for the FTSE/ATHEX-20 market and 0.00821% (= 20/2,435.5) for the FTSE/ATHEX Mid-40 market. Therefore, an investor with a mean-variance utility function would prefer the VECM-GARCH hedging strategy to the OLS strategy, as φ = 0.0001399 < 0.000215 in the FTSE/ASE-20 market and φ = 0.0000821 < 0.000308 in the FTSE/ASE Mid-40 market.

4.2.2. Out-of-Sample Hedge Ratios

While the in-sample performance of the alternative hedging strategies provides an indication of their historical performance, investors are more concerned about how well they can do in the future. In order to investigate that, we use an initial portion of the sample for estimation and reserve the remaining sample for out-of-sample forecasting. Following Tashman (2000), nonoverlapping independent out-of-sample N-period ahead forecasts are generated over the forecast period. In order to avoid the bias induced by serially correlated overlapping forecast errors, the estimation period is augmented recursively by N-periods ahead every time (where N corresponds to the number of steps ahead). In the case of non-independent forecasts, a shock in a specific

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forecast horizon may affect all other forecasting horizons. Thus, non-overlapping independent observations desensitise forecast error measures to special events and specific phases of business (Geppert, 1995).

Thus, for the weekly data, we withhold 30 weekly observations at the end of the sample (that is, 19 November 2003 to 7 June 2004, representing a period of seven months), and estimate the conditional models using only the data up to 12 November 2003. Then, we perform one-step (one-week)-ahead forecasts of the covariance and the variance for 19 November 2003. These are then used to estimate the one-step-ahead forecast of the hedge ratio for that week. Then, the following week, the observation of 19 November 2003 is added to the sample and the exercise is repeated, again forecasting the hedge ratio of the following week. We continue updating the models by adding one weekly observation at a time to the estimated model and forecasting the hedge ratios until the end of our data set. The results for the out-of-sample hedging effectiveness are presented in Table 3, panels C and D. The same table reports the results of the corresponding exercise using daily data. For that, in both markets, we withhold 135 daily observations at the end of the sample; that is, 19 November 2003 to 7 June 2004, representing again a period of seven months, and estimate the conditional models using only the data up to 19 November 2003. We then follow the same procedure as described above for the weekly data, but this time by adding one daily observation at a time.

In contrast to the in-sample results, in the FTSE/ATHEX-20 market the VECM-GARCH model seems to perform the worst (81.17% and 72.77% variance reductions for weekly and daily data, respectively), compared with the alternative constant hedging strategies, in terms of both reducing the variability and increasing the expected utility of the returns of the hedged portfolio. The highest variance reduction is achieved by the conventional (OLS) model (94.73% and

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89.45% for weekly and daily data, respectively), followed at marginally lower values by the VECM (94.43% and 89.37%), the naïve model (94.13% and 87.53%) and the VECM-GARCH. These results are qualitatively the same when using the UI measure as the model selection criterion.

In the FTSE/ATHEX Mid-40 market also the VECM-GARCH-X model has the worst performance (VR 96.15% and 55.97% for weekly and daily data, respectively), compared with the alternative constant conventional (OLS) and VECM hedging strategies, in reducing the variability of the returns of the hedged portfolio. The highest variance reduction is achieved by the OLS model for weekly data (96.20%) and by the VECM model for daily data (91.04%)10. When the UI measure is used as the model selection criterion the results are the same, as the OLS and the VECM models have the highest utility increase for weekly and daily hedging frequencies, respectively. It can be argued that when the speculative components are taken into account in the optimal hedge ratio calculations (like in the case of the FTSE/ATHEX Mid-40 market), investors can increase the utility they derive from their hedges beyond the level obtained through risk minimisation strategies. This is because the increased benefits from speculation outweigh the costs, for the particular level of risk aversion of the assumed investor (k = 3).

These results seem to be in accordance with the literature: Lence (1995) argues that the benefits of sophisticated estimation techniques of the hedge ratio are small and that hedgers may do better by focusing on simpler and more intuitive hedge models. Lien et al. (2002) by examining ten cash and futures markets, covering currency futures, commodity futures and stock index futures in the NYSE and S&P500 markets, report that OLS hedge ratios perform better than the vector GARCH hedge ratios. However, there seems to be a conflict in the literature about GARCH

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hedge ratios, as some studies report that they outperform, in terms of risk reduction, the constant hedge ratios (see Gagnon and Lypny, 1997). But these gains are market specific and vary across different contracts while, occasionally, the benefits in terms of risk reduction seem to be minimal (Lien and Tse, 2002). Therefore, the differences in the performance of GARCH models in hedge ratio estimation appear to be a consequence of different hedging horizons, different market conditions, different futures contract specifications, different levels of trading and other marketspecific and/or contract-specific economic differences. In the end, whether constant or time varying hedge ratios are more appropriate is a matter of empirical evidence, and this paper contributes to answering this empirical question, from the Greek markets’ point of view.

It may be argued that time-varying VECM-GARCH hedge ratios need to be updated and rebalanced as new information, regarding cash and futures prices, appears in the market. However, in a thin trading futures market, like the ATHEX market, information arrival and assimilation is not as fast as in other well-developed and liquid derivatives markets, which would justify the use of time-varying hedge ratio models in the latter markets. Overall, the results, from both daily and weekly data, reveal that in the FTSE/ATHEX-20 and FTSE/ATHEX Mid-40 markets, the relationship between cash and futures prices is quite stable and market agents can use simple constant hedge models in order to obtain out-of-sample optimum hedge ratios.

