The Effect of Asymmetries on Optimal Hedge Ratios
The Effect of Asymmetries on Optimal Hedge Ratios
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Author(s): , Ólan T. Henry and
Source: The Journal of Business , Vol. 75, No. 2 (April 2002), pp. 333-352
Published by: The University of Chicago Press
Stable URL: https://www.jstor.org/stable/10.1086/338484
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https://www.jstor.org/stable/10.1086/338484
(Journal of Business, 2002, vol. 75, no. 2)
� 2002 by The University of Chicago. All rights reserved.
0021-9398/2002/7502-0005$10.00
University of Reading
Ólan T. of Melbourne
University of Bristol
The Effect of Asymmetries on
Optimal Hedge Ratios*
I. Introduction
Over the past two decades, increases in the availability
and usage of derivative securities has allowed agents
who face price risk the opportunity to reduce their
exposure. Although there are many techniques avail-
able for reducing and managing risk, the simplest and
perhaps the most widely used is hedging with futures
contracts. A hedge is achieved by taking opposite po-
sitions in spot and futures markets simultaneously, so
that any loss sustained from an adverse price move-
ment in one market should, to some degree, be offset
by a favorable price movement in the other. The ratio
of the number of units of the futures asset that are
purchased relative to the number of units of the spot
asset is known as the hedge ratio.
Since risk in this context is usually measured as the
volatility of portfolio returns, an intuitively plausible
strategy might be to choose the hedge ratio that min-
* This article was written while the second author was on study
leave at the ISMA Centre, University of Reading. The development
of this article benefited from comments by the anonymous referees
and discussions with , , and –
mers. The responsibility for any errors or omissions lies solely with
the authors.
There is widespread evi-
dence that the volatility
of stock returns displays
an asymmetric response
to good and bad news.
This article considers the
impact of asymmetry on
time-varying hedges for
financial futures. An
asymmetric model that
allows forecasts of cash
and futures return volatil-
ity to respond differently
to positive and negative
return innovations gives
superior in-sample hedg-
ing performance. How-
ever, the simpler symmet-
ric model is not inferior
in a hold-out sample. A
method for evaluating the
models in a modern risk-
management framework
is presented, highlighting
the importance of allow-
ing optimal hedge ratios
to be both time-varying
and asymmetric.
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334 Journal of Business
imizes the variance of the returns of a portfolio containing the spot and futures
position; this is known as the optimal hedge ratio. There has been much
empirical research into the calculation of optimal hedge ratios (see, e.g., Cec-
chetti, Cumby, and Figlewski 1988; Myers and Thompson 1989; Baillie and
Myers 1991; Kroner and Sultan 1991; Lien and Luo 1993; Lin, Najand, and
Yung 1994; Strong and Dickinson 1994; Park and Switzer 1995).
The general consensus is that the use of multivariate generalized autore-
gressive conditionally heteroscedastic (MGARCH) models yields superior per-
formances, evidenced by lower portfolio volatilities, than either time-invariant
or rolling ordinary least squares (OLS) hedges. Cecchetti et al. (1988), Myers
and Thompson (1989), and Baillie and Myers (1991), for example, argue that
commodity prices are characterized by time-varying covariance matrices. As
news about spot and futures prices arrives to the market, the conditional
covariance matrix and, hence, the optimal hedging ratio become time-varying.
Baillie and Myers (1991) and Kroner and Sultan (1991), inter alia, employ
MGARCH models to capture time variation in the covariance matrix and the
resulting hedge ratio.
On the other hand, there is also evidence that the benefits of a time-varying
hedge are substantially diminished as the duration of the hedge is increased
(e.g., Lin et al. 1994). Moreover, there is evidence that the use of volatility
forecasts implied by options prices can further improve hedging effectiveness
(Strong and Dickinson 1994).
This article has three main aims. First, we link the concept of the optimal
hedge with Kroner and Ng’s (1998) notion of the “news impact surface.” The
hedging surface of the OLS model is independent of news arriving to the
market and therefore could be suboptimal. Second, we extend the models of
Cecchetti et al. (1988), Myers and Thompson (1989), and Baillie and Myers
(1991) to allow for time variation and asymmetry across the entire variance-
covariance matrix of returns. This means that the hedge ratio will be sensitive
to the size and sign of the change in prices resulting from information arrival.
Third, we adapt the methods used by Hsieh (1993) to show how the effec-
tiveness of hedges can be evaluated by the calculation of the minimum capital
risk requirements (MCRRs). Such a procedure allows the hedging performance
of the various models to be assessed using a relevant economic loss function
as well as on purely statistical grounds.
