1
Dynamic currency hedging for international stock portfolios
Christine Brown
Monash University, Australia
Jonathan Dark
The University of Melbourne, Australia
Wei Zhang*
Deakin University, Australia
Abstract
The paper studies dynamic currency risk hedging of international stock portfolios using
a currency overlay. A dynamic conditional correlation (DCC) multivariate GARCH
model is employed to estimate time-varying covariance among stock market returns and
currency returns. The conditional covariance is then used in the estimation of
risk-minimizing conditional hedge ratios. The study considers 7 developed economies
over the period January 2002 to April 2010, and estimates daily conditional hedge ratios
for portfolios of various stock market combinations. Conditional hedging is shown to
dominate traditional static hedging and unconditional hedging in terms of risk reduction
both in-sample and out-of-sample, especially during the recent global financial crisis.
Conditional hedging also proves to consistently reduce portfolio risk for various levels
of foreign investments.
* Corresponding author. This paper is part of Wei’s PhD dissertation at the Univeristy of Melbourne. Wei
can be contacted via [email protected] . The authors thank Stephen Brown, George Milunovich,
Bernardo da Veiga, Sheng-Yung Yang, Chris Bilson, Spencer Martin, Ming-hua Liu and Michael Chng
for helpful comments.
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1. Introduction
International investing has become popular in the past few decades, attributable to
globalization and investors’ gradual recognition of the benefit from international
diversification. Investors include international assets in their portfolios to add return or/and to
diversify risk, which automatically expose them to the movements in foreign exchange rates.
Such exposure could reduce or even eliminate the benefit from international diversification if
not managed properly. A large volume of evidence can be found in the literature in favour of
hedging currency risk arising in an international asset portfolio, but there is no consensus
over how much exposure should be hedged and what strategies should be followed in
conducting hedges.
This study presents empirical tests of a number of competing currency hedging strategies for
the purpose of portfolio risk management. We consider portfolio investors holding an
internationally diversified equity portfolio such as an international index fund. Each asset in
the portfolio represents a stock index in a particular currency. We define currency hedging as
the practice of using foreign currencies to achieve a hedger’s objective, defined as portfolio
variance minimization. The exposure to a foreign currency can be hedged by taking an
appropriate short position in a forward contract written on the currency. The hedge ratio of a
foreign currency is defined as the amount of exposure being hedged with respect to the total
underlying exposure.
A number of static strategies have been proposed in the past. For example, Perold and
Schulman (1988) advocate for the currency exposure to be fully hedged based on their
famous “free lunch” claim that currency returns average to zero over the long run and
substantial risk reduction can be achieved through currency hedging at no loss of return. At
the other extreme, Froot (1993) claims that currency exposure should be left unhedged for
long term investors based on the assumption that purchasing power parity holds in the long
run and exchange rates display mean reversion. In between the extremes is Black’s (1989,
1990) universal hedge ratio, which he claims should always be less than 1 (full hedge) based
3
on Siegel’s paradox.2 The universal hedge ratio has an estimate of 0.77 based on historical
data. Another popular strategy commonly adopted by practitioners is to hedge half of the
currency exposure. This is supported by Gastineau (1995) who argues that the rule can be
used both for passive currency management and as a reasonable starting point for active
currency management. The most commonly adopted static strategies including no hedge, half
hedge and full hedge generally do not account for the correlations among currencies,
underlying assets and the cross-correlations between currencies and underlying assets. These
strategies are shown to be outperformed by hedging strategies that do take these correlations
into account. Black’s universal hedge ratio, which is derived in the setting of the international
CAPM, does account for the correlation between currencies. However the model has been
criticized as having little practical use because it only holds under unrealistic assumptions.
Jorion (1994) examines the role that currency plays in a global portfolio and proposes three
possible ways of including currency in the global mean-variance optimization: First, in a joint,
full portfolio optimization, the correlations between currencies and underlying asset classes
are accounted for in the optimization process, and the positions in assets and currencies are
determined simultaneously. Second, in a two-stage partial portfolio optimization, the
underlying asset portfolio is first determined without any currency considerations, currency
positions are then optimized given the core portfolio. Third, in a separate portfolio
optimization, currency positions are determined separately from underlying asset allocations
via two separate optimization processes. The currency management approaches in the partial
and separate optimizations are classified as a ‘currency overlay’ strategy.
The joint optimization fully exploits the correlations and cross-correlations among all assets
and currencies, and is shown using ex-post data to outperform an overlay strategy in which
currency positions and asset positions are not determined simultaneously. Though
theoretically sound, the same is not found in ex-ante studies mainly due to high estimation
risk. For example, Eun and Resnick (1988) and Larsen and Resnick (2000) find low accuracy
2 Siegel’s paradox describes the situation that for a pair of currencies, the expected percentage movement of
each currency measured in units of the other (discrete return) do not sum up to zero.
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in estimating the input parameters (especially the mean) out of sample contributes to the
ex-ante poor performance of the joint optimization. On the other hand, hedging for the
purpose of risk minimization mitigates estimation risk as the covariance structure is found to
be estimated with relative precision (e.g. Jorion, 1985). This approach is adopted in a recent
study by Campbell, Medeiros and Viceira (2010) who examine the unconditional demand for
currencies for risk management purpose by an investor with a given portfolio of stocks or
bonds, using an overlay strategy for hedging.
Papers discussed up to this point largely rely on models that are developed on the strong
assumption that the variances of, and the covariances between, the changes in underlying
assets and currency forward returns are constant over time. Yet empirical evidence suggests
otherwise (e.g. Bolleslev, Chou & Kroner, 1992; Longin & Solnik, 1995; Sheedy, 1998),
which consequently fuelled the development of a number of dynamic hedging models. This
branch of research in the literature of currency hedging attempts to develop hedging
strategies that are based on times-series modelling of conditional mean, variances and
covariances, thereby producing time-varying conditional hedge ratios. For example, Gagnon,
Lypny and McCurdy (1998) base their study on a BEKK trivariate GARCH model; Guo
(2003) adopts a multivariate GARCH model with time-varying correlations proposed by Tse
and Tsui (2002); Hautsch and Inkmann (2003) employ the Dynamic Conditional Correlation
(DCC)-GARCH model developed by Engle (2002). Research in this area commonly
documents the ex-post superior performance of a conditional hedging strategy relative to that
of unconditional strategies that are time-invariant, and relative to strategies adopting fixed
hedge ratios. But little has been said about the performance of these more complicated
dynamic hedging strategies ex ante.3
To fill the gap in the literature, we develop a conditional hedging strategy within the standard
3 One exception is the study by Kroner and Sultan (1993) which includes an ex-ante study of the effectiveness
of the univariate hedging strategy adopted in the paper. However, this strategy ignores the correlations among
currencies, and the cross-correlations between currencies and underlying assets, therefore is sub-optimal.
5
framework that has been adopted by Campbell et al. (2010) and allow both the mean and the
covariance structure of the international portfolio to be time-varying. The strategy employs a
vector autoregression (VAR) to model the conditional mean and the DCC-GARCH to model
the conditional covariance structure on a daily basis. The study investigates the conditional
hedging strategy implemented via a currency overlay to minimize overall portfolio risk of a
given international stock portfolio. Performance of the strategy is compared to that of static
hedging (hedge ratios of 0, 0.5 and 1) and unconditional hedging (hedge ratios estimated with
OLS regression) both in-sample and out-of-sample. The in-sample period is Jan 2002-Dec
2005 and the out-of-sample period is Jan 2006-Apr 2010. Our method is not new, however,
this is the first time the performance of such a conditional currency hedging strategy is tested
out-of-sample.
The study considers seven countries, Australia, Canada, Japan, the U.K., Switzerland,
Germany and the U.S. We take the perspective of a US investor and explore a number of
portfolios investing in two countries and all seven countries respectively, in order to
investigate the effect of multicurrency diversification on the effectiveness of hedging. For the
majority of the portfolios examined, the conditional hedging strategy outperforms the other
strategies in terms of risk reduction within sample. For portfolios invested in the U.S. and one
foreign market, hedging of investments denominated in AUD, CAD and JPY appears to be
especially rewarding for U.S.-based investors under all hedging strategies. On the other hand,
static hedging of investments made in GBP, CHF and EUR tends to add to portfolio risk,
though the conditional strategy manages to reduce portfolio risk. In contrast, all hedging
strategies help reduce risk for a portfolio consisting of all seven stock markets, with the
conditional strategy achieving the highest level of risk reduction. We also demonstrate the
consistent dominance of the conditional hedging strategy over the alternative strategies for
various levels of foreign investments during the in-sample period.
Out of sample, the conditional hedging strategy clearly dominates other hedging strategies in
terms of risk reduction. This result is more pronounced for the period covering the GFC when
hedging is needed the most. The dominance of conditional hedging over other strategies is in
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many cases statistically significant, although conditional hedging is not significantly different
from unconditional hedging for the portfolio composed of all seven stock markets. Our
findings confirm the benefit of implementing a conditional strategy such as the one employed
in our study, and raise questions about the common practice of adopting the naïve static
hedging strategies such as full hedge and half hedge.4
The rest of the paper is organized in the following way. Section 2 specifies the theoretical
model and econometric method used to derive the risk-minimizing conditional hedge ratio.
Section 3 describes the data. Section 4 presents the in-sample hedging results for selected
portfolios. The effect of different levels of foreign investments on hedging performance is
reported in Section 5. Section 6 presents the out-of-sample hedging results. Section 7
concludes.
