48 (2008) 153–174
Do the pure martingale and joint normality
hypotheses hold for futures contracts?
Implications for the optimal hedge ratios
Sheng-Syan Chen
a,∗, Cheng-few Lee
b,c, Keshab Shrestha
daDepartment of Finance, College of Management, National Taiwan University, No. 85, Sec. 4, Roosevelt Rd., Taipei, Taiwan
bDepartment of Finance and Economics, School of Business, Rutgers University, Piscataway, NJ, USA
cGraduate Institute of Finance, College of Management, National Chiao Tung University, Hsinchu, Taiwan dDivision of Banking and Finance, Nanyang Business School, Nanyang Technological University, Singapore
Received 4 October 2005; received in revised form 18 October 2005; accepted 26 October 2005 Available online 27 January 2006
Abstract
It is well known that the optimal hedge ratios derived based on the mean-variance approach, the expected utility maximizing approach, the mean extended-Gini approach, and the generalized semivariance approach will all converge to the minimum-variance hedge ratio if the futures price follows a pure martingale process and if the spot and futures returns are jointly normal. In this paper, we perform empirical tests to see if the pure martingale and joint normality hypotheses hold using 25 different futures contracts and five different hedging horizons. Our results indicate that the pure martingale hypothesis holds for all commodities and all hedging horizons except for three stock index futures contracts. As for joint normality, we propose two new tests based on the generalized method of moments, which allow for calculating multivariate test statistics that take account of the contemporaneous correlation across spot and futures returns. Our findings show that the joint normality hypothesis generally does not hold except for a few contracts and relatively long hedging horizons.
© 2005 Board of Trustees of the University of Illinois. All rights reserved. JEL classification: G13
Keywords: Hedge ratio; Pure martingale hypothesis; Joint normality hypothesis
∗Corresponding author. Tel.: +886 2 33661083; fax: +886 2 23640881. E-mail address: [email protected] (S.-S. Chen).
1. Introduction
Derivative securities such as futures contracts have been extensively used by practitioners in hedging risk exposure to commodity prices, exchange rates, interest rates, and prices of other financial securities. In the past, both academicians and practitioners have shown great interest on the issue of hedging with futures, which is evident from a large number of articles written in this area. One of the main issues in futures hedging involves the determination of the optimal hedge ratio. However, the optimal hedge ratio depends critically on the particular objective function to be optimized. Many different objective functions are currently being used. For example, one of the most widely used optimal hedge ratios is the so-called minimum-variance (MV) hedge ratio. This MV hedge ratio is derived by minimizing the variance of the hedged portfolio and it is quite simple to understand and estimate (e.g., seeJohnson, 1960; Ederington, 1979;Myers & Thompson, 1989). However, the MV hedge ratio completely ignores the expected return of the hedged portfolio. Therefore, this strategy is in general inconsistent with the mean-variance framework unless the individuals are infinitely risk-averse or the futures price follows a pure martingale process (i.e., expected futures price change is zero).
Other objective functions used in the derivation of the optimal hedge ratio include some lin-ear combinations of the expected return and risk (variance) of the hedged portfolio, where the expected return is maximized and at the same time the risk is minimized (e.g., seeHoward & D’Antonio, 1984;Cecchetti, Cumby, & Figlewski, 1988;Hsin, Kuo, & Lee, 1994). These objec-tive functions are consistent with the mean-variance framework. The optimal hedge ratios based on these objective functions can be referred to as the optimal mean-variance hedge ratios. It is important to note that if the futures price follows a pure martingale process, then the optimal mean-variance hedge ratio will be the same as the MV hedge ratio.
Another aspect of the mean-variance-based strategies is that even though they are an improve-ment over the MV strategy, for them to be consistent with the expected utility maximization principle, either the utility function needs to be quadratic or the returns should be jointly normal. If neither of these assumptions is valid, then the hedge ratio may not be optimal with respect to the expected utility maximization principle. Some researchers have solved this problem by deriving the optimal hedge ratio based on the maximization of the expected utility (e.g., seeCecchetti et al., 1988). However, this approach requires the use of a specific utility function and specific return distribution.
Attempts have been made to eliminate these specific assumptions regarding the utility function and return distributions. Some of them involve the minimization of the mean extended-Gini (MEG) coefficient, which is consistent with the concept of stochastic dominance (e.g., seeCheung, Kwan, & Yip, 1990;Kolb & Okunev, 1992, 1993;Lien & Luo, 1993;Shalit, 1995;Lien & Shaffer, 1999).
Shalit (1995)shows that if the spot and futures returns are jointly normally distributed, then the MEG-based hedge ratio will be the same as the MV hedge ratio.
Hedge ratios based on the generalized semivariance (GSV) or lower partial moments have been recently proposed (e.g., seeDe Jong, De Roon, & Veld, 1997;Lien & Tse, 1998, 2000;Chen, Lee, & Shrestha, 2001). These hedge ratios are also consistent with the concept of stochastic dominance. Furthermore, these GSV-based hedge ratios have another attractive feature in that they measure portfolio risk by the GSV, which is consistent with the risk perceived by managers, because of its emphasis on the returns below the target return (seeCrum, Laughhunn, & Payne, 1981; Lien & Tse, 2000).Lien and Tse (1998)show that if the spot and futures returns are jointly normally distributed, then the minimum-GSV hedge ratio will be equal to the MV hedge ratio.
It is clear from the above discussion that the optimal hedge ratios which are derived based on the mean-variance approach, the expected utility maximizing approach, the MEG approach, and the GSV approach will all converge to the MV hedge ratio if the futures price follows a pure martingale process and if the spot and futures returns are jointly normal. Because the MV hedge ratio is easy to understand, simple to compute, and most widely used, it is important to investigate if these two conditions hold. If these two conditions hold, then we do not have to compute various hedge ratios, because all of them will converge to the same MV hedge ratio.
In this paper, we perform empirical tests to see if the pure martingale and joint normality hypotheses hold using 25 different futures contracts. T-tests are used to test the pure martingale hypothesis that the expected return on futures is equal to zero. As for the joint normality of spot and futures returns, we develop two new tests based on the generalized method of moments (GMM).
The GMM approach is proposed byHansen (1982)and implemented byRichardson and Smith
(1993)in their study of multivariate normality in stock returns. The two new tests developed in this study allow for calculating multivariate test statistics that take account of the contemporaneous correlation across spot and futures returns. To see if the results of our tests depend on the length of hedging horizon, we also perform tests using five different hedging horizons.
We find that the pure martingale hypothesis holds for all the 25 commodities except for SP500, TSE35, and FTSE100. This is true for all the five different hedging horizons. The results suggest that ignoring the expected return in the derivation of the optimal hedge ratio does not significantly change the optimal hedge ratio except for the three stock index futures. Therefore, with the exception of a few futures contracts, the mean-variance hedge ratio would be approximately the same as the MV hedge ratio. The empirical tests for the joint normality of spot and futures returns show that the joint normality hypothesis tends to be rejected for all the 25 commodities when the length of hedging horizon is relatively short. For longer hedging horizons, the joint normality hypothesis holds only for a few futures contracts according to our two tests based on the GMM approach. Our results suggest that the hedge ratios which are derived based on the expected utility maximizing approach, the MEG approach, and the GSV approach will not converge to the MV hedge ratio for most futures contracts and for shorter hedging horizons.
The remainder of this paper is organized as follows. In Section2, alternative theories for deriving the optimal hedge ratios are reviewed. Section3 develops two new tests for the joint normality of spot and futures returns. The empirical results are presented in Section4. The paper concludes in the final section.
