Bealieu maturity effect

Ž .Journal of Empirical Finance 5 1998 177–195

Time to maturity in the basis of stock marketindices: Evidence from the S&P 500 and the

MMI

Marie-Claude Beaulieu )

´Departement de finance et assurance and CREFA, UniÕersite LaÕal, Quebec, Canada G1K 7P4´ ´ ´

Abstract

This paper focuses on the behaviour of the basis in stock market index futures contractsover the lifetime of futures contracts. The model in this paper relaxes the cost of carrymodel assumptions of constant interest rate and known dividend yield over the lifetime offutures contracts. This allows for a test of the presence of time to maturity in the conditionalvariance of the model using GARCH. The empirical evidence reveals that, consistent with

Ž .Samuelson’s 1995 analysis, time to maturity is a determinant of the conditional varianceof the basis. Furthermore, it implies that time to maturity cannot be accounted for bytransaction costs or cost of carry. q 1998 Elsevier Science B.V. All rights reserved.

JEL classification: G13

Keywords: Basis; Stock market index; Intertemporal risk; Nonsynchronous trading; Time to maturity;One-step ahead forecasts

1. Introduction

Previous empirical studies of the basis in stock market index futures contractsŽCastelino and Francis, 1982; MacKinlay and Ramaswamy, 1988; Duan and Hung,

) Tel.: q1-418-656-2926; fax: q1-418-656-2624; e-mail: [email protected].

0927-5398r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.Ž .PII S0927-5398 97 00017-0

( )M.-C. BeaulieurJournal of Empirical Finance 5 1998 177–195178

.1991 have suggested the existence of a relation between the remaining time tomaturity of the contracts and the basis which could lead to arbitrage opportunities 1

or to a predictable variance of the basis. The traditional approach to pricing futuresŽ .contracts is the cost of carry model. MacKinlay and Ramaswamy 1988 analyze

the S&P 500 futures prices defining a mispricing series as the difference betweenthe actual futures price and an estimate of those prices constructed with the cost ofcarry model. They find that the absolute value of mispricing depends on the timeto expiration. In this paper, I use a model of the intertemporal change of the basisto gauge the presence of time to maturity in the conditional variance of the basis.

Futures prices are typically collected daily for each contract until its expiration;data collection then continues with a new contract. The fact that futures prices arenot collected at times corresponding to constant maturity explains why remainingtime to maturity can influence the conditional variance of the time series of futuresprices. Assuming that spot prices follow a stationary autoregressive process,

Ž .Samuelson 1965 defines futures prices as the expected spot price at maturity ofthe contract. He shows that the conditional variance of the futures price changesper unit of time increases as time to maturity decreases.

Ž .Castelino and Francis 1982 build on Samuelson’s analysis of futures prices tostudy the effect of time to maturity on the basis. They show that the conditionalvariance of the change in the basis decreases when time to maturity decreases. Ascontract maturity approaches, futures prices evolve into spot prices due to thereduction of interest rate risk. Therefore the arrival of new information is morelikely to affect spot and futures prices in the same manner if it arrives close tomaturity, causing a reduction in the basis variance.

As pointed out by MacKinlay and Ramaswamy, the unanticipated interestearnings arising from financing or reinvesting the marking to market cash flows toand from the futures position may explain why the absolute value of mispricingdefined in terms of the cost of carry model diminishes with time to maturity 2. For

Ž .instance, French 1983 is critical of approaches that ignore marking to marketsince he finds significant differences between futures and forward prices in copperand silver. MacKinlay and Ramaswamy also suggest that transaction costs mayexplain the presence of a maturity effect in their analysis. Indeed, FiglewskiŽ .1984 claims that large transaction costs to acquire the S&P 500 stocks encour-age the use of hedging portfolios that do not incorporate all the component stocksin the index. In that case, the expected number of transactions is greater further

1 Even though the basis in futures contracts shrinks with approaching maturity because the futuresŽ .price must equal the spot price at maturity see Fig. 1 , time to maturity should not be a characteristic

Ž .of the mean of the basis adjusted for the cost of carry if futures and spot prices simultaneously reflectŽ .all available information. In that case, the basis adjusted for the cost of carry today contains all the

relevant information about the expected basis tomorrow.2 Ž .This idea is clear in the characterization of Cox et al. 1981 of futures prices since interest rate

uncertainty decreases as the maturity date gets closer.

( )M.-C. BeaulieurJournal of Empirical Finance 5 1998 177–195 179

away from maturity because the incomplete hedge portfolio will need to beadjusted over time.

The model presented in this paper relaxes the assumption of constant cost ofcarry 3 and allows me to infer whether a stochastic interest rate eliminates thetime to maturity effect in the variance of the basis. Furthermore, I compare results

Ž .from the Standard and Poor’s 500 S&P 500 index basis and the Major MarketŽ .Index MMI basis. This comparison is important because of possible differences

in the extent of nonsynchronous trading in the index. From the observed rate ofŽ .change on the S&P 500 index and the MMI, Stoll and Whaley 1990 and Chan

Ž .1992 find evidence of a higher degree of serial correlation in the return on theS&P 500 index than in the return on the MMI. They interpret this result asevidence that the S&P 500 index is more subject to nonsynchronous trading than

Žthe MMI, since the MMI is a subset of twenty blue-chip stocks more actively. 4traded than those typical of the S&P 500 from the S&P 500 index . A

comparison of the results for the basis in the S&P 500 and in the MMI allows meto gauge whether transaction costs can explain the relevance of time to maturity inthe conditional variance. Suppose hedgers hold an approximate instead of an exactreplica of the index. Then, ex ante, the expected cost of revising the portfolio willbe higher for the S&P 500 because of its greater diversification.

