Description
This paper provides a critique of minimum
variance hedging using futures. The paper
develops the conventional minimum variance
hedge ratio (MVHR) and discusses its
estimation. A review of the wide variety of
alternative methods used to construct MVHRs
is then performed. These methods highlight
many of the potential limitations in the
conventional framework. The paper argues that
the literature should focus more on the
assumptions underlying the conventional
MVHR, rather than improving the techniques
used to estimate the conventional MVHR.
Accounting Research Journal
A Critique of Minimum Variance Hedging
J onathan Dark
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ACCOUNTING RESEARCH JOURNAL VOLUME 18 NO 1 (2005)
40
A Critique of Minimum Variance Hedging
Jonathan Dark
Department of Econometrics and Business Statistics
Monash University
Abstract
This paper provides a critique of minimum
variance hedging using futures. The paper
develops the conventional minimum variance
hedge ratio (MVHR) and discusses its
estimation. A review of the wide variety of
alternative methods used to construct MVHRs
is then performed. These methods highlight
many of the potential limitations in the
conventional framework. The paper argues that
the literature should focus more on the
assumptions underlying the conventional
MVHR, rather than improving the techniques
used to estimate the conventional MVHR.
1. Introduction
The importance of managing risk has seen a
voluminous futures hedging literature develop
over the last half a century. Much of the
hedging literature seeks to minimise the risk
exposure associated with a given position in the
spot market. The literature commonly estimates
minimum variance hedge ratios (MVHRs)
based on the early work of Ederington (1979).
This approach will be referred to as the
conventional MVHR.
Much of the recent literature that employs
the conventional approach, focuses on
improving the methods used to estimate
minimum variance hedge ratios. These
methods typically allow for cointegration
between the spot and futures markets,
conditional information and conditional
heteroscedasticity. This paper argues that the
literature should focus more on the assumptions
underlying the conventional MVHR, rather
than improving the techniques used to estimate
the MVHR.
This paper is based on Chapter 5 of my Phd dissertation.
Acknowledgment: The author thanks Robert Faff for
encouragement and helpful comments.
Section 2 will outline the conventional
approach to hedge ratio determination and
illustrate the use of the more sophisticated
estimation procedures. Section 3 will discuss
the limited improvements obtained when
employing the more sophisticated estimation
methods. The section will then discuss the
alternative approaches used to determine hedge
ratios. These alternative approaches will
highlight many of the possible shortcomings in
the conventional approach. Section 4 will
examine the conditions required for the MVHR
to be utility maximizing. This will include a
brief discussion of the limitations in the mean
variance framework and the use of alternative
measures of risk. Section 5 will conclude.
2. Conventional Hedge Ratio
determination
Hedging combines spot and futures positions to
form a hedged portfolio. Define
s
X and
f
X
as the spot and futures positions, time t as the
commencement date of the hedge, time t+r as
the reversal date of the hedge,
t
S and
t
F as
the spot and futures prices at time t, and
t t t
B F S = ? as the basis at time t. Following
Ederington (1979) the hedge ratio, ? is
defined as
f
s
X
X
?
? =
. (1)
Given that
s
X and
f
X are usually of
opposite signs, ? is usually positive. To
simplify notation it is assumed that the hedger
has a fixed spot position of one unit, that is
s
X =1.
Assuming that the spot and futures prices
are equal on expiration of the futures contract,
risk can be eliminated completely by taking a
fully offsetting futures position at time t
( ?=1) and reversing the position on the
futures expiry date. Unless the reversal of the
hedge coincides with the futures expiry date,
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A Critique of Minimum Variance Hedging
41
the hedge is exposed to basis risk (defined as
the variance of the basis). Basis risk means that
the gain or loss on reversal of the hedge is
uncertain. It is this type of hedge which has
received the most attention in the literature, and
is the focus of this paper.
2.1 Naïve hedging and Working’s approach
The naïve approach sees the primary
motivation for hedging as risk reduction and
sets ?=1. If the value of the change in the
spot equals the value of the change in the
futures (
t r t t r t
S S F F
+ +
? = ? ), there is no
basis risk and the hedger’s wealth remains
unchanged. Alternatively, the hedger’s wealth
remains unchanged given that the gain or loss
on the hedge is equal to the change in the basis
( ) ( )
t t t r t r
F S F S
+ +
? ? ? ! "
# $
which is equal
to zero. Under these circumstances, the spot
price risk has been eliminated completely and
the hedger has “locked in” the spot price at
time t. If the spot and futures markets do not
move together perfectly, the hedger is exposed
to basis risk, with the change in the basis
resulting in a change in wealth. Naïve hedging
therefore completely eliminates spot price risk
and replaces it with basis risk. Risk reduction
only occurs if the variance of the basis is less
than the variance of the spot.
Working (1953a, 1953b, 1961) was critical
of the naïve approach, given its failure to
incorporate changes in the basis into the
hedging decision. Working assumed that
hedges were motivated by the desire to profit
from favourable changes in the basis, with risk
reduction being incidental. Assuming a long
spot position, if the basis was expected to fall,
the hedger would set ?= 1. If the basis was
expected to rise, the hedger would not hedge at
all, with ?= 0.
1
2.2 The conventional approach
Ederington (1979), Figlewski (1986) and
Castelino (1992) overview the development of
the conventional approach which originated
from the work of Johnson (1960) and Stein
(1961). The conventional approach allows for
futures bias and partial hedging. Futures bias
means that the futures are biased predictors of
the spot. If futures are unbiased the futures
1 See Ederington (1979), Castelino (1992) and Brown
(1985) for further discussion.
follow a martingale process where
2
( ) 0
t t r t
E F F
+
? =
. (2)
Partial hedging ( 0 1 < ? < ) allows the
hedger to determine the optimal tradeoff
between spot price risk and basis risk.
The hedge is viewed as a two security
portfolio consisting of spot and futures
positions. It is assumed that the hedger has an
expected mean variance utility function. This
preference function assumes either quadratic
preferences or normally distributed returns. It is
assumed that the hedger has a given spot
position of one unit and seeks to maximise
expected profit adjusted for risk at time (t+r).
The hedger therefore seeks to maximise the
following objective function (?)
( ) ( )
2
t r
t t r
Max E
?
? ??
+
+
? = ?
(3)
where ? represents the risk aversion
parameter,
( )
t t r
E ?
+
represents the expected
profit from the hedge between time t and time
t+r
( ) ( ) ( )
t t r t t r t t t r t
E E S S E F F ?
+ + +
= ? ?? ?
(4)
and
2
t r
?
?
+
represents the variability in hedged
portfolio returns
2 2 2 2
2
t r
s f sf ?
? ? ? ?
+
= + ? ? ? (5)
where
2
s
? is the spot variance,
2
f
? is the
futures variance and
sf
? is their covariance.
Equations 4 and 5 reveal that different
combinations of risk and return can be
generated by varying the hedge ratio, ?
3
. The
hedge ratio is found by maximising Equation 3
with respect to ? (Sephton, 1993)
( )
2 2
2
sf t t r t
f f
E F F ?
? ??
+
?
? = ?
