applied financial economics the hedging effectiveness of stock index fixtures (holmes & 2001)

Applied Financial Economics, 2001, 11, 57 ± 68 The hedging eVectiveness of stock index ® xtures: evidence for the FTSE-100 and FTSE-mid250 indexes traded in the UK D A R R E N B U T T E R W O R T H and P H I L H O L M E S * { Financial Services Team, London Economics Ltd, 66 Chiltern St, London W1M 1PR, UK and {Department of Economics and Finance, University of Durham, 23/26 Old Elvet, Durham, DH1 3HY, UK E-mail: darren.butterworth @londonecon.co.u k and P.R.Holmes@ durham.ac.uk This study provides the ® rst investigation of the hedging eŒ ectiveness of the FTSEMid250 stock index futures contract In contrast to previous studies, the portfolios to be hedged are actual diversi® ed portfolios in the form of investment trust companies (ITCs) Furthermore, in addition to using the well established hedging strategies, consideration is also given to hedge ratios estimated on the basis of the Least Trimmed Squares approach Despite relatively thin trading, the FTSE-Mid250 contract is shown to be an important additional hedging instrument Surprisingly, the new contract is more eŒ ective for hedging ITCs than is the established FTSE-100 contract The study also demonstrates that previous studies overstate the hedging eŒ ectiveness of UK stock index futures, in that they assume the portfolio to be hedged is one which underlies a broad market index I INTRODUCTION Futures contracts play an important practical role by expanding the investor opportunity set through the introduction of negative correlation not typically found in cash markets The existence of stock index futures contracts allows investors to avoid market risk, not easily avoided using cash assets alone, due to short selling restrictions Stock index futures were introduced in the UK in May 1984 when trading began in the FTSE-100 contract on the London International Financial Futures Exchange (LIFFE) This provided investors with the means to hedge the risk associated with a broadly diversi® ed stock portfolio However, since the contract relates to an index comprising the 100 highest market capitalization ® rms, it is questionable whether it is suitable for hedging risk associated with portfolios of smaller companies stocks To expand the opportunity set further and allow the hedging of the risk of such stocks, futures trading on the FTSE- Mid250 (hereafter, Mid250) index began in February 1994 To date trading has been thin compared to that in the FTSE-100 contract For example, open interest in the Mid250 contract is frequently only about 5% of that in the FTSE-100, while the value of each Mid250 contract typically has been less than 50% of that of the FTSE-100 Trading volume in the new contract suggests that while it oŒ new risk reduction opportunitie s in principle, its pracers tical signi® cance is limited in that it provides little extra hedging opportunit y compared to the FTSE-100 contract This study investigates whether the new contract is eŒ ective for hedging a range of stock portfolios and compares its hedging performance with that of the FTSE-100 contract In addition to providing the ® rst test of hedging eŒ ectiveness of the new contract, the study oŒ improveers ments on previous studies of stock index futures hedging In particular, a wide range of cash portfolios is used for assessing hedging performance, by using not only portfolios which mirror indexes underlying futures contracts, but *Corresponding author Applied Financial Economics ISSN 0960± 3107 print/ISSN 1466± 4305 online # 2001 Taylor & Francis Ltd http://www.tandf.co.uk/journals 57 58 also a spread of investment trust companies (ITCs).1 Examining hedging eŒ ectiveness over such a range of cash portfolios makes it possible to determine whether the new contract adds substantially to investors’ opportunity sets by markedly enhancing hedging performance Previous studies of stock index futures hedging have examined performance for cash portfolios which underlie broad market indexes or portfolios constructed speci® cally for the analysis By utilizing ITCs, the contracts’ performance is assessed for hedging actual diversi® ed portfolios, rather than portfolios constructed by the researcher Further-more, in addition to using the well established hedging strategies, consideration is also given to hedge ratios estimated on the basis of the Least Trimmed Squares approach Further features are that the ® rst analysis of daily hedging using UK futures is provided and, unlike previous studies, consideration is given to using a combination of contracts (the Mid250 and the FTSE-100) to hedge The rest of the paper is organized as follows The next section brie¯ y reviews previous empirical studies to identify research issues requiring further investigation A section setting out the data and method used follows and the results are then presented A ® nal section provides a summary and conclusions II PREVIOUS EMPIRICAL STUDIES AND RESEARCH ISSUES The ® rst analysis of hedging eŒ ectiveness of stock index futures was by Figlewski (1984) and considerable work followed Researchers have concentrated on three hedge strategies: the traditional one-to-one hedge; the beta hedge; and the minimum variance hedge, proposed by Johnson (1960).2 With all three strategies there is a need to determine the hedge ratio, h, which measures the ratio of the number of units traded in the futures market to the number of units traded in the cash market The traditional strategy involves hedgers adopting a futures position equal in magnitude but opposite in sign to the cash position, i.e h ˆ ¡1 Implicit in such a strategy is the view that futures and cash prices move closely together Indeed, if proportionate price changes in one market exactly match those in the other market, then price risk is eliminated The beta hedge strategy is very similar, but recognizes that the cash portfolio to be hedged may not match the portfolio under1 D Butterworth and P Holmes lying the futures contract With the beta hedge strategy, h is calculated as the negative of the beta of the cash portfolio Thus, for example, if the cash portfolio beta is 1.5, the hedge ratio will be ¡1.5, since the cash portfolio is expected to move by 1.5 times the movement in the futures contract Where the cash portfolio is that which underlies the futures contract, the traditional strategy and the beta strategy yield the same value for h In practice, price changes in the two markets not move exactly together and, therefore, the traditional or beta hedge will not minimize risk The minimum variance hedge ratio (MVHR) takes account of this imperfect correlation and identi® es the hedge ratio which minimizes risk (as measured by