a study of excess volatility of gold and silver
TRANSCRIPT
IIMA-IGPC Conference on Gold and Gold Markets
12 January 2018
A Study of Excess Volatility of Gold and Silver
Parthajit Kayal
Madras School of Economics, Chennai
S. Maheswaran
Institute of Financial Management & Research, Chennai
This Working Paper was presented by Parthajit Kayal at the Conference on
Gold and Gold Markets organised by the India Gold Policy Centre (IGPC) at the
Indian Institute of Management, Ahmedabad (IIMA), India with full logistics support for
presentation and dissemination from IGPC, IIMA.
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A Study of Excess Volatility of Gold and Silver
Parthajit Kayal1, S. Maheswaran
2
Abstract
This paper discusses the case of strong path dependency in asset prices from the theoretical
and empirical standpoints. Specifically, it demonstrates persistence of excess volatility in
the gold spot price data that engenders excessive path dependence, whereas it is not the
same with silver. For this study, we use the extreme value estimator proposed by Rogers
and Satchell (1991) and the VRatio proposed by Maheswaran et al (2011). The data for
the study is for the period from January, 2001 to December, 2016. We use multiple-days‘
time horizons for examining the excess volatility with a better approximation of Brownian
motion in the data. We capture the excess volatility in the gold data using the Binomial
Markov Random Walk model. In this paper, we also utilize the Expected Lifetime
Shortfall (ELS) ratio, as a measure of risk to test for the presence of mean reversion in
asset prices. Using this ratio, one can observe that the strong mean-reverting characteristic
in gold makes it a better investment choice than silver, in general, in the medium term.
JEL Classification: G11, G12, G14, G15, G17, F37, Q02
Key Words: Volatility, Commodity Market, precious metals, random walk, Brownian
motion, simulation, extreme value estimator, and market efficiency
Last Updated: January 2018
1 Parthajit Kayal, (corresponding author), Lecturer, Madras School of Economics (MSE), Gandhi Mandapam Road,
Behind Government Data Centre, Kottur, Chennai 600025, India. e-mail: [email protected]
2 S. Maheswaran, Professor, Institute for Financial Management and Research, 24 Kothari Road, Nungambakkam,
Chennai 600034, India. e-mail: [email protected]
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A Study of Excess Volatility of Gold and Silver
1. Introduction
Precious metals have been popular as a medium of exchange and stores of value for
millennia (Fergal at al., 2015). Many investors invest in gold or silver, in various forms
(Jewelleries, bars, coins, exchange traded funds etc.) in their portfolios. During the crisis time like
equity market turmoil, geopolitical tensions, weaknesses of US dollar etc. gold and silver become
a major choice of investment due to its safe haven asset status (Lucey and Li, 2014). Gold, being
the main precious metal, is very sensitive to geopolitical crises (Hammoudeh et al, 2010). Demand
for gold spikes in the immediate wake of any bad events like economic slowdowns, debt crises,
macroeconomic policy changes, increase in expected deflation etc. We also see the similar trend
in case of demonetization in India (World Gold Council, 2017). These events cause sharp changes
in prices and result in high volatility in gold prices. Silver also suffers associated volatility being
the precious metal alternative for gold. In fact, silver and gold prices have a very strong positive
correlation (Garbade and Silber, 1983a; Garbade and Silber, 1983b; Ma, 1985 etc.). However, if
we consider the correlation of changes in daily prices, it is difficult to establish a directional
causality (The Silver Institute, 2017). This is because a price change in gold and silver is not
proportional. According to the report of The Silver Institute (2017), there exist even an inverse
correlation during certain periods (recently observed in 2016 Q3 and 2017 Q1). Similar trend was
first highlighted by Escribano and Granger (1998) as they pointed that gold and silver prices
begun diverging in 1990‘s. They suspected this emerging trend is due to silver‘s increasing
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importance as in industrial metal while gold was widely held used for investment purpose. The
gold-silver price ratio tends to vary due to the same reason. Historically, this ratio is very high
during the periods like major wars, market crisis, economic recession etc. If compared with gold,
silver has relatively lower market liquidity levels and its demand fluctuates between industrial use
and as a financial asset, which makes its price notoriously volatile quite unlike gold (Clark, 2013).
