CENTRAL BANK OF CYPRUS

EUROSYSTEM

WORKING PAPER SERIES

The Systematic Risk of Gold at Different Timescales

Antonis Michis

March 2018

Working Paper 2018-1

2

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3

The Systematic Risk of Gold at Different Timescales

Antonis A. Michis*

Abstract

Gold is frequently cited by investors as a financial asset that can be associated with a

negative beta coefficient. I investigate this hypothesis by estimating the beta coefficient of

gold at different timescales and examining the associated implications for investors with

different planning horizons. Estimation is performed using maximal overlap discrete

wavelet transforms of gold and stock market returns in four major currencies. The results

suggest that gold tends to be associated with a negative beta coefficient when considering

long-term investment horizons, and this finding is consistent across markets and

currencies.

JEL classification: G11; G12; G15

Keywords: gold returns; systematic risk; timescales; wavelets

* Statistics Department, Central Bank of Cyprus, 80 Kennedy Avenue, CY-1076, Nicosia, Cyprus. Email: AntonisMichis@centralbank.gov.cy, https://sites.google.com/site/antonismichis/ The opinions expressed in this paper are those of the author and do not necessarily reflect the views of the Central Bank of Cyprus or the Eurosystem.

4

1. Introduction

In recent years, there has been a renewed interest in the use of gold as an investment asset.

Gold has good diversification properties since it does not correlate with most other

financial assets and is considered valuable due to its ability to preserve wealth (Heidorn

and Demidova-Menzel, 2008). Currently, a range of gold investment products are

available, including forward contracts, futures, options, warrants, gold accumulation

plans, gold certificates, exchange traded funds, gold-linked bonds and structured notes. In

addition to improving liquidity, these instruments have also provided investors with

valuable risk management tools.

In addition to diversification purposes, gold is also widely held by central banks as

a reserve asset since it can provide valuable liquidity in times of financial strain. For

example, during the Asian financial crisis of 1997, the Republic of South Korea

purchased large amounts of domestic private gold in exchange for debt instruments

denominated in its national currency. These gold reserves were subsequently sold for US

dollars to alleviate the country’s external debt (see Cai et. al. 2008).

The price of gold exhibits different characteristics in the short term than those in the

long term. In the long term, the price of gold is frequently associated with expectations

regarding US dollar exchange rates, and it is also considered a long-term hedge against

US inflation. In the short-term, the price of gold is highly volatile due to changes in

supply and demand. Short-term supply is primarily influenced by price changes and

changes in the lease rate of gold. In addition to inflation and exchange rate movements,

short-term demand is also influenced by credit risk, political instability, per capita income

and gold’s beta coefficient (see Heidorn and Demidova-Menzel, 2008). The high price

volatility of gold in the short term suggests that it is more appropriate for a long-term

strategy.

Gold is also frequently cited in the finance literature as an asset that can be

associated with a negative beta coefficient (see, for example, Ehrhardt and Brigham,

2011, p. 249). Since this is an important coefficient for investors, the purpose of this study

is to estimate the beta coefficient of gold at different timescales. The results, based on

four major currencies and ten years of market data, suggest that gold tends to be

5

associated with a negative beta coefficient when considering long-term investment

horizons.

The rest of the article is organised as follows. Section 2 provides a short review of

the relevant literature, and Section 3 presents the multi-period version of the Capital Asset

Pricing Model (CAPM) that is appropriate for estimating the beta coefficient of gold at

different timescales. Section 4 presents the wavelet estimator of the beta coefficient

suggested by Gencay et. al. (2005) for estimating the systematic risk of an asset at

different timescales and in Section 5 the beta coefficient of gold is estimated using data

from four major currencies. The main conclusions of the study are summarized in Section

6.

2. Gold as an investment asset

Gold is frequently considered a hedge or safe haven asset that can improve portfolio

performance in times of financial distress (Scott-Ram, 2002, p. 137). However, these

properties of gold vary depending on the financial asset class used and the market under

consideration. A study by Baur and McDermott (2010) showed that gold is a good hedge

and a safe haven for stocks in the US and the major European markets, but the same is not

true for other important markets, such as Japan and Brazil. Using wavelet multiscale

analysis and data from the US, UK and German equity and bond markets, Bredin et. al.

(2015) showed that gold can be considered mainly as a short-term hedge for all assets (for

time intervals of up to one year) and a long-term safe haven for stocks.