2. Methodology
In this section, we first develop a currency-overlay style unconditional hedging strategy
(using currency forward contracts) for a risk-minimizing investor. We then specify an
econometric model that estimates a dynamic variance-covariance matrix that is subsequently
used in the estimation of conditional hedge ratios.
We develop the following strategy from the perspective of a US investor, though the strategy
can be easily applied to other base currencies as well. Define Si,t as the spot USD price of
foreign currency i at time t, and define Pi,t as the foreign currency asset value inclusive of
dividend reinvestments at time t.
For investments denoted in one foreign currency, unhedged discrete return measured in USD
is defined as
, 1 , 1
,
, ,
1i t i tuh
i t
i t i t
P SR
P S
(2.1)
4 Michenaud and Solnik (2008) suggests that 39% of investors adopt a no-hedging policy, 34% choose a 50%
hedging policy, 14% select a 100% hedging policy and 13% opt for some other hedge ratios.
7
Return on a long forward contract normalized by the current spot exchange rate is defined as
, 1 ,
,
,
i t i t
i t
i t
S Ff
S
(2.2)
where Fi,t denotes the one period forward dollar price of foreign currency i. The hedged
return on investment in country i is then given by5
, , , ,
h uh
i t i t i t i tR R h f (2.3)
where hi,t is the hedge ratio of the investment in country i at time t. We assume that the
investor sells forward hi,t dollars worth of currency i per dollar invested in the stock market of
country i at the time the investment was made. For instance hi,t = 0 indicates that the
investment in currency i is left unhedged,6 and hi,t =1 implies that the investment is fully
hedged. In the case that hi,t is negative, the investment’s exposure to currency i is increased
by buying currency i forward. The investment is said to be “over hedged” if hi,t >1. This
occurs when the amount of currency i sold forward is greater than the underlying exposure in
currency i. At this stage we have not imposed any constraint on the hedge ratios, though in
practice, currency managers are commonly restricted from taking speculative positions in
currencies,7 meaning that the position taken in any currency forward contract cannot exceed
the investment exposure to that currency (hedge ratio>1) or exaggerate the exposure (hedge
ratio<0). In the empirical analysis, we will explore both constrained ( 0 hedge ratio 1 ) and
unconstrained hedge ratios, and compare the hedging results.
We now assume a US investor invests in assets denominated in multiple currencies with
predetermined portfolio weights, and wishes to manage the foreign currency exposure
embedded in his/her portfolio with a currency overlay. In this set-up, it is assumed that the
investor invests in N+1 stock markets (including the domestic market), and is exposed to N
foreign currencies with the USD as the base currency.
5 Note that country i and foreign currency i will be used interchangeably.
6 The interpretation of the hedge ratio here is consistent with Glen and Jorion (1993) and Jorion (1994), but
different from the interpretation of the hedging demand used in Campbell et al. (2010). 7 See for example Gardner and Stone (1995), Clarke and kritzman (1996) and Xin (2003). This excludes active
currency managers whose goal is to add return.
8
Let 1, 2, , 1,[ , , , , ] 'uh uh uh
t t N t N tR R R R R be an (N+1)×1 vector of unhedged asset returns in USD
from all countries, with RN+1 being the return from the U.S. W denotes an (N+1)×1 vector of
portfolio weights wi with wN+1 being the weight in the US stock market. Let f denote an N×1
vector of forward currency returns fi,t. h is an N×1 vector of hedge ratios hi, t. The hedged
gross portfolio return is given by
' '( )h
pR W R h w f (2.4)
where w is an N×1 vector of portfolio weights wi excluding the weight in the U.S., and
⊙represents element-by-element multiplication. So the conditional variance of the hedged
portfolio return is given by
1 1 1
1
var ( ) I var ( ' ) I var [ '( )] I
2cov [ ' , '( )] I
h
t p t t t t t
t t
R
W R h w f
W R h w f (2.5)
Assume the investor’s objective is to minimize the variance of the hedged portfolio return with
respect to a vector of hedge ratios
1min var ( ) Ih
t p tR h
(2.6)
Then the first order condition is given by
1 1[var ( )] I cov ( ' , ) I 0t t t t w f h W R w f (2.7)
The vector of time-varying (conditional) optimal hedge ratios is therefore given by
1
1 1var ( ) I cov ( ' , ) It t t t
h w f W R w f (2.8)
Under the assumption that the variances and covariances in equation (2.8) are constant over
time, the vector of unconditional optimal hedge ratios can then be written as
1var ( )cov( ' , ) h w f W R w f (2.9)
Equation (2.9) implies that the vector of unconditional optimal hedge ratios can be generated
by the following OLS regression
' '( ) W R β w f (2.10)
where β is an N×1 vector of coefficients.
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Econometric model
The DCC-GARCH model first proposed by Engle (2002) is used for the estimation of the
conditional variance-covariance matrix. The DCC model is chosen for its apparent
advantages over other multivariate GARCH models. The model can be estimated in two steps
– the first is a series of univariate GARCH estimates and the second is the correlation
estimate. This two-step estimation procedure enables the estimation of large correlation
matrices since the number of parameters to be estimated in the correlation process is
independent of the number of series to be correlated. However, this estimation procedure
does require the standard errors of the parameters to be modified for consistency and
efficiency.
We first model the return series [ , ]'tX R f with a Vector Autoregression (VAR)8
1 1 t t- 2 t-2 p t-p tX a b X b X b X e (2.11)
where 1I ( , )t Nt te 0 Σ , 1It
is the information set at time t-1, and tΣ is the
variance-covariance matrix of the asset returns and currency forward returns at time t,
conditioning on the information available at time t-1. The VAR structure allows each return
series to be explained by a constant term and lagged values of all return series. The error
terms are then used in the estimation of the covariance structure for the multivariate return
series.
Following Engle (2002), the conditional variance-covariance matrix is defined as:
t t t tΣ D Λ D (2.12)
where Dt is the (2N+1)×(2N+1) diagonal matrix of time varying standard deviations from
univariate GARCH models with ,i t on the ith
diagonal, and Λt is the time varying correlation
8 All the return series are stationery. This is verified by applying Augmented Dickey-Fuller test to each return
series. The Johansen cointegration test result suggests that country stock indices are not cointegrated at price
levels. Since the return on a forward contract is calculated with both the spot and forward exchange rates as has
been illustrated in equation (2.2), the cointegration test is not performed for the currencies.
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matrix. The elements of Dt are estimated by univariate GARCH models,9 so the specification
for the conditional variance is:
, , ,
1 1
2 2 2, ,
i i
i t i t p i t q
P Q
p q
i i p i qe
(2.13)
where i , ,i p and ,i q are nonnegative, and 1 1
, , 1i iP Q
p q
i p i q
. The dynamic correlation
structure proposed by Engle (2002) is:
1 1 1 1
(1 ) ( )M L M L
m l m l
m l m l
t t-m t-m t-lQ Q ε ε' Q (2.14)
where -1
t t tε D e is a vector of residuals standardized by their conditional standard
deviations, and Q is the unconditional covariance of the standardized residuals resulting
from the first stage estimation. m and l are nonnegative scalar parameters satisfying
1 1
1M L
m l
m l
.
Define t
*Q to be a diagonal matrix composed of the square root of the diagonal elements
oftQ , so we have (with k = 2N+1)
11
22
0 0
0 0
0 0
t
kk
q
q
(2.15)
Therefore
t t
*-1 *-1
t tΛ Q Q Q (2.16)
with the ijth
element of tΛ being
,
,
ij t
ij t
ii jj
q
q q .
9 A number of models namely EGARCH, GJR, PARCH and GARCH have been explored to determine whether
there is asymmetry in currency return volatility. We found little evidence of asymmetric volatility, and in
majority of the cases both AIC and SC suggest that GARCH is the best model among the four.
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The log-likelihood of the DCC estimator can be written:
1
1 1 1
1
1
1
1((2 1) log(2 ) log( ) )
2
1((2 1) log(2 ) log( ) )
2
1((2 1) log(2 ) 2log( ) log( ) )
2
T
t
T
t
T
t
L N
N
N
t t
t t t t
t t
' -1
t t
'
t t t t
'
t t t
Σ e Σ e
D Λ D e D Λ D e
D Λ ε Λ ε
(2.17)
We model the covariance matrix using a DCC(1,1)-GARCH(1,1) model.10
Estimating the
GARCH and DCC parameters separately result in loss of efficiency relative to a maximum
likelihood estimation of all the parameters at once, although consistency is less of an issue.
Following the two-step estimation procedure, the standard errors of all the GARCH and DCC
parameters are modified according to the theorems provided in Engle and Sheppard (2001).
This ensures that the standard errors are consistent and asymptotically efficient.
3. Data
The empirical analysis of this study considers 7 countries: the U.S., the U.K., Canada,
Australia, Switzerland, Japan and Germany. Throughout the study, the U.S. is considered as
the domestic country and the US dollar is the base currency. This study uses daily data11
over
the period January 2002-April 2010. Morgan Stanley Capital International (MSCI) country
stock indices12
in local currencies are used to measure country stock market performance.
