Is gold a safe haven or a hedge for the US dollar implications for risk management

For a wide set of currencies, our empirical evidence revealed 1 positive and significant average dependence between gold and USD depreciation, consistent with the fact that gold can act a

Is gold a safe haven or a hedge for the US dollar? Implications for risk

management

Universidade de Santiago de Compostela, Departmento de Fundamentos del Análisis Económico, Avda Xoán XXIII, s/n, 15782 Santiago de Compostela, Spain

a r t i c l e i n f o

Article history:

Received 5 December 2012

Accepted 23 March 2013

Available online 18 April 2013

JEL classification:

C52

C58

F3

G1

Keywords:

Gold

Exchange rates

Hedge

Safe haven

Copulas

a b s t r a c t

We assess the role of gold as a safe haven or hedge against the US dollar (USD) using copulas to charac-terize average and extreme market dependence between gold and the USD For a wide set of currencies, our empirical evidence revealed (1) positive and significant average dependence between gold and USD depreciation, consistent with the fact that gold can act as hedge against USD rate movements, and (2) symmetric tail dependence between gold and USD exchange rates, indicating that gold can act as an effective safe haven against extreme USD rate movements We evaluate the implications for mixed gold-currency portfolios, finding evidence of diversification benefits and downside risk reduction that confirms the usefulness of gold in currency portfolio risk management

Ó 2013 Elsevier B.V All rights reserved

1 Introduction

For many years strengthened gold prices in combination with

US dollar (USD) depreciation has attracted the attention of

inves-tors, risk managers and the financial media The fact that when

the USD goes down as gold goes up suggests the possibility of using

gold as a hedge against currency movements and as a safe-haven

Some studies have examined the usefulness of gold as a hedge

against inflation (Chua and Woodward, 1982; Jaffe, 1989; Ghosh et al.,

2004; McCown and Zimmerman, 2006; Worthington and Pahlavani,

2007; Tully and Lucey, 2007; Blose, 2010; Wang et al., 2011and

refer-ences therein), whereas other studies have examined gold’s safe-haven

status with respect to stock market movements (Baur and McDermott,

2010; Baur and Lucey, 2010; Miyazaki et al., 2012) and oil price changes

(Reboredo, 2013a).2However, few studies have considered the role of

gold as hedge or investment safe haven against currency depreciation Beckers and Soenen (1984) studied gold’s attractiveness for investors and its hedging properties, finding asymmetric risk diversification for gold’s holding positions for US and non-US investors.Sjasstad and Scaccia-villani (1996) and Sjasstad (2008)found that currency appreciations or depreciations had strong effects on the price of gold.Capie et al (2005) confirmed the positive relationship between USD depreciation and the price of gold, making gold an effective hedge against the USD More re-cently,Joy (2011)analysed whether gold could serve as a hedge or an investment safe haven, finding that gold has been an effective hedge but a poor safe haven against the USD This paper contributes in two ways

to the existing literature on gold as a hedge and/or safe haven against cur-rency depreciation

First, we study the dependence structure for gold and the USD by using copula functions, which provide a measure of both average dependence and upper and lower tail dependence (joint extreme movements) This information is crucial in determining gold’s role

as a hedge or an investment safe haven, provided the distinction be-tween a hedge and safe-haven asset is made in terms of dependence under different market circumstances (see, e.g.,Baur and McDermott, 2010; Joy, 2011) Previous studies have examined the behavior of the correlation coefficient between gold and the USD exchange rate (Joy,

2011), but only provide an average measure of dependence Other studies have examined the marginal effects of stock prices on gold prices using a threshold regression model, with the threshold given

0378-4266/$ - see front matter Ó 2013 Elsevier B.V All rights reserved.

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E-mail address: juancarlos.reboredo@usc.es

1

Pukthuanthong and Roll (2011) showed that the price of gold is related with

currency depreciation in every country O’Connor and Lucey (2012) analyse the

negative correlation between returns for gold and traded-weighted exchange returns

for the dollar, yen and euro.

2

Other studies analyse the relationship between gold, oil and exchange rates (see,

e.g., Sari et al., 2010; Kim and Dilts, 2011; Malliaris and Malliaris, 2013 ) and between

these variables and interest rates ( Wang and Chueh, 2013 ).

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Journal of Banking & Finance

