short-run deviations and optimal hedge ratio: evidence from stock futures

Short-run deviations and optimal hedge ratio:evidence from stock futures

Taufiq Choudhry *

School of Management, University of Southampton, Bradford, Bradford BD9 4JL, UK

Received 26 May 2001; accepted 25 February 2002

Abstract

This paper investigates the effects of the long-run relationship between stock cash index and

futures index on the hedging effectiveness of six stock futures markets. Effectiveness of five

different hedging ratios depending on different estimation procedures is investigated. The

unhedged, the traditional hedge and the minimum variance hedge ratios are all constant while

the bivariate GARCH and GARCH-X hedge ratios are time varying. The effectiveness of the

hedge ratio is compared by investigating the total sample and the out-of-sample performance

of the five ratios. The total sample period consists of daily returns from January 1990 to

December 1999. Two out-of-sample periods used are from January 1998 to December 1999 (2

years) and from January 1999 to December 1999 (1 year). Results show that the time-varying

hedge ratio outperforms the constant hedge ratio.

# 2002 Elsevier Science B.V. All rights reserved.

Keywords: G1; G13; G15

Keywords: Hedge ratio; Bivariate GARCH; Variance

1. Introduction

According to Figlewski (1984) and Fortune (1989), the introduction of stock index

futures is the most important financial innovation affecting the quality of

information about future common stock prices and the possibility of unbundling

the market and non-market components of risk and return. Silber (1985) suggests

* Tel.: �/44-1274-234363; fax: �/44-1274-235680

E-mail address: [email protected] (T. Choudhry).

J. of Multi. Fin. Manag. 13 (2003) 171�/192

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that futures markets provide two main functions: risk transfer and price discovery.

Risks are transferred to those willing to bear them as hedgers reduce their risk by

paying a premium to speculators.1 In other words, stock index futures can be used to

hedge market risk caused by cash (spot) price fluctuations.2 The existence of futures

trading also enhances the ability of investors to form judgements about expected

forward cash prices (Fortune, 1989).

This paper investigates and compares the risk-reducing effectiveness of differentoptimal (minimum risk) hedge ratios for the stock futures of Australia, Germany,

Hong Kong, Japan, South Africa and the United Kingdom. The main contribution

of the paper is to investigate the effects of short-run deviations from the long-run

relationship between cash index and futures index on the optimal hedge ratio. An

optimal hedge ratio is defined as the proportion of a cash position that should be

covered with an opposite position on a futures market.

In this paper, the optimal hedge ratios are estimated by OLS regressions and the

generalized autoregressive conditional heteroscedasticity (GARCH) model. Theoptimal hedge ratio estimated by means of the OLS regressions are constant, while

the GARCH model, taking into consideration the time-varying distribution of the

cash and futures price changes, provides changing hedge ratio.3 Park and Switzer

(1995) claim that if the joint distribution of stock index and futures prices is changing

over time, estimating a constant hedge ratio may not be appropriate. In other words,

the hedge ratios will certainly vary over time as the conditional distribution between

cash and futures prices changes (Baillie and Myers, 1991). This paper is motivated by

the claims of Park and Switzer (1995) and Butterworth and Holmes (1996) whoindicate the lack of research in this field for stock market futures, especially for

markets other than the United States.

2. Optimal hedge ratios

This section describes the optimal hedge ratio, relying heavily on Cecchetti et al.

(1988) and Baillie and Myers (1991). The returns on the portfolio of an investortrying to hedge some proportion of the cash position in a futures market can be

represented by

1 According to Fortune (1989), risk transfer and price discovery also take place in the absence of

futures markets but these two factors are enhanced in the presence of futures markets. This happens

because the costs of futures transactions are considerably less than the cost of cash (spot) transactions.2 According to Figlewski (1984), risk minimization will depend upon the behaviour of the difference

between the futures prices and the cash prices. Thus some measure of risk will be imposed on hedging a

position in stock, the risk being that the change in the futures price over time will not track exactly the

value of the cash position.3 Park and Switzer (1995) also apply the GARCH method to estimate optimal hedge ratios for the

United States stock futures market. Baillie and Myers (1991) and Myers (1991) study the commodities

futures market using the GARCH method. Park and Switzer (1995) provide citations of other studies that

apply the GARCH model.

T. Choudhry / J. of Multi. Fin. Manag. 13 (2003) 171�/192172

rt�rct �bt�1rf

t ; (1)

where rt is the return holding the portfolio of cash and futures position between t�/1

and t , rct the return on holding the cash position for the same period, rf

t the return on

holding the futures position for the same period and bt�1 the hedge ratio. Thevariance of the return on the hedged portfolio is give by

Var(rt=Vt�1)�Var(rct=Vt�1)�b2

t�1Var(rft =Vt�1)�2bt�1Cov(rc

t ; rft =Vt�1); (2)

where Vt�1 is the information available over the last period. As indicated by

Cecchetti et al. (1988), the return on a hedged position will normally be exposed torisk caused by unanticipated changes in the relative price between the position being

hedged and the futures contract. This ‘basic risk’ ensures that no hedge ratio

completely eliminates risk. The hedge ratio that minimizes risk may be obtained by

setting the derivative of Eq. (2) with respect to b equal to zero. The hedge ratio bt�1

can then be expressed as

bt�1��

Cov(rct ; rf

t=Vt�1)

Var(rft=Vt�1)

�: (3)

The value of bt�1 which minimizes the conditional variance of the hedged portfolio

return is the optimal hedge ratio (Baillie and Myers, 1991).4 Commonly, the value of

the hedge ratio is less than unity so that the hedge ratio that minimizes risk in the

absence of basic risk turns out to be dominated by b when basic risk is taken intoconsideration.5

There are three hedge strategies involving constant hedge ratios: the traditional

one-to-one hedge, the beta hedge and the minimum variance hedge. The traditional

hedge strategy involves adopting a futures position that is equal in magnitude but

opposite in sign to the established cash position (Butterworth and Holmes, 1996).