The paper is organized as follows. Section 2 derives the model of theintertemporal change in stock market index. Section 3 presents data sources andrelated descriptive statistics. Section 4 reports empirical results of estimation of themodel of the intertemporal change in the basis. The one-step-ahead forecastingproperties of two different time to maturity specifications in the conditionalvariance of the univariate estimation are also investigated at various horizons.Section 5 concludes.

2. Exposition of the model

The model of the intertemporal change in the basis developed in this paper isbased on the equilibrium valuation of the basis in foreign exchange futures

3 Constant cost of carry refers to two important assumptions limiting the explanatory power of thecost of carry model. First, the interest rate is assumed constant and second the dividend yield on thestock is assumed known over the lifetime of the futures contract.

4 Ž .Kleidon 1992 distinguishes between nonsynchronous trading or nontrading and stale pricing.According to his definitions, nonsynchronous trading is the event where the recorded price of a stock isfor the last trade which occurred previously, while stale pricing occurs when a trade is executed at aprice set by a limit order issued much earlier than the moment it arrives at the market and so does notincorporate current information. With the exception of October 19, 1987, nonsynchronous trading willbe the dominant factor in the analysis and a comparison between the basis in the S&P 500 index andthe MMI should reveal the effects of different degrees of nonsynchronous trading in the two indicesand not of stale pricing.


Ž .contracts, first presented by McCurdy and Morgan 1993 . The intertemporalchange in the basis is defined as the basis today minus the basis yesterday adjustedfor the net cost of carry. It represents the return on a position in the basis betweenty1 and t.

Let F be the price at time t of a futures contract to deliver one unit of thet

index at time T , S be the spot price at time t of one unit of the index, R bet ty1

one plus the U.S. riskless rate of interest from time ty1 to time t, D be thet

dividend, announced at time ty1 and received at time t from one unit of theindex, d be the dividend yield on the index from time ty1 to time t orty1

D rS , M be the nominal benchmark variable at time t and E be thet ty1 t ty1

expectation operator conditional on the information set available at time ty1.To adapt McCurdy and Morgan’s model for stock index data, one has to take

into account the return obtained from a long position in the stock index net of theborrowing cost and the payoff from a short position in futures contracts. Thestrategy is to borrow at ty1 to buy one unit of the index on the spot market,obtaining a capital gain or loss plus the dividend yield d at t. At the same time,ty1

one goes short one futures contract on the index. This strategy requires no netinvestment at ty1. At t, the holder of the unit of index will have S qD sS qt t t

d S and will owe S R on it. At t, to eliminate the short position inty1 ty1 ty1 ty1

futures contracts, the holder will go long one futures contract. The resulting payoffof this strategy is

S qd S yS R y F yF , 1Ž . Ž .t ty1 ty1 ty1 ty1 t ty1

which has a net present value of zero. Using the intertemporal valuation operatorŽ .Richard and Sundaresan, 1981 , M , and the definition of covariance, thet

valuation model can be written as

F yS F F ySt t ty1 t tE y yR qd sycov M R , .ty1 ty1 ty1 ty1 t ty1S S Sty1 ty1 ty1

2Ž .The model of the intertemporal change in the basis leads to better properties of thetime series than the cost of carry model. As described by MacKinlay and

Ž .Ramaswamy 1988 , the time series of the level of the basis is highly seriallycorrelated creating inefficient estimates. To get around that problem, Miller et al.Ž .1994 use first differences in the basis. By eliminating overnight price changes,they can ignore the opportunity cost of holding the basis position from one periodto the next, as well as any systematic risk in holding the basis position. The modelof the intertemporal change in the basis provides a theoretical framework for theintertemporal valuation of the basis that is consistent with cash dividend payments,the interest cost of carrying the index, and the systematic risk in the position.


3. Data and descriptive statistics

3.1. Data

The data used in this paper come principally from two sources: the ChicagoŽ .Mercantile Exchange CME for the S&P 500 and the Chicago Board of Trade

Ž .CBOT for the MMI. The S&P 500 data set contains daily futures and spot pricesfrom September 30, 1985 to December 31, 1991. The S&P 500 index is avalue-weighted index composed of 500 widely held stocks. The index is updatedcontinuously using the most recent prices as reported of the component 500 stocks.The index series contains some prices based on nonsynchronous trading, especiallyfor the thinly traded stocks. The daily futures prices database consists of transac-tion data for the S&P 500 futures contracts 5. Up to March 1987, the futures

Žcontract’s expiration was the third Friday of the delivery month March, June,.September and December . In order to avoid problems due to the triple witching

hour, the CME then changed the last day of trading of the contract to the lastbusiness day prior to the third Friday of the delivery month of the contract.

