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1、THE JOURNAL OF FINANCE VOL.XXXIX,NO.3 JULY 1984Hedging Performance and Basis Risk in StockIndex FuturesSTEPHEN FIGLEWSKI*IN EARLY 1982,TRADING BEGAN at three different exchanges in futures contractsbased on stock indexes.Stock index futures were an immediate success,andquickly led to a proliferation
2、 of new futures and options markets tied to variousindexes.One reason for this success was that index futures greatly extended therange of investment and risk management strategies available to investors byoffering them,for the first time,the possibility of unbundling the market andnonmarket compone
3、nts of risk and return in their portfolios.Many portfoliomanagement and other hedging applications in investment banking and securitytrading have been described elsewhere ranging from use by a passive fundmanager to reduce risk over a long time horizon to use by an underwriter tohedge the market ris
4、k exposure in a stock offering for one or two days.In considering the potential applications of index futures,it is clear that innearly every case a cross-hedge is involved.That is,the stock position that isbeing hedged is different from the underlying portfolio for the index contract.This means tha
5、t return and risk for an index futures hedge will depend upon thebehavior of the basis,i.e.,the difference between the futures price and the cashprice.Hedging a position in stock will necessarily expose it to some measure ofbasis riskrisk that the change in the futures price over time will not track
6、exactly the value of the cash position.Basis risk can arise from a number of different sources,and is a more significantproblem for stock index contracts than for other financial futures,like Treasurybills and bonds.The most apparent cause of basis risk is the nonmarketcomponent of return on the cas
7、h stock position.Since the index contract is tiedto the behavior of an underlying stock market index,nonmarket risk cannot behedged.This is the essential problem of a cross-hedge.However,basis risk canbe present even when the hedge involves a position in the index portfolio itselfand there is no non
8、market risk.For one thing,returns to the index portfolioinclude dividends,while the index,and the index future,only track the capitalvalue of the portfolio.Any risk associated with dividends on the portfolio willbecome basis risk in a hedged position.Still,dividends are fairly low and alsoquite stab
9、le,so this may not be a terribly important shortcoming.Much more important than dividend risk is the fact that the futures price isnot directly tied to the underlying index,except for the final settlement price on*New York University.I would like to thank the Interactive Data Corporation for providi
10、ng dataand computer support for this project and my colleagues in the NYU Finance Department for helpfulcomments and suggestions.Thanks also to Steven Freund for able research assistance.See,for example,Figlewski and Kon 1982.In fact,in the case of the Value Line futures contract traded on the Kansa
11、s City Board of Trade,there is no stock portfolio which exactly duplicates the index.See Elton,Gniber,and Rentzler 1983 or Hoag and Labarge 1983.657658 The Journal of Financethe expiration date.Day to day fluctuations in the difference between theminduce fluctuations in the returns on a hedged posit
12、ion.The magnitude of thisrisk,which is in addition to nonmarket and dividend risk,is limited by thepossibility of arbitrage between cash and futures markets.In markets wheretransactions costs are small and arbitrage is straightforward,as in Treasury bills,basis risk may be negligible.But for stock i
13、ndex futures,a perfect arbitrageappears to be infeasible.The Standard and Poors 500 index,for instance,contains 500 stocks in precise proportions.It is impossible to assemble such aportfolio in a reasonable size and to buy or sell all of the stocks simultaneouslyin order to capitalize on short run d
14、eviations of the index futures price from itstheoretical level,especially if it is necessary to sell them all short.Instead,arbitrage as it is done in this market is essentially risk-arbitrage.A tradingportfolio consisting of a small subset(perhaps fifty or so)of the stocks in theindex is selected a
15、nd traded against the futures contract when discrepanciesbecome too large.Because the trade is not risk free and there are sizabletransactions costs,the range within which the futures price can move fairly freelywithout inducing arbitrage trading is broad enough to allow substantial basisrisk.This p
16、aper examines the basis and the different sources of basis risk on theStandard and Poors 500 index contract.We develop a number of results aboutthe use and usefulness of stock index futures in hedging and about the behaviorof this new market as it has evolved.The next section presents a simplifiedth
17、eory of hedging in the presence of basis risk and displays the risk-returncombinations that could have been achieved in practice by hedging severalbroadly diversified stock portfolios with S&P 500 futures.Section 3 discusses thesources of basis risk in a hedge involving the Standard and Poors portfo