The article is laid out in six sections. Section II presents the theoretical
framework for deriving the hedge ratios, while Section III describes the data.
Section IV presents the empirical evidence on the performance of each hedging
model, while Section V outlines the bootstrap methodology used to calculate
the MCRR for each of the portfolios. Section VI concludes.
II. The Derivation of Optimal Hedge Ratios
Let and represent the logarithms of the stock index and stock indexC Ft t
futures prices, respectively. The actual return on a spot position held from
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Effect of Asymmetries 335
time is ; similarly, the actual return on a futurest � 1 to t DC p C � Ct t t�1
position is . However, at time , the expected return,DF p F � F t � 1t t t�1
, of the portfolio comprising one unit of the stock index and b unitsE (R )t�1 t
of the futures contract may be written as
E (R ) p E (DC ) � b E (DF ), (1)t�1 t t�1 t t�1 t�1 t
where is the hedge ratio determined at time , for employment inb t � 1t�1
period t.1 The variance of the portfolio may be written as
2h p h � b h � 2b h , (2)p, t C, t t�1 F, t t�1 CF, t
where , , and represent the conditional variances of the portfolioh h hp, t F, t C, t
and the spot and futures positions, respectively, and represents the con-hCF, t
ditional covariance between the spot and futures position. If the agent has the
two-moment utility function,
U(E R , h ) p E (R ) � wh , (3)t�1 t p, t t�1 t p, t
then the utility maximizing agent with degree of risk aversion w seeks to
max U(E R , h )t�1 t p, t
2( ) ( )p E DC �b E DF �w(h � b h � 2b h ). (4)t�1 t t�1 t�t C, t t�1 F, t t�1 CF, t
Solving equation (4) with respect to b under the assumption that is aFt
martingale process such that E (DF ) p E (F ) � F p F � F p 0t�1 t t�1 t t�1 t�1 t�1
yields , the optimal number of futures contracts in the investor’s portfolio,∗bt�1
hCF, t∗b p � . (5)t�1
If the conditional variance–covariance matrix is time-invariant (and if andCt
are not cointegrated), then an estimate of b*, the constant optimal hedgeFt
ratio, may be obtained from the estimated slope coefficient b in the regression
DC p a � bDF � u . (6)t t t
The OLS estimate of is also valid for the multiperiod hedge inb p h /hCF F
the case where the investors’ utility function is time separable.
However, it has been shown by numerous studies (see Sec. I above) that
the data do not support the assumption that the variance-covariance matrix
of returns is constant over time. Therefore, we follow recent literature by
employing a bivariate generalized autoregressive conditional heteroscedastic-
ity (GARCH) model, which allows the conditional variances and covariances
used as inputs to the hedge ratio to be time-varying.
1. Note that we are not requiring at this stage that the hedge ratio, , be time-varying but,bt�1
rather, that it is determined using information to time .t � 1
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336 Journal of Business
In the absence of transactions costs, market microstructure effects, or other
impediments to their free operation, the efficient markets hypothesis and the
absence of arbitrage opportunities imply that the spot and corresponding fu-
tures markets react contemporaneously and identically to new information.
There has been some debate in the literature as to whether this implies that
the two markets must be cointegrated. Ghosh (1993), for example, suggests
that market efficiency should imply that cash and futures are cointegrated,
while Baillie and Myers (1991) suggest that, as a consequence of possible
nonstationarity of the risk-free proxy employed in the cost-of-carry model,
this need not be the case. We do not wish to enter into this debate from a
theoretical viewpoint, but suffice to say that in all of our ensuing analysis,
we allow for, but do not impose, a [�1, 1] cointegrating vector for the two
series. The conditional mean equations of the model employed in this article
are a bivariate Vector Error Correction Mechanism (VECM), which may be
written as
DY p m � GDY � Pv � ��t i t�i t�1 t
(F) (F)⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤F m G G p �t F i, F i, C F F, tY p ;mp ;G p ;Pp ;� p . (7)⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥t i (C) (C) tC m G G p �⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦t C i, F i, C C C, t
Under the assumption , where et represents the innovation�FQ ∼ (0, H )t t t
