A comparison of foreign exchange forward and futures prices

Journal

of Banking

and

Finance

IS (1991)

1057-1080.

North-Holland

A comparison of foreign exchange forward and futures prices Michael A. Polakoffr Syracuse

University,

Syracuse,

New

York

13244-2130.

L;SA

Paul C. Grier University

Center ar Binghamton,

Received

May 1989, final version

New

York 13901, USA

received

October

1990

In theory, futures resettlement should create systematic pricing differences between futures and forward contracts; however, previous empirical studies do not demonstrate such difierences in the foreign exchange markets. On the other hand, inferences from these studies are suspect since they do not directly implement theortical measures of resettlement eflects and are unadjusted for sub-Gaussian data properties. In this paper, these limitations are addressed through the use of the Multivariate Autoregressive Moving-Average model. Our lindings provide evidence of systematic pricing influences arising from futures resettlement.

1. Introduction

The fundamental distinction between forward and futures contracts is that the latter are marked-to-market while the former are not [Black (1976)l.l This institutional arrangement, also known as resettlement, has been the subject of several theoretical price studies. Cox, Ingersoll, and Ross [hereafter CIR (1981)] demonstrate that, when capital markets are perfect and interest rates are stochastic, resettlement should cause forward and futures prices to *This paper is based on my doctoral thesis. I am very grateful to my dissertation committee, Paul Grier (Chairman), Charles Bischolf, and Haim Ofek, for their support and guidance. I would also like to acknowledge the exceptionally substantial contributions made by Murray Polakoff, Charles BischolT, and two anonymous referees of this journal. In particular, comments and suggestions made by one of the anonymous referees played an especially integral role in the development of this presentation. An earlier version of this paper received an outstanding paper award (1988 Southwestern Finance Association) sponsored by the Chicago Board of Trade. ‘Marking-to-market stipulates that open futures positions are to be revalued daily. That is, settlement prices of futures contracts from successive days are contrasted. With open futures positions artificially offset in this manner, the difference between successive settlement prices gives rise to losses or gains. These losses or gains are reconciled before the next trading session to limit position risk to a single day. As a result, actual cash prolits and losses occur on a daily basis. 0378-l266/91/%03.50

0

1991-Elsevier

Science Publishers

B.V. (North-Holland)

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P.C. Grier. Foreign uchange

forward and futures prices

diverge. If markets are efficient and futures prices are stochastic, Morgan (1981) concludes that resettlement must result in forward and futures price differences. Levy (1989) states that the differential is related only to the uncertainty about the next day’s interest rate, which is sufftciently small to reduce forward-futures pricing differentials to a matter of empirical inconsequence. While systematic forward-futures price differentials have been documented by empirical studies of Treasury bills, these differences have been attributed to factors other than resettlement. Such findings are consistent with the numerous imperfections arising between the Treasury bill forward and futures markets. Segmented markets [Branch (1978)-J, a default risk premium in the futures contract [Lang and Rasche (1978)], asymmetric execution costs [Allen and Thurston (1988), Cappoza and Cornell (1979)], and differential liquidity and default risk premia [Kamara (1988)] have been cited as causes underlying Treasury bill forward-futures price spreads. In comparison with Treasury bills, there is a relative lack of imperfections between the foreign exchange (FX) forward and futures markets.’ For this reason, FX is a logical area in which to isolate resettlement effects [Cornell and Reinganum (1981)]. Interestingly, previous direct comparisons of FX forward and futures contracts reveal systematically homogeneous prices [Park and Chen (1983 Cornell and Reinganum (1981)]. If resettlement cannot be shown to affect FX prices, there is probably little reason to expect that this feature exerts pricing influences elsewhere. As compelling a factor as resettlement may be from a theoretical perspective, empirical tests do not support its practical significance. A re-examination of this issue is, however, in order. First of all, statistical inferences from earlier studies are suspect since these investigations were conducted under the premise that FX data exhibit Gaussian properties. This condition is not applicable to FX as is well-established in the FX literature [Hall, Brorsen and Irwin (1989), So (1987), and Westerfield (1977)]. More importantly, the standard t-statistic [Park and Chen (1983, Cornell and Reinganum (1981)] and unadjusted ordinary least squares (OLS) [Park and Chen (1985)] variables employed in earlier studies do not empirically represent daily futures resettlement as set forth by underlying theoretical 2Additionally, Treasury bill futures contracts permit variation in the maturity of the deliverable asset. On the other hand, maturity-matched FX forward and futures contracts have identical terms to their closing trade date. Cornell and Reinganum (1981) state that FX forwards and futures contracts are taxed symmetrically when their maturities are less than one year. They further note that, unlike the players that predominate in the Treasury bill market, the major contracting players in the FX forward and futures markets are ‘major banks, corporations, and governmental bodies’. The default risk of these large institutions is small and comparable to the default risk associated with futures (clearinghouse) exchanges [Cornell and Reinganum (1981)]. Thus, on the whole, the FX forward and futures markets tend to have fewer structural disparities than exist elsewhere and thus are better suited for isolation of marking-to-market effects.

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models. Hence, the pricing effects of proceeds arising from daily resettlement cash flows are not tested directly. This paper undertakes the task of replicating daily resettlement and attempts to demonstrate how attendant uncertainty can be incorporated suitably into theoretical expressions of forward-futures price divergence. As a result, we are able to implement a direct causal model of price differences arising between FX forward and futures contracts. We also employ a statistical approach consonant with representations posited by theory. Denis (1976) finds a ‘high degree of efficiency’ between the FX forward and futures markets3 Our findings provide considerable evidence of systematic divergence between the prices of FX forward and futures contracts. The remainder of the paper is organized as follows. Section 2 reviews resettlement as a theoretical basis for forward and futures price differences. Section 3 examines earlier testing methods. In section 4, the structural form of the Multivariate Autoregressive Moving-Average (MARMA) model presented is developed formally. The MARMA model is extended into testable form in section 5, and results of estimation are provided in section 6. A summary of our methodology and findings concludes the paper in section 7.

2. The theoretical case for price divergence The theoretical factors dictating differences between forward and futures prices can be identified through a brief comparison of the instruments’ payout schedules. Suppose that a pair of forward and futures contracts are purchased at the same price and are held to maturity. They represent equal units of the same underlying commodity, have identical delivery specitications, default risk, tax treatment, share a single maturity date, and trade in efficient markets devoid of transactions costs. With same-day delivery, the commodities underlying the maturing contracts must converge to a single future spot price. Disregarding the issue of resettlement for the moment, these forward and futures contracts would be arbitraged so as to generate the same net cash flow since their opening and liquidation prices are the same. Unlike the forward, however, a futures contract is marked-to-market each 3Denis (1976) linds ‘extremely high correlation coefficients between contemporaneous price levels demonstrated in the interbank FX forward market and those found on the IMM (International Monetary Market, where FX futures trades are conducted). He also suggests that forward prices may lead those found in the futures market, but acknowledges that statistical evidence of this condition is weak and may be due to sampling error. It should be noted that IMM trading volume was far lower at the time of Denis’ study than was the case several years later. This reduced level of trading volume may have contributed to any marginal lead-lag relationships.