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / j b f

by a specific quantile of the stock returns distribution (Baur and

McDermott, 2010; Baur and Lucey, 2010; Wang and Lee, 2011; Ciner

et al., 2012); however, the correlation coefficient is insufficient to

de-scribe the dependence structure (Embrechts et al., 2003)—especially

when the joint distribution of gold and exchange rates is far from

the elliptical distribution—and the marginal effects captured by the

threshold regression do not fully account for joint extreme market

movements Therefore, we propose the use of copulas to test gold’s

hedge and safe-haven ability, as they fully describe the dependence

structure and allow more modeling flexibility than parametric

bivar-iate distributions

Second, since knowledge of gold and USD co-movement is useful for

portfolio managers who want portfolio diversification and investment

protection against downside risk, we investigated the implications of

gold-USD market average and tail dependence for risk management by

comparing the risk for gold-USD portfolio holdings to the risk for simple

currency portfolios We also evaluated whether an investor could achieve

downside risk gains from a portfolio composed of gold and currency by

studying the value-at-risk (VaR) performance

Our empirical study of the hedge and safe-haven properties of

gold against USD exchange rates covered the period January

2000–September 2012 and evaluated the USD exchange rate with

a wide set of currencies and a USD exchange rate index We

mod-eled marginal distributions with an autoregressive moving average

(ARMA) model with threshold generalized autoregressive

condi-tional heteroskedasticity (TGARCH) errors and different copula

models with tail independence, symmetric and asymmetric tail

dependence We provide empirical evidence of positive average

dependence and symmetric tail dependence between gold and

USD depreciation, with the Student-t copula as the best performing

dependence model This evidence is consistent with the role of

gold as a hedge and safe-haven asset against currency movements

We also address the risk management consequences of the links

between gold and USD depreciation, providing evidence for gold’s

usefulness in a currency portfolio—given that it shows evidence of

hedging effectiveness in reducing portfolio risk—and for a VaR

reduction and better performance in terms of the investor’s loss

function with respect to a portfolio composed only of currency

The rest of the paper is laid out as follows: in Section2we

out-line the methodology and test our hypothesis In Sections3 and 4

we present data and results, respectively, and we discuss the

impli-cations in terms of portfolio risk management in Section5 Finally,

Section6concludes the paper

2 Methodology

The role of gold as a hedge or safe haven with respect to

cur-rency movements depends on how gold and curcur-rency price

changes are linked under different market circumstances

Baur and Lucey (2010) and Baur and McDermott (2010), the

dis-tinctive feature of an asset as a hedge or safe haven is as follows:

– Hedge: an asset is a hedge if it is uncorrelated or negatively

cor-related with another asset or portfolio on average

– Safe haven: an asset is a safe haven if it is uncorrelated or

neg-atively correlated with another asset or portfolio in times of

extreme market movements

The crucial distinction between the two is whether dependence

holds on average or under extreme market movements.3To

distin-guish between hedge and safe-haven properties we need to measure dependence between two or more random variables in terms of aver-age movements across marginals and in terms of joint extreme market movements

We used copulas to flexibly model the joint distribution of gold and the USD and then linked information on average and tail dependence arising from copulas to the hedge and safe-haven properties of gold against the USD A copula4 is a multivariate cumulative distribution function with uniform marginals U and V, C(u,v) = Pr[U 6 u, V 6v], that capture dependence between two ran-dom variables, X and Y, irrespective of their marginal distributions,

FX(x) and FY(y), respectively.Sklar’s (1959)theorem states that there exists a copula such that

FXYðx; yÞ ¼ CðFXðxÞ; FYðyÞÞ; ð1Þ

where FXY(x, y) is the joint distribution of X and Y, u = FX(x) and

v= FY(y) C is uniquely determined on RanFXx RanFYwhen the mar-gins are continuous Likewise, if C is a copula, then the function FXY

in Eq.(1)is a joint distribution function with margins FXand FY The conditional copula function (Patton, 2006) can be written as:

FXYjWðx; yjwÞ ¼ CðFXjWðxjwÞ; FYjWðyjwÞjwÞ; ð2Þ

where W is the conditioning variable, FXjW(xjw) is the conditional distribution of XjW = w, FYjW(yjw) is the conditional distribution of YjW = w and FXYjW(x, yjw) is the joint conditional distribution of (X, Y)jW = w

Consequently, the copula function relates the quantiles of the marginal distributions rather than the original variables This means that the copula is unaffected by the monotonically increasing trans-formation of the variables Copulas can also be used to connect mar-gins to a multivariate distribution function, which, in turn, can be decomposed into its univariate marginal distributions and a copula that captures the dependence structure between the two random variables Thus, copulas allow the marginal behavior of the random variables and the dependence structure to be modeled separately and this offers greater flexibility than would be possible with para-metric multivariate distributions Moreover, modeling dependence structure with copulas is useful when the joint distribution of two variables is far from the elliptical distribution In those cases, the tra-ditional dependence measure given by the linear correlation coeffi-cient is insufficoeffi-cient to describe the dependence structure (see Embrechts et al., 2003) Furthermore, some measures of concor-dance (Nelsen, 2006) between random variables, like Spearman’s rho and Kendall’s tau, are properties of the copula

A remarkable feature of the copula is tail dependence, which measures the probability that two variables are in the lower or upper joint tails of their bivariate distribution This is a measure of the pro-pensity of two random variables to go up or down together The coef-ficient of upper (right) and lower (left) tail dependence for two random variables X and Y can be expressed in terms of the copula as:

kU¼ lim

u!1Pr X P F1

X ðuÞjY P F1Y ðuÞ

¼ lim

u!1

1  2u þ Cðu; uÞ

1  u ; ð3Þ

kL¼ lim

u!0Pr X 6 F1

X ðuÞjY 6 F1Y ðuÞ

¼ lim

u!0

Cðu; uÞ

where F1

x and F1

kU, kL2 [0, 1] Two random variables exhibit lower (upper) tail dependence if kL> 0 (kU> 0), which indicates a non-zero probability

of observing an extremely small (large) value for one series together with an extremely small (large) value for another series

The copula provides information on both dependence on average and dependence in times of extreme market movements Depen-dence on average (given by linear correlation, Spearman’s rho or

3 Baur and McDermott (2010) draw a distinction between strong and weak hedges

and safe havens on the basis of the negative value or null value of the correlation, 4