As gold has an important monetary component and demand for gold is stronger, investors
tend to perceive that gold would be less volatile than silver (Hecht, 2016). The demand for silver
is very sensitive to price as it is more commodity-driven than gold and its monetary element is not
very strong like gold (Batten et al., 2014). A visual comparison in Figure 1 shows that gold seems
less volatile than silver. A similar observation made by Morales and Andreosso-O‘Callaghan
(2011) showed that silver‘s daily returns has a standard deviation which is more than twice of that
of gold.
Figure 1: Gold and silver normalized spot prices Price of 2
nd January 2001 is taken as 100
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Year
Gold and Silver Spot Prices)
Gold Silver
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High volatility in the prices of gold and silver alter the portfolio choice of individual and
institutional investors. Although both the metals are considered to be safe haven assets, their price
changes are always not unidirectional and proportional as highlighted before. Changes in gold
prices are also affected by psychological barriers (Aggarwal and Lucey, 2007). Silver being the
low cost precious metal with a significant industrial demand doesn‘t reflect any such barriers
(Lucey and O'Connor, 2016). Therefore, we would not be wrong to expect that gold and silver
will react differently to same news or the magnitudes of reaction will be different. For the same
reason the price changes of gold and silver would exhibit a dissimilar pattern. It gets reflected in
their volatility through price changes dynamics. Moreover, this price changes dynamics can be
captured more accurately using daily extreme prices than just closing prices. This study focuses on
this issue by examining and comparing the volatility of both metals. However, a volatility
comparison study of different assets can have some shortcomings: (a) the comparison between
different assets cannot be used unless we have an acceptable benchmark (b) sometimes high
volatility can be due to high uncertainty in market, in general, and hence, it is not wise to compare
volatility during a particular time period with average volatility of other periods. To achieve this,
we need a different approach which allows us to compare an asset with itself and in the same time
period. We adapt the VRatio proposed by Maheswaran et al (2011). This method compares two
different and independent measurements of volatility (high-to-low and open-to-close) for the same
asset, for the same time period. We discuss the estimation and advantages of VRatio in section 3.
Unlike our initial visual observations in Figure 1, we find after our analysis, quite
contrarily, the presence of excess volatility in gold prices rather than silver. This leads us to our
second objective, viz., to capture the increasing excess volatility by using the Binomial Markov
Random Walk model and allowing for the strong form of path dependence. In this work, we also
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utilize a new measure of risk called the Expected Lifetime Shortfall (ELS) ratio, to test for the
presence of mean reversion in asset prices. A third objective is to draw implications of the
estimated results on volatility for recommending the choice of gold or silver in portfolio design
and hedging strategies.
This paper is organized as follows. We present a brief review of the literature on excess
volatility and extreme value estimators in Section 2. Section 3 provides the data and describes the
methodology used in this study. Section 4 describes the empirical results. In Section 5, we
capture the volatility in the gold data using the Binomial Markov Random Walk model and
conclude this work in Section 6. A more detailed presentation on the ELS ratio is presented in the
Appendix A.
2. Review of Literature
The recent fall and then rise of commodity markets and the associated volatility in precious
metals prices offer a strong motivation to examine if gold and silver are still good choices for
investment. This study aims to draw implications of investments in precious metals with reference
to excess volatility in gold and silver prices.
Size of the literatures on gold and silver is large and growing. A significant part of these
literatures investigate the volatility aspects of these metals using spot, future, and ETF (exchange
traded funds) prices. However, these literatures mostly focus on determinants aspects of volatility
(Hammoudeh and Yuan, 2008; Batten et al., 2010; etc.), volatility spillovers (Antonakakis and
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Kizy, 2015; Kang et al., 2017; etc.), linkage with other assets (Sarwar, 2016; Mensi et al., 2015;
etc.), hedging and portfolio diversification abilities (Pierdzioch et al., 2016; Bruno and
Chincarini, 2010; Hillier, 2006; Jaffe, 1989; etc.), safe haven status (Lucey and Li, 2015; Ciner et
al., 2013; etc.) etc. There are not many studies which investigate the price change dynamics of
these metals.
