Price Discovery and Hedging Properties of Gold
and Silver Markets
Isabel Figuerola-Ferretti
Business Department, Universidad Carlos III de Madrid
Jesus Gonzalo
Economics Department, Universidad Carlos III de Madrid
January 19, 2010
Abstract
In this paper we introduce a non linear price discovery equilibrium
model that captures the long term heding properties of gold and silver
markets. Price discovery is regime dependent and is determined by the
relative number of participants in the gold and silver markets. Using
COMEX gold and silver prices, the VIX implied volatility index and the
dollar-euro exchange rate, we show that gold and silver are cointegrated
only under weak dollar and high volatility conditions. In this state, gold
dominates silver indicating that is most e¢ cient in re‡ecting their common
hedging characteristics. Under low volatility and strong dollar conditions
gold and silver can not be regarded as substitutes for hedging similar types
of risks.
1
Introduction
For centuries gold and silver were perceived to be important hedging devices in
times of economic turmoil. Following the collapse of Bretton Woods, both prices
were deregulated and the markets changed signi…cantly. In this paper we use
recently developed time series techniques to analyze the long term relationship
and price discovery between gold and silver COMEX future prices. The main
objective is to see whether there is a stable or semistable relationship between
both markets, and to determine whether and under which circumstances gold
and silver prices share a common stochastic trend that allows us to determine
price discovery.
Documenting the long term relations and leadership among …nancial markets
can help trading strategies. Silver and gold prices are usually considered as
substitutes to reduce similar types of risk in portfolios. They are also tied
1
via a popular intercommodity spread strategy known as the gold silver spread
(formally de…ned as the di¤erence in the gold and silver prices). The strategy
involves simultaneously taking a long position on one and a short position on
the other. The rationale of this type of trading lies on the belief that the long
run di¤erence between the two contracts is stable. One explanation for this is
that there are many commonalities in the underlying economic, …nancial and
political factors that drive the prices of the two commodities and that makes
them appropriate investment devices for diversi…cation purposes
However, gold and silver have distinct important uses for which they are
no substitutes. For instance, silver has been extensively used as metal for electronics, X rays and photography because of its unique physical and chemical
properties. On the other hand demand for gold is largely determined by the
actions of Central Banks, as industrial nations maintain a steady proportion of
gold in their foreign exchange reserves. On the private side, the demand for
gold consumption is dominated by jewelry applications . It can therefore be
argued that separate fundamentals determine the prices of gold and silver and
therefore they do not share a common stochastic trend.
Price discovery is one of the social bene…ts provided by future markets. It
refers to the use of future prices for pricing cash market transactions (Working
(1948), Wiese (1978)). In general, price discovery is the process of uncovering
an asset’s full information or permanent value. The unobservable permanent
component re‡ects the underlying fundamentals. It di¤ers from the observable
price which measures the permanent or e¢ cient price with some transitory error.
The contribution of markets to price discovery has been a relevant subject of
research for some time. Several studies investigate the informational advantages
of markets for single underlying assets (Garbade and Silver 1983, Harris et
al. 1998, Hasbrouck 1995). Generally the price discovery literature applies
two methodologies (see Lehman 2002, special issue on the Journal of Financial
Markets) the information shares of Hasbrouck (1995) (IS thereafter) and the
Gonzalo Granger (1995) Permanent-Transitory decomposition (PT thereafter).
The use of the PT measure has been justi…ed by Figuerola-Gonzalo (2007) (FG
thereafter) who develop a general equilibrium model to measure price discovery
that leads to the PT decomposition. Their theoretical …nding states that price
discovery depends on the relative number of players in each market. This …nding
is consistent with the literature, where the consensus has generally been that
price discovery takes place in the most liquid markets where trading volume is
concentrated.
Price discovery in the gold and silver market has been studied by Bahram
et al. (2000) who use intraday data and an Error Correction Model (ECM) to
conclude that the silver contract bears the majority of the burden of convergence
to the gold-silver spread. Prior work on the gold silver relationship focuses on
cointegration analysis. Wahab Chon and Lashgri (1994) use daily cash and
future prices and …nd that there is cointegration between the gold and silver
in both the spot and futures markets. Escribano and Granger (1998) analyze
monthly prices of gold and silver using a sample ranging from 1971 to mid 1990s.