For an introduction to copulas, see Joe (1997) and Nelsen (2006) For an overview

Kendall’s tau) can be obtained from the dependence parameter of the

copula; dependence in times of extreme market movements can be

obtained through the copula tail dependence parameters given by

Eqs.(3) and (4) On the basis of copula dependence information,

we can formulate two hypotheses in order to determine whether

gold can serve as a hedge or as a safe haven against USD depreciation:

Hypothesis 1 :qG;CP0ðgold is a hedgeÞ;

Hypothesis 2 : kU>0 ðgold is a safe havenÞ;

whereqG,Cis the measure of average dependence between the value

of gold and USD depreciation Thus, gold can act as a hedge if we do

not find evidence against Hypothesis 1 Similarly, if Hypothesis 2 is

not rejected, gold can serve as a safe-haven asset against extreme

market movements in the USD depreciation; in other words, gold

pre-serves its value when the USD depreciates (there is co-movement

be-tween gold and exchange rates at the upper tail of their joint

distribution) By considering kL instead of kUin Hypothesis 2, we

can test gold’s safe-haven property in the case of extreme downward

market movements, which is of interest for investors holding short

positions in the USD In this case, gold can act as a safe-haven asset

against extreme downward market movements provided Hypothesis

2 is not rejected for kL

The specification of the copula function is crucial to

determin-ing the role of gold as a hedge or safe haven against the USD We

considered different copula function specifications in order to

cap-ture different patterns of dependence and tail dependence,

whether tail independence, tail dependence, asymmetric tail

dependence or time-varying dependence The bivariate Gaussian

copula (N) is defined by CN(u,v;q) =U(U1(u),U1(v)), whereU

is the bivariate standard normal cumulative distribution function

U1(v) are standard normal quantile functions The Gaussian

cop-ula has zero tail dependence, kU= kL= 0 The Student-t copula is

gi-ven by CSTðu;v;q;tÞ ¼ T t1

t ðuÞ; t1

t ðvÞ

, with T as the bivariate Student-t cumulative distribution function with a correlation

coef-ficientq, and where t1

t ðuÞ and t1

t ðvÞ are the quantile functions of the univariate Student-t distribution withtas the

degree-of-free-dom parameter The appealing feature of the Student-t copula is

that, since it allows for symmetric non-zero dependence in the tails

(seeEmbrechts et al., 2003), large joint positive or negative

kU¼ kL¼ 2tt þ1  ffiffiffiffiffiffiffiffiffiffiffiffi

p ffiffiffiffiffiffiffiffiffiffiffiffi

1 q p

= ffiffiffiffiffiffiffiffiffiffiffiffi

1 þq p

>0, where tt+1() is the cumulative distribution function (CDF) of the Student-t

distribu-tion Tail dependence relies on both the correlation coefficient

and the degree-of-freedom parameter The Clayton copula is given

by CCLðu;v;aÞ ¼ maxfðu aþv a 1Þ1=a;0g It is asymmetric, as

dependence is greater in the lower tail than in the upper tail,

where it is zero: kL= 21/a (kU= 0) The Gumbel copula is also

asymmetric but has greater dependence in the upper tail than in

the lower tail, where it is zero: kU= 2  21/d(kL= 0) The Gumbel

copula is given by CG(u,v; d) = exp (((logu)d+ (logv)d)1/d) Note

that, when d = 1, the two variables are independent The

symme-trized Joe–Clayton copula (seePatton, 2006) allows upper and

low-er tail dependence and symmetric dependence as a special case

when kU= kL This copula is defined as:

CSJCðu;v;kU;kLÞ ¼ 0:5ðCJCðu;v;kU;kLÞ þ CJCð1  u; 1 v;kU;kLÞ

where CJC(u,v; kU, kL) is the Joe–Clayton copula, defined as:

CJCðu;v;kU;kLÞ ¼ 1  1  ½1  ð1  uÞ  j c

þ½1  ð1 vÞjc 11= c1=j

where j= 1/log2(2  kU),c= 1/log2(kL), and kL2 (0, 1), kU2 (0, 1) With a view to considering possible time variation in the conditional copula—and thus in gold and exchange rate dependence—we will as-sume that the copula dependence parameters vary according to an evolution equation FollowingPatton (2006), for the Gaussian and Student-t copulas, we specify the linear dependence parameterqt

so that it evolves according to an ARMA (1,q)-type process:

qt¼K w0þ w1qt1þ w2

1 q

Xq j¼1

U1ðutjÞ U1ðvtjÞ

!

whereK(x) = (1  ex)(1 + ex)1is the modified logistic transforma-tion to keep the value ofqtin (1, 1) The dependence parameter is explained by a constant,w0, by an autoregressive term,w1, and by the average product over the last q observations of the transformed variables,w2 For the Student-t copula,U1(x) is substituted by t1

t ðxÞ The above copula parameters are estimated by maximum like-lihood (ML) using a two-step procedure called the inference func-tion for margins (IFMs) method (Joe and Xu, 1996) The bivariate density function is decomposed into the product of the marginal densities and the copula density according to Eqs (1) and (2)

We first estimate the parameters of the marginal distributions sep-arately by ML and then estimate the parameters of a parametric copula by solving the following problem:

h¼ arg max

h

XT t¼1

ln cð^ut; ^vt;hÞ; ð8Þ

where h are the copula parameters, ^ut¼ FXðxt; ^axÞ and ^vt¼ FYðyt; ^ayÞ are pseudo-sample observations from the copula.5