The MMI database contains daily futures and spot prices over the same timeperiod. It is a price-weighted index for which the price of each stock at each pointin time is adjusted for stock splits and stock dividends. The last day of trading ofthese futures contracts is the third Friday of the delivery month. In the case of theMMI, the futures contract delivery months are the first three consecutive monthsŽ .e.g. October, November and December plus the next three months in the March,June, September and December cycle. In order to make results comparable for theS&P 500 and the MMI indices, I consider the contracts on a three-month to

Ž .maturity cycle only March, June, September and December . Furthermore, be-cause the basis is deterministic on the expiration day of futures contracts, the lastobservation of the basis in the series for each contract is dropped.

The dividends for the stocks in the S&P 500 index are estimated by therealized daily dividend yield of the value-weighted index of all NYSE stocks

Ž .supplied by the Center for Research in Security Prices CRSP . To get thedividend yield, I subtracted VWRET from VWRETD in the CRSP file. Thedividends for the MMI are the actual dividends paid on the 20 component stocksof the index obtained from CBOT. To construct the dividend yield, I divided the

5 Because the model uses a modified lagged endogenous variable, it is important to construct theseries consistently when a contract change occurs. In this event, I have to compare the value of thebasis for the new contract with the lagged value of the basis of the same contract. In other words, whena contract change has occurred, the data for the basis and its lagged value must both refer to the newcontract.


Table 1Ž .Estimated serial correlation coefficients. Observed rates of return of the S&P 500 index R , the MMIs

Ž . Ž .R , and rates of change of the S&P 500 index futures contract R , and the MMI futures contractm sfŽ .Rmf

Lag R R R Rs sf m mf

Ž . Ž . Ž . Ž .r t r r t r r t r r t rk k k k k k k k

1 0.0561 2.24 y0.0146 y0.58 0.0192 0.77 0.0273 1.092 y0.0267 y1.06 y0.0404 y1.62 y0.0471 y1.88 y0.0579 y2.323 y0.1390 y5.56 y0.1810 y7.24 y0.1270 y5.08 y0.1380 y5.524 y0.0327 y1.31 y0.0464 y1.86 y0.0194 0.78 y0.0515 y2.065 y0.0205 y0.82 y0.0046 y0.18 y0.0091 0.36 y0.0057 y0.236 0.0213 0.85 0.0488 1.95 0.0133 0.53 0.0379 1.527 y0.0191 y0.76 0.0210 0.84 y0.0158 y0.63 y0.0015 y0.068 0.0360 1.44 0.0181 0.72 0.0286 1.14 0.0201 0.809 y0.0003 y0.01 y0.0192 y0.77 y0.0209 y0.84 y0.0140 y0.56

10 y0.0179 y0.72 y0.0346 y1.38 y0.0186 y0.74 y0.0241 y0.96

The sample includes data from September 30, 1985 to December 31, 1991. The estimated serialcorrelation coefficients are for the residuals from the regression of the specified rate of return or rate ofchange on a constant and a dummy variable which takes the value of one for Mondays, holidays andtwo other dummy variables that take the value of one for the market crashes of October 19, 1987 and

Žof October 13, 1989 and zero otherwise. The t-ratio corresponds to the null hypothesis of r thek.estimated serial correlation coefficient equals zero, where k is the number of lags.

sum of those dividends by the index price the day before the ex-dividend date andadjusted this ratio with the MMI divisor 6.

The interest rates are the overnight effective Federal funds rate, adjusted forweekends and holidays, so that the rate for a normal weekend is approximatelythree times that of a typical weekday. When any interest rate was missing becauseof holidays, I used the previous day rate. Finally, the benchmark portfolio is the

Ž .daily excess return on the Morgan Stanley Capital International MSCI worldindex in U.S. dollars over the Federal funds rate for a maturity of one day. MSCIcalculates their index from closing values of the 19 country component indices.

3.2. DescriptiÕe statistics

Table 1 shows the autocorrelation function of the rate of change of the spot andfutures prices for the S&P 500 and MMI indices 7. Daily rates of change are not

6 I am thankful to Barry Schachter from the Comptroller of the Currency for making the MMIdividends available to me and to Louis Gagnon from Queen’s University for providing the MMIdivisor.

7 Each index price series is converted into a rate of return and each futures price series is convertedinto a rate of change. Then all rates are prewhitened by a regression on a constant, a dummy variablethat takes the value of one on the first trading day of the week, a dummy for the market crash ofOctober 19, 1987 and a dummy variable for the market crash of October 13, 1989.


Žas highly autocorrelated as returns over five-minute intervals Stoll and Whaley,.1990 but the return on the S&P 500 index is positively and significantly

autocorrelated at its first lag. This is not the case for the return on the MMI; thet-statistic on the first lag is not significantly different from zero at a five percent

Ž .level of significance. This supports Lo and MacKinlay’s 1990 model according

Fig. 1. Basis defined as in Table 2 in both indices over remaining days to maturity of the futurescontract. The sample includes data from September 30, 1985 to December 31, 1991.


to which nonsynchronous trading can be best represented by an autoregressiveprocess of order one.