According to this strategy, the cash price and the futures price move closely together

and if the proportionate price changes in one market exactly match proportionate

price changes in the other market, then price risk will be eliminated.6 The traditionalhedge will not minimize risk because the cash and the futures market may not move

perfectly together. The minimum variance hedge ratio takes account of this lack of

perfect correlation and identifies the hedge ratio which minimizes risk. The minimum

variance hedge ratio is estimated as the slope coefficient of the following regression:

4 Similar analysis is provided in Figlewski (1984) and Myers (1991).5 According to Cecchetti et al. (1988), the optimal hedge ratio b can be expressed as rsc/sf, where r is

the correlation between futures price and cash price, sc the cash standard deviation and sf the futures

standard deviation. Thus, if the futures have the same or higher price volatility than the cash, the hedge

ratio can be no greater than the correlation between them, which will be less than unity.6 With the beta hedge strategy, the hedge ratio is calculated as the beta of the cash portfolio based on

the market model (Figlewski, 1984). This strategy takes account of the fact that the cash portfolio to be

hedged may not match the portfolio on which the futures contract is written (Butterworth and Holmes,

1996). Further, according to Figlewski (1984), the beta hedge ratio is only optimal when the position is to

be held until maturity of the futures. This paper does not investigate the beta hedge.

T. Choudhry / J. of Multi. Fin. Manag. 13 (2003) 171�/192 173

rct �a�brf

t �ot; (4)

where rct ; rf

t and b are defined as before and ot is an error term.7 If the joint

distribution of cash and futures prices is changing through time, regression

employing past data will not correctly estimate the current risk minimizing hedgeratio (Cecchetti et al., 1988). Under time-varying distribution, the bivariate GARCH

models may be applied to estimate time-varying hedge ratios. According to Cecchetti

et al., another problem with the regression estimated hedge ratio is that it does not

take into consideration the effect on expected returns. Hedging away the risk must

also hedge away the expected return to bearing that risk. This is the cost of hedging,

and the reward to risk bearing in the futures market. This paper estimates the

traditional, the minimum variance and the time-varying hedge ratios, and compares

their hedging effectiveness for six stock futures market.

3. The bivariate GARCH, GARCH-X models and time-varying hedge ratio

3.1. Bivariate GARCH

As shown by Baillie and Myers (1991) and Bollerslev et al. (1992), weak

dependence of successive asset price changes may be modelled by means of the

GARCH model. According to the GARCH model, the conditional variance of a

time series depends upon the squared residuals of the process (Bollerslev, 1986). TheGARCH model has the advantage of incorporating heteroscedasticity into the

estimation procedure and it also captures the tendency for volatility clustering in

financial and economic data. The GARCH model may be applied both in univariate

and multivariate forms. Simplicity is the main advantage of a univariate GARCH

model, but it utilizes only the information in one market’s own history (Wahab,

1995). The multivariate GARCH model uses information from more than one

market’s history. According to Conrad et al. (1991), multivariate models provide

more precise estimates of the parameters because they utilize information in theentire variance�/covariance matrix of the errors. Further, the generated regressor

problem associated with univariate models is avoided in multivariate models because

it estimates all parameters jointly (Pagan, 1984). According to Engle and Kroner

(1995), multivariate GARCH models are useful in multivariate finance and

economic models, which require the modelling of both variance and covariance.

7 As pointed out by the referee, lead�/lag effect is important in a relationship between stock cash and

stock futures markets. Frino et al. (2000) provides a good discussion of the lead�/lag relationship between

stock cash and futures market. In this paper, Eq. (4) is also estimated with six leads and lags of the returns

in the future market as dependent variables. Results from the lead�/lag tests are not provided in order to

save space but are available on request.


Multivariate GARCH models allow the variance and covariance to depend on the

information set in a vector ARMA manner (Engle and Kroner, 1995). This, in

turn, leads to the unbiased and more precise estimate of the parameters (Wahab,

1995).

The following bivariate MA(1)-GARCH(p ,q) model may be used to represent the

returns from the stock cash and futures markets:

yt�m�ot�uot�1; (5)

ot=Vt�1�N(0;Ht); (6)

vech Ht�C�Xp

j�1

Aj vech o2t�j�

Xq

j�1

Bj vech Ht�j; (7)

where yt� (rct ; rf

t ) is a (2�/1) vector containing stock returns from the cash (spot) and

futures markets, Ht is a (2�/2) conditional covariance matrix, C is a (3�/1)

parameter vector (constant), Aj and Bj are (3�/3) parameter matrices and vech is thecolumn stacking operator that stacks the lower triangular portion of a symmetric

matrix. The moving average (MA) term uot�1 is included to capture the effect of

non-synchronous trading. According to Susmel and Engle (1994), non-synchronous

trading induces negative serial correlation and the MA term allows for autocorrela-

tion induced by discontinuous trading in the asset (as suggested by Scholes and

Williams, 1977).

Engle and Kroner (1995) and Wahab (1995) state that various restrictions may be

imposed in this parameterization to make estimation easier. A parsimoniousrepresentation can be obtained by imposing a diagonal restriction on the multi-

variate GARCH parameter matrices so that each variance and covariance element

depends only on its own past values and prediction errors (Bollerslev et al., 1988). In

other words, this presentation is obtained by assuming that matrices Aj and Bj

are diagonal. According to Engle and Kroner (1995) and Baillie and Myers (1991),

the stated restriction seems plausible since it implies that each variance and

covariance depends only on its own past values and prediction errors. The following

equations present a diagonal vech bivariate GARCH(1,1) conditional varianceequation(s):

H11;t�C1�A11(o1;t�1)2�B11(H11;t�1); (7a)

H12;t�C2�A22(o1;t�1o2;t�1)�B22(H12;t�1); (7b)

H22;t�C3�A33(o2;t�1)2�B33(H22;t�1): (7c)

In the bivariate GARCH(1,1) system, the diagonal vech parameterization involves

nine conditional variance parameters. To ensure a positive conditional variance, the

values of C , A11, A33, B11 and B33 are restricted to zero or greater. The ARCHprocess in the residuals from the cash equation is shown by the coefficient of

(o1,t�1)2, (A11), while the coefficient of (o2,t�1)2, (A33), represents the ARCH process

in the futures equation residuals. The parameters A22 and B22 represent the

covariance GARCH parameters, which account for the conditional covariance

between cash and futures prices. Significant covariance parameters imply strong


interaction between the cash and futures prices.8 As advocated by Baillie and Myers

(1991, p. 116), it is vital to let the conditional covariance be time-dependent, as in the

bivariate GARCH model, rather than be a constant. This ability of the bivariate

GARCH model to have time-dependent conditional variance makes it ideal to

provide a time-variant hedge ratio.

