股指期货研究中外权威论文-14.pdf

Joumat of Business Finance Junkus and Lee 1985 Peters 1986 Malliaris and Urrutia 1991 and Lindahl 1992 In particular atten tion has focused on the impact of hedge duration and time to expiration on hedging effectiveness and the stability of optimal hedge ratios through time All of the studies cited use the simple OLS approach for estimating hedge ra tios Other estimation techniques have also been used to construct hedge ra tios For example Ghosh 1993 proposes and uses an error correction model ECM arguing that the standard approach to estimating hedge ratios is mis specified because it ignores lagged values while Cecchetti Cumby and Fig lewski 1988 use an Autoregressive Conditional Heteroskedasticity ARCH model to estimate hedge ratios within a maximum expected utility framework In the UK stock index futures were introduced in May 1984 with the ad vent of futures trading on the FTSE 100 index This contract has proved to be extremely popular with investors For example for the September 1992 con tract daily volume reached a peak of over 28 000 contracts and open interest a peak of 44 000 contracts The mean value of each contract was approximately 60 000 However in spite of the value of trading in the FTSE 100 contract and the length of time since this contract was introduced the hedging effec tiveness of the FTSE 100 contract has not been examined previously This is particularly surprising given the central role of hedging within futures mar kets In this paper an attempt to fill this gap in the literature is offered by exam ining the hedging effectiveness of the FTSE 100 stock index futures contract for the period from July 1984 to June 1992 The paper focuses on four areas of particular interest the determination of the appropriate econometric techni que for estimating a hedge ratio which minimises the variance of returns how hedge ratios and hedging effectiveness vary with the duration of a hedge the impact of time to expiration of the contract on hedge ratios and hedging effec tiveness and the stability of hedge ratios through time The question of the appropriate econometric technique to use is addressed by estimating mini mum variance hedge ratios using the standard OLS technique an error cor rection model ECM and the Generalised ARCH GARCH approach The issue of which technique to employ has important implications for practi tioners wishing to employ this contract for hedging Unlike previous studies the fourth issue is examined both by estimating annual hedge ratios and by testing the stationarity of hedge ratios using the tests of unit root suggested by Dickey and Fuller The question of the stability of hedge ratios is particularly important given that hedgers are likely to use hedge ratios estimated in one period to hedge positions in the coming period The paper proceeds as follows The next section briefly discusses theoretical issues The third section sets out Blackwell Publishers Ltd 1996 HEDGE RATIO ESTTIMATION AND HEDGING EFFECTIVENESS 65 the data used and outlines the methodology adopted Results are presented in the fourth section which is followed by a summary and concluding remarks THEORETICAL ISSUES In considering the use of futures contracts to hedge an established spot position the investor must decide on the hedge ratio h to be employed The hedge ratio is the ratio of the number of units traded in the futures market to the number of units traded in the spot market The particular hedging strategy adopted will depend crucially on the investor s objectives In this paper we will make the standard assumption that the objective of hedging is to reduce risk Thus the essentially speculative view of hedging proposed by Working 1953 will not be examined Three hedging strategies will be considered the classic one to one hedge the beta hedge strategy and the minimum variance hedge strategy proposed by Johnson 1960 The classic hedge strategy emphasises the potential for futures contracts to be used to reduce risk The strategy is very simple involving the hedger in tak ing up a futures position which is equal in magnitude but opposite in sign to the spot market position i e a hedge ratio of 1 If proportionate price changes in the spot market exactly match those in the futures market then price risk will be eliminated However in practice there is unlikely to be per fect correlation between spot and futures returns and hence the hedge ratio which minimises the variance of returns will differ from 1 see Figlewski 1984 This arises for two reasons firstly the underlying portfolio of assets which is to be hedged may not exactly mirror the index on which the futures contract is written and secondly basis risk exists Hence two alternative hedge ratios have been proposed The beta hedge has the same objective as the classic 1 1 hedge namely es tablishing a futures position which is equal in size but opposite in sign to the spot position However with the beta hedge it is recognised that the stock portfolio to be hedged may differ from that underlying the futures contract to be used to hedge Lindahl points out that in this case the number of futures contracts for full hedge coverage needs to be adjusted by the portfolio s beta a statistic that describes the portfolio s tendency to rise or fall in value compared to the market Lindahl 1992 p 35 In many cases the portfolio to be hedged will be a subset of the portfolio under lying the futures contract and hence the beta will deviate from unity and the beta hedge ratio from 1 On the other hand if the portfolio to be hedged is that underlying the futures contract the beta hedge ratio will be the same as the classic hedge ratio The second alternative hedge ratio attempts to deal with the problem of ba sis risk Figlewski 1984 identifies two sources of such risk Firstly futures con e Blackwcll Publishers Ltd 1996 66 HOLMES tracts do not yield dividend income whereas the spot position will However Figlewski demonstrates that dividend income is