Furthermore, the results reveal that futures contracts serve their risk management function through hedging, as they provide considerable variance reductions in comparison to staying unhedged. In the FTSE/ATHEX-20 market the greatest variance reduction is 94.73% when using weekly data, while in the FTSE/ATHEX Mid-40 market this figure is 96.20%. These figures are approximately 5% and 3% lower when using daily data. Coupling these lower hedging effectiveness figures when using daily data, with the higher transactions costs, points to using

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weekly rather than daily rebalancing of portfolios by market practitioners. The magnitude of the hedging effectiveness measures is more or less comparable with other studies in the literature (most of which use weekly data) on stock index futures contracts, which show variance reductions of: 90% for the Swiss Market Index futures (Stulz et al., 1990), 87% for the Hang Seng futures (Yau, 1993), 94% for the FTSE-100 futures (Lee, 1994), 97.91% and 77.47% for the S&P500 and the Canadian stock index futures contracts (Park and Switzer, 1995), 88% for the Spanish Ibex 35 futures (Lafuente and Novales, 2003), 96.41% and 96.70% for the S&P500 and NASDAQ-100 stock index futures, respectively (Chiu et al. 2005), 96.82% for the Nikkei 225 futures (Lee, 2006), and 97% for the Swedish OMX-index futures (Norden, 2006). It can be noticed that hedging effectiveness seems to be slightly higher in more recent studies.

4.2.3. Utility Maximisation Hedging Effectiveness Comparison

In order to measure the sensitivity of the utility results to different values of the risk aversion coefficient, k, utility comparisons associated with alternative hedge ratios for a range of different risk aversion coefficients are made, for daily and weekly frequencies, both in-sample and out-ofsample. For a range of values of k (0.1, 0.5, 1, 1.5, 2, 3, and 4) the utility levels for unhedged and hedged portfolios are estimated using the portfolio’s mean and variance of returns following estimation of UMHRs. Table 4 panel A presents the weekly and daily in-sample utilities for the FTSE/ATHEX-20 market. With the exception of daily data for risk aversion levels at the low range of 0.1 and 0.5, where the naïve hedge ratio is preferred, the results, for both weekly and daily data, indicate that the VECM-GARCH model is the most preferred across all degrees of risk aversion, as it provides the greatest level of utility.

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For example, when k = 1 under weekly data, the VECM-GARCH yields a marginal utility gain of -0.000139 (= -0.000239 – 0.000378) over the OLS model. The out-of-sample results, presented in panel B of the same table, indicate that under weekly data the VECM-GARCH model is still the most preferred one, but when daily data are used the naïve (for levels of risk aversion 0.1, 0.5, 1.0 and 1.5) and the conventional (OLS) models (for levels of risk aversion 2.0, 3.0 and 4.0) are the most preferred. Therefore, for the FTSE/ATHEX-20 market, irrespective of the level of the risk aversion coefficient, it seems that the VECM-GARCH model is the most preferred using weekly data (both in-sample and out-of-sample), while when using daily data the VECM-GARCH model is preferred in-sample and the naïve or the OLS is preferred out-ofsample.

The in-sample results for the FTSE/ATHEX Mid-40 market, presented in panel C of Table 4, indicate that when using either weekly or daily data the VECM-GARCH-X model is the most preferred, irrespective of the risk aversion level. Finally, panel D of the same table, presents the out-of-sample utility comparison results for the FTSE/ATHEX Mid-40 market. For both weekly and daily data, and for values of the risk aversion coefficient from 0.1 to 2, the naïve model is the most preferred as it provides the greatest utility levels. For higher levels of risk aversion (3 and 4), the OLS and the VECM models are preferred for weekly and daily data, respectively.

Overall, it seems that the risk aversion coefficient of the expected utility influences the hedge ratio model selection process, according to the data frequency (daily or weekly), the hedging forecast horizon (in-sample or out-of-sample) and the futures contract used (FTSE/ATHEX-20 or FTSE/ATHEX Mid-40).

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Next, we create a mean-standard deviation portfolio opportunity frontier, which describes the return and risk trade-offs from hedging with the FTSE/ATHEX Mid-40 futures contract, as an example. Table 5 reports the daily out-of-sample results for the portfolio returns (Equation 1) and standard deviations (Equation 2) for the UMHRs (γ**) of the FTSE/ATHEX Mid-40 futures contract between 0 and 1.6. These results are also graphed in Figure 1. The highest risk (standard deviation) is related with the unhedged portfolio (γ** = 0), whereas the minimum risk portfolio corresponds to point 0.8, with an associate return of -0.000062 and a standard deviation of 0.003527 (variance of 0.000012). For values of γ** between 0.8 and 1.6, there are portfolios corresponding to higher risk. Portfolios on the negatively sloped part of the opportunity frontier (for values of γ** between 0.8 to 0) can be eliminated, as these inefficient portfolios, for the same risk level, offer lower return than the portfolios on the positively sloped part of the frontier.

Figure 1 demonstrates the principal investment decision of the hedger; that is, if it worth taking lower hedge ratios (for example, lower than 1.6) that will result in lower returns, but at the same time will result in lower risk as well. This decision is influenced not only by the level of risk aversion but also from the costs. Thus, the hedger should be rewarded for the decrease in return by an adequate decrease in risk. By comparing the returns and the variances of the unhedged and hedged positions, for the various values of the γ**, as showed in Table 5, the following cost of hedging measure can be estimated:

Cost of Hedging =

% Reduction in Return % Reduction in Variance

(12)

where, % Reduction in Return = 1 – [(Hedged Return)/(Unhedged Return)] and % Reduction in Variance = 1 – [(Hedged Variance)/(Unhedged Variance)]. The cost of hedging estimates, for the specific period under evaluation, can be seen in the last column of Table 5, with larger estimates

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implying higher costs of risk reduction. The minimum-standard deviation portfolio corresponds to a cost of 0.95, which implies that a 1 per cent reduction in risk will result in a 0.95 per cent reduction in return. A hedger then, has to decide whether this risk reduction cost is sensible enough for his risk aversion level.

4.2.4. Optimal Rebalancing Frequency

An investor’s anticipated trading frequency or investment horizon is seen as one of the most important factors affecting the asset allocation and investment decisions for financial asset holding (Douglas Van Eaton and Conover, 2002). In the case of active asset management, the optimal holding horizon, which determines a specific trading strategy, represents how actively (frequently) investors should trade to maximise post-transaction-cost profits. However, so far there is no literature attempting to find the optimal trading frequency for a financial asset or a portfolio and there is no consensus on the selection of the data frequency relative to the expected holding horizon (Dunis and Miao, 2004). Mian and Adam (2001), among others, argue that the appropriate sampling frequency depends on the particular context. For example, if long-term forecasts are needed, a low-frequency data estimated model would be the appropriate one.

