Model Specification for the Estimation of the Optimal Hedge Ratio with Stock Index Futures: an Application to the Italian Derivatives Market

Transcription

1 Model Specification for the Estimation of the Optimal Hedge Ratio with Stock Index Futures: an Application to the Italian Derivatives Market by Agostino Casillo 1 ABSTRACT Several techniques to estimate the hedge ratio with index futures contracts have been proposed in the literature. While these techniques hold theoretical appeal, there is no univocal evidence as to their effectiveness. This research provides an empirical comparison of four different econometric techniques in the context of hedging the market risk using the FIB 30 index futures contract of the Italian Derivatives Market. Specifically, the OLS regression model, the bivariate vector autoregressive model (BVAR), the vector error correction model (VECM) and the multivariate GARCH model are employed. Then the hedging effectiveness is measured in terms of ex-post and ex-ante variance minimization. It is generally found that only the GARCH model is able to outperform the OLS model and the improvement is not really consistent. KEYWORDS: Optimal hedge ratio; Hedging; Stock Index Futures; Italian Derivatives Market. JEL Classification: G10, G13. 1 University of Birmingham and Associazione Guido Carli

2 1. Introduction To implement a hedge, it is necessary to calculate the corresponding hedge ratio that is defined by Hull (2003: 78) as the ratio of the size of the portfolio taken in futures contracts to the size of the exposure. Researchers have distinguished three hedge strategies: the traditional one to one hedge, the beta hedge, and the minimum variance hedge. All three strategies require determining the optimal hedge ratio h* but the traditional strategy emphasizes the potential for futures contract to be used to reduce risk. The strategy is very simple and involves the adoption of a fixed hedge h=-1 which consists of taking a futures position that is equal in magnitude, but opposite in sign, to the spot position. If price changes in the futures market exactly match those in the future markets the adoption of a one to one strategy will be enough to eliminate the price risk. The beta hedge strategy is quite similar but it is used when the cash portfolio to be hedged does not exactly match the portfolio underlying the futures contract. For this reason the optimal hedge ratio is calculated this time as the negative of the beta of the cash portfolio. Naturally, if the cash portfolio is that which underlies the futures position, the two strategies will give the same value for the hedge ratio. However, in practice the prices on the spot and futures markets do not move exactly together and a hedge ratio derived from the traditional or beta hedge strategy would not minimize the risk. In particular Peters (1986) has shown that mispricing adds 20% to the volatility of the futures contract. Since the futures contract is more volatile than the underlying index the use of a portfolio beta as a sensitivity adjustment would result in a portfolio being over-hedged. This imperfect correlation is taken in consideration instead by the minimum variance hedge ratio. The model has been proposed by Johnson (1960) and Stein (1961) and developed by Ederington (1979), who employs portfolio theory to identify the hedge ratio (h*) as: 2

3 σ * S, h t 1 = - 2 σ F, t F, t (1.3) where σ 2 F,t represents the conditional variance of the futures contract and σ 2 SF,t is the conditional covariance between the index futures and the spot position on the same index. The negative sign just means that the hedging of a long stock position requires a short position in the corresponding futures contract. In order to show how the above can be derived let S t and F t be the logarithms of the stock index and index futures prices respectively. The starting point necessary to derive the model is represented by the two equations that express the actual returns offered from the two assets at time t, S t = S t S t-1 =R s,t and F t = F t F t-1 =R f,t. Hence, the expected return at time t-1 of a portfolio (P) consisting in a long position in Q s unit of the stock index and Q f units of the corresponding futures contract, is given by: E t-1 (P t ) = Q s E t-1 ( S t ) Q f E t-1 ( F t ) (1.4) where the price changes are assumed to be stochastic, and the spot position Q s is given. The resulting variance associated to this portfolio is σ p,t =Q σ S, t +Q σ F,t 2Q s,t-1 Q f,t-1 σ S,F, t (1.5) s, t 1 f, t 1 with σ 2 S, t and σ 2 F,t representing the conditional variance of the cash and futures positions. If the goal is to maximize investor s expected utility we need to define its utility function, that is the risk-return trade-off, in order to go on. Following Kahl s formulation, Brooks et al. (2002: 335) show that if the investor has the following linear utility function, U ( E t-1 (P t ), σ 2 p,t) = E t-1 (P t ) - ψ σ 2 p,t (1.6) where ψ is the degree of risk aversion, then agent s maximisation problem can be expressed as, max U( E t-1 (P t ), σ 2 p,t ) = Q s E t-1 ( S t ) Q f E t-1 ( F t ) ψ ( σ S, t +Q σ F,t 2Q s,t-1 Q f,t-1 σ S,F, t, ) (1.7). Q s, t 1 f, t 1 3