In contrast, Baur and Lucey (2010) showed that gold should not be considered as a

safe haven for US, UK and German bonds. This finding has also been supported in a

recent study by Agyei-Ampomah et al. (2014), who found evidence of positive co-

movements between gold, UK bonds and German bonds during periods of increased

market volatility. Interestingly, in the same study, gold was found to provide an effective

hedge for bonds from countries with excessive debt problems.

Gold is also frequently cited as a good hedge for US inflation, but the same is not

true for the euro zone countries (see Heidorn and Demidova-Menzel, 2008). However,

gold can be considered a good hedge for US inflation only when considering very long

6

investment horizons. In the short term, the price of gold is highly volatile, rendering gold

less appropriate for inflation hedging (Fraser-Sampson, 2011, p. 169).

The safe haven properties of gold with respect to US dollar exchange rate

movements have been demonstrated by Joy (2011) and Reboredo (2013). In this case too,

investments in gold are more meaningful as long-term prospects because periods of

extreme US dollar exchange rate movements are more closely linked with

macroeconomic conditions in the US economy that typically vary over business cycle

frequencies.

Barro and Misra (2016) developed an asset pricing model for rare disasters to

examine the hedging properties of gold against macroeconomic declines. The model was

then matched to US data on gold returns for the period of 1836-2011. Gold’s average real

rates of price changes were found to be low and similar during disaster and non-disaster

periods. Therefore, gold did not serve well as a hedge against macroeconomic decays.

These properties align with a model characterised by a high elasticity of substitution

between gold services and ordinary consumption, as well as an expected real rate of gold

returns that is close to the risk-free rate.

In contrast, Long et. al. (2016) reached a different conclusion after examining the

explosive behaviour of gold prices during the period 1968-2013. Using a generalized sup

ADF test, the authors identify several explosive periods in the gold market, including the

period associated with the 2007-2008 subprime mortgage crisis. All detected periods

coincide with events that negatively affect investors’ confidence, such as economic

recessions, political instability and inflationary pressures in the economy. According to

the authors, these findings provide further evidence that gold can be considered as a safe

haven by investors holding risky assets.

Another area of research concerns the efficiency of the gold market. This has been

investigated by Tschoegl (1980), Solt and Swanson (1981) and Aggrawal and Soenen

(1988), with most studies suggesting a relatively efficient market for gold in the US. In a

recent study, Smales (2014) examined the relationship between news sentiment and

returns in the futures market for gold. The author found evidence of an asymmetric

response to news releases since negative news was found to exert greater influence on

returns than positive news. Furthermore, the magnitude of this “news sentiment – returns”

7

relationship is influenced by the macroeconomic environment (phase of the business

cycle) and the credit and position constraints imposed on futures traders (e.g.,

requirements for inventory in physical gold, position limits and margin calls).

Due to its high price volatility, gold is more appropriate for investors adopting a

passive, long-term investment strategy. Using a portfolio consisting of gold, stocks, 10-

year government bonds and 3-month Treasury bills, Michis (2014) demonstrated that gold

provides the lowest contribution to portfolio risk only when considered over medium- and

long-term investment horizons (timescales). Over short-time horizons, gold’s contribution

to portfolio risk was found to be similar to that of stocks.

Since gold exhibits low correlation with most other financial assets, it is frequently

associated with a near zero (and occasionally negative) beta coefficient (Ehrhardt and

Brigham, 2011, p. 249). Using gold returns and S&P 500 data for the period 1977-2006,

Heidorn and Demidova-Menzel (2008) estimated a 3-year rolling beta coefficient for gold

using a rolling regression technique. Their results confirmed the aforementioned assertion

that during periods when the stock market was underperforming, the beta coefficient of

gold was negative.

Most of the aforementioned empirical studies suggest that the diversification

properties of gold differ by timescale and that gold is more appropriate for investors

adopting a “buy and hold” passive investment strategy with a long-term orientation. Since

the beta coefficient is frequently calculated by investors and financial analysts, the

purpose of this study is to estimate the beta coefficient of gold at different timescales and

to examine the associated implications for investors.

3. The CAPM in a multi-period setting

Fama (1970) and Elton and Gruber (1974) provided the following set of conditions for

which the CAPM can be extended to a multi-period setting: (i) consumers’ tastes (for

goods and services) are independent of future events, (ii) consumption opportunities

(goods and prices) are known at the beginning of each period and (iii) the distribution of

all asset returns is known at the beginning of each decision period. Elaborating on these

assumptions, Hansen and Jagannathan (1991) developed a multi-period equilibrium asset

pricing model that yields the CAPM.