Given the bivariate GARCH model of the cash and the futures stock returns

presented above, the time-varying hedge ratio can be expressed as

bt�H̄12;t

H̄22;t

; (8)

where H̄12; t is the estimated conditional variance between the cash and futures stock

returns and H̄22; t the estimated conditional variance of the futures returns from thebivariate GARCH model. Given that conditional covariance is time-dependent, the

optimal hedge ratio will be time-dependent.

3.2. Bivariate GARCH-X

An extension of the GARCH model linked to an error-correction model of

cointegrated series has been put forward by Lee (1994). This model is known as the

GARCH-X model. Cointegration implies that in a long-run relationship betweentwo or more nonstationary variables, it is required that these variables should not

move too far apart from each other. Such nonstationary variables might drift apart

in the short-run, but in the long-run they are constrained. Thus, cointegration means

that one or more linear combinations of these variables is stationary even though

individually they are not. The error-correction term, which represents the short-run

deviations from the long-run cointegrated relationship, has important predictive

powers for the conditional mean of the cointegrated series (Engle and Yoo, 1987).

According to Lee (1994), if short-run deviations affect the conditional mean, theymay also affect conditional variance, and a significant positive effect may imply that

the further the series deviate from each other in the short-run, the harder they are to

predict. If the error-correction term (short-run deviations) from the cointegrated

relationship between cash price and futures price affects the conditional variance

(and conditional covariance), then conditional heteroscedasticity may be modelled

with a function of the lagged error correction term. If shocks to the system that

propagate on the first and the second moments change the volatility, then it is

reasonable to study the behaviour of conditional variance as a function of short-rundeviations (Lee, 1994). Given that short-run deviations may affect the conditional

variance and conditional covariance, then it will also influence the time-varying

optimal hedge ratio as defined in Eq. (8).

The following bivariate MA(1)-GARCH(p ,q )-X model may be used to represent

the returns from the stock cash and futures markets:

8 Bera and Higgins (1993) and Engle and Kroner (1995) provide detailed analysis of multivariate

GARCH models.


yt�m�d(zt�1)�ot�uot�1; (9)

ot=Vt�1�N(0;Ht); (10)

vech Ht�C�Xp

j�1

Aj vech(ot�j)2�

Xq

j�1

Bj vech Ht�j

�Xk

j�1

Dj vech(zt�1)2; (11)

where zt is the error-correction term from the cointegration relationship between

stock cash index and futures index.9 The remaining variables are as described earlier.The error-correction term in the returns equation (Eq. (9)) measures the affect of the

short-run deviations on the cash and futures returns. The squared error term in the

conditional variance and covariance equations (Eq. (11)) measure the influences of

the short-run deviations on conditional variance and covariance.

The following equations present a diagonal vech bivariate MA(1)-GARCH(1,1)-X

conditional variance equation(s) with the squared error-correction term (zt) lagged

once:

H11;t�C1�A11(o1;t�1)2�B11(H11;t�1)�D11(zt�1)2; (11a)

H12;t�C2�A22(o1;t�1o2;t�1)�B22(H12;t�1)�D22(zt�1)2; (11b)

H22;t�C3�A33(o2;t�1)2�B33(H22;t�1)�D33(zt�1)2: (11c)

As advocated by Lee (1994, p. 337), the square of the error-correction term (z)

lagged once should be applied in the GARCH(1,1)-X model. The parameters, D11

and D33 indicate the effects of the short-run deviations between the cash and the

futures prices from a long-run cointegrated relationship on the conditional variance

of the residuals of the cash and futures returns, respectively. The parameter D22

shows the effect of the short-run deviations on the conditional covariance between

cash and futures returns. As stated above, if short-run deviations between cash price

and futures price affect the conditional variance of the cash and futures returns, andthe conditional covariance between the two returns, then optimal hedge as defined in

Eq. (8) will also be affected. In other words, if D33 and D22 are significant then H12

(conditional covariance) and H22 (conditional variance of futures returns) are going

to differ from the standard GARCH model H12 and H22. In such a case, the

GARCH-X time-varying hedge ratio will be different from the GARCH time-

varying hedge ratio. If the two time-varying hedge ratios are different, then the

9 The following cointegration relationship is investigated by means of the Engle�/Granger method:

St�h�a6

i��6giFt�i�zt; where St and Ft are log of cash index and futures index, respectively. In this test

also six leads and lags of the future price index are added in order to investigate the lead/lags effect. This

was suggested by the referee.


interesting empirical question arises; which one is more effective? All the above

methods of estimating the hedge ratios (including the constant ratios) are applied

and their effectiveness is compared in this paper.

4. The data and the basic statistics

Daily stock returns from the cash (spot) and the futures markets of Australia,

Germany, Hong Kong, Japan, South Africa and the United Kingdom are used in the

empirical tests. All the data range from January 1990 to December 1999 except for

Germany and South Africa where the data are from January 1991 to December

1999. All futures price indices are continuous series.10 The Australian spot stock

index is based on the All Ordinary price index and the futures prices are based on the

All Ordinary futures index. The All Ordinary share index contains 307 Australian

stocks. The cash and futures prices of the German indices are based on the Dax 30

index and the Eurex-Dax futures index. The Dax 30 contains 30 German stocks. The

Hang Seng price index and Hang Seng index futures represent the Hong Kong cash

and futures prices, respectively. The Hang Seng index contains 33 stocks on the

Hong Kong stock exchange. In the case of Japan, the Nikkei 225 price index is used

for the cash price index and Nikkei stock average futures prices are used for the

futures index. The Nikkei 225 contains 225 Japanese stocks. The JSE industrial index

and the Industrial 25 index represent the South African cash and futures prices.

These indices include 25 South African stocks. The United Kingdom cash and

futures prices are represented by the FTSE-100 index and the FTSE-100 futures

index, respectively. The United Kingdom indices include 100 stocks. All data are

obtained from DATASTREAM .

Stock returns are defined as the first difference in the log of price indices (both

cash and futures). Table 1 shows some of the basic statistics of the 12 stock returns.11

As expected, all series are found to have significant and positive kurtosis implying

higher peaks and fatter tails. Most series are also skewed except the cash returns of

Hong Kong and the futures returns of the United Kingdom. Significant mean is

found only for South Africa and the United Kingdom.