Although our study compares the gold and silver in terms of their volatility, it is also
necessary to discuss the literature which link gold and silver in a common thread as they are
considered to be closed substitutes. In early studies, Garbade and Silber (1983a, 1983b)
established a strong linkage between gold and silver prices. Later, Ma (1985) and Wahab et al.
(1994) found similar results. The findings of Koutsoyiannis (1983), Ciner (2001), and Escribano
and Granger (1998) are in contrary to long run relationship of gold and silver prices. This
difference was due to choice of different time periods. Gold-silver parity weakened over the 1990s
due to increased importance of silver as an industrial metal (Lucey and Tully, 2006). In a recent
study Batten et al. (2013) showed the existence of mean reversion in gold and silver price spread.
To name few of other studies which also investigate the volatility of gold and silver prices in
different context are Adrangi et al. (2000), Chatrath et al. (2001), Liu and Chou (2003), Chng and
Foster (2012), Batten et al. (2010), Batten et al. (2014) etc.
This study is more related to the literature of overreaction and excess volatility as we
investigate the volatility through daily extreme prices changes. Arrival of unexpected news causes
overreaction in the market that leads to immediate volatility during trading hours (Ederington and
Lee 1993) before it settles down. This arises due to initial informational asymmetry (Barclay and
Hendershott 2003) and investors‘ sentiment driven by short-run factors (Kleidon, 1981). This
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study aims to capture this volatility caused by overreaction. The literature on excess volatility and
overreaction grew from the late 70‘s and is not very vast yet. This literature contradicts the
common hypotheses maintained in the market efficiency literature including the study of Fama
(1970). Seminal work by Shiller (1979) investigated the excess volatility on bonds markets.
Using expectation models, he found that yields of long-term bonds are excessively volatile. In a
different paper, Shiller (1981) explored the applicability of efficient markets hypothesis model on
real stock data and argued that efficient markets model was at best an ―academic" model and that
it did not serve as a good fit for unobservable data. In the same year, LeRoy and Porter (1981)
found evidence of excess volatility on a long time series data of US stock prices and later De
Bondt and Thaler (1985) also confirmed the same. Short-run factors drive the sentiments of
investors that lead to overreaction in prices and cause excess volatility (Kleidon, 1981). Similar
findings are also observed in different stock markets. For examples Cuthbertson and Hyde (2002)
found clear evidence of excess volatility in French and German stock markets, De Long and
Grossman (1993) observed excess volatility in British stock prices in the pre-World War I period.
In a recent study Kayal and Maheswaran (2017) showed evidence of excess volatility in the EUR-
INR and GBP-INR currency pairs.
There is sufficient scope to discover more knowledge about excess volatility and
understand its manifestations and implications in various contexts, i.e., stock markets, currency
markets, commodities markets, etc., and using different methodological approaches. Precious
metals are a major part of investment portfolios. Most of the studies focused on precious metals
volatility are based on closing prices while ignoring other extreme prices like Open, High, and
Low. Hence, there is sufficient scope to contribute further knowledge about excess volatility in
precious metals considering all extreme prices. This study is an attempt to study excess volatility
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in precious metals using the extreme value estimator. This study provides an opportunity to
compare high-to-low volatility with open-to-close volatility and also recommend a possible way to
capture the increasing excessive volatility in data. In the case of efficiency, extreme value
volatility estimators are much better in comparison with the return-based volatility estimators
(Parkinson, 1980; Garman and Klass, 1980; Rogers and Satchell, 1991; Yang and Zhang, 2000).
Precious metals are a major part of investment portfolios. Most of the studies focused on
precious metals volatility are based on closing prices while ignoring other extreme prices like
Open, High, and Low. Hence, there is sufficient scope to contribute further knowledge about
excess volatility in precious metals considering all extreme prices. This study is an attempt to
study excess volatility in precious metals using the extreme value estimator. This study provides
an opportunity to compare high-to-low volatility with open-to-close volatility and also recommend
a possible way to capture the increasing excessive volatility in data.