variance) as h* in Equation 1: h* ˆ ¡ Cov…Rs ; Rf † Var…Rf † …1† Again, the negative sign re¯ ects that to hedge a long stock position requires selling futures Using the MVHR as the basis for hedging implicitly assumes investors are in® nitely risk averse, i.e they will forgo an in® nite amount of expected return in exchange for an in® nitely small risk reduction While such an assumption about the risk-return trade-oŒ is unrealistic, the MVHR provides an unambiguous benchmark against which to assess hedging performance The majority of research on stock index futures hedging relates to the USA, although more recently work has been published in relation to the FTSE-100 contract This section aims to outline the main themes of previous research, and identify shortcomings and areas where further development is required Figlewski (1984) examined hedging eŒ ectiveness for the S&P 500 futures contract in relation to portfolios underlying ® ve major stock indexes for the period June 1982 to September 1983 While all ® ve indexes represented diversi® ed portfolios, two included only the largest capitalization stocks, two included smaller companies and one contained only 30 stocks of very large ® rms Figlewski included dividend payments in the return series, but found that their inclusion did not alter the results Consequently, and given the relatively stable and predictable nature of dividends, subsequent studies have excluded dividends Figlewski showed that ex post MVHRs can be estimated by OLS using historical data He found that for all indexes hedge performance was less good using the beta hedge ratio than when the MVHR was used With large capitalization portfolios, risk was reduced An ITC is a company formed for the purpose of holding investments It is a closed end fund which raises capital by issuing shares and uses that capital to buy shares in other companies The ITC is managed full-time by a specialist In the context of the USA, an ITC is similar to a mutual fund Other researchers associated with this approach include Stein (1961) and Ederington (1979) While some studies have incorporated expected returns into hedging decisions and developed risk-return measures of hedging eŒ ectiveness (see, for example, Howard and D’Antoniou (1984, 1987) and Chang and Shanker (1987)), such models suŒ from the same er shortcoming in that they require a subjective assessment to be made in relation to investor preferences Hedging eVectiveness for cross hedges by 70% ± 80% using the MVHR For smaller stocks portfolios, hedging eŒ ectiveness was considerably reduced Hedging performance was less good for overnight hedges than for one week and four week hedges Figlewski (1985) investigated hedging performance for three stock index futures for holding periods ranging from one day to three weeks As with his earlier analysis, eŒ ectiveness improved as the duration of the hedge increased from days to weeks Once again, portfolios of small stocks were hedged less eŒ ectively than were those comprising large stocks Junkus and Lee (1985) tested hedging eŒ ectiveness of three USA stock index futures exchanges using four commodity hedging models They found the MVHR was most eŒ ective at reducing the risk of a cash portfolio comprising the index underlying the futures contract Peters (1986) also con® rmed the superiority of the MVHR over the beta hedge Graham and Jennings (1987) were ® rst to examine hedging eŒ ectiveness for cash portfolios not matching an index Random sampling techniques were used to form 90 equity portfolios each comprising ten stocks Interestingly this study found that stock index futures were less than half as eŒ ective at hedging non-index portfolios as they were at hedging cash indexes Finally for the USA, Lindahl (1992) examined hedge duration and hedge expiration eŒ ects for the MMI and S&P 500 futures contracts Lindahl’s results suggested that both hedge ratios and hedging eŒ ectiveness increase as hedge duration increases However, there was no obvious pattern in terms of risk reduction in relation to time to expiration Hedging eŒ ectiveness for the UK was ® rst examined by Holmes (1995) , for the FTSE-100 contract Ex ante MVHRs for the period 1984-199 were used and the cash portfolio hedged was that underlying the futures contract His results showed that even using ex ante hedge ratios, the contract enabled risk reduction of more than 80% Holmes (1996) investigated ex post hedging eŒ ectiveness for the same contract and the same cash portfolio as in the earlier paper and showed that standard OLS provided MVHR estimates superior to those estimated by GARCH or using an error correction method His results suggested eŒ ectiveness increased with hedge duration, in line with Figlewski and Lindahl for the USA, but that there was no strong discernible pattern in expiration eŒ ects The impact of portfolio composition on systematic risk and hedging eŒ ectiveness was examined by Holmes and Amey (1995) They constructed portfolios of UK stocks and considered the FTSE-100 contract As the number of stocks in portfolios increased from 1, through 5, 10, 15 and 20, to 25 hedging eŒ ectiveness increased markedly While previous studies suggested the FTSE-100 contract removed approximately 80% of cash portfolio risk when the portfolio was the underlying index, risk reduction was only about 60% for portfolios comprising 25 stocks 59 There are a number of points to draw from the studies considered First, the MVHR provides superior hedging performance in terms of risk reduction Second, a duration eŒ is evident, with longer hedges more eŒ ect ective In contrast, there is no strong evidence of expiration eŒ ects Third, the nature of the portfolio hedged is an important determinant of hedging performance For example, Figlewski (1984) found hedging eŒ ectiveness was less for portfolios comprising small stocks, Graham and Jennings (1987) found the hedging of portfolios comprising only ten stocks was much less eŒ ective than for portfolios matching an index and Holmes and Amey (1995) found similar results for the UK While the composition of the cash portfolio is clearly important, previous studies have failed to address true hedging eŒ ectiveness by examining performance for actual stock portfolios Portfolios used for examining hedging eŒ ectiveness have been either market indexes or constructed by the researcher In addition, to date no analysis has been undertaken of the eŒ ectiveness of UK stock index futures when hedging small capitalization stocks Furthermore, in relation to the UK, no consideration has been given to performance over very short durations It is also worth noting that as yet no attention has been given to the hedging eŒ ectiveness of the Mid250 contract and to