10 The continuous series is a perpetual series of futures prices. It starts at the nearest contract month

which forms the first values for the continuous series, until either the contract reaches its expiry date or

until the first business day of the actual contract month. At this point, the next trading contract month is

taken.11 The stochastic structure of the data is checked by means of several unit root tests. Results from the

augmented Dickey�/Fuller test, the Phillips�/Perron test and the KPSS test show all nine logs of price

indices to be nonstationary at levels and stationary after first difference. Thus all cash and futures returns

are stationary in levels. These unit root test results are available on request.


Table 1

Basic statistics of cash and futures stock returns total period

Australia Germany Hong Kong Japan South Africa United Kingdom

Cash Futures Cash Futures Cash Futures Cash Futures Cash Futures Cash Futures

Mean 0.00025 0.00025 0.00068 0.00068 0.00069 0.00068 �/0.00028 �/0.00028 0.00048** 0.00047*** 0.00040** 0.00040***

Variance 0.00007 0.00011 0.00014 0.00016 0.00029 0.00038 0.00022 0.00022 0.000095 0.00019 0.00008 0.00011

Skewness �/0.263* �/0.215* �/0.599* �/0.505* 0.060 0.448* 0.339* 0.252* �/1.275* �/0.905* 0.084*** 0.022

Kurtosis 4.660* 4.390* 6.270* 7.810* 11.300* 12.470* 4.210* 2.720* 16.210* 19.310* 2.271* 1.604*

Obs: observations.

* Significance at the 1% level.

** Significance at the 5% level.

*** Significance at the 10% level.

T.

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Table 2

Standard bivariate MA(1)-GARCH(1,1) results

Australia Germany Hong Kong Japan South Africa U.K.

m1 0.00032** (2.35) 0.00078* (4.48) 0.0012* (5.39) 0.00022 (1.28) 0.00080* (4.93) 0.00060* (3.61)

u1 0.0776* (4.74) 0.2092* (11.65) 0.1257* (8.48) 0.2658* (17.86) �/0.0936* (�/4.72) 0.1151* (7.47)

m2 0.00032** (2.12) 0.00083* (4.94) 0.00117* (5.35) 0.000192 (1.06) 0.00076* (3.66) 0.00054* (3.61)

u2 0.2151* (13.87) 0.3267* (19.09) 0.2422* (17.08) 0.2842* (18.79) 0.0812* (4.29) 0.1901* (13.04)

C1 1.70�/10�5* (9.42) 0.47�/10�5* (9.89) 0.01�/10�5* (17.28) 2.08�/10�5* (13.00) 0.81�/10�5* (17.72) 0.27�/10�5* (10.75)

A11 0.1264* (15.31) 0.0723* (13.75) 0.1088* (19.98) 0.1444* (17.19) 0.1740* (19.49) 0.0575* (14.58)

B11 0.6203* (20.13) 0.8951* (130.01) 0.8557* (147.05) 0.7687* (70.69) 0.7120* (62.97) 0.9030* (192.59)

C2 1.87�/10�5* (10.89) 0.47�/10�5* (11.75) 0.98�/10�5* (15.87) 2.16�/10�5* (12.11) 0.81�/10�5* (18.55) 0.30�/10�5* (11.20)

A22 0.1219* (15.21) 0.0611* (13.45) 0.0941* (20.76) 0.1352* (16.37) 0.1320* (19.70) 0.0560* (14.36)

B22 0.6417* (24.97) 0.9046* (147.94) 0.8710* (174.84) 0.7678* (67.12) 0.7600* (88.32) 0.9050* (200.99)

C3 2.42�/10�5* (11.20) 0.68�/10�5* (11.29) 1.04�/10�5* (13.41) 2.33�/10�5* (11.07) 1.44�/10�5* (17.71) 0.03�/10�5* (11.09)

A33 0.1270* (14.78) 0.0638* (11.99) 0.0884* (21.70) 0.1393* (16.08) 0.1230* (21.56) 0.0556* (13.83)

B33 0.6631* (27.57) 0.8965* (117.72) 0.8810* (192.40) 0.7610* (61.05) 0.7780* (86.41) 0.9093* (200.44)

L 24023.02 20187.04 21643.50 22698.18 20676.72 25166.51

/ot=H1=2t /

LB cash 7.92 3.02 7.89 7.22 8.28 9.69

LB future 3.28 8.07 9.05 7.55 5.50 10.39

/o2t =Ht/

LB cash 2.30 0.80 2.66 5.49 5.19 2.30

LB futures 3.49 0.39 5.51 10.55 5.57 7.34

LB: Ljung�/Box statistics for serial correlation of the order 9. t -Statistics in the parentheses; L : log-likelihood function value.


** Significance at the 5% level. ***Significance at the 10% level.

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5. The empirical results

5.1. The bivariate GARCH, cointegration, GARCH-X and OLS results

Table 2 shows the results from the standard bivariate MA(1)-GARCH(1,1) model

for the total period.12 The bivariate MA(1)-GARCH(1,1) results are quite standard.

The ARCH coefficients (A11 and A33) are all positive and significant thus implying

volatility clustering both in the cash returns and the futures returns. The ARCH

coefficients are also less than unity in all cases. The sign and significance of thecovariance parameters (A22 and B22) indicate positive and significant interaction

between the two prices for all six cases. The MA coefficient (u ) is also positive and

significant in all markets except for the South African cash returns. The significant

MA term may be due to non-synchronous trading.13 To assess the general

descriptive validity of the model, a battery of standard specification tests is

employed. Specification adequacy of the first two conditional moments is verified

through the serial correlation test of white noise. These tests employ the Ljung�/Box

Q -statistics on the standardized (normalized) residuals (ot=H1=2t ) and standardized

squared residuals (ot=H2t ): All series are found to be free of serial correlation (at the

5% level). Absence of serial correlation in the standardized squared residuals implies

the lack of need to encompass a higher order ARCH process (Giannopoulos, 1995).