. (6)
Hedge ratio determination requires an
expected futures price
( )
t t r
E F
+
, plus a
measure of risk aversion, ? . The MVHR
overcomes these issues by minimising the
variability in the expected hedged return
2 If futures are biased the equality in Equation 2 does not
hold. Allowing for futures bias is important, given that it
may result in significant hedging losses. To illustrate,
under a long spot/short futures hedge, if the futures are
downward biased, there will be a loss on reversal of the
futures position.
3 This may also be represented diagrammatically, see
Ederington (1979) and Cecchetti et al (1988).
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ACCOUNTING RESEARCH JOURNAL VOLUME 18 NO 1 (2005)
42
( )
t t r
E ?
+
.
4
Minimising Equation 5 with
respect to ? yields
2
sf
f
?
?
? =
(7)
where the MVHR is typically obtained via the
estimate of ? in the following ordinary least
squares (OLS) regression
5
t t t
S F ? ? ? = + + . (8)
Therefore the risk averse investor uses a
combination of the MVHR (the risk component
of the hedge) and a futures bias term (the return
component) to determine the optimal hedge
ratio. The hedge ratio in Equation 6 equals the
MVHR if the futures follow a martingale
process, where
( )
0
t t r t
E F F
+
? = , or if the
hedger is extremely risk averse, where
? ?? (Kahl, 1983; Sephton, 1993).
Therefore if futures are unbiased, the MVHR
applies to all hedgers, irrespective of their
degree of risk aversion.
It is well known that unbiasedness is a joint
hypothesis of market rationality and risk
neutrality. Both hypotheses are controversial.
The controversy applies to commodities
(Brenner and Kroner, 1995; Chowdhury, 1991;
Beck, 1994; Graham-Higgs et al, 1999),
currencies (Bollerslev et al, 1992; Frankel and
Froot, 1987; Ito, 1990) and interest rates
(Hegde and MacDonald, 1986; Cole and
Reichenstein, 1994). See Kellard et al (1999)
for a review. Consequently conclusions
supporting unbiasedness must be treated with
caution. Unless the hedger exhibits extreme
risk aversion, the use of the MVHR may not
maximise the hedger’s objective function.
The following section discusses some of the
popular methods used to estimate MVHRs.
These more sophisticated methods seek to
address some of the limitations in the OLS
approach discussed above (Equation 8).
2.3 The focus of recent research -
improving MVHR estimation
The literature now typically employs more
sophisticated methods to estimate MVHRs.
4 See Castelino (1992) for an excellent reconciliation of
the MVHR and Working’s approach.
5 Myers and Thompson (1989) and Castelino (1990a)
note that there is no consensus on whether Equation 8
should be in levels, first differences or returns.
Estimation methods commonly allow for
conditional information, conditional
heteroscedasticity and cointegration between
the spot and futures markets.
Conditional information may be
accommodated through the re-specification of
Equation 8. Myers and Thompson (1989) and
Viswanath (1993) perform a regression
between the spot and futures that includes
regressors additional to those in Equation 8.
Myers and Thompson (1989) include lagged
spot and futures variables, whilst Viswanath
(1993) includes the current level of the basis.
By arguing that it is the intercept in Equation 8
that should be a function of conditional
information, these approaches do not address
the finding by Bell and Krasker (1986), that the
intercept and the slope are a function of
conditional information. The approaches also
do not allow for conditional heteroscedasticity,
imposing constant hedge ratios.
6
Another approach that allows for conditional
information and conditional heteroscedasticity
estimates time varying covariances. Kroner and
Sultan (1993) assume that the hedger seeks to
minimise the variability in the hedged return
conditional on the information available. This
results in the following MVHR
, 1
2
, 1
sf t
t
f t
?
?
+
+
? =
(9)
where the time subscript t, is used to denote the
use of conditional rather than unconditional
moments. As a consequence, the hedge ratio is
dynamic, changing through time in response to
new information.
The bivariate GARCH family of processes
(Bollerslev et al, 1992) allow for conditional
information and time varying covariances and
are therefore a very popular way of estimating
dynamic MVHRs. MVHRs are therefore
constructed by making one period ahead
forecasts of Equation 9 over the life of the
hedge. See Baillie and Myers (1991), Myers
(1991) and Sephton (1993).
Ghosh (1993) finds that the MVHR
estimated via Equation 8 is outperformed by a
MVHR that allows for cointegration. Lien
(1996) provides a theoretical justification for
this result. The more recent literature therefore
6 See Lien and Luo (1994) for further discussion.
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A Critique of Minimum Variance Hedging
43
estimates MVHRs via a bivariate error
correction model (ECM)
, 1 1 1 1, , 1, , ,
1 1
, 2 2 1 2, , 2, , ,
1 1
k k
s t t i s t i i f t i s t
i i
k k
f t t i s t i i f t i f t
i i
R a b z c R d R
R a b z c R d R
?
?
? ? ?
= =
? ? ?
= =
= + + + +
= + + + +
% %
% %
(10)
where
1 t
z
?
is the error correction term,
, s t
R
and
, f t
R are the returns in the spot and futures
markets respectively, k represents the number
of lags, and the MVHR is estimated as
( )
( )
, ,
,
cov
var
s t f t
f t
? ?
?
? =
(11)
where
( )
, ,
cov
s t f t
? ? and
( )
,
var
f t
? are
either time invariant or time varying (as in
Equation 9). If one employs a time invariant
strategy, a bivariate ECM is estimated and
unconditional estimates of
( )
, ,
cov
s t f t
? ? and
( )
,
var
f t
? are used to construct ?. Dynamic
MVHR estimation typically estimates time
varying
( )
, ,
cov
s t f t
? ? and
( )
,
var
f t
? via a
bivariate error correction GARCH model.
7
See
Kroner and Sultan (1993), Park and Switzer
(1995), Koutmos and Pericli (1998) and Lien
and Tse (1999).
The above has discussed the derivation and
estimation of the conventional MVHR and the
conditions when it will be utility maximizing.
Much of the recent research has accepted the
assumptions of the conventional MVHR,
focusing on improving the methods of
estimation. The rest of the paper argues that the
literature should focus more on the assumptions
underlying the conventional MVHR. Section
3.1 will demonstrate that the more
sophisticated estimation approaches have
provided marginal improvements in the level of
risk reduction. Section 3.2 will argue that the
assumptions of the conventional MVHR may
be inappropriate, given that they fail to allow
for multiple exposures, hedges over multiple
periods, basis convergence, estimation risk and
the possibility of using options.
7 There is no consensus on what the findings of
cointegration imply for futures unbiasedness.
MacDonald and Taylor (1988), Wahab and Lashgari
(1993) and Lien (1996) argue that cointegration implies
market inefficiency. Krehbiel and Adkins (1993),
Chowdhury (1991) and Pizzi et al (1998) suggest that
cointegration implies market efficiency.
3. Limitations in the conventional
approach
3.1 The performance of the more
sophisticated estimation methods
Table 1 documents the literature which
examines the performance of alternative
methods of MVHR estimation using the
conventional approaches. The Table reports the
risk reduction, where the data set is divided into
estimation and forecast periods, with the
forecast period being used to assess hedging
effectiveness.