In finance literature, extreme value volatility estimators based on the high and the low
prices have been accepted as being highly efficient estimators. The first set of extreme value
volatility estimators are known as the Method of Moments (MM) estimators (see Garman and
Klass, 1980; Parkinson, 1980; Rogers and Satchell, 1991; and Kunitomo, 1992) and the other set
of estimators are Maximum Likelihood (ML) estimators (see Ball and Torous, 1984; Magdon-
Ismail and Atiya, 2003; and Horst, Rodriguez, Gzyl, and Molina, 2012). The ML estimators are
considered less advantageous in comparison with the MM estimators due to the intricacy of the
joint density functions, and being non-expressive in the closed-form (Maheswaran and Kumar,
2013). Also, it is very difficult to assess the sensitivity of the ML estimators to outlier
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observations. Hence, the MM estimators are better than the ML estimators for extreme value
volatility estimation.
Among the different MM estimators, the extreme value estimator RS proposed by Rogers
and Satchell (1991) is appealing because it estimates the unconditional variance and is unbiased
for any value of drift. The Yang and Zhang (2000) estimator is an improvement over that of RS as
they incorporated opening jumps in their method. In this study, the RS estimator is more suitable
over that of Yang and Zhang as we are studying excess volatility by comparing high-to-low
volatility with open-to-close volatility. We checked for the effect of the opening jumps in our data
and found it to be negligible.3 We use multiple-days‘ time horizons helps in improving the
Brownian motion approximation, which is the major assumption of our model. Hence, this
method can give better insights about the volatility. The VRatio proposed by Maheswaran et al
(2011) is the ratio of high-low volatility to open-close volatility and is hence a scale free measure.
This ratio is independent of the level of volatility and allows us to distinguish between other
aspects of stochastic price movements. This measure enables us to examine the structure of
volatility. All other estimators do not satisfy the unbiasedness property if the mean return or the
drift element is non-zero.
Volatility in financial assets is an important topic for study. Precious metals are a major
part of investment portfolios. Most of the studies focused on precious metals volatility are based
on closing prices while ignoring other extreme prices like Open, High, and Low. Hence, there is
sufficient scope to contribute further knowledge about excess volatility in precious metals
3 The volatility of overnight returns is found to be around 1.22 percent of daily returns (close-to-close returns) for gold
and 2.44 percent for silver.
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considering all extreme prices. This is probably the first study on the volatility of gold and silver
spot prices using extreme value estimators. This study provides an opportunity to compare high-
to-low volatility with open-to-close volatility and also recommend a possible way to capture the
increasing excessive volatility in data.
3. Data and Methodology
We utilized daily time series Open, High, Low and Close (OHLC) data of both spot prices
(gold and silver). The gold (XAGUSD) and silver (XAUUSD) spot prices are measured in US
dollars per troy ounce. We have used daily time series data (five working days per week) from
January 2001 to December 2016, a period of about 16 years. We also checked the results dividing
the sample as data from the pre- and the post-financial crisis of 2008. All the data described above
have been collected from the Bloomberg database.
Table 1
Summary Statistics
This table reports summary statistics of daily close-to-close logarithm
returns of gold and silver for the period of January 2001 to December 2016.
Gold Silver
Mean % 0.041 0.049
Median % 0.050 0.117
Max % 10.788 14.088
Min % -9.074 -18.442
Standard Deviation % 1.138 1.950
Kurtosis 5.692 8.117
Skewness -0.183 -0.912
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Table 1 summarizes the descriptive statistics of the daily close-to-close logarithm returns
of gold and silver for the period of January 2001 to December 2016. Gold price has an average
daily log return of is 0.04% with a standard deviation of 1.14%. For the silver price, the average
daily log return is 0.05% with a higher standard deviation (1.95%) than gold. Both the metals
exhibit negative skewness. However silver returns are more skewed and has more kurtosis than
those of gold. We describe the methodologies and analyze the data below. We also measure the
volatility of overnight returns for both the metals. We find it to be insignificant as it is around
1.22% of daily returns (close-to-close returns) for gold and 2.44% for silver.