Unhedged country stock market return measured in US dollar is computed from the country
stock index using equation (2.1). Spot and forward exchange rates are quoted in terms of the
US dollar, and the return on a long position in a one-day currency forward contract is
computed using equation (2.2). All data are sourced from DataStream. Data from January
2002 to December 2005 are used for in-sample analysis, and data from January 2006 to April
10
Results generated using higher orders are similar. We try to keep our model simple as the more
heavily-parameterised models tend to forecast poorly out of sample. 11
Using daily data exposes the study to a non-synchronous trading problem as stock markets from different
time-zones open and close at different times. As a robustness check, the same analyses are also performed using
weekly data. The results obtained using weekly data are similar to the results reported in the paper, and are
available on request. 12
These series include dividend reinvestments after deducting withholding tax.
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2010 are used for out-of-sample analysis.
Summary statistics of daily stock market returns and currency forward returns for the
in-sample period Jan 2002- Dec 2005 are reported in Table 1. Standard deviation reported in
rows 3 and 7 shows that the risk of unhedged USD foreign stock returns is in many cases
much higher than the risk of the corresponding local currency returns. This demonstrates the
contribution of currency risk to the overall risk of holding foreign assets. However the
average currency forward return is positive for all countries considered. This is attributable to
the weakening of USD against almost all the major currencies during the sample period. This
in turn explains the fact that for all countries, unhedged USD returns are on average higher
than the corresponding local currency returns within sample.
Table 1 Summary statistics
Statistics of stock market returns and forward currency hedge returns for the period January 2002 to December
2006. Mean local currency return is the average daily stock market return measured in local currency. Mean
USD return is the average daily unhedged stock market return13
measured in US dollars. Mean currency
forward return14
is the average daily return on a long position in a one-day currency forward contract. Daily
data on MSCI country stock indices, spot exchange rates and one-day forward exchange rates are obtained from
DataStream. Mean and standard deviation are in percentage terms.
% Australia Canada Japan UK Switzerland Germany USA
mean stock return (local currency) 0.044 0.045 0.051 0.025 0.031 0.019 0.019
standard deviation (local currency) 0.645 0.827 1.144 1.104 1.207 1.650 1.082
mean currency forward return 0.023 0.027 0.020 0.008 0.029 0.026 -
standard deviation 0.663 0.513 0.586 0.515 0.649 0.582 -
mean stock return (USD) 0.082 0.076 0.063 0.041 0.052 0.045 0.019
standard deviation (USD) 0.977 0.982 1.302 1.095 1.148 1.565 1.082
Table 2 reports unconditional correlations of daily stock market returns denominated in US
dollars. It illustrates that there is high correlation among the stock market returns of
Switzerland, Germany and the U.K. This is expected given the close linkage between
13
This is computed using equation (2.1). 14
This is computed using equation (2.2).
13
European economies. The correlation is also high among the US, Canadian and German stock
markets. But most of the correlations in the table are far from perfect, which suggests that
benefit can be derived from international diversification.
Table 2 USD stock market return correlations
Unconditional correlations among unhedged daily country stock market returns measured in US dollars for the
period January 2002 to December 2005.
Australia Canada Japan UK Switzerland Germany USA
Australia 1.00
Canada 0.33 1.00
Japan 0.43 0.23 1.00
UK 0.34 0.47 0.20 1.00
Switzerland 0.32 0.43 0.22 0.75 1.00
Germany 0.23 0.55 0.17 0.72 0.71 1.00
USA 0.04 0.60 0.08 0.41 0.37 0.61 1.00
Table 3 reports unconditional correlations among daily returns of long forward contracts on
the selected currencies. We observe very high correlation among the forward returns of
‘European currencies’, namely the Euro, Swiss franc and British pound. This again reflects
the close relationship between the European countries, and also indicates that cross hedging15
can be effective for investments denominated in these currencies. High correlations are also
reported for some other currency pairs. But the imperfect correlations imply that hedging
with a portfolio of currencies could reduce the aggregate risk of the hedge instruments.
15
Cross hedging occurs when the exposure to currency A is hedged with forward contracts written on currency
B, where currencies A and B are highly correlated.
14
Table 3 Currency forward return correlations
Unconditional correlations among daily currency forward returns of various currencies for the period January
2002 to December 2005.
AUD CAD JPY GBP CHF EUR
AUD 1.00
CAD 0.54 1.00
JPY 0.45 0.32 1.00
GBP 0.56 0.39 0.47 1.00
CHF 0.55 0.45 0.56 0.74 1.00
EUR 0.60 0.47 0.54 0.75 0.95 1.00
The unconditional correlations among stock market returns and currency forward returns are
reported in Table 4. With a few exceptions, the correlations are quite low. It is therefore
reasonable to leave some currency exposure unhedged for diversification purposes, because
natural hedges exist in portfolios. Depending on portfolio composition, it might even be
attractive to increase the portfolio’s exposure to certain currencies, given the negative
correlation between the US stock market and the currency forward return on the three
European currencies. On the other hand, high correlation between AUD and the Australian
stock market probably indicates that most if not all of the exposure in AUD should be
hedged.
Table 4 USD stock market return and currency forward return correlations
Unconditional correlations between unhedged stock market return and currency forward returns for the period
January 2002 to December 2005. All returns are measured in US dollar and are on daily basis. The underlined
correlation coefficients are not significant at the 5% level.
Australia Canada Japan UK Switzerland Germany USA
AUD 0.75 0.32 0.28 0.27 0.26 0.19 -0.01
CAD 0.42 0.54 0.20 0.19 0.22 0.17 0.04
JPY 0.29 0.14 0.48 0.04 0.13 0.03 -0.05
GBP 0.34 0.17 0.15 0.22 0.16 0.06 -0.10
CHF 0.31 0.13 0.14 0.02 0.19 -0.01 -0.16
EUR 0.37 0.16 0.17 0.06 0.20 0.04 -0.15
4. Currency overlay
This section presents in-sample hedging results of currency overlay strategies. The analysis
15
examines portfolios of various stock market combinations,16
and estimates both
unconditional and conditional hedge ratios for the currencies to which a certain stock
portfolio is exposed. A number of static hedge ratios are also included as benchmarks for
performance comparison. Initially all portfolios considered are equally weighted (this will be
relaxed later in the paper).
The unconditional hedge ratios are estimated using OLS regression based on equation (2.9).
The conditional hedge ratios are computed, based on equation (2.8), from conditional
covariance matrix estimated with the DCC-GARCH model. Other static hedging strategies
considered include: i) Full hedge, under which 100% of the portfolio’s currency exposure is
hedged by taking a long position in currency forward contracts. ii) Half hedge, under which
only 50% of the currency exposure embedded in the portfolio is hedged. And iii) No hedge,
under which the currency exposure of the portfolio is left unhedged. All hedges are
implemented as a currency overlay from a US perspective.
Given the fact that in practice, portfolio managers are often prohibited from taking
speculative positions17
in currency forward contracts when implementing hedging strategies,
constrained hedge ratios (hedge ratios lie within the range [0,1]) are also computed for
completeness. Hedging results of all the overlay strategies are compared for each stock
portfolio considered.
4.1 Two-asset portfolios
We examine hedging strategies for six two-asset-portfolios each containing the US stock
index and the stock index of another country. Each stock portfolio is exposed to only one
foreign currency and the exposure is hedged by taking a short position in forward contracts
written on that currency. The unconditional hedge ratios for the six currencies are reported in
16
We examined a number of two-asset, four-asset and seven-asset portfolios. For the consideration of space,
only the results for two-asset and seven-asset portfolios are reported in the paper, results for four-asset portfolios
are available on request. 17
This results from a short position taken in a currency forward contract that exceeds the underlying exposure
to that currency.
16
panel A of Table 5.
To demonstrate how to interpret the result, a hedge ratio of 1.09 for AUD implies that for
every dollar invested in Australia, 1.09 dollars worth of AUD should be sold forward. A
hedge ratio of -0.19 for EUR means that for every dollar invested in Germany, 0.19 dollar
worth of EUR should be bought forward. Overall, the result suggests that exposure in AUD,
CAD and JPY should be close to fully hedged, whereas the majority of the exposure in GBP
and CHF should be left unhedged and the exposure in EUR should be magnified by buying
forward contracts written on EUR. This is not surprising given the negative unconditional
correlation between currency forward returns on the three European currencies and US stock
returns, as well as the relatively high correlation between currency forward returns on AUD,
CAD and JPY and US stock returns during the sample period.
We also computed the correlation between the six equally weighted portfolios with the return
on the corresponding currency forward contracts. The forward return on EUR has an
unconditional correlation of -0.05 with the unhedged portfolio composed of Germany and the
U.S. This explains why it is desirable to increase the exposure to EUR. In contrast, the
correlation between the forward return on AUD and the portfolio consisting of Australia and
the U.S. is 0.49, hence the large hedge ratio for AUD. All hedge ratios are significant at the
5% level except that for Swiss franc and Euro. These results are largely consistent with
Campbell et al. (2010).
To estimate time-varying correlations using DCC-GARCH, a VAR model is first fitted to the
stock index return and currency forward return series of each portfolio.18
The residuals are
then used in the univariate GARCH estimation and the dynamic correlation estimation. The
estimation result is included in Appendix A. The conditional hedge ratios are then computed
18
A VAR model with 3 lags (VAR(3)) is fitted for the portfolio containing Australia (Australia-US portfolio); a
VAR(3) is fitted for the Canada-US portfolio; a VAR(1) is fitted for the Japan-US portfolio; a VAR(8) is fitted
for the UK-US portfolio; a VAR(3) is fitted for the Switzerland-US portfolio; and a VAR(2) is fitted for the
Germany-US portfolio; Lag length is selected using the AIC criteria and residual correlograms.