Fig. 2 plots the intertemporal change in the basis against time to maturity forthe two indices. Comparing Fig. 2 with Fig. 1, one can see that the former series is

Fig. 2. Intertemporal change in the basis defined as in Table 3 in both indices over remaining days tomaturity of the futures contract. The sample includes data from September 30, 1985 to December 31,1991.


more evenly spread around zero. Therefore, the intertemporal change in the basisproduces a series which does not depend as strongly on time to maturity as doesthe basis itself.

Table 2 presents summary statistics of the basis and Table 3 of the intertempo-ral change in the basis. Comparing the two, it appears that the series in the lattertable are not as autocorrelated as those of the basis, probably because theintertemporal change in the basis approximates a first difference. The autocorrela-tion function of the intertemporal change shows a significant coefficient at its firsttwo lags for the S&P 500 index but not for the MMI, for which the autocorrela-tion function has only one significant coefficient at its first lag. This is anindication that the S&P 500 basis has a more complicated structure than that inthe MMI.

Ž . Ž .Furthermore, consistent with Miller et al. 1994 , Beaulieu and Morgan 1996show that the first difference of the return on the basis depends on an autoregres-

Ž .sive term f that captures the nonsynchronous trading resulting from the basisposition if futures and spot prices are perfectly correlated and of equal variance.

Table 2Descriptive statistics of the basis

Descriptive statistics

S&P 500 index MMI

Sample size 1582 1582y3 y3Mean 3.89=10 3.19=10y3 y3Std. deviation 5.56=10 3.66=10y2 y2Minimum y9.49=10 y3.56=10y2 y2Maximum 2.95=10 1.51=10

Estimated serial correlation coefficients

Lag S&P 500 index MMI

Ž . Ž .r t r r t rk k k k

1 0.607 24.18 0.631 25.102 0.403 12.18 0.535 15.883 0.401 11.12 0.459 11.864 0.376 9.69 0.414 9.865 0.317 7.73 0.379 8.526 0.271 6.36 0.340 7.317 0.230 5.28 0.290 6.038 0.116 2.61 0.234 4.769 0.103 2.31 0.218 4.37

10 0.149 3.33 0.215 4.26

The sample includes data from September 30, 1985 to December 31, 1991. The basis is defined asŽ .F yS rS for each index. The estimated serial correlation coefficients are for the basis of botht t ty1

Ž .indices. The t-ratio corresponds to the null hypothesis of r the estimated serial coefficient equalsk

zero, where k is the number of lags.


Table 3Descriptive statistics of the intertemporal change in the basis

Descriptive statistics

S&P 500 index MMI

Sample size 1582 1582y5 y5Mean 2.31=10 3.07=10y3 y3Std. deviation 4.82=10 3.01=10y2 y2Minimum y8.66=10 y2.49=10y2 y2Maximum 9.49=10 2.73=10

Estimated serial correlation coefficients

Lag S&P 500 index MMI

Ž . Ž .r t r r t rk k k k

1 y0.257 y10.23 y0.426 y16.972 y0.270 y10.27 y0.028 y0.983 0.032 1.12 y0.035 y1.214 0.039 1.40 y0.002 y0.075 0.002 0.07 0.002 0.066 y0.012 y0.44 0.026 0.877 0.106 3.72 y0.001 y0.048 y0.111 y3.87 y0.037 y1.279 0.091 y3.15 y0.037 y1.24

10 0.101 3.46 0.105 3.56

The sample includes data from September 30, 1985 to December 31, 1991. The intertemporal change inwŽ . x wŽ . Ž . xthe basis is defined as F yS rS y F r S y R qd for each index. The esti-t t ty1 ty1 ty1 ty1 ty1

mated serial correlation coefficients are for the intertemporal change in the basis of both indices. TheŽ .t-ratio corresponds to the null hypothesis of r the estimated serial coefficient equals zero, where k isk

the number of lags.

These assumptions also imply that futures and spot prices are cointegrated. In thiscase, the first-order serial correlation of the intertemporal change in the basis is

1yfr 1 sy . 3Ž . Ž .

2

Again in Table 3, the coefficients on the first-order serial correlation of theintertemporal change in the basis is closer y1r2 for the MMI than for the S&P500 index. One can therefore interpret the smaller first-order autocorrelation

Ž .coefficient in absolute value on the S&P 500 index basis as evidence of largerproblems due to nonsynchronous trading in this index than in the case of the MMI.

4. Empirical results

Ž .This paper utilizes GARCH Engle, 1982; Bollerslev, 1986 to estimate themodel of the basis. The basis, like many financial series, is subject to the presence


of heteroscedasticity and leptokurtosis. By using conditional second moments,Ž .GARCH helps taking these factors into account Bollerslev et al., 1992 .

4.1. Estimation of the model

The bivariate estimation of the model consists of a joint estimation of the basisequation, a benchmark equation and a conditional covariance matrix. The positive

Ž .definite parameterization follows that proposed by Engle and Kroner 1995Ž .BEKK .