The literature generally supports the
estimation of dynamic MVHRs using a
bivariate error correction GARCH model. The
earlier results (Cechetti et al, 1988; Baillie and
Myers, 1991; Sephton, 1993) rejected the time
invariance in MVHRs, finding that bivariate
GARCH models achieved greater risk
reduction than the OLS MVHR (Equation 8).
Subsequent results further illustrated the
benefits of allowing for cointegration and
conditional heteroscedasticity (Kroner and
Sultan, 1993; Park and Switzer, 1995; Koutmos
and Pericli, 1998; Lien and Tse, 1999). These
results also appear to be unaffected by the
inclusion of transaction costs (Kroner and
Sultan, 1993; Park and Switzer, 1995; Koutmos
and Pericli, 1998). Unfortunately the
incremental benefits provided by these more
sophisticated estimation methods are often
quite small. Lien and Luo (1994) and Myers
(1991) show that even though the bivariate
GARCH models statistically outperform the
other estimation approaches, this does not
necessarily result in superior risk reduction.
3.2 Relaxing the assumptions of the
conventional MVHR
This section discusses the literature that relaxes
some of the assumptions of the conventional
MVHR, taking into account: i) multiple
exposures; ii) multiple periods; iii) basis
convergence; iv) estimation risk; and v) the use
of options and futures. MVHRs estimated using
the conventional approach may therefore over
or understate the number of futures contracts
that are required to hedge a given exposure.
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ACCOUNTING RESEARCH JOURNAL VOLUME 18 NO 1 (2005)
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Table 1
Summary of the literature investigating the performance of MVHRs
Reference Data MVHRs Dynamic
strategy
Comments regarding risk
reduction
Cecchetti
et al (1988)
20 yr T-bond
Monthly, 1/78-
12/83
Biv-ARCH(3) Single
period
Time invariant MVHR
inappropriate.
Baillie and
Myers
(1991)
Commodities
Daily, 82-86
(periods vary)
OLS,
Biv-GARCH
Single
period
Time invariant MVHR
inappropriate.
GARCH outperforms OLS.
Myers
(1991)
Wheat Weekly,
6/77-5/83
OLS,
Biv-GARCH
Single
period
MVHRs are time varying, however
the GARCH model’s performance
is only marginally better. Suggested
that once transaction costs are taken
into account, the OLS strategy is
probably the preferred strategy.
Sephton
(1993)
Commodities
Daily, 5/88-
5/89
OLS,
Biv-GARCH
Single
period
Time invariant MVHR
inappropriate.
GARCH outperforms OLS.
Kroner and
Sultan
(1993)
Currencies vis-
a vis USD
Weekly, 2/85-
2/90
Naïve,OLS,
Biv-ECM,
Biv-EC-GARCH
Single
period
Biv-EC-GARCH provides best
performance (with and without
transaction costs).
Lien and
Luo (1994)
Currencies
vis-à-vis USD,
Weekly, 3/80-
12/88
OLS,
Biv ECM,
Biv-EC-GARCH
Multi-
period
OLS and Biv ECM hedge
outperform the Biv-EC-GARCH.
Park and
Switzer
(1995)
S&P500,
Toronto 35
Weekly, 6/88-
12/91
Naïve,
OLS, ECM,
Biv-EC-GARCH
Single
period
Biv-EC-GARCH hedge provides
superior performance (with and
without transaction costs).
Koutmos
and Pericli
(1998)
Commercial
paper with T-
bill futures,
Weekly,
1/85- 3/96
OLS,
Biv-GARCH,
Biv-EC-GARCH
Single
period
Biv-EC-GARCH provides best
performance (with and without
transaction costs). Both
cointegration and time varying
moment estimation improves
hedging performance.
Lien and
Tse (1999)
Nikkei 225
Daily, 1/89-
8/97
OLS, VAR,
ECM,
Biv-VAR/EC-
GARCH
Single
period
Including GARCH improves
hedging performance. EC-GARCH
is the dominant strategy. OLS
provides the worst performance.
OLS = ordinary least squares estimation via Equations 7 & 8. ECM = error correction model estimation,
Equations 7 & 10. Biv-GARCH/Biv-EC-GARCH = dynamic estimation of Equation 9 using bivariate
GARCH/bivariate error correction GARCH.
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A Critique of Minimum Variance Hedging
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3.2.1 Multiple exposures
Each of the above approaches assume that the
hedger only has a single asset in the spot
market which is exposed to price uncertainty.
This is unrealistic given that most hedgers have
multiple exposures. A portfolio manager for
example, is likely to have domestic and
overseas, bond, equity and currency exposures.
Figlewski (1986) defines the methods of hedge
ratio determination in Section 2 as micro
hedging strategies, given that each asset or
liability is considered in isolation. In contrast,
macro hedging considers assets or liabilities in
groups, hedging the net exposure.
Gagnon et al (1998) derive a MVHR for a
portfolio of currencies, where futures contracts
are available for each currency. The MVHR for
each exposure takes into account the
covariance between the futures and the spot, as
well as all the other futures and spot positions
in the portfolio. The approach however
assumes that there is no quantity risk, with the
number of units of currency fixed. This is
unrealistic, given that a portfolio manager is
typically exposed to currencies where the
quantity (determined by the change in market
values denominated in foreign currency) as
well as the price (determined by the change in
exchange rates) is uncertain.
Giaccotto et al (2001) address the
limitations in Gagnon et al (1998), by allowing
for multiple price and quantity exposures. The
hedge ratio is a function of the full covariance
structure of changes in spot prices, quantities
and futures prices, and therefore takes into
account any of the natural hedges that may
exist. It is demonstrated that failure to account
for each of these covariances will lead to
systematic over or under hedging, supporting
the use of a macro hedging framework.
3.2.2 Multiple periods
The conventional dynamic MVHR (Equation
9) seeks to minimise the conditional variation
in portfolio returns period by period. Howard
and D’Antonio (1991), Vukina and Anderson
(1993), Lien and Luo (1994) and Lee (1999)
derive MVHRs that seek to minimise risk over
the life of the hedge. Howard and D’Antonio
(1991) do not allow for conditional information
and Lien and Luo (1994) require sophisticated
estimation procedures. Vukina and Anderson
(1993) do not allow for conditional
heteroscedasticity, and the approach cannot be
easily generalised to a hedge over a large
number of periods.
These approaches also fail
to allow for multiple exposures. A superior
approach is developed by Lee (1999) who
derives a multi-asset, multi-period dynamic
MVHR which allows for conditional
information and conditional heteroscedasticity.
The approach captures the interperiod
dependencies over the life of the hedge, and
reduces the volatility commonly associated
with dynamic MVHR estimation. This is
intentional given that a hedging strategy that
reflects short lived volatility fluctuations is
unstable, costly and ineffective when hedging
over the long term (Lee, 1999).
3.2.3 Basis convergence
All of the above methods fail to impose basis
convergence, and therefore ignore information
that could be used when estimating hedge
ratios. Castelino (1989, 1990a, 1990b, 1992)
considers the impact of basis convergence on
MVHR determination. As highlighted earlier, a
hedge ratio of unity will be risky if the hedge is
reversed prior to contract expiration (given
basis risk). Castelino therefore develops a
MVHR that adjusts the hedge ratio away from
unity as the hedge reversal date differs from the
contract expiration date.