They split their sample and …nd evidence of cointegration between 1971 and
2
1990. In their out of sample analysis they argue that the relationship between
gold and silver is diminishing after 1990. This is con…rmed by Ciner (2004) who
uses gold and silver prices traded on the Tokyo Commodity Exchange (TOCOM)
for the 1994-1998 period and …nd no evidence of cointegration between gold and
silver . Cointegration based techniques were also applied to the spreads by ShiMiin and Chih-Hsie (2003) who use a fractional ECM analysis to forecast spot
and future spot and silver spreads. They claim that signi…cant riskless pro…ts
can be earned based on ECM forecasting of the changes of spot and future
spreads.
Another popular area of research has focused on the casual relationship
between gold and silver. Of particular relevance are the papers of Chan and
Montain (1988) and Frank and Stengos (1989), who provided a an analysis
of the pricing relationship between gold and other precious metals. Frak and
Stengos revealed some non linear dependence between silver and gold prices,
whereas Chan and Montain found a "simultaneous relationship" between the
price of gold, the price of silver and the treasury bill rate.
The literature on the role of gold and other precious metals in …nancial
markets also looks at the diversi…cation properties of precious metals when
combined with stock market investments in …nancial portfolios (see Hiller et
al. (2006) and references therein). The consensus is that there is a generally
observed diversifying e¤ect for gold. Baur and Lucey (2007) considered the role
of gold as a save heaven.
This paper contributes to the price discovery literature of gold and silver taking account of their hedging statistical properties and the presence of non linearities in their spread. In particular we extend the model proposed by FiguerolaGonzalo (2007) to the two regime case to capture the hedging properties of
gold and silver in times of adverse market conditions. This allows us to establish which metal is most reliable as a hedging investment in the two di¤erent
regimes. Our theoretical model suggests that discovery or reliability as hedging
asset will be determined by the relative number of participants in each market
for the di¤erent regimes.
Precious metals are classi…ed as investment as well as consumption assets.
Non commercial positions in these metals are generally taken as means of portfolio diversi…cation. The relevance of precious metals as hedging instruments
increased signi…cantly over the last 7 years and accelerated with the consequences subprime crises such as general fear, high volatility and weak dollar.
Gold prices increased by 30% over the last year and reached a maximum of
$1226.56 in December. Silver prices rose by 109% mainly due to increased investor demand re‡ected in higher non commercial positions. Investors increase
their exposure to gold and silver for similar fundamental reasons such as weakening of the dollar or an increased market liquidity. Given that silver is cheaper
than gold, market participants can substitute into the less expensive alternative. In this paper we use the VIX volatility index and the euro dollar exchange
rate as threshold variables in a non linear price discovery model to address the
following two questions: are gold and silver substitute assets under times of
economic turmoil? if this is the case, which market is more e¢ cient as a hedg3
ing instrument? Once we are able to determine which price is most e¢ cient in
processing information, investors can evaluate the risk of holding the precious
metal and make appropriate decisions.
The paper is organized as follows. In section 2 we describe the threshold
equilibrium model with …nite elasticity of supply of arbitrage services for gold
and silver COMEX prices. We demonstrate that price discovery is determined
by the relative number of participants in each of the regimes. Section 3 presents
empirical estimates of the model developed in section 2, and its linear counterpart. This requires i) testing for cointegration ii) Estimating a non linear
VECM and iii) obtaining the price discovery parameters. Section 4 concludes.
Graphs are collected in the appendix.
2
Theoretical Framework: A threshold equilibrium model for Price Discovery in Gold and
Silver Markets
The goal of this section is to capture the hedging properties and characterize
price discovery of silver and gold prices by re-examining the equilibrium non arbitrage framework developed by FG and its extension to the non linear case. We
develop a non linear VECM to test whether price discovery changes according
to the magnitude of some threshold variable re‡ecting the existence of di¤erent
market conditions. The main objective is to establish the extent to which gold
and silver provide hedging capabilities in times of "abnormal" market behavior.
A growing body of research in recent time series literature has incorporated
non linear behavior into conventional linear reduced form speci…cations such
as autoregresive and moving average models. The motivation from moving
away from the traditional linear model with constant parameters arises from the
observation that many economic and …nancial time series are often characterized
by regime-speci…c behavior and asymmetric responses to shocks. For such series
the linearity and parameter constancy restrictions are typically inappropriate
and may lead to misleading inferences about their dynamics.