17
from equation (2.8) using conditional correlations. The average hedge ratios are reported in
panel B of Table 5. The variation of the conditional hedge ratios is not shown in this table, but
on average the conditional hedge ratios do not deviate much from the unconditional hedge
ratios.
Table 5 Hedge ratios for two-asset portfolios
This table shows both unconditional and conditional hedge ratios for equally-weighted two-asset portfolios over
the sample period January 2002 to December 2005. Each portfolio contains the US stock index and the index of
a foreign country with equal weights, the first row of the table indicates with which foreign country’s stock
index the portfolio is formed. Unconditional hedge ratios are generated by regressing the unhedged portfolio
return (in USD) onto the currency forward return. Daily conditional hedge ratios are computed using equation
(2.8), with conditional covariance matrix estimated from the DCC-GARCH model.
Australia Canada Japan UK Switzerland Germany
unconstrained 1.09 1.10 0.97 0.26 0.06 -0.19
constrained 1.00 1.00 0.97 0.26 0.06 0.00
unconstrained 0.99 1.17 0.99 0.31 0.16 -0.09
constrained 0.94 0.94 0.86 0.46 0.39 0.34
Panel A: unconditional hedge ratios
Panel B: average conditional hedge ratios
Using two 2-asset portfolios as examples,19
Figure 1 demonstrates the way conditional
correlation, constrained and unconstrained conditional hedge ratios vary over time. It is clear
that correlations among currency forward returns and stock market returns are not constant
over time. In fact some are rather volatile during the sample period. For example, in Figure
1(i), the CHF had a correlation of about 0.4 with the Swiss stock market at the beginning of
the sample period, which dropped to around -0.2 in the following year, and then increased to
above 0.5 in the subsequent two years. A similar degree of variation is observed for
correlations among other assets as well. The instability of correlations has important
implications for hedge ratios, since the risk-minimizing hedge ratio is directly derived from
the covariance matrix of the assets and currencies underlying the portfolio. Having a static
hedge ratio can hardly be justified in view of the fact that correlations among assets are
highly volatile.
19
Only the results for 2 out of 6 portfolios are presented here in order to preserve space. Full results are
available on request.
18
Figure 1 Conditional correlations and hedge ratios (two-asset)
(i) Switzerland-U.S.
0 200 400 600 800 1000 1200-2
-1
0
1
2
conditio
nal hedge r
atios
0 200 400 600 800 1000 1200-0.5
0
0.5
1
days
conditio
nal corr
ela
tions
CHF conditional hedge ratios
Constrained conditional hedge ratios
Switzerland (stock)-CHF (currency)
U.S. (stock)-CHF (currency)
Switzerland (stock)-U.S. (stock)
end-2003end-2002end-2004 end-2005
(ii) Germany-U.S.
0 200 400 600 800 1000 1200
-2
0
2
conditio
nal hedge r
atios
0 200 400 600 800 1000 1200-0.5
0
0.5
1
days
conditio
nal corr
ela
tions
EUR conditional hedge ratios
Constrained conditional hedge ratios
Germany (stock)-EUR (currency)
U.S. (stock)-EUR (currency)
Germany (stock)-U.S. (stock)
end-2002 end-2003 end-2004 end-2005
19
As can be seen in the figure, the hedge ratios conditional on the time-varying correlations
fluctuated during the sample period in response to the changes in the correlations. For
example, Figure 1(ii) depicts an increase in the correlation between returns on German and
US stock market at the beginning of the sample period. While in the same period, a dramatic
decline is evidenced for the correlation between the US stock returns and EUR forward
returns as well as for the correlation between the German stock returns and EUR forward
returns. This implies that while the German stock market moves more in line with the US
stock market in this period, the Euro tends to appreciate when either stock market falls (since
both correlations became negative in mid-2002). Therefore having a naked (unhedged)
position in EUR serves as a natural hedge against unfavourable movements in the stock
markets. In fact, the exposure in EUR should be magnified to fully exploit the benefit of the
negative correlations, as has been suggested by the negative hedge ratios for that period. The
hedge ratio climbed to around 1 in the subsequent year when the correlations changed.
Patterns in the movement of conditional hedge ratios of other currencies can be explained by
changes in correlations in a similar manner. The effectiveness of conditional hedging,
measured as the percentage portfolio risk reduction relative to no hedge is reported in Table 6,
along with hedging results of unconditional hedging and static hedging. The table illustrates
that there is a sizable reduction in risk from conditional hedging for portfolios containing
Australia (11.9%), Canada (4.6%) and Japan (6.1%) respectively. But the risk reduction is not
superior to that of a simple 100% hedge, probably due to the fact that investments in these
currencies are recommended to be fully hedged if not over hedged, therefore a 100% hedge
could prove to be just as effective and contains no estimation error.20
On the other hand, risk
reduction from conditional hedging for portfolios containing the U.K. (2.2%), Switzerland
(3.3%) and Germany (4.8%) respectively is relatively low. But conditional hedging has
clearly outperformed other hedging strategies, as the 100% hedging strategy resulted in added
20
The percentage risk reduction from conditional hedging is slightly lower than that from 100% hedging for
portfolios containing Australia and Canada, though the difference is not statistically significant. This is likely to
be caused by specification error of the conditional hedge model, but the phenomenon disappears when more
assets are included in the portfolio.
20
risk for all three portfolios, and unconditional hedging had virtually no risk reduction. Also,
constraining the hedge ratios to lie between 0 and 1 results in a deterioration of the
performance of both conditional and unconditional hedging.
We also conducted an F-test for equal variance analysis firstly using conditional hedging and
then using no hedge as the benchmark. Conditional hedging is chosen as one of the
benchmarks because we are interested in comparing the performance of conditional hedging
with that of alternative strategies. The F-test result shows that the variance reduction achieved
with conditional hedging relative to no hedging is generally not statistically significant except
for AUD and JPY. The difference between conditional hedging and other hedging strategies is
also not statistically significant, except for the difference between conditional hedging and
100% hedging for CHF and EUR, as well as the difference between conditional hedging and
50% hedging for EUR (as shown in the stdev column). On the other hand, the risk reduction
achieved by all hedging strategies relative to no hedge is significant at the 1% level for the
portfolio containing Australia (as shown in the %down column). However, the risk reduction
produced by various strategies lacks statistical significance for the rest of the portfolios
except the one invested in Japan.
21
Table 6 Hedging results comparison for two-asset portfolios
The table reports the mean and the standard deviation of hedged portfolio returns under different hedging
strategies: No hedge (h=0), half hedge (h=0.5), full hedge (h=1), unconditional hedge (hedge ratio generated by
OLS regression), constrained unconditional hedge, conditional hedge (hedge ratio based on dynamic conditional
covariance matrix) and constrained conditional hedge. All constrained hedge ratios are bounded by 0 and 1.
Each portfolio contains the US stock index and the index of a foreign country with equal weights, the first row
of the table indicates with which foreign country’s stock index the portfolio is formed. All calculations are based
on daily observations over the period January 2002 to December 2005. Mean and standard deviation are in
percentage terms. An F-test of equal variance (standard deviation) is performed for the conditional hedge against
every other hedge. In the “stdev” column, ***, ** and * respectively represent 1%, 5% and 10% significance
levels at which the null of equal variance can be rejected. The best performing hedge with the lowest standard
deviation is in bold. %down = percentage change in standard deviation relative to no hedge.21
The statistical
difference between no hedge and every other hedge is shown by the significance indicators (*** etc) in the
“%down” column.
hedging
strategy mean stdev %down mean stdev %down mean stdev %down
no hedge 0.050 0.742*** 0.0 0.047 0.924 0.0 0.041 0.879** 0.0
half hedge 0.043 0.677 8.7*** 0.039 0.894 3.3 0.036 0.844 4.0
full hedge 0.037 0.649 12.5*** 0.032 0.881 4.7 0.031 0.832 5.3*
unconditional 0.036 0.649 12.6*** 0.031 0.880 4.8 0.031 0.832 5.3*
const'd uncon 0.037 0.649 12.5*** 0.032 0.881 4.7 0.031 0.832 5.3*
conditional 0.038 0.654 11.9*** 0.031 0.882 4.6 0.028 0.825 6.1**
const'd con 0.038 0.654 11.9*** 0.033 0.882 4.6 0.032 0.830 5.6*
hedging
strategy mean stdev %down mean stdev %down mean stdev %down
no hedge 0.030 0.914 0.0 0.035 0.924 0.0 0.032 1.192 0.0
half hedge 0.028 0.916 -0.1 0.027 0.936 -1.3 0.024 1.208** -1.3
full hedge 0.026 0.933 -2.1 0.020 0.975** -5.4* 0.018 1.241*** -4.1
unconditional 0.030 0.914 0.1 0.033 0.925 -0.1 0.033 1.191 0.1
const'd uncon 0.030 0.914 0.1 0.033 0.925 -0.1 0.031 1.192 0.0
conditional 0.032 0.895 2.2 0.040 0.894 3.3 0.044 1.135 4.8
const'd con 0.032 0.903 1.2 0.034 0.912 1.3 0.032 1.181 1.0
Australia Canada
GermanyUK Switzerland
Japan
4.2 Seven-asset portfolio
This sub-section repeats the above analysis for an equally weighted portfolio formed with all
7 stock markets and 6 currencies. The MVGARCH estimation is reported in Appendix B. The
hedge ratios are reported in Table 7. Similar to the results in the previous section, the
exposure in AUD, CAD and JPY should be over hedged. The exposure in EUR is also over
21
For example, %down for full hedge = (stdev no hedge - stdev full hedge)/stdev no hedge *100
22
hedged, but this is offset by a greater and negative hedge position in CHF. The exposure in
GBP is partially hedged. Campbell et al. (2010) also investigate unconditionally hedging the
same set of currencies for an equally-weighted seven-stock-market portfolio similar to our
seven-asset portfolio. They find that exposure to AUD, CAD, JPY and GBP should be over
hedged. And exposure to EUR and CHF should be increased. The difference in hedge ratio
for EUR could result from the fact that, exposure to EUR in their portfolio comes from
investing in a basket of European countries referred to as ‘Euroland’, which is replaced by
Germany in our study. Figure 2 shows the large unconstrained hedge positions taken in CAD,
CHF and EUR over the sample period. Hedge ratios are volatile even when they are bounded
between 0 and 1.