2. Alternative derivations of the optimal hedge ratio
Consider a hedged portfolio consisting of Cs units of a long spot position and Cfunits of a
short futures position.1Let Stand Ftdenote the spot and futures prices at time t, respectively. The
return on the hedged portfolio, Rh, is then given by: Rh=
CsStRs− CfFtRf
CsSt = R
s− hRf, (1)
where (h = (CfFt/CsSt)) is the so-called hedge ratio, and (Rs= ((St+1− St)/St) = St/St) and
(Rf= ((Ft+1− Ft)/Ft) = Ft/Ft) are so-called one-period returns on the spot and futures
posi-tions, respectively. Sometimes, the hedge ratio is discussed in terms of price changes (profits)
instead of returns. In this case, the profit on the hedged portfolio, VH, and the hedge ratio, H,
are respectively, given by:
VH= CsSt− CfFtand H= Cf
Cs
. (2)
The optimal hedge ratio (either h or H) will depend on a particular objective function to be optimized. In this section, we briefly discuss the optimal hedge ratios derived based on the MV approach, the mean-variance approach, the expected utility maximizing approach, the MEG approach, and the GSV approach.
2.1. Minimum-variance hedge ratio
The most common hedge ratio is the MV hedge ratio.Johnson (1960)derives this hedge ratio by minimizing the variance of changes in the value of the hedged portfolio as follows:
Var(VH)= Cs2Var(S)+ C 2
fVar(F )− 2CsCfCov(S, F ). (3)
The MV hedge ratio, in this case, is given by:
HJ∗= Cf Cs =
Cov(S, F )
Var(F ) . (4)
We can alternatively derive the MV hedge ratio by minimizing the variance of the return on the hedged portfolio (Var(Rh)), which is given by:
Var(Rh)= Var(Rs)+ h2Var(Rf)− 2hCov(Rs, Rf). (5)
In this case, the MV hedge ratio is given by:
h∗J = Cov(Rs, Rf)
Var(Rf) = ρ
σs σf
(6) where ρ is the correlation coefficient between Rsand Rf, and σsand σfare standard deviations of Rsand Rf, respectively.
The MV hedge ratio is easy to understand and simple to compute. However, in general the MV hedge ratio is not consistent with the mean-variance framework since it ignores the expected return on the hedged portfolio. The MV hedge ratio would be consistent with the mean-variance framework only if either the investors are infinitely risk-averse or the expected return on the futures contract is zero.
2.2. Optimum mean-variance hedge ratio
In order to make the hedge ratio consistent with the mean-variance framework, we need to explicitly include the expected return on the hedged portfolio in the objective function. For exam-ple,Hsin et al. (1994)derive the optimal hedge ratio that maximizes the following utility function:
Max
Cf V (E(Rh), σh; A)= E(Rh)− 0.5Aσ 2
where σh2= Var(Rh) and A represents the risk aversion parameter. In this case, the optimal hedge
ratio is given by:
h2= − C∗fF CsS = − E(Rf) Aσ2 f − ρσs σf . (8)
It can be seen from Eqs.(6) and (8)that if A→ ∞ or E(Rf) = 0, then h2 would be equal to
the MV hedge ratio h∗J. Therefore, the MV hedge ratio would be the same as the optimal mean-variance hedge ratio if the expected return on the futures contracts is zero (i.e., futures prices follow a simple martingale process).
2.3. Sharpe hedge ratio
Another way of making the hedge ratio consistent with the mean-variance framework is incor-porating the risk-return tradeoff (Sharpe measure) in the objective function. For example,Howard and D’Antonio (1984)consider the optimal level of futures contracts by maximizing the ratio of the portfolio’s excess return to its volatility:
Max
Cf θ =
E(Rh)− RF σh
, (9)
where RFrepresents the risk-free interest rate. In this case, the optimal number of futures positions, C∗ f, is given by: C∗ f = −Cs (S/F )(σs/σf)[(σs/σf)(E(Rf)/(E(Rs)− RF))− ρ] [1− (σs/σf)(E(Rf)ρ/(E(Rs)− RF))] . (10)
From the optimal futures position, we can obtain the following optimal hedge ratio:
h3= −
(σs/σf)[(σs/σf)(E(Rf)/(E(Rs)− RF))− ρ]
[1− (σs/σf)(E(Rf)ρ/(E(Rs)− RF))]
. (11)
Again, if E(Rf) = 0, then h3reduces to: h3= σ s σf ρ, (12)
which is the same as the MV hedge ratio h∗J.
2.4. Maximum expected utility hedge ratio
Another class of hedge ratios is based on the maximization of the expected utility derived from the hedged portfolio. For example,Cecchetti et al. (1988)derive the hedge ratio that maximizes the expected utility where the utility function is assumed to be the logarithm of terminal wealth. Specifically, they derive the optimal hedge ratio that maximizes the following expected utility function: Rs Rf log[1+ Rs− hRf]f (Rs, Rf)dRsdRf, (13)
where the density function f(Rs, Rf) is assumed to be bivariate normal. If the returns on the spot
the expected return and variance of return of the hedged portfolio. Therefore, if the futures prices also follow the pure martingale process, then the expected utility-based hedge ratio will be the same as the MV hedge ratio.
2.5. Minimum mean extended-Gini coefficient hedge ratio
Another approach to deriving the optimal hedge ratio is based on the mean extended-Gini coefficient, which can be shown to be consistent with the concept of stochastic dominance. For example,Cheung et al. (1990),Kolb and Okunev (1992),Lien and Luo (1993),Shalit (1995), andLien and Shaffer (1999)all consider this approach. It minimizes the MEG coefficient Γν(Rh)
defined as follows:
Γν(Rh)= −νCov(Rh, (1 − G(Rh))ν−1), (14)
where G is the cumulative probability distribution and ν is the risk aversion parameter. Note that 0≤ ν < 1 implies risk seekers, ν = 1 implies risk-neutral investors, and ν > 1 implies risk-averse investors.Shalit (1995)shows that if the futures and spot returns are jointly normally distributed, then the minimum-MEG hedge ratio would be the same as the MV hedge ratio.
2.6. Optimum mean-MEG hedge ratio
Instead of minimizing the MEG coefficient,Kolb and Okunev (1993)alternatively consider
maximizing the utility function defined as follows:
U(Rh)= E(Rh)− Γv(Rh). (15)
Unlike the minimum-MEG hedge ratio, this optimum mean-MEG hedge ratio incorporates the expected return on the hedged portfolio. Again, if the futures price follows a martingale process (i.e., E(Rf) = 0) and if the futures and spot returns are jointly normally distributed, then
the optimum mean-MEG hedge ratio would be the same as the MV hedge ratio.
2.7. Minimum generalized semivariance hedge ratio
The hedge ratio based on the generalized semivariance has been recently proposed (seeDe
Jong et al., 1997;Lien & Tse, 1998, 2000;Chen et al., 2001). In this case, the optimal hedge ratio is obtained by minimizing the GSV given below:
Vδ,α(Rh)=
δ
−∞(δ− Rh) α
dG(Rh), α > 0, (16)
where G(Rh) is the probability distribution function of the return on the hedged portfolio Rh. The
parameters δ and α (which are both real numbers) represent the target return and risk aversion, respectively. The risk is defined in such a way that the investors consider only the returns below the target return (δ) to be risky. It can be shown (see Fishburn, 1977; Bawa, 1978) that α < 1 represents a risk-seeking investor and α > 1 represents a risk-averse investor.
Lien and Tse (1998)show that the GSV hedge ratio, which is obtained by minimizing the GSV, would be the same as the MV hedge ratio if the futures and spot returns are jointly normally distributed.