There are two different categories of techniques used to find the optimal trading frequency. First, fundamental analysis is used, despite the fact that the process of finding the right parameters for fundamental models could be time-consuming and indecisive. Second, technical rules, like volatility filters and model switching strategies, are used, but their results are influenced from the selection of parameters and with the addition of certain filters. Thus, results coming from trading rules are sensitive in terms of the selected data frequencies and thus, different trading frequencies can be derived.

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Dunis and Miao (2004) apply technical trading rules to investigate the optimal trading frequency question of seven futures contracts between January 1998 to March 2004.11 The results indicate that technical trading rules perform poorly in periods when market volatility is high. The two used volatility filters that were proposed, namely a no-trade filter, where all market positions are closed in volatility periods and a reverse filter where signals from a simple Moving Average Convergence and Divergence (MACD) system are reversed if the market volatility is higher than a given threshold, improve the futures performance in most cases when applied not to single assets but to portfolios of assets. Dunis and Miao (2004) report that the results for the optimal trading frequencies differ for the different assets under review. More specifically, the results for stock index futures indicate that the optimal trading frequency is around 2-4 trades per year. For Aluminium, Copper and Brent oil futures the optimal periods are 12-18, 6-7 and 32-42, respectively. Finally, for the bond futures the optimal trading frequencies are 5-8 and 11-18 trades per year for the 30-year T-bond and for the 10-year Bund futures, respectively. Thus, it seems that the optimal rebalancing frequency depends on the specific market characteristics of each financial asset.

However, generally speaking, mean-variance expected utility-maximising investors prefer timevarying hedge strategies to conventional ones and only rebalance their portfolios when the increased expected utility from rebalancing is greater than the transactions costs incurred from updating the hedge (see Kroner and Sultan, 1993, amongst others). The optimal rebalancing frequency, in the presence of transactions costs, is of high importance to the decision-maker, and thus the hedger’s utility function can dictate the economic benefits of hedging. In this setting, let φ represent the transactions costs in the futures market. Therefore, a mean-variance expected

utility-maximising investor will rebalance at time t if and only if:

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R H,t – φ – kVar(R H,t ) > R 'H,t – kVar(R 'H,t )

(13)

where, R H,t = (∆St – γ *t ∆Ft) and Var(R H,t ) = [Var(∆St) + γ *t 2 Var(∆Ft ) – 2γ *t Cov(∆St, ∆Ft )] are the expected return and variance of the rebalanced portfolio (estimated with the use of a timevarying VECM-GARCH hedge ratio model), respectively, while R 'H,t = (∆St – γ *t ' ∆Ft) and Var(R 'H,t ) = [Var(∆St ) + γ *t '2 Var(∆Ft ) – 2 γ *t ' Cov(∆St, ∆Ft)] are the return and variance of the non-rebalanced portfolio (estimated with the use of a constant OLS hedge ratio model).

Table 6 panel A presents the number of portfolio rebalances made by the investor, under weekly frequencies, for different levels of the risk aversion coefficient (k). The results indicate that in the FTSE/ATHEX-20 futures market (assuming φ = 0.01399%) the investor will choose to rebalance his portfolio 214 times on average, out of a total of 249 weeks, in-sample for high levels of the risk aversion coefficient (1 – 4). This indicate that 214 times on average the left-hand-side estimate of Equation (13), derived from the use of time-varying (VECM-GARCH) hedge ratios, was found larger than the right-hand-side estimate, derived from the use of constant (OLS) hedge ratios. Thus, a time-varying hedge ratio model, with rebalancing transactions costs, on average is more preferable than a constant hedge ratio model with no rebalancing transactions costs. However, the same investor out-of-sample will choose to rebalance his portfolio only 30 times on average, indicating that constant (conventional – OLS) models may be more preferable. The daily frequency results, presented in the same table panel B, are in line with the weekly results. That is, the investor will rebalance his portfolio 1,151 times on average, out of a total of 1,185 days, in-sample for high levels of the risk aversion coefficient, but he will rebalance his portfolio only 135 times on average out-of-sample.

32

In the FTSE/ATHEX Mid-40 futures market (assuming φ = 0.00821%) the in-sample results indicate that under weekly frequencies (panel A), the investor rebalances his portfolio 187 times for φ = 0.1 down to 46 times for φ = 4, out of a total of 226 weeks and under daily frequencies (panel B) the investor rebalances his portfolio 1,069 times for φ = 0.1 down to 187 times for φ = 4, out of a total of 1,081 days. However, from the out-of-sample results it is apparent that the constant (OLS) models are preferable for all levels of the risk aversion coefficient (k), as the right-hand-side estimate of Equation (12) is found to be larger than the left-hand side estimate and consequently, the investor chooses not to rebalance at all. Overall, these results are in line with the previous results of Table 3, where in-sample the time-varying VECM-GARCH-X models are found to be the most preferable, but out-of-sample simple constant regression models are sufficient12.

5. CONCLUSION

This paper examined the hedging effectiveness in terms of both variance reduction and expected utility increase, of investors taking hedges in the FTSE/ATHEX-20 and FTSE/ATHEX Mid-40 stock index futures contracts in Greece. In both markets, cash and futures prices are cointegrated, thus standing in a long-run relationship between them. Using weekly and daily data, both insample and out-of-sample hedging performances are examined in these markets, considering alternative models that allow for the estimation of both constant and time-varying optimal hedge ratios. Results from in-sample tests indicate that time-varying hedge ratios marginally outperform alternative specifications in reducing market risk in some cases, while those from out-of-sample tests indicate that constant hedge ratios, coming from simpler models provide maximum variance reduction and utility increase hedge ratios. Utility and variance comparisons between weekly and daily data point to weekly rebalancing choices, providing better results for the hedger, compared to daily ones.

33

Overall, the results reveal that the two stock index futures contracts on ADEX serve their risk management function through hedging, as they provide considerable variance reductions/utility increases in comparison to unhedged positions. Investors who are interested in the Greek stock index market can benefit from these results by developing appropriate hedge ratios in each market, in order to reduce their price risk more efficiently. This paper has contributed in making this issue clear for the Greek stock market.