Table 7 Hedge ratios for seven-asset portfolio
This table presents both unconditional and conditional hedge ratios for equally-weighted seven-asset portfolios
over the sample period January 2002 to December 2005. Unconditional hedge ratios are generated by regressing
the unhedged portfolio return (in USD) onto the currency forward returns. Daily conditional hedge ratios are
computed using equation (2.8), with conditional covariance matrix estimated from the DCC-GARCH model.
AUD CAD JPY GBP CHF EUR
unconstrained 2.95 2.47 1.14 0.67 -3.47 1.19
constrained 1.00 1.00 1.00 0.67 0.00 1.00
unconstrained 2.26 2.18 1.36 1.00 -2.45 1.07
constrained 1.00 0.99 0.95 0.79 0.00 0.71
Panel A: Unconditional hedge ratios
Panel B: Average conditional hedge ratios
23
Figure 2 Conditional hedge ratios for seven-asset portfolio
0 200 400 600 800 1000 1200-5
0
5
10
conditio
nal hedge r
atios
0 200 400 600 800 1000 12000
0.2
0.4
0.6
0.8
1
days
constr
ain
ed c
onditio
nal hedge r
atios
AUD
CAD
JPY
GBP
CHF
EUR
AUD
CAD
JPY
GBP
CHF
EUR
The hedging results presented in Table 8 show that conditional hedging which reduces
portfolio risk by 14% is superior to all other hedging strategies in terms of risk reduction.
The full hedge realizes the least amount of risk reduction. Also, the performance of
various hedging strategies tends to improve as diversification increases. For example,
the full hedge helps in reducing risk for the seven-asset portfolio, even though it tends to
add risk to some less diversified two-asset portfolios we examined before. This is
attributable to increased currency risk due to the increase in the level of foreign
investments, which make currency hedging more beneficial. Also, increased
diversification provides more natural hedges among currencies given that correlations
between currencies are not perfect. The F-test result illustrates that conditional hedging
is statistically significantly better than other hedging strategies except for unconditional
hedging, and both conditional and unconditional hedging significantly reduce portfolio
risk. Besides, the effectiveness of hedging is greatly reduced when the hedge ratio is
constrained to lie between 0 and 1. This is true for both unconditional hedging and
conditional hedging.
24
Table 8 Hedging results comparison for seven-asset portfolio
This table presents mean and standard deviation of hedged portfolio returns under different hedging
strategies: No hedge (h=0), half hedge (h=0.5), full hedge (h=1), unconditional hedge (hedge ratio
generated by OLS regression), constrained unconditional hedge, conditional hedge (hedge ratio based on
dynamic conditional covariance matrix) and constrained conditional hedge. All constrained hedge ratios
are bounded by 0 and 1. The portfolio contains all seven stock indices. All calculations are based on daily
observations over the period January 2002 to December 2005. Mean and standard deviation are in
percentage terms. An F-test of equal variance (standard deviation) is performed for the conditional hedge
against every other hedge. In the “stdev” column, ***, ** and * respectively represent 1%, 5% and 10%
significance levels at which the null of equal variance can be rejected. The best performing hedge with the
lowest standard deviation is in bold. % down = percentage change in standard deviation relative to no
hedge. The statistical difference between no hedge and every other hedge is shown by the significance
indicators (*** etc) in the “%down” column.
hedging
strategy mean stdev % down
no hedge 0.054 0.812*** 0.0
half hedge 0.044 0.776*** 4.5
full hedge 0.035 0.790*** 2.8
unconditional 0.040 0.714 12.1***
const'd uncon 0.043 0.752** 7.4**
conditional 0.039 0.698 14.0***
const'd con 0.040 0.753** 7.3**
Aus-Can-Jap-UK-Swit-Ger-US
In summary, the conditional hedging strategy generally outperforms alternative hedging
strategies in terms of portfolio risk reduction within sample, though conditional hedging
is not found to be statistically significantly different from unconditional hedging in all
cases. We find that the hedge ratios for the same currencies differ across portfolios
depending on portfolio composition, and the size of the hedge positions increases as the
portfolio becomes more diversified. In addition, the performance of various hedging
strategies differs across portfolios. For instance, in the two-asset cases, conditional
hedging, unconditional hedging as well as the full hedge perform equally well for
portfolios containing Australia, Canada and Japan. While conditional hedging reduces
risk for portfolios containing Switzerland, Germany and the U.K., the full hedge
introduces more risk to all three portfolios. When it comes to the seven-asset portfolio,
all hedging strategies manage to reduce risk, and conditional hedging significantly
outperforms all hedging strategies except for unconditional hedging. Finally, restricting
the hedge ratios to lie between 0 and 1 significantly reduces the effectiveness of both the
25
conditional and unconditional hedging strategies.
5. The effect of portfolio weights on hedging performance
The analyses done in section 4 are based on the simple assumption that all equity
portfolios are equally weighted. But what is often observed in practice is that
international equity portfolios tend to have a home bias22
with high concentration in the
domestic equity market. In other cases, investors may choose to have more or less
weight in foreign investments for strategic reasons. This section is therefore devoted to
exploring the relationship between hedging effectiveness and the proportion of foreign
investments held in a portfolio. To keep the analysis simple, the weight in the US market
is set to vary between portfolios at 10% increments. Once the weight in the US is
determined for a particular portfolio, the remaining portfolio weight is then spread
equally among foreign stock markets. If we let WUS denote the portfolio weight in the
US market where WUS 0.1,0.2,...,0.9 , and let N denote the number of foreign
markets included in the portfolio. Then the weight in any foreign market is simply (1-
WUS)/N. Therefore for each asset combination explored, there are 9 portfolios23
that
have the same underlying assets but different portfolio weights in those assets.
5.1 Two-asset portfolios
We first examine the six two-asset combinations each invests in the US market and one
foreign market. Figure 3 shows the percentage risk reduction (same as %down from
section 4) under various hedging strategies relative to the unhedged portfolio risk. For
example, Figure 3(i) shows that for portfolio 1 which allocates 10% of the portfolio in
the U.S. and 90% in Australia, the half hedge can achieve about 20% risk reduction
relative to no hedge whereas other hedging strategies can realise about 30% risk
22
See French and Poterba (1991), Cooper & Kaplanis (1994), Tesar & Werner (1995), Strong & Xu
(2003), Li (2004) and Van Nieuwerburgh & Veldkamp (2009) among others. 23
For example, portfolio 1 of a certain asset combination has 10% of the portfolio invested in the US
market, and 90% of the portfolio value spread equally among the foreign markets. Similarly, portfolio 9
has 90% value invested in the US market and 10% value spread among foreign assets.
26
reduction. For portfolio 9 which has 90% of its investments made in the U.S. and only
10% in Australia, no hedging strategy seems to help reducing risk.
Figure 3 Portfolio allocation and hedging effectiveness (two-asset portfolios)
1 2 3 4 5 6 7 8 90
10
20
30
(i) Australia-U.S.
1 2 3 4 5 6 7 8 90
5
10
15
20(ii) Canada-U.S.
1 2 3 4 5 6 7 8 90
5
10
15(iii) Japan-U.S.
perc
enta
ge r
isk r
eduction
1 2 3 4 5 6 7 8 9-5
0
5
10(iv) U.K.-U.S.
1 2 3 4 5 6 7 8 9-5
0
5
10(v) Switzerland-U.S.
portfolio
1 2 3 4 5 6 7 8 9-4
-2
0
2
4(vi) Germany-U.S.
portfolio
no hedge
half hedge
full hedge
unconditional hedge
constrained unconditional hedge
conditional hedge
constrained conditional hedge
note on portfolio weights:
portfolio1: 10% in the U.S. and 90% in foreign markets
portfolio2: 20% in the U.S. and 80% in foreign markets
...
...