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business day.4 As stated earlier, this feature represents the fundamental distinction between forward and futures contracts. Under stochastic conditions, resettlement of the futures position will give rise to daily cash flows. These cash flows will be invested or financed, depending on whether a profit or loss occurs.’ Unless the net interest payout arising from resettlement cash flows is zero, the futures contract cannot have the same total payout as its forward counterpart. By the law of one price, this inequality should generate price differences between the two instruments. There is, however, an additional issue to be considered. All resettlement interest gains or losses are determined after a futures position has been initiated. Hence, there is uncertainty associated with the future net interest payout. Since this payout plays a crucial role in determining differences between forward and futures prices, treatment of expectational processes is critical to modeling relative price differentials between these instruments. If expected changes in futures prices are zero and futures prices and interest rates are independent, expected resettlement proceeds must also be zero, and a zero net interest payout is expected as well. Under this circumstance and in the absence of a risk premium, forward and futures prices should be identical. In contrast to CIR (1981), Levy (1989) maintains that the relation between forward and futures prices does not depend on stochastic interest rates that will occur over the life of a futures contract. In his framework, forward and futures prices must be equal if only the next day’s interest rate can be known with certainty. Acknowledging that this is not possible, Levy (1989) nevertheless states that the uncertainty associated with one-day interest forecasts should be sufficiently small to reduce forward-futures pricing differentials to a matter of empirical inconsequence. CIR (1981) incorporate resettlement assuming perfectly divisible contracts in continuous time and conclude that the covariance between stochastic interest rates and changes in futures prices (the uncertain resettlement flow) determines whether there is a forward-futures differential. Thus, stochastic interest rates over the life of the contract are necessary for a differential. Levy (1989) maintains that only the next day’s uncertainty in interest rates is relevant because an arbitrage position with fractional futures contracts can be created one day and rebalanced on the following day. Morgan (1981) also indicates the importance of stochastic interest rates, but arrives at a 4Futures trades cannot be initiated without the commitment of original margin, but these funds can be made available in the form of a Treasury bill. As a result, no opportunity cost need be incurred in the process of opening new positions. Therefore, original margin is ignored in our analysis. However, variation margin payments associated with resettlement must be made available in ‘good funds’ (cash equivalent). It is this condition that is relevant to the pricing process. ‘If resettlement losses are fmanced by savings, an opportunity cost is incurred. Therefore, the concept of resettlement expense is not compromised by this possibility.

M.A.

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somewhat different characterization of the differential. Polakoff (1991) reconciles the lack of a theoretical consensus in the literature by focusing on the issue of contract divisibility. Futures contracts are indivisible as an outgrowth of the standardization of contracts. Organized futures exchanges provide standardization as a service to the economy thereby reducing transactions costs and enhancing welfare [Telser and Higinbotham (1977)]. Thus in practice, the precise portfolio that Levy (1989) describes cannot be constructed. Although forward contracts, in principle, can be tailored to the needs of the participants, forwards are also standardized to some extent, though less so than futures. If forward contracts could be obtained in infinitely divisible units, then for a fixed quantity of futures contracts an appropriate fractional amount of forwards could be used to create a riskless portfolio as CIR (1981) or Levy (1989) (modified) would require. However, again, practical constraints may not allow containment of such a perfect markets ideal. The forward contracts quoted in the market (and typically employed in futures market research) have fixed delivery dates as well as standard or minimum denominations. Because the participants are acutely aware of the imperfections in these markets, they are not willing to transact in every arbitrary denomination, at least not without some price concessions. Thus, while there is greater flexibility in forward exchange markets than in futures markets, the divisibility will not be infinite, leading to an inability to construct perfectly riskless arbitrage portfolios. In Polakoff’s (1991) view, the CIR (1981), Morgan (1981), and Levy (1989) models are equivalent when accounting for the indivisibility of the contracts with Morgan’s (1981) inferences emerging as the most appropriate premise for empirical testing. The extent to which indivisibilities referred to by Polakoff (1991) affect futures and forward prices is ultimately an empirical question that we attempt to answer in this study. In view of the above, it is not surprising that actual FX forward and futures data consistently reflect differences between the contracts’ prices. On the other hand, previous studies have not found these differences to be statistically significant. The conclusiveness of these findings must be judged on the strength of the methodologies employed, and it is to this matter we now turn. 3. Statistical characteristics

of the data and previous tests of price divergence

Examined separately, daily FX forward price distributions have been identified as non-stationary by Westerfield (1977). Hall, Brorsen and Irwin (1989) find that time series of daily FX futures price distributions correspond to the mixture of normals hypothesis. However, the forward-futures pricedifferential component analyzed in this study (and those discussed below)

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does not represent a daily difference. Unfortunately, direct comparisons of FX forward and futures prices cannot be properly conducted on such a basis. Because the maturities for forward and futures contracts are reported on different bases, there is at best one day during the life of the futures when its maturity can be matched to a sequence of otherwise comparable forwards. To illustrate, suppose that two sequences of daily three-month forward and futures prices are compared. Each successive forward contract matures in three months. Each successive futures contract declines in maturity until its settlement date is reached. When three-month contracts or those of any term are used, five comparable forward-futures maturities at most can be retrieved from a multi-year list of rates (inasmuch as the standardized futures delivery dates were March, June, July, September, and December throughout much of our data sample). Were it not for the July settlement date, the differentials in this example would represent a three-month time series. This restriction on available data points is a constraint to which all direct comparisons of FX forward and futures prices are subject. Proceeding under the constraint dictated by maturity matching, t-statistic measurements conducted by Park and Chen (1985) reveal, out of 20 sets tested, only four FX forward-futures price differentials that were rejected for insignificance at the 5% level. Also employing an event methodology based on the use of an ordinary r-statistic, Cornell and Reinganum (1981) tested 20 sets of differences and found that only two were rejected for insignificance at the 5% level. Based on these results, both sets of authors conclude that FX forward and futures contracts do not demonstrate systematic price divergence. The t-statistic used in the aforementioned studies is used to determine whether price differentials of maturity-matched FX forward and futures rates differ significantly from zero. Mean differences are obtained by first subtracting forward from futures rates. While this process may result in distributional stationarity for the differenced series, there is evidence that autocorrelation remains. We further note that these t-test applications are not causal implementations of theoretical models specified by CIR (1981) or Morgan (1981). In their investigations of Treasury bill forward and futures prices, Vignola and Dale (1979), Rendleman and Carabini (1979), and Lang and Rasche (1978) note the inappropriateness of using the ordinary t-statistic as a measure when classical distributional assumptions are violated. They cite the frequent sign reversals that occur when bill forward and futures rates are subtracted from each other for the purpose of obtaining a mean differential. These reversals are also common when maturity-matched FX forward and futures rates are subtracted and suggest autocorrelation, a condition not accounted for by the ordinary c-statistic. As a result, inferences from ordinary f-tests used in the FX studies cited above must be viewed as suspect. In its continuous form, the CIR (1981) treatment of forward and futures