For the basic methodology we use the VRatio proposed by Maheswaran et al. (2011) in
multiple days‘ framework and bootstrap simulation.. However, we also employ the Binomial
Markov Random Walk model (see section 5) and utilize a new measure of risk called the ELS
ratio (see appendix) in this paper.
3.1. VRatio calculation using Extreme value estimator (RS estimator) of
variance
The choice of methodology is motivated by the relative efficiency and advantages which
are discussed above in the literature review section. As highlighted before, among all MM
estimators, the extreme value estimator RS proposed by Rogers and Satchell (1991) is appealing
because it is an unconditional variance estimator which is unbiased for any drift value. The
assumption for this model is that the normalized price series follows a Brownian motion. In the
context of Brownian motion, the RS estimator is unbiased. OHLC prices of tradable financial
assets are easily accessed nowadays and reveal more information about price movements when
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compared to just closing data. Before we calculate the RS estimators, the data are put through the
transformation described below to get the Brownian motion in the price series.
Using this daily OHLC prices, we define is the price of an asset at time t, on day n.
Now let represents logarithm of the price . Here, when markets open and
when markets close. When n=1, it denotes the starting day of the sample and n=N the last day of
the sample. N is the actual of the sample size.
( )
Now we can get the intra-day (log) price on day n when normalized by the opening price.
Then we normalise the High (maximum), Low (Minimum) and Closing (Terminal) prices by the
opening price (see equations 3.2, 3.3, and 3.4).
From the daily OHLC prices, we calculate three different prices series namely
which are induced by the Brownian motion with drift μ and variance . Now, to identify the
volatility in the series, we aim to estimate the parameters μ and .
Further to determine the RS estimator proposed by Rogers and Satchell (1991), we first
estimate the extreme value price series and in equations 3.5 and 3.6 respectively.
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Here and . Now, we take the simple arithmetic average of
these two extreme values and and define it as prices n equation 3.7.
Hence, we can compute the daily prices using the daily OHLC prices. Now, if we
take the simple arithmetic average of the daily prices over the entire sample time period,
we will get the RS estimator in equation 3.8.
∑
Under the Brownian motion assumption, the extreme value estimator, RS, is unbiased and
uncorrelated with the usual sample variance (Maheswaran et al., 2011). The usual sample
variance is calculated from the normalized closing prices‘ daily returns in equation 3.9
∑ ̂
The VRatio is the ratio of the RS estimator and the sample variance (see equation 3.10).
The VRatio compares the high-low volatility with the open to close volatility.
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If the VRatio is more than one, the RS estimator suffers from an upward bias4 relative to
the usual variance of the normalized terminal prices, , i.e., there is excess volatility in the
data of asset prices. The possible reasons of excess volatility in the data could be due to a
negative correlation among intra-day price changes and the presence of path dependency.
3.2. Multiple-days’ Time Horizons
In this section, we use multiple-days‘ time horizons to observe how the VRatio changes
over different time horizons. We can then investigate whether the values observed in the assets
show any sign of excess volatility. It may be difficult to observe an appropriate level of Brownian
motion in real data. The unbiasedness property of the RS estimator rests on the extent of
Brownian motion approximation of the actual data. With multiple-days‘ time horizons, say, for k-
days, it may be feasible to get a better approximation of the Brownian motion. With a proper
approximation of Brownian motion, it is possible to see that the VRatio converges to 1 as we
allow k-days‘ time windows. This also takes care of the opening jumps5 by accounting for them
in the model.
Suppose T is the number of trading days in our sample, where, T = 1, 2, 3… N, where N
refers to the total number of days in the sample. Here, we take k = 1, 2, 3… 20. Now, using the
daily OHLC data, we get the k-days‘ OHLC prices for each k. Given T = 1, 2, 3… N and k = 1, 2,
3… 20, we generate k-days‘ OHLC samples in the following way:
4 The RS estimator could also be severely downward biased in the presence of the random walk effect in data (Kumar
and Maheswaran , 2014). 5 RS estimator assumed no opening jumps (Yang and Zhang , 2000).