Table 1
Conditions for various hedge ratios to be equal to the minimum-variance (MV) hedge ratio
Hedge ratio Required conditions
Optimum mean-variance hedge ratio Pure martingale
Sharpe hedge ratio Pure martingale
Maximum expected utility hedge ratio Pure martingale and jointly normality Minimum MEG coefficient hedge ratio Joint normality
Optimum mean-MEG hedge ratio Pure martingale and joint normality
Minimum GSV hedge ratio Joint normality
Optimum mean-GSV hedge ratio Pure martingale and joint normality Note: This table summarizes specific conditions required for each hedge ratio to converge to the MV hedge ratio. The pure martingale condition refers to the condition that the futures price follows a pure or simple martingale process. The joint normality condition refers to the joint normality of returns on spot and futures positions.
2.8. Optimum mean-generalized semivariance hedge ratio
Chen et al. (2001)extend the GSV hedge ratio to a mean-GSV (M-GSV) hedge ratio by incorporating the mean return in the derivation of the optimal hedge ratio. The M-GSV hedge ratio is obtained by maximizing the following mean-risk utility function, which is similar to the conventional mean-variance based utility function (see Eq.(7)):
U(Rh)= E(Rh)− Vδ,α(Rh). (17)
Chen et al. (2001)show that the M-GSV hedge ratio would be the same as the MV hedge ratio if both the pure martingale and joint normality hypotheses hold.
From our discussion of various optimal hedge ratios above, it is clear that if the both the pure martingale and joint normality conditions hold, all the hedge ratios will be the same as the MV hedge ratio. Specific conditions required for each hedge ratio to converge to the MV hedge ratio are summarized inTable 1.
3. New tests for the joint normality of spot and futures returns
In this section, we develop new tests for the joint normality of spot and futures returns. Our new tests are based on the generalized method of moments approach, which is proposed by
Hansen (1982)and implemented byRichardson and Smith (1993)in their study of multivariate normality in stock returns. Let R1 and R2, respectively, denote the return on spot and futures
positions, σ1 and σ2, respectively, denote the standard deviations of R1and R2, and ρ denote
the correlation coefficient between R1and R2. If the returns on the spot and futures are jointly
normally distributed, then the following moment conditions must hold:
E[h(R, θ)] = 0, (18) where R = R1 R2 ,
h(R, θ) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ (R1− μ1) (R2− μ2) (R1− μ1)2− σ12 (R2− μ2)2− σ22 (R1− μ1)(R2− μ2)− σ1σ2ρ (R1− μ1)3 (R2− μ2)3 (R1− μ1)4− 3σ14 (R2− μ2)4− 3σ24 (R1− μ1)2(R2− μ2) (R1− μ1)(R2− μ2)2 (R1− μ1)2(R2− μ2)2− σ12σ22(1+ 2ρ2) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , and θ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ μ1 μ2 σ2 1 σ2 2 ρ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .
Note that θ is the vector of the unknown parameters. In this case, we have twelve moment equations and five unknown parameters. The GMM involves the estimation of the five unknown parameters by setting the following five linear combinations of the moment equations to zero:
AgT(θ)= 0, (19) where A = dtS−1, d= E ∂h(R, θ) ∂θ , S = E[h(R, θ)h(R, θ)t], gT(θ)= 1 T T t=1 h(Rt, θ),
and T denotes the sample size.Hansen (1982)provides the following asymptotic distribution for the estimators of parameters as well as moments:
√
T ( θ − θ)asy∼N(0, [dtS−1d]−1), (20)
√
where
V = S − d[dtS−1d]−1dt.
The expressions for the matrices d, S, A, and V are provided in theAppendix A.
We then discuss about the specific characteristics of the GMM applied to the current situation.
For example, the value of the optimal matrix A is such that solving Eq.(19)for the unknown
parameters is equivalent to solving the first five moment equations for the unknown parameters. This leads to the following estimators for the five unknown parameters:
ˆ μ1= 1 T T t=1 R1t, μˆ2= 1 T T t=1 R2t, ˆσ21= 1 T T t=1 (R1t− ˆμ1)2, ˆσ22= 1 T T t=1 (R2t− ˆμ2)2, and ρ = (1/T )Tt=1(R1t−μ 1)(R2t−μ 2) σ2 1 σ2 2 .
These estimators are exactly the same as those obtained byRichardson and Smith (1993). Since we use the first five moment conditions to estimate the parameters, we can use the remaining seven moments to test for the joint normality hypothesis. As most of the empirical normality tests are performed using standardized moments, we use the following standardized sample moments,
g∗T(ˆθ), instead of the moments given in Eq.(18): g∗T(ˆθ)= [g1T( θ), g2T( θ), g3T( θ), g4T( θ), g5T( θ), g6T( θ), g7T( θ)]t, g1T( θ) = (1/T ) T t=1(R1t−μ 1)3 ( σ21)3/2 , g2T( θ) = (1/T ) T t=1(R2t−μ 2)3 ( σ22)3/2 , g3T( θ) = (1/T ) T t=1(R1t−μ 1)4 ( σ21)2 − 3, g4T ( θ) = (1/T ) T t=1(R2t−μ 2)4 ( σ22)2 − 3, g5T( θ) = (1/T ) T t=1(R1t−μ 1)2(R2t−μ 2) σ2 1( σ2 2) 1/2 , g6T( θ) = (1/T ) T t=1(R1t−μ 1)(R2t−μ 2)2 σ2 2( σ2 1) 1/2 , and g7T( θ) = (1/T ) T t=1(R1t−μ 1)2(R2t−μ 2)2 σ2 1 σ2 2 ,
The first two sample standardized moments, g1Tand g2T, are the sample skewness of R1and R2,
respectively. Furthermore, the third and fourth sample moments, g3Tand g4T, are the standardized
sample kurtosis of R1and R2, respectively. It can be shown that the asymptotic distribution of the
standardized sample moments is given by: √
where V∗= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 6 6ρ3 0 0 6ρ 6ρ2 0 6ρ3 6 0 0 6ρ2 6ρ 0 0 0 24 24ρ4 0 0 24ρ2 0 0 24ρ4 24 0 0 24ρ2 6ρ 6ρ2 0 0 4ρ2+ 2 4ρ + 2ρ3 0 6ρ2 6ρ 0 0 4ρ+ 2ρ3 4ρ2+ 2 0 0 0 24ρ2 24ρ2 0 0 4+ 16ρ2+ 4ρ4 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .
The joint normality test can then be performed based on the Wald statistic given below:
W∗= T (g∗T)t(V∗)−1g∗T, (23)
where the Wald statistic has an asymptotic χ2distribution with seven degrees of freedom. In addition to the above test for the joint normality, we can also perform the GMM test based only on the skewness and kurtosis. This amounts to using the following four moments, instead of seven moments, in the Wald test:
g∗∗T (ˆθ)= [g1T( θ), g2T( θ), g3T( θ), g4T( θ)]t.
In this case, the Wald statistic, which has an asymptotic χ2distribution with four degrees of freedom, is given by:2
W∗∗= T (g∗∗T )t(V∗∗)−1g∗∗T , (24) where V∗∗= ⎡ ⎢ ⎢ ⎢ ⎣ 6 6ρ3 0 0 6ρ3 6 0 0 0 0 24 24ρ4 0 0 24ρ4 24 ⎤ ⎥ ⎥ ⎥ ⎦.
In this study, the test based on the Wald statistic given by Eq.(24)is referred to as the GMM test and that based on the Wald statistic given by Eq.(23)is referred to as the Extended GMM test. The relative performance of these two tests for various sample sizes and correlations will be analyzed based on the Monte Carlo simulation with 20,000 replications. There are at least two reasons why we examine two GMM-based tests even though(24)is the special case of(23). First, the GMM test is simpler than the Extended GMM tests. Therefore, if the performance of the GMM test is similar to that of the Extended GMM test, then we should use the simpler one, the
2If we assume that the two series are independent, then the W**statistic is given by: W∗∗= T g2 1T 6 + g2 3T 24 + T g2 2T 6 + g2 4T 24 → χ2(2)+ χ2(2)= χ2(4).