8

To see this model, consider the representative investor’s utility maximization

problem subject to a given budget constraint, where jtc is consumption in (future) period

j and is the discount rate:

0j

jt

j

t cUEmax .

The first-order conditions of this maximization problem, for all assets i and periods j ,

have the following form:

jt

'

jt,it

j

t

' cUr1EcU .

Dividing both sides with the marginal utility of consumption generates the following

important equation:

j,tjt,it mr1E1 where

t

'

jt

'

j

j,tcU

cUm .

The stochastic discount factor jtm , represents the intertemporal rate of substitution.

Dropping the time subscripts for simplicity, the equality above can also be written in the

following format:

mrrEmEr11 iii .

Using the risk-free rate ( fr ), the expectation of the stochastic discount factor becomes

fr1/1mE , which gives rise to the following asset pricing equation after

substitution into the previous equation:

mrrEr1rr iiffi .

9

When asset returns and the stochastic discount factor follow the multivariate normal

distribution, the conditional expectation of the discount factor can be written as a linear

function of asset returns, including the return of the efficient (or market) portfolio ( Mr ):

M

* rbam . Substituting this linear function into the previous equation and allowing

for 2

MffM r1/rrb can be shown to generate the CAPM†:

fMifi rrrr .

This equation suggests that the excess return of asset i is proportional to the market

premium. The proportionality factor (beta or systematic risk) is:

2

M

Mii

)r,r(Cov

(1)

In this multi-period setting, Gencay et. al. (2005) suggested an estimator consisting of

wavelet variances and covariances to estimate the systematic risk of an asset at different

timescales. In Section 5, this estimator is used in the context of a non-decimated wavelet

transform to examine the systematic risk of gold at different times-scales.

4. Multiscale estimation of beta

Wavelet analysis is based on new families of orthonormal bases functions that enable the

breakdown of time series (or signals) into different timescales. They are particularly

suited for the study of financial time series, since such series usually consist of multiple

frequencies, determined by the activities of numerous market participants operating over

different time horizons (Masset, 2015).

† See Elton et. al. (2014, p. 306) for the detailed derivation. Cochran (2005, p. 155) proposed an alternative

derivation of the multi-period CAPM using a quadratic value function and assuming i.i.d asset returns.

10

Using a mother wavelet basis function , a set of wavelets kj , can be obtained by

dilating (expansion in the range) and translating (shift in the range) the basis function as

follows (see, In and Kim, 2013, p. 17):

)2(2)( 2/

, ktt jj

kj .

The dilation and translation operations simultaneously analyse signals at different points

in time and frequencies. This time-frequency analysis is not possible with a Fourier

transform, which only operates in the frequency domain. For smaller values of the index j

wavelets become finer and taller scale objects, oscillate more quickly, have less duration

and are packed closer. In this way, they cover a wider set of frequencies (see Nason and

Sapatinas, 2002).

The wavelets generated from the dilation and translation operations constitute an

orthonormal basis in the space of square-integrable functions. Any signal in this space can

be represented using a linear combination of wavelets:

, , ( )t j k j k

j Z k Z

q t

(2)

with wavelet coefficients‡:

, , ( )j k t j kR

q t .

The representation in equation (2) provides a deconstruction of the signal into different

timescales proportional to j2 (a multiresolution analysis). For a signal consisting of

72128 observations, the coefficients at scale 1 ( 1q ) capture variation over cycles of 2-4

months. Similarly, the coefficients at scales 2 ( 2q ) and 3 ( 3q ) are associated with

variations over cycles of 4-8 and 8-16 months, respectively, and this is the case up to

scale 7.

‡ Z and R represent the sets of integer and real numbers respectively.

11

Financial time series are most commonly analysed using two wavelet transforms:

(i) the discreet wavelet transform (DWT) and (ii) the maximal overlap discreet wavelet

transform (MODWT). The MODWT is not exactly orthogonal (as is the case with the

DWT); however, it is more efficient§ and, unlike the DWT, has the following important

properties: (i) it generates vectors of wavelet coefficients that have lengths equal to that of

the actual time series, (ii) it can be applied to time series of any length and (iii) the

generated wavelet coefficients are not affected by circular shifts in the actual data (see

Kim and In, 2010).

Property (iii) is important for financial time series since the generated coefficients

can be aligned to match key features and dynamics in the actual time series such as jumps

and changes in regime**. When working with non-dyadic length time series, Gencay et. al.