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Open = Open (T)
High = Max (High (T: T+k-1))
Low = Min (Low (T: T+k-1))
Close = Close (T+k-1)
Now, for all T+k-1<N+1, we have new OHLC data for each k = 1, 2, 3… 20. Hence, for
each sample of assets, we have 20 different sets of OHLC data with respect to k-days. Using the
method described earlier, we construct normalized extreme values ( ) for each sample to
determine the RS estimator and . Taking the ratio of both the estimators, we calculate the
VRatios for all samples. Hence, for each asset, we have 20 different VRatios for k = 1, 2, 3… 20.
3.3. Bootstrap Simulation
In order to determine if our findings are robust enough, we undertake simulation by
generating bootstrap replications of each random sample with replacement. This method enables
us to produce new samples of similar size and determines the standard error (SE). In this
bootstrap simulation, we generate 1000 new samples for each actual k-days sample. That is to
say, for a given k, we create 1000 new bootstrap samples of ( ). Now, we use each
sample to calculate the VRatio and finally get 1000 VRatios for each k. The average of all the
VRatios is defined as the Boot Mean, and the standard error of all the VRatios is defined as Boot
SE. The same process is run for each k to generate 20 separate Boot Means and Boot SEs to gain
multiple-days‘ time horizons results for each asset. The same process is repeated for all the assets.
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4. Empirical Findings
The VRatio compares the high-low volatility with the open to close volatility. If the
VRatio is more than 1, the RS estimator suffers from an upward bias relative to the usual variance
of the normalized closing prices, Var , i.e., there is excess volatility in the data. Multiple-days‘
time horizons allows us to observe how the VRatio changes over different time horizons. Hence,
we can investigate whether the values observed in the assets show any sign of excess volatility.
With proper approximation of Brownian motion it is possible to see that the VRatio converges to 1
as we allow multiple-days‘ time windows6.
We find that for silver, the VRatio starts at 1.06 and converges to 1 when we use the period
2001-2015. In pre-crisis period, the VRatio for silver is at 1.16 and decreases over the k-days to
1.09, and during the post-crisis period, it is almost hovering around 1.0. This result is almost in
conformity with the Random walk theory and it shows that there is not much excess volatility in
silver prices. In the case of gold, the VRatio starts at 0.99 and increases to reach 1.20 in the 2001
to 2015 period. When we consider the pre-crisis period, we see that the VRatio starts at 1.05 and
reaches 1.26 over k-days. The post crisis period also shows similar results, starting at just 0.92
and then increasing to 1.16 when k=20.
The findings show that the volatility of silver is more stable in comparison with that of
gold (see Figures 2 and 3 or Tables 2 and 3). In the case of gold, the excess volatility becomes
larger and it keeps increasing as we move to multiple-days‘ time windows. The increasing excess
volatility in gold prices is puzzling. We show that this can be captured through the negative
6 It can be shown easily using simulation of random numbers.
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correlation among the price changes and the presence of path dependency. In the next section, we
try to capture the gold prices data using a binomial Markov Random model.
Figure 2: VRatio for silver over Multiple-days’ time horizons.
Figure 3: VRatio for gold over Multiple-days’ time horizons
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
VR
atio
K-days VRatio for Silver Spot
Period 2001-16 Period 2001-07 Period 2009-16
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
VR
atio
K-days VRatio for Gold Spot
Period 2001-16 Period 2001-07 Period 2009-16
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Table 2
The VRatio for gold spot prices for Multiple-days’ time horizons
This table reports the actual VRatio, Boot Mean VRatio, standard error and t-statistics for the gold spot prices over k-
days‘ time horizons for all the three time periods (2001-16, 2001-07, and 2009-16).