ENDNOTES 1

Detailed contract specifications of the two futures contracts can be found on the ADEX website (www.adex.ase.gr), while the names of the companies, comprising each of the underlying indices, the ATHEX-20 and the ATHEX Mid-40 may be found on the ATHEX website (www.ase.gr). 2 Return-variance maximization is equivalent to variance minimization, provided expected returns are zero. 3 Several other specifications are also used, such as a bivariate VECM-EGARGH (Nelson, 1991) and a VECM-GJRGARCH (Glosten et al., 1993), but yield inferior results judged by the evaluation of the log-likelihood values and from diagnostic tests on standardised residuals. These are available from the authors on request. 4 The use of the lagged squared ECT specification, instead of the lagged level or the lagged absolute value specifications is justified in the empirical work as most of the times it provides uniformly superior results (see Lee, 1994). It should be noted that the lagged squared ECT specification is used, instead of the lagged squared basis, as the restrictions placed on the cointegrating vector to represent the exact lagged basis are rejected in both markets (not shown). 5 Cointegration test results are available from the authors on request. 6 Results from daily data are qualitatively the same as those with weekly data and can be obtained from the authors on request. 7 Results from daily data are qualitatively the same as those with weekly data and can be obtained from the authors on request. 8 Kroner and Sultan (1993) report this parameter to be 4; Chou (1988) reports it to be 4.5 and Poterba and Summers (1986) report it to be 3.5. For more details regarding the empirical estimation of the parameter k see Poterba and Summers (1986). 9 The brokers commission for an order (buy or sell) in the FTSE/ATHEX-20 market ranges from 3€ (for large investors) to 20€ and for the FTSE/ATHEX Mid-40 market it ranges from 3€ (for large investors) to 25€. 10 The VECM-GARCH-X model, for the FTSE/ATHEX Mid-40 market is also estimated without the inclusion of a lagged squared ECT in the variance equation, in order to verify that the results are not affected by this term. The results are qualitatively the same and are available from the authors upon request. 11 The S&P500 futures trading at the Chicago Mercantile Exchange (CME), the Euro STOXX50 and the 10-year Bund futures trading at EUREX, the 30-year T-Bond futures trading at the Chicago Board of Trade (CBT), the Copper and Aluminium futures trading at the London Mercantile Exchange (LME), and the Brent Oil futures trading at Intercontinental Exchange (ICE). 12 Comparisons are also made between the time-varying VECM-GARCH-X models and the constant VECM models, yielding qualitatively similar results (not reported).

34

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Lee, H-T. (2006): “Dynamic Futures hedging with an Asymmetric Markov regime Switching BEKK GARCH Model,” Conference Proceedings, AsianFA/FMA Meeting, 10 -12 July, Auckland, New Zealand. Lence, S. H. (1995): “The Economic Value of Minimum-Variance Hedges,” American Journal of Agricultural Economics, 77: 353-364. Lien, D. and Tse, Y. K. (2002): “Some Recent Developments in Futures Hedging,” Journal of Economic Surveys, 16(3): 357-396. Lien, D., Tse, Y. K. and Tsui, A. K. C. (2002): “Evaluating the Hedging Performance of the ConstantCorrelation GARCH Model,” Applied Financial Economics, 12: 791-798. Ljung, M. and Box, G. (1978): “On a Measure of Lack of Fit in Time Series Models,” Biometrica, 65: 297303. Markowitz, H. (1959): “Portfolio Selection”, New York: John Wiley & Sons. Mian, G. M. and Ada,, C. M. (2001): “Volatility Dynamics in High Frequency Financial Data: An Empirical Investigation of the Australian Equity Returns,” Applied Financial Economics, 11: 341-352. Nelson, D. B. (1991): “Conditional Heteroskedasticity in Asset Returns: A New Approach,” Econometrica, 59: 347-370. Norden, L. (2006): “Does an Index Futures Split Improve Hedging Effectiveness of the Futures Contract?,” Conference Proceedings, Financial Management Association Annual Conference, 7-10 June, Stockholm, Sweden. Park, T. and Switzer, L. (1995): “Time-Varying Distribution and the Optimal Hedge Ratios for Stock Index Futures,” Applied Financial Economics, 5: 131-137. Phillips, P. C. B. and Perron, P. (1988): “Testing for a Unit Root in Time Series Regressions,” Biometrica, 75: 335-346. Poterba, J. and Summers, L. (1986): “The Persistence of Volatility and Stock Market Fluctuations,” American Economic Review, 76: 1142-1151. Schwartz, G. (1978): “Estimating the Dimension of a Model,” Annals of Statistics, 6: 461-464. Shanno, D. F. and Phua, K. H. (1980): “Remark on Algorithm 500, A Variable Metric Method for Unconstrained Minimization,” ACM Trans. Math. Software, 6: 618-622. Simons, K. (1997): “Model Error,” New England Economic Review, November/December, 17-28. Stulz, R.M., Wasserfallen, W. And Stucki, T. (1990): “Stock Index Futures in Switzerland: Pricing and Hedging Performance,” Review of Financial Markets, 9(3): 576-592. Tashman, L. J. (2000): “Out-of-Sample Tests of Forecast Accuracy: An Analysis and Review,” International Journal of Forecasting, 16: 437-450. Yau, J. (1993): “The Performance of the Hong Kong Hang Seng Index Futures Contract in Risk-Return Management,” Pacific Basin Finance Journal, 1(1): 381-406. Zellner, A. (1962): “An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests of Aggregation Bias,” Journal of the American Statistical Association, 57: 500-509.