4 Substituting, we can derive the solution that permits to maximize the utility function, that is the condition for an optimum, Q f σ sf δf - = 2-2 Q σ νσ s f Q s f (1.8), that can also be written as δf Q f = 2 νσ f σ sf - Q s 2 σ f (1.9). Assuming that ψ is indefinitely high 2 the investor futures position does not depend from δf the risk parameter because the ratio 2 νσ f becomes insignificant. In that case the hedge ratio is: Q f σ sf h* = = - 2 Q σ s f σ s =- ρ (1.10), σ f where ρ is the correlation coefficient between S and F. Substituting this expression into the portfolio variance equation we obtain the variance of return for the minimum risk hedge ratio, σ = σ (1- ρ ) (1.11) MIN S Therefore, the return on a hedged position will normally be subject to the risk caused by unanticipated changes in the relative price between the position being hedged and the futures contract. The formula clearly expresses that only when there is a perfect correlation the risk can be completely eliminated by hedging. However, as recognized by Cecchetti et al. (1988) and Castellino (1990) the Minimum Variance (MV) hedge ratio is in general inconsistent with the mean-variance framework. In order to be consistent the MV hedge ratio has to implicitly assume that either expected returns on the futures contract needs to be zero or that investors are infinitely risk averse, as shown above, which means that they will renounce an infinite amount of expected return in exchange for an indefinitely small risk reduction. Such an assumption is undoubtedly unrealistic, SF 2 i.e. ψ 4

5 nevertheless the minimum variance hedge ratio is useful in order to provide a benchmark for the hedging performances. Many studies have been proposed on the empirical estimation of the optimal hedge ratio employing various techniques. If the spot and futures prices are not cointegrated and the conditional variance-covariance matrix is time invariant, it has been shown that a constant optimal hedge ratio can be obtained from the slope coefficient h in the regression: S t = α + h* F t + ε t. (1.12). Hence, the simple method of the Ordinary Least Squares regression in which the coefficient estimate for the future price gives the hedge ratio by regressing the spot on the future price, has been employed in the past by various studies, like Ederington (1979), Malliaris and Urrutia (1991) and Benet (1992). However, this method has suffered various criticisms. It has been shown (Pindyck, 1984; Poterba and Summers, 1986; Bollerslev, 1986; Baillie and De Gennaro, 1990) that stocks returns typically exhibit time-varying conditional heteroscedasticity and hence the data do not support the assumption that the variance-covariance matrix of return is constant over time. Thus, in order to improve the estimation of the hedge ratio, it is necessary to consider the possible time-varying nature of the second moments. For this reason the more recent literature (Baillie and Myers, 1991; Myers, 1991; Sephton, 1993; Park and Switzer, 1995) has proposed the use of hedging strategies based on the GARCH (generalized autoregressive conditional heteroscedasticity) class of models which allow the conditional variances and covariances used as inputs to the hedge ratio to be time-varying. Another issue addressed by a large number of researchers is the important role played by the theory of cointegration between futures and spot market in determining the optimal hedge ratio. In fact, the presence of the efficient market hypothesis and the absence of arbitrage opportunity imply that spot and futures markets are cointegrated and an error correction representation must exist. 5

6 Finally, other researchers have proposed more complex techniques and some special case of the above techniques for the estimation of the OHR. Among these we mention the random coefficient autoregressive offered by Bera et al. (1997), the Fractional Cointegrated Error Correction model by Lien and Tse (1999), the Exponentially Weighted Moving Average Estimator by Harris and Shen (2002), and the asymmetric GARCH by Brooks et al. (2002). Nevertheless, whether a distinctive superior hedge ratio estimation methodology exists is a question that is still under debate. 1.2 Research structure In spite of the copious literature on the subject there is no general consensus about a technique that could systematically provide a better estimation of the optimal hedge ratio for stock index futures. Indeed, if the OLS model shows various inadequacies from a statistical point of view, more complex models have offered conflicting results from an empirical point of view. Thus the objective of this study is to offer a contribution to this question applying various models to the calculation of the hedge ratio of the stock index futures and the underlying cash index for the MIB 30 market. Moreover this work expands the range of the markets that has been tested because to the best of our knowledge no study in the international literature has been made on the FIB 30 stock index futures of the Italian Derivatives Market. The rest of the paper is organized as follows. In the next section a detailed review of previous studies specifically conducted on stock index futures is presented. Then section 3 provides an overview of the data sample while section 4 describes the methodology of the research, analysing in-depth the statistical tests and econometric models that are employed in the estimation. Finally section 5 presents the results of the statistical tests and of the hedge ratio estimation and compares the hedging performances of the various models. 6

7 2. Literature review The great part of studies on stock index futures hedging still relates to the United States of America, although more recently various works have been published in relation to the UK, Australia and Japan. This section aims to outline the main themes of previous works and identify the most recent trend of research. The review clearly focuses on studies specifically conducted on stock index futures but works on optimal hedge ratios estimation that refer to other types of futures contracts are also reported when their results are considered important in order to understand the development of the research. In fact, since the existing literature is quite voluminous, a selection is necessary in order to convey the main themes. 2.1 The early studies The hedging of basis risk is grounded in the mean variance framework of Markowitz (1952), and is firstly applied to futures hedging by Working (1953), Johnson (1960), and Stein (1961) but Ederington (1979) properly formalizes these ideas. He argues that a minimum variance hedge ratio can be defined as the ratio of the covariance between the futures and spot price to the variance of the future price. Then he shows that the minimum variance hedge ratio is just the slope coefficient from an OLS regression where spot and future prices are respectively the dependent and independent variable. Ederington s theories have been followed by numerous studies including Hill and Schneeweis (1981, 1982); Benninga et al. (1983); Figlewski (1984, 1985); Witt et al. (1987); Myers and Thompson (1989); Castellino (1990); Myers (1991). These studies have tried in to analyse in particular the question of whether levels, changes, or return models should be used in the regression approach to optimal hedge ratio estimation. For example, Witt et al. (1987) 7