whether this contract provides market participants with another important means by which to hedge stock portfolio risk Finally, consideration needs to be given to the way in which hedge ratios are estimated In particular, while the OLS estimation of the MVHR has many desirable characteristics it is associated with the unattractive property of being sensitive to outliers Therefore, in order to allow for this problem and take account of the fact that futures prices are often characterized by kurtosis, it may be desirable to generate hedge ratios using an approach which minimizes the impact of outliers One such approach is the Least Trimmed Squares (LTS) method employed by Knez and Ready (1997) The LTS approach trims a proportion of the most extreme observations and then ® ts the remaining observations using ordinary least squares Thus the LTS coe cient represents the value that minimizes the sum of the squared residuals where the sum is taken over all the observation s which are not trimmed If the MVHR is considerably diŒ erent to the hedge ratio generated using LTS, then this would strongly indicate that the MVHR is being driven by a small sample of extreme observations and raises concerns over the possible biasedness of the MVHR This would have important implications when ex ante hedge ratios are determined on the basis of estimations using historical data This study addresses shortcomings of previous work in a number of important ways: The ® rst assessment of hedging eŒ ectiveness of the Mid250 contract is presented In addition, compari- 60 D Butterworth and P Holmes sons are made between the performance of this contract and that of the FTSE-100 contract for a number of diŒ erent portfolios Given that one aim of the introduction of the new contract is to enable more eŒ ective hedging of small capitalization stocks, this is clearly important In addition to assessing hedging performance for cash portfolios mirroring broad indexes, cross hedging performance is analysed by examining the hedging of actual cash portfolios held by professional managers in the form of ITCs Since returns on ITCs represent the returns on professionally managed, well diversi® ed, portfolios, evaluation of hedging eŒ ectiveness in relation to these portfolios provides new insights into the capabilities for hedging actual portfolios Consideration is given here not only to the hedging eŒ ectiveness of the FTSE-100 and Mid250 when used separately, but also to their use in combination Not only are the well established hedging strategies used, but consideration is also given to an alternative method for generating hedge ratios by using the LTS approach By comparing hedge ratios estimated by OLS with those determined using LTS it should be possible to determine the importance of outliers on the estimated hedge ratios The ® rst investigation of hedging eŒ ectiveness of stock index futures in the UK over short periods is provided, by examining daily hedges I I I D A T A A N D M ET H O D Hedging performance is examined for the FTSE-100 and Mid250 index futures contracts traded on LIFFE by using cash and futures return data for February 1994 (date of introduction of the Mid250 contract) to December 1996 The FTSE-100 represents the 100 largest companies traded on the London Stock Exchange The Mid250 represents the next 250 largest companies (i.e numbers 101 to 350 by market capitalization) Both indexes are weighted by market capitalization Thirty-six cash portfolios comprising four indexes and 32 investment trusts are used The four indexes are the FTSE-100, the Mid250, the FTSE-350 (comprising the largest 350 companies) and the FT Investment Trust (FTIT) index The ITCs were chosen to provide a range of portfolios which diŒ substantially in their composition Seven er categories of ITCs are used:4 (1) General funds: at least 80% of the assets are in UK registered companies; (2) Capital Growth funds: at least 80% of the assets are in UK registered companies, with stocks chosen to accentuate capital growth; (3) Income Growth funds: at least 80% of their assets are in UK equities whose policy is to accentuate income growth; (4) High Income funds: at least 80% of assets are in equities and convertibles; the aim is to achieve a yield in excess of 125% of that of the FT Actuaries All-Share Index; (5) Smaller Company (SC) funds: at least 50% of assets are in smaller and medium sized companies; (6) Venture and Development Capital (VDC) funds: a signi® cant portion of the trusts’ portfolio is invested in securities of unquoted companies; and (7) Property funds: at least 80% of the assets of these funds are in listed property equities For each of the ® rst six categories, returns on ® ve ITCs are used to analyse hedge eŒ ectiveness In the case of Property funds, only two ITCs were used due to a lack of appropriate funds with su ciently long returns series To alleviate any problems arising from thin trading only funds with a market capitalization in excess of £20 million at the beginning of the period under investigation are included The funds provide a broad range of portfolios, which diŒ er in terms of objectives and composition In particular, the SC funds and the VDC funds represent investments in relatively low capitalization stock Hedging such funds is expected to be less eŒ ective with the FTSE-100 contract, given the composition of the underlying index It is therefore of interest to determine if this is the case and whether the Mid250 contract adds markedly to hedging performance for such portfolios Analysis is carried out for two diŒ erent hedge durations: daily and weekly Hedge durations of longer than a week are not considered due to problems of sample size After removing non-trading days the daily series consists of 715 observations, the weekly series 148 observations The returns series for each cash portfolio and each futures contract is calculated as the logarithmic price change: Rt ˆ log Pt Pt¡1 ´ …2† where, Rt is the daily or weekly return on either the cash or futures position and Pt is the price at time t Price is the daily or weekly closing price All data were obtained from Datastream Four hedging strategies are considered First, the traditional hedge is examined Second, the MVHR, as shown in Equation 1, is used Figlewski (1984) showed the MVHR can be estimated by regressing cash returns on futures The de® nitions of these categories of ITCs are taken from the Association of Investment Trust Companies’ monthly report 61 Hedging eVectiveness for cross hedges returns using historical information, with h* the negative of the slope coe cient, b, in the following equation: RSt ˆ a ‡ bRF t ‡ et …3† where RSt is the return on the cash portfolio in time period t; RFt is the return on the