For the marginal distribution, we considered an ARMA (p, q)

et al (1993)with the aim of accounting for the most important stylized features of gold and exchange rate return marginal distri-butions, such as fat tails and the leverage effect.6As a result, the marginal model for the gold or exchange rate return, rt, can be spec-ified as:

rt¼ /0þXp

j¼1

/jrtjþetXq

i¼1

where p and q are non-negative integers and where / and h are the

AR and MA parameters, respectively It is assumed that the white noise processetfollows a Student-t distribution:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit

r2

tðt 2Þ

r

withtdegrees of freedom, and wherer2

tis the conditional variance

ofetevolving according to:

r2

j¼1

bjr2 tjþXm i¼1

aje2 tiþXm j¼1

wherexis a constant;r2

tjis the previous period’s forecast error var-iance, the generalized autoregressive conditional heteroskedasticity

periods, the autoregressive conditional heteroskedasticity (ARCH) component; Itj= 1 if etj< 0, otherwise 0; and wherec captures leverage effects Forc> 0, the future conditional variance will in-crease proportionally more following a negative shock than follow-ing a positive shock of the same magnitude Leverage or inverse

5 Under standard regularity conditions, this two-step estimation is consistent and the parameter estimates are asymptotically efficient and normal (see Joe, 1997 ).

6

We also modeled marginal distributions using a more general GARCH specifica-tion; namely, the general class of power ARCH models as proposed by Ding et al (1993) and Hentschel (1995) The empirical results were similar to those presented

leverage effects have been found in some commodity prices (see, e.g.,

Mohammadi and Su, 2010; Bowden and Payne, 2008; Reboredo,

2011; Reboredo, 2012b) and in some exchange rates (Reboredo,

2012a) The number of p, q, r and m lags for each series was selected

using the Akaike information criterion (AIC)

The performance of the different copula models was evaluated

et al (2001) and Rodriguez (2007)

3 Data

We empirically investigated the hedge and safe-haven

proper-ties of gold against the USD using weekly data from 7 January

2000 to 21 September 2012 The starting sample period was

deter-mined by the introduction of the euro as a currency in financial

markets from 1999 Also, the use of weekly data is more

appropri-ate for our purpose of characterizing the dependence structures

between gold and the USD; this is because daily or high-frequency

data may be affected by drifts and noise that could mask the

dependence relationship and complicate modeling of the marginal

distributions through non-stationary variances, sudden jumps or

long memory Data for gold prices—measured in USD per ounce—

and the USD rate—measured as USD per unit of foreign currency

(an exchange rate increase means USD depreciation)—were

down-loaded from the website of the Bank of England (

http://www.bank-ofengland.co.uk) Exchange rate data was collected for currencies

as follows: the Australian dollar (AUD), the Canadian dollar

(CAD), the euro (Germany, France, Italy, Netherlands,

Belgium/Lux-embourg, Ireland, Spain, Austria, Finland, Portugal, Greece,

Slove-nia, Cyprus, Slovakia and Malta), the British pound (GBP), the

Japanese yen (JPY), the Norwegian krone (NOK) and the Swiss franc

(CHF) The set of countries used for this study includes the vast

majority of market traders in international exchange Additionally,

to examine the relationship between gold and the USD aggregate

exchange rate, we considered the Broad Trade Weighted Exchange

Index (TWEXB) of the US Federal Reserve (these data were

www.frbstlouis.com) Fig 1 displays gold price-exchange rate

dynamics for the different currencies considered throughout the

sampling period Consistent trends can be observed: gold prices

rose exponentially, whereas the USD depreciated against the main

currencies With the intensification of the global financial crisis

after 2008, gold prices and USD depreciation with respect to most

currencies analysed also moved in lock-step

Descriptive statistics and stochastic properties for the return

data for gold and USD rates are reported inTable 1 The mean

re-turns were close to zero for all rere-turns series and were small

rela-tive to their standard deviations, which would indicate no

significant trend in the data The difference between the maximum

and minimum values shows that gold prices were more volatile

than the USD Negative values for skewness were common for all

series and all returns show excess kurtosis—ranging from 4.1 to

14.5—confirming thus the presence of fat tails in the marginal

dis-tributions or relatively frequent extreme observations The Jarque–

Bera test for normality of the unconditional distribution strongly

rejected the normality of the unconditional distribution for all

the series Furthermore, the values of the Ljung-Box statistic for

uncorrelation up to 20th order in the squared returns suggested

the existence of serial correlation for all the series Also, the

La-grange multiplier for ARCH (ARCH-LM) statistic for serially

corre-lated squared returns indicated that ARCH effects were likely to

be found in all the return series with the exception of the Swiss

franc The linear correlation coefficient indicates that gold and

USD exchange rates were positively dependent; hence, the value

of gold and the USD value move in opposite directions, opening

up the possibility of using gold as a hedge

We firstly examined the dependence structure between gold and the USD by obtaining the empirical copula table for the re-turns in the following way For each pair of gold and USD rere-turns,

we ranked each series in ascending order and separated observa-tions uniformly into 10 bins in such a way that bin 1 included observations with the lowest values and bin 10 included observa-tions with the highest values We then counted the number of observations that shared each (i, j) bin for i, j = 1, , 10 through the sample period, for t = 1, , T, and included this number in

a 10  10 matrix in such a way that the matrix rows included the bins of one series in ascending order from top to bottom and the matrix columns included the bins of the other series in ascending order from left to right If the two series were perfectly positively (negatively) correlated we would see most observations lying on the diagonal connecting the upper-left corner with the lower-right corner (the lower-left corner with the upper-right corner) of the 10  10 matrix; and if they were independent we would expect the numbers in each cell to be about the same In addition, if there was lower tail dependence between the two ser-ies we would expect more observations in cell (1, 1); and if there was upper tail dependence we would expect more observations

in cell (10, 10)