8 study the theoretical and practical differences among these three specifications frequently used to estimate hedge ratios in relation to the hedging in different agricultural commodities. They find that hedge ratios based on price-level regressions are as statistically significant as those by the other two procedures, in disagreement with the opinion of some previous studies, and argue that theoretically the proper hedge ratio estimation technique depends upon the objective function of the hedger and the type of hedge being considered. Myers and Thompson (1989) try to offer a generalized approach in order to solve the same issue. However also in this case none of these regressions seems to provide an absolute answer to the question and every one of them appear appropriate under special circumstances. Moreover, their application to the hedging in corn, soybeans and wheat futures markets shows that the simple regression model is the more appropriate also if its estimates are close to those obtained with the generalized approach. 2.2 The development of the research The first researcher to focus on the use of stock index is Figlewski. Since then numerous works have followed with varying conclusions. Figlewski (1984) investigates hedging performances for the Standard & Poor s 500 futures contract related to a portfolio underlying the five major stock indexes for the period He includes dividend payments in the return series but finds that this choice does not alter the obtained results because of their predictable nature. Consequently, following studies have excluded this component from the calculation of the returns. He confirms that ex-post minimum variance hedge ratios can be estimated by OLS using historical data and shows that in all cases hedging performances are worse using the beta hedge ratio instead of the minimum variance hedge ratio. Also Junkus and Lee (1985) examining hedging effectiveness of three USA stock indexes futures 8

9 exchanges, confirm that the minimum variance hedge ratio is superior to all the others in reducing the risk associated to the cash portfolio. However Ghosh (1993), through empirical calculation based on several stock portfolios hedged with Standard & Poor s 500 index futures, finds the hedge ratios obtained from traditional models to be underestimated because the cointegration relation is ignored. In fact, the theory of cointegration developed by Engle and Granger (1981) shows that if two series are cointegrated there must exist an error correction representation that permits to include both the short-run dynamics and the long-run information. In accordance Lien and Luo (1993) prefer an error correction model in order to estimate the hedge ratios for stock index markets, because of the relationship between spot and futures markets. These considerations are analytically confirmed by Lien (1996) that proves that the hedger makes a mistake if the hedging decision is based on the hedge ratio obtained from a first difference model that does not include an error correction term. Also Chou et al. s work (1996) agrees with the previous conclusions investigating the hedging behavior of the Nikkei Stock Index futures. This study estimates and compares the hedge ratios of the conventional and error correction model revealing that the error correction model is superior in a statistical sense. At the same time, other studies were supporting the possibility of improving the estimation of minimum risk variance ratio through the adoption of time-dependent conditional variance models such as the GARCH framework developed by Engle (1982) and Bollerslev (1986). In this way Baillie and Myers (1991) and Myers (1991), who examining commodity futures report improvements in hedging performances over the constant hedge approach by following a dynamic strategy based on a GARCH model. However Myers (1991) underlines that despite significant theoretical advantages, in practice the GARCH model performs only slightly better than a simple constant hedge ratio. For this reason, he asserts that assuming constant optimal hedge ratios and using linear regression approaches, may be an acceptable approximation. 9

10 Later on, Kroner and Sultan (1993) find similar results with five currency futures over the period Park and Switzer (1995) are the first to apply dynamic hedging to the stock index futures, and in particular to the daily data from the Standard and Poor s 500 Index Futures and the Toronto 35 Index Futures. They argue that in this case, and also considering transaction costs, the GARCH model gives an improved hedging strategy in comparison with the traditional strategy, the OLS hedge, and the OLS with cointegration between spot and futures. However Lien and Luo (1994) do not agree about the supposed superiority of GARCH to all the other models in the hedge ratio estimation. In fact in their opinion however GARCH may characterize the price behavior of financial assets, the cointegration relation, and hence the error correction term, is the only truly indispensable element in order to optimise the hedging performances of the obtained ratio. Fackler and McNew (1994) go beyond, deeply criticizing the negative features of the GARCH model. In fact, in their opinion GARCH methodology is a less than ideal technique for estimating time-varying hedge ratios (1994: 620). This is because the estimation of GARCH models requires the use of nonlinear optimization algorithms and the imposition of inequality restrictions on model parameters. Holmes study (1995) is the first to investigate ex-ante hedging effectiveness for the UK in relation to FTSE-100 contract. Considering the period , the results show the possibility to achieve a risk reduction of more than 80% in comparison with the unhedged portfolio. In a following article Holmes (1996) examines ex-post hedging effectiveness for the same contract as in the earlier study. He shows that in terms of risk reduction a hedge strategy based on a minimum variance hedge ratio estimated through a OLS slightly outperforms a strategy based on a minimum variance hedge ratio estimated using more advanced techniques such as the ECM and the GARCH approach but also by simpler techniques like the beta hedge and the unhedged portfolio. 10