Ž .The intertemporal capital asset pricing model CAPM , which I use for thevaluation of the intertemporal change in the basis, involves the intertemporalmarginal rate of substitution between periods as the benchmark for asset pricing. I

Ž .replace the benchmark variable M McCurdy and Morgan, 1991 with a bench-tŽ .mark portfolio the return on a worldwide index hypothesized to be conditionally

Ž .mean-variance efficient Hansen and Richards, 1984 . This is an attractive alterna-tive to the benchmark based on consumption because of measurement problems in

Ž .consumption data Breeden et al., 1989 . This approach has been found empiri-Ž . Ž .cally useful Mark, 1988 but is, of course, subject to the critique of Roll 1977 .

Two instrumental variables are used to predict market excess return. AnŽ .indicator variable for Mondays is used for the weekend effect French, 1980 .

Second, the excess market return is assumed to follow an autoregressive processŽ .Lo and MacKinlay, 1990 . The conditional variance of the market excess return

Ž .follows a GARCH process with an asymmetric term Glosten et al., 1993 .Ž .The basis equation is as specified in Eq. 2 after invoking rational expectations

Ž .and replacing the expectations in Eq. 2 by the ex post values of the variables atŽ .time t. Note that a moving average MA term might be necessary if the level of

Žthe basis is stationary, that is if futures and spot prices are cointegrated Beaulieu.and Morgan, 1996 . The conditional variance of the basis follows a GARCH

process with asymmetric matrices A, E and B and is a function of time tomaturity. The model also includes dummy variables for the market crash ofOctober 1987 for the S&P 500 basis and the MMI basis in the conditionalvariance of the basis and that of the market portfolio’s excess return.

Let R be the rate of return on the benchmark portfolio from time ty1 tom t

time t, r be the riskless rate of return from time ty1 to time t, m be anf t t

indicator variable that takes the value of one when t is the first trading day of theweek and zero otherwise, X be a set of instruments known at time ty1,m , ty1

Sy be an indicator variable in the conditional variance of the market excessty1

return equation that takes the value of one when ´ is negative and zerom t

otherwise, ´ be an error term at time t for the market excess return equation,m t

´ be an error term at time t for the intertemporal change in the basis, u be ab t t

vector with components ´ ,´X, m be a coefficient on basis risk which takesŽ .m t b t

the value of one if the conditional CAPM model is appropriate, H be thet

variance–covariance matrix of the system of mean equations with asymmetric


components, h be the variance of the benchmark return on the H matrix, hm t t bm t

be the covariance between the benchmark return and the intertemporal change inthe basis in the H matrix, d be a dummy variable taking the value of one ont m t

October 19 and 21, 1987 and zero otherwise, d be a dummy variable that takescc t

the value of one at the beginning of each contract and zero otherwise, l be at

vector with components 0,dX, f be a vector with components d ,d

X, KŽ . Ž .cc t t m t m t t

be a matrix equal to L L with L sL sL s0 and L s Ty t , L beŽ .i j i jt 11 t 12 t 21 t 22 t t

a function of remaining days to maturity in spike form such as introduced byŽ .Baillie and Bollerslev 1989 and extended to the multivariate BEKK model by

Ž .Morgan and Neave 1992 . The equivalent formulation for the time to maturityspecification is L sK y AXK AqBXK BqEXK E , C be an upperŽ .t t ty1 ty1 ty1

triangular matrix for the constants in the variance, A be an asymmetric matrix forthe ARCH terms, E be an asymmetric matrix for the asymmetric ARCH termsand B be an asymmetric matrix for the GARCH terms.

The estimated system is

R yr sg qhm qg R yr q´ 4Ž . Ž .m t f t 0m t 1m m ty1 f ty1 m t

F yS F ht t ty1 bm t Xy yR qd sg qm g X qC´Ž .ty1 ty1 0b m m ty1 b ty1S S hty1 ty1 m t

q´ 5Ž .b t

with either

H sCXCqAX u uX AqBXH X BqEXSXy u uX Sy EqDXl l

X Dt ty1 ty1 ty1 ty1 ty1 ty1 ty1 t t

qFXf f

X F 6Ž .t t

or

H sCXCqAX u uX AqBXH X BqEXSXy u uX Sy Et ty1 ty1 ty1 ty1 ty1 ty1 ty1

qL qFXf f

X F . 6XŽ .t t t

Ž .In Eq. 6 , the maturity effect in the conditional variance of the basis isparameterized with a dummy variable that takes the value of one at the beginningof new contracts and zero otherwise. Through the recursive form of the GARCH

Ž X .variance, the time to maturity effect will gradually become smaller. Eq. 6 , as inŽ .Morgan 1995 , uses a spike function for time to maturity. In this form, the

remaining days to maturity variable is introduced at time t in the conditionalvariance in such a way that the conditional variance at time tq1 does not dependon the variable introduced at time t. This insures that this source of change invariance does not influence the conditional variance in following periods.