vector in equation (6) and represents an error correction term, and byut�1
defining ht as vech(Ht), where vech(.) denotes the vector-half operator that
stacks the lower triangular elements of an matrix into anN # N [N(N �
vector, the bivariate VECM(p) GARCH(1,1) vech model may be1)/2] # 1
written as
′vec(H ) p h p h pC � A vec(� � ) � B h , (8)t t CF, t 0 1 t�1 t�1 1 t�1⎢ ⎥
⎡ ⎤ ⎡ ⎤ ⎡ ⎤c a a a b b b11 11 12 13 11 12 13
c c ;A p a a a ;B p b b b .0 12 1 21 22 23 1 21 22 23⎢ ⎥ ⎢ ⎥ ⎢ ⎥
c a a a b b b⎣ ⎦ ⎣ ⎦ ⎣ ⎦22 31 32 33 31 32 33
Restricting the matrices A1 and B1 to be diagonal gives the model proposed
by Bollerslev, Engle, and Wooldridge (1988), where each element of the
conditional variance-covariance matrix hij, t depends on past values of itself
and past values of . There are 21 parameters in the conditional var-� � ′t�1 t�1
iance-covariance structure of the bivariate GARCH(1,1) vech model (eq. [8])
to be estimated, subject to the requirement that Ht be positive-definite for all
values of in the sample. The difficulty of checking, let alone imposing such�t
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Effect of Asymmetries 337
a restriction, led Engle and Kroner (1995) to propose the Bollerslev, Engle,
Kroner, and Kraft (BEKK) parameterization:
′ ′ ′∗ ∗ ∗ ′ ∗ ∗ ∗H p C C � A � � A � B H B . (9)t 0 0 11 t�1 t�1 11 11 t�1 11
The BEKK parameterization requires estimation of only 11 parameters in
the conditional variance-covariance structure and guarantees Ht to be positive
definite. It is important to note that the BEKK and vec models imply that
only the magnitude of past return innovations is important in determining
current conditional variances and covariances. This assumption of symmetric
time-varying variance-covariance matrices must be considered tenuous given
the existing body of evidence documenting the asymmetric response of equity
volatility to positive and negative innovations of equal magnitude (see Engle
and Ng 1993; Glosten, Jagannathan, and Runkle 1993; Kroner and Ng 1998,
inter alia).
Defining , for , the BEKK model iny p min {� , 0} j p futures, cashj, t t
equation (9) may be extended to allow for asymmetric responses as
′ ′ ′ ′∗ ∗ ∗ ′ ∗ ∗ ∗ ∗ ′ ∗H p C C � A � � A � B H B � D y y D , (10)t 0 0 11 t�1 t�1 11 11 t�1 11 11 t�1 t�1 11
∗ ∗ ∗ ∗⎡ ⎤ ⎡ ⎤c c a a∗ 11 12 ∗ 11 12C p ;A p ;⎢ ⎥ ⎢ ⎥0 ∗ 11 ∗ ∗0 c a a⎣ ⎦ ⎣ ⎦22 21 22
∗ ∗ ∗ ∗⎡ ⎤ ⎡ ⎤ ⎡ ⎤b b d d y∗ 11 12 ∗ 11 12 F, tB p ;D p and y p . (11)⎢ ⎥ ⎢ ⎥ ⎢ ⎥11 ∗ ∗ 11 ∗ ∗ tb b d d y⎣ ⎦ ⎣ ⎦ ⎣ ⎦21 22 21 22 C, t
The symmetric BEKK model (eq. [9]) is given as a special case of equation
(10) for , for all values of m and n.d p 0m, n
III. Data Description
The data employed in this study comprise 3,580 daily observations on the
FTSE 100 stock index and stock index futures contract spanning the period
January 1, 1985–April 9, 1999.2 Days corresponding to U.K. public holidays
are removed from the series to avoid the incorporation of spurious zero returns.
The FTSE 100 comprises the 100 U.K. companies, quoted on the London
Stock Exchange, with the largest market capitalization and accounting for
73.2% of the market value of the FTSE All Share Index on December 29,
1995 (Sutcliffe 1997). The FTSE 100 futures contracts are quoted in the same
units as the underlying index, except that the decimal is rounded to the nearest
2. Since these contracts expire four times per year—March, June, September, and Decem-
ber—we obtain a continuous time series by using the closest-to-maturity contract unless the next
closest has greater volume, in which case we switch to this contract.
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338 Journal of Business
0.5.3 The price of a futures contract (contract size) is the quoted number
(measured in index points) multiplied by the contract multiplier, which is £25
for the contract. There are four delivery months: March, June, September,
and December. Trading takes place in the 3 nearest delivery months, although
volume in the “far” contract is very small. Each contract is therefore traded
for 9 months. The FTSE100 futures contracts are cash-settled as opposed to
physical delivery of the underlying. All contracts are marked to market on
the last trading day, which is the third Friday in the delivery month, at which
point all positions are deemed closed. For the FTSE100 futures contract, the
settlement price on the last trading day is calculated as an average of minute-
by-minute observations between 10:10 a.m. and 10:30 a.m., rounded to the
nearest 0.5.