11 2.3 Recent issues in the hedge ratio estimation Recent researches in optimal hedge ratio estimation have proposed the adoption of increasingly complex estimation techniques. These studies certainly offer an important theoretical contribution but it is not yet completely clear if these models are able to offer a substantial improvement from a practical point of view. Lypny and Powalla (1998) study the hedging effectiveness of a dynamic hedging strategy based on a GARCH (1,1) covariance structure, combined with an error correction of the mean returns, for the German stock index DAX futures. They explain that the adoption of this model gives significant improvements over a simple constant hedge and over a dynamic hedge with the error correction but without the GARCH (1,1) covariance structure, which is consistent with the study by Park and Switzer (1995). Also Chakraborty and Barkoulas (1999) test the GARCH (1,1) model for future contracts of five leading currencies. However, in their case despite the empirical evidence supports time-varying optimal hedge ratios the dynamic model significantly outperforms the static model just in one case out of five. A fractionally integrated error correction is applied by Lien and Tse (1999) for estimating the hedge ratio. The results from this model are compared with that obtained from the OLS, VAR, EC and ARFIMA-GARCH approaches using daily data for the Nikkei Stock Average Index over the period 1989 to They obtain that the OLS estimation of the minimum variance hedge ratio provides the worst outcomes for hedging periods longer than 5 days as compared to the other methods. Afterward Miffre (2001) proposes a new methodology, the GMM, that tries to estimate time-dependent minimum variance hedge ratios minimizing hedging rule as a set of conditional moments restrictions. However, the following tests suggest that the traditional and the OLS approaches can even result in better hedging performances than the proposed methodology. 11

12 Recently, Yang s study (2001) describes and estimates four different models to calculate optimal hedge ratios for the All Ordinaries Share Price Index (AOI) and the corresponding index futures contract (SPI) on the Australian market. In this case the Error Correction model is found to perform better off than the OLS regression model, the bivariate vector autoregressive model (BVAR) and the multivariate diagonal Vec GARCH model. This result is consistent with Ghosh (1993) and Lien s (1996) works which had proved that the lack of consideration for the cointegration between spot and futures markets leads to a smaller than optimal position. Also Sim and Zurbruegg s (2001) study agrees on the importance of utilizing an estimation technique of the hedge ratio that takes into account the cointegrative relationship between spot and futures markets but it is considered as much as important as considering the time-varying nature of market risk. Their arguments are based on an empirical application to the FTSE-100 spot and futures contracts that shows the advantages of employing a time-varying hedge ratio rather than a simpler constant measure. The paper by Harris and Shen (2002) offers an alternative to traditional approaches to the OHR. The researchers adopt an OHR robust to leptokurtosis and compare it to the rolling window and the EWMA approach in relation to the daily data for the FTSE 100 index futures for the period The results show that the variance of the new OHR is as much as 70% lower than the variance of the OHRs obtained from the traditional methods, implying a substantial reduction in the transaction costs. Another paper by Lien et al. (2002) uses a constant correlation model in relation to ten spot and futures markets covering currency, commodity and stock index futures. It asserts that also if conditional heteroscedasticity is a characteristic of many financial series there is no definitive conclusion about the superior hedging performances of the GARCH model. Therefore, the performances of the MV hedge ratio from an OLS and a CC-GARCH are obtained and compared showing that the GARCH strategy is not able to outperform the OLS 12

13 strategy. Moreover, given the high transaction costs associated with the GARCH strategy the authors argue that it should not be considered for hedging purposes but just for data description. Brooks et al. (2002) focus on the impact of asymmetries on the hedging of stock-index position through index futures. The outcome of their research, based on 3850 daily observations of the FTSE-100 stock index and stock index futures contract spanning the period January 1985 to April 1999, shows that asymmetric models in which positive and negative prices changes affect volatility forecast differently permit, in general, to improve the hedging effectiveness. In the same year Laws and Thompson (2002) compare the hedge ratios obtained through the Naïve, OLS, GARCH (1,1), Egarch in Mean and EWMA methods. Their study, based on the FTSE-100 stock index futures, points out that in this case the EWMA method results in being superior to the others. However a recent work by Moosa (2003) questions the theory that the choice of the model used to estimate the hedge ratio can really affect the effectiveness of hedging. His study analyses this proposition employing four different models: a level model, a first difference model, a simple error correction and a general error correction model. These models are compared in relation to two different sets of data. The first is a set of monthly observations on cash and futures prices of Australian stocks while the second sample consisted of quarterly observations covering the period on the spot exchange rates of the pound and the Canadian dollar against the US dollar. The results show that the model specification does not make any significative difference for hedging effectiveness in both cases. For this reason Moosa concludes that Although the theoretical arguments for why model specification does matter are elegant what really matters for the success or failure of a hedge is the correlation between the prices of the unhedged position and the hedging instrument (2003: 13