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prices assigns a higher price to the futures contract when percentage changes in futures and default-free discount bond prices have local covariances less than zero over a given time frame. Conversely, the futures price will be lower than that of the forward when percentage changes in futures and default-free discount bond prices have local covariances greater than zero over a specified time interval. The unanticipated portion of these changes has been tested in the FX markets by use of a standard OLS model [Park and Chen (1985)]. There are several problems associated with this application of the continuous-time OLS framework . First, local covariances between unanticipated percentage changes in FX futures and bond prices do not always display the same sign over time. When they do not, the continuous version of the CIR (1981) relationship is violated. Second, resettlement is a discrete process that gives rise to daily cash flows. Models based on a continuoustrading framework do not directly analyze these cash flows and therefore must be considered an approximate measure of the marking-to-market effect. Third, the use of OLS is predicated on the assumption that residuals are uncorrelated and homoskedastic. So (1987) demonstrates that this assumption is violated in the case of daily FX futures prices, thereby leading to inefficient estimators and incorrect confidence intervals when OLS is applied to such data. As underscored by the discussion presented in section 5 of this paper, there is evidence of residual correlation for OLS applications to nondaily, maturity-matched FX forward-futures price differentials, a condition that similarly leads to questionable statistical inferences. 4. The MARMA

model

4.1. Fundamental

basis

We now begin our explicit discussion as to how a MARMA model can be used to reflect daily futures resettlement, incorporate uncertainty associated with resettlement, and address sub-Gaussian statistical properties. Our treatment starts with eq. (l), which is essentially a restatement of Morgan’s (1981) eq. (4).6 This relationship depicts the cash flows that would arise from offsetting positions simultaneously taken in forward and futures contracts sharing the same delivery date. ’ Assume that a long position in a single forward contract is taken against a corresponding futures sale. With the futures contract marked-to-market at the end of each business day, 6The notation used in eq. (1) corresponds to the widely known notation employed by CIR (1981) in their seminal treatment. As noted earlier, the Morgan (1981) and CIR (1981) models yield identical inferences when the latter approach is adjusted to reflect futures indivisibility. ‘This arbitrage relationship is used by Morgan (1981) and CIR (1981) to illustrate the factors underlying forward-futures price differences, and it has an institutional counterpart. Class B members of the IMM engage exclusively in arbitrage between the IMM futures and interbank forward markets.

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positive (negative) resettlements are invested (financed) over the term remaining until both contracts mature. We note that the term on the left-hand side of the equation equals the value of the right-hand side and that the righthand term is stochastic. In circumstances of risk-neutral valuation (as under the CIR assumptions), the right-hand side of (1) would correspond solely to the expected value of the summation. Let s-1

P(r)C-G(r)+H(r)]=V

r=j

- 1 [H(j+l)-H(j)][R(j+l)]

1, (1)

where P(r)

GO) H(r) H(j) s t

N.i+ 1)

P(N - G(t)+ H(t)1 -xfZf ;xfr:

[H(j+l)-H(j)] CH(j+ I)-H(j)lCNj+

t

l)]

= the price at time c of a default-free discount bond, = the forward price at period t, = the futures price at period t, = the futures price at period j, = the maturity date of the forward and futures contracts, = the present time period, =a term percentage rate-of-return from period j+ 1 to s- 1, = the current value of the forwardfutures price differential, = the futures resettlement mechanism, = the net interest payout, = represents the valuation function that incorporates riskiness as well as expected return, =j, j+l,..., s-l.

As set forth by eq. (l), forward and futures prices ‘must differ by virtue of the daily resettlement’ [Morgan (1981)]. In other words, forward and futures prices will diverge by the value of the net interest payout, a value that is uncertain at time r and will be nonzero in the general case. 4.2. The operational

incorporation

of uncertainty

In the context of the Morgan (1981) model, arbitrage represents the fundamental basis for differential pricing between forward and futures contracts. The forward payout is certain, however, the precise value of the net interest portion of the futures payout cannot be arbitraged away since its value is not known with certainty when the futures position is initiated. Since this payout plays a crucial role in determining differences between forward

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and futures prices, treatment of traders’ expectational, or forecasting, processes represents a critical step in extending the stochastic relationship set forth in eq. (1) into a suitable empirical form. In order to emphasize the role of expectations as they influence relative forward-futures price relationships, we restore eq. (1) in the following form: Let s-

p(t)C-G(O+W)l=E,

-

1

1 &CWi+ I)-JW)lCW+ 111, t=j

(2)

1

where E, = the expectations operator, 0, =discount factor for risk. If FX forward-futures arbitrageurs are indeed risk averse, the potential dispersion, or total risk, surrounding non-arbitrageable expected interest payouts arising from resettlement should also be considered as a possible influence on the contracts’ relative values. The discount for riskiness may be stationary over time or may vary with the level of expected resettlements. Even if the latter condition is the case and the expected level of resettlements played no role by itself (or if the expectation was zero), resettlement would still have an influence on the forward-futures differential. To the best of our knowledge, Denis’ (1976) is the only direct analysis of the state of cross-market efficiency implied by price levels of FX forward and futures contracts. His findings are consistent with a state of efficiency. We have performed Granger causality tests on price changes taking place in the FX forward and futures markets to determine if significant lead-lag relationships emerge.’ Systematic evidence of such relationsips is not demonstrated, a condition again consistent with a state of cross-market efficiency. It has been well established that the econometric structure of MARMA time-series models is consistent with rational expectations forecasts of stochastic processes;9 therefore, linear combinations of historical values corresponding to the model’s endogenous and exogenous variable may be used to reflect the first term on the right-hand side of (2).” For the purpose of this investigation, no harm is done by modeling the risk as a constant sThe results of these tests are available from the authors upon request. ‘The structural consistency between rational expectations and Box-Jenkins (1970) or MARMA, processes has been addressed by the literature [Zellner and Palm (1974)] and succinctly restated by Kokkelenberg and Bischoff (1986): ‘the Box-Jenkins method of time series analysis to ascertain and represent the stochastic processes of concern . . . is consistent with rational expectations’. “Under the rational expectations approach, all information potentially is useful within the context of an economic model [Kokkelenberg and Bischoff (1986)]. Accordingly, information provided by past values of the endoaenous variable should not be eliminated methodoloaicallv From the modeling process.