futures contract in time period t; et is an error term and a, b are regression parameters, where ¡b is the MVHR, h* Third, the LTS hedge ratio is investigated In order to generate the LTS hedge ratio the residual series from the estimated equation (et in Equation 3) is collected Both the cash and futures returns are then ranked in relation to the absolute size of their associated residual term The ® rst observation in both the cash and futures return series are associated with the smallest residual in absolute size and the ® nal observation in both the cash and futures returns series is associated with the largest residual in absolute size In view of the ® ndings of Knez and Ready (1997) and the number of observations in our daily and weekly samples we adopt a trimming coe cient of 10% This trims away the 10% of cash and futures returns which are associated with the largest residuals, measured in absolute size This produces a trimmed daily sample of 643 observations and a trimmed weekly sample of 132 observations Having trimmed away the extreme outliers from both samples, the LTS hedge ratios are then estimated by employing OLS to the remaining 90% of observations Finally, given that cross hedges are being considered, the beta hedge is used The beta hedge ratio is calculated as the negative of ­ in the following equation:6 RSt ˆ ¬ ‡ ­ RINDt ‡ "t …4† where RINDt is the return on the index underlying the futures contract; "t is an error term and all other terms are as previously de® ned Consideration is given to mean and standard deviation of returns of the unhedged and the hedged positions In addition, the degree of risk reduction will be determined as: Risk reduction ˆ ¼u ¡ ¼h £ 100 ¼u …5† where ¼u is the standard deviation of returns on the unhedged (i.e cash) position; ¼h is the standard deviation of returns on the hedged position The eŒ ectiveness of the four strategies is investigated using the FTSE-100 and the Mid250 contracts individually In addition, for the MVHR and LTS strategies composite hedges are examined, where the two futures contracts are combined into a `synthetic’ FTSE-350 contract Returns on the synthetic futures are the weighted average of returns on the FTSE-100 and Mid250 contracts, with the weights attached to the two contracts varying from : ¡1 to ¡1 : Weights always sum to and change at intervals of 0.25 Thus, 13 composite hedges are considered for each of the thirty-six cash portfolios.7 I V EM P I R I C A L R E S U L T S Stock market indexes Empirical analysis begins by investigating whether the new contract adds markedly to the ability to hedge broad based cash portfolios Therefore, the reduction in risk achieved by the FTSE-100 and Mid250 futures when the cash portfolio is an index is examined The four indexes described above are considered Results using traditional and beta hedge strategies for daily and weekly hedge durations are presented in Tables and respectively For each table, panel A shows the mean and standard deviation of returns for cash portfolios;8 panel B shows results when hedging with the FTSE-100 contract; and panel C shows results when using the Mid250 contract In panels B and C the hedge ratio, mean and standard deviation of returns and percentage reduction in the standard deviation from the unhedged position are shown In relation to daily data, Table 1, panel A shows that the four cash portfolios diŒ considerably in terms of their er risk-return pro® les over the sample period For example, the FTSE-100 index gave an annual mean return of 7.7% , with a standard deviation of returns of 10.9% , compared to ® gures for the Mid250 of 4.5% and 7.0% respectively In terms of hedging, the traditional hedge is very eŒ ective when the cash portfolio is that which underlies the contract under consideration, as expected For example, panel B shows that hedging the FTSE-100 cash index with the FTSE-100 contract achieves risk reduction of over 64% , while using the Mid250 contract to hedge the Mid250 index achieves risk reduction of 45.2% (see panel C) These Knez and Ready generate separate regressions using ordinary least squares and then LTS for various trimming coe cients within the range of 5% to 50% of the sample They show that LTS slopes are similar using either 50% or 95% of the data This implies that the tendency for extreme observations to be in¯ uential is explained by a small percentage of the observations We therefore choose a 10% trimming coe cient In the remainder of the paper the hedge ratio will be referred to as a positive number for convenience, even though in practice hedging an established spot position is likely to require selling futures By creating various weighted `synthetic’ FTSE 350 contracts, the panel approach provides a detailed picture of the impact on hedging eŒ ectiveness arising from changes in the contribution made by the Mid 250 contract to the composition of the `synthetic’ hedge All mean and standard deviation ® gures reported in the tables and the text have been annualized to allow more convenient comparison between hedges of diŒ erent durations 62 D Butterworth and P Holmes Table The hedging eVectiveness of the FTSE-100 and FTSEMid 250 contracts: daily data Hedge ratio Cash portfolio (A) Unhedged FTSE 100 FTSE 250 FTSE 350 FTIT (B) Hedging with the FTSE100 contract Traditional hedge FTSE 100 FTSE 250 FTSE 350 FTIT Beta hedge FTSE 100 FTSE 250 FTSE 350 FTIT (C) Hedging with the FTSE-Mid 250 contract Traditional hedge FTSE 100 FTSE 250 FTSE 350 FTIT Beta hedge FTSE 100 FTSE 250 FTSE 350 FTIT Mean return 7.669 4.464 6.937 1.691 S.D of returns Table The hedging eVectiveness of the FTSE-100 and FTSEMid 250 contracts: weekly data Decrease in S.D.* Hedge ratio Mean return S.D of returns 7.687 4.474 6.953 1.695 11.169 9.276 10.469 9.027 1.000 1.000 1.000 1.000 ¡0.020 ¡0.466 ¡0.122 ¡0.851 2.500 7.398 3.266 7.222 77.613 20.254 68.800 19.991 1.000 0.695 0.932 0.678 ¡0.020 ¡0.134 ¡0.048 ¡0.501 2.500 5.606 2.752 5.270 77.613 39.566 73.711 41.621 1.000 1.000 1.000 1.000 0.458 0.012 0.356 ¡0.373 5.717 2.610 4.523 4.789 48.811 71.868 56.794 46.952 1.008 1.000 1.006 0.843 0.453 0.012 0.352 ¡0.278 5.720 2.610 4.527 4.424 48.800 71.868 56.755 50.994 Cash portfolio (A) Unhedged FTSE 100 FTSE 250 FTSE 350 FTIT 10.912 6.999 9.737 8.138 1.000 1.000 1.000 1.000 ¡0.146 ¡3.351 ¡0.878 ¡6.124 3.924 9.296 4.805 9.112 64.038 ¡32.816 50.651 ¡11.971 1.000 0.503 0.888 0.586 ¡0.146 0.533 ¡0.003 ¡2.889 3.924 5.005 3.755 5.884 64.038 28.491 61.436 27.689 1.000 1.000 1.000 1.000 3.293 0.087 2.560 ¡2.686 6.938 3.836 5.681 5.506 36.422 45.192 41.651 32.337 1.223 1.000 1.172 0.961 2.317 0.087 1.808 ¡2.515 7.064 3.836 5.875 5.410 35.264 45.192 39.657 33.520 (B) Hedging with the FTSE 100 contract Traditional hedge FTSE 100 FTSE 250 FTSE 350 FTIT Beta hedge FTSE 100 FTSE 250 FTSE 350 FTIT (C) Hedging with the FTSE-Mid 250 contract Traditional hedge FTSE 100 FTSE 250 FTSE 350 FTIT Beta hedge FTSE 100 FTSE 250 FTSE 350 FTIT Decrease in S.D.