Table 4 presents the principal coefficient estimates of the basis with bothindices. The estimates reveal that all coefficients are positive and significantlydifferent from zero, except for the constant in the MMI basis equation. Withrespect to the time to maturity coefficient, results imply that the conditional


Table 4Selected coefficient estimates

S&P 500 MMI

contract change spike contract change spike

Parameters for the market equationŽ . Ž . Ž . Ž .g 0.009 0.002 0.009 0.002 0.009 0.002 0.009 0.0020mŽ . Ž . Ž . Ž .h y0.014 0.005 y0.014 0.005 y0.017 0.005 y0.016 0.005Ž . Ž . Ž . Ž .g 0.176 0.030 0.183 0.036 0.196 0.030 0.193 0.0271m

Parameters for the basis equationŽ . Ž . Ž . Ž .g y0.005 0.002 y0.004 0.001 0.001 0.001 0.001 0.0010bŽ . Ž . Ž . Ž .T y t 0.052 0.014 0.072 0.032 0.058 0.015 0.054 0.016Ž . Ž . Ž . Ž .C y0.790 0.021 y0.790 0.022 y0.786 0.021 y0.784 0.018

The sample includes data from September 30, 1985 to December 31, 1991. Contract change refers tothe maturity effect in the conditional variance of the basis parameterized with a dummy variable thattakes the value of one at the beginning of new contracts and zero otherwise. Spike refers to theremaining days to maturity variable introduced at time t in the conditional variance of the intertempo-ral change in the basis so that the conditional variance at time tq1 is made independent of the variable

Ž . Ž .introduced at time t. The selected coefficient estimates come from systems of Eqs. 4 – 6 and Eqs.Ž . Ž X .4 – 6 . g is the constant in the market portfolio equation. h is the coefficient for the indicator0m

variable for the first trading day of the week and g is the coefficient on the first lag of the market1m

portfolio excess return over the riskfree rate. g is the constant in the intertemporal change in the0b

basis equation, T y t refers to coefficient estimates of the different time to maturity specificationsŽ . Ž X .presented for the conditional variance in Eq. 6 and Eq. 6 and C is the coefficient on the first-orderŽ .moving average term. Robust standard errors Bollerslev and Wooldridge, 1992 are in parentheses.

variance of the basis decreases as the futures contract approaches expiration.Furthermore, there is not much difference across indices and time to maturityspecifications. Overall, the size of the maturity effect is comparable in both basesand since the cost of approximating the MMI is almost certainly lower than thatfor the S&P 500 index, this is an indication that transaction costs are not a validexplanation of the presence of a maturity effect in the conditional variance of thebasis.

Table 5 presents results of tests for the S&P 500 index and the MMI basis. Thebenchmark portfolio test is a joint test of the conditional CAPM and of the

Ž .conditional efficiency of the benchmark portfolio Mark, 1988 . As described byŽ .McCurdy and Morgan 1993 , one can treat M , the true benchmark portfolio, as at

) wŽlatent variable and project it onto a constant, R sR yr and D B s F ymt m t f t t t. x wŽ . xS rS y F rS yR qd . Let c be the coefficient for the formert ty1 ty1 ty1 ty1 ty1 1

and c the coefficient for the latter. The test equation for the benchmark portfolio2

according to those authors is

c rc Var D B qCov DB , R)Ž . Ž . Ž .2 1 ty1 t ty1 t m t)E DB s E R 7Ž .ty1 t ty1 m t

) )c rc Cov D B , R qVar RŽ . Ž . Ž .2 1 ty1 t m t ty1 m t


Table 5Specification tests for the bivariate analysis

S&P 500 MMI


Ž . Ž . Ž . Ž .BT 3.61 0.01 y1.08 0.14 1.01 0.16 y1.11 0.13Ž . Ž . Ž . Ž .ms0 1.90 0.17 2.03 0.15 0.31 0.58 0.01 0.99Ž . Ž . Ž . Ž .ms1 22.45 0.00 18.97 0.00 0.29 0.53 0.01 0.99

The sample includes data from September 30, 1985 to December 31, 1991. Contract change refers tothe maturity effect in the conditional variance of the basis parameterized with a dummy variable thattakes the value of one at the beginning of new contracts and zero otherwise. Spike refers to theremaining days to maturity variable introduced at time t in the conditional variance of the intertemporalchange in the basis so that the conditional variance at time tq1 is made independent of the variable

Ž .introduced at time t. Tests on BT, the benchmark test statistic, adapted from Mark 1988 ; p-values forthis test, in parentheses, are for the unit normal distribution. Tests on m are for the coefficient in

Ž . Ž . Ž . Ž X .systems of equations Eqs. 4 – 6 and Eqs. 4 – 6 . ms0 tests the null hypothesis that the basissystematic risk is zero, ms1 tests the null hypothesis the CAPM restriction that m is not significantly

Ž 2Ž . .different from 1 p-values in both cases are for the x 1 distribution .