14 19). In substance, a low correlation necessarily implies a poor hedging performance while a high correlation produces a good hedging performance. 2.4 Summary of the literature review It is apparent from previous studies that many researchers have proposed the estimation of time-varying optimal hedge ratios for stock index futures based on GARCH models or extensions of them (Baillie and Myers, 1991; Myers, 1991; Kroner and Sultan, 1993; Park and Switzer, 1995; Sim and Zurbruegg, 2001; Brooks, 2002). Moreover, is also generally recognized that when the presence of cointegration is detected the inclusion of an error correction term becomes appropriate (Ghosh, 1993; Lien, 1993; Kroner and Sultan, 1993; Lien and Luo, 1994). More complex alternatives to the traditional GARCH model have also been offered in recent years (Lien and Tse, 1999; Harris and Shen, 2000; Miffre, 2001; Brooks, 2002). However, some authors have pointed out that despite the fact that the more advanced techniques have the potential to exhibit a better performance there are some drawbacks that should be considered. First of all, some of these methods can be too difficult to estimate and can produce significative transaction costs (Lien, 2002). Moreover, more simple methods like the OLS one can reach similar levels of performance many times and can be more than satisfactory for hedging purposes (Myers, 1991; Holmes, 1995; Chakrabothy and Barkoulas, 1999; Miffre, 2001). Conclusively, there is not the evidence necessary in order to suggest the establishment of a unique optimal hedge ratio. 14

15 3. Data 3.1 The Italian Derivatives Market The Italian Derivatives Market (IDEM) is the Italian market of stock derivatives run by Borsa Italiana Stock Exchange of Milan. The IDEM market was born in November 1994, when the telematic trading of FIB 30, the futures contract on the MIB 30 index, began. The MIB 30 index is an arithmetic average, weighted by market capitalization, based on the 30 most highly capitalized companies listed in Borsa Italiana and accounts for more than 60% of the total market value of the Italian Stock Exchange. The members of the MIB 30 index are subject to revision by Borsa Italiana two times a year, in March and September. The index is computed using the transaction prices, rather than using the best bid and ask prices quotation as it happens for the FTSE 100 Index in the U.K, and is updated every minute as long as the Borsa Italiana is open. It has basis =10,000 which means that by common consent it has been fixed that the 31 st December 1994 the index had that value. The FIB 30 futures contract is quoted in the same units as the underlying index, except that the decimal is rounded to the nearest 0.5. There are four delivery months: March, June, September, and December. Trading takes place in the three nearest delivery months, although volume in the far contract is usually very small. Each contract is therefore traded for 9 months. The FIB30 contracts are cash-settled as opposed to the physical delivery of the underlying. All contracts are marked to market on the last trading day, which is the third Friday in the delivery month, at which point all the positions are closed. For the FIB 30 the settlement price on the last trading day is calculated as an average of minute-by-minute observations between 9.30 am and am, when the trading in an expiring contract ceases. To the best of our knowledge in international literature no study has been conducted about the 15

16 model specification for the estimation of the OHR for the FIB 30 contract hence it seems plausible to choose it as the subject of our study. 3.2 Data The data used in this study has been obtained from Datastream. It includes the MIB 30 index and the corresponding index futures contract FIB 30 daily closing prices on a daily basis for the period of 28 th November th June After removing non-trading days the series consist of 2,489 observations. Only the first 2,459 observations are used in the test, leaving the last 30 observations, starting from 11 th May 2004, for an ex-ante hedge ratio performance comparison. The return series for the cash portfolio and futures contract are calculated as the logarithmic price change: S t = R S,t = log PS, t (3.1), PS, t 1 and F t = R F,t = log P PF F, t, t 1 (3.2), where S t and F t are the daily returns on the cash and futures position and P S and P F are the spot and futures prices at time t. Both logarithmic series are represented in Figure 1, which shows a strong correlation between them. Summary statistics of prices changes of the MIB 30 spot and futures index returns are also provided in Table 1. 16

17 4. Research methodology The literature review in the second section of this work has widely shown that there are a variety of methods available to obtain the hedge ratio. In this study four different models are described and estimated to calculate optimal hedge ratios. In the following paragraphs we present and discuss each model while the last paragraph explains the methodology used in order to compare the resulting hedge ratios. 4.1 The conventional regression model As mentioned in the first section, the simplest technique to estimate a hedge ratio is represented by a linear regression of change in spot prices on change in futures price. This method has been widely applied in the literature. Witt et al. (1988) report three basic specifications of this model: a price level, a price change and a percentage change. However, the use of a price level model could be inappropriate from both a theoretical and a statistical point of view. In fact Hill and Schneeweis (1981) Brown (1985) and Wilson (1987) argue that theoretically hedgers attempt to reduce the risk derived from unexpected price changes from the time the hedge begin until it is lifted. Moreover, Benninga et al. (1984) point out that statistically price differences are more appropriate because cash and futures prices can be highly correlated if corresponding trends exist between the two of them. In that case estimating the hedge ratios by using price levels may result in an autocorrelation in the residuals leading to a violation of the OLS assumption and inefficient estimates of hedge ratios. At the same time the use of a price difference equation seems to be preferable to a percentage change model. In fact the percentage change model could be useful when cash and 17