18、lio itself.We consider the effects of dividend risk,the length of the holding period,andthe time remaining to expiration of the futures contract.In Section 4,we examinethe movement in the basis over time,which determines the return to a hedgedportfolio.We begin with a discussion of the equilibrium f
19、utures price,based onarbitrage with the cash market,and then examine empirically how well the theorydescribes the level and dynamics of actual futures prices.We find a cleardistinction between the behavior of the market in its early months and currently.The final section summarizes the results and s
20、uggests some implications aboutcontract design for stock index futures.II.Hedging with Stock Index FuturesIndividual stocks and all stock portfolios,except for those specifically designedto have zero beta,are exposed to some market risk.In this section we will firstdiscuss in theoretical terms how a
21、 single futures contract based on a broad marketindex can be used to hedge market risk due to price fluctuation.We will thenexamine the returns and risk on actual hedged portfolios.*Other strategies,like placing newly available funds selectively into stocks or a combination ofstock index futures and
22、 Treasury bills according to the relative pricing of the two are also employed,especially by institutional investors.We restrict the analysis to strategies involving a constant hedge ratio.The evidence presentedbelow suggests that hedge performance may be improved further by use of a dynamic strateg
23、y.Risk in Stock Index Futures 659Let us begin by defining the random variable returns on the portfolio to behedged,Rp,the spot index,Ri,and the index futures contract,Rp,assuming aholding period of length T.where VQ and VT denote the beginning and ending market values for the portfolio.Dp represents
24、 the cumulative value as of T of the dividends paid out on theportfolio during the period,assuming reinvestment at the riskless rate of interestfrom the date of payout until T.The dividend payout is a random variablebecause the amount,its timing,and the reinvestment rate are all uncertain as oftime
25、0.The return on the index portfolio is,2,where variables are defined analogously to(1).The rate of return on a futures contract is not a well-defined concept,sincetaking a futures position does not require an initial outlay of capital.Forexpository convenience we will define the rate of return on fu
26、tures as the changein the futures price divided by the initial level of the spot index:(3)Expressing this in terms of the basis,i.e.,the futures price minus the spot index:p T-/o+6i Dj FT-IT)-Fp-Ip)ti=+Ip Ip-BQio(4)The rate of return on a stock index futures contract is equal to the total returnon t
27、he underlying index portfolio,minus the dividend yield on the index,plusthe change in the basis over the period as a fraction of the initial index.Now consider the return on a hedged portfolio in which futures contracts onN index shares have been sold short against the long portfolio of stocks.Anind
28、ex share is defined to be an amount of the index portfolio whose market valueis equal to$1 times the spot index.Most currently traded stock index futureshave contract sizes of 500 index shares.-Vp+Dp)-NFT-Fp)/MO /FT-FoVol IpRH=Rp hRp,(5)*The initial margin deposit to open a futures position does not
29、 represent an investment of capitalsince it can be posted in the form of interest bearing Treasury bills.660 The Journal of Financewhere h,the hedge ratio,is the current value of the index shares sold forward asa fraction of the current value of the portfolio being hedged,h determines theoverall ris
30、k and return characteristics of the hedged position,which are given bythe customary portfolio formulas:Rh=Rp-hRf(6)ol=al+haj-2hapF.(7)Bars represent expectations,a with a single subscript denotes a variance and awith two subscripts a covariance.To find the constant hedge ratio which minimizes risk,w
31、e set the derivativeof(7)with respect to h equal to zero and obtainh*=upp/ol(8)This is easily computed in practice by simply running a regression of Rp on RFusing historical data.The slope coefficient in the equation is h*.One might thinkof it as the beta of the portfolio with respect to the futures
32、 contract.Substituting into(7)yields the variance of returns for the minimum risk hedge,riN=4-PIF)(9)where PPF is the correlation coefficient between the returns on the stock portfolioand the futures contract.It is apparent that only with perfect correlation canrisk be completely eliminated by hedgi
33、ng.Looking back to the definition of Rp in eq.(4),we see that the variance offutures returns is infiuenced by three random variables:total returns on themarket index portfolio,dividends on the market,and the change in the basisbetween the future and its underlying index.These will naturally all affe