For this equation to be consistent with the conditional CAPM, c rc has to equal2 1

zero 8. The benchmark test statistics presented in Table 5 test whether thatconstraint holds. A rejection can imply that the benchmark used in the estimationŽ ) .R is not conditionally mean variance efficient or that the conditional CAPMm t

does not hold. The results of the benchmark test reveal that for the S&P 500 basis,only in the case where the dummy variable for the contract change is used can onereject the hypothesis that c rc is equal to zero. In no case is the benchmark2 1

rejected for the MMI.The evidence on the S&P 500 and the MMI basis risk provides an interesting

contrast. For the S&P 500 basis, regardless of the specifications used for theŽmaturity effect and of the choice of conditional distribution, m which is equal to

.one if the conditional CAPM holds , the coefficient on basis risk, is neversignificantly different from zero. When m is constrained to one, the OPG–LMstatistic has a p-value of 0.00, clearly rejecting the hypothesis of m equal to one.Similar evidence is not found in the MMI basis since neither hypothesis, m equalto one or m equal to zero, is rejected. Therefore, the risk premium in the S&P 500basis is not consistent with the conditional CAPM while the conditional CAPMcannot clearly be rejected using the MMI. However, it is possible that basis risk isa component of the S&P 500 basis. Yet some feature of the data set, possibly

8 Ž .Note that according to Mark 1988 , the benchmark portfolio test is for the hypothesis of c not2

significantly different from zero. In order to make the estimation procedure more parsimonious andreduce the estimation by one coefficient, using simple algebra one can simplify the original benchmarkportfolio test to the hypothesis of c rc not significantly different from zero, where c rc is2 1 2 1

estimated as a single coefficient.


Table 6Conditional moment tests and likelihood ratio tests

S&P 500 MMI


Conditional moment testsŽ . Ž . Ž . Ž .T y t in mean 0.57 0.45 0.40 0.53 1.45 0.23 0.45 0.50Ž . Ž . Ž . Ž .T y t in covariance 0.76 0.38 0.29 0.59 0.01 0.96 0.49 0.48

Likelihood ratio testsŽ . Ž . Ž . Ž .T y t in variance 77.55 0.00 99.73 0.00 54.60 0.00 73.22 0.00

The sample includes data from September 30, 1985 to December 31, 1991. Contract change refers tothe maturity effect in the conditional variance of the basis parameterized with a dummy variable thattakes the value of one at the beginning of new contracts and zero otherwise. Spike refers to theremaining days to maturity variable introduced at time t in the conditional variance of the intertempo-ral change in the basis so that the conditional variance at time tq1 is made independent of the variable

Ž .introduced at time t. Conditional moment tests evaluate whether T y t belongs in the mean of theintertemporal change in the basis or in the covariance with the market portfolio; while the likelihoodratio test statistic evaluates how much explanatory power is lost in the model of the intertemporalchange in the basis when time to maturity is excluded from the conditional variance of the basis

2Ž .equation. P-values, in parentheses, are for the x 1 distribution.

nonsynchronous trading, is interfering with the estimation causing m to besignificantly different from one.

Ž .Finally, Table 6 presents results of the likelihood ratio LR tests and condi-Ž .tional moment CM tests when m is constrained to one. Test statistics for time to

maturity in the mean of the basis equation and in the conditional covariance of thebasis are insignificant. For the mean, this implies that the intertemporal change inthe basis is consistent with market efficiency and that it is not possible to use timeto maturity to take advantage of potential arbitrage opportunities. In the case of thecovariance, it means that the systematic risk of the basis position is also indepen-dent of remaining days to maturity and that the risk premium is unpredictable inthat respect. Table 6 also reports LR tests for the influence of time to maturity onthe conditional variance of both indices. The results indicate that the conditionalvariance of the intertemporal change in the basis in both indices is clearly afunction of time to maturity. That is, a model that incorporates marking to marketdoes not seem to resolve the question of why time to maturity influences theconditional variance of the basis.

4.2. Out-of-sample forecasts

In order to assess the importance of time to maturity in the variance of the basisin stock market index futures contracts, I present out-of-sample forecasts atvarious horizons. Since basis risk does not appear to be a very large component ofthe basis, and to facilitate calculations, I assume it to be equal to zero. This


simplification allows me to produce forecasts from a univariate model of theintertemporal change in the basis with conditional variance h in which Lt t

s Ty t . The model isŽ .F yS Ft t ty1

y yR qd sg qC´ q´ 8Ž .ty1 ty1 0b b ty1 b tS Sty1 ty1

h scqa´ 2 qbh qDd q ff 9Ž .t b ty1 ty1 t t

or

h scqa´ 2 qbh qg L y aqb L q ff 9XŽ . Ž .Ž .t b ty1 ty1 t ty1 t

The parameterizations for time to maturity correspond to those in the bivariateŽ . Ž .analysis. Eq. 9 corresponds to Eq. 6 ; d is an indicator variable for a contractt

change but D is now a scalar coefficient. Given the recursive form of the GARCHconditional variance, the influence of time to maturity gradually becomes smaller

Ž X . Ž X.over the life of the futures contract. Eq. 9 corresponds to Eq. 6 and uses aspike function of time to maturity. In this form, the remaining days to maturityvariable is introduced at time t in the conditional variance in such a way that theconditional variance at time tq1 is made independent of the variable at time t.