The Wald test is equivalent to theJarque-Bera (1987)normality test when the two series are independent. Therefore, the Jarque-Bera normality test is a just special case of the Wald test.
Table 2
Summary of 25 futures contracts
Commodity Sample period Sample size
1 SP500 June 1, 1982–December 31, 1997 4066
2 TSE35 March 1, 1991–December 31, 1997 1783
3 Nikkei 225 September 5, 1988–December 31, 1997 2432
4 TOPIX September 5, 1988–December 31, 1997 2432
5 FTSE100 May 3, 1984–December 31, 1997 3564
6 CAC40 March 1, 1989–December 31, 1997 2305
7 All ordinary January 3, 1984–December 31, 1997 3651
8 Soybean oil January 2, 1979–December 31, 1997 4956
9 Soybean January 2, 1979–December 31, 1997 4956
10 Soy meal January 2, 1979–December 31, 1997 4956
11 Corn January 2, 1979–December 31, 1997 4956
12 Wheat March 30, 1982–December 31, 1997 4111
13 Cotton January 3, 1980–December 31, 1997 4694
14 Cocoa November 1, 1983–December 31, 1997 3696
15 Coffee January 2, 1979–December 31, 1997 4956
16 Pork bellies January 2, 1979–December 31, 1997 4956
17 Hogs March 30, 1982–December 31, 1997 4111
18 Crude oil April 4, 1983–December 31, 1997 3847
19 Silver January 2, 1979–December 31, 1997 4956
20 Gold January 2, 1979–December 31, 1997 4956
21 Japanese yen January 2, 1986–December 31, 1997 3129
22 Deutsche mark January 2, 1986–December 31, 1997 3129
23 Swiss franc January 2, 1986–December 31, 1997 3129
24 British pound January 2, 1986–December 31, 1997 3129
25 Canadian dollar November 30, 1987–December 31, 1997 2632 Note: This table lists the commodities, sample periods, and sample sizes for the 25 different futures contracts used for empirical analyses in this study. The data are obtained from Datastream.
GMM test. Second, the Extended GMM test imposes more moment restrictions than the GMM test. This enables us to see if including more moment conditions improves the test performance.3
4. Empirical analysis
This article analyzes 25 different futures contracts where the futures prices are associated with nearest-to-maturity contracts. A list of the futures contracts, sample periods, and sample sizes are given inTable 2. The data are obtained from Datastream. To see if the results of our tests depend on the length of hedging horizon, we also perform tests using five different hedging horizons (1 day, 1 week, 4 week, 8 week, and 12 week).
We first test if the futures price follows a pure martingale process. T-tests are used to examine the pure martingale hypothesis that the expected return on futures is zero.Table 3presents the results. We find that the pure martingale hypothesis holds for all the 25 futures contracts except for SP500, TSE35, and FTSE100. This is true for all the five different hedging horizons considered in
3 The joint normality test imposes much more moment restrictions than those considered in this study. We can derive many different GMM tests that use various combinations of moment restrictions. Therefore, it is important to know if using more moment conditions improves the test statistic. In this paper, we partially answer this question. We would like to pursue this in our future research.
Table 3
Mean returns over various holding periods for different types of futures contracts
Commodity Holding period
1 Day 1 Week 4 Week 8 Week 12 Week
SP500 Sample size 4066 813 203 101 67 Mean return (%) 0.0535 0.2678 1.0548 2.1270 3.1631 T-ratio 2.9051*** 3.0611*** 3.6122*** 3.6652*** 3.7557*** TSE35 Sample size 1783 356 89 44 29 Mean return (%) 0.0354 0.1708 0.6832 1.3617 2.1386 T-ratio 1.9734** 1.9629* 2.0725** 2.2758** 2.2400** Nikkei 225 Sample size 2432 486 121 60 40 Mean return (%) −0.0245 −0.1291 −0.4544 −0.9070 −1.3604 T-ratio −0.8768 −0.8945 −0.7645 −0.9152 −0.6719 TOPIX Sample size 2432 486 121 60 40 Mean return (%) −0.0238 −0.1261 −0.4712 −0.8664 −1.2995 T-ratio −0.9026 −0.9112 −0.8292 −0.9702 −0.6933 FTSE100 Sample size 3564 712 178 89 59 Mean return (%) 0.0424 0.2082 0.8329 1.6659 2.4686 T-ratio 2.3882** 2.3228** 2.3540** 2.3732** 2.2792** CAC40 Sample size 2305 461 115 57 38 Mean return (%) 0.0285 0.1426 0.5378 1.0494 1.5741 T-ratio 1.1388 1.1909 1.1989 1.2520 1.1203 All ordinary Sample size 3651 730 182 91 60 Mean return (%) 0.0328 0.1624 0.6374 1.2748 2.0016 T-ratio 1.3499 1.4353 1.3724 1.3238 1.4075 Soybean oil Sample size 4956 991 247 123 82 Mean return (%) 0.0001 0.0017 0.0063 0.0452 0.0678 T-ratio 0.0043 0.0157 0.0148 0.0510 0.0555 Soybean Sample size 4956 991 247 123 82 Mean return (%) −0.0005 −0.0010 0.0109 0.0565 0.0847 T-ratio −0.0276 −0.0106 0.0298 0.0768 0.0765 Soy meal Sample size 4956 991 247 123 82 Mean return (%) 0.0015 0.0094 0.0685 0.2015 0.3022 T-Ratio 0.0750 0.0886 0.1788 0.2550 0.2566 Corn Sample size 4956 991 247 123 82 Mean return (%) 0.0027 0.0142 0.0791 0.1437 0.2155 T-ratio 0.1428 0.1437 0.1904 0.1565 0.1619 Wheat Sample size 4111 822 205 102 68 Mean return (%) −0.0030 −0.0143 −0.0350 −0.0625 −0.0937 T-ratio −0.1321 −0.1262 −0.0857 −0.0817 −0.0801 Cotton Sample size 4694 938 234 117 78 Mean return (%) −0.0023 −0.0106 −0.0411 −0.0823 −0.1234 T-ratio −0.0875 −0.0819 −0.0776 −0.0753 −0.0837 Cocoa Sample size 3696 739 184 92 61 Mean return (%) −0.0047 −0.0228 −0.0992 −0.1984 −0.2962 T-ratio −0.1594 −0.1540 −0.1807 −0.1859 −0.1937
Table 3 (Continued )
Commodity Holding period
1 Day 1 Week 4 Week 8 Week 12 Week
Coffee Sample size 4956 991 247 123 82 Mean return (%) 0.0037 0.0201 0.1237 0.1190 0.1785 T-ratio 0.1183 0.1263 0.1906 0.0874 0.0881 Pork bellies Sample size 4956 991 247 123 82 Mean return (%) −0.0028 −0.0103 −0.0153 0.0566 0.0849 T-ratio −0.0736 −0.0522 −0.0187 0.0389 0.0409 Hogs Sample size 4111 822 205 102 68 Mean return (%) −0.0041 −0.0204 −0.0607 −0.0830 −0.1244 T-ratio −0.1583 −0.1578 −0.1171 −0.0778 −0.0783 Crude oil Sample size 3847 769 192 96 64 Mean return (%) −0.0133 −0.0668 −0.2471 −0.4941 −0.7412 T-ratio −0.3639 −0.3712 −0.3921 −0.3387 −0.3535 Silver Sample size 4956 991 247 123 82 Mean return (%) −0.0003 0.0017 −0.0434 −0.1672 −0.2508 T-ratio −0.0099 0.0115 −0.0668 −0.1281 −0.1127 Gold Sample size 4956 991 247 123 82 Mean return (%) 0.0051 0.0266 0.0972 0.2654 0.3981 T-ratio 0.2866 0.2922 0.2730 0.3554 0.3557 Japanese yen Sample size 3129 625 156 78 52 Mean return (%) 0.0136 0.0696 0.2847 0.5693 0.8540 T-ratio 1.0721 1.1480 1.0934 1.0399 0.9939 Deutsche mark Sample size 3129 625 156 78 52 Mean return (%) 0.0095 0.0498 0.1987 0.3974 0.5962 T-Ratio 0.7440 0.8280 0.7743 0.7151 0.6991 Swiss franc Sample size 3129 625 156 78 52 Mean return (%) 0.0107 0.0569 0.2264 0.4528 0.6792 T-ratio 0.7536 0.8489 0.7900 0.7122 0.7358 British pound Sample size 3129 625 156 78 52 Mean return (%) 0.0041 0.0218 0.0855 0.1710 0.2564 T-ratio 0.3225 0.3547 0.3342 0.3358 0.2830 Canadian dollar Sample size 2632 526 131 65 43 Mean return (%) −0.0032 −0.0175 −0.0587 −0.1192 −0.1223 T-ratio −0.5701 −0.6035 −0.5281 −0.6014 −0.4088
Note: This table presents the results for the mean returns over various holding periods for each of the futures contracts listed inTable 2.