Chen et al (1999) allows for convergence,
conditional information and conditional
heteroscedasticity. Chen et al (1999) model the
basis and spot as a bivariate GARCH process
with a maturity effect. By specifying the mean
and variance of the basis as a function of time
to maturity, the maturity of the contract
influences the behaviour of the basis. The
model is therefore able to impose the condition
that at maturity, the basis and its conditional
variance are zero. Chen et al (1999) derive the
MVHR as a function of time to maturity.
Estimated MVHRs are inversely related to the
time to maturity and therefore support the
insight of Castelino (1989, 1990a, 1990b,
1992).
3.2.4 Estimation risk
Lence and Hayes (1994) are critical of the
conventional MVHR given that it employs a
parameter certainty equivalent (PCE) approach.
The PCE approach derives the MVHR under
the assumption that the probability density
function (PDF) and its parameters are known
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ACCOUNTING RESEARCH JOURNAL VOLUME 18 NO 1 (2005)
46
with certainty. This ignores estimation risk,
which arises from less than perfect information
about the functional form of the PDF or its
parameter values. Failure to recognise the
estimation risk means that under the PCE
approach, slight changes in data sets can result
in large changes in the estimated MVHR.
Lence (1995) extends this further by arguing
that the conventional MVHR ignores
estimation risk, commissions, margins and the
lumpiness of contracts. It also fails to allow for
simultaneous borrowing, lending or investing
in other assets. The MVHR is determined via
the Lagrangian technique, where end of period
wealth is maximised subject to a number of
linear constraints. The magnitude of the hedge
ratio is shown to be very sensitive to the
relaxation of the assumptions in the
conventional MVHR.
3.2.5 Hedging with futures and options
The above approaches assume that the only
derivative available for hedging is a futures
contract. Lence et al (1994), Moschini and
Lapan (1995), Sakong et al (1993) and Froot et
al (1993), demonstrate that hedging a non-
linear payoff in the spot, requires the use of
options and futures. This is because a position
in futures and options can create an offsetting
non linear payoff, in contrast to futures, that
only provide offsetting linear payoffs.
Moschini and Lapan (1995) and Sakong et
al (1993) show how a non-linear spot payoff
can be a result of the interaction between price
and production yield uncertainty (a quantity
risk). These results are based on a one period
model, given the assumption that the firm is
only concerned with a single production cycle.
Lence et al (1994) allow for two production
cycles which is appropriate for firms that
exhibit forward looking behaviour. It is argued
that output price changes in one period will
change the perceived relationship between next
period’s input and output prices. Lence et al
(1994) show that under these circumstances
and non stochastic production, there will be a
non linear payoff in the subsequent period that
can be hedged with futures and options. Froot
et al (1993) also employ a two period approach
to examine the impact of hedging on optimal
financing and investment decisions. A number
of situations are presented where a non linear
hedging strategy is required to hedge the
internal cashflows used to finance an
investment project.
In summary there is a vast literature that
seeks to address the limitations in the
conventional MVHR. Unfortunately an
approach that simultaneously addresses all of
these limitations has not been forthcoming. The
conventional MVHRs use of the mean variance
framework is one further possible limitation
and is the subject of the next section.
4. Limitations in the mean variance
framework
The conventional approach employs a mean-
variance framework assuming that
maximisation of the objective function
(Equation 3) results in utility maximisation.
This however is not necessarily the case, given
that the MVHR equals the utility maximising
hedge ratio only if the hedger has a quadratic
utility function or normally distributed profits
(Kahl, 1983).
Arrow (1971) argues that quadratic utility is
highly implausible given that it implies
increasing absolute risk aversion. This suggests
that as an individual becomes wealthier, they
will decrease the amount of risky assets held.
The normality requirement also appears
unlikely in most financial markets, with return
distributions exhibiting leptokurtosis and
skewness. Nonetheless Levy and Markowitz
(1979) argue that regardless of the utility
function or the distribution of returns, the
maximisation of a mean-variance objective
function may provide a reasonable
approximation of the true objective function.
The MVHR is therefore utility maximising
if: a) at least one of the conditions for the
conventional approach to be utility maximising
is met; and b) the futures are unbiased or the
hedger is extremely risk averse. For example,
Giaccotto et al (2001) show that the MVHR
equals the utility maximising hedge ratio if
utility is represented as a general von Neuman-
Morgenstern utility function (a more general
utility function than the quadratic utility
function), the variables are normally
distributed, and futures prices follow a
martingale process.
Given that the conditions required for the
conventional MVHR to be utility maximizing
are quite restrictive, other approaches may
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A Critique of Minimum Variance Hedging
47
provide superior outcomes (given that they are
utility maximising). Consequently the hedging
outcomes using the conventional MVHR may
be dominated stochastically. Stochastic
dominance is based on the von Neumann-
Morgenstern utility functions and applies
selection rules that are based on pairwise
comparisons between distributions that require
knowledge of the complete distribution. This is
in contrast to the mean-variance approach
which only requires knowledge of the mean
and variance. See Bawa (1975, 1978), Fishburn
(1977), Yitzhaki (1982), Shalit and Yitzhaki
(1984) for further details.
Given that the conventional approach may
not be utility maximising, alternative measures
of risk have been used to derive hedge ratios.
Chen et al (2001) discuss the alternative risk
measures used in the hedging literature, namely
the mean- extended Gini coefficient (MEG)
and the generalised semivariance (GSV).
The mean Gini coefficient is a measure of
variability first applied to finance by Yitzhaki
(1982). Hedge ratios estimated using this
measure of risk (Kolb and Okunev, 1992,
1993) are consistent with second order
stochastic dominance (Chen et al, 2001),
however estimation can be difficult (see Kolb
and Okunev, 1992 for details).
The variance and MEG measures treat risk
as being two sided. This may be inappropriate
given that agents are generally more concerned
with managing downside risk (Crum et al,
1981; Lien and Tse, 2000). Here the returns
below a target return are considered risky,
whilst returns above the target are not. This risk
can be captured via the GSV. The GSV hedge
ratio therefore finds a value of
t
? that
minimises the variability in hedged returns
below a target value, not the total variability in
hedge returns. See Lien and Tse (1998, 2000)
and Chen et al (2001).
In summary, the assumptions of the mean
variance framework may mean that the
conventional MVHR is inappropriate and
alternative measures of risk like the MEG and
GSV are required.
5. Conclusion
This paper has performed a critique of
minimum variance hedging using futures. The
paper developed the conventional approach to
hedge ratio determination and discussed some
of the methods used to estimate MVHRs. The
paper then highlighted some of the weaknesses
in the approach. The conventional approach
does not allow for multiple exposures, multiple
periods, basis convergence, estimation risk, or
the use of futures and options. It was shown
that if a hedger is not extremely risk averse and
uses the conventional MVHR, this may not
maximise the hedger’s objective function.