* Note: *This measures the percentage of the standard deviation of returns of the unhedged portfolio that is removed by hedging Note: *As Table results indicate the new contract is not as eŒ ective at hedging its underlying index as is the more established contract, using the naive strategy The new contract is also less eŒ ective at hedging the FTSE-350 index Given the composition of the FTSE-350 index, these results are not surprising Results for other cross hedges are of more interest First, panel B shows that the FTSE-100 contract was not eŒ ective at hedging either the Mid250 or the FTIT indexes using the traditional hedge For both hedges the standard deviation of returns is higher and mean returns lower for the hedged position than for the unhedged position In contrast, the Mid250 contract oŒ an eŒ ers ective means by which to cross-hedge Table 1, panel C demonstrate s that using this contract for a traditional hedge, when the cash portfolio is the FTSE-100, achieves risk reduction of over 36% Similarly, risk reduction in relation to the FTIT cash portfolio is about one third Now consider the beta hedge The traditional and beta hedges are identical when the cash portfolio is that under- lying the contract For cross-hedging, the FTSE-100 contract is superior when hedging the FTSE-350 (risk reduction of 61.4% for the FTSE-100 contract, compared to 39.7% for the Mid250 contract) However the Mid250 contract again is superior for cross-hedging other indexes The FTSE-100 contract achieves risk reduction of below 29% when the cash portfolio is the Mid250 or the FTIT In contrast, for the FTSE-100 and FTIT cash portfolios, the Mid250 index achieves risk reduction in excess of one third with the beta strategy The results suggest that for daily hedging the new contract provides an important additional hedging vehicle for some broadly diversi® ed portfolios Table shows results for traditional and beta weekly hedges Results are very similar to those for daily data, although the new contract’s value is more marked When the cash portfolio is that underlying the contract, risk reduction is substantial with both contracts (over 70% ), as in previous studies for the FTSE-100 (see Holmes, 1995, 1996) Thus, hedging eŒ ectiveness improves as 63 Hedging eVectiveness for cross hedges Table Hedging eVectiveness using the MVHR and LTSHR strategies: daily data Hedge ratio Mean return S.D of returns 7.669 4.464 6.937 1.691 10.911 6.998 9.736 8.137 0.803 0.391 0.710 0.451 1.392 1.411 1.390 ¡1.831 2.960 4.784 2.944 5.612 72.875 31.638 69.765 31.034 0.810 0.361 0.712 0.415 1.341 1.646 1.372 ¡1.550 2.961 4.800 2.944 5.632 72.864 31.409 69.764 30.793 1.050 0.766 0.985 0.779 3.069 1.111 2.630 ¡1.711 6.926 3.343 5.682 5.209 36.513 52.239 41.648 35.987 1.099 0.784 1.028 0.735 2.856 1.031 2.445 ¡1.520 6.937 3.346 5.692 5.221 36.411 52.192 41.545 35.842 1.245 1.111 1.211 ¡2.201 2.882 3.343 2.609 4.892 73.589 52.239 73.205 39.887 1.220 1.031 1.184 ¡1.971 2.882 3.346 2.609 4.907 73.587 52.192 73.201 39.701 Cash portfolio (A) Unhedged FTSE 100 FTSE 250 FTSE 350 FTIT (B) Hedging with the FTSE 100 contract MVHR FTSE 100 FTSE 250 FTSE 350 FTIT LTSHR FTSE 100 FTSE 250 FTSE 350 FTIT (C) Hedging with the FTSE Mid 250 contract MVHR FTSE 100 FTSE 250 FTSE 350 FTIT LTSHR FTSE 100 FTSE 250 FTSE 350 FTIT (D) Composite hedges** MVHR FTSE 100 0.924 FTSE 250 0.766 FTSE 350 0.823 FTIT 0.743 LTSHR FTSE 100 0.927 FTSE 250 0.784 FTSE 350 0.827 FTIT 0.699 Decrease in S.D.* Note: *As Table **Results are reported for the optimal combination of FTSE-100 and FTSE-Mid 250 contract in terms of maximum risk reduction The optimal combinations are 0.75 : 0.25, 0:1, 0.75 : 0.25 and 0.25 : 0.75 respectively hedge duration rises For cross hedges, the new contract again achieves superior risk reduction for the FTIT (47% compared to 20% for the FTSE-100 contract) Tables and report daily and weekly results respectively for the mean and standard deviation of returns using the MVHR and LTS hedge ratio (LTSHR) Results relate to the same cash portfolios as in Tables and For convenience panel A again shows details of unhedged positions Panels B and C show results for hedging with the FTSE-100 and Mid250 contract respectively Panel D reports results for the `synthetic’ FTSE-350 contract In all cases the results for the MVHR are reported ® rst, followed by the results for the LTSHR Results are also reported for the optimal combination of the two contracts.9 The optimal combinations of the FTSE-100 and Mid250 contracts for daily data for the four cash portfolios are 0.75 : 0.25 (FTSE-100), : (Mid250), 0.75 : 0.25 (FTSE-350) and 0.25 : 0.75 (FTIT) Thus, for example, in constructing a synthetic futures which minimizes the return variance when hedging the FTIT portfolio, the optimal mix involves a weighting of 0.25 in the FTSE100 contract and 0.75 in the new contract First, the MVHR results for daily and weekly hedges are discussed and then these are compared with the LTSHR results In relation to the MVHR, Table 3, panel B shows the FTSE-100 contract greatly reduces risk for the FTSE-100 (73% ) and FTSE-350 (70% ) cash portfolios for daily hedges For the other portfolios risk reduction of only about 30% is achieved The Mid250 contract is less successful at reducing risk for the FTSE-100 and FTSE-350 cash portfolios, as expected, given that the FTSE-100 dominates these indexes by market capitalization However, for the other portfolios the new contract is superior for hedging Risk reduction of 52% and 36% is achieved for the Mid250 and FTIT cash portfolios Thus, for portfolios with smaller capitalization the new contract is a signi® cant additional hedging facility Results in relation to the construction of a synthetic futures are very interesting (see panel D) In all cases, the optimal combination involves some use of the new contract, while for the Mid250 cash portfolio the FTSE-100 contract should not be used Thus, even for the FTSE-100 cash portfolio, the introduction of the new contract adds to hedging eŒ ectiveness.10 In Table 4, once again hedging performance improves as hedge duration rises to a week However, the main results are unchanged: for the FTSE-100 and FTSE-350 portfolios, the FTSE-100 contract provides higher risk reduction than the Mid250 contract, with the new contract superior for other cash portfolios Optimal combinations for the synthetic contract are the same as for daily data Thus, All other combinations identi® ed in the previous section were used to identify the optimal mix