36

Table 1. Descriptive Statistics of Weekly and Daily Logarithmic First-Differences of Cash and Futures Prices Data series are measured in logarithmic first differences. Numbers in parentheses below the headers Cash and Fut. 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

ˆ 3 ~ N(0,6) and T ( αˆ 4 – 3) ~ N(0,24), respectively. Q(i) and Q2(i) are the Ljung-Box (1978) Q statistics on the first i (for i = 4, 12 and 24) lags of the sample the null are T α 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-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 1st Diffs correspond to price series in log-levels and log-first differences, respectively. The 95% critical value for the ADF and PP tests is –2.88. 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 Q2(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. Panel A: FTSE/ATHEX-20 Weekly Cash and Futures Price Series (09/99 to 06/04) Mean Skew Kurt Q(4) Q(12) Q(24) Q2(4) Q2(12) Q2(24) ARCH ARCH ARCH J-B ADF PP ADF PP (4) (12) (24) Lev Lev 1st Diffs 1st Diffs Cash -0.0033 0.254 1.389 959.26 2,671.1 4,766.8 860.80 2,662.0 2,888.3 1,293.1 368.8 182.7 21.356 -1.855 -1.737 -11.284 -15.923 (249) [0.1892] [0.103] [0.000] [0.000] [0.000] [0.000] [0.000] [0.000] Q2(12) [0.000] [0.000] [0.000] [0.000] (1) (5) (1) (10) Fut. -0.0032 0.191 1.049 959.94 2,672.9 4,760.9 863.76 2,665.0 2,890.1 1,266.3 354.2 167.3 11.681 -1.707 -1.697 -11.711 -15.078 (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) Mean Skew Kurt Q(4) Q(12) Q(24) Q2(4) Q2(12) Q2(24) ARCH ARCH ARCH J-B ADF PP ADF PP (4) (12) (24) Lev Lev 1st Diffs 1st Diffs Cash -0.0064 -0.007 1.771 80.32 2,080.1 3,415.3 622.72 2,066.6 2,671.4 1,208.2 427.3 140.7 27.556 -2.189 -2.346 -13.379 -13.311 (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) Fut. -0.0066 -0.057 1.521 804.32 2,053.3 3368.0 606.63 2,026.7 2,556.7 966.7 346.8 125.2 20.427 -2.203 -2.376 -13.585 -13.521 (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 (1,081) Fut. (1,081)

Mean

Skew

Kurt

Q(4)

Q(12)

Q(24)

Q2(4)

Q2(12)

Q2(24)

-0.0007 [0.1552] -0.0006 [0.1824]

0.183 [0.010] 0.166 [0.019]

3.546 [0.000] 3.347 [0.000]

4,706.4 [0.000] 4,706.6 [0.000]

13,867 [0.000] 13,874 [0.000]

27,033 [0.000] 27,044 [0.000]

4,587.0 [0.000] 4,589.9 [0.000]

12,961 [0.000] 13,017 [0.000]

23,951 [0.000] 24,089 [0.000]

ARCH (4) 33,947 [0.000] 31,138 [0.000]

ARCH (12) 11,043 [0.000] 10,205 [0.000]

ARCH (24) 5,850.8 [0.000] 5,240.4 [0.000]

ARCH (12) 13,053 [0.000] 7,920 [0.000]

ARCH (24) 5,925.6 [0.000] 3,.494.6 [0.000]

J-B 620.81 [0.000] 552.31 [0.000]

ADF Lev -1.730 (0) -1.691 (1)

PP Lev -1.685 (1) -1.653 (3)

ADF 1st Diffs -29.789 (0) -31.659 (0)

PP 1st Diffs -29.734 (5) -31.663 (1)

Panel D: FTSE/ATHEX Mid-40 Daily Cash and Futures Price Series (02/00 to 06/04) Cash (1,081) Fut. (1,081)

Mean

Skew

Kurt

Q(4)

Q(12)

Q(24)

Q2(4)

Q2(12)

Q2(24)

-0.0013 [0.0183] -0.0014 [0.0412]

-0.203 [0.006] -0.126 [0.090]

3.192 [0.000] 4.092 [0.000]

4,234.5 [0.000] 4,226.1 [0.000]

12,171 [0.000] 12,131 [0.000]

22,868 [0.000] 22,738 [0.000]

4,030.6 [0.000] 3,993.6 [0.000]

10,582 [0.000] 10,431 [0.000]

17,738 [0.000] 17,306 [0.000]

ARCH (4) 41,183 [0.000] 29,228 [0.000]

J-B 460.66 [0.000] 748.13 [0.000]

ADF Lev -1.991 (1) -2.060 (0)

PP Lev -1.950 (9) -2.008 (6)

ADF 1st Diffs -27.181 (0) -30.950 (0)

PP 1st Diffs -27.214 (8) -30.967 (7)

Table 2. Maximum Likelihood Estimates of VECM-GARCH-X Models (FTSE/ATHEX-20: 1999:09-2004:06, FTSE/ATHEX Mid-40:2000:02-2004:06) The maximum-likelihood estimates of the preferred VECM-GARCH and VECM-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 Q2(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

∆St =

∑a

S,i∆St-i

p−1

+

i=1

∑a

F,i∆St-i

i=1

S,i∆Ft-i

+ aSXt-1 + εS,t

F,i∆Ft-i

+ aFXt-1 + εF,t

i=1

p−1

∆Ft =

∑b p−1

+

∑b

;

i=1

 εS , t   | Ωt-1 ~ t-dist(0,Ht, v) ε F , t  

εt = 

Ht = A'A + B'Ht-1B + C'εt-1εt-1'C + G'(X t −1 )2G Coefficients

VECM-GARCH(1,1) Cash Futures FTSE/ATHEX-20 FTSE/ATHEX-20

Panel A: Conditional Mean Parameters aj, j = S, F 0.263 (0.909) [0.363] aj,1, j = S, F -0.707 (-2.386) [0.017] bj,1, j = S, F 0.712 (2.453) [0.014]

0.719 (2.401) [0.016] -0.525 (-1.731) [0.083] 0.561 (1.875) [0.061]

Panel B: Conditional Variance Parameters a11 0.0270 (8.151) [0.000] a21 0.0257 (6.166) [0.000] a22 0.0244 (6.168) [0.000] bkk , k = 1, 2 0.268 (1.359) 0.431 (2.514) [0.174] [0.012] ckk , k = 1, 2 0.332 (3.403) 0.323 (3.566) [0.001] [0.000] gkk , k = 1, 2 v

5.773 (3.941) [0.000]