Limitations in the mean variance framework
may also mean that the use of alternative risk
measures are required. Given the limited
benefits from employing more sophisticated
estimation methods, the literature should
probably focus more of its attention on the
assumptions underlying the MVHR, rather than
improving the estimation techniques.
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doc_670695457.pdf
This paper provides a critique of minimum
variance hedging using futures. The paper
develops the conventional minimum variance
hedge ratio (MVHR) and discusses its
estimation. A review of the wide variety of
alternative methods used to construct MVHRs
is then performed. These methods highlight
many of the potential limitations in the
conventional framework. The paper argues that
the literature should focus more on the
assumptions underlying the conventional
MVHR, rather than improving the techniques
used to estimate the conventional MVHR.
Accounting Research Journal
A Critique of Minimum Variance Hedging
J onathan Dark
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ACCOUNTING RESEARCH JOURNAL VOLUME 18 NO 1 (2005)
40
A Critique of Minimum Variance Hedging
Jonathan Dark
Department of Econometrics and Business Statistics
Monash University
Abstract
This paper provides a critique of minimum
variance hedging using futures. The paper
develops the conventional minimum variance
hedge ratio (MVHR) and discusses its
estimation. A review of the wide variety of
alternative methods used to construct MVHRs
is then performed. These methods highlight
many of the potential limitations in the
conventional framework. The paper argues that
the literature should focus more on the
assumptions underlying the conventional
MVHR, rather than improving the techniques
used to estimate the conventional MVHR.
1. Introduction
The importance of managing risk has seen a
voluminous futures hedging literature develop
over the last half a century. Much of the
hedging literature seeks to minimise the risk
exposure associated with a given position in the
spot market. The literature commonly estimates
minimum variance hedge ratios (MVHRs)
based on the early work of Ederington (1979).
This approach will be referred to as the
conventional MVHR.
Much of the recent literature that employs
the conventional approach, focuses on
improving the methods used to estimate
minimum variance hedge ratios. These
methods typically allow for cointegration
between the spot and futures markets,
conditional information and conditional
heteroscedasticity. This paper argues that the
literature should focus more on the assumptions
underlying the conventional MVHR, rather
than improving the techniques used to estimate
the MVHR.
This paper is based on Chapter 5 of my Phd dissertation.
Acknowledgment: The author thanks Robert Faff for
encouragement and helpful comments.
Section 2 will outline the conventional
approach to hedge ratio determination and
illustrate the use of the more sophisticated
estimation procedures. Section 3 will discuss
the limited improvements obtained when
employing the more sophisticated estimation
methods. The section will then discuss the
alternative approaches used to determine hedge
ratios. These alternative approaches will
highlight many of the possible shortcomings in
the conventional approach. Section 4 will
examine the conditions required for the MVHR
to be utility maximizing. This will include a
brief discussion of the limitations in the mean
variance framework and the use of alternative
measures of risk. Section 5 will conclude.
2. Conventional Hedge Ratio
determination
Hedging combines spot and futures positions to
form a hedged portfolio. Define
s
X and
f
X
as the spot and futures positions, time t as the
commencement date of the hedge, time t+r as
the reversal date of the hedge,
t
S and
t
F as
the spot and futures prices at time t, and
t t t
B F S = ? as the basis at time t. Following
Ederington (1979) the hedge ratio, ? is
defined as
f
s
X
X
?
? =
. (1)
Given that
s
X and
f
X are usually of
opposite signs, ? is usually positive. To
simplify notation it is assumed that the hedger
has a fixed spot position of one unit, that is
s
X =1.
Assuming that the spot and futures prices
are equal on expiration of the futures contract,
risk can be eliminated completely by taking a
fully offsetting futures position at time t
( ?=1) and reversing the position on the
futures expiry date. Unless the reversal of the
hedge coincides with the futures expiry date,
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the hedge is exposed to basis risk (defined as
the variance of the basis). Basis risk means that
the gain or loss on reversal of the hedge is
uncertain. It is this type of hedge which has
received the most attention in the literature, and
is the focus of this paper.
2.1 Naïve hedging and Working’s approach
The naïve approach sees the primary
motivation for hedging as risk reduction and
sets ?=1. If the value of the change in the
spot equals the value of the change in the
futures (
t r t t r t
S S F F
+ +
? = ? ), there is no
basis risk and the hedger’s wealth remains
unchanged. Alternatively, the hedger’s wealth
remains unchanged given that the gain or loss
on the hedge is equal to the change in the basis
( ) ( )
t t t r t r
F S F S
+ +
? ? ? ! "
# $
which is equal
to zero. Under these circumstances, the spot
price risk has been eliminated completely and
the hedger has “locked in” the spot price at
time t. If the spot and futures markets do not
move together perfectly, the hedger is exposed
to basis risk, with the change in the basis
resulting in a change in wealth. Naïve hedging
therefore completely eliminates spot price risk
and replaces it with basis risk. Risk reduction
only occurs if the variance of the basis is less
than the variance of the spot.
Working (1953a, 1953b, 1961) was critical
of the naïve approach, given its failure to
incorporate changes in the basis into the
hedging decision. Working assumed that
hedges were motivated by the desire to profit
from favourable changes in the basis, with risk
reduction being incidental. Assuming a long
spot position, if the basis was expected to fall,
the hedger would set ?= 1. If the basis was
expected to rise, the hedger would not hedge at
all, with ?= 0.
1
2.2 The conventional approach
Ederington (1979), Figlewski (1986) and
Castelino (1992) overview the development of
the conventional approach which originated
from the work of Johnson (1960) and Stein
(1961). The conventional approach allows for
futures bias and partial hedging. Futures bias
means that the futures are biased predictors of
the spot. If futures are unbiased the futures
1 See Ederington (1979), Castelino (1992) and Brown
(1985) for further discussion.
follow a martingale process where
2
( ) 0
t t r t
E F F
+
? =
. (2)
Partial hedging ( 0 1 < ? < ) allows the
hedger to determine the optimal tradeoff
between spot price risk and basis risk.
The hedge is viewed as a two security
portfolio consisting of spot and futures
positions. It is assumed that the hedger has an
expected mean variance utility function. This
preference function assumes either quadratic
preferences or normally distributed returns. It is
assumed that the hedger has a given spot
position of one unit and seeks to maximise
expected profit adjusted for risk at time (t+r).
The hedger therefore seeks to maximise the
following objective function (?)
( ) ( )
2
t r
t t r
Max E
?
? ??
+
+
? = ?
(3)
where ? represents the risk aversion
parameter,
( )
t t r
E ?
+
represents the expected
profit from the hedge between time t and time
t+r
( ) ( ) ( )
t t r t t r t t t r t
E E S S E F F ?
+ + +
= ? ?? ?
(4)
and
2
t r
?
?
+
represents the variability in hedged
portfolio returns
2 2 2 2
2
t r
s f sf ?
? ? ? ?
+
= + ? ? ? (5)
where
2
s
? is the spot variance,
2
f
? is the
futures variance and
sf
? is their covariance.
Equations 4 and 5 reveal that different
combinations of risk and return can be
generated by varying the hedge ratio, ?