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term because any relationship with forecast levels will be captured by the independent variable and indicate a role for resettlement. As an alternative to methods employed in the past, we use a MARMA process to test FX forward and futures price differences. The model is a system of stochastic difference equations that is used both to reflect daily resettlement as it occurs under actual trading conditions and as it is fundamentally represented by economic theory. The informational conditions corresponding to a state of efficiency are theortically tantamount to a rational expectations premise [Sheffrin (1983)], and the MARMA model is consistent with rational expectations models proposed by Sargent (1987) and Zellner and Palm (1974). In the context in which it is applied in this paper, the rational expectations treatment of informational uncertainty is appropriate and provides a basis for bridging difficulties involved in empirically representing theoretical models of forward-futures price divergence. For MARMA processes, Box and Jenkins (1970) specify procedures that transform sub-Gaussian data into series which satisfy the classical linear assumptions. As a result, straightforward statistical inference of parameter estimates is ensured. The Box-Jenkins procedures may be applied in a manner that does not compromise fundamental theoretical relationships. Thus, we believe the MARMA approach to measuring FX forward and futures price differentials affords a harmonious integration of measurement technique with pricing and uncertainty relationships. The bivariate stochastic reduced form set forth in eq. (3) serves as an operational version of eq. (2) and is the assumed expression for our rational expectations treatment of uncertainty. Lag orders on the net interest payout are substituted for the expectations operator on the net interest payout depicted in eq. (2). The basis of the model’s structural lag orders are dictated by an econometric implementation of rational expectations forecasts of future net interest payouts. l1 We have also incorporated a constant term in order to detect the presence of risk-aversion.” Let 1’,=c,-A(L)_‘B(L)x,+u,,

(3)

“Box and Jenkins (1970) specify statistical procedures that can be implemented for the purpose of identifying lag structures on the dependent and independent variables of a MARMA function. This process can result in the elimination of recent lag orders. We did not employ this portion of the Box-Jenkins (1970) methods to identify ARMA orders on structural variables. Because the rational expectations premise dictates that all information is important, short-term lags should not be precluded statistically from economic consideration. Thus, only when purely statistical issues are dealt with, as is the case in section 5, do we make statistical adjustments in the manner articulated by Box and Jenkins. The preceding approach provides a firmer economic basis for MARMA modeling [Pindyck and Rubinfeld (1981)]. “Park and Chen (1985) include the same term in their regression analysis for precisely the same rationale as ours. In theory, this term can be negative. With respect to the total risk premium under investigation in their paper and ours, no systematic support for its presence is found.

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where Y,

=J’(t)C-G(t)+WOl,

cf

= the resettlement risk premium,

B(L)x,

: the weighted lagged value of interest payouts,

4

an estimate/proxy = the error term, = the lag operator,

L A(L) B(L) GP 6s

A(L), B(L)

employed as for the expected resettlement payout,

=(l-ac(1L-a2L2--~~-a,L’), =(Bo-BlL-lj2L*--..--B,L”), = lag Farameters,

= lag orders, : the generalized lag structures associated with MARMA timeseries forecasts of uncertainty processes.

On the left-hand side of eq. (3), y, corresponds to values of actual forward-futures price differentials. As noted in section 3, maturity matching of forward and futures contracts precludes comparison of these instruments’ prices on a daily basis. On the right-hand side of eq. (3), x, corresponds to the net interest payout for a given time period. This term replicates futures resettlement and is compiled on a daily basis. The time series of summed x, terms is maturity matched against corresponding yt terms. To illustrate, suppose y, represents a price differential with each forward and futures component having a three month term-to-delivery from time j to s- 1. If x, could be known with certainty, it would correspond to the sum of daily resettlement proceeds, or net interest payout, occurring for time j to s- 1. Because this knowledge is not attainable at time j, B(L)x, represents a forecast of the net interest payout for j to s- 1, based on the sums of threemonth values of x, from past periods of j to s- 1. The B(L) lag structure on x, is the operator by which expected values of the net interest payouts are extrapolated from historical data into the current period j to s- 1. Embodied in past actual values, information provided by the data characteristics of the historical x, series is extrapolated forward as a basis for expectations of x, that will occur during the present time period j to s- 1. From a theoretical perspective, this treatment represents a rational expectations forecast of x, under conditions of uncertainty. Econometrically, the timeseries pattern of past values of x, is being used to explain behavior in the current forward-futures price differential, or y,. In reduced form, the A(L)-’ lag structure appearing on the right-hand side of eq. (3) is to be interpreted as a lag order on y,. From an econometric perspective, lag orders on y, account for variance (in y,) not otherwise explained by x,. More importantly, however, consistency with a rational expectations treatment of uncertainty dictates the use of all information provided within the context of an economic model [Kokkelenberg and

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Bischoff (1986)]. Therefore, while our primary concern is demonstrating the pricing influence of net interest payouts arising from resettlement, information provided by past forward-futures price relationships is also modeled in order to retain theoretical consonance. Together, lag orders on x, and y, provide the most information attainable within the context of economic theory; that is, application of the rational expectations treatment of uncertainty to the Morgan (1981) framework. Hence, eq. (3), the structural portion of our model, represents the current value of differences between equilibrium forward and futures prices as functions of the expected value of the net interest payout, uncertainty surrounding this expected value, and the informational content provided by recent forward-futures price relationships.

5. Extension 5.1.

into measurement

form

The data

Forward and futures prices were obtained from the International Monetary Market (IMM) for the period May 1976December 1984. For this period, forward prices were provided to the IMM by Continental Illinois Bank. FX futures trading was initiated by the IMM on May 16, 1972; however, Cornell and Reinganum (1981) observe that volume was relatively low until mid-1974. Trading volume was considerably greater by 1976, therefore we chose to begin our sample at this later date.i3 Thirty, 60, 90, and 180-day maturities were used for British pound, German mark, Japanese yen, and Swiss franc forward and futures contracts, resulting in estimation of 16 separate forward-futures price relationships. Consistent with Cornell and Reinganum (1981), one-year contracts were not included due to the potential for tax-induced pricing distortions.‘4 Following the procedure used by Cappoza and Cornell (1979), the expense 13The Park and Chen (1985) and Cornell and Reinganum (1981) studies are the most noteworthy examinations of comparable FX forward-futures price differences. Systematic evidence of signiticant differentials is not found in either case. Had our data set covered a very different time frame from the Park and Chen (1985) paper, we feel it would be more difficult to conclude that our results do not arise from the use of a significantly different data set. Therefore, we constrained our data sample to the period represented by the more recent Park and Chen (1985) paper, subject to a desire for the data to cover a period of higher trading volume. iAMoreover, we adopted an adjustment employed by Cornell and Reinganum (1981) (who also use IMM data) to avoid overstating forward-futures price differentials. The mean forward bid-ask spread was computed from periods when this information was available. One half of this spread was added to the bid side of the forward price. In the rare instances where price limits were reached in the futures market, these data were excluded from the computation of bid-ask means. Cornell and Reinganum note that IMM forward quotes are for 1:00PM, Chicago time, while IMM futures quotes range from 1: 15I:25 PM, Chicago time. Their view is that these reporting discrepancies will not give rise to systematic biases.