18 futures prices do not change linearly with respect to one another, for example in cross hedging when different commodities are being compared. On the contrary since in this case we are considering direct hedging a price differences model should be preferred to a percentage change model because it is more parsimonious and eases hedge ratio interpretation. Hence defining the actual returns offered from the two assets at time t as S t and F t the price model can be obtained regressing S t on F t. The linear regression used is the following S t = α + β F t + ε t (4.1) where α is the constant term while ε t is the error term from the Ordinary Least Squares (OLS) model. The slope of the regression β provides an estimate for the minimum variance hedge ratio h*. The OLS technique is quite robust and simple to use but in order to be valid and efficient there are specific assumptions that must be satisfied. In particular, we are going to use the Breusch-Godfrey test in order to examine whether serial correlation between the error terms is present. In fact the Classical Linear Regression Model assumes that cov(ε i, ε j )=0 3 and the presence of autocorrelation could produce inefficient coefficient estimates implying wrong estimates for the standard errors. The Breusch-Godfrey test is preferable to the Durbin- Watson (DW) because it is not subject to the DW restrictions and permits to detect a wider range of residual autocorrelation forms. The hypothesis that there is no serial correlation between the current value and the previous value of each variable of the model (S and F) will also be tested using the Ljung-Box statistic which permits to verify if k of the correlation coefficients are jointly equal to 0. Next, we are going to apply the Bera-Jarque test to examine that disturbances are normally distributed. The Bera-Jarque test is the most commonly applied statistic to this end and exploits the properties that characterize the skewness and kurtosis of a normal distribution in order to do that. 3 i.e. the errors are independent with one another. 18

19 Finally we are going to test if errors do not have a constant variance, that is in other words the presence of heteroscedasticity. The tests necessary to detect the existence of heteroscedasticity and ARCH effects are presented later together with the introduction of the GARCH model. 4.2 The Bivariate Vector Autoregressive Model Despite the traditional approach being undoubtedly inviting for its easy application, it suffers from a number of limitations. As earlier noted by Herbst et al. (1993) one weakness of the above model is represented by the fact that estimation of the minimum variance hedge ratio could suffer from problems of serial correlation in OLS residuals. Moreover this model does not take account of the fact that movements in the spot and futures markets may influence the current price movement. These problems can be solved through a bivariate Vector autoregressive model (VAR) that is a systems regression model where there are two variables, each of whose current value depends on a combination of previous value of both the variables. For this reason, in order to eliminate the serial correlation and take in consideration independent variables that can influence the dependent variable we repropose the following model already tested by Yang (2001) for the Australian market: S t s k = α + β S + λ F + ε i= 1 si t i k i= 1 si t i st (4.2a) F t f k = α + β S + λ F + ε i= 1 fi t i k i= 1 fi t i ft (4.2b) where α is the intercept, β s, β f, λ s and λ f are positive parameters, ε st and ε ft are independently identically distributed error terms. The optimal lag length k that permits eliminating the serial correlation from the system equation is defined using the multivariate versions of the 19

20 Akaike s and Schwarz s Bayesian information criteria. Possible disagreements between the two criteria are solved through the likelihood ratio test. If we let σ 2 ( ε ft ) = σ ff, and cov (ε st, ε ft ) = σ sf we can easily define the optimal hedge ratio from the model as: σ h*= σ sf ff (4.3). 4.3 The Vector Error Correction Model The presence or the absence of a cointegration relationship between the variables is an important investigation to conduct since standard inference procedures do not apply to regression where the dependent variable and the regressors are integrated. In fact Ghosh (1993), and Kroner and Sultan (1993) have shown that the regression given by an equation like the standard VAR is misspecified when spot and future index prices are cointegrated because it ignores the error correction, excluding the impact of last period s equilibrium error. Hence Ghosh (1993) and Lien (1996) show that the existence of a cointegration relationship between spot and future markets will typically produce a downwardly biased hedge ratio if the error correction term is not considered in the estimated equation. Following the theory of cointegration developed by Engle and Granger (1987: 235) we can define as integrated of order d, denoted x t ~ I (d), a series with no deterministic component which has a stationary, invertible, ARMA representation after differencing d times. If two variables, x t and y t, are both I(1), it is generally true that a linear combination of them z t such that z t = x t + α y t (4.4) will also be I (1). However it is possible that z t ~ I (0) when a special constraint operates on the long-run components of the two series, in other words if the two series are cointegrated. 20

21 To formalize these ideas, let v t be a n x 1 vector of variables, then the components of v t are said to be cointegrated of order (d, b) if: (i) all the components of v t are I(d); (ii) there exist at least one vector with coefficient α ( 0) such that zt = α v t ~ I(d-b), with b>0. Ordinary least squares (OLS) method is inappropriate if x t and y t are non-stationary because the standard errors are not consistent. However, Engle and Granger show that if two series are non-stationary but a linear combination of them is stationary an error correction representation, not subject to spurious results, that links these variables must exist. The theoretical relationship that links spot and futures index price is represented by the cost-of-carry model: F t = S t e (r-d) (T t) (4.5) where F t is the stock index futures price at time t, S t is the stock index price at time t and T represents the future expiration date, so that (T-t) is the time to maturity of the future contract. Finally, (r-d) is the net cost of carrying the underlying stock in the index, which is the time value cost of wealth tied up in the stock index investment, offset by the flow of dividends from the underlying basket of stocks Taking logarithms of both sides the relation gives log F t = log S t + (r-d) (T-t), (4.6) implying that the long-term relationship between the logs of the spot and futures price should be equal to one and consequently the basis should be stationary. If, on the contrary, the future prices could wander without bound an arbitrage opportunity would arise offering the traders the possibility of a riskless profit such that the relationship would be soon brought back to equilibrium. Because cointegration is an econometric way of identifying linear combinations of variables that are bound by some relationship in the long run, spot and futures are supposed to be cointegrated since they are the price of the same asset at different points in time. Testing for cointegration and estimating cointegrating vectors is therefore an important issue in the specification of the model for calculating the hedge ratio. 21