Table 7One-step ahead out-of-sample variance forecasts

Forecasting horizon S&P 500 MMI

LR test MAE LR test MAE

Contract changeŽ . Ž . Ž . Ž .May 22, 1990 to December 19, 1991 10.75 0.00 0.69 0.25 17.58 0.00 0.18 0.43Ž . Ž . Ž . Ž .May 22, 1990 to March 7, 1991 3.34 0.07 0.15 0.44 17.10 0.00 0.28 0.39Ž . Ž . Ž . Ž .March 8, 1991 to December 19, 1991 7.69 0.01 0.19 0.42 0.48 0.51 y0.72 0.24

SpikeŽ . Ž . Ž . Ž .May 22, 1990 to December 19, 1991 11.81 0.00 1.13 0.13 21.09 0.00 0.72 0.23Ž . Ž . Ž . Ž .May 22, 1990 to March 7, 1991 5.51 0.02 0.05 0.48 16.98 0.00 0.13 0.45Ž . Ž . Ž . Ž .March 8, 1991 to December 19, 1991 6.30 0.01 y1.02 0.15 4.11 0.04 y0.32 0.37

Contract change refers to the maturity effect in the conditional variance of the basis parameterized witha dummy variable that takes the value of one at the beginning of new contracts and zero otherwise.Spike refers to the remaining days to maturity variable introduced at time t in the conditional varianceof the intertemporal change in the basis so that the conditional variance at time tq1 is madeindependent of the variable introduced at time t. The LR test statistic evaluates how much explanatorypower is lost in the model of the intertemporal change in the basis when time to maturity is excludedfrom the conditional variance of the basis equation. MAE is a t-test statistic on the difference betweenthe mean absolute error of a model that includes time to maturity in its conditional variance and that fora model that does not include it. The time period from May 22, 1990 to December 19, 1991 contains400 one-step-ahead out-of-sample forecasts. Time periods from May 22, 1990 to March 7, 1991 andfrom March 8, 1991 to December 19, 1991 each contain 200 one-step-ahead out-of-sample forecasts.P-values are in parentheses.


In order to forecast the conditional variance for the error term at tq1, I useobservations 1 through t. Then, I add the observation for tq1, update the modelparameter estimates and forecast the variance for tq2. I follow the sameprocedure until I have reached the last observation of my forecasting horizon.

I have chosen two criteria for comparison of the results of the out-of-sampleforecasts of the models including and excluding time to maturity. The first

Ž .criterion is the difference in mean absolute error MAE between the two models.The second criterion is a likelihood ratio test evaluating the gain in explanatorypower resulting from the inclusion of time to maturity in forecasts of theconditional variance of the basis. The results of the LR tests, reported on Table 7,show that overall for sets of 200 and 400 of one-step ahead forecasts, the inclusionof time to maturity adds to the explanatory power of forecasts of the variance ofthe intertemporal change in the basis in both indices, independently of theparameterization used. Furthermore, time to maturity specifications in the variancehave more explanatory power over longer time periods. Nonetheless, over shortersubperiods, only in two cases out of eight the explanatory power gained is notsignificantly different from zero at a five percent level of significance. Theevidence with MAE is weaker.

5. Summary and conclusions

The model of the intertemporal change in the basis relaxes the cost of carrymodel assumptions of constant interest rate and known dividend yield over thelifetime of the futures contract. Using this model, in two different parameteriza-tions, I present evidence that the conditional variance of the basis decreases as thematurity date gets closer. This result is consistent with Castelino and Francis’Ž .1982 modelling of the conditional variance of the basis and, ultimately, with

Ž .Samuelson’s 1965 hypothesis. Furthermore, the evidence also suggests that inout-of-sample forecasts, the model of the conditional variance of the basis withtime to maturity is a better predictor of the conditional variance of the basis thanthe model that does not incorporate time to maturity. This evidence is consistentwith the fact that interest rate risk cannot explain the presence of a maturity effectin the conditional variance of the basis in stock market index futures contracts.

The comparison of the S&P 500 and the MMI basis reveals interesting featuresof the data. First, it appears that time to maturity in the conditional variance of thebasis is of similar importance in both indices. This implies that the presence oftime to maturity in the conditional variance of the basis cannot be explained bytransaction costs. Furthermore, the analysis finds no evidence supporting thepresence of basis risk in the S&P 500 basis. This contrasts with the MMI basismodel in which case I cannot reject the CAPM and positive basis risk. Since theMMI is a subset of the S&P 500 index and that the analysis is the same for bothbases, nonsynchronous trading problems in the S&P 500 index is the most likelyexplanation for this phenomenon.


Acknowledgements

This paper is based on my doctoral dissertation presented at Queen’s Univer-sity, March 1994. My thanks to I.G. Morgan who has directed this research. I alsowish to thank A.C. MacKinlay, T.H. McCurdy and E.H. Neave and an anonymousreferee. A previous version of this paper was presented at the 1993 NorthernFinance Association conference. Funding from Queen’s University School ofBusiness and School of Graduate Studies as well as the Fonds pour la formation

Ž .de jeunes chercheurs et l’aide a la recherche FCAR is gratefully acknowledged.`The usual disclaimer applies.

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Education

Bealieu maturity effect