The results for the other combinations are available from the authors on request 10 This ® nding can, in part, be explained by the fact that the composition of the two cash indexes is revised on a regular basis re¯ ecting changes in market capitalization When changes are made, some stocks move out of the FTSE-100 into the Mid250 and others make the move in the opposite direction 64 D Butterworth and P Holmes Table Hedging eVectiveness using the MVHR and LTSHR strategies: daily data Hedge ratio Mean return S.D of returns 7.687 4.474 6.953 1.695 11.169 9.276 10.469 9.027 0.882 0.601 0.819 0.595 0.778 ¡0.233 0.538 ¡2.966 2.026 5.483 2.367 5.166 81.860 40.895 77.390 42.765 0.881 0.575 0.808 0.563 0.784 ¡0.033 0.622 ¡2.718 2.026 5.492 2.371 5.181 81.859 40.795 77.352 42.603 0.985 0.918 0.970 0.809 3.366 0.447 2.698 ¡1.854 5.715 2.483 4.514 4.411 48.827 73.228 56.885 51.134 0.984 0.928 0.995 0.764 3.369 0.402 2.587 ¡1.655 5.715 2.486 4.521 4.433 48.827 73.205 56.820 50.894 1.001 0.447 0.678 ¡2.477 1.975 2.483 1.794 4.173 82.317 73.228 82.863 53.775 1.017 0.402 0.705 ¡2.464 2.013 2.486 1.795 4.173 81.975 73.205 82.858 53.774 Cash portfolio (A) Unhedged FTSE 100 FTSE 250 FTSE 350 FTIT (B) Hedging with the FTSE 100 contract MVHR FTSE 100 FTSE 250 FTSE 350 FTIT LTSHR FTSE 100 FTSE 250 FTSE 350 FTIT (C) Hedging with the FTSE Mid 250 contract MVHR FTSE 100 FTSE 250 FTSE 350 FTIT LTSHR FTSE 100 FTSE 250 FTSE 350 FTIT (D) Composite hedges** MVHR FTSE 100 0.959 FTSE 250 0.918 FTSE 350 0.900 FTIT 0.795 LTSHR FTSE 100 0.957 FTSE 250 0.928 FTSE 350 0.896 FTIT 0.792 Decrease in S.D.* Note: *As Table **Results are reported for the optimal combination of FTSE-100 and FTSE-Mid 250 contract in terms of maximum risk reduction The optimal combinations are 0.75:0.25, 0:1, 0.75:0.25 and 0.25:0.75 respectively the new contract improves hedging eŒ ectiveness even when the cash portfolio is that underlying the FTSE-100 contract 11 Tables and also provide an opportunity to compare the daily and weekly hedging results for the MVHR and the LTSHR when the cash portfolios consist of stock market indexes It is clear that when the cash portfolios are broad based market indexes trimming the sample to remove the largest 10% of outliers tends to result in very small changes to the size of the optimal hedge ratio and levels of risk reduction For instance, in the case of the hedge between the FTSE 100 contract and the FTSE 100 index (Table 3, panel B), the hedge ratio changes from 0.803 to 0.810 and the level of risk reduction falls from 72.875% to 72.864% For the cross hedges involving the FTSE 100 contract, the diŒ erences between the MVHR and LTSHR remain small with the level of risk reduction being achieved by the LTSHR being within 0.3% of the MVHR In the case of the Mid250 contract (Table 3, panel C), the diŒ erence between the MVHRs and the LTSHRs are of a similar magnitude to those involving the FTSE 100 contract with the diŒ erences in the size of the hedge ratios being less than 0.05 and diŒ erence in the levels of risk reduction being less than 0.2% When hedges of weekly duration are considered (Table 4), the MVHRs and the LTSHRs are extremely similar, with the LTSHRs approaching those of the MVHR strategy For the hedge between the FTSE 100 contract and FTSE 100 index, the MVHR and LTSHR are 0.882 and 0.881, and levels of risk reduction are 81.860% and 81.859% respectively Similarly, in the case of the hedge between the Mid250 contract and the Mid250 index, the MVHR and LTSHR are 0.918 and 0.928, and levels of risk reduction are 73.228% and 73.0% respectively Hence it is clear that when the cash and futures series are highly correlated removing the largest 10% of outliers makes little impact on hedging performance, demonstrating that the hedge ratios estimated by OLS are indeed robust Investment trust companies The hedging eŒ ectiveness of the two contracts when the cash portfolios are ITCs is now examined Given the superiority of the MVHR and LTSHR strategies, only those strategies are considered Rather than report results for each of the thirty-two portfolios, average results for each category of ITCs11 are reported Tables and report results for ITCs for daily and weekly hedges respectively The format of the tables is similar to Tables and However, in addition to showing average risk reduction for each category of ITC, maximum and minimum standard deviations for each category are also shown Panel A in both tables demonstrates that the cash portfolios vary substantially in terms of mean and standard Results in relation to traditional and beta hedge strategies and those relating to individual investment trust companies are available on request 65 Hedging eVectiveness for cross hedges Table Hedging investment trusts portfolios using the MVHR and LTSHR strategies: daily data Standard deviation of returns Average hedge ratio Cash portfolio (A) Unhedged portfolio General Capital growth Income growth High income Small company Venture/development Property (B) Hedging with the FTSE 100 contract MVHR General Capital growth Income growth High income Small company Venture/development Property LTSHR General Capital growth Income growth High income Small company Venture/development Property (C) Hedging with the FTSE Mid 250 contract MVHR General Capital growth Income growth High income Small company Venture/development Property LTSHR General Capital growth Income growth High income Small company Venture/development Property (D) Composite hedgesy MVHR General Capital growth Income growth High income Small company Venture/development Property LTSHR General Capital growth Income growth High income Small company Venture/development Property Average mean return Minimum Maximum Average Decrease* 5.905 3.362 0.553 ¡5.079 1.880 11.535 ¡7.579 8.651 8.260 7.664 8.927 6.743 6.707 13.496 13.974 12.525 11.924 13.896 12.409 9.602 13.883 10.675 9.555 9.878 10.681 9.742 8.111 13.690 0.459 0.237 0.343 0.278 0.233 0.159 0.186 2.319 1.512 ¡2.131 ¡7.252 0.060 10.293 ¡9.036 7.721 8.014 7.251 8.532 6.641 6.610 13.019 11.291 11.684 10.804 13.037 11.037 9.345 13.821 8.767 8.969 8.719 10.037 9.202 7.786 13.420 17.369 5.911 11.118 5.963 5.050 3.777 1.995 0.405 0.184 0.285 0.206 0.167 0.117 0.166 2.739 1.925 ¡1.677 ¡6.686 0.573 10.618 ¡8.876 7.784 8.024 7.273 8.553 6.670 6.623 13.033 11.331 11.759 10.823 13.063 11.054 9.363 13.821 8.799 9.004 8.758 10.086 9.245 7.806 13.427 17.051 5.563 10.700 5.493 4.587 3.539 1.940 0.832 0.497 0.658 0.571 0.478 0.332 0.490 2.263 1.191 ¡2.326 ¡7.570 ¡0.208 10.082 ¡9.719 7.231 7.467 7.059 8.334 6.447 6.472 12.660 10.811 11.338 10.536 12.465 10.817 9.079 13.512 8.299 8.631 8.252 9.638 8.924 7.615 13.086 21.945 9.551 15.781 9.638 7.968 5.815 4.435 0.783 0.403 0.551 0.421 0.344 0.237 