Panel C: Diagnostic Tests on Standardised Residuals Log-Likelihood 1,130.58 Skewness 0.127 [0.449] 0.140 [0.405] Kurtosis 1.389 [0.000] 1.579 [0.000] Q(12) 8.455 [0.672] 7.865 [0.725] Q2(12) 8.826 [0.638] 7.516 [0.756] ARCH(12) 0.658 [0.790] 0.510 [0.907] AIC -2,233.17 SBIC -2,185.98

(7) (9)

VECM-GARCH(1,1)-X Cash Futures FTSE/ATHEX-40 FTSE/ATHEX-40

0.488 (1.898) [0.058] 0.213 (1.801) [0.072] -0.38 (-1.851) [0.063]

0.862 (3.130) [0.002] 0.427 (1.769) [0.069] -0.233 (-1.879) [0.075]

0.0015 (0.663) [0.507] 0.0029 (1.243) [0.214] 0.0058 (2.339) [0.019] -0.938 (-41.411) -0.922 (-30.123) [0.000] [0.000] 0.188 (2.457) 0.207 (2.459) [0.014] [0.013] 0.982 (5.166) 1.092 (4.559) [0.000] [0.000] 6.410 (3.504) [0.000]

925.62 0.177 [0.325] 0.212 [0.240] 2.092 [0.000] 2.645 [0.000] 15.743 [0.151] 12.824 [0.305] 6.578 [0.832] 5.985 [0.874] 0.699 [0.751] 0.601 [0.839] -1,819.23 -1,767.53

Table 3. Portfolio Risk-Return and Hedge Ratios The hedged and (unhedged) portfolio variances and returns are estimated, along with the respective hedge ratios for five different hedge ratio models: They are VECM-GARCH-X, VECMGARCH, VECM, OLS, naïve strategy by taking a futures position, which is the same size as the cash position (γ* = 1). In the FTSE/ATHEX-20 market the Biast+1 = 0 in the corresponding futures market, hence the MVHR is appropriate; in the FTSE/ATHEX Mid-40 market the UMHR is appropriate. However, MVHRs are also presented. Expected utilities (Equation 3) for k = 3, VRs (Equation 10) and UIs (Equation 11) are also shown in the table. * Denotes the model with the greatest variance reduction (utility increase). Hedged Portfolio Return

RH,t = ∆St - γ t ∆Ft

Hedged Portfolio Variance

2 σ H,t

Expected Utility

EtU(RH,t+1) = Et(RH,t+1) – k Vart (RH,t+1) ;

= Vart(∆St – γt∆Ft) = Vart(∆St ) + γt Vart(∆Ft ) – 2γtCovt(∆St, ∆Ft ) (2) Variance Reduction (%) of the Unhedged position VR = Var ( ∆ S t ) − Var ( R H, t ) x 100

MVHR Return (γ*) (RH,t) Panel A: Daily In-Sample Unhedged 0 -0.000724 Naïve, γ*=1 1 -0.000012 OLS 0.868 -0.000107 VECM 0.871 -0.000104 GARCH 0.876 -0.000057 Panel B: Weekly In-Sample Unhedged 0 -0.003502 Naïve, γ*=1 1 0.000018 OLS 0.953 -0.000148 VECM 0.959 -0.000126 GARCH 0.962 0.000058 Panel C: Daily Out-of-Sample Unhedged 0 0.001373 Naïve, γ*=1 1 0.000039 OLS 0.868 0.000216 VECM 0.870 0.000213 GARCH 0.878 0.000247 Panel D: Weekly Out-of-Sample Unhedged 0 0.005781 Naïve, γ*=1 1 0.000222

OLS VECM GARCH

0.954 0.959 0.965

(1)

0.000478 0.000451 -0.002975

2

Var ( ∆ S t )

FTSE/ATHEX-20 Variance Expected (σ2Η,t) Utility

k=3

(3) Utility Increase (%) of the Unhedged position

Variance Reduction

Utility Increase

MVHR (γ*)

FTSE/ATHEX Mid-40 Return Variance Variance (RH,t) (σ2Η,t) Reduction

UMHR (γ**)

UI = E t U (R

H ,t

)− E

t

U (∆ S t

)

FTSE/ATHEX Mid-40 Return Variance Expected (RH,t) (σ2Η,t) Utility

(10)