Within this setting, a general class of models that has been particularly popular are the family of the threshold models, characterized by piecewise linear
processes separated according to the magnitude of a threshold variable that triggers regime changes. When each linear regime follows an autoregresive process,
we have well known threshold autoregresive models the properties of which have
been extensively investigated (see for example Gonzalo and Montesinos 2000 and
Gonzalo and Pitarakis 2002). The concept of threshold cointegration has attracted considerable attention from practitioners interested in uncovering non
linear adjustment patterns in relative prices and other variables. The goal of
this section is to link non linear cointegration behavior to the hedging properties
of gold and silver prices in order to determine price discovery.
4
Let gt be the natural logarithm of the one month futures price of a gold in
period t and let st be the natural log of the one month futures silver price .
Let rt be the arbitrage cost applicable to the gold and silver markets. In order
to …nd the non-arbitrage equilibrium condition the following set of standard
assumptions apply in this section:
(a.1) No taxes or transaction costs
(a.2) No limitations on borrowing
(a.3) No cost other than arbitrage risk cost
(a.4) No limitations on short sale of the commodity of the spot market
(a.5) Arbitrage risk costs are given by the process
rt = r + I(0)
_
Where ris the mean of rt and I(0) is an stationary process with mean zero and
…nite positive variance.
(a.6) gt and st are I(1).
If rt , is the continuously compounded arbitrage risk cost applicable to gold
silver spread , by the above assumptions a1-a4, non-arbitrage equilibrium conditions imply
st = gt + rt
(1)
. For simplicity and without loss of generality for the rest of the paper it will
be assumed T1=1From (a.5) and (a.6), equation (1) implies that gt and st are
cointegrated . The following Econometric relationship will be used to represent
the constant long run spread
gt =
2 st
+
3
(2)
Gold and silver prices are therefore closely linked via low risk spread arbitraging, where 3 = rt and 2 st + 3 represents the silver price free of arbitrage
risk cost.
2.1
Equilibrium prices with …nitely elastic supply of arbitrage services under the presence of two regimes
Consider two regimes in the economy speci…ed by the threshold variable qt d the
indicator function I(:)and the threshold parameter . We de…ne two regimes in
the economy so that when I(qt 1 > ) we are in regime 1 and when I(qt 1
)
and we are in regime 2, where d is the known lag length.
To describe the interaction between gold and silver prices we must …rst
specify the behavior of agents in the marketplace. There are N1;G participants
5
in gold market in regime 1 and N2;G participants in the gold market in regime
2. Accordingly there are N1;S participants in the silver market in regime 1 and
N2;S participants in silver market in regime 2. Let the demand for arbitrage
services be H1 in regime 1 and H2 in regime 2. Let Ei;t be the endowment of the
ith participant immediately prior to period t and Rit the reservation price at
which the that participant is willing to hold the endowment Ei;t: , under regime
1 and 2. Then the demand schedule of the ith participant in the gold market in
period t in regime 1 will be given by
Ei;t
A(gt
Ri;t )
A > 0; i = 1; :::N1;G under regime 1
(3)
Ei;t
A(gt
Ri;t )
A > 0; i = 1; :::N2;G under regime 2
(4)
The aggregate gold market demand for arbitrageurs is
H1 ((
2 st
+
3)
gt ) ; H1 > 0 under regime 1
(5)
H2 ((
2 st
+
3)
gt ) ; H2 > 0 under regime 2
(6)
Equivalent relationships apply to the silver market.