not a major cause of basis risk in practice Secondly and more importantly while the activities of arbitra geurs will ensure that the futures price equals the spot price at expiration at other times spot and futures prices may diverge beyond the cost of carry rela tionship For there to be no such divergence perfect arbitrage would be neces sary meaning that arbitrageurs must buy or sell all stocks in the index whenever the futures price deviates from its theoretical level Transactions costs will prevent this from occurring hence there will be a range in which futures prices can deviate from their theoretical levels Given the lack of perfect correlation the value of A which minimises risk will differ from 1 even when the portfolio to be hedged mirrors that underlying the futures contract Johnson 1960 proposed the minimum variance hedge ratio mvhr as an alternative to the classic hedge The strategy based on the mvhr is consistent with the traditional approach in that it emphasises the risk reduction properties of futures However unlike the traditional approach it does not make naive assumptions about movements in the basis Rather John son defines hedging as minimising the price risk associated with holding a pre determined spot position The mvhr h is shown by Johnson to be h Cov R Rf YB x Rf 1 where R is the return on the spot position and Rjis the return on the futures position The negative sign reflects the fact that to hedge a long stock position it is necessary to sell futures contracts In practice h is computed by regressing the returns on the spot position against the returns on the futures contract using historical data The estimated slope coefficient is multiplied by 1 to obtain the hedge ratio When this is done the coefficient of determination I is an appropriate measure of hedging effectiveness see Johnson 1960 By comparing the minimum variance hedge ratio with unity a direct com parison is made between h and the classic hedge Such a comparison is made by Figlewski 1984 and 1985 He demonstrates that for the USA the mvhr is superior to the classic or beta hedge ratio in terms of risk reduction The above discussion raises two important points Firstly given that hed ging with futures replaces price risk with basis risk an important point to con sider is the effectiveness of a futures market in reducing risk for a particular asset Secondly in practice the minimum variance hedge ratio will depart from unity This is due in large part to changes in the basis Given that such changes depend largely on the behaviour of arbitrageurs and that this beha viour may change over time the minimum variance hedge ratio may also change over time Important empirical points to consider therefore relate to the value of h at a particular time for a particular asset and whether h is stable over time In addition two factors have been identified as being of potential influence Blackwell Publishera Ltd 1996 HEDGE RATIO ESTTIMATION AND HEDGING EFFECTIVENESS 67 on the value of the minimum variance hedge ratio namely the duration of the hedge and the length of time to expiration of the futures contract Both of these are considered here The extent to which the spot and futures markets are related at any particu lar time will depend crucially upon the extent to which prices deviate from their equilibrium relationship before arbitrageurs are enticed to enter the market to earn excess returns Due to the activities of arbitrageurs discrepan cies between spot and futures prices cannot become arbitrarily large How ever the variance of returns will increase with the length of time considered and thus the fraction of total risk accounted for by basis risk will decrease as the holding period of any hedge increases Thus hedging effectiveness should in crease as the duration of the hedge increases Similarly it is to be expected that the duration of a hedge will affect the value of A with longer duration hedges being associated with higher absolute values of i This occurs because as the duration of the hedge increases and basis risk falls as a proportion of total risk the correlation coefficient between returns in the spot and futures markets will move closer to unity and thus A is expected to increase towards unity as the duration of the hedge increases In addition the attractiveness of an arbitrage opportunity depends upon the length of time that the position must be held to yield the profit Since the equilibrium relationship between futures and spot prices must exist at expira tion the level of deviations from equilibrium might be expected to fall as con tract expiration approaches Hence as expiration approaches the minimum variance hedge ratio may be expected to approach unity However this as sumes that the volatility of futures prices falls as expiration approaches In practice this may not occur due to there being a fixed cost component in trans actions costs which may lead to the arbitrage window widening as expiration approaches Thus there may be no tendency for the basis to shrink until the moment of expiration The extent to which there is an expiration effect is therefore an empirical issue From the above discussion of hedging it is clear that in assessing the hedging role of stock index futures it is necessary to determine the extent of risk reduc tion which these contracts allow In addition the value of the minimum var iance hedge ratio needs to be determined and the stability ofthis ratio and the effectiveness of hedges in relation to time to contract expiration and hedge duration must also be examined Furthermore in the light of recent studies which have adopted techniques other than OLS to estimate hedge ratios it is important to determine the most appropriate method by which to estimate mvhrs DATA AND METHODOLOGY In this paper the hedging performance of the FTSE 100 futures contract is ex amined using data relating to the period July 1984 to June 1992 and using the Blackwell Publishers Ltd 1996 68 HOLMES methodology