(11) Utility Increase

0.002761 0.000440 0.000383 0.000382 0.000366

-0.009007 -0.001332 -0.001256 -0.001250 -0.001155

84.06 86.13 86.16 86.74*

. 0.00767 0.00775 0.00776 0.00785*

1 0.766 0.774 0.789

-0.001345 0.000012 -0.000306 -0.000294 -0.000306

0.000348 0.000088 0.000061 0.000061 0.000062

74.79 82.50* 82.49 82.14

1.462 1.228 1.237 1.252

-0.001345 0.000639 0.000321 0.000333 0.000735

0.000348 0.000298 0.000165 0.000169 0.000226

-0.002388 -0.000255 -0.000175 -0.000175 0.000057

0.00213 0.00221 0.00221 0.00245*

0.001519 0.000106 0.000102 0.000102 0.000099

-0.008059 -0.000300 -0.000454 -0.000432 -0.000239

93.02 93.29 93.29 93.48*

0.00776 0.00760 0.00763 0.00782*

1 0.897 0.927 0.921

-0.005162 -0.000083 -0.000608 -0.000455 -0.000332

0.001933 0.000126 0.000105 0.000107 0.000102

93.46 94.55 94.49 94.75*

1.378 1.274 1.304 1.299

-0.005162 0.001835 0.001310 0.001462 0.002608

0.001933 0.000610 0.000414 0.000466 0.000744

-0.010962 0.000004 0.000067 0.000063 0.000375

0.01097 0.01103 0.01103 0.01134*

0.001355 0.000169 0.000143 0.000144 0.000369

-0.002692 -0.000468 -0.000213 -0.000219 -0.000860

87.53 89.45* 89.37 72.77

0.00222 0.00248* 0.00247 0.00183

1 0.763 0.772 0.312

-0.000457 0.000037 -0.000080 -0.000075 -0.000331

0.000143 0.000018 0.000013 0.000013 0.000063

87.16 90.90 91.04* 55.97

1.425 1.188 1.198 0.738

-0.000457 0.000247 0.000130 0.000135 -0.000242

0.000143 0.000083 0.000038 0.000039 0.000034

-0.000886 -0.000001 0.000015 0.000016 -0.000343

0.00089 0.00090 0.00090* 0.00054

0.000664 0.000039

0.003789 0.000105

94.13

1

-0.002785 0.000121

0.000858 0.000041

95.19

1.482

-0.002785 0.001523

0.000858 0.000365

-0.005919 0.000426

0.00635

0.000035 0.000037 0.000125

0.000373 0.000340 -0.003350

94.73* 94.43 81.17

-0.00368 0.00342* -0.00345 -0.00714

0.896 0.926 0.936

-0.000181 -0.000094 -0.000019

0.000033 0.000033 0.000033

96.20* 96.16 96.15

1.378 1.408 1.418

0.001220 0.001308 0.000780

0.000256 0.000285 0.000116

0.000452 0.000452 0.000432

0.00637* 0.00637 0.00635

Table 4. Utility Maximisation Hedging Effectiveness Comparison This table presents the hedging effectiveness comparison from the expected utility maximization Equation (3) for a range of values of the risk aversion coefficient (k = 0.1, 0.5, 1.0, 1.5, 2.0, 3.0, and 4.0), using both daily and weekly data, over both in-sample and out-of-sample periods. The utility levels for unhedged and hedged portfolios are estimated using the portfolio’s mean and variance of return. Panel A: FTSE/ATHEX-20 In-Sample Expected Utilities Weekly k = 0.1 k = 0.5 k=1 k = 1.5 k=2 k=3 k =4 Unhedged, γ* = 0 -0.003403 -0.004010 -0.004770 -0.005529 -0.006289 -0.008059 -0.009327 Naïve, γ* = 1 -0.000521 -0.000563 -0.000616 -0.000669 -0.000722 -0.000300 -0.000934 OLS -0.000286 -0.000327 -0.000378 -0.000429 -0.000480 -0.000454 -0.000684 VECM -0.000263 -0.000304 -0.000355 -0.000406 -0.000457 -0.000432 -0.000661 VECM-GARCH -0.000150 -0.000189 -0.000239 -0.000288 -0.000338 -0.000239 -0.000536 Daily k = 0.1 k = 0.5 k=1 k = 1.5 k=2 k=3 k =4 Unhedged, γ* = 0 -0.000959 -0.002064 -0.003444 -0.004825 -0.006205 -0.009007 -0.011727 Naïve, γ* = 1 -0.000061 -0.000237 -0.000457 -0.000677 -0.000897 -0.001332 -0.001777 OLS -0.000150 -0.000303 -0.000494 -0.000686 -0.000877 -0.001256 -0.001643 VECM -0.000147 -0.000300 -0.000491 -0.000682 -0.000873 -0.001250 -0.001637 VECM-GARCH -0.000093 -0.000239 -0.000422 -0.000605 -0.000788 -0.001155 -0.001520 Panel B: FTSE/ATHEX-20 Out-of-Sample Expected Utilities Weekly k = 0.1 k = 0.5 k=1 k = 1.5 k=2 k=3 k =4 Unhedged, γ* = 0 -0.003317 -0.003583 -0.003915 -0.004247 -0.004579 0.003789 -0.005907 Naïve, γ* = 1 -0.000624 -0.000639 -0.000659 -0.000678 -0.000698 0.000105 -0.000776 OLS -0.000547 -0.000561 -0.000578 -0.000596 -0.000613 0.000373 -0.000683 VECM -0.000566 -0.000580 -0.000599 -0.000617 -0.000636 0.000340 -0.000710 VECM-GARCH -0.000262 -0.000312 -0.000374 -0.000437 -0.000499 -0.003350 -0.000749 Daily k = 0.1 k = 0.5 k=1 k = 1.5 k=2 k=3 k =4 Unhedged, γ* = 0 -0.000819 -0.001361 -0.002038 -0.002716 -0.003393 -0.002692 -0.006103 Naïve, γ* = 1 -0.000035 -0.000102 -0.000187 -0.000271 -0.000356 -0.000468 -0.000694 OLS -0.000077 -0.000134 -0.000205 -0.000277 -0.000348 -0.000213 -0.000634 VECM -0.000076 -0.000134 -0.000206 -0.000278 -0.000350 -0.000219 -0.000638 VECM-GARCH -0.000154 -0.000301 -0.000486 -0.000670 -0.000855 -0.000860 -0.001593 Panel C: FTSE/ATHEX Mid-40 In-Sample Expected Utilities Weekly k = 0.1 k = 0.5 k=1 k = 1.5 k=2 k=3 k =4 Unhedged, γ** = 0 -0.005355 -0.006128 -0.007095 -0.008062 -0.009028 -0.010962 -0.012895 Naïve, γ** = 1 0.001774 0.001529 0.001224 0.000919 0.000614 0.000004 -0.000607 OLS 0.001268 0.001103 0.000895 0.000688 0.000481 0.000067 -0.000348 VECM 0.001415 0.001229 0.000996 0.000763 0.000529 0.000063 -0.000403 VECM-GARCH-X 0.002534 0.002236 0.001864 0.001492 0.001119 0.000375 -0.000370 Daily k = 0.1 k = 0.5 k=1 k = 1.5 k=2 k=3 k =4 Unhedged, γ** = 0 -0.001372 -0.001511 -0.001684 -0.001858 -0.002031 -0.002377 -0.002724 Naïve, γ** = 1 0.000609 0.000490 0.000341 0.000192 0.000043 -0.000255 -0.000553 OLS 0.000305 0.000239 0.000156 0.000073 -0.000009 -0.000175 -0.000340 VECM 0.000316 0.000248 0.000164 0.000079 -0.000006 -0.000175 -0.000344 VECM-GARCH-X 0.000713 0.000622 0.000509 0.000396 0.000283 0.000057 -0.000170 Panel D: FTSE/ATHEX Mid-40 Out-of-Sample Expected Utilities Weekly k = 0.1 k = 0.5 k=1 k = 1.5 k=2 k=3 k =4 Unhedged, γ** = 0 -0.002871 -0.003214 -0.003643 -0.004072 -0.004501 -0.005359 -0.006217 Naïve, γ** = 1 0.000426 0.000061 0.001486 0.001340 0.001157 0.000975 0.000792 OLS 0.001195 0.001092 0.000964 0.000836 0.000708 0.000197 0.000452 VECM 0.001279 0.001165 0.001022 0.000880 0.000737 0.000452 0.000166 VECM-GARCH-X 0.000768 0.000722 0.000664 0.000606 0.000548 0.000432 0.000316 Daily k = 0.1 k = 0.5 k=1 k = 1.5 k=2 k=3 k =4 Unhedged, γ** = 0 -0.000471 -0.000528 -0.000600 -0.000672 -0.000743 -0.000886 -0.001030 Naïve, γ** = 1 0.000239 0.000206 0.000164 0.000123 0.000082 -0.000001 -0.000083 OLS 0.000126 0.000111 0.000092 0.000073 0.000054 0.000015 -0.000023 VECM 0.000131 0.000115 0.000095 0.000075 0.000056 0.000016 -0.000023 VECM-GARCH-X -0.000245 -0.000259 -0.000276 -0.000293 -0.000309 -0.000343 -0.000377