***1% significance level. ** 5% significance level. * 10% significance level.
this study. The results suggest that with the exception of SP500, TSE35, and FTSE100, ignoring the expected return in the derivation of the optimal hedge ratio does not significantly change the optimal hedge ratio. Therefore, the mean-variance hedge ratio would be approximately the same as the MV hedge ratio for most futures contracts.
We now test the joint normality of spot and futures returns. As mentioned above, we use two tests for the joint normality hypothesis. The first test is the GMM test, which is
Fig. 1. Relationship between the root mean squared error and sample size for the correlation of 90% for the GMM and extended GMM tests.
based on the Wald statistic with four degrees of freedom (Eq. (24)). The second test is the
Extended GMM test, which is based on the Wald statistic with seven degrees of freedom (Eq.(23)).
The empirical probabilities of rejecting a true null hypothesis of joint normal distribution (i.e., the empirical sizes) for four levels of critical value from the asymptotic distribution (nominal size of 1, 5, 10, and 20%) are given inTable 4. The empirical sizes are based on Monte Carlo simulations with 20,000 replications and are computed for various correlation coefficients between the two random variables (ranging from 0 to 98%) and for various sample sizes (ranging from 50 to 10,000). It is clear fromTable 4that for smaller sample sizes, the Extended GMM test performs better than the GMM tests.
SinceTable 4contains a large number of empirical sizes for different values of sample size and correlation, it would be worthwhile to summarize the results. We use the root mean squared error (RMSE) to summarize the deviation of the empirical size from the nominal size (i.e., the size distortion). The RMSE is computed as follows:
RMSE=
(e0.01− 0.01)2+ (e0.05− 0.05)2+ (e0.10− 0.10)2+ (e0.20− 0.20)2
4 ,
where e␣represents the empirical size at the level α. The root mean squared errors for various
sample sizes and correlations are summarized inTable 5. The relationship between the RMSE
and sample size for a correlation of 90% is shown inFig. 1for the GMM and Extended GMM
test statistics.4
It is interesting to note fromTable 5andFig. 1that the RMSE (i.e., the size distortion) of the GMM and Extended GMM tests depends on the sample size and decreases as the sample size is increased. This is not surprising due to the fact that the GMM-based tests are asymptotic tests. It is also interesting to note that for smaller sample sizes, the Extended GMM test (the Wald test based on seven moments) performs better than the GMM test (the Wald test based on four
4We do not find any pattern in the relationship between RMSE and the correlation. Furthermore, the sample correlations between the futures and spot returns are very high (normally more than 90%). Therefore, we use a 90% correlation in
Table 4
Empirical sizes of the GMM and extended GMM tests with 20,000 replications
N ρ GMM percentile Extended GMM percentile
1% 5% 10% 20% 1% 5% 10% 20% 50 0.00 0.022 0.044 0.066 0.108 0.026 0.050 0.073 0.120 50 0.10 0.023 0.048 0.068 0.107 0.025 0.050 0.075 0.120 50 0.20 0.022 0.044 0.065 0.108 0.025 0.050 0.073 0.116 50 0.50 0.022 0.044 0.065 0.107 0.024 0.050 0.073 0.123 50 0.80 0.023 0.047 0.070 0.115 0.024 0.052 0.079 0.129 50 0.90 0.024 0.047 0.070 0.112 0.025 0.050 0.074 0.121 50 0.98 0.025 0.051 0.074 0.116 0.027 0.053 0.080 0.126 100 0.00 0.023 0.050 0.078 0.131 0.025 0.056 0.086 0.141 100 0.10 0.024 0.054 0.081 0.135 0.026 0.058 0.088 0.147 100 0.20 0.025 0.052 0.078 0.132 0.027 0.057 0.087 0.145 100 0.50 0.025 0.051 0.077 0.131 0.026 0.057 0.088 0.147 100 0.80 0.023 0.052 0.079 0.135 0.026 0.057 0.087 0.143 100 0.90 0.026 0.055 0.083 0.141 0.026 0.058 0.090 0.150 100 0.98 0.023 0.051 0.080 0.136 0.026 0.057 0.088 0.145 200 0.00 0.023 0.052 0.089 0.157 0.023 0.059 0.094 0.165 200 0.10 0.023 0.055 0.087 0.156 0.025 0.061 0.096 0.166 200 0.20 0.022 0.052 0.084 0.155 0.023 0.059 0.091 0.163 200 0.50 0.024 0.053 0.087 0.153 0.026 0.059 0.094 0.169 200 0.80 0.021 0.053 0.086 0.150 0.022 0.059 0.093 0.163 200 0.90 0.023 0.056 0.089 0.157 0.025 0.059 0.094 0.166 200 0.98 0.022 0.053 0.087 0.161 0.025 0.059 0.095 0.168 500 0.00 0.019 0.053 0.094 0.180 0.020 0.057 0.098 0.185 500 0.10 0.019 0.054 0.093 0.177 0.020 0.055 0.097 0.179 500 0.20 0.019 0.052 0.092 0.176 0.019 0.056 0.097 0.183 500 0.50 0.019 0.052 0.093 0.175 0.019 0.055 0.098 0.182 500 0.80 0.019 0.052 0.094 0.175 0.019 0.055 0.098 0.181 500 0.90 0.018 0.056 0.096 0.176 0.018 0.056 0.099 0.190 500 0.98 0.020 0.053 0.093 0.176 0.020 0.057 0.098 0.182 1000 0.00 0.017 0.052 0.097 0.187 0.016 0.054 0.099 0.187 1000 0.10 0.014 0.051 0.095 0.188 0.015 0.053 0.099 0.193 1000 0.20 0.015 0.049 0.092 0.182 0.016 0.053 0.098 0.188 1000 0.50 0.016 0.055 0.101 0.191 0.018 0.057 0.103 0.193 1000 0.80 0.015 0.051 0.095 0.186 0.016 0.052 0.095 0.187 1000 0.90 0.015 0.051 0.095 0.183 0.017 0.053 0.097 0.190 1000 0.98 0.015 0.052 0.097 0.187 0.015 0.055 0.099 0.192 5000 0.00 0.012 0.051 0.101 0.202 0.012 0.052 0.101 0.201 5000 0.10 0.012 0.052 0.101 0.198 0.012 0.051 0.101 0.200 5000 0.20 0.013 0.053 0.101 0.201 0.011 0.051 0.098 0.198 5000 0.50 0.011 0.049 0.100 0.196 0.012 0.053 0.101 0.200 5000 0.80 0.011 0.051 0.100 0.199 0.010 0.048 0.100 0.196 5000 0.90 0.010 0.049 0.097 0.197 0.011 0.049 0.096 0.196 5000 0.98 0.012 0.052 0.100 0.199 0.012 0.053 0.102 0.201 10000 0.00 0.010 0.051 0.099 0.199 0.010 0.050 0.099 0.197 10000 0.10 0.012 0.053 0.103 0.201 0.011 0.054 0.102 0.204 10000 0.20 0.011 0.052 0.100 0.195 0.010 0.051 0.104 0.200 10000 0.50 0.011 0.052 0.100 0.200 0.010 0.051 0.101 0.202 10000 0.80 0.011 0.051 0.100 0.195 0.010 0.051 0.102 0.199 10000 0.90 0.011 0.049 0.099 0.199 0.011 0.051 0.101 0.200 10000 0.98 0.011 0.050 0.101 0.200 0.012 0.052 0.099 0.198
Note: This table shows the empirical probabilities of rejecting a true null hypothesis of joint normal distribution (i.e., the empirical sizes) for four levels of critical value from the asymptotic distribution (nominal size of 1, 5, 10, and 20%). The empirical sizes are based on Monte Carlo simulations with 20,000 replications and are computed for various correlation coefficients between the two random variables (ranging from 0 to 98%) and for various sample sizes (ranging from 50 to 10,000). N denotes sample size and ρ denotes the correlation coefficient between the two simulated random variables.