0.427 2.478 1.603 ¡1.859 ¡6.911 0.377 10.495 ¡9.443 7.267 7.556 7.105 8.401 6.499 6.502 12.672 10.832 11.364 10.556 12.500 10.865 9.171 13.520 8.316 8.669 8.312 9.718 8.990 7.655 13.096 21.763 9.123 15.099 8.853 7.275 5.334 4.363 0.804 0.490 0.625 0.562 0.473 0.328 0.507 2.048 1.115 ¡2.573 ¡7.633 ¡0.270 10.057 ¡9.427 7.216 7.465 7.060 8.334 6.429 6.469 12.660 10.544 11.334 10.458 12.465 10.689 9.075 13.412 8.142 8.610 8.205 9.636 8.896 7.599 13.036 23.277 9.746 16.210 9.660 8.228 6.006 4.796 0.708 0.404 0.506 0.405 0.337 0.231 0.410 2.493 1.497 ¡2.010 ¡6.913 0.327 10.477 ¡9.112 7.225 7.554 7.106 8.401 6.493 6.505 12.671 10.583 11.351 10.471 12.501 10.731 9.167 13.454 8.215 8.645 8.283 9.729 8.967 7.641 13.063 22.539 9.344 15.322 8.753 7.467 5.498 4.603 Notes: * The decrease in the S.D of returns relates to a comparison of the average S.D of returns for the hedged position with that of the unhedged position y The optimal combinations of the FTSE-100 and FTSE-Mid 250 contract range from 0.25:0.75 to ¡0.5 to 1.5 For no portfolio did the weight given to the FTSE-Mid 250 contract fall below 0.75 66 D Butterworth and P Holmes Table Hedging investment trusts portfolios using the MVHR and LTSHR strategies: weekly data Standard deviation of returns Average hedge ratio Cash portfolio (A) Unhedged portfolio General Capital growth Income growth High income Small company Venture/development Property (B) Hedging with the FTSE 100 contract MVHR General Capital growth Income growth High income Small company Venture/development Property LTSHR General Capital growth Income growth High income Smallcompany Venture/development Property (C) Hedging with the FTSE Mid 250 contract MVHR General Capital growth Income growth High income Small company Venture/development Property LTSHR General Capital growth Income growth High income Smallcompany Venture/development Property (D) Composite hedgesy MVHR General Capital growth Income growth High income Small company Venture/development Property LTSHR General Capital growth Income growth High income Small company Venture/development Property Average mean return Minimum Maximum Average Decrease* 5.919 3.370 0.554 ¡5.091 1.885 11.561 ¡7.596 10.870 10.403 9.992 10.764 8.890 7.961 16.393 13.604 13.546 14.124 15.406 14.456 10.857 17.599 11.816 11.568 11.894 12.785 11.990 9.347 16.996 0.623 0.381 0.532 0.486 0.435 0.259 0.509 1.037 0.385 ¡3.616 ¡8.901 ¡1.521 9.531 ¡11.583 6.785 9.290 8.228 9.583 8.591 7.485 15.585 9.865 12.004 11.014 13.660 12.633 9.930 15.766 8.784 10.471 9.746 11.226 10.640 8.684 15.676 25.429 9.343 17.882 12.262 10.837 6.794 7.632 0.594 0.329 0.487 0.449 0.367 0.225 0.357 1.266 0.793 ¡3.263 ¡8.605 ¡0.993 9.796 ¡10.394 6.800 9.290 8.276 9.629 8.628 7.515 15.699 9.869 12.099 11.042 13.662 12.671 9.991 15.879 8.803 10.503 9.775 11.247 10.683 8.706 15.789 25.255 9.086 17.630 12.086 10.486 6.571 6.966 0.864 0.586 0.785 0.705 0.672 0.391 0.776 2.127 0.798 ¡2.887 ¡8.182 ¡1.063 9.846 ¡11.000 6.920 8.754 7.852 9.215 8.060 7.290 15.153 9.960 11.762 10.074 13.227 12.205 9.725 15.190 8.243 9.941 9.052 10.746 10.009 8.486 15.171 30.279 13.910 23.639 16.031 16.263 8.953 10.618 0.792 0.470 0.664 0.576 0.534 0.308 0.523 2.446 1.309 ¡2.357 ¡7.616 ¡0.458 10.210 ¡9.889 6.954 8.853 7.934 9.288 8.184 7.362 15.290 9.960 11.909 10.233 13.243 12.276 9.771 15.463 8.291 10.022 9.150 10.831 10.111 8.529 15.376 29.857 13.231 22.823 15.325 15.392 8.462 9.397 0.853 0.588 0.779 0.693 0.673 0.385 0.773 1.556 0.867 ¡3.165 ¡8.359 ¡1.059 10.022 ¡11.091 6.268 8.596 7.750 9.215 7.902 7.290 15.110 9.617 11.706 10.055 13.205 12.174 9.708 15.136 8.036 9.868 8.960 10.735 9.971 8.556 15.123 31.974 14.525 24.431 16.122 16.662 8.282 10.905 0.829 0.520 0.694 0.662 0.607 0.310 0.610 1.667 1.103 ¡2.796 ¡8.198 ¡0.806 10.326 ¡10.481 6.276 8.684 7.757 9.216 8.002 7.381 15.116 9.622 11.715 10.102 13.217 12.180 9.744 15.379 8.045 9.907 9.021 10.747 10.010 8.599 15.248 31.890 14.191 23.913 16.027 16.290 7.779 10.144 Note: * The decrease in the S.D of returns relates to a comparison the average S.D of returns for the hedged position with that of the unhedged position y The optimal combinations of the FTSE-100 and FTSE-Mid 250 contract range from 0.5:0.5 to ¡0.75 to 1.75 For no portfolio did the weight given to the FTSE-Mid 250 contract fall below 0.5 67 Hedging eVectiveness for cross hedges deviations of returns Thus, the 32 portfolios under consideration cover a broad range of cash portfolios, providing an opportunity for a thorough assessment of true hedge eŒ ectiveness As can be seen in Table 5, panels B and C, in all cases average standard deviation of returns is lower when a MVHR strategy using the Mid250 contract is used compared to using the FTSE-100 contract The same is true for weekly hedges (see Table 6), providing strong evidence in support of the usefulness of the Mid250 contract for hedging actual cash portfolios However, while the results show the superiority of the Mid250 contract for hedging these cash portfolios, risk reduction is far less than when cash portfolios are broad market indexes For example, for daily hedges, in no case does average risk reduction exceed 22% for the ITC portfolios and even when using the Mid250 contract, average reduction is below 10% for ® ve categories This compares with risk reduction for broad market indexes of 36% ± 52% when the Mid250 contract is used and up to 73% for the FTSE-100 contract The results for SC funds and VDC funds are particularly weak, suggesting that while the new contract relates to an index covering smaller companies, it is still not suitable for hedging portfolios comprising very low value stocks Results for weekly hedges are markedly better In only one case is average risk reduction below 10% when using the Mid250 contract (three when using the FTSE-100 contract) and average risk reduction of almost one third is achieved for the General funds using the new contract Results for the optimal synthetic futures show minor improvement over results for the Mid250 contract In all cases the optimal combination involves using the Mid250 contract, again supporting the view that the new contract has an important role to play, particularly in relation to hedging portfolios which are not broadly diversi® ed The results in Tables and allow us to compare the hedging eŒ ectiveness of the MVHR and LTSHR when the cash portfolios are ITCs It is apparent that when the cash portfolios are ITCs, removing the largest 10% of outliers has a more signi® cant eŒ on the optimal hedge ratio ect than in the case where the cash portfolios are stock market indexes For both the FTSE 100 and Mid250 contracts (Tables and 6, panels B and C), trimming away the 10% most extreme observations alters the average size of the hedge ratio