3
. The
hedge ratio is found by maximising Equation 3
with respect to ? (Sephton, 1993)
( )
2 2
2
sf t t r t
f f
E F F ?
? ??
+
?
? = ?
. (6)
Hedge ratio determination requires an
expected futures price
( )
t t r
E F
+
, plus a
measure of risk aversion, ? . The MVHR
overcomes these issues by minimising the
variability in the expected hedged return
2 If futures are biased the equality in Equation 2 does not
hold. Allowing for futures bias is important, given that it
may result in significant hedging losses. To illustrate,
under a long spot/short futures hedge, if the futures are
downward biased, there will be a loss on reversal of the
futures position.
3 This may also be represented diagrammatically, see
Ederington (1979) and Cecchetti et al (1988).
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( )
t t r
E ?
+
.
4
Minimising Equation 5 with
respect to ? yields
2
sf
f
?
?
? =
(7)
where the MVHR is typically obtained via the
estimate of ? in the following ordinary least
squares (OLS) regression
5
t t t
S F ? ? ? = + + . (8)
Therefore the risk averse investor uses a
combination of the MVHR (the risk component
of the hedge) and a futures bias term (the return
component) to determine the optimal hedge
ratio. The hedge ratio in Equation 6 equals the
MVHR if the futures follow a martingale
process, where
( )
0
t t r t
E F F
+
? = , or if the
hedger is extremely risk averse, where
? ?? (Kahl, 1983; Sephton, 1993).
Therefore if futures are unbiased, the MVHR
applies to all hedgers, irrespective of their
degree of risk aversion.
It is well known that unbiasedness is a joint
hypothesis of market rationality and risk
neutrality. Both hypotheses are controversial.
The controversy applies to commodities
(Brenner and Kroner, 1995; Chowdhury, 1991;
Beck, 1994; Graham-Higgs et al, 1999),
currencies (Bollerslev et al, 1992; Frankel and
Froot, 1987; Ito, 1990) and interest rates
(Hegde and MacDonald, 1986; Cole and
Reichenstein, 1994). See Kellard et al (1999)
for a review. Consequently conclusions
supporting unbiasedness must be treated with
caution. Unless the hedger exhibits extreme
risk aversion, the use of the MVHR may not
maximise the hedger’s objective function.
The following section discusses some of the
popular methods used to estimate MVHRs.
These more sophisticated methods seek to
address some of the limitations in the OLS
approach discussed above (Equation 8).
2.3 The focus of recent research -
improving MVHR estimation
The literature now typically employs more
sophisticated methods to estimate MVHRs.
4 See Castelino (1992) for an excellent reconciliation of
the MVHR and Working’s approach.
5 Myers and Thompson (1989) and Castelino (1990a)
note that there is no consensus on whether Equation 8
should be in levels, first differences or returns.
Estimation methods commonly allow for
conditional information, conditional
heteroscedasticity and cointegration between
the spot and futures markets.
Conditional information may be
accommodated through the re-specification of
Equation 8. Myers and Thompson (1989) and
Viswanath (1993) perform a regression
between the spot and futures that includes
regressors additional to those in Equation 8.
Myers and Thompson (1989) include lagged
spot and futures variables, whilst Viswanath
(1993) includes the current level of the basis.
By arguing that it is the intercept in Equation 8
that should be a function of conditional
information, these approaches do not address
the finding by Bell and Krasker (1986), that the
intercept and the slope are a function of
conditional information. The approaches also
do not allow for conditional heteroscedasticity,
imposing constant hedge ratios.
6
Another approach that allows for conditional
information and conditional heteroscedasticity
estimates time varying covariances. Kroner and
Sultan (1993) assume that the hedger seeks to
minimise the variability in the hedged return
conditional on the information available. This
results in the following MVHR
, 1
2
, 1
sf t
t
f t
?
?
+
+
? =
(9)
where the time subscript t, is used to denote the
use of conditional rather than unconditional
moments. As a consequence, the hedge ratio is
dynamic, changing through time in response to
new information.
The bivariate GARCH family of processes
(Bollerslev et al, 1992) allow for conditional
information and time varying covariances and
are therefore a very popular way of estimating
dynamic MVHRs. MVHRs are therefore
constructed by making one period ahead
forecasts of Equation 9 over the life of the
hedge. See Baillie and Myers (1991), Myers
(1991) and Sephton (1993).
Ghosh (1993) finds that the MVHR
estimated via Equation 8 is outperformed by a
MVHR that allows for cointegration. Lien
(1996) provides a theoretical justification for
this result. The more recent literature therefore
6 See Lien and Luo (1994) for further discussion.
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estimates MVHRs via a bivariate error
correction model (ECM)
, 1 1 1 1, , 1, , ,
1 1
, 2 2 1 2, , 2, , ,
1 1
k k
s t t i s t i i f t i s t
i i
k k
f t t i s t i i f t i f t
i i
R a b z c R d R
R a b z c R d R
?
?
? ? ?
= =
? ? ?
= =
= + + + +
= + + + +
% %
% %
(10)
where
1 t
z
?
is the error correction term,
, s t
R
and
, f t
R are the returns in the spot and futures
markets respectively, k represents the number
of lags, and the MVHR is estimated as
( )
( )
, ,
,
cov
var
s t f t
f t
? ?
?
? =
(11)
where
( )
, ,
cov
s t f t
? ? and
( )
,
var
f t
? are
either time invariant or time varying (as in
Equation 9). If one employs a time invariant
strategy, a bivariate ECM is estimated and
unconditional estimates of
( )
, ,
cov
s t f t
? ? and
( )
,
var
f t
? are used to construct ?. Dynamic
MVHR estimation typically estimates time
varying
( )
, ,
cov
s t f t
? ? and
( )
,
var
f t
? via a
bivariate error correction GARCH model.
7
See
Kroner and Sultan (1993), Park and Switzer
(1995), Koutmos and Pericli (1998) and Lien
and Tse (1999).
The above has discussed the derivation and
estimation of the conventional MVHR and the
conditions when it will be utility maximizing.
Much of the recent research has accepted the
assumptions of the conventional MVHR,
focusing on improving the methods of
estimation. The rest of the paper argues that the
literature should focus more on the assumptions
underlying the conventional MVHR. Section
3.1 will demonstrate that the more
sophisticated estimation approaches have
provided marginal improvements in the level of
risk reduction. Section 3.2 will argue that the
assumptions of the conventional MVHR may
be inappropriate, given that they fail to allow
for multiple exposures, hedges over multiple
periods, basis convergence, estimation risk and
the possibility of using options.
7 There is no consensus on what the findings of
cointegration imply for futures unbiasedness.
MacDonald and Taylor (1988), Wahab and Lashgari
(1993) and Lien (1996) argue that cointegration implies
market inefficiency. Krehbiel and Adkins (1993),
Chowdhury (1991) and Pizzi et al (1998) suggest that
cointegration implies market efficiency.
3. Limitations in the conventional
approach
3.1 The performance of the more
sophisticated estimation methods
Table 1 documents the literature which
examines the performance of alternative
methods of MVHR estimation using the
conventional approaches. The Table reports the
risk reduction, where the data set is divided into
estimation and forecast periods, with the
forecast period being used to assess hedging
effectiveness.