...
portfolio 9: 90% in the U.S. and 10% in foreign markets
From the graphs, it is apparent that hedging makes sense for Australia-U.S., Canada-U.S.
and Japan-U.S. combinations. All hedging strategies except for the 50% hedge are
almost equally effective in terms of risk reduction for these combinations under various
portfolio weights. Besides, risk reduction declines monotonically as the weight put in
27
the U.S. increases, this is true for all three asset combinations. The difference between
conditional hedging and no hedge becomes statistically insignificant when more than
50% of the portfolio is invested in the U.S.24
In contrast, conditional hedging outperforms all other hedging strategies for the
U.K.-U.S., Switzerland-U.S. and Germany-U.S. combinations. But the percentage risk
reduction is not as large as in the previous case, especially when the weight in the U.S.
increases to 70% or above, the amount of risk reduction reduces to merely 2-3% which
is not statistically significant. It is also worth noting that despite the strong performance
of the simple full hedge in the previous case, the strategy actually adds risk to the
portfolios regardless of the portfolio weights assumed (with the one exception of a
portfolio investing 70% or more in the U.K.), and a 50% hedge does not do much better.
This implies that the effectiveness of a naïve static hedging strategy can be greatly
affected by the currencies to which the portfolio has exposure.
5.2 Seven-asset portfolio
In this sub-section we examine the relationship between portfolio weights and hedging
effectiveness for portfolios composed of all stock markets and currencies. From Figure 4,
it is observable that conditional hedging is more effective in reducing portfolio risk than
alternative strategies. Though the effect of hedging diminishes as weight in the US
market increases, considerable risk reduction can still be realized under conditional
hedging for all portfolio compositions. At the extremes, a portfolio with 90% of its value
invested in foreign assets can reduce more than 17% of the portfolio risk using
conditional hedging. And a portfolio with only 10% invested in foreign assets is able to
achieve a risk reduction of about 4%, though this reduction is not statistically
significant.25
24
For all portfolios and various portfolio weights, an F-test of equal variance is performed for conditional
hedging against every other strategy, full results of the tests are available on request. 25
The risk reduction achieved by conditional hedging becomes statistically insignificant when 70% or
more of the portfolio is invested in the US stock market.
28
Figure 4 Portfolio allocation and hedging effectiveness (seven-asset portfolio)
1 2 3 4 5 6 7 8 9-5
0
5
10
15
20
portfolio
pe
rce
nta
ge
ris
k r
ed
uctio
n
no hedge
half hedge
full hedge
unconditional hedge
constrained unconditional hedge
conditional hedge
constrained conditional hedge
note on portfolio weights:
portfolio1: 10% in the U.S. and 90% in foreign markets
portfolio2: 20% in the U.S. and 80% in foreign markets
...
...
...
portfolio 9: 90% in the U.S. and 10% in foreign markets
In contrast, it is not worthwhile to implement the naïve half hedge or full hedge if less
than 35% of the portfolio is held in foreign currencies, as virtually no risk reduction is
provided by either hedging strategy. In fact a home biased portfolio is better off being
unhedged than fully hedged (half hedged) since hedging tends to add risk when more
than 67% (77%) of the portfolio is invested in the base currency. Moreover, the
constrained hedge ratios are less effective in reducing risk compared to the
unconstrained ones, and are only marginally better than the fixed hedge ratios (0.5 and
1). This is true for both the conditional and unconditional hedge ratios. Lastly,
unconditional hedging closely tracks the performance of conditional hedging for this
particular asset combination, and it is not statistically significantly different from
conditional hedging regardless of the level of foreign investment.
To conclude, conditional hedging is shown to consistently reduce risk for various asset
portfolios of various levels of foreign investments within sample, while the performance
29
of full hedge and half hedge differ significantly across portfolios. Nevertheless, as
variances and correlations of international stock markets are not constant over time, the
documented results in this section may not hold for other periods, especially during
market downturns which are characterized by abnormally high volatility and
correlations.
6. Out-of-sample test results
In order to compare the performance of conditional hedging with the other hedging
strategies out of sample, we make one-step-ahead forecasts of the covariance structure
of asset and currency returns using VAR and DCC-GARCH parameters estimated
within sample. Daily conditional hedge ratios for the out-of-sample period are then
calculated from the forecasted covariance matrix for various portfolios. Given that the
out-of-sample period (Jan 2006-Apr 2010) covers the 2008 global financial crisis (GFC),
the performance of the conditional hedging strategy could be affected by a potential
structural break in the data. Therefore the same analysis is also conducted for a
pre-crisis period Jan 2006 to Dec 2006 for comparison, as the crisis arguably started
around mid-2007.26
6.1 Pre-crisis period (Jan 2006 – Dec 2006) analysis
The hedging result for two-asset portfolios is presented in Table 9. It appears that no
hedging strategy is consistently better than the other strategies during this period.
Conditional hedging performs the best among all hedging strategies for three out of the
six portfolios examined, but it is outperformed by unconditional hedging for two
portfolios. For the portfolio containing the U.K., both conditional hedging and
unconditional hedging underperform a simple 100% hedge. However, the difference
between conditional hedging and alternative hedging strategies lack statistical
26
The GFC is commonly believed to have begun in mid-2007 as a result of losses on mortgage backed
securities (see Mizen, 2008, Caballero & Kurlat, 2009, Brunnermeier, 2009 among many others). For
example, In June 2007, two hedge funds run by Bear Stearns that had large holdings of subprime
mortgages sustained great losses and filed for bankruptcy.
30
significance. The difference between conditional hedging and no hedge is significant at
5% level only for the portfolio invested in Australia. These results are largely
comparable to the in-sample results for two-asset portfolios.
Table 9 Hedging results comparison for two-asset portfolios (2006)
This table reports mean and standard deviation of hedged portfolio returns under different hedging
strategies: No hedge (h=0), half hedge (h=0.5), full hedge (h=1), unconditional hedge (hedge ratio
generated by OLS regression within-sample), constrained unconditional hedge, conditional hedge (hedge
ratio estimated with forecasts of dynamic conditional covariance matrix based on in-sample
VAR-DCC-GARCH parameters) and constrained conditional hedge. The constrained hedge ratios are
bounded by 0 and 1. Each portfolio contains the US stock index and the index of a foreign country with
equal weights, the first row of the table indicates with which foreign country’s stock index the portfolio is
formed. All calculations are based on daily return observations over the period January 2006 to December
2006. Mean and standard deviation are in percentage terms. An F-test of equal variance (standard
deviation) is performed for the conditional hedge against every other hedge. In the “stdev” column, ***,
** and * respectively represent 1%, 5% and 10% significance levels at which the null of equal variance
can be rejected. The best performing hedge with the lowest standard deviation is in bold. %down =
percentage change in standard deviation relative to no hedge. The statistical difference between no hedge
and every other hedge is shown by the significance indicators (*** etc) in the “%down” column.
hedging
strategy mean stdev %down mean stdev %down mean stdev %down
no hedge 0.082 0.623** 0.0 0.061 0.728 0.0 0.043 0.763 0.0
half hedge 0.075 0.569 8.7 0.060 0.688 5.5 0.038 0.717 6.0
full hedge 0.069 0.541 13.1** 0.058 0.668 8.2 0.034 0.693 9.2
unconditional 0.067 0.540 13.4** 0.058 0.667 8.4 0.034 0.694 9.1
const'd uncon 0.069 0.541 13.1** 0.058 0.668 8.2 0.034 0.694 9.1
conditional 0.066 0.544 12.7** 0.056 0.674 7.4 0.039 0.691 9.4
const'd con 0.068 0.543 12.9** 0.056 0.669 8.2 0.038 0.695 8.9
hedging
strategy mean stdev %down mean stdev %down mean stdev %down
no hedge 0.080 0.650 0.0 0.076 0.652 0.0 0.089 0.758 0.0
half hedge 0.067 0.619 4.9 0.064 0.616 5.4 0.076 0.724 4.5
full hedge 0.054 0.612 5.9 0.053 0.611 6.3 0.063 0.708 6.5
unconditional 0.074 0.631 3.0 0.074 0.646 0.9 0.094 0.775 -2.3
const'd uncon 0.074 0.631 3.0 0.074 0.646 0.9 0.089 0.758 0.0
conditional 0.061 0.615 5.4 0.060 0.608 6.7 0.068 0.707 6.7
const'd con 0.061 0.615 5.4 0.060 0.608 6.7 0.069 0.708 6.5
Australia Canada
GermanyUK Switzerland
Japan
Now we shift our focus to a portfolio consists of all seven stock markets. Table 10
documents the performance of the hedging strategies for the pre-crisis period. All
strategies help reduce portfolio risk which, in comparison with the two-asset portfolios,
31
highlights the effect of multicurrency diversification on the performance of hedging
strategies out of sample. Moreover, conditional hedging manages to lower the risk of
the portfolio by 22.2%, higher than the risk reduction achieved by any other strategy.
However, conditional hedging is only statistically significantly better than no hedge, but
not other hedging strategies. Also, putting a constraint on the hedge ratio leads to less
favourable hedging results.