Table 5
Root mean squared errors of the GMM and extended GMM tests
Sample size Correlation GMM test Extended GMM test
50 0.00 0.050 0.043 50 0.10 0.049 0.042 50 0.20 0.050 0.045 50 0.50 0.050 0.041 50 0.80 0.046 0.038 50 0.90 0.047 0.042 50 0.98 0.044 0.039 100 0.00 0.037 0.031 100 0.10 0.035 0.029 100 0.20 0.036 0.030 100 0.50 0.037 0.029 100 0.80 0.035 0.030 100 0.90 0.032 0.027 100 0.98 0.034 0.029 200 0.00 0.023 0.020 200 0.10 0.024 0.020 200 0.20 0.024 0.021 200 0.50 0.025 0.018 200 0.80 0.027 0.020 200 0.90 0.023 0.019 200 0.98 0.022 0.018 500 0.00 0.011 0.010 500 0.10 0.013 0.012 500 0.20 0.013 0.010 500 0.50 0.014 0.010 500 0.80 0.014 0.011 500 0.90 0.013 0.007 500 0.98 0.013 0.011 1000 0.00 0.008 0.007 1000 0.10 0.007 0.005 1000 0.20 0.010 0.007 1000 0.50 0.006 0.007 1000 0.80 0.008 0.008 1000 0.90 0.009 0.006 1000 0.98 0.007 0.005 5000 0.00 0.001 0.002 5000 0.10 0.002 0.001 5000 0.20 0.002 0.002 5000 0.50 0.002 0.002 5000 0.80 0.001 0.002 5000 0.90 0.002 0.003 5000 0.98 0.001 0.002 10000 0.00 0.001 0.002 10000 0.10 0.002 0.003 10000 0.20 0.003 0.002 10000 0.50 0.001 0.001 10000 0.80 0.003 0.001 10000 0.90 0.001 0.001 10000 0.98 0.001 0.002
Note: This table summarizes the root mean squared errors of the GMM and Extended GMM tests for various sample sizes and correlations.
Table 6
The GMM and extended GMM tests of joint normality for different types of futures contracts
Commodity Holding period
1 Day 1 Week 4 Week 8 Week 12 Week SP500 Correlation (%) 93.96 97.33 98.75 99.39 99.73 GMM test 7194823.86 258023.46 1190.90 319.98 155.83 Extended GMM test 9100003.91 371834.42 1212.43 324.80 174.29 TSE35 Correlation (%) 87.13 95.88 96.57 98.95 98.44 GMM test 2706.24 5.97 1.90 0.99 7.45 Extended GMM test 6726.16 18.01 22.46 4.58 12.75 Nikkei 225 Correlation (%) 92.71 97.30 99.04 99.22 99.79 GMM test 3362.80 225.63 14.03 4.86 22.76 Extended GMM test 5479.85 522.86 22.28 7.03 46.96 TOPIX Correlation (%) 88.09 96.53 99.15 99.40 99.79 GMM test 4439.33 251.18 31.78 0.28 22.95 Extended GMM test 16672.01 359.44 41.95 17.62 30.30 FTSE100 Correlation (%) 91.83 96.56 97.96 99.10 99.53 GMM test 89098.73 18242.27 1186.55 196.12 11.70 Extended GMM test 204594.66 19371.95 1195.95 196.84 19.13 CAC40 Correlation (%) 95.09 98.11 99.16 99.20 99.51 GMM test 1417.93 26.04 13.80 17.24 14.89 Extended GMM test 1854.57 27.82 15.76 17.63 16.86 All ordinary Correlation (%) 29.09 87.26 96.06 95.88 98.54 GMM test 10888707.70 81539.62 15772.48 3004.62 930.78 Extended GMM test 11123966.19 81830.43 15782.74 3301.60 996.38 Soybean oil Correlation (%) 80.97 91.47 95.55 97.09 97.13 GMM test 1252.00 149.79 33.68 28.03 5.23 Extended GMM test 3984.27 236.39 51.56 37.95 21.05 Soybean Correlation (%) 85.72 91.87 93.48 94.99 95.33 GMM test 4689.86 567.05 37.67 24.13 11.33 Extended GMM test 21707.21 2659.71 54.37 34.62 28.70 Soy meal Correlation (%) 79.16 88.04 88.69 88.69 91.17 GMM test 12810.59 351.47 76.66 11.12 23.73 Extended GMM test 29195.01 854.18 206.62 52.84 32.08 Corn Correlation (%) 73.65 81.40 83.64 86.91 90.96 GMM test 123853.48 2585.63 250.69 186.19 29.99 Extended GMM test 293273.80 3460.75 523.97 398.92 36.42 Wheat Correlation (%) 62.28 77.16 79.48 83.74 84.65 GMM test 221619.84 2275.95 166.03 39.07 1.16 Extended GMM test 424795.71 3808.78 308.97 47.24 1.95 Cotton Correlation (%) 37.49 40.31 48.16 92.45 92.07 GMM test 351397138.62 3110061.61 37520.48 2268.92 334.01 Extended GMM test 407677749.96 3786817.86 54402.60 2288.36 373.03 Cocoa Correlation (%) 71.66 89.40 91.44 90.95 89.04 GMM test 5677.15 51.51 5.31 26.90 19.66 Extended GMM test 9452.43 214.86 34.47 52.41 21.61
Table 6 (Continued )
Commodity Holding period
1 Day 1 Week 4 Week 8 Week 12 Week Coffee Correlation (%) 57.65 69.95 76.62 86.18 91.70 GMM test 513042.92 10148.36 287.26 30.44 11.57 Extended GMM test 674689.61 15644.33 620.82 81.13 13.65 Pork bellies Correlation (%) 25.57 48.49 67.00 66.49 61.03 GMM test 813014.98 3069.79 36.64 1.99 12.02 Extended GMM test 833588.14 3914.48 74.72 13.33 14.69 Hogs Correlation (%) 14.03 35.71 52.93 52.50 65.22 GMM test 77435.64 613.29 5.52 1.71 0.99 Extended GMM test 77604.08 722.23 9.54 4.86 3.09 Crude oil Correlation (%) 75.94 91.20 97.66 98.85 99.50 GMM test 361983.66 4705.67 178.98 360.55 449.66 Extended GMM test 403914.46 5258.44 231.13 360.82 457.62 Silver Correlation (%) 52.63 81.11 95.85 98.09 99.71 GMM test 8610770.32 20301.95 527.68 317.90 244.51 Extended GMM test 8649436.52 49177.18 4322.66 2569.89 248.49 Gold Correlation (%) 42.83 88.66 97.34 98.57 99.10 GMM Test 53625.90 3326.17 861.15 559.55 91.41 Extended GMM Test 56730.04 4034.65 887.98 792.72 96.86 Japanese yen Correlation (%) 93.26 97.97 99.21 99.60 99.91 GMM test 3351.01 105.11 2.78 4.75 11.34 Extended GMM test 4670.46 331.67 35.92 10.63 73.44 Deutsche mark Correlation (%) 94.21 98.42 99.36 99.67 99.89 D. GMM test 745.05 14.05 3.71 1.72 4.42 Extended GMM test 765.41 42.53 5.00 6.26 14.90 Swiss franc Correlation (%) 94.28 98.66 99.46 99.74 99.90 GMM test 546.76 24.67 1.60 2.33 1.70 Extended GMM test 572.31 91.32 6.56 5.49 5.02 British pound Correlation (%) 92.23 95.85 99.12 99.48 99.85 GMM test 1831.78 370.38 55.39 4.36 144.97 Extended GMM test 4343.07 1049.84 110.27 17.68 327.63 Canadian dollar Correlation (%) 89.70 94.62 97.16 98.38 99.38 GMM test 1546.11 26.55 7.09 6.56 1.47 Extended GMM test 3477.71 214.68 10.61 11.30 23.95 Note: This table presents the results of the joint normality tests over various holding periods for each of the futures contracts listed inTable 2. The GMM test statistics have a chi-square distribution with 4 degrees of freedom and their critical values at the 1, 5, and 10% levels are 13.277, 9.488, and 7.779, respectively. The Extended GMM test statistics have a chi-square distribution with 7 degrees of freedom and their critical values at the 1, 5, and 10% levels are 18.475, 14.067, and 12.017, respectively. The figures in bold face indicate that the joint normality hypothesis cannot be rejected at the 5% level.