Table 2displays the empirical copula table for all the gold-USD exchange rate pairs Evidence of positive dependence is indicated

by the fact that the number of observations along the upper-left/ lower-right diagonal is greater than the number of observations

in the other cells Hence, the USD value and gold prices move in opposite directions Likewise, in comparing the lowest and highest 10th percentiles, there are no significant differences in the joint ex-treme frequencies, which is evidence of potential symmetric tail dependence Furthermore, frequencies at the upper and lower quantiles are greater than for the remaining quantiles Overall, the results inTable 2are fully consistent with the positive depen-dence shown by the unconditional correlation coefficient displayed

inTable 1

4 Empirical results 4.1 Results for the marginal models The marginal distribution model described in Eqs.(9)–(11)was estimated for gold and all the exchange rates by considering differ-ent combinations of the parameters p, q, r and m for values ranging from zero to a maximum lag of two.Table 3reports the results The most suitable model was, according to the AIC values, an ARMA (0,0)-TGARCH (1,1) specification with the exception of gold, where lags 1 and 5 were included in the mean specification, and the yen, where a TGARCH (2,2) volatility specification was preferred Vola-tility was quite persistent in all the series and the leverage effect was significant for gold and two exchange rates; this is consistent with previous empirical results for gold and exchange rates (see, e.g.,McKenzie and Mitchell, 2002; Reboredo, 2012a) In addition, the last two rows ofTable 3also show that neither autocorrelation nor ARCH effects remained in the residuals

The goodness-of-fit assessment of the marginal models is cru-cially important given that the copula is mis-specified when the marginal distribution models are also mis-specified, that is, when the probability transformations ^ut¼ FXðxt; ^axÞ and ^vt¼ FYðyt; ^ayÞ are not i.i.d uniform (0, 1) Therefore, we tested the

goodness-of-fit of the marginal models by testing the i.i.d uniform (0, 1) of ^ut and ^vtin two steps (seeDiebold et al., 1998)

First, we tested for the serial correlation of ð^ut uÞk and ð^vt vÞkon h = 20 lags for both variables for k = 1, 2, 3, 4 and used

Fig 1 Gold prices and USD exchange rates (7 January 2000–21 September 2012).

the LM statistic, defined as (T  h)R2where R2is the coefficient of

determination for the regression, to test the null of serial

indepen-dence The LM statistic is distributed asv2(h) under the null

Ta-ble 4reports the results of this test for the marginal distribution

models; the i.i.d assumption could not be rejected at the 5% level

Second, we tested if ^utand ^vtwere uniform (0, 1) using the

Kol-mogorov–Smirnov, Cramer–von Mises and Anderson–Darling tests,

which compare the empirical distribution and the specified

theoret-ical distribution function P values for all these tests are reported in

the last three rows ofTable 4; for all the marginal models we were

unable to reject the null of correct specification of the distribution

function at the 5% significance level To sum up, the

goodness-of-fit tests for our marginal distribution models indicated that these

were not mis-specified; as a result, the copula model can correctly

capture co-movement between gold and exchange rate markets

4.2 Copula estimates of dependence

Before providing estimates for the parametric copulas described

above, we first obtained a non-parametric estimate of the copula

This estimate, proposed byDeheuvels (1978), at points i

, is gi-ven by

b

C i

T;

j

T

¼1

T

XT

k¼1

1fu k 6 u ðiÞ ;vk 6vðjÞ g; ð12Þ

where u(1)6u(2)6   6 u(T)andv(1)6v(2)6   6v(T)are the order

statistics of the univariate samples and where 1 is the usual

indica-tor function.Fig 2, which depicts non-parametric density estimates

for bivariate density for gold and USD depreciation, indicates (a)

po-sitive dependence between gold and the USD depreciation against a

wide set of currencies; (b) upper and lower tail dependence,

mean-ing that gold and USD exchange rate markets boom and crash

to-gether; and (c) a low probability of disjoint extreme market

movements, so extreme upward (downward) gold price movements

are not in lock-step with extreme downward (upward) USD

depre-ciation movements This graphical evidence is consistent with the

empirical copula results shown inTable 2and has, obviously,

impli-cations for the role of gold as a safe-haven asset (discussed below)