Accordingly the gold market will clear at the value of gt that solves
N1;G
X
N1;G
Ei;t =
i=1
i=1
N2;G
X
i=1
X
fEi;t
A(gt
Ri;t )g+H1 ((
2 st
+
3)
gt ) when 1(qt
1
> ) and we are in regime 1
(7a)
N2;G
Ei;t =
X
i=1
fEi;t
A(gt
Ri;t )g+H2 ((
2 st
+
3)
gt ) when 1(qt
1
) and we are in regime 2
(7b)
The silver market will clear at the value of st such that
6
N1;S
X
N1;S
Ei;t =
i=1
i=1
N2;S
X
i=1
X
fEi;t
A(st
Ri;t )g+H1 ((
2 st
+
3)
st ) when 1(qt
1
> ) and we are in regime 1
(7c)
N2;S
Ei;t =
X
i=1
fEi;t
A(st
Ri;t )g+H2 ((
2 st
+
3)
st ) when 1(qt
1
) and we are in regime 2
(7d)
Solving system 7 as a function of the mean reservation price for gold market
PNG
1
participants in regime 1 RtG = N1;G
i=1 Ri;t and the mean reservation price
PN1;s
1
for silver market participants in regime 1 RtS = N1;S
i=1 Ri;t we obtain
g1;t =
s1;t =
(AN1;s + H1 2 ) NG RtG + H1 N1;s 2 RtS + H1 Ns
(H1 + AN1;G ) N1;S + H1 N1;G 2
AN1;G RtG + (H1 + AN1;G )N1;S RtS + H1 N1;G
(H1 + AN1;G ) N1;S + H1 N1;G 2
3
3
(8)
(9)
Equilibrium prices in regime 2 will therefore be given by
g2;t =
s2;t =
(AN s + H2 2 ) NG RtG + H2 N2;S 2 RtS + H2 NS
(H2 + ANG ) N2;S + H2 N2;G 2
AN2;s RtG + (H2 + AN2;G )N2;s Rts + H2 N2;G
(H2 + AN2;G ) N2;s + H2 N2;G 2
3
3
(10)
(11)
To derive the dynamic price relationships, the model in equation must be
characterized with a description of the evolution of reservation prices. Following
FG It is assumed that in regime 1, immediately after market clearing in period
t 1 the ith gold market participant was willing to hold an amount Eit at a
price g1;t 1 . Accordingly the mean reservation price in each market in period t
will be
7
RtG
RtS
wt1;G
= g1;t 1 + vt + wtG
= s1;t 1 + vt + wtS
with
N1;S
NG
X
X
1;S
1
1
= N1;G
wi;t , wt = N1;S
wi;t
i=1
(12)
i=1
Equivalent dynamics apply to regime 2. Substituting equation 12 into the
equations (8-11) the following VAR model is obtained under regime 1
gt
st
gt 1
uG
H1 3
N1;S
t
+ (M1 )
+
N1;G
d1
uSt
st 1
where
v G + wtG
= M1 tS
vt + wtS
1
N1;G (H1 2 )
H1 2 N1;G
=
H1 N1;G
(H1 + AN1;s )N1;G
d1
(13)
=
uG
t
uSt
M1
Taking …rst di¤erences of eq 13 and taking account of equilibrium prices in
both regimes, we may write the non linear threshold model within the general
equilibrium model as
gt
st
=
H1 N1;S
1
1
d1 N1;G
H2 N2;s
1
1
+
d2 N2;G
gt
st
3
3
1
(qt
1
gt
st
1
1
d
(qt
> )+
(14)
) + ut
d
Applying the PT decomposition, our non linear price discovery metric will
be de…ned in terms of the relative number of participants in each of the observed
regimes.
N1;S
N1;G
gt +
st
N1;G + N1;S
N1;G + N1;S
when
(qt
d
> )
(15)
N2;G
N2;S
gt +
st
N2;G + N2;S
N2;G + N2;S
when
(qt
d
)
(16)
This implies that price discovery will depend only on the relative number of
players in each of the regimes and it is not dependent on the spread´s long term
parameters or the elasticities of arbitrage services.
8
3
Empirical Price Discovery in gold and silver
markets
The data include daily observations of gold and silver 1 month future COMEX
prices . Prices are available from January 1990 to October 2009. Threshold
variable data include daily observations for the S&P 500 VIX implied volatility
index quoted in the Chicago Board Options Exchange, and the dollar-euro exchange rate. The data source is Ecowin for all series. Figures in the appendix
depict gold and silver COMEX prices together with the VIX index (Figure 1)
and the dollar-euro exchange rate (Figure 2). Gold and silver COMEX prices
seem to be a¤ected by the VIX index and dollar-euro exchange rate data.
As a preliminary Analysis we report linear cointegration and price discovery
results. Table 1A in the appendix presents a linear Augmented Dickey Fuller
cointegration analysis. Results suggest that gold and silver are I(1) series and
are cointegrated, suggesting that they can be regarded as substitutes investment
assets. Table 2A in the appendix shows results from estimating a linear VECM.