Table 3 shows the cointegration test results between the log of cash price index and

the log of futures price index for the total period.14 Tests are conducted based on the

Engle and Granger (1987) two-step method.15 Description of the Engle and Granger

method is not provided here since it is available in numerous articles and books. In

the cointegration tests, six leads and lags of the futures price index are also added tocheck for possible lead�/lag effect on the relationship and the time-varying hedge

ratio.16 Cointegration between the two price series is found in all six tests. Both the

augmented Dickey�/Fuller tests with and without the trend indicate stationary

12 In a GARCH(p ,q ) model, different combinations of p and q may be applied but, as indicated by

Bollerslev et al. (1992, p. 10), p�/q�/1 is sufficient for most financial and economic series. Bollerslev

(1988) provides a method of selecting the length of p and q in a GARCH model. Tests in this paper were

also conducted with different combinations of p and q with p�/q�/2 being the maximum lag length.

Results based on log-likelihood function and likelihood ratio tests indicate that the best combination is

p�/q�/1. These results are available on request.13 The significant MA term may also be due to different news observed by different investors or the

same news being interpreted differently by investors. This could create a negative serial correlation, as a

result of a process of price adjustment where the price bounces back and forth between centres with

different information.14 As required by any cointegration tests, the stochastic structure of the individual variables has to be

checked. Standard unit root tests such as ADF and KPSS showed that logs of all price indices are

nonstationary in levels but stationary after first difference. Thus the cash and the futures returns are

stationary in levels. These results are available on request.15 Other forms of cointegration tests such as the multivariate Johansen and Juselius (1990) method were

also applied. Results obtained are the same as when using the Engle and Granger method. Johansen tests

results are not presented to save space but are available on request.16 As stated earlier, this was suggested by the referee.


Table 3

Cointegration tests between cash price index and futures price index

Con Ft�6 Ft�5 Ft�4 Ft�3 Ft�2 Ft�1 Ft Ft�1 Ft�2 Ft�3 Ft�4 Ft�5 Ft�6 H1 H2 ADF-T ADF-NT

Australia �/0.048 0.021 0.001 �/0.005 0.018 0.004 0.034 0.669 0.130 0.014 0.021 0.011 0.004 0.083 7.22*** 92.67*** �/0.091* (�/8.48) {4} �/0.090* (�/8.48) {4}

Germany �/0.050 0.023 �/0.002 �/0.002 0.020 �/0.021 �/0.006 0.767 0.204 0.019 �/0.005 0.001 �/0.004 0.010 4.80*** 77.25*** �/0.237* (�/9.16) {12} �/0.232* (�/9.08) {12}

HK 0.017 0.023 �/0.010 �/0.016 0.027 0.003 0.017 0.798 0.124 �/0.001 0.000 0.011 �/0.009 0.030 6.37*** 45.29*** �/0.107* (�/8.32) {4} �/0.105* (�/8.29) {4}

Japan 0.228 �/0.009 0.000 0.002 0.004 �/0.005 0.020 0.931 0.037 �/0.009 �/0.001 0.005 �/0.013 0.012 0.94 2.87*** �/0.152* (�/7.86) {8} �/0.134* (�/7.68) {8}

SA �/0.326 �/0.044 �/0.008 �/0.014 �/0.014 0.001 0.021 0.591 0.107 0.045 0.035 0.029 0.023 0.267 1.44 38.73*** �/0.012** (�/3.30) {12} �/0.010** (�/3.10) {12}

UK �/0.076 0.025 �/0.004 0.004 0.009 �/0.008 0.013 0.828 0.077 0.015 0.002 0.009 �/0.005 0.043 5.33*** 34.54*** �/0.055* (�/6.79) {4} �/0.055* (�/6.77) {4}

Number of lags in brackets. H1: null hypothesis that all lead coefficients are equal to zero. H2: null hypothesis that all lag coefficients are equal to zero. ADF-

T: ADF tests with trend. ADF-NT: ADF tests with no trend. t -Statistics in the parentheses.

* Rejection of the null at the 1% level.

** Rejection of the null at the 5% level.

*** Rejection of the unit root at the 5% level or above.

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residuals (error term) in levels from the OLS regression between the two prices.17 In

all tests, cointegration is indicated at the 5% level or above.18 A similar result has

been provided by Ghosh (1993b) for the United States stock futures. Results indicate

a significant effect of the lags in all tests and a significant effect of leads in all cases

except Japan and South Africa. These stationary error terms from the cointegration

test represent the short-run deviations between the cash price and the futures prices.

In the GARCH-X model, these error terms are included in the mean, conditional

variance and covariance equations.

Table 4 shows the bivariate GARCH(1,1)-X model results for the total period.19

These results are similar to the standard GARCH results shown in Table 2. The

ARCH coefficient is significant and less than unity for both the cash and futures

returns in all six tests. Once again the MA coefficient is positive and significant in all

cases except the South African cash returns. All markets again show evidence of

significant interaction between the cash returns and the futures returns. The short-

run deviations between the cash price and future prices have a positive and

significant effect (d1) on the cash returns in all six tests. In the case of the futures

return, the deviations (d2) only significantly influence the German futures return.

The important part of the GARCH-X results is the influence of the short-run

deviations between the cash price and the futures price on the conditional variance

and covariance. The parameters measuring the effects of the short-run deviations on

the conditional variance of cash returns (D11) and futures returns (D33) are mostly

found to be positive and significant. A positive and significant effect of the short-run

deviations on the conditional variance implies that as the deviation between the cash

and future prices gets larger, the volatility of cash and futures returns increases and

prediction becomes more difficult. The short-run deviation effect on conditional

variance is only insignificant in the case of German cash returns. In the case of

German futures returns, the short-run deviations impose a significant negative effect.

The parameter (D22) that measures the effect of the short-run deviations on the

conditional covariance is found to be significant and positive in all cases except for

the United Kingdom. These results clearly show that short-run deviations do indeed

impose a significant effect on the conditional variance of the cash and futures

returns, and also on the conditional covariance between the two returns. The

question to be answered is whether these effects of the short-run deviations also

influence the effectiveness of the time-varying hedge ratio. Once again the

standardized residuals and the standardized squared residuals are found to be free

of serial correlation (at the 5% level). Absence of serial correlation in the

17 The lag length applied in the cointegration ADF test is based on the AIC evidence. The maximum

number of lags applied is 12. Results with the maximum lags required to remove serial correlation are

presented.18 Results from the two smaller samples are similar. These results are not provided in order to save

space but are available on request.19 Once again log-likelihood function and likelihood ratio tests indicate that the best combination is

p�/q�/1.