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and P.C. Grier, Foreign exchange forward and futures prices

1069

portion of the interest payouts was obtained by applying Treasury bill yields plus a 50 basis point spread against the negative proceeds of daily futures revaluations.15 The interest earned component of the payout was obtained by applying Treasury bill rates against positive daily futures revaluations. Interest rate data were obtained from the Board of Governors of the Federal Reserve System and the Wall Street Journal. Trading fees for major players in the FX forward market consist of the bid-ask spread, which typically ranges from 0.08% to 0.13% of the value of European currencies [Denis (1976)]. On a dollar basis, the level of this range is slightly greater than the $45 commission charged by IMM exchange members for processing customers’ round-turn FX futures trades [Denis (1976)]. However, IMM clearing members are assessed an even lower charge by the clearing corporation for trade processing. It appears that the futures market offers lower transactions costs for major players, a condition consistent with theoretical treatment addressing the rationale for futures markets [Silber (1985), Telser (1977)]. While it is reasonabe to suspect that transactions cost disparities could affect FX forward-futures pricing differentials, quantification of these transactions cost differentials is not a straightforward matter,16 and prior studies do not treat this issue as material. 5.2. Empirical form of the model If appropriate statistical inferences are to be made subsequent to estimation, the characteristics of the underlying data must be adjusted for subGaussian characteristics. In this section, the model presented in eq. (3) is rsIn assuming that markets are perfect, CIR (1981) use Treasury bills to represent the financing (reinvestment) costs (returns) arising from futures resettlement. In actual practice, Cappoza and Cornell (1979) note that government securities dealers charge 50 basis points over the Treasury bill rate for financing costs, although Allen and Thurston (1988) maintain that the term repo rate is a more accurate reflection of financing costs in the government bill market. We have opted to use Cappoza and Cornell (1979) proxy for linancing spreads due to the fact that FX market participants typically do not hold large inventories of government bills available for repos or reverses. Our spread proxy aligns mean interest payouts more closely with mean forward-futures differentials. To illustrate, the mean net interest payout corresponding to the time series of resettled 90-day British pound futures prices is 0.00096 when the 50 basis point factor is expunged from negative resettlements. Inclusion of this spread results in a mean net interest payout of 0.00068. The mean of 90-day British pound forward-futures price diNerentials used in our sample is 0.00058. i61nasmuch as tests of FX forward-futures price differences have not revealed systematic divergence, it is implicit that execution costs disparities are also statistically insignificant. While the results of these tests are open to question on measurement grounds, there is difficulty in appropriately measuring transactions costs differentials in the FX forward and futures markets [Denis (1976)]. Forward spreads fluctuate as do commissions for processing futures trades. Moreover, the concept of a forward spread is elusive. In order for the spread to be ‘locked-in’, a buyer and seller would have to be simultaneously matched for the same curency, maturity date, and price. According to Denis (1976). such precision matching is seldom possible.

J.B.F.-B

1070

M.A.

Polakoff

and P.C. Grim,

Foreign

exchange forward

and futures

prices

extended into a form that accounts for properties exhibited by the FX data under consideration. As pointed out previously in section 3, y, (the forward-futures price differential) must be maturity matched, a process that cannot properly be conducted on a daily basis. When y, data points are obtained, subject to the maturity-matching constraint, we find that these points are stationary, but often are correlated. Representing a summed series of daily futures resettlements applied against Treasury bill yields (plus a 50 basis point spread in the case of negative resettlements), each x, term corresponds to the net interest payout for a given time period. From a statistical standpoint, the futures resettlements necessary to obtain x, result in distributional stationarity. Stationarity in the x, and yI series was confirmed through checks against the 10% critical level of the Box-Pierce Q-statistic.” Although the x, and y, series are stationary, correlation in each series frequently is demonstrated. This condition must be addressed if efficient estimators are to be obtained. Box and Jenkins (1970) suggest that the efficiency of MARMA estimation can be improved if the correlated x, series is first prewhitened by an ARMA filter statistically determined by its partialand full-autocorrelation functions. We denote the prewhitening of x, in eq. (4).

P(+, = dL)e,

(4)

(5) where p and q are autoregressive respectively, e, is white noise, L is the lag operator.

(AR) and moving-average

(MA) lag orders,

To preserve equational consonance and as specified by Box and Jenkins (1970), the remainder of eq. (3) is multiplied by q-‘(L)p(L) to yield

4(L) - lP(L)Y, = 4(L) - 'Pv-k, - A(L) - l WML) +

“Statistical upon request.

support

- 'PWX, (6)

t?(L)- 'Pw49

for conditions

not presented

in this section

are available

from the authors

M.A. Polokoff and P.C. Grier, Foreign exchange forward and furures prices

yj=cj-A(L)-‘B(L)xj+o,,

1071

(7)

where , YI

=4(L) - 'P(QY,* =4(L) - ‘P(mt7

ct -A(L) 4

- ‘B(L)xj

=

-A(L)_‘B(L)q(L)_‘p(L).u,,

= q(L)_

‘P(m4.

Hall, Brorsen and Irwin (1989) find that the variance of FX futures price series is serially correlated. They note that ‘classical statistical methods may be validly applied to most daily futures price series if an adjustment for heteroskedasticity is made to the data’. Typically, a generalized least squares (GLS) technique is employed for the purpose of addressing conditions of heteroskedasticity. The procedure undertaken in eqs. (4)-(6) above is analogous to the application of a GLS technique when confronting heteroskedastic or correlated disturbances in the covariance matrix of the standard linear model [Johnston (1984)]. When GLS is applied due to the presence of a non-diagonal covariance matrix, the structural coefficients of the underlying standard linear model are not compromised [Theil (1971)]. Similarly, the structural coefficients of eq. (3) corresponding to the x, term are not compromised by the prewhitening process when the remainder of the equation is treated by the identical filter [Johnston (1984)]. In addition to exhibiting stationarity for the individual y, and x, series, the relationship between these terms was revealed as bivariate stationary for all currencies and maturities tested. This condition was confirmed by BoxPierce Q-statistic checks against the crosscorrelation function. As required, an ARMA process (of order m, n) was also applied to correlated u, terms. Based on eq. (7), the estimated reduced forms exhibit white-noise processes with respect to disturbances and non-correlation of residuals with the exogenous series, again, as verified through the use of the Q-statistic. Filter and residual-correction orders are presented in table 2 for each reduced-form relationship estimated for marking-to-market effects.

6. Structural coefficients of estimation The full results of estimation are set forth in table 3. In cases where parameter estimates (other then risk premium terms) were small in relation to their standard errors, they were dropped. This omission does not affect the accuracy of the remaining parameter estimates [Box and Jenkins (1970)], and

M.A. Polakoff and P.C. Grier, Foreign exchange forward and futures prices

1072

Table Crosscorrelation To lag

Chi square

Degrees of freedom

Critical Q-value

5 11

9.37 12.90

6 12

10.64

check

1

between

x, and y, series.’