22 4.3.1 Tests for unit roots A stationary series can be defined as one with a constant mean, constant variance and constant autocovariances for each given lag (Brooks, 2002: 367). The implications of stationarity in series are profound since it affects their own properties and behaviour. In a pioneering work Granger and Newbold (1974) have shown that regressing non-stationary variables onto each other leads to potentially misleading inferences about the estimated parameters and the degree of association. In fact the use of non-stationary data can imply that the regression of one variable on the other could show significant coefficient estimates and a high value for the R 2 measure even when the two variables are not related to each other. This kind of model is known as spurious regression and in this case the standard assumption for asymptotic analysis are not valid. Therefore, the first step involves testing for the order of integration of the individual variables under investigation. Various methods have been proposed in order to test for stationarity. They are based on the fact that a non-stationary series is characterized by a unit root. Indeed, the data-generating process can be written as (1-L) y t = α +ε t (4.7). Thus the characteristic equation has a single root equal to one, hence the name. The early work on testing for a unit root in time series is proposed by Dickey and Fuller (Fuller, 1976; Dickey and Fuller, 1979). The aim of their test, in its basic version, is to examine the null hypothesis that in the equation y = α y 1 + ε (4.8) t o t t the value of the parameter α 0 is equal to 0, i.e. the series has a unitary root for the characteristic equation and hence is not stationary, versus the one-sided alternative hypothesis that α 0 <0. If spot and futures prices are found to be non-stationary, the test should be repeated to detect the order of non-stationarity. In that case the null hypothesis will be that the series has two unit roots against the alternative hypothesis of just one unit root. However not 22

23 many series in finance are found to have more than a unit root. The test statistics for the original DF test do not follow a standard distribution and critical values are obtained through simulation experiments. The Dickey-Fuller test is based on the assumption that the error term ε t is white noise and hence not autocorrelated. However if this is not the case the test will be oversized, implying that the portion of times in which a correct null hypothesis is incorrectly rejected will be higher than the nominal size chosen. For this reason, Dickey and Fuller, Dickey and Said (1984), Phillips (1987), Phillips and Perron (1988), and others developed modifications of the Dickey-Fuller test that apply when ε t is not white noise. Basically, in that case the solution is to augment the test adding p lags of the dependent variable of the regression ( y). Here we apply the following augmented Dickey-Fuller (ADF) test for the presence of unit root: y = a p 0 y t 1 + α i yt i + ε t (4.9) i= 1 where y t = y t y t 1. Lagged differences ensure that any dynamic structure present in the dependent variable is erased and that εt is a stochastic term that has mean equal to 0, a constant variance and follows the assumption of absence of autocorrelation. The test is subject to the same critical values of the Dickey-Fuller test. The number of lags p is chosen in order to optimise the trade-off between the removal of all the autocorrelation and the increase in the coefficient standard errors. The optimal number of lags p of the dependent variable is selected using two information criteria the Akaike information criterion (AIC) and the Schwartz s Bayesian information criterion (SBIC), based on a maximum lags of 7. Really, it has been explained by Kim and Schmidt (1993) that the accuracy of the Dickey- Fuller test decreases in presence of ARCH effects, implying the rejection of the null hypothesis of a unit root too often. However the same paper concludes that the effect is not great and the test can still be considered quite reliable. An alternative test that utilizes non- 23

24 parametric correction for serial correlation has been proposed by Phillips and Perron (1988). Their test is quite similar to the ADF one, and suffers from more or less the same limitations, but includes a correction to the ADF procedure to allow for autocorrelated residuals. Also if the two tests give the same conclusion often, Leybourne and Newbold (1999) report that it has been noted that if the process is stationary but with a root close to the non-stationary boundary these tests can give different outcomes and that is why we consider both of them in order to have more robust results. Once the presence of unit roots is detected it is possible to test for cointegration. Cointegration is analysed by the two-step procedure of Engle and Granger (1987). The Engle and Granger test is a single equation technique that basically uses the ADF test on the residuals ε t of the regression y = α + ε (4.10) t x t t to see whether they are stationary or not. However since this test is conducted on the residuals of an estimated model, then the critical values are different from those specified for an ADF test on series of raw data. Engle and Granger (1987) have tabulated these new values giving their name to this application Model specification The finding that S t and F t are cointegrated implies that the cointegrating vector must be incorporated in the VAR obtaining a Vector Error Correction Model. Denoting the error correction term as E the VECM can be written as: S t = c s + k k si St i + λsi Ft i α s Et 1 i= 1 i= 1 β + ε st (4.11a) F t = c f + k k fi St i + λ fi Ft i + α f Et 1 i= 1 i= 1 β + ε ft (4.11b) 24