Table 4

Bivariate MA(1)-GARCH(1,1)-X results

Australia Germany Hong Kong Japan South Africa UK

m1 0.00034** (2.40) 0.00113* (5.80) 0.0010* (4.26) 0.00062* (3.13) 0.00064* (3.67) 0.00061* (4.39)

d1 0.085* (4.55) 0.537* (18.372) 0.188* (6.49) 0.259* (6.51) 0.0135* (2.873) 0.0500** (2.14)

u1 0.047* (2.73) 0.055* (2.66) 0.060* (3.64) 0.137* (7.79) �/0.098* (�/4.96) 0.095* (5.71)

m2 0.00032*** (1.91) 0.0012* (5.87) 0.0012* (4.82) 0.00027 (1.32) 0.0007* (3.25) 0.0005* (3.53)

d2 �/0.022 (�/0.96) 0.164* (7.48) �/0.0088 (�/0.29) �/0.021 (�/0.53) 0.0032 (0.006) �/0.0204 (�/0.79)

u2 0.154* (8.53) 0.138* (5.00) 0.158* (9.33) 0.163* (9.43) 0.072* (3.72) 0.162* (9.76)

C1 1.27�/10�5* (8.19) 7.10�/10�6* (9.01) 8.40�/10�6* (10.34) 1.18�/10�5* (10.56) 7.55�/10�6* (15.11) 8.27�/10�6* (13.53)

A11 0.111* (15.11) 0.096* (12.33) 0.107* (19.61) 0.118* (16.17) 0.1550* (15.80) 0.105* (17.00)

B11 0.676* (24.76) 0.845* (78.17) 0.853* (142.72) 0.823* (87.54) 0.7170* (57.84) 0.778* (121.64)

C2 1.41�/10�5* (9.57) 7.51�/10�6* (12.81) 8.27�/10�6* (10.17) 1.20�/10�5* (9.81) 7.03�/10�6* (17.09) 9.45�/10�6* (14.63)

A22 0.107* (14.27) 0.084* (12.53) 0.093* (19.61) 0.108* (14.02) 0.1190* (16.25) 0.104* (17.07)

B22 0.694* (30.39) 0.862* (100.42) 0.868* (167.72) 0.828* (83.27) 0.7660* (78.07) 0.776* (186.05)

C3 1.88�/10�5* (10.16) 1.07�/10�6* (15.96) 8.84�/10�6* (9.60) 1.28�/10�5* (8.88) 1.15�/10�5* (14.79) 1.05�/10�5* (14.51)

A33 0.113* (13.22) 0.087* (12.46) 0.087* (19.71) 0.108* (14.02) 0.1140* (19.36) 0.103* (15.71)

B33 0.707* (32.44) 0.859* (103.91) 0.880* (180.09) 0.827* (75.74) 0.7810* (76.83) 0.789* (151.48)

D11 0.0175** (2.51) 0.0069 (0.635) 0.0247* (7.15) 0.040** (2.24) 0.0009* (5.46) 0.0119*** (1.66)

D22 0.0264** (2.57) �/0.0445* (�/7.78) 0.015* (3.65) 0.034*** (1.76) 0.0025* (7.29) 0.0136 (1.49)

D33 0.0173** (2.44) �/0.0178** (�/2.55) 0.018* (5.04) 0.032*** (1.85) 0.0011* (5.37) 0.0134*** (1.65)

L 24096.55 20458.51 21763.13 22896.82 20715.78 25097.99

/ot=H1=2t /

LB Cash 4.39 1.94 8.21 4.31 6.61 7.43

LB Futures 5.34 1.88 3.66 9.26 4.65 8.93

/o2t =Ht/

LB cash 3.88 0.76 2.43 2.99 5.52 11.41

LB futures 11.07 0.66 3.90 5.30 6.52 10.29


** Significance at the 5% level.

*** Significance at the 10% level.

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standardized squared residuals implies the lack of need to encompass a higher-order

ARCH process.20

Table 5 shows the OLS estimation of Eq. (4) for the total period. It shows the

results from the regression between the cash returns and the futures returns for all six

countries.21 The size of the coefficient on the futures returns shows the constant

minimum variance hedge ratio. In all tests, the coefficient on the future returns is

positive and significantly different from zero at the 1% level. Also, in all cases the

hedge ratio is found to be significantly less than unity by means of the F -test. Thus

results fail to indicate a one-to-one relationship between the cash and the futures

returns. The largest coefficient is found in the case of Japan (0.93) and the lowest for

South Africa (0.5998). These results are quite standard and are found in other studies

also.22

Fig. 1 shows the time-varying hedge ratios from the standard bivariate

GARCH(1,1) and the bivariate GARCH(1,1)-X and the constant minimum variance

hedge ratio for Hong Kong. The top graph shows the standard GARCH time-

varying ratio against the constant ratio and the bottom graph represents the time-

varying ratio from the GARCH-X against the constant ratio. Both time-varying

ratios are clustered around the constant hedge ratio. A close inspection of both time-

varying ratios indicates that they are very similar but not identical. The large dip at

the end is possibly due to the Asian crisis of 1997 and 1998. Graphs of other five

markets also convey the same story and are not provided to save space but are

available on request.

20 Both the GARCH and GARCH-X models were estimated by means of the Berndt et al. (1974)

algorithm.21 As noted earlier in footnote 11, all cash and futures return series are found to be stationary in levels.

Thus their application in OLS regressions is quite safe.22 In the lead�/lags tests, lags had a significant effect in all six tests but leads were only significant in the

cases of Australia and Hong Kong. These results are available on request.

Table 5

OLS regression between cash and futures returns minimum variance hedge ratio

Country Constant Futures returns Diagnostic F -test

Australia 0.000087 (1.083) 0.6529* (86.60) R2�/0.742; DW�/2.43 2118.89**

Germany 0.00016 (1.124) 0.7632* (67.50) R2�/0.660; DW�/2.80 438.72**

Hong Kong 0.00015 (1.086) 0.7885* (110.78) R2�/0.824; DW�/2.70 883.03**

Japan �/0.00001 (�/0.113) 0.9350* (132.27) R2�/0.870; DW�/2.67 97.66**

South Africa 0.00019 (1.741) 0.5998* (74.57) R2�/0.703; DW�/2.04 2475.01**

UK 0.00007 (1.426) 0.8260* (164.59) R2�/0.912; DW�/2.40 1201.71**

DW: Durbin�/Watson. t -Statistics in the parentheses.