Summary statistics for the data are displayed in panel A of table 1. Using
unit root tests, it is not possible to reject the null hypothesis
of nonstationarity for the cash and futures price series. This nonstationarity
of the price series is consistent with weak-form efficiency of the cash and
futures markets. The return series are calculated as and100 # (C /C )t t�1
, respectively. The returns are skewed to the left, leptokurtic,100 # (F /F )t t�1
and stationary. These features are entirely in accordance with expectations
and results presented elsewhere. In the absence of a long-run relationship
between , optimal inference based on asymptotic theory requires theC and Ft t
use of returns rather than price data in calculating the estimation of dynamic
hedge ratios.
Results for both Engle and Granger (1987) and Johansen (1988) tests for
cointegration are displayed in table 1. The Engle and Granger results of table
1, panel B, clearly demonstrate that the null of nonstationarity in the residuals
of the cointegrating regression is strongly rejected, for the test both with and
without a constant term. Moreover, the estimated slope coefficient is very
close to unity, whether the spot or futures price is the dependent variable.
Similarly, the Johansen test statistics, for both the trace and the max forms,
reject the null of no cointegrating vector but do not reject the null of one
cointegrating vector. A restriction of the cointegrating relationship between
the series to be [1, �1] was marginally rejected at the 5% level. However,
after normalizing the estimated cointegration vector on Ct, the estimated co-
efficient on Ft was �1.006, suggesting that this rejection may not be eco-
nomically important. On close examination of the short-run components of
the VECM, it appears that the futures prices are weakly exogenous. A like-
lihood ratio test supports this restriction. Thus, while the cointegrating equi-
librium is defined by both cash and futures prices, equilibrium is restored
through the cash markets. A test of the joint hypothesis that futures prices
are weakly exogenous and that the parameters of the cointegration vector are
[1,�1] was not rejected at the 5% level of significance. Baillie and Myers
3. The reason for this is that the minimum price movement (known as tick) for the futures
contract is £12.50, i.e., a change of 0.5 in the index.
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TABLE 1 Summary Statistics and Cointegration Tests
Unit Root Tests ADF(m) ADF
Ft �1.7028 1.9982
Ct �1.0082 2.2269
Summary Statistics
Mean Variance Skewness Excess Kurtosis
Ft .0392 1.1424 �1.6081 25.3160
Ct .0389 .8286 �1.6602 25.6852
Engle- Tests
f0 f1 ADF(m) ADF
Ft as dependent variable �.0327
�8.3846 �8.3859
g0 g1 ADF(m) ADF
Ct as dependent variable .0386
�8.4026 �8.4039
r p 0 91.75 92.58
r p 1 .83 .83
H0: b’ p [�1, 1] H0: a p [1, 0] F FH : b’ p [�1, 1] a p [1, 0]0
Likelihood ratio tests 4.4800
Note.—ADF p Augmented .
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340 Journal of Business
TABLE 2 Estimates of the Multivariate Asymmetric GARCH Model
Conditional Mean Equations
DY p m � GDY � Pv � ��t i t�i t�1 t
(F) (F)F m G Gt F i,F i,CY p ;m p ;G p ;t i (C) (C)[ ] [ ] [ ]C m G Gt C i,F i,C
P p ;� pt[ ] [ ]p �C C, t
�.0078 �.0225⎡ ⎤
( ) ( ).0060 .0072
G p1 ⎢ ⎥.0759 .0257
( ) ( ).0053 .0061⎣ ⎦
�.1499 .1399⎡ ⎤
( ) ( ).0089 .0110
G p2 ⎢ ⎥.0272 .0238
( ) ( ).0080 .0092⎣ ⎦
�.1225 .1083⎡ ⎤
( ) ( ).0111 .0149
G p3 ⎢ ⎥�.0352 .0293
( ) ( ).0114 .0117⎣ ⎦
�.0699 .0084⎡ ⎤
( ) ( ).0227 .0256
G p4 ⎢ ⎥.0141 �.0032
( ) ( ).0182 .0232⎣ ⎦
m p ⎢ ⎥.0523
( ).0050⎣ ⎦
P p ⎢ ⎥�.1719
( ).0142⎣ ⎦
Note.—SEs are displayed in parentheses.
(1991) argue that a perfect one-to-one association does not exist in a com-
modity futures hedge because of the cost of carry, although this does not
preclude some other cointegrating relationship from existing. On balance, the
data appear to be cointegrated with a [1,�1] cointegrating vector.
IV. Hedging Model Estimates, Tests, and Performance
Given the evidence of a long-run or cointegrating relationship between
, the conditional mean equations are parameterized as a VECM ratherC an
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