Table 5reports results for the parametric copula models

de-scribed above Examining the elliptical copulas, for all exchange

rates the dependence parameter in the Gaussian and Student-t

copulas (i.e., the correlation coefficient) was positive, strongly

sig-nificant and consistently close to the linear correlation coefficient

for the data The strength of dependence was very similar across

currencies, for correlation coefficients ranging between 0.37 and

0.51 The degrees of freedom for the Student-t copula were not

very low (ranging from 9 to 18), indicating the existence of tail

dependence for all the currencies By considering asymmetric tail

dependence, parameter estimates for the Clayton and Gumbel cop-ulas were significant and reflected positive dependence between gold and exchange rates Tail dependence was also different from zero and the lower and upper tail dependence parameters of the Clayton and Gumbel copulas had similar values Additionally, the estimated values of kLand kUfor the symmetrized Joe–Clayton cop-ula were significant in most of the cases, indicating similar depen-dence in the lower and upper tails (with the exception of CAD and JPY) Finally, time-varying dependence results for the normal and Student-t copulas also indicated positive dependence, as the corre-lation coefficients had positive values throughout the sample per-iod, displaying good results in terms of the AIC for the time-varying Gaussian copula for the yen

The comparison of the estimated copula models is essential to test the two hypotheses regarding gold’s hedge or safe-haven sta-tus against the USD; different copula models have different aver-age and tail dependence characteristics, so we need to choose the copula that most adequately represents the dependence struc-ture of gold and the USD exchange rate For the AIC adjusted for small-sample bias, the Student-t copula offered the best perfor-mance for all the exchange rates, except for CAD and JPY where the symmetrized Joe-Clayton copula and the time-varying Gauss-ian copula, respectively, performed better.7Hence: (a) Hypothesis

1 cannot be rejected since the correlation coefficient is significant and positive for the whole sample period, meaning that gold is a hedge against the USD (when the USD value falls/the USD exchange rate rises, the gold price rises and vice versa); (b) Hypothesis 2 can-not be rejected for both kL and kU because the Student-t copula exhibits upper and lower tail dependence, so gold is a safe haven against USD movements

However, the results for Hypothesis 2 were slightly different for the CAD and the JPY For the CAD, lower tail dependence was sig-nificant, although not upper tail dependence, indicating gold as a strong safe haven against the USD-CAD exchange rate in market downturns, but not in market upturns For the JPY, there was tail independence since the Gaussian copula was preferred, meaning that market movements between gold and the JPY were indepen-dent under extreme market circumstances

5 Implications for risk management Evidence regarding strengthened gold prices and USD deprecia-tion presented above through copulas is crucially relevant for cur-rency investors hedging their exposure to curcur-rency price movements and downside risk The portfolio implications were

Table 1

Descriptive statistics for gold and USD exchange rate returns.

3971.2 ⁄

1025.7 ⁄

24.47 ⁄

715.4 ⁄

159.7 ⁄

32.64 ⁄

803.40 ⁄

52.89 ⁄

Q(20) 434.40 ⁄

166.64 ⁄

404.97 ⁄

95.44 ⁄

454.87 ⁄

53.02 ⁄

112.30 ⁄

21.25 ⁄

153.97 ⁄

ARCH-LM 10.93 ⁄

5.70 ⁄

12.27 ⁄

3.58 ⁄

13.81 ⁄

2.71 ⁄

2.75 ⁄

Notes Weekly data for the period 7 January 2000–21 September 2012 JB is thev2

statistic for the test of normality Q(k) is the Ljung–Box statistics for serial correlation in the squared returns computed with k lags ARCH-LM is Engle’s LM test for heteroskedasticity, computed using 20 lags Corr Gold is the Pearson correlation for each series with gold.

⁄

Indicates rejection of the null hypothesis at the 5% level.

7

Similar results were obtained using the goodness of fit test proposed by Genest

considered in order to determine whether the use of gold could re-duce currency-related risks and losses Hence, to evaluate the attractiveness of gold in terms of currency risk management, we considered different kind of portfolios against a benchmark portfo-lio, called portfolio 1, composed only of currency

First, we considered a portfolio, called portfolio 2, obtained by minimizing the risk of a currency-gold portfolio without reducing the expected return According toKroner and Ng (1998), the opti-mal weight of gold in portfolio 2 at time t is given by:

xG

C

t hGCt

under the restriction that xG

t ¼ 1 ifxG

t >1 andxG

t ¼ 0 if xG

and where hGt, hCt, and hGCt are the conditional volatility of gold, the conditional volatility of currency and the conditional covariance between gold and currency at time t, respectively By construction, the weight of the currency in the portfolio is equal to 1 xG

t

The optimal portfolio at each time t resulted from using the relevant

copula model fit (the Student-t copula for most of the exchange rates) Second, we considered an equally weighted portfolio called portfolio 3, with good out-of-sample performance according to DeMiguel et al (2009) Third, we considered a hedged portfolio called portfolio 4, obtained from a variance minimization hedging strategy consisting of holding a short position of an amount of b fu-tures and a long position in the spot market (seeHull, 2011) We considered a long position of one USD on the currency market hedged by a short position of b USD on the gold market, given by:

bt¼h

GC t

The risk reduction effectiveness of each portfolio was evaluated

by comparing the percentage reduction in the variance of a portfo-lio with respect to portfoportfo-lio 1:

REvariance¼ 1 VariancePortfolio j

VariancePortfolio 1

where j = 2, 3, 4 and variancePortfolio jand variancePortfolio 1are vari-ances in the returns for the portfolio j and portfolio 1, respectively

A higher risk reduction effectiveness ratio means greater variance reduction.Table 6reports the risk reduction effectiveness results for gold and currency portfolios 2–4 by considering different cur-rencies with respect to the USD The results indicate consistent risk reduction effectiveness for gold in portfolios 2 and 4, where weights were obtained optimally However, when the weights were not de-rived optimally (i.e., they were determined exogenously and main-tained constant over time), as happened with portfolio 3, there were

no gains from including gold in the portfolio This evidence was common to the different currencies, with generally better results for portfolio 4 than for portfolio 2 (with the exception of the CAD and the JPY) These results confirm the usefulness of gold in reduc-ing risk in a currency portfolio

Table 2

Empirical copula for gold and USD exchange rate returns.