* Significant at the 1% level.

** Reject the null at the 5% level or above.


Fig. 1. Constant and GARCH (top) and constant and GARCH-X (bottom) time-varying hedge ratio.


5.2. Total sample hedge ratios comparison result

As indicated by Baillie and Myers (1991) and Park and Switzer (1995), comparison

between the effectiveness of different hedge ratios is made by constructing portfolios

implied by the computed ratios and then comparing the variance of these

constructed portfolios. The portfolios are constructed as (rct �bt�r

ft ); where rc

t is the

cash (spot) returns, rft the futures returns and bt�the estimated optimal hedge ratio.23

The variance of these constructed portfolios is estimated and compared. The change

in variance is calculated as (Varothers�/VarGARCH)/Varothers. Since the main theme of

this paper is to investigate the effectiveness of the time-varying hedge ratio,

comparison of changes in variance is only conducted between the time-varying

ratio portfolios (estimated by means of the GARCH model and the GARCH-X

model) and constant ratio portfolios.

Table 6 shows the variance of the portfolio estimated using the different types of

optimal hedge ratios and the percentage change in the variance of the portfolios

constructed using the time-varying hedge ratio from the bivariate GARCH(1,1)

model, GARCH(1,1)-X model and the three constant hedge ratio portfolios. A

comparison between the standard bivariate GARCH(1,1) and the constant hedge

ratio portfolios indicates that time-varying hedge ratio portfolios have the lowest

variance in only two cases, Australia and Hong Kong. The reduction in the variance

is quite substantial compared with the unhedged portfolio. In the case of Japan, both

the traditional hedge and the minimum variance hedge ratio portfolios outperform

the GARCH time-varying ratio portfolio.

23 In the case of the constant ratio, the time subscript does not exist.

Table 6

Total period (January 1990�/December 1999) portfolio variance and percentage change in the variance


Unhedged 0.0000656 0.000142 0.000286 0.000219 0.0000953 0.0000806

Traditional 0.0000307 0.0000574 0.0000671 0.0000294 0.0000581 0.0000103

Minimum 0.0000169 0.0000484 0.0000500 0.0000284 0.0000283 0.0000071

GARCH 0.0000153 0.0000494 0.0000481 0.0000298 0.0000284 0.0000074

GARCH-X 0.0000152 0.0000490 0.0000484 0.0000297 0.0000282 0.0000076

Percentage change in the portfolio variance between GARCH and other methods (excluding GARCH-X)

Unhedged 76.68 65.21 83.18 86.39 70.20 90.82

Traditional 50.16 13.94 28.32 �/1.36 51.12 28.16

Minimum 9.47 �/2.07 3.80 �/4.93 �/0.35 �/4.23

Percentage change in the portfolio variance between GARCH-X and other methods

Unhedged 76.83 65.55 83.31 86.44 70.41 90.57

Traditional 50.49 14.63 38.64 �/1.02 51.46 26.21

Minimum 10.06 �/1.24 3.20 �/3.37 0.35 �/7.04

GARCH 0.65 0.82 �/0.62 0.34 0.70 �/2.70


Portfolios created using hedge ratios from the GARCH-X models provide similar

results. Once again in only two cases, Australia and South Africa, the GARCH-X

time-varying ratio provides the lowest variance. As expected, the GARCH-X time-

varying ratio outperforms the unhedged and the traditional hedge in all cases, except

Japan where the traditional hedge does better by a small margin. The minimum

variance hedge ratio does do better in three of the cases, Germany, Japan and the

United Kingdom. The GARCH-X does better than the standard GARCH in allcases, except for Hong Kong and the United Kingdom. In the case of Hong Kong,

the reduction is by less than 1%. For the remaining four countries, the GARCH-X

ratio portfolio produces the lowest variance, though the difference between the two

time-varying ratios is small. There is a substantial drop in the variance between the

constant ratio portfolios and the GARCH-X portfolios especially for Australia,

Hong Kong and South Africa. The inconsistent performance of both GARCH

model ratios may be attributed to the complexity of the GARCH model (Baillie and

Myers, 1991).

5.3. Out-of-sample hedge ratios comparison result

Park and Switzer (1995) and Baillie and Myers (1991) further claim that the more

reliable measure of hedging effectiveness is the hedging performance of different

methods for out-of-sample periods. This paper compares the hedging effectiveness ofthe five different methods during two different out-of-sample time periods. The out-

of-sample periods used are from January 1998 to December 1999 and from January

1999 to December 1999. In order to investigate the out-of-sample hedging

effectiveness of the five hedging methods, the bivariate GARCH, the GARCH-X

models and the minimum variance equations are estimated for the periods January

1990�/December 1997 and January 1990�/December 1998, and then the estimated

parameters are applied to compute the hedge ratios and the portfolios for the two

out-of-sample periods.24 Once again hedging effectiveness is compared by comparingthe variance of these portfolios and the change in the variance.

Table 7 shows the variance of the out-of-sample portfolios and the percentage

change in variance of the portfolios from January 1998 to December 1999.

Comparing the change in the variance between the standard GARCH ratio

portfolios and the constant ratio portfolios indicates the superior performance of

the time-varying hedge ratio. The standard GARCH ratio outperforms all three

constant ratios for all markets. The decline in the variance of the portfolios is again

quite substantial especially for Australia, Hong Kong and the United Kingdom. TheGARCH-X time-varying hedge ratio outperforms all the constant hedge ratios

except in the case of South Africa but fails to provide the lowest variance compared

with the GARCH time-varying ratio in three cases, though the difference between

24 The bivariate GARCH and the minimum variance equation estimations of the period 1990�/1997

and 1990�/1998 are not provided in order to save space but are available on request. These parameters are

similar to the ones estimated for the whole sample period.


the two time-varying hedge ratio is not much. In the case of Australia, Hong Kong

and Japan, the GARCH-X does better than the standard GARCH. For the

remaining three countries, the standard GARCH outperforms the GARCH-X.