BP30 crosscorrelations 18.55

- 0.046 -0.019 DM30

5 11

4.20 8.93

6 12

10.64 18.55

0.067 -0.007

0.176 -0.305

0.147 0.034

0.108 -0.010

- 0.079

0.159 -0.125

0.123 -0.214

- 0.079

0.043 0.112

0.284 0.090

0.011 - 0.009

-0.140

-0.144 0.271

0.079 0.251

0.232 - 0.099

-0.185

-0.124 0.005

0.091 0.103

-0.019

0.155 - 0.076

0.110 - 0.008

-0.024 0.250

-0.113

0.005 0.056

- 0.062 -0.101

0.111 0.255

0.030

-0.041 -0.117

0.380 -0.043

- 0.022 0.052

0.411

-0.030 0.030

0.097 - 0.270

- 0.045 0.174

0.178

0.309 0.032

-0.181 0.091

0.114 0.042

- 0.050

0.058 0.250

0.114 0.156

-0.077 0.103

-0.099 0.060

0.104 - 0.082

0.020 -0.066

0.139

0.251 0.146

0.027 -0.181

- 0.054

-0.126 0.007

0.060 -0.034

-0.071

crosscorrelations

0.088 0.05 1

0.074 0.184

0.02 1 0.198

JY30 crosscorrelations 5 11

3.09 5.32

6 12

10.64 18.55

-0.041 -0.097

- 0.047 -0.134

SF30 crosscorrelations 5 11

5.20 13.02

6 12

10.64 18.55

0.054 0.218

0.149 -0.044

BP60 crosscorrela:ions -0.235 0.060

5 11

11.63 13.34

6 12

10.64 18.55

5 11

2.37 5.29

6 12

IO.64 18.55

-0.107 0.042

IO.64

JY60 crosscorrelations 0.046 0.204 -0.196 0.590

DM60

5 11

2.32 17.69

6 12

18.55

0.387 0.094

-0.218 -0.164

crosscorrelations -0.026 0.090

SF60 crosscorrelations 5 11

10.41 18.39

6 12

10.64 18.55

-0.239 0.189

0.244 0.139

BP90 crosscorrelations 5 11

5.36 10.76

6 12

10.64 18.55

-0.265 - 0.045 DM90

5 11

12.38 13.53

6 12

10.64 18.55

5 11

2.01 5.81

6 12

10.64 18.55

5 11

5.26 7.09

6 12

10.64 18.55

5 11

7.77 10.31

6 12

10.64 18.55

- 0.454 0.089

0.272 0.144

crosscorrelations 0.075 -0.116

JY90 crosscorrelations -0.144 0.042

-0.065 -0.113

-0.001

SF90 crosscorrelations 0.131 -0.132 BP180 -0.416 -0.148 DM180 5 11

4.61 14.38

6 12

10.64 18.55

-0.158 -0.547

-0.276 0.048

crosscorrelations -0.117 -0.052

0.013 -0.020

crosscorrelations -0.107 0.003

-0.171 0.018

M.A. Polakoff and P.C. Grier, Foreign exchange forward and futures prices Table

1073

1 (continued)

To lag

Chi square

Degrees of freedom

Critical Q-value

5 11

0.08 17.33

6 12

10.64 18.55

JY 180 crosscorrelations -0.021 -0.010 -0.024 0.017 -0.756 0.032

0.010 0.012

0.038 -0.029

-0.025

10.64 18.55

SF180 crosscorrelations 0.093 - 0.074 0.039 -0.018 -0.051 -0.019

0.063 0.03 1

-0.060 - 0.029

- 0.022

5 I1

0.76 0.94

6 12

=BP = British pound, DM = German mark, JY = Japanese yen, SF = Swiss franc, 30, 60, 90, and 180 refer to model maturities, 10% is used as the cutoff for critical Q-values.

Table 2 Statistical

BP30 DM30 JY30 SF30 BP60 DM60 JY60 SF60 BP90 DM90 JY90 SF90 BP180 DM180 JY180 SF180

adjustments

on reduced eq. (7)’

forms

represented

by

ARMA order (p q) Filter on x,

ARMA order (m n) Error correction on u,

None (1 0) None None None (1 0) None None (0 2) (2 0) None (1 0) (1 0) (1 0) None (0 1)

(3 0) (3 2) (0 1) (0 1) None None None None None (5 0) (6 0) None None None None None

‘BP = British pound, Japanese yen, SF=Swiss model maturities.

DM = German mark, JY = franc, 30, 60, 90, and 180 refer to

it accounts for the model forms presented below.‘s Estimation execution was conducted using the method of maximum likelihood. Investigation of the marking-to-market effect calls for close scrutiny of the ‘*In implementing a structurally based inclusion of current and recent information, as dictated by the use of the rational expectations assumption, we initially extended to five lags the order structures on x; and y;. Greater lags were excluded. Box and Jenkins (1970) state that such truncation procedures are an appropriate approximation of the entire structure. On the initial

1074

M.A.

Polakoff

and P.C. Grier,

Forrifn

crrhange

forward

andjbrures

prices

influences exerted by the net interest-payout and risk-premium terms.” Of the I6 estimations undertaken, interest payouts are signi~cant in I2 cases: nine at the 5% and three at the 107; confidence levels. Examination of the significant lag orders associated with the interest payouts reveals wide disparities across the differing currencies and maturities tested. Rational expectations based forecasts for different currencies and maturities do not have to be formed in an identical fashion, so the lag-order heterogeneity encountered is to be expected. If expectations were identical with respect to factors generating movements in 90-day Swiss franc and British pound futures prices, identical or equivalent lag structures (our rational expectations forecast for future net resettlement payouts) should emerge for the two currencies’ forward-futures rate differentials. However, it is not the case that the two currencies are identically affected by the same economic factors. Neither do identical participants conduct transactions across the range of major currencies traded. Thus, homogeneous expectations for differing currencies by distinct sets of players would be implied by a preponderance of similar, structural lag orders. Given the heterogeneous currencies and maturities under examination, expectation homogeneity does not appear to be an intuitively reasonable assumption. Fifteen of the 16 estimations exhibit insignificant risk premia. A significant risk premium emerges for the 90-day Japanese yen model. While time series studies of daily FX forward and futures prices document the systematic presence of risk premia [Hodrick and Srivastava (1984, 1987); Samuelson (1965)], these risk premia are not comparable to those considered in analyses that directly contrast forward and futures prices. Because of the maturitymatching constraint discussed earlier (section 3), separate examinations of daily forward and futures time series cannot be used for direct forwardfutures price comparisons. In the context of forward-futures arbitrage

five-order lag-structures, those parameter estimates that emerged as sig~i~cant, or nearly significant, were retained. Those that were quite small in relation to their standard errors were dropped. Box and Jenkins (1970) maintain that this step can be instituted ‘without aflecting the estimates of the remaining parameters to the accuracy considered: The significant structural lag orders reported demonstrate the pervasiveness of resettlement pricing effects, but we note that these orders are not unique. While Tiao and Tsay (1988) propose methods for fmding unique parameterizations of vector stochastic processes, their approach does not admit underlying economic relationships. instead, a pureIy statistical procedure is employed. Even if such an approach were deemed economically appropriate for our application, there is no procedure yet developed that affords unique structural parameterizations for MARMA models. Finally, we observe that rational expectations theory does not provide an intuitive basis for interpreting the importance or magnitude of one significant information subset (lag-order coefficient) relative to another. lgThe time series of forward-futures price differentials was subjected to the standard r-test as previous researchers had done. The results were comparable in terms of lack of sign&ant differences, indicating that the different results obtained here are not attributable to a different data set.

