Dynamic currency hedging for international stock portfolios · The paper studies dynamic currency...

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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.

mailto:[email protected]

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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

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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.

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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.

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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.

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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)

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22

0 0

0 0

0 0

t

kk

q

qQ

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

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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).

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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

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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

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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


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


...

...

...

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:



...

...

...

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


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


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


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


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)






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



hedging


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***


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)






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


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


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


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)






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



hedging


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***


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

References

Batini, N., and T. Dowling, 2011, Interpreting currency movements during the crisis:

What is the role of interest differentials? IMF working paper WP/11/14.

Black, F., 1989, Universal hedging: optimizing currency risk and reward in international

equity portfolios, Financial Analysts Journal 45, 16-22.

Black, F., 1990, Equilibrium exchange rate hedging, Journal of Finance 45, 899-907.

Bolleslev, T., R. Chou, and K. Kroner, 1992, ARCH modeling in finance: A review of

the theory and empirical evidence, Journal of Econometrics 52, 5-59.

Brunnermeier, M.K., 2009, Deciphering the liquidity and credit crunch 2007-2008,

Journal of Economic Perspectives 23, 77-100.

Caballero, R. J., and P. D. Kurlat, 2009, The ‘surprising’ origin and nature of financial

crises: A macroeconomic policy proposal, MIT Department of Economics working

paper No. 09-24.

Campbell, J., K. de Medeiros, and L. Viceira, 2010, Global currency hedging, Journal

of Finance LXV, 87-122.

Clarke, R., and M. Kritzman, 1996, Currency management: Concepts and practices, the

Research Foundation of the Institute of Chartered Financial Analysts.

Cooper, I., and E. Kaplanis, 1994, Home bias in equity portfolios, inflation hedging,

and international capital market equilibrium, The Review of Financial Studies 7, 45-60.

Engle, R., and K. Sheppard, 2001, Theoretical and empirical properties of dynamic

conditional correlation multivariate GARCH. NBER Working Paper 8554.

Engle, R., 2002, Dynamic conditional correlation: a simple class of multivariate

generalized autoregressive conditional heteroskedasticity model. Journal of Business

and Economic Statistics 20, 339-350.

Eun, C., and B. Resnick, 1988, Exchange rate uncertainty, forward contracts and

international portfolio selection, Journal of Finance 43, 197-216.

Froot, K., 1993, Currency hedging over long horizons, NBER Working Paper 4355.

French, C., and J. Poterba, 1991, Investor diversification and international equity

markets, The American Economic Review 81, 222-226.

41

Gagnon, L., T., McCurdy, and G., Lypny, 1998, Hedging foreign currency portfolios,

Journal of Empirical Finance 5, 197-220.

Galati, G., and M. Melvin, 2004, Why has FX trading surged? Explainning the 2004

triennial survey, BIS Quarterly Review, December 67-74.

Gardner, G., and D. Stone, 1995, Estimating currency hedge ratios for international

portfolios, Financial Analysts Journal 51, 58-64.

Gardner, G., and T. Wuilloud, 1995, Currency risk in international portfolios: How

satisfying is optimal hedging? Journal of portfolio management 21, 59-67.

Gastineau, G., 1995, The currency hedging decision: A search for synthesis in asset

allocation, Financial Analysis Journal 51, 8-17.

Glen, J., and P. Jorion, 1993, Currency hedging for international portfolios, Journal of

Finance 48, 1865-1886.

Guo, B., 2003, Currency risk hedging with time-varying correlations, SSRN working

paper.

Hautsch, N., and J. Inkmann, 2003, Optimal hedging of the currency exchange risk

exposure of dynamically balanced strategic asset allocations, Journal of Asset

Management 4, 173-189.

Hazuka, T., and L. Huberts, 1994, A valuation approach to currency hedging, Financial

Analysts Journal 50, 55-59.

Jorion, P., 1985, International portfolio diversification with estimation risk, The Journal

of Business 58, 259-278.

Jorion, P., 1994, Mean/variance analysis of currency overlays, Financial Analysts

Journal 50, 48-56.

Kohler, M., 2010, Exchange rates during financial crisis, BIS Quarterly Review, March,

39-50.

Kroner, K., and J. Sultan, 1993, Time-varying distributions and dynamic hedging with

foreign currency futures, The Journal of Financial and Quantitative Analysis 28,

535-551.

Larsen, G., and B. Resnick, 2000, The optimal construction of internationally

diversified equity portfolios hedged against exchange rate uncertainty, European

Financial Management 6, 479-514.

42

Li, K., 2004, Confidence in the familiar: An international perspective, Journal of

Financial and Quantitative Analysis 39, 47-68.

Longin, F., and B. Solnik, 1995, Is the correlation in international equity returns

constant: 1960-1990? Journal of International Money and Finance 14, 3-26.

McCauley, R.N., and P. McGuire, 2009,Dollar appreciation in 2008: safe haven, carry

trades, dollar shortage and overhedging, BIS Quarterly Review, December 85-93.

Michenaud, S., and B. Solnik, 2008, Applying regret theory to investment choices:

Currency hedging decisions, Journal of International Money and Finance 27, 677-694.

Mizen, P., 2008, The credit crunch of 2007-2008: A discussion of the background,

market reactions and policy responses, Federal Reserve Bank of St. Louis Review 90,

531-367.

Perold, A., and E. Schulman, 1988, The free lunch in currency hedging: Implications for

investment policy and performance standards, Financial Analysts Journal 44, 45-50.

Sheedy, E., 1998, Correlation in currency markets: A risk-adjusted perspective, Journal

of Financial Markets, Institutions and Money 8, 59-82.

Strong, N., and X. Xu, 2003, Understand the equity home bias: Evidence from survey

data, The review of Economics and Statistics 85, 307-312.

Tesar, L. L., and I. M. Werner, 1995, Home bias and high turnover, Journal of

International Money and Finance 14, 467–492.

Tse, Y., and A. Tsui, 2002, A multivariate generalized autoregressive conditional

heteroskedasticity model with time-varying correlations, Journal of Business and

Economics Statistics 20, 351-362.

Van Nieuwerburgh, S., and L. Veldkamp, 2009, Information immobility and the home

bias puzzle, The Journal of Finance 64, 1187-1215.

Xin, H., 2003, Currency overlay, a practical guide (risk books, Incisive Financial

Publishing Ltd)

43

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

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