25 where the coefficients α s and α f can be interpreted as speed of adjustment factors and measure how quickly each market reacts to the deviation from the long run equilibrium relationship. Hence if the cointegrating vector between F t and S t were denoted as (1, -β) the error correction term E is equal to (s t βf t ). In this way, the error correction term permits correcting a proportion of last period s equilibrium error. Therefore if this equilibrium is positive because the spot rate is too high in relation to the value of the futures price, the error correction term should push down the spot price back to equilibrium and α s will have a negative value. At the same time the error correction term should push up F t in the second equation, implying a negative sign for α F. A zero coefficient for one error correction term would imply that however the two markets move following a long run relationship one market does all the readjustment, in other words it is a market to follow the other. On the contrary, it is not possible that α s =α f =0 since otherwise the model would be a standard VAR in differences and the two variables would not be cointegrated. The constant hedge ratio can again be calculated as shown in equation (4.3). 4.4 The Bivariate GARCH with error correction model The problem with the models previous offered is that they implicitly assume that the risk in spot and futures markets is constant over time, implying that the minimum risk hedge ratio will be the same all the time irrespective of when the hedging is undertaken. However this assumption clearly contrasts with the reality since as Bollerslev (1990) and Kroner and Sultan (1991) have shown the availability of new information produces changes in the risk of the various assets. Hence the risk minimizing hedge ratios should be time varying. Therefore conventional models cannot produce risk-minimizing hedge ratios, raising important concerns regarding the risk reduction properties associated with the traditional methods. 25

26 There are various statistical tests useful in detecting the presence of a non-constant variance, or heteroscedasticity, in the data. Most of the tests are applied to the ordinary least square residuals which will mimic the heteroscedasticity of the true disturbances. However to formulate most of the available tests it is necessary to specify the nature of the heteroscedasticity. For this reason one of the most popular methodologies is represented by White s general test which makes few assumptions about the likely form of the heteroscedasticity. Moreover, the presence of ARCH effect is tested as well. Actually, the volatility clustering is a well-known characteristic of many series of asset return. This particular specification of heteroscedasticity refers to the observation that in many financial time series the current level of volatility tends to be positively correlated with its level during the previous periods. Therefore the Lagrange Multiplier test of Engle (1982) is going to be used to detect the presence of this feature. If the data sample is characterized by sudden spikes and periods of increased volatility as usually happens with spot and futures data, the assumption of a constant variance results inappropriate. The development of the Autoregressive Conditional Heteroscedasticity (ARCH) model introduced by Engle (1982) and generalized by Bollerslev (1986) and its enormous extension and derivations in the early 1980 s has proved to be very useful in providing an econometric method to calculate time-varying hedge ratios based on the conditional variance and covariance. ARCH and GARCH models allow for the conditional variance of a series to be dependent upon lagged squared error residuals and its own lagged values. In this way the models should afford better predictions of changes in the basis by internalising the temporal variability of the covariance matrix of spot and futures price changes and by allowing shocks to volatility to persist. In general GARCH models are preferable to ARCH models since they are more parsimonious and avoid overfitting leading consequently to a smaller probability to breach the non-negativity constraints that characterize the conditional variance equation. 26

27 Unfortunately, there are several futures of the GARCH methodology that may make it a less than ideal technique for estimating time-varying hedge ratios. From a practical standpoint, the adoption of a GARCH model requires the use of nonlinear optimization algorithms and possibly the imposition of inequality restrictions on model parameters. Moreover, the methodology can be quite taxing and the only evidence that supports the effort for this kind of model comes from simulation that compare the risk reduction benefits of an estimated non-constant hedge ratio to those of an estimated constant hedge ratio. The model tested herein is a multivariate GARCH with error correction that takes account for the ARCH effects in the residuals of the error correction model. This kind of model has already been employed in Brooks et al. (2002). The conditional mean equation of the model is represented by the following Vector Error Correction model: S t = c 0 + γ 0 E t-1 + ε st (4.12a) F t = c 1 + γ 1 E t-1 + ε ft (4.12b) Under the assumption ε t Ω t ~(0, H t ), and by defining h t as vech(h t ), where vech(.) denotes the vector-half operator that stacks the lower triangular elements of a N N matrix into an [N(N 1)/2] 1 vector, the bivariate VECM (k) GARCH (1,1) may be written as: that can be expanded as: h vec(h T ) = h t = h h ss, t sf, t ff, t = c0 + A 1 vec(ε s,t-1; ε f,t-1 )+B 1 h t-1 (4.13) h h h ss, t sf, t ff, t c = c c ss, t sf, t ff, t a + a a a a a a a a ε s, t 1 ε s, t 1, ε f 2 ε f, t 1, t 1 b + b b b b b b b b h h h ss, t 1 sf, t 1 ff, t 1 where h ss and h ff represent the conditional variance of the errors (ε s,t, ε f,t ) from the mean equations, while h sf represents the conditional covariance between spot and futures prices. However there are 21 parameters in the conditional variance-covariance structure of the above bivariate GARCH (1,1) vech model to be estimated, subject to the constraint that H t be 27