Notes: For each series there are 663 observations Gold returns are ranked along the horizontal axis and in ascending order (from top to bottom) and oil returns are ranked along the vertical axis and in ascending order (from left to right) Each box includes the number of observations that belongs to the respective quantiles of the gold and oil series.

In addition, we evaluated the usefulness of gold in providing

protection against downside risk and possibly dangerous tail-risk

events, by estimating the VaR of a portfolio composed of gold

and currencies The VaR is defined as the maximum loss in

portfo-lio value for a given time period and a given confidence level The

VaR at time t for an asset or a portfolio with a return rtis

charac-terized, for a (1  p) confidence level, as:

wherewt1is the information set at t  1 So, the VaR is simply the

loss associated with the pth percentile of the returns distribution for

a given period It can be computed as:

VaRtðpÞ ¼lt t1

t ðpÞ ffiffiffiffiffi

ht

p

whereltand ffiffiffiffiffi

ht

p

are the conditional mean and standard deviation for the asset returns and where t1

t ðpÞ denotes the p quantile of the Student-t distribution withtdegrees of freedom, since gold and

ex-change rate returns followed this distribution

A risk measure related to VaR is the expected shortfall (ES),

de-fined as the expected size of the loss if the VaR is exceeded, that is:

ES ¼ E½rtjrt<VaRtðpÞ: ð18Þ

Given a portfolio composed of gold and currencies, we compute

the single-period log returns as:

rt¼ log xG

ter G

t þ 1 xG

t

er C t

where rG

for gold, for the currencies and for the fraction of the portfolio in-vested in gold, respectively We used Monte Carlo simulation to cal-culate the portfolio VaR and ES from the marginal distribution functions and the copula function information as follows: (1) from estimated copula functions we simulated two innovations for each time t; (2) we transformed these simulated values into standardized residuals by inverting the marginal cumulated distribution function for each index; and (3) we used the simulated standardized residu-als to compute gold and currency returns from the estimated mar-ginal models and, for given portfolio weights, computed the portfolio returns in Eq.(19) We repeated this process 1000 times for t = 1, , T The VaR was obtained as the value of the pth percen-tile in the distribution of the portfolio returns The ES was measured

as the mean value for situations in which portfolio returns exceeded the VaR

We evaluated downside risk gains as follows First, the accuracy

of the VaR for each portfolio was tested using the likelihood ratio

(1998), which takes independence and unconditional coverage into account (see, e.g.,Jorion, 2007) Second, we considered the VaR and

ES reductions for portfolios 2–4 compared to those for portfolio 1

Table 3

Estimates of the marginal distribution models for gold and exchange rate returns.

Mean

(3.99) ⁄

(2.77) ⁄

Variance

(1.97) ⁄

(2.45) ⁄

(2.11) ⁄

(1.46) (2.36) ⁄

(2.45) ⁄

(3.22) ⁄

(1.28) (2.71) ⁄

(1.69)

(20.14) ⁄

(11.77) ⁄

(28.84) ⁄

(29.94) ⁄

(12.19) ⁄

(3.19) ⁄

(24.16) ⁄

(11.91) ⁄

(21.92) ⁄

(2.46) ⁄

(1.97) ⁄

(1.76) (0.17)

(2.45) ⁄

(3.88) ⁄

(2.29) ⁄

(1.24)

Notes: This table reports the ML estimates and z statistic (in brackets) for the parameters of the marginal distribution model defined in Eqs (9)–(11) The lags p, q, r and m were selected using the AIC for different combinations of values ranging from 0 to 2 For the JPY series a TGARCH (2,2) specification was selected (reported values are for the first lag) LogLik is the log-likelihood value LJ is the Ljung–Box statistic for serial correlation in the model residuals computed with 20 lags ARCH is Engle’s LM test for the ARCH effect in the residuals up to 10th order P values (in square brackets) below 0.05 indicate rejection of the null hypothesis.

⁄

Indicates significance at the 5% level.

Table 4

Goodness-of-fit test for the marginal distribution models.

Notes: This table reports the p values for the LM statistic for the null of no serial correlation for the first four moments of the variables u t andvt from the marginal models presented in Table 4 , where ð^ u t   uÞ k and ð^vt  vÞkare regressed on 20 lags for both variables for k = 1, 2, 3, 4 and the LM statistic is distributed asv2 (20) under the null P values below 0.05 indicate rejection of the null hypothesis that the model is correctly specified K–S, C–vM and A–D denote the Kolmogorov–Smirnov, Cramer–von Mises and Anderson–Darling tests (for which p values are reported) for the adequacy of the distribution model.