set out below The spot portfolio to be hedged is that underlying the FTSE 100 index Thus it is assumed that the portfolio to be hedged moves exactly with movements in the FTSE 100 index and hence the beta hedge is equivalent to the classic hedge Given the widespread use of index funds by portfolio managers this assumption is reasonable In line with previous studies no adjustment is made for dividends Hedge ratios are calculated by regressing the natural log of spot price changes against the natural log of the futures price changes In all estimations the futures contract nearest to expiration is used The data used for both spot and futures relate to closing prices on a weekly basis Hedging effectiveness for hedges of 1 2 and 4 weeks duration is examined The FTSE 100 futures con tract trades in a cycle of March June September December There are thus 416 observations for one week hedges 8 years x 4 quarterly expiration dates X 13 weeks per quarter 192 observations for two week hedges 8x4x6 and 96 observations for four week hedges 8 x 4 x 3 All prices are obtained from Datastream In examining hedging effectiveness minimum variance hedge ratios are es timated and comparisons drawn with the risk return properties of the beta hedge strategy The examination of hedging performance and hedging effec tiveness is undertaken in four stages The first stage of the analysis is concerned with the appropriate econometric technique to use when estimating hedge ratios with the objective of minimis ing the variance of returns In the first instance simple OLS regressions are run for nonoverlapping 1 2 and 4 week hedges The hedges are lifted at between zero and twelve weeks prior to expiration The OLS regression which is esti mated is Et 2 where DSt The one week return on the spot index for one week hedges the two week return on the spot index for two week hedges and the four week return on the spot index for four week hedges DFi the one week return on the futures price for one week hedges the two week return on the futures price for two week hedges and the four week return on the futures price for four week hedges a b regression parameters where b is the minimum variance hedge ratio A and et a residual term Returns are calculated as log P Pj i If the spot and futures price data are found to be integrated of order one 1 with the first differences being 7 0 then there exists an ECM represen tation of equation 2 Hedge ratios can therefore be estimated using an ECM as in equation 2a DSt a bDFt c St i dFt i et 2a Blackwell Publishcra Ltd 1996 HEDGE RATIO ESTIMATION AND HEDGING EFFECTIVENESS 69 where St i Ft i the one period lagged values of the spot and futures prices in logs c d regression parameters c St i dFt i the error correction mechanism and all other terms are as in equation 2 In addition given that there is widespread evidence that asset returns exhi bit heteroskedasticity violating one of the assumptions underlying OLS hedge ratios are estimated using a GARCH 1 1 representation as in equa tion 2b DSt a bDFt d et N 0 G 2b Comparisons are then made of the hedging effectiveness associated with each hedging strategy by examining the means and standard deviations of returns for hedged portfolios based on the mvhr estimations by the simple OLS the ECM and the GARCH techniques the beta hedge portfolio and also the un hedged portfolios for each of the different hedge durations This allows deter mination of the most appropriate technique for estimating mvhrs The results from the estimation of equation 2 also allow the issue of hedge duration to be examined This constitutes the second stage of the analysis By comparing the mvhrs from equation 2 and the s for hedges of different durations the impact of hedge duration on hedging effectiveness and the size of the hedge ratios can be determined The issue of the stability of the minimum variance hedge ratios with respect to time to contract expiration constitutes the third stage of the analysis This issue is examined by means of multiple regression using dummy variables to represent different subsets of the data based on weeks to expiration Equations are estimated separately for hedge durations of 1 2 and 4 weeks and are of the form a 6o DFt Do bi DFf Di bi DFf D2 bn DFt Dn et where a bi regression parameters where the b s are minimum variance hedge ratios for hedges with i weeks to expiration and Di dummy variables relating to one for hedges with i weeks to ex piration and zero otherwise For hedge durations of 1 week i 0 1 11 2 weeks j 0 2 10 4 weeks i 0 4 8 Comparison of the estimated A s are made to examine the impact of time to expiration on mvhrs Blackwell Publishers Ltd 1996 70 HOLMES Since changes in the behaviour of arbitrageurs over time may lead to the mvhr changing independent of changes in hedge duration and time to con tract expiration it is of interest to examine the stability of hedge ratios further This is done in the fourth stage of the analysis Two different approaches are used to analyse this issue Firstly rather than estimate hedge ratios for the whole period optimal hedge ratios are estimated on a yearly basis Thus mvhrs are estimated for each year from July to June for the period 1984 to 1992 Comparisons are then made of the annual hedge ratios and hedging ef fectiveness in each year Such analysis is only undertaken for hedge durations of 1 and 2 weeks due to small sample sizes for hedges of 4 weeks duration Secondly mvhrs are estimated using a moving window or rolling regres sion procedure and tests are carried out to determine whether the generated hedge ratio series are stationary For this analysis equation 2 is estimated using the first j observations and then subsequently re estimated for every group of J consecutive observations by dropping the first observation and add ing the next Different values dfj the size of the window are used to determine whether this influences the results For 1 week hedges windows of 4 8 and 13 weeks are used The size of