M.A.

Polakoj’and

P.C. Grier.

Foreign

Table Structural Parameters:

c;

pi-1

BP30 Estimate Std. error f-stat. y; Obs.: 35

0.07 0.27 0.27

3.38 1.36 2.48*

DM30 Estimate Std. error f-stat. y; Obs.: 33

0.003 0.07 0.04

3.48 1.90 1.s4**

JY30 Estimate Std. error t-stat. y: obs.: 33

0.0002 0.008 0.03

5.31 3.60 1.47

SF30 Estimate Std. error t-stat. y: Obs.: 34

7.29E-‘2.36E-s 0.0013 1.35E-* 0.06 1.75;;

BP60 Estimate Std. error t-stat. y; Obs.: 34

0.0014 0.0027 0.52

-0.004 0.07 0.06

0.86 0.44 1.94**

JY60 Estimate Std. error t-stat. y; Obs.: 33

-0.002 0.008 -0.22

0.96 0.40 2.38’

0.0006 0.0010 0.60

BP90 Estimate Std. error t-stat. y: Obs.: 34

0.0009 0.0019 0.47

DM90 Estimate Std. error t-stat. y; Obs.: 33

0.034 0.036 0.95

Pi-.,

prices

of estimation.’ z;_,

z;_?

z;_,

-0.82 0.12 - 6.67

- 1.23 0.59 - 2.07

-0.78 0.08 - 10.14

-1.04 0.07 - 14.10

-0.74 0.09 - 8.05

0.26 0.10 2.56

0.79 0.27 2.94

0.47 0.23 2.08*

0.45 0.17 2.62.

-0.61 0.23 -2.61*

0.61 0.18 3.49*

-0.81 0.14 - 5.61

1075

3

p;_‘% p;_>

0.75 0.34 2.21*

0.38 0.13 2.87’

and futures

0.92 0.30 3.07

1.23 0.59 2.07*

DM60 Estimate Std. error t-stat. y; Obs.: 32

SF60 Estimate Std. error t-stat. y: Obs.: 33

K-2

coelkients

exchange forward

1.14 0.12 9.82

0.60 0.10 5.50

z;-.$

z;-s

WA. PD~uko~ and P.C. Grier. Foreigrz exchange forward

1076

c;

JY90 Estimate Std. error r-slat. y; Ohs.: 34

- 0.0009 0.0043 -2.18’

SF90 Estimate Std. error f-stat. y; Ohs.: 34

-0.ooQ7 0.0023 -0.30

BP180 Estimate Std. error t-stat. y; Ubs.: 3% DMt8O Estimate Std. error t-stat. y: Obs.: 32 JY180 Estimate Std. error t-stat. y: Gbs.: 30 SF180 Estimate Std. error t-stat. y; Obs.: 32

prices

Table 3 (continued)

-Parameters:

a~df~rares

/?;_ $

O.o#&

-0.13

-0.4t 2.25 -0.18

O”oQ2 0.011 0.18

fz;_,

iz

1.13 0.10 11.28

- 0.23 O.lQ - 2.30

( 5-j

--z;_.$ Lx;_, _.I-

1.57 0.73 2.14%

0.0014 0.24 0.0026 0.11 0.54 2.1@

-o.Qmi

g-2 ------yrz:: - it; - J

0.05 0.02

2.47’

-0.98 0.13 -7.52

- 1.01 0.19 - 5.43

- 1.17 0.25 -4.61

- 1.05 0.32 -3.31

-0.93 0.33 -2.86

-0.97 0.24 - 7.02

- 1.23 0.44 - 2.80

- 0.99 0.44 - 2.28

*c~=r~e~tleme~t risk-premium parameter at t =0, ~~=parameter on lagged net interest payouts {to the fifth lag order on x;), z;=parameter on lagged current valued forward-futures price differences (to the !ifth lag order on yi), yb= total observations of current valued forward-futures price differences at r=O, * -5% level of significance, ** = IOP;;,level of significance, y;=coefIicients are not highlighted, BP=British pound, DM= German mark, JY =Japanese yen, SF=Swiss franc, 30, 60, 90, and 180 refer to model maturities.

relationships presented by Morgan (19811, the risk premium analyzed in this study represents a potential risk-averse pricing influence that may result from uncertainty surrounding the value of future net interest payouts. These uncertain payouts are generated by futures resettlement. From an arbitrage perspective, it is the net interest payout that represents the sole characteristic not shared by comparable forward and futures contracts.

M.A.

Polakqfand

P.C. Grier, Foreign

exchange forward

and futures

prices

1077

We note that Kamara (1988) has documented the presence of differential liquidity and default risk premia arising between the Treasury bill forward and futures markets. He attributes this disparity to the influence of distinct trading structures found in the separate markets. This condition does not apply in the case of FX [Cornell and Reinganum (1981)]. As a result, Cornell and Reinganum (1981) conclude that the FX forward and futures markets have comparable default risk. If systematic evidence of risk premia was found in our analysis, it might be argued that the origin of these terms should be attributed to factors considered by Kamara (1988). Since no systematic evidence of risk premia is uncovered, the basis of such a discussion appears moot. Insight into the general absence of resettlement risk premia may be provided by the work of Dusak (1973), whose capital asset pricing framework demonstrates zero systematic risk for wheat, soybean, and corn futures contracts. The risk premium of this study has been addressed from the totalrisk perspective dictated by arbitrage arguments. If Dusak’s (1973) findings apply to FX futures contracts, it is entirely consistent that evidence of systematic risk premia would fail to materialize in the course of our estimations.

7. Summary and conchsions There are well established theoretical bases upon which to expect statistically significant pricing differences between comparable FX forward and futures contracts; however, empirical evidence of FX resettlement effects has eluded prior testing approaches. As an alternative to these earlier tests, this paper employs a MARMA process to investigate the pricing influences of marking-to-market. Inasmuch as earlier empirical models do not directly test resettlement effects and ignore important statistical properties of the data under examination, there are several advantages associated with the use of a MARMA framework. The proceeds from daily resettlement can be represented as they are dealt with theoretically and as they occur under actual trading conditions. The apparent state of cross-market efficiency demonstrated between the FX forward and futures markets is consistent with modeling uncertainty accompanying resettlement-generated, net interest payouts according to a rational expectations premise; and, in turn, the econometric form of the MARMA process is consistent with rational expectations models developed in the economic literature. For MARMA processes, Box and Jenkins (1970) specify procedures that transform sub-Gaussian data into series which satisfy

1078

M.A. Polakoff and P.C. Grier, Foreign

e.vhange_fortiard

and futures

prices

As a result, straightforward statistical the classical linear assumptions. inference of parameter estimates is ensured without compromise of fundamental structural relationships dictated by theory. It has been our intention to implement a procedure that jointly integrates the economic and statistical issues of concern that arise in the area of FX forward-futures price measurement. To this end, we believe a MARMA application represents an appropriate methodological alternative to earlier approaches. Using foreign currencies, contract maturities, and sample periods similar to those analyzed in previous studies, the methodology applied in this paper yields very different results. Twelve of our 16 estimations demonstrate

significant disparities in price between FX forward and futures contracts. By way of contrast and as stated earlier, Park and Chen (1985) found four statistically significant FX forward-futures price differentials out of 20 tested. Cornell and Reinganum (198 1) found two out of 20 such differences. Using a similar methodology as those studies, comparable results were obtained. Thus, it is fair to say that our findings are generated by the use of a differing methodology and not a qualitatively distinct sample set. On the basis of the results demonstrated in this paper, there is now empirical support for theoretical treatment of the resettlement issue.