Table 10 Hedging results comparison for seven-asset portfolio (2006)
This table presents mean and standard deviation of hedged portfolio returns under different hedging
strategies: No hedge (h=0), half hedge (h=0.5), full hedge (h=1), unconditional hedge (hedge ratio
generated by OLS regression within-sample), constrained unconditional hedge, conditional hedge (hedge
ratio estimated with forecasts of dynamic conditional covariance matrix based on in-sample
VAR-DCC-GARCH parameters) and constrained conditional hedge. The constrained hedge ratios are
bounded by 0 and 1. The portfolio contains all seven stock indices. All calculations are based on daily
observations over the period January 2006 to December 2006. Mean and standard deviation are in
percentage terms. An F-test of equal variance (standard deviation) is performed for the conditional hedge
against every other hedge. In the “stdev” column, ***, ** and * respectively represent 1%, 5% and 10%
significance levels at which the null of equal variance can be rejected. The best performing hedge with
the lowest standard deviation is in bold. %down = percentage change in standard deviation relative to no
hedge. The statistical difference between no hedge and every other hedge is shown by the significance
indicators (*** etc) in the “%down” column.
hedging
strategy mean stdev % down
no hedge 0.084 0.728*** 0.0
half hedge 0.070 0.644 11.5***
full hedge 0.055 0.603 17.2***
unconditional 0.077 0.608 16.5***
const'd uncon 0.073 0.629 13.5***
conditional 0.061 0.566 22.2***
const'd con 0.067 0.614 15.7***
Aus-Can-Jap-UK-Swit-Ger-US
To sum up, for the year 2006, conditional hedging does not consistently outperform
other hedging strategies for the two-asset portfolios, while it continues to dominate
other strategies for the seven-asset portfolio. However, in general, the difference
between conditional hedging and alternative hedging strategies lacks statistical
significance. These results are generally comparable to the in-sample results. The
similarity in the results could be due to similar market conditions shared by the
32
pre-crisis period and the in-sample period, thus we do not expect much change in asset
correlations and volatilities in 2006.
6.2 Analysis for the period covering the crisis (Jan 2006-Apr 2010)
Section 6.1 documents some important results for the out-of-sample performance of
conditional hedging under normal market condition. It is interesting to see how the
strategy holds up during the recent global financial crisis. Table 11 reports the hedging
performance of various strategies for two-asset portfolios during the period covering the
GFC. In contrast to the result for the pre-crisis period, conditional hedging consistently
outperforms other hedging strategies for all six portfolios. Conditional hedging is
statistically significantly better than unconditional hedging for portfolios containing
Japan, the U.K. and Germany. It is also statistically significantly better than both no
hedge and half hedge for four of the portfolios. Similar to the result for the in-sample
period, all hedging strategies manage to reduce risk for portfolios containing Australia
and Canada. However, this time the risk reductions achieved by all hedging strategies
have strong statistical significance.
It is interesting to note that all hedging strategies increase the risk of the portfolio
containing Japan over this period with the exception of conditional hedging, which
reduces the portfolio risk by 4.3%. Although conditional hedging is not statistically
significantly better than no hedge, it is significantly better than half hedge, full hedge
and unconditional hedging at 5% level.
From undocumented descriptive statistics of the data, JPY strengthened27
against USD
during this period and had a correlation of -0.28 with the US stock market during this
period. Its correlation with the Japanese stock market also dropped significantly
compared to the in-sample period. This helps explain why the static and unconditional
27
There are two possible explanations for the movements in the yen exchange rate. One is yen’s ‘safe
haven’ status that causes international flight to quality during the crisis. The other is the unwinding of
currency carry trades that use yen as the funding currency (e.g. Kohler, 2010; Batini & Dowling, 2011).
33
hedging strategies that try to reduce the exposure to JPY end up adding risk to the
portfolio. It also emphasises the value of conditional hedging especially during the
times of market turbulence, as conditional hedging constantly adjusts the hedge ratios as
new market data becomes available.
Table 11 Hedging results comparison for two-asset portfolios (2006-2010)
This table presents mean and standard deviation of hedged portfolio returns under different hedging
strategies: No hedge (h=0), half hedge (h=0.5), full hedge (h=1), unconditional hedge (hedge ratio
generated by OLS regression within-sample), constrained unconditional hedge, conditional hedge (hedge
ratio estimated with forecasts of dynamic conditional covariance matrix based on in-sample
VAR-DCC-GARCH parameters) and constrained conditional hedge. The constrained hedge ratios are
bounded by 0 and 1. Each portfolio contains the US stock index and the index of a foreign country with
equal weights, the first row of the table indicates with which foreign country’s stock index the portfolio is
formed. All calculations are based on daily observations over the period January 2006 to April 2010.
Mean and standard deviation are in percentage terms. An F-test of equal variance (standard deviation) is
performed for the conditional hedge against every other hedge. In the “stdev” column, ***, ** and *
respectively represent 1%, 5% and 10% significance levels at which the null of equal variance can be
rejected. The best performing hedge with the lowest standard deviation is in bold. %down = percentage
change in standard deviation relative to no hedge. The statistical difference between no hedge and every
other hedge is shown by the significance indicators (*** etc) in the “%down” column.
hedging
strategy mean stdev %down mean stdev %down mean stdev %down
no hedge 0.037 1.451*** 0.0 0.032 1.643*** 0.0 0.006 1.144 0.0
half hedge 0.033 1.268*** 12.6*** 0.027 1.551*** 5.6* -0.003 1.174 -2.6**
full hedge 0.029 1.123 22.6*** 0.023 1.479 10.0*** -0.011 1.231*** -7.6**
unconditional 0.028 1.102 24.1*** 0.023 1.467 10.7*** -0.011 1.227*** -7.2**
const'duncon 0.029 1.123 22.6*** 0.023 1.479 10.0*** -0.011 1.227*** -7.2**
conditional 0.023 1.080 25.6*** 0.018 1.399 14.8*** 0.016 1.095 4.3
const'd con 0.024 1.125 22.4*** 0.022 1.479 10.0*** 0.004 1.141 0.2
hedging
strategy mean stdev %down mean stdev %down mean stdev %down
no hedge 0.017 1.499*** 0.0 0.019 1.314 0.0 0.023 1.534*** 0.0
half hedge 0.019 1.416** 5.5* 0.012 1.289 1.9 0.020 1.472** 4.0
full hedge 0.022 1.354 9.7*** 0.005 1.289 1.9 0.016 1.426 7.0**
unconditional 0.018 1.454*** 3.0 0.018 1.310 0.3 0.025 1.561*** -1.8
const'd uncon 0.018 1.454*** 3.0 0.018 1.310 0.3 0.023 1.534*** 0.0
conditional 0.026 1.304 13.0*** 0.000 1.275 2.9 0.001 1.387 9.6***
const'd con 0.020 1.358 9.4*** 0.007 1.272 3.1 0.013 1.429 6.9**
Australia Canada
GermanyUK Switzerland
Japan
Figure 5 illustrates how the estimated conditional correlations and hedge ratio vary over
the out-of-sample period for the two-asset portfolio containing Japan. The correlation
34
between JPY and the US stock market declines gradually over this period and is
negative for most of the period. On the other hand, there is a sudden drop in the
correlation between JPY and the Japanese stock market during the second half of year
2008, when the GFC intensified. These results are consistent with the aforementioned
unconditional correlations estimated for this period. The conditional hedge ratio for JPY
changes accordingly over the period. It plummets in the second half of 2008 as a
reflection of the drastic changes in the correlations during that time. These results
demonstrate the ability of conditional hedging to adapt to the changes in market
conditions especially asset correlations during the GFC.28
It also points out the fact that
the performance of static and unconditional hedging strategies is sample-specific and is
very sensitive to changes in market conditions.
28
The DCC-GARCH model forecasts dramatic changes in correlations among all stock markets and
currencies during the second half of 2008. Most of the correlations increased during the GFC with a few
exceptions like the correlations between JPY and foreign stock markets.
35
Figure 5 Conditional correlations and hedge ratio for portfolio: Japan-U.S.
0 200 400 600 800 1000 1200-3
-2
-1
0
1
2co
nditi
onal
hed
ge r
atio
s
0 200 400 600 800 1000 1200-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
days
cond
ition
al c
orre
latio
ns
Japan (stock)-JPY (currency)
U.S. (stock)-JPY (currency)
Japan (stock)-U.S. (stock)
end-2007 end-2008
All in all, in the case of two-asset portfolios, there is clear evidence of the superior
ex-ante performance of the conditional hedging strategy relative to other strategies
during the period covering the GFC. In many cases, the dominance of conditional
hedging is statistically significant.
The hedging result presented in Table 12 for a seven-asset portfolio shows that all
hedging strategies manage to reduce portfolio risk when the portfolio is more
diversified. This is comparable to the result documented for the pre-crisis period as well
as the in-sample period, and serves to re-emphasise the importance of diversification for
its impact on hedging effectiveness of all the strategies out of sample. In addition,
conditional hedging generates an impressive portfolio risk reduction of 35.3%, which is
significantly better than the risk reduction of 10% and 17.6% achieved with the half
36
hedge and full hedge respectively, despite the fact that static hedging is more widely
used in practice. Conditional hedging also outperforms unconditional hedging, although
the difference between the two is not statistically significant. Moreover, it appears that
the success of conditional and unconditional hedging comes from their ability to take
large hedge positions that are often a few times the underlying exposure. When the
hedge ratios are constrained to lie between 0 and 1, both the conditional and
unconditional hedging strategies come close to a simple full hedge.