The literature generally supports the
estimation of dynamic MVHRs using a
bivariate error correction GARCH model. The
earlier results (Cechetti et al, 1988; Baillie and
Myers, 1991; Sephton, 1993) rejected the time
invariance in MVHRs, finding that bivariate
GARCH models achieved greater risk
reduction than the OLS MVHR (Equation 8).
Subsequent results further illustrated the
benefits of allowing for cointegration and
conditional heteroscedasticity (Kroner and
Sultan, 1993; Park and Switzer, 1995; Koutmos
and Pericli, 1998; Lien and Tse, 1999). These
results also appear to be unaffected by the
inclusion of transaction costs (Kroner and
Sultan, 1993; Park and Switzer, 1995; Koutmos
and Pericli, 1998). Unfortunately the
incremental benefits provided by these more
sophisticated estimation methods are often
quite small. Lien and Luo (1994) and Myers
(1991) show that even though the bivariate
GARCH models statistically outperform the
other estimation approaches, this does not
necessarily result in superior risk reduction.
3.2 Relaxing the assumptions of the
conventional MVHR
This section discusses the literature that relaxes
some of the assumptions of the conventional
MVHR, taking into account: i) multiple
exposures; ii) multiple periods; iii) basis
convergence; iv) estimation risk; and v) the use
of options and futures. MVHRs estimated using
the conventional approach may therefore over
or understate the number of futures contracts
that are required to hedge a given exposure.
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Table 1
Summary of the literature investigating the performance of MVHRs
Reference Data MVHRs Dynamic
strategy
Comments regarding risk
reduction
Cecchetti
et al (1988)
20 yr T-bond
Monthly, 1/78-
12/83
Biv-ARCH(3) Single
period
Time invariant MVHR
inappropriate.
Baillie and
Myers
(1991)
Commodities
Daily, 82-86
(periods vary)
OLS,
Biv-GARCH
Single
period
Time invariant MVHR
inappropriate.
GARCH outperforms OLS.
Myers
(1991)
Wheat Weekly,
6/77-5/83
OLS,
Biv-GARCH
Single
period
MVHRs are time varying, however
the GARCH model’s performance
is only marginally better. Suggested
that once transaction costs are taken
into account, the OLS strategy is
probably the preferred strategy.
Sephton
(1993)
Commodities
Daily, 5/88-
5/89
OLS,
Biv-GARCH
Single
period
Time invariant MVHR
inappropriate.
GARCH outperforms OLS.
Kroner and
Sultan
(1993)
Currencies vis-
a vis USD
Weekly, 2/85-
2/90
Naïve,OLS,
Biv-ECM,
Biv-EC-GARCH
Single
period
Biv-EC-GARCH provides best
performance (with and without
transaction costs).
Lien and
Luo (1994)
Currencies
vis-à-vis USD,
Weekly, 3/80-
12/88
OLS,
Biv ECM,
Biv-EC-GARCH
Multi-
period
OLS and Biv ECM hedge
outperform the Biv-EC-GARCH.
Park and
Switzer
(1995)
S&P500,
Toronto 35
Weekly, 6/88-
12/91
Naïve,
OLS, ECM,
Biv-EC-GARCH
Single
period
Biv-EC-GARCH hedge provides
superior performance (with and
without transaction costs).
Koutmos
and Pericli
(1998)
Commercial
paper with T-
bill futures,
Weekly,
1/85- 3/96
OLS,
Biv-GARCH,
Biv-EC-GARCH
Single
period
Biv-EC-GARCH provides best
performance (with and without
transaction costs). Both
cointegration and time varying
moment estimation improves
hedging performance.
Lien and
Tse (1999)
Nikkei 225
Daily, 1/89-
8/97
OLS, VAR,
ECM,
Biv-VAR/EC-
GARCH
Single
period
Including GARCH improves
hedging performance. EC-GARCH
is the dominant strategy. OLS
provides the worst performance.
OLS = ordinary least squares estimation via Equations 7 & 8. ECM = error correction model estimation,
Equations 7 & 10. Biv-GARCH/Biv-EC-GARCH = dynamic estimation of Equation 9 using bivariate
GARCH/bivariate error correction GARCH.
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3.2.1 Multiple exposures
Each of the above approaches assume that the
hedger only has a single asset in the spot
market which is exposed to price uncertainty.
This is unrealistic given that most hedgers have
multiple exposures. A portfolio manager for
example, is likely to have domestic and
overseas, bond, equity and currency exposures.
Figlewski (1986) defines the methods of hedge
ratio determination in Section 2 as micro
hedging strategies, given that each asset or
liability is considered in isolation. In contrast,
macro hedging considers assets or liabilities in
groups, hedging the net exposure.
Gagnon et al (1998) derive a MVHR for a
portfolio of currencies, where futures contracts
are available for each currency. The MVHR for
each exposure takes into account the
covariance between the futures and the spot, as
well as all the other futures and spot positions
in the portfolio. The approach however
assumes that there is no quantity risk, with the
number of units of currency fixed. This is
unrealistic, given that a portfolio manager is
typically exposed to currencies where the
quantity (determined by the change in market
values denominated in foreign currency) as
well as the price (determined by the change in
exchange rates) is uncertain.
Giaccotto et al (2001) address the
limitations in Gagnon et al (1998), by allowing
for multiple price and quantity exposures. The
hedge ratio is a function of the full covariance
structure of changes in spot prices, quantities
and futures prices, and therefore takes into
account any of the natural hedges that may
exist. It is demonstrated that failure to account
for each of these covariances will lead to
systematic over or under hedging, supporting
the use of a macro hedging framework.
3.2.2 Multiple periods
The conventional dynamic MVHR (Equation
9) seeks to minimise the conditional variation
in portfolio returns period by period. Howard
and D’Antonio (1991), Vukina and Anderson
(1993), Lien and Luo (1994) and Lee (1999)
derive MVHRs that seek to minimise risk over
the life of the hedge. Howard and D’Antonio
(1991) do not allow for conditional information
and Lien and Luo (1994) require sophisticated
estimation procedures. Vukina and Anderson
(1993) do not allow for conditional
heteroscedasticity, and the approach cannot be
easily generalised to a hedge over a large
number of periods.
These approaches also fail
to allow for multiple exposures. A superior
approach is developed by Lee (1999) who
derives a multi-asset, multi-period dynamic
MVHR which allows for conditional
information and conditional heteroscedasticity.
The approach captures the interperiod
dependencies over the life of the hedge, and
reduces the volatility commonly associated
with dynamic MVHR estimation. This is
intentional given that a hedging strategy that
reflects short lived volatility fluctuations is
unstable, costly and ineffective when hedging
over the long term (Lee, 1999).
3.2.3 Basis convergence
All of the above methods fail to impose basis
convergence, and therefore ignore information
that could be used when estimating hedge
ratios. Castelino (1989, 1990a, 1990b, 1992)
considers the impact of basis convergence on
MVHR determination. As highlighted earlier, a
hedge ratio of unity will be risky if the hedge is
reversed prior to contract expiration (given
basis risk). Castelino therefore develops a
MVHR that adjusts the hedge ratio away from
unity as the hedge reversal date differs from the
contract expiration date.