34、ct therisk minimizing hedge ratio as well.In the special case where dividends are not random and the hedge is to be helduntil the futures contracts expire,so that the change in the basis is alsononstochastic,the effect of these terms disappears,leavingh*=api/a=0p.(10)The risk minimizing hedge ratio
35、in this special case is the portfolios betacoefficient with respect to the market index.Earlier discussion of hedging withindex futures suggested using the portfolios beta as the appropriate hedge ratio.While dividends tend to be relatively stable,the same is not true of the basis,which is quite vol
36、atile over short periods.Hence using beta as the hedge ratio isunlikely to be optimal,except when the position is to be held until maturity ofthe futures.To examine how effective stock index futures hedges would have been inreality,we have calculated the risk and return combinations which could have
37、been achieved by selling Standard and Poors 500 futures against the underlyingportfolios of five major stock indexes over one week holding periods.The sampleperiod was from June 1,1982 through September 30,1983.The indexes were theStandard and Poors 500 index itself,the New York Stock Exchange compo
38、site,the American Stock Exchange composite,the National Association of SecuritiesRisk in Stock Index Futures 661Dealers Automated Quotation System(NASDAQ)index of over-the-counterstocks,and the Dow Jones Industrials index.These are all diversified portfolios-meaning nonmarket risk is substantially s
39、maller than for individual stocksbutthey are different in character from one another.The first two are market value weighted portfolios containing,respectively,500 and about 1500 of the largest capitalization stocks.Either is a good proxyfor the market portfolio of financial theory and should contai
40、n very littlenonmarket risk.The AMEX and OTC indexes are also value weighted and welldiversified,within their segments of the market,but both contain stocks ofsmaller companies which move somewhat independently of the S&P index.Finally,the Dow Jones portfolio contains only 30 stocks of very large fi
41、rms,weighted by their market prices.But despite its different composition,the Dowportfolio is significantly more closely correlated with the S&P than are theAMEX or OTC.The first,and most difficult,step was to construct series of weekly returns,including dividends,on the five portfolios.Dividends ar
42、e a problem because thepayout pattern is very lumpy within a quarter.This might impart potentiallyimportant,and unhedgeable,variation in returns over short holding periods.Dividends were treated differently for each portfolio.For the S&P portfolio itself we constructed a dividend inclusive series fr
43、omdata on the actual dividends paid on all 500 stocks over the sample period.Thisseries is analyzed below when we look at the importance of systematic dividendrisk.Dividends for the NYSE portfolio were assumed to follow the same timepattern as on the S&P,and were scaled to give the same dividend yie
44、ld.TheAMEX index includes dividends,treating them as if they were reinvested in theportfolio as they accrued,so no dividend adjustment was necessary.Without dataon the payout pattern for the OTC index,we were forced to assume dividendswere paid continuously over the sample period at an annual rate e
45、qual to 87percent of the rate on the NYSE index,the historical value for the relativepayouts on these two indexes.Finally,for the Dow we used a series which tookthe actual payout on the index portfolio and smoothed it evenly over the quarter.Thus,dividend risk has been eliminated from the OTC and Do
46、w portfolios,butdividend yield remains.The futures contract was in all cases the Standard and Poors 500 futurenearest to expiration,assuming a rollover to the next contract at expiration.Weconfined the analysis to the near contract because preliminary research showedthat there was not very much diff
47、erence between the hedging properties of thenearest and the second contract.Also nearly all trading volume is in the nearmonth so that liquidity is much greater in that contract.Returns on futures werecomputed as in equation(3).Finally,since in theory the return to a fully hedged portfolio(or more g
48、enerally,one containing only unsystematic risk)should be equal to the riskless interestrate,for comparison we constructed a series of holding period yields on a risklessasset.For this purpose we chose three month bank CDs,assuming they were See Ibbotsen and Fall 1979.662 The Journal of Financeheld f
49、or one week and sold in the secondary market at a price that yielded themarket interest rate at that time.Table 1 shows the results of one week hedges taking both the risk minimizingh*from eq.(8)and the portfolios beta as hedge ratios.The first two columnsshow annualized mean returns and standard de
50、viations on the unhedged portfo-lios.The next three give the hedge ratio,mean return and standard deviation forthe minimum risk hedged position,while the final columns provide the sameinformation for positions with the portfolio beta as hedge ratio.Note that bothh*and beta have been calculated from