by 6± 28% for daily hedges and by 5± 33% for weekly hedges In all but ® ve cases the diŒ erence is 10% or greater Generally, the percentage variations in the estimated hedge ratios are higher for daily hedges than weekly hedges 12 However, while the MVHRs are noticeably larger than the LTSHRs, the levels of risk reduction achieved by 12 both strategies remain similar For all ITC categories, the LTSHR achieves levels of risk reduction which are within 0.7% of the MVHR when the FTSE 100 contract is used, and within 1.3% of the MVHR when the Mid 250 contract is used However, the results for ITCs suggest that outliers impact on estimated hedge ratios Given that there is no reason to expect outliers in one period to be repeated in a subsequent period, it is important to give consideration to the way in which hedge ratios are estimated for portfolios which not mirror a broad market index In particular, when ex ante hedge ratios are to be determined on the basis of estimations using historical data, it may be desirable to consider using the LTS approach.13 V SUMMARY AND CONCLUSIONS This paper provides the ® rst assessment of hedging performance of the Mid250 futures contract Given the low level of trading in this contract hedging eŒ ectiveness may be limited The paper also provides the ® rst examination of hedging eŒ ectiveness of stock index futures when the cash portfolio to be hedged is an actual portfolio, rather than a broad market index or a portfolio speci® cally constructed for the purposes of research Thus, it provides the ® rst true assessment of hedging eŒ ectiveness in practice Results demonstrate that in spite of low trading volume, the Mid250 contract provides an important additional hedging instrument The ® ndings in relation to hedging broad market indexes show the superiority of the new contract over the FTSE-100 contract in relation to cash portfolios mirroring the Mid250 and the FTIT indexes When considering actual cash portfolios in the form of ITCs, the results clearly demonstrate the bene® ts to be gained from using the new contract In all cases, the average standard deviation of returns is lower when the Mid250 contract is used as compared to the use of the FTSE-100 contract Results also show that previous studies of hedging eŒ ectiveness have greatly exaggerated the risk reduction which can be achieved While previous studies for the UK have found risk reduction of 60% to 80% , this study shows that for many portfolios, including those comprising smaller stocks, risk reduction of below 20% is achieved Thus, while the new contract does signi® cantly add to the ability to hedge risk, for many portfolios there is still no satisfactory means by which to achieve substantial risk reduction Finally, the ® ndings in the paper indicate that when the cash indexes are broad stock market indexes the MVHR is a robust estimator and that hedge eŒ ectiveness is not strongly aŒ ected by the presence of outliers However, The exceptions to this are the results for the property funds for both contracts and the general fund for the Mid250 contract The results presented here suggest that an examination of ex ante hedging eŒ ectiveness when using MVHRs and LTSHRs is worthy of consideration Such an investigation is beyond the scope of this current paper 13 68 when ITCs are considered, outliers impact noticeably on estimated hedge ratios and as a result consideration should be given to using the LTS method of estimation A C K N O W LE D G E M E N TS The authors gratefully acknowledge the helpful comments of Professor Denis O’ Brien, Professor Ron Smith of Birbeck College, Jonathan Rougier, Alberto Carparni of London Economics and an anonymous referee from Applied Financial Economics They are also grateful to Austin McCarthy for help with the data collection The usual disclaimer applies R E F E R E N C ES Chang, J S K and Shanker, L (1987) A risk-return measure of hedging eŒ ectiveness, Journal of Financial and Quantitative Analysis, 22, 372± Ederington, L H (1979) The hedging performance of the new futures markets, Journal of Finance, 34, 157± 70 Figlewski, S (1984) Hedging performance and basis risk in stock index futures, Journal of Finance, 39, 657± 669 Figlewski, S (1985) Hedging with stock index futures: theory and application in a new market, Journal of Futures Markets, 5, 183± 99 Graham, D and Jennings, R (1987) Systematic risk, dividend yield and the hedging performance of stock index futures, Journal of Futures Markets, 7, 1± 13 D Butterworth and P Holmes Holmes, P (1995) Ex ante hedge ratios and the hedging eŒ ectiveness of the FTSE-100 stock index futures contract, Applied Economics Letters, 2, 56± Holmes, P (1996) Stock index futures hedging: hedge ratio estimation, duration eŒ ects, expiration eŒ and hedge ratio stabiects lity, Journal of Business Finance and Accounting, 23, 63± 78 Holmes, P and Amey, M (1995) Portfolio composition, diversi® able risk and the hedging eŒ ectiveness of the FTSE-100 stock index futures contract, University of Durham Working Paper, No 149 Howard, C T and D’Antoniou, L J (1984) A risk-return measure of hedging eŒ ectiveness, Journal of Financial and Quantitative Analysis, 19, 101± 12 Howard, C T and D’Antoniou, L J (1987) A risk-return measure of hedging eŒ ectiveness: a reply, Journal of Financial and Quantitative Analysis, 22, 377-381 Johnson, L (1960) The theory of hedging and speculation in commodity futures, Review of Economic Studies, 27, 139± 51 Junkus, J C and Lee, C F (1985) Use of three stock index futures in hedging decisions, Journal of Futures Markets, 5, 201± 22 Knez, P and Ready, M (1997) On the robustness of size and book-to-market in cross-sectional regressions, Journal of Finance, 52, 1355± 82 Lindahl, M (1992) Minimum variance hedge ratios for stock index futures: duration and expiration eŒ ects, Journal of Futures Markets, 12, 33± 53 Peters, E (1986) Hedged equity portfolios: components of risk and return, in Advances in Futures and Options Research, JAI Press, 1, part B, pp 75± 91 Stein, J L (1961) The simultaneous determination of spot and futures prices, American Economic Review, 51, 1012± 25  ... underlying index as is the more established contract, using the naive strategy The new contract is also less eŒ ective at hedging the FTSE-350 index Given the composition of the FTSE-350 index, these... Notes: * The decrease in the S.D of returns relates to a comparison of the average S.D of returns for the hedged position with that of the unhedged position y The optimal combinations of the FTSE-100... by the LTSHR being within 0.3% of the MVHR In the case of the Mid250 contract (Table 3, panel C), the diŒ erence between the MVHRs and the LTSHRs are of a similar magnitude to those involving the