moments). However, the difference in performance between the two tests disappears when the sample size is large (greater than 1000). This is an important result due to the fact that when developing GMM-based tests, we usually have an option in choosing the number of moments as well as specific moments to be included in the tests. The results indicate that for smaller sample sizes, it is better to choose more moments.
We now discuss the results of the joint normality tests for the 25 futures commodities listed in
Table 2.Table 6presents the results. Note that the GMM statistics have a chi-square distribution with four degrees of freedom. The critical values at the 1, 5, and 10% levels for the GMM test statistics are 13.277, 9.488, and 7.779, respectively. However, the Extended GMM test statistics have a chi-square distribution with seven degrees of freedom and their critical values at the 1, 5, and 10% levels are 18.475, 14.067, and 12.017, respectively. In order to see if the joint normality hypothesis holds for various data frequencies, we perform the GMM and Extended GMM tests for daily, 1-, 4-, 8-, and 12-week returns. It is important to note that we use non-overlapping returns here. This avoids the serial correlation problem caused by the use of overlapping returns.
Table 6shows that when daily data is used, the joint normality of spot and futures returns is rejected for all the 25 commodities considered in this study. The results for one-week hedging periods are similar for all the futures contracts. The only exception is TSE 35 when the GMM test is employed to test for the joint normality. However, the joint normality hypothesis is rejected for TSE 35 when the Extended GMM test is used. Our findings suggest that the joint normality hypothesis tends to be rejected for all the futures contracts when the hedging horizon is relatively short.
Table 6also shows that for hedging horizons equal to or longer than four weeks, both the GMM and Extended GMM test statistics are insignificant for the following commodities: TSE35 (8 and 12 week), Nikkei 225 (8 week), wheat (12 week), pork bellies (8 week), hogs (4, 8, and 12 week), Japanese yen (8 week), Deutsche mark (4 and 8 week), Swiss franc (4, 8, and 12 week), and Canadian dollar (4 and 8 week). That is, for those commodities, the joint normality of spot and futures returns is supported for some hedging horizons. Our findings suggest that, except for a few commodities and longer hedging horizons, the joint normality hypothesis generally does not hold. Therefore, the hedge ratios derived based on the expected utility maximizing approach, the MEG approach, and the GSV approach will not converge to the MV hedge ratio for most futures contracts and for relatively short hedging horizons.
5. Conclusions
The optimal hedge ratios which are derived based on the mean-variance approach, the expected utility maximizing approach, the mean extended-Gini approach, and the generalized semivariance approach will all converge to the minimum-variance hedge ratio if the futures price follows a pure martingale process and if the spot and futures returns are jointly normal. Because the minimum-variance hedge ratio is easy to understand, simple to compute, and most widely used, it is important to investigate if these two conditions hold. In this paper, we perform empirical tests to see if the pure martingale and joint normality hypotheses hold using 25 different futures contracts and five different hedging horizons.
In order to test for the joint normality of spot and futures returns, we develop two new tests, the GMM and Extended GMM tests, based on the generalized method of moments, which allow for calculating multivariate test statistics that take account of the contemporaneous correlation across spot and futures returns. The GMM test uses fewer moment conditions than the Extended GMM test. Since the relative performances of those two tests are not known, we analyze their relative performances based on Monte Carlo simulations with various correlation coefficients and sample sizes. For large sample sizes, we find no difference in performances between the two tests. However, for small sample sizes, the Extended GMM test is found to perform better than the GMM test in terms of size distortion. The results suggest that for smaller sample sizes, it is better to include more moment conditions.