References Allen, L. and T. Thurston, 1988, Cash-futures arbitrage and forward-futures spreads in the treasury bill market, The Journal of Futures Markets 8, Oct., 563-573. Baille, R. and T. Bollerslev, 1989, Common stochastic trends in a system of exchange rates, Journal of Finance 19, March, 167-181. Black, F., 1976, The pricing of commodity contracts, Journal of Financial Economics 3, Jan., 167-179. Box, G. and G. Jenkins, 1970, Time series analysis: Forecasting and control (Holden-Day, London). Branch, B., 1978, Testing the unbiased expectations theory of interest rates, The Financial Review 13, Fall, 51-66. Cappoza, D. and B. Cornell, 1979, Treasury bill pricing in the spot and futures markets, The Review of Economics and Statistics 61, Nov., 5 1)_520. Chow, B. and Brophy, D., 1978, The U.S. treasury bill futures market and hypotheses regarding the term structure of interest rates, The Financial Review 13, Fall, 36-50. Cornell, B. and M. Reinganum, 1981, Forward prices and futures prices: Evidence from the foreign exchange market, The Journal of Finance 36, Sept., 1035-1045. Cox, J., J. Ingersoll Jr. and S. Ross, 1976, The valuation of options for alternative stochastic processes, Journal of Financial Economics 3, Jan., 145-166. Cox, J., J. Ingersoll Jr. and S. Ross, 1981, The relation between forward prices and futures prices, Journal of Financial Economics 9, Dec., 321-346. Denis, J., 1976, How well does the IMM track the interbank forward market?, Financial Analysts Journal, Jan., 50-54. Dusak, K., 1973, Futures trading and investor returns: An investigation of commodity market risk premiums, Journal of Political Economy 81, Dec., 1387-1406. Hail, J., B. Brorsen and S. Irwin, 1989, The distribution of futures prices: A test of the stable Paretian and mixture of normals hypothesis, Journal of Financial and Quantitative Analysis 24, 105-116.

M.A.

Polakojf and

P.C. Grier,

Foreign

exchange forward

and futures

prices

1079

Hodrick, R. and S. Srivastava, 1984. An investigation of risk and return in forward foreign exchange. Journal of International Money and Finance 3, 5-29. Hodrick, R. and S. Srivastava, 1987, Foreign currency futures, Journal of International Economics 22. l-24. Jarrow, R. and Cl. Oldlield, 1981, Forward contracts and futures contracts, Journal of Financial Economics 9, Dec., 373-382. Johnston, J., 1984, Econometric methods, third edition (McGraw-Hill, New York). Kamara, A., 1982, Issues in futures markets: A survey, The Journal of Futures Markets 2. 261-294. Kamara, A., 1988, Market trading structures and asset pricing: Evidence from the treasury-bill markets, The Review of Financial Studies 1, Winter, 357-375. Kane, E., 1980, Market incompleteness and divergence between forward anf futures interest rates, The Journal of Finance 35, May, 221-234. Kokkelenberg, E. and C. Bischoff, 1986, Expectations and factor demand, The Review of Economics and Statistics 68, Aug., 423-431. Lang, R. and R. Rasche. 1978, A comparison of yields on futures contracts and implied forward rates, The Federal Reserve Bank of St. Louis Review 60, Dec., 21-30. Levy, A., 1989. A note on the relationship between forward and futures contracts, Journal of Futures Markets 9, April, 171-173. Miller, E., 1980. Tax-induced bias in markets for futures contracts, The Financial Review 15, Spring, 35-38. Morgan, G., 1981, Forward and futures pricing of treasury bills, Journal of Banking and Finance 5, Dec., 483496. Park, H. and A. Chen. 1985, DiNerences between forward and futures prices: A further investigation of the marking-to-market effects, The Journal of Futures Markets 5, Spring, 77-83. Pindyck, R. and D. Rubinfeld, 1981, Econometric models and economic forecasts, second edition (McGraw-Hill, New York). Polakoff, M., 1991, A note on the role of futures indivisibility: Reconciling the theoretical literature, The Journal of Futures Markets, forthcoming. Polakoff, M., 1987, Forward-futures price divergence in the market for foreign exchange, Ph.D. dissertation, State University of New York at Binghamton, Binghamton, New York. Poole, W., 1978. Using T-bill futures to gauge interest-rate expectations, The Economic Review of the Federal Reserve Bank of San Francisco, Spring, 7-19. Puglisi, D., 1978, Is the futures market for treasury bills efficient?, The Journal of Portfolio Management 4, Winter, 64-67. Rendleman Jr., R. and C. Carabini, 1979, The efficiency of the treasury bill futures market, The Journal of Finance 34, Sept., 895-914. Richard, S. and M. Sundaresan, 1981, A continuous time equilibrium model of forward prices and futures prices in a multigood economy, Journal of Financial Economics 9, Dec., 347-371. Samuelson, P., 1965, Proof that properly anticipated prices fluctuate randomly, Industrial Management Review 6, Spring, 41-49. Sargent, T.. 1987, Macroeconomic theory, second edition (Academic Press, Boston). Sheffrin, S., 1983, Rational expectations (Cambridge University Press, Cambridge). Silber, W., 1985, The economic role of financial futures, in: Anne E. Peck, ed., Futures markets: Their economic role, American Enterprise Institute for Public Policy Research, 83-l 14. So, J., 1987, The sub-Gaussian distribution of currency futures: Stable Paretian or nonstationary?, The Review of Economics and Statistics 69, 100-107. Telser, L. and H. Higinbotham, H., 1977, Organized futures markets: Costs and benefits, Journal of Political Economy 85, Oct., 969-IOOO. The& H., 1971, Principles of econometrics (Wiley, New York). Tiao, G. and R. Tsay, 1988, Model specification in multivariate time series, Working paper, Carnegie Mellon University, Jan. 1988. Vignola, A. and C. Dale, 1979, Is the futures market for treasury bills efficient?, The Journal of Portfolio Management 5, Winter, 78-81.

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M.A. Polakofl and P.C. Grier, Foreign exchange forward and futures prices

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