(2002, p. 144) suggested first padding the data with the last value of the series (mirroring

the last observations) to increase their length to the next power of two and then applying

the MODWT. In a second stage, the coefficients affected by this boundary value solution

are removed from the generated vectors. Even though other methods exist for addressing

this boundary value problem, Masset (2015) considers this method the most suitable for

stock returns since it can account for the presence of volatility clustering.

In practice, discreet wavelet transforms are computed using an iterative pyramid

algorithm that filters the data using a discreet high-pass (wavelet) filter with coefficients

( 10 ,..., Lhh ) and a discreet low-pass (scaling) filter with coefficients ( 10 ,..., Lgg ). When

using the MODWT, the pyramid algorithm starts by filtering the actual time series using

the rescaled filters j

jj hh 2/~

and j

jj 2/gg~ as follows:

1

0

mod1,1

~~L

l

Nltt hq and

1

0

mod1,1~~

L

l

Nltt gv .††

§ The MODWT uses more information and is therefore superior in terms of asymptotic relative efficiency

(see, Percival 1995). ** This essentially means that the wavelet coefficients are not characterised by phase shifts where peaks and

troughs have a different timing than in the original time series. †† The modulus operator, Nlt mod , is the remainder from the division of lt with N .

12

In the second stage of the algorithm, the rescaled filters are applied to the vector of

scaling coefficients 1~v , which generates the second level of wavelet and scaling

coefficients vectors 2~q and 2

~v . Similarly, in the third stage of the algorithm the rescaled

filters are applied on the vector 2~v and the procedure is repeated J times to form the

matrix of wavelet coefficient vectors '

1 ]~,~,...,~[~JJ vqqq .

To estimate the systematic risk of an asset at different timescales using equation

(1), it is necessary to obtain estimates of the associated wavelet variances and

covariances. The wavelet variance of time series t , at timescale j , can be estimated

using the following unbiased estimator (see Gencay et. al., 2002, p. 241):

1

1

2

,

2 )~(~1

)(T

Lt

tj

j

j

j

qT

where 1)1)(12( LL j

j is the length of the wavelet filter. The number of

coefficients unaffected by the boundary is 1~

jj LTT ; therefore, all the wavelet

coefficients affected by the periodic boundary conditions are excluded from this

estimator. Accordingly, the following unbiased wavelet covariance estimator can be used

to breakdown the covariance between two time series (t and

ty ) into different

timescales (see Kim and In, 2010):

1

1

,,

2

, )~)(~(~1

)(T

Lt

y

kjkj

j

jy

j

qqT

.

5. Estimation results

Gold returns were calculated as log differences using a dataset obtained from the World

Gold Council. The dataset consisted of monthly gold prices (per troy ounce) for the period

May 2003 - December 2013, denominated in four major currencies: dollar, euro, sterling

and yen. The dataset was also augmented by data from the stock market and 10-year

government bond returns of the respective countries, available from the OECD statistical

13

database for the same period. The stock market data enable the estimation of the beta

coefficient at different timescales using equation (1).

Table 1 includes summary statistics for the aforementioned gold, stock and bond

returns, separated by country/currency. On average, 10-year government bonds provided

the highest returns over the period examined, with gold returns being slightly better than

stock market returns. These findings are consistent across markets and currencies. In

contrast, gold exhibits the highest standard deviation in all currencies, and it is also

associated with the highest maximum values, which suggests a very high price volatility

in the short term. With respect to the other instruments, the lowest standard deviation is

associated with 10-year government bonds. These findings are also illustrated in Chart 1,

which exhibits returns in dollars.

To estimate the beta coefficient of gold at different timescales, the asset returns

were deconstructed using the Daubechies least asymmetric wavelet filter with length 8 in

the context of the MODWT. This filter has good frequency localisation properties and

generated good resolutions of the time series considered in this study. It was also used by

Gencay et. al. (2005), Kim and In (2010) and Michis (2015, 2014) for the analysis of

similar financial data. Following the wavelet transforms, separate beta coefficients were

estimated by timescale and currency using equation (1) and the wavelet variances (for

stock market returns) as well as covariances (between gold and stock market returns)

presented in Section 4. The estimation results are included in Table 2.

The beta coefficient of gold measures how movements in gold returns are related to

movements of the stock market index. Negative beta values suggest that gold returns

move in the opposite direction from stock market returns, and gold is therefore an

effective portfolio diversifier. Accordingly, the investor demand for gold should be

inversely related to beta.