Table 5. Portfolio Risk-Return Trade-offs of FTSE/ATHEX Mid-40 Futures Daily out-of-sample portfolio return and risk estimates are calculated using Equations (1) and (2), respectively, for different levels of the UMHR (γ** = 0, … , 1.6). The implied risk aversion parameter is estimated in every case using k = -(Bias t+1 )/2(γ ** – γ * ) Var(∆F t+1 ). The % Reduction in Return = 1 – [(Hedged Return)/(Unhedged Return)] and the % Reduction in Variance = 1 – [(Hedged Variance)/(Unhedged Variance)]. Bold indicates the minimumstandard deviation portfolio. Optimal Implied k Portfolio Portfolio St. Portfolio % Reduction % Reduction Cost of UMHR, γ** Return Deviation Variance in Return in Variance Hedging 0 -0.00000025 -0.000457 0.011970 0.000143 0.1 -0.00000028 -0.000407 0.010649 0.000113 10.8 20.9 0.52 0.2 -0.00000033 -0.000358 0.009347 0.000087 21.6 39.0 0.55 0.3 -0.00000040 -0.000309 0.008075 0.000065 32.4 54.5 0.60 0.4 -0.00000051 -0.000259 0.006850 0.000047 43.2 67.3 0.64 0.5 -0.00000070 -0.000210 0.005700 0.000032 54.1 77.3 0.70 0.6 -0.00000111 -0.000160 0.004684 0.000022 64.9 84.7 0.77 0.7 -0.00000264 -0.000111 0.003906 0.000015 75.7 89.4 0.85 0.8 0.00000687 -0.000062 0.003527 0.000012 86.5 91.3 0.95 0.9 0.00000149 -0.000012 0.003672 0.000013 97.3 90.6 1.07 1 0.00000084 0.000037 0.004290 0.000018 108.1 87.2 1.24 1.1 0.00000058 0.000086 0.005215 0.000027 118.9 81.0 1.47 1.2 0.00000045 0.000136 0.006312 0.000040 129.7 72.2 1.80 1.3 0.00000036 0.000185 0.007508 0.000056 140.5 60.7 2.32 1.4 0.00000030 0.000234 0.008761 0.000077 151.4 46.4 3.26 1.5 0.00000026 0.000284 0.010051 0.000101 162.2 29.5 5.50 1.6 0.00000023 0.000333 0.011364 0.000129 173.0 9.9 17.53

Table 6. Optimal Rebalancing Frequency, VECM-GARCH-X vs. OLS The number of portfolio rebalances made by the investor are presented, when the potential gains in utility from the reduced variance offset the transactions costs that are incurred, for different levels of the risk aversion coefficient (k). Equation (13) is used to estimate the mean-variance portfolios of the VECM-GARCH-X and OLS models. It is estimated that the transactions costs (φ) are 0.01399% and 0.00821% in the FTSE/ATHEX-20 and FTSE/ATHEX Mid-40 markets, *

**

respectively. In the FTSE/ATHEX-20 and FTSE/ATHEX Mid-40 markets, the γ t and the γ t

hedge ratios are used, respectively. In the FTSE/ATHEX-20 market there are 249 hedging weeks and 1,185 hedging days, whereas in the FTSE/ATHEX Mid-40 market there are 226 hedging weeks and 1,081 hedging days. Panel A: Weekly Frequencies FTSE/ATHEX-20 k 0.1 0.5 1 1.5 2 2.5 3 4 0 0 162 214 214 214 214 214 In-Sample 0 0 0 24 30 30 30 30 Out-of-Sample FTSE/ATHEX Mid-40 k 0.1 0.5 1 1.5 2 2.5 3 4 187 183 165 148 119 93 68 46 In-Sample 0 0 0 0 0 0 0 0 Out-of-Sample Panel B: Daily Frequencies k In-Sample Out-of-Sample

0.1 0 0

0.5 0 0

k In-Sample Out-of-Sample

0.1 1,069 0

0.5 1,012 0

FTSE/ATHEX-20 1 1.5 2 2.5 239 1,149 1,151 1,151 3 59 130 135 FTSE/ATHEX Mid-40 1 1.5 2 2.5 809 662 530 392 0 0 0 0

1

3 1,151 135

4 1,151 135

3 294 0

4 187 0

Figure 1. Return-Risk trade-offs from Hedging with FTSE/ATHEX Mid-40 Futures

The numbers on the portfolio opportunity frontier refer to UMHR (γ**). The estimate 0.8 is the minimumstandard deviation portfolio. 0.00040 0.00030

Portfolio Return

0.00020 0.00010

1 0.9 0.8 0.7

0.00000 -0.00010 -0.00020

1.2

1.1

0.6

0.5

-0.00030

1.4

1.3

0.4

0.3

-0.00040 -0.00050 0.0000

0.0020

0.0040

0.0060

0.0080

Portfolio Standard Deviation

2

1.6

1.5

0.2

0.0100

0.1

0 0.0120

0.0140