The shorter out-of-sample period of January 1999�/December 1999 results are

presented in Table 8. Once again the standard GARCH ratio does better than the

constant ratios. There is no difference in the minimum variance ratio and the time-

Table 7

Out-of-sample period (January 1998�/December 1999) portfolio variance and percentage change in the

variance


Unhedged 0.0000710 0.000257 0.000501 0.000217 0.000202 0.000147

Traditional 0.0000187 0.0000589 0.000147 0.0000273 0.0000768 0.0000106

Minimum 0.0000134 0.0000603 0.000112 0.0000253 0.0000445 0.0000101

GARCH 0.0000121 0.0000539 0.0000964 0.0000251 0.0000433 0.0000088

GARCH-X 0.0000118 0.0000546 0.000096 0.0000250 0.0000450 0.0000091


Unhedged 82.96 79.03 80.76 88.43 78.56 94.01

Traditional 35.29 8.49 34.42 8.06 43.62 16.98

Minimum 9.70 10.61 13.93 0.79 2.70 12.87


Unhedged 83.38 78.75 80.84 88.48 77.72 93.81

Traditional 36.90 7.30 34.69 8.42 41.41 14.15

Minimum 11.94 9.45 14.29 1.20 �/1.12 9.90

GARCH 2.48 �/1.30 0.41 0.40 �/3.93 �/3.41

Table 8

Out-of-sample period (January 1999�/December 1999) portfolio variance and percentage change in the

variance


Unhedged 0.0000578 0.000186 0.000268 0.000156 0.0000980 0.000123

Traditional 0.0000126 0.0000266 0.000187 0.0000207 0.0000370 0.00000627

Minimum 0.00000990 0.0000300 0.000137 0.0000195 0.0000156 0.00000620

GARCH 0.00000897 0.0000254 0.000103 0.0000195 0.0000156 0.00000531

GARCH-X 0.0000090 0.0000268 0.000104 0.0000197 0.0000154 0.00000560


Unhedged 84.43 86.34 61.57 87.50 84.80 95.69

Traditional 28.57 4.51 31.34 5.80 57.84 14.52

Minimum 9.09 15.33 24.82 0.00 0.00 14.53


Unhedged 84.43 85.59 61.19 87.37 84.29 95.45

Traditional 28.57 �/0.75 44.39 4.83 58.38 11.11

Minimum 10.00 11.94 24.09 �/1.03 1.28 9.68

GARCH 0.00 5.51 �/0.97 �/1.03 1.28 �/5.66


varying ratio from the GARCH model results for Japan and South Africa. Reducing

the size of the out-of-sample size does not change the GARCH ratio versus the

constant ratio results by much. Once again the decline in the variance is quite large in

some cases. The GARCH-X ratio outperforms the unhedged portfolio in all six tests

but fails against the traditional hedge for Germany and against the minimum

variance hedge for Japan. The traditional hedge and the minimum variance hedge

only outperform the GARCH-X hedge by 1% or less. Comparison between the twotime-varying hedge indicates a better performance by the GARCH-X for Germany

and South Africa. The standard GARCH is just as good as the GARCH-X in the

case of Australia and does better for Hong Kong, Japan and the United Kingdom.

Once again the inconsistent performance of the GARCH time-varying ratios may be

attributed to the complexity of the GARCH model. The out-of-sample results do

provide evidence of superiority of the time-varying hedge ratio of the GARCH

model.

6. Conclusion

It is a well-documented claim in the futures market literature that optimal hedge

ratio should be time-varying and not constant (see Baillie and Myers, 1991; Myers,

1991). This paper investigates the hedging effectiveness of six stock futures. The

paper further studies the effects of short-run deviations from a long-run relationship

between the cash index and futures index on the optimal hedge ratio. The markets

under study are Australia, Germany, Hong Kong, Japan, South Africa and theUnited Kingdom during the period January 1990�/December 1999. Five different

hedging ratios, depending on different estimation procedures, are compared for

hedging effectiveness. Three of these five ratios are constant: the unhedged, the

traditional hedge and the minimum variance hedge. The fourth and the fifth ones,

the standard bivariate GARCH hedge ratio and the bivariate GARCH-X ratio, are

time-varying as they take into consideration the time-varying distribution of the cash

and futures returns. The GARCH-X model further takes into consideration the

effects of the long-run cointegration between the cash price and the futures price.The hedging effectiveness is compared by checking the variance of the portfolios

created using these hedge ratios. The lower the variance of the portfolio, the higher is

the hedging effectiveness of the hedge ratio.

The effectiveness of the hedge ratio is investigated by comparing the total sample

(January 1990�/December 1999) and of out-of-sample performance of the different

hedge ratios for two periods January 1998�/December 1999 (two years) and January

1999�/December 1999 (one year). The parameters of the GARCH model, GARCH-

X model and other models are estimated for the periods January 1990�/December1997 and January 1990�/December 1998, and then hedge ratios are forecast for the

out-of-sample periods January 1998�/December 1999 and January 1999�/December

1999 based upon the estimated parameters.

During the total period, the constant minimum variance hedge ratio does better

than the time-varying GARCH hedge ratio in most cases. In turn the GARCH-X


does better than the standard GARCH in most of these tests. Using the longer out-

of-sample period (January 1998�/December 1999), both versions of the time-varying

ratios do better than the constant ratios. A comparison between just the two time-

varying GARCH model ratios shows that the standard bivariate GARCH model

performs better than the bivariate GARCH-X model in three cases. The smaller out-

of-sample (January 1999�/December 1999) shows that the standard GARCH model

hedge ratio is again better than the constant ratios in all cases. Considering theGARCH-X, it seems to perform better than the constant ratio in four cases. Once

again the GARCH hedge ratio performance is better than the GARCH-X hedge

ratio in three cases.

These results provide some indication of the superior performance of the time-

varying hedge ratio as compared with the constant ratios. Results also indicate that

taking into consideration the short-run deviations between the cash price and the

futures price may also improve the hedging effectiveness of the time-varying hedge

ratio. Results in this paper advocate further research in this field. Further researchmay be conducted using different frequency of the data, time period, type of futures

markets, etc.

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short-run deviations and optimal hedge ratio: evidence from stock futures

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