Estimation output shows that whereas gold does not react signi…catively to
changes in the equilibrium error while silver does, suggesting that gold leads
silver in the linear case, implying that is more e¢ cient as a hedging instrument.
3.1
Non Linear Price Discovery with the VIX index as a
threshold variable
The second purpose of this analysis is to use the non linear framework to determine whether gold and silver hedging role is regime dependent. Particularly
we want to establish whether theirs substitutability is greater during periods of
"abnormal" market conditions. The investment and diversi…cation properties of
precious metals has been widely discussed in the literature. Hillier et al (2004)
provide a review of this literature. Other authors have concentrated in gold and
have analyzed its role as a potential hedging variable (see Davidson et al. 2003)
and a as a save heaven (see Baur and Lucey 2007). The consensus of these
studies is that there is a diversifying e¤ect in gold.
In this section we consider the price discovery properties of precious metals taking a proxy for market volatility as the threshold variable. Hillier et al.
(2004) look at the hedging properties of precious metals considering GARCH(1,1)
errors of S&P 500 market index returns. We follow Hillier et al. (2004) in using volatility to promote understanding of the investment properties of precious
metals. To accommodate conditional diversifying properties of gold and silver
markets, we consider general market volatility indicator, the S&P 500 implied
volatility VIX index, which reports implied volatilities of CBOT traded options
on market index. This is constructed as a weighted average from the implied
volatilities obtained from out of the money options whose underlying asset are
included in the S&P 500. The level of the VIX implied volatility index is taken
as threshold variable. The ptimal threshold level is chosen using the AIC and
BIC criteria. We divide the data on the threshold variable into deciles and …nd
9
that the optimal threshold level is at the 80% decile for which the VIX index
takes a value of 25.24. The threshold variable q takes the value of 1 when the
volatility index is greater than 25.24 and zero otherwise. The optimal threshold
lag lenght is 2. And the optimal number of lags in the system, according to the
BIC criteria is 2. Results from estimating the non linear VECM for the VIX index are reported in Table 1. We take those periods where the VIX index reaches
a value greater than 25. 24 as times of "abnormal" volatility conditions. Estimation output reported in table 1 suggest that gold and silver COMEX prices
are only cointegrated under times of "abnormal volatility" conditions. Under
this regime the silver price reacts signi…catively to changes in the equilibrium
error but gold prices do not react, to euquilibrium error changes, indicating that
the gold market is dominant in terms of price discovery. Gold futures are more
liquid that silver COMEX futures. If we take volumes traded as a proxy for
the number of participants we can state that the results on price discovery are
consistent with with the theoretical prediction.
Table 1. Estimation of the non linear VECM with VIX
as threshold variable (t stats in brackets)
Sample June 1990 - October 2009
N1;S
gt 1
gt
1
(qt d
)zt 1
+
=H
d1
N1;G
st 1
st
N2;S
gt 1
+
(q
< )zt 1
N2;F 0 t d
st 1
1
0:0004
B (0:041) C
gt
gt 1
C
=B
@ 0:004 A (V IXt 2 25:24)zt 1 st 1 +
st
(1:730)
1
0
0:0003
B ( 0:247) C
gt 1
C
+B
@ 0:002 A (V IXt 2 < 25:24)zt 1 st 1
(0:782)
zt = gt 1:22 0:74st k(AIC) = 2;
3.2
The exchange rate as a threshold variable
In this section we focus on the relationship between gold and silver prices and
the exchange rate. It is commonly believed that since the dissolution of the
Bretton Woods International monetary system, ‡oating exchange rates among
the major currencies have been a major source of price instability in the gold
market. Before the euro was formally introduced, Sjaastad and Scacchiavillani
(1996) argued that the European countries heavily dominate the international
market for gold and hence movements in the European exchange rates against
the US dollar impact heavily on the dollar price of gold. 1
1 While a 10% appreciation of the Deutche Mark (against all other currencies) increases
the dollar price of gold by 6.5% (and viceversa) a 10% appreciation of the dollar against the
10
Following this argument, we look at the relationship of precious metal prices
taking the $/e exchange rate as the threshold variable in a non linear VECM
framework. The consensus in the literature is that gold prices are negatively
related to the exchange rate (measured in units of foreign currency), see Sherman (1983). Capie et al. (2005) …nd negative signi…cant correlations between
the change in the log price and the change in the pound-dolar and yen-dolar
exchange rate providing statistical con…rmation of the hedging properties of
gold.