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

0.5

1.0

1.5

0.5

1.0

1.5

0.2 0.4 0.6 0.8

0.2 0.4 0.6 0.8 CAD

Gold

0.2 0.4 0.6 0.8

0.2 0.4

0.6

0.8

x

0.4 0.6 0.8

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1.0 1.2 1.4

x y

0.2 0.4 0.6 0.8

0.2 0.4

0.6

0.8

0.4

0.6

0.8

1.0

1.2

1.4

JPY

Gold

0.2 0.4 0.6 0.8 0.2

0.4 0.6 0.8 NOK

Gold

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

CHF

0.4 0.6 0.8

0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0

TWEXB

Gold

0.5

1.0

1.5

0.5 1.0 1.5

0.5 1.0 1.5

Fig 2 Empirical non-parametric density estimates for gold and the USD exchange rates.

Third, we considered a VaR-based investor loss function (seeSarma

et al., 2003; Reboredo, 2013b; Reboredo et al., 2012) given by:

lt¼ E½rt VaRtðpÞ21fr t VaR t ðpÞg; ð20Þ

where 1 is the usual indicator function and where the quadratic term

takes into account the magnitude of the failure, penalizing large

deviations more than small deviations We compared portfolios 2–

4 with portfolio 1 considering the loss differential, zt¼ lt l1t We

tested the null of a zero median loss differential against the

alterna-tive of a negaalterna-tive median loss differential by employing the

one-sided sign test defined as: S ¼ PT

t¼11fz t P0g 0:5T

ð0:25TÞ0:5 This test was asymptotically distributed as a standard normal and the null

could be rejected when S < 1.645

Table 7reports the risk evaluation results for a 99% confidence

level using the best fitting copula, the Student-t copula (with the

exception of the CAD and the JPY).8The conditional coverage test

indicated that portfolios composed of gold and currencies performed equally well in terms of the VaR, since the null of correct conditional coverage was not rejected at the 5% significance level, except for portfolio 2 with the JPY and portfolio 3 with the AUD Conditional coverage results for portfolio 1 were less positive, since half of the currency portfolios did not have correct conditional coverage at the 5% significance level, although they did at 10% (with the excep-tion of the EUR) By examining the effect of the VaR reducexcep-tion of including gold in the currency portfolio, we found evidence of VaR reduction only in the portfolios configured for optimal weights Hence, the expected maximum loss in portfolio value was greater

in the currency portfolios than in the mixed gold and currency port-folios Consistent with the increase in average risk reported above, there was no reduction in VaR for the equally weighted portfolio The ES was also reduced for portfolios 2 and 3, and was, in general, slightly larger for portfolio 4 Finally, evidence provided by the one-sided sign test indicated that the optimal weight and equally weighted portfolios outperformed the currency portfolio These re-sults support the usefulness of including gold in a currency portfolio for risk management purposes

Table 5

Estimates for the copula models.

(0.028) ⁄

(0.031) ⁄

(0.028) ⁄

(0.030) ⁄

(0.036) ⁄

(0.026) ⁄

(0.027) ⁄

(0.025) ⁄

(0.030) ⁄

(0.032) ⁄

(0.029) ⁄

(0.031) ⁄

(0.036) ⁄

(0.028) ⁄

(0.024) ⁄

(0.027) ⁄

(3.515) ⁄

(4.783) ⁄

(3.169) ⁄

(4.612) ⁄

(4.672) ⁄

(13.751) (6.974) ⁄

(1.068) ⁄

(0.063) ⁄

(0.061) ⁄

(0.062) ⁄

(0.061) ⁄

(0.058) ⁄

(0.063) ⁄

(0.067) ⁄

(0.066) ⁄

(0.043) ⁄

(0.039) ⁄

(0.045) ⁄

(0.041) ⁄

(0.032) ⁄

(0.044) ⁄

(0.043) ⁄

(0.046) ⁄

(0.067) ⁄ (0.064) (0.059) ⁄ (0.062) ⁄ (0.004) (0.058) ⁄ (0.059) ⁄ (0.056) ⁄

(0.041) ⁄

(0.043) ⁄

(0.046) ⁄

(0.046) ⁄

(0.054) ⁄

(0.044) ⁄

(0.043) ⁄

(0.047) ⁄

(0.116) (0.043) (0.759) (0.157) ⁄

(0.343) (0.230) (0.212) ⁄

(0.097) ⁄

(0.031) (0.024) ⁄

(0.136) (0.159) ⁄

(0.062) (0.084) ⁄

(0.195)

(0.082) ⁄

(0.296) ⁄

(0.077) ⁄

(1.777) (0.037) ⁄

(0.743) ⁄

(0.527) ⁄

(0.105) ⁄

(0.678) (0.283) ⁄

(0.440) ⁄

(0.268) (1.278)

(0.148) (0.071) (0.121) (0.080) (0.156) ⁄

(0.108) (0.049) (0.069)

(2.421)

Notes: The table shows the ML estimates for the different copula models for gold and the USD Standard error values (in brackets) and the AIC values adjusted for small-sample bias are provided for the different copula models The minimum AIC value (for gold) indicates the best copula fit For the TVP Gaussian and Student-t copulas, q in Eq.

(7) was set to 10.

⁄

Indicates significance at the 5% level.

Table 6

Risk reduction effectiveness for gold and currency portfolios.

Notes: This table reports the results of risk reduction effectiveness for portfolios composed of gold and currencies with respect to a portfolio composed only of currencies according to the risk effectiveness ratio in Eq (15) Portfolio 2 weights are given by Eq (13) , portfolio 3 has equal weights and portfolio 4 weights are given by Eq (14)

8

For reasons of brevity, we do not report the results for 95% and 99.9% They are,