GER JAP UK US GER JAP UK US GER JAP UK US

Mean 0.712 0.362 0.416 0.557 3.160 1.274 3.864 3.541 0.831 0.897 0.968 0.996

St. Dev 4.901 5.273 3.734 4.285 0.980 0.350 1.046 1.003 5.297 5.549 5.478 5.646

Min -23.391 -24.795 -20.067 -25.472 1.200 0.490 1.640 1.530 -15.987 -26.741 -15.730 -19.095

Max 12.976 9.442 8.503 11.914 4.560 1.960 5.430 5.110 13.693 12.125 16.013 13.026

Stock market index 10-year gov. bonds Gold

Table 1 Summary statistics of returns

14

The beta coefficients in Table 2 vary by timescale even when considering each

currency in isolation. For example, in the case of Germany, the beta coefficient is positive

when considering short term cycles of 2-4 weeks (timescale d1), but it becomes negative

in timescales d2, d3 and d4 that are associated with cycles of 4-8, 8-16 and 16-32 weeks,

respectively. However, there are two common findings across currencies: (i) at short-term

cycles of 2-4 weeks, (d1) the beta coefficients are positive in all currencies and receive

their maximum values, and (ii) the beta coefficients associated with long-term cycles in

the data (d7) are negative for all currencies.

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3M

ay-

20

03

Au

g-2

00

3

No

v-2

00

3

Feb

-20

04

Ma

y-2

00

4

Au

g-2

00

4

No

v-2

00

4

Feb

-20

05

Ma

y-2

00

5

Au

g-2

00

5

No

v-2

00

5

Feb

-20

06

Ma

y-2

00

6

Au

g-2

00

6

No

v-2

00

6

Feb

-20

07

Ma

y-2

00

7

Au

g-2

00

7

No

v-2

00

7

Feb

-20

08

Ma

y-2

00

8

Au

g-2

00

8

No

v-2

00

8

Feb

-20

09

Ma

y-2

00

9

Au

g-2

00

9

No

v-2

00

9

Feb

-20

10

Ma

y-2

01

0

Au

g-2

01

0

No

v-2

01

0

Feb

-20

11

Ma

y-2

01

1

Au

g-2

01

1

No

v-2

01

1

Feb

-20

12

Ma

y-2

01

2

Au

g-2

01

2

No

v-2

01

2

Feb

-20

13

Ma

y-2

01

3

Au

g-2

01

3

No

v-2

01

3

Chart 1 Asset returns in US dollars

Gold

Stocks

Bonds

Scale GER JAP UK US

d1 0.290 0.398 0.379 0.488

d2 -0.042 0.307 0.193 0.322

d3 -0.259 0.193 -0.259 -0.035

d4 -0.038 0.390 -0.175 0.258

d5 0.064 0.257 -0.007 -0.029

d6 0.035 0.014 0.005 0.027

d7 -0.191 -0.013 -0.203 -0.048

Table 2 Estimates of systematic risk by time-scale

15

The results in Table 2 suggest that the usefulness of gold in reducing portfolio risk

varies by timescale. Gold is not an effective diversifier in the short term (e.g., cycles

associated with timescale d1) since it exhibits positive correlation with stock market

returns. In contrast, it has good diversification properties over long time horizons

(timescale d7) since it exhibits negative correlation with stock market returns. More

importantly, these results are consistent across the four major currencies examined in this

study. Therefore, gold is more appropriate for investors adopting a long-term “buy and

hold” investment strategy, irrespective of whether their investment positions are

denominated in dollars, euros, sterlings or yen.

6. Conclusions

The price of gold exhibits different characteristics in the short term than in the long term.

Accordingly, the systematic risk of gold should be expected to vary by timescale with

different implications for investors trading over different time horizons and currencies.

The empirical results in this study confirm this assertion. When considering cycles in the

range of 4 to 128 weeks (d2-d6), the sign and magnitude of the beta coefficient of gold

varies by timescale and currency. The same is not true for timescales d1 and d7 that are

associated with short term (2-4 months) and long-term (in excess of 128 months) cycles in

the return series. The beta coefficient at timescale d1 is always positive and receives its

maximum value per currency. In contrast, it is always negative when evaluated over

timescale d7. Consequently, in the long term, gold returns move in the opposite direction

from stock market returns, and gold is therefore an effective portfolio diversifier.

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