Capie et al. (2005) are concerned by the role of precious metals as a hedge
against changes in the external purchasing power of the dollar. They argue that
If gold were a perfect external hedge its dollar (i.e. nominal ) price would rise
at the same rate and time as the number of units of foreign currency per dollar
fell. In this paper we investigate the extent to which the exchange rate hedging
properties of precious metals a¤ect price discovery. Figure 2 suggest that the
rise of silver and gold prices over the past 7 years coincides with an increase
in the price of both metals and an increase in the dollar-euro exchange rate,
suggesting that both gold and silver act as a hedge against low dollar. As in
the previous analysis, we choose the optimal threshold level using the AIC and
BIC criteria. The optimal threshold is set at its 70% decile level which is 1.27.
The optimal threshold lag length is 1. We use the BIC criteria for choosing
the optimal system lag length which is set at 2.
Table 2 reports results from estimating a non linear VECM with the threshold variable q taking the value of 1 when the dollar-euro exchange rate is greater
than 1.27 and zero otherwise. Note that the sample now starts in January 2000,
so that it is consistent with the introduction of the euro.
Table 2. Estimation of the non linear VECM with
dollar-euro exchange as threshold variable
Sample June 2000 - October 2009
N1;S
gt 1
gt
1
(qt d
)zt 1
+
=H
d1
N1;G
st 1
st
N2;S
gt 1
+
(q
< )zt 1
N2;F 0 t d
st 1
1
0:003
B (1:367) C
gt
gt 1
C
=B
@ 0:001 A (exratet 1 1:27)zt 1 st 1 +
st
0
1 (3:541)
0:001
B (0:717) C
gt 1
C
+B
@ 0:004 A (exratet 1 < 1:27)zt 1 st 1
(0:810)
zt = gt 0:377 0:869st k(AIC) = 2;
Deutche Mark depresses the price of gold by about 8%
11
Results reported in table 2 are consistent with those in table 1. They show
that gold and silver are cointegrated under "abnormal" exchange rate conditions, re‡ected in a weak dollar-euro exchange rate. In this state while gold
does not react signi…catively to changes in the long term equilibrium, silver
does, suggesting that gold is information dominant when gold and silver are
cointegrated. Again this is consistent with our theoretical predictions.
4
Conclusions
In this paper we introduce a non linear price discovery equilibrium model that
captures the long term hedging properties of gold and silver markets. In consensus with the literature, the theoretical predictions of the model show that price
discovery takes place where volumes are concentrated. Using one month future
COMEX gold and silver daily price data, we show that gold and silver future
prices are only cointegrated under weak dollar and high volatility conditions.
This …nding should be of relevance to participants in gold and silver markets
as it indicates that under normal (strong dollar and low volatility) conditions
these two markets should be approached as separate markets, and changes in
the gold to silver ratio should not be used to predict prices in the future. It also
suggests that under normal conditions gold and silver should not be regarded as
substitutes to hedge similar type of risks. Gold and silver act as substitutes in
times of generalized fears of economic turmoil, and that, in this state, the gold
market is most important in re‡ecting their hedging statistical properties.
5
Appendix
Table 1A: Linear Dickey Fuller and Cointegration test results
Aug DF test Levels Aug DF test (…rst di¤erences)
gt
0:067
71:59
st
0:544
72:23
zt
2:594
12
Table 2A: Linear VECM estimation results (t ratios in brackets)
Sample Jan 1990-October 2009
g
N1;s
gt 1
gt
H
(zt 1 )
+ k lags of gstt 11 + uuts
st = d
t
N1;G
st 1
zt = gt
3
2 st k(AIC) = 2;
Gold and
0 Silver 1
0:002
B (0:833) C
g
gt 1
gt
C
B
+ k lags of gstt 11 + uuts
st = @ 0:003 A (zt 1 )
t
st 1
(2:27)
zt = gt 1:22 0:75st k(AIC) = 2;
6
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