Our tests of the pure martingale hypothesis show that the expected return on futures is generally zero except for a few futures contracts. This is true for all the five different hedging horizons. The results suggest that ignoring the expected return in the derivation of the optimal hedge ratio does not significantly change the optimal hedge ratio for most commodities. Therefore, with the exception of a few commodities, the mean-variance hedge ratio would be approximately the same as the minimum-variance hedge ratio. We also find that the joint normality hypothesis tends to be rejected for all the 25 commodities when the length of hedging period is short. For longer hedging horizons, the joint normality hypothesis holds only for a few commodities. Our results suggest that the hedge ratios which are derived based on the expected utility maximizing approach, the mean extended-Gini approach, and the generalized semivariance approach will not converge to the minimum-variance hedge ratio except for a few contracts and relatively long hedging horizons. Finally, it is important to note that even though the test method suggested byRichardson and Smith (1993)and the one suggested in this paper are both based on the GMM method, these methods use different sets of moments in the implementation of normality tests. Furthermore, the results obtained byRichardson and Smith (1993)have implications for asset pricing whereas the results obtained in this paper have implications for the optimal hedge ratios under different approaches. Appendix A dt= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ −1 0 0 0 0 −3σ2 1 0 0 0 −2σ1σ2ρ −σ 2 2 0 0 −1 0 0 0 0 −3σ2 2 0 0 −σ 2 1 −2σ1σ2ρ 0 0 0 −1 0 −σ2 2ρσ1 0 0 −6σ 2 1 0 0 0 −σ 2 2(1+ 2ρ 2) 0 0 0 −1 −σ1 2σ2ρ 0 0 0 −6σ 2 2 0 0 −σ 2 1(1+ 2ρ 2) 0 0 0 0 −σ1σ2 0 0 0 0 0 0 −4σ2 1σ 2 2ρ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ S = [S1, S2] S1= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ σ2 1 σ1σ2ρ 0 0 0 3σ1 3σ1ρ σ1σ2ρ σ2 2 0 0 0 3σ2ρ 3σ2 0 0 2σ4 1 2σ 2 1σ 2 2ρ2 2σ 3 1σ2ρ 0 0 0 0 2σ2 1σ 2 2ρ 2 2σ4 2 2σ1σ 3 2ρ 0 0 0 0 2σ3 1σ2ρ 2σ1σ 3 2ρ σ 2 1σ 2 2(1+ ρ 2) 0 0 3σ1 3σ2ρ 0 0 0 15 ρ(9 + 6ρ2) 3σ1ρ 3σ2 0 0 0 ρ(9 + 6ρ2) 15 0 0 12σ2 1 12σ 2 2ρ 2 12σ1σ2ρ 0 0 0 0 12σ2 1ρ 2 12σ2 2 12σ1σ2ρ 0 0 3σ1ρ σ2(1+ 2ρ2) 0 0 0 15ρ 3(1+ 4ρ2) σ1(1+ 2ρ2) 3σ2ρ 0 0 0 3(1+ 4ρ2) 15ρ 0 0 2σ2 1(1+ 5ρ 2) 2σ2 2(1+ 5ρ 2) σ1σ2ρ(8 + 4ρ2) 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
S2= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 3σ1σ2ρ σ1(1+ 2ρ2) 0 0 0 σ2(1+ 2ρ2) 3σ2ρ 0 12σ2 1 12σ 2 1ρ 2 0 0 2σ2 1(1+ 5ρ 2) 12σ2 2ρ 2 12σ2 2 0 0 2σ 2 2(1+ 5ρ 2) 12σ1σ2ρ 12σ1σ2ρ 0 0 σ1σ2ρ(8 + 4ρ2) 0 0 15ρ 3(1+ 4ρ2) 0 0 0 3(1+ 4ρ2) 15ρ 0 96 ρ2(72+ 24ρ2) 0 0 12+ 84ρ2 ρ2(72+ 24ρ2) 96 0 0 12+ 84ρ2 0 0 3(1+ 4ρ2) ρ(9 + 6ρ2) 0 0 0 ρ(9 + 6ρ2) 3(1+ 4ρ2) 0 12+ 84ρ2 12+ 84ρ2 0 0 8+ 68ρ2+ 20ρ4 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 σ2 1(ρ2− 1) −σ ρ 1σ2(ρ2− 1) 0 0 0 0 0 0 0 0 0 0 −σ ρ 1σ2(ρ2− 1) 1 σ2 2(ρ2− 1) 0 0 0 0 0 0 0 0 0 0 0 0 1 2σ4 1(ρ2− 1) 0 − ρ 2σ3 1σ2(ρ2− 1) 0 0 0 0 0 0 0 0 0 0 1 2σ4 2(ρ2− 1) − ρ 2σ1σ32(ρ2− 1) 0 0 0 0 0 0 0 0 0 ρ σ2 1(1− 2ρ2+ ρ4) ρ σ2 2(1− 2ρ2+ ρ4) −σ 1+ ρ2 1σ2(1− 2ρ2+ ρ4) 0 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ V = 0 5×5 0 5×7 0 7×5 V1 7×7 V1= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 6σ6 1 6σ 3 1σ 3 2ρ 3 0 0 6σ5 1σ2ρ 6σ14σ 2 2ρ 2 0 6σ3 1σ 3 2ρ 3 6σ6 2 0 0 6σ 2 1σ 4 2ρ 2 6σ 1σ52ρ 0 0 0 24σ8 1 24σ 4 1σ 4 2ρ 4 0 0 24σ6 1σ 2 2ρ 2 0 0 24σ4 1σ 4 2ρ 4 24σ8 2 0 0 24σ 2 1σ 6 2ρ 2 6σ5 1σ2ρ 6σ12σ 4 2ρ 2 0 0 4σ4 1σ 2 2ρ 2+ 2σ4 1σ 2 2 4σ 3 1σ 3 2ρ + 2σ 3 1σ 3 2ρ 3 0 6σ4 1σ 3 2ρ 2 6σ 1σ25ρ 0 0 4σ 3 1σ 3 2ρ + 2σ 3 1σ 3 2ρ 3 2σ2 1σ 4 2+ 4σ 2 1σ 4 2ρ 2 0 0 0 24σ6 1σ 2 2ρ 2 24σ2 1σ 6 2ρ 2 0 0 4σ4 1σ 4 2+ 16σ 4 1σ 4 2ρ 2+ 4σ4 1σ 4 2ρ 4 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ References
Bawa, V. S. (1978). Safety-first, stochastic dominance, and optimal portfolio choice. Journal of Financial and Quantitative Analysis, 13, 255–271.
Cecchetti, S. G., Cumby, R. E., & Figlewski, S. (1988). Estimation of the optimal futures hedge. Review of Economics and Statistics, 70, 623–630.
Chen, S. S., Lee, C. F., & Shrestha, K. (2001). On a mean-generalized semivariance approach to determining the hedge ratio. Journal of Futures Markets, 21, 581–598.
Cheung, C. S., Kwan, C. C. Y., & Yip, P. C. Y. (1990). The hedging effectiveness of options and futures: A mean-Gini approach. Journal of Futures Markets, 10, 61–74.
Crum, R. L., Laughhunn, D. L., & Payne, J. W. (1981). Risk-seeking behavior and its implications for financial models. Financial Management, 10, 20–27.
De Jong, A., De Roon, F., & Veld, C. (1997). Out-of-sample hedging effectiveness of currency futures for alternative models and hedging strategies. Journal of Futures Markets, 17, 817–837.
Ederington, L. H. (1979). The hedging performance of the new futures markets. Journal of Finance, 34, 157–170. Fishburn, P. C. (1977). Mean-risk analysis with risk associated with below-target returns. American Economic Review,
67, 116–126.
Hansen, L. P. (1982). Large sample properties of generalized method of moment estimators. Econometrica, 50, 1029–1054. Howard, C. T., & D’Antonio, L. J. (1984). A risk-return measure of hedging effectiveness. Journal of Financial and
Quantitative Analysis, 19, 101–112.
Hsin, C. W., Kuo, J., & Lee, C. F. (1994). A new measure to compare the hedging effectiveness of foreign currency futures versus options. Journal of Futures Markets, 14, 685–707.
Jarque, C. M., & Bera, A. K. (1987). A test for normality of observations and regression residuals. International Statistical Review, 55, 163–172.
Johnson, L. L. (1960). The theory of hedging and speculation in commodity futures. Review of Economic Studies, 27, 139–151.
Kolb, R. W., & Okunev, J. (1992). An empirical evaluation of the extended mean-Gini coefficient for futures hedging. Journal of Futures Markets, 12, 177–186.
Kolb, R. W., & Okunev, J. (1993). Utility maximizing hedge ratios in the extended mean Gini framework. Journal of Futures Markets, 13, 597–609.
Lien, D., & Luo, X. (1993). Estimating the extended mean-Gini coefficient for futures hedging. Journal of Futures Markets, 13, 665–676.
Lien, D., & Shaffer, D. R. (1999). A note on estimating the minimum extended Gini hedge ratio. Journal of Futures Markets, 19, 101–113.
Lien, D., & Tse, Y. K. (1998). Hedging time-varying downside risk. Journal of Futures Markets, 18, 705–722. Lien, D., & Tse, Y. K. (2000). Hedging downside risk with futures contracts. Applied Financial Economics, 10, 163–170. Myers, R. J., & Thompson, S. R. (1989). Generalized optimal hedge ratio estimation. American Journal of Agricultural
Economics, 71, 858–868.
Richardson, M., & Smith, T. (1993). A test for multivariate normality in stock returns. Journal of Business, 66, 295–321. Shalit, H. (1995). Mean-Gini hedging in futures markets. Journal of Futures Markets, 15, 617–635.