Chen et al (1999) allows for convergence,
conditional information and conditional
heteroscedasticity. Chen et al (1999) model the
basis and spot as a bivariate GARCH process
with a maturity effect. By specifying the mean
and variance of the basis as a function of time
to maturity, the maturity of the contract
influences the behaviour of the basis. The
model is therefore able to impose the condition
that at maturity, the basis and its conditional
variance are zero. Chen et al (1999) derive the
MVHR as a function of time to maturity.
Estimated MVHRs are inversely related to the
time to maturity and therefore support the
insight of Castelino (1989, 1990a, 1990b,
1992).
3.2.4 Estimation risk
Lence and Hayes (1994) are critical of the
conventional MVHR given that it employs a
parameter certainty equivalent (PCE) approach.
The PCE approach derives the MVHR under
the assumption that the probability density
function (PDF) and its parameters are known
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with certainty. This ignores estimation risk,
which arises from less than perfect information
about the functional form of the PDF or its
parameter values. Failure to recognise the
estimation risk means that under the PCE
approach, slight changes in data sets can result
in large changes in the estimated MVHR.
Lence (1995) extends this further by arguing
that the conventional MVHR ignores
estimation risk, commissions, margins and the
lumpiness of contracts. It also fails to allow for
simultaneous borrowing, lending or investing
in other assets. The MVHR is determined via
the Lagrangian technique, where end of period
wealth is maximised subject to a number of
linear constraints. The magnitude of the hedge
ratio is shown to be very sensitive to the
relaxation of the assumptions in the
conventional MVHR.
3.2.5 Hedging with futures and options
The above approaches assume that the only
derivative available for hedging is a futures
contract. Lence et al (1994), Moschini and
Lapan (1995), Sakong et al (1993) and Froot et
al (1993), demonstrate that hedging a non-
linear payoff in the spot, requires the use of
options and futures. This is because a position
in futures and options can create an offsetting
non linear payoff, in contrast to futures, that
only provide offsetting linear payoffs.
Moschini and Lapan (1995) and Sakong et
al (1993) show how a non-linear spot payoff
can be a result of the interaction between price
and production yield uncertainty (a quantity
risk). These results are based on a one period
model, given the assumption that the firm is
only concerned with a single production cycle.
Lence et al (1994) allow for two production
cycles which is appropriate for firms that
exhibit forward looking behaviour. It is argued
that output price changes in one period will
change the perceived relationship between next
period’s input and output prices. Lence et al
(1994) show that under these circumstances
and non stochastic production, there will be a
non linear payoff in the subsequent period that
can be hedged with futures and options. Froot
et al (1993) also employ a two period approach
to examine the impact of hedging on optimal
financing and investment decisions. A number
of situations are presented where a non linear
hedging strategy is required to hedge the
internal cashflows used to finance an
investment project.
In summary there is a vast literature that
seeks to address the limitations in the
conventional MVHR. Unfortunately an
approach that simultaneously addresses all of
these limitations has not been forthcoming. The
conventional MVHRs use of the mean variance
framework is one further possible limitation
and is the subject of the next section.
4. Limitations in the mean variance
framework
The conventional approach employs a mean-
variance framework assuming that
maximisation of the objective function
(Equation 3) results in utility maximisation.
This however is not necessarily the case, given
that the MVHR equals the utility maximising
hedge ratio only if the hedger has a quadratic
utility function or normally distributed profits
(Kahl, 1983).
Arrow (1971) argues that quadratic utility is
highly implausible given that it implies
increasing absolute risk aversion. This suggests
that as an individual becomes wealthier, they
will decrease the amount of risky assets held.
The normality requirement also appears
unlikely in most financial markets, with return
distributions exhibiting leptokurtosis and
skewness. Nonetheless Levy and Markowitz
(1979) argue that regardless of the utility
function or the distribution of returns, the
maximisation of a mean-variance objective
function may provide a reasonable
approximation of the true objective function.
The MVHR is therefore utility maximising
if: a) at least one of the conditions for the
conventional approach to be utility maximising
is met; and b) the futures are unbiased or the
hedger is extremely risk averse. For example,
Giaccotto et al (2001) show that the MVHR
equals the utility maximising hedge ratio if
utility is represented as a general von Neuman-
Morgenstern utility function (a more general
utility function than the quadratic utility
function), the variables are normally
distributed, and futures prices follow a
martingale process.
Given that the conditions required for the
conventional MVHR to be utility maximizing
are quite restrictive, other approaches may
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A Critique of Minimum Variance Hedging
47
provide superior outcomes (given that they are
utility maximising). Consequently the hedging
outcomes using the conventional MVHR may
be dominated stochastically. Stochastic
dominance is based on the von Neumann-
Morgenstern utility functions and applies
selection rules that are based on pairwise
comparisons between distributions that require
knowledge of the complete distribution. This is
in contrast to the mean-variance approach
which only requires knowledge of the mean
and variance. See Bawa (1975, 1978), Fishburn
(1977), Yitzhaki (1982), Shalit and Yitzhaki
(1984) for further details.
Given that the conventional approach may
not be utility maximising, alternative measures
of risk have been used to derive hedge ratios.
Chen et al (2001) discuss the alternative risk
measures used in the hedging literature, namely
the mean- extended Gini coefficient (MEG)
and the generalised semivariance (GSV).
The mean Gini coefficient is a measure of
variability first applied to finance by Yitzhaki
(1982). Hedge ratios estimated using this
measure of risk (Kolb and Okunev, 1992,
1993) are consistent with second order
stochastic dominance (Chen et al, 2001),
however estimation can be difficult (see Kolb
and Okunev, 1992 for details).
The variance and MEG measures treat risk
as being two sided. This may be inappropriate
given that agents are generally more concerned
with managing downside risk (Crum et al,
1981; Lien and Tse, 2000). Here the returns
below a target return are considered risky,
whilst returns above the target are not. This risk
can be captured via the GSV. The GSV hedge
ratio therefore finds a value of
t
? that
minimises the variability in hedged returns
below a target value, not the total variability in
hedge returns. See Lien and Tse (1998, 2000)
and Chen et al (2001).
In summary, the assumptions of the mean
variance framework may mean that the
conventional MVHR is inappropriate and
alternative measures of risk like the MEG and
GSV are required.
5. Conclusion
This paper has performed a critique of
minimum variance hedging using futures. The
paper developed the conventional approach to
hedge ratio determination and discussed some
of the methods used to estimate MVHRs. The
paper then highlighted some of the weaknesses
in the approach. The conventional approach
does not allow for multiple exposures, multiple
periods, basis convergence, estimation risk, or
the use of futures and options. It was shown
that if a hedger is not extremely risk averse and
uses the conventional MVHR, this may not
maximise the hedger’s objective function.
Limitations in the mean variance framework
may also mean that the use of alternative risk
measures are required. Given the limited
benefits from employing more sophisticated
estimation methods, the literature should
probably focus more of its attention on the
assumptions underlying the MVHR, rather than
improving the estimation techniques.
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