Table 12 Hedging results comparison for seven-asset portfolio (2006-2010)
This table presents mean and standard deviation of hedged portfolio returns under different hedging
strategies: No hedge (h=0), half hedge (h=0.5), full hedge (h=1), unconditional hedge (hedge ratio
generated by OLS regression within-sample), constrained unconditional hedge, conditional hedge (hedge
ratio estimated with forecasts of dynamic conditional covariance matrix based on in-sample
VAR-DCC-GARCH parameters) and constrained conditional hedge. The constrained hedge ratios are
bounded by 0 and 1. The portfolio contains all seven stock indices. All calculations are based on daily
observations over the period January 2006 to April 2010. Mean and standard deviation are in percentage
terms. An F-test of equal variance (standard deviation) is performed for the conditional hedge against
every other hedge. In the “stdev” column, ***, ** and * respectively represent 1%, 5% and 10%
significance levels at which the null of equal variance can be rejected. The best performing hedge with
the lowest standard deviation is in bold. %down = percentage change in standard deviation relative to no
hedge. The statistical difference between no hedge and every other hedge is shown by the significance
indicators (*** etc) in the “%down” column.
hedging
strategy mean stdev % down
no hedge 0.028 1.378*** 0.0
half hedge 0.021 1.240** 10.0**
full hedge 0.014 1.136 17.6***
unconditional 0.022 0.970 29.6***
const'd uncon 0.019 1.204** 12.6**
conditional 0.030 0.891 35.3***
const'd con 0.022 1.108 19.6***
Aus-Can-Jap-UK-Swit-Ger-US
In summary, we find strong evidence that conditional hedging dominates the alternative
hedging strategies ex ante. This result is more pronounced for the period covering the
GFC, when hedging is needed the most. The findings reinforce our in-sample results
and confirm the superiority of conditional hedging over static and unconditional
hedging strategies.
37
7. Conclusion
This paper investigates the practice of currency hedging for international stock
portfolios for the purpose of risk-minimization. The performance of various hedging
strategies are examined for a number of portfolio compositions both in-sample and
out-of-sample. Our data spans the period January 2002 to April 2010. The in-sample
period is from 2002 to end 2005 and the out-of-sample period is from 2006 to 2010.
Our focus is on investigating whether a conditional hedging strategy is better at
achieving a hedger’s objective compared to unconditional hedging and static strategies
such as no hedge, half hedge and full hedge. Our aim is to ascertain if the conditional
hedging strategy that relies heavily on time-series modeling of conditional mean and
covariance structure of the returns provides consistent benefit for hedging and if it is
worth implementing in practice.
For the purpose of risk minimization, conditional hedging generally outperforms
alternative hedging strategies in terms of portfolio risk reduction within sample, though
conditional hedging is not found to be statistically significantly different from
unconditional hedging in all cases. We find that the hedge ratios for the same currencies
differ across portfolios depending on portfolio composition, and the performance of
various hedging strategies also differs across portfolios. In general, the full hedge is
found to be undesirable for CHF, EUR and GBP though highly effective for AUD, CAD
and JPY. Such a strategy adds risk to two-asset portfolios that have exposure to the
European currencies, and is dominated by conditional and unconditional hedging for a
seven-asset portfolio. Portfolio managers in practice often are prohibited from taking
speculative currency positions when hedging. We find that restricting the hedge ratios to
lie between 0 and 1 significantly reduces the effectiveness of both conditional and
unconditional hedging strategies.
Although we assume all portfolios are equally weighted, we are aware of the home-bias
of international equity portfolios in practice. Therefore the relationship between
38
hedging effectiveness and the proportion of foreign investments held in a portfolio is
also explored in the study by varying the fraction of portfolios allocated to foreign
assets. Over the period 2002-2005, the conditional hedging strategy is shown to
consistently dominate an unconditional hedging strategy, no hedge, half hedge and full
hedge for all portfolio combinations and also for different level of foreign investments.
We also show that the full hedge tends to add risk to portfolios containing the Swiss, the
German and the UK stock markets for various levels of foreign investments. This result
confirms that our findings for equally weighted portfolios apply to other portfolio
allocations as well.
Out of sample, conditional hedging outperforms alternative strategies over the pre-crisis
period for most of the portfolios considered. However in all cases, conditional hedging
is not statistically significantly different from the other hedging strategies except for no
hedge. Over the period covering the GFC, there is clear evidence of the consistent
dominance of conditional hedging over alternative hedging strategies. The dominance
in many cases has strong statistical significance. The outstanding ex-ante performance
of conditional hedging reinforces our in-sample results and confirms the superiority of
conditional hedging over static and unconditional hedging strategies. The exceptional
performance of conditional hedging over the GFC also highlights the ability of
conditional hedging to adapt to changes in market conditions and effectively adjust
conditional hedge ratios. Our findings confirm the benefit of implementing a
conditional strategy such as the one employed in our study, and raise questions about
the common practice of adopting the naïve static hedging strategies such as full hedge
and half hedge.
Finally, we summarize below the limitations of the current study and their implications
for future research:
1) The DCC model restricts all correlation processes to have the same dynamic
structure, which is undesirable for large systems. Various generalizations of the
DCC model can be used to allow for more flexible correlation structure, but this
39
is usually at the cost of lower estimation efficiency.
2) The out-of-sample forecasts of mean, variances and covariance are based on
fixed parameters estimated within sample. Updating (re-estimating) the model
parameters during the out-of-sample period as new data become available could
produce more relevant results for the period.
3) The analyses done in the thesis assume daily rebalancing of hedging positions
and no transactions costs. Future study on conditional hedging relative to less
costly hedging strategies could incorporate transactions costs for a more
accurate comparison.
4) The study examines only seven developed countries for portfolio diversification
and hedging. These countries are all in a relatively mature stage of their
economic development, their economies are highly co-integrated and are likely
to be affected in a similar way by common shocks to the global economy. Future
study can include in the investment set emerging markets, whose economies are
less correlated with that of the developed markets, and such investments will
provide benefit in diversification and possibly in hedging performance.
40
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Appendix A Estimated DCC GARCH parameters for two-asset portfolios
Conditional standard deviation is estimated by univariate GARCH model, and a set of GARCH
parameters ω,α,β are returned for each serie. δ and η are DCC parameters from the conditional
correlation estimation. The standard error of each parameter is reported next to the parameter.
Australia std errors Canada std errors Japan std errors
ω 3.04E-06 8.49E-07 6.32E-06 3.48E-06 1.19E-05 6.88E-06
α 0.0151 0.0112 0.0839 0.0222 0.1081 0.0282
β 0.9433 0.0178 0.8487 0.0530 0.8162 0.0627
USA USA USA
ω 1.16E-06 5.57E-07 1.12E-06 5.48E-07 1.05E-06 5.47E-07
α 0.0855 0.0143 0.0874 0.0140 0.0889 0.0149
β 0.9048 0.0160 0.9036 0.0157 0.9035 0.0165
AUD CAD JPY
ω 1.14E-06 5.38E-07 5.57E-07 2.66E-07 5.80E-06 4.64E-06
α 0.0347 0.0100 0.0558 0.0128 0.0143 0.0458
β 0.9389 0.0190 0.9242 0.0187 0.8170 0.1111
δ 0.0202 0.0050 0.0178 0.0056 0.0246 0.0102
η 0.9697 0.0098 0.9715 0.0095 0.9512 0.0242
UK std errors Switzerland std errors Germany std errors
ω 1.77E-06 8.58E-07 2.32E-06 1.04E-06 2.28E-06 1.29E-06
α 0.0870 0.0206 0.0880 0.0190 0.1022 0.0263
β 0.8935 0.0270 0.8914 0.0239 0.8886 0.0292
USA USA USA
ω 1.36E-06 7.71E-07 1.21E-06 5.80E-07 1.09E-06 5.57E-07
α 0.0899 0.0169 0.0831 0.0136 0.0868 0.0141
β 0.8991 0.0208 0.9066 0.0157 0.9048 0.0158
GBP CHF EUR
ω 8.86E-07 4.39E-07 1.46E-06 6.40E-07 7.69E-07 1.33E-06
α 0.0591 0.0195 0.0186 0.0102 0.0264 0.0013
β 0.9086 0.0281 0.9469 0.0184 0.9505 0.0023
δ 0.0158 0.0046 0.0188 0.0058 0.0139 0.0034
η 0.9805 0.0069 0.9788 0.0072 0.9833 0.0047
44
Appendix B Estimated DCC GARCH parameters for 7-asset portfolio
Conditional standard deviation is estimated by univariate GARCH model, and a set of GARCH
parameters ω,α,β are returned for each serie. δ and η are DCC parameters from the conditional
correlation estimation. The standard error of each parameter is reported next to the parameter.
Australia std errors AUD std errors
ω 4.09E-05 8.66E-06 3.42E-07 3.33E-07
α 0.0467 0.0156 0.0275 0.0092
β 0.3810 0.1256 0.9640 0.0141
Canada CAD
ω 0.0000 0.0000 1.46E-07 2.37E-07
α 0.0672 0.0218 0.0365 0.0101
β 0.8975 0.0531 0.9584 0.0156
Japan JPY
ω 0.0000 0.0000 0.0000 0.0000
α 0.0801 0.0249 0.0144 0.0017
β 0.8853 0.0445 0.9096 0.0021
UK GBP
ω 1.07E-06 8.31E-07 4.39E-07 1.96E-07
α 0.0701 0.0206 0.0470 0.0129
β 0.9166 0.0264 0.9373 0.0149
Switzerland CHF
ω 8.69E-07 8.01E-07 1.68E-06 1.65E-06
α 0.0539 0.0141 0.0326 0.0053
β 0.9372 0.0175 0.9283 0.0031
Germany EUR
ω 0.0000 0.0000 7.30E-07 1.28E-06
α 0.0642 0.0163 0.0310 0.0018
β 0.9292 0.0175 0.9472 0.0018
USA
ω 0.0000 0.0000
α 0.0518 0.0134
β 0.9434 0.0151
δ 0.0080 8.69E-04
η 0.9842 0.0018