k-
days’
VRatio Boot
Mean
Boot
SE
t-
stat VRatio
Boot
Mean
Boot
SE
t-
stat VRatio
Boot
Mean
Boot
SE
t-
stat
Period 2001-16 Period 2001-07 Period 2009-16
k=1 0.99 0.99 0.04 -0.27 1.05 1.05 0.06 0.85 0.92 0.93 0.05 -1.47
k=2 1.04 1.04 0.04 0.91 1.15 1.15 0.05 2.89 0.97 0.97 0.06 -0.48
k=3 1.05 1.06 0.04 1.51 1.15 1.15 0.05 3.27 1.01 1.01 0.05 0.18
k=4 1.05 1.05 0.04 1.42 1.10 1.10 0.04 2.36 1.04 1.04 0.05 0.85
k=5 1.04 1.04 0.04 1.26 1.07 1.07 0.04 1.69 1.05 1.05 0.05 0.95
k=6 1.04 1.04 0.04 1.07 1.07 1.08 0.05 1.73 1.03 1.03 0.05 0.63
k=7 1.05 1.05 0.03 1.56 1.08 1.09 0.04 2.05 1.02 1.03 0.04 0.61
k=8 1.07 1.07 0.03 2.23 1.11 1.11 0.04 2.76 1.03 1.03 0.04 0.79
k=9 1.09 1.09 0.03 2.82 1.12 1.13 0.04 2.93 1.04 1.04 0.04 0.99
k=10 1.09 1.10 0.03 3.06 1.14 1.15 0.04 3.35 1.04 1.04 0.04 1.01
k=11 1.10 1.11 0.03 3.32 1.16 1.16 0.05 3.46 1.04 1.04 0.04 0.98
k=12 1.13 1.13 0.03 4.23 1.18 1.18 0.04 4.03 1.05 1.05 0.04 1.26
k=13 1.15 1.15 0.03 4.75 1.19 1.19 0.05 4.17 1.07 1.07 0.04 1.72
k=14 1.16 1.17 0.03 5.26 1.22 1.22 0.05 4.75 1.07 1.07 0.04 1.64
k=15 1.17 1.17 0.03 5.41 1.23 1.23 0.05 4.46 1.08 1.09 0.04 2.02
k=16 1.18 1.18 0.03 5.51 1.25 1.25 0.05 4.94 1.10 1.11 0.04 2.55
k=17 1.19 1.19 0.03 5.70 1.26 1.26 0.05 4.95 1.12 1.12 0.04 2.88
k=18 1.19 1.20 0.03 5.84 1.26 1.26 0.05 4.73 1.14 1.14 0.04 3.43
k=19 1.20 1.20 0.03 5.84 1.26 1.26 0.06 4.55 1.15 1.15 0.04 3.67
k=20 1.20 1.20 0.03 6.09 1.25 1.25 0.06 4.48 1.16 1.16 0.04 3.69
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Table 3
The VRatio for silver spot prices for Multiple-days’ time horizons
This table reports the actual VRatio, Boot Mean VRatio, standard error and t-statistics for the silver spot prices over k-
days‘ time horizons for all the three time periods (2001-16, 2001-07, and 2009-16)
k-
days’
VRatio Boot
Mean
Boot
SE t-stat VRatio
Boot
Mean
Boot
SE t-stat VRatio
Boot
Mean
Boot
SE t-stat
Period 2001-16 Period 2001-07 Period 2009-16
k=1 1.06 1.07 0.05 1.42 1.18 1.18 0.08 2.20 0.98 0.99 0.06 -0.25
k=2 1.06 1.07 0.05 1.38 1.17 1.17 0.07 2.27 1.00 1.00 0.07 -0.01
k=3 1.06 1.06 0.04 1.46 1.14 1.15 0.07 2.03 1.01 1.02 0.06 0.23
k=4 1.06 1.06 0.04 1.41 1.09 1.09 0.06 1.45 1.05 1.05 0.06 0.89
k=5 1.05 1.05 0.04 1.38 1.06 1.06 0.05 1.21 1.06 1.07 0.06 1.19
k=6 1.04 1.04 0.03 1.08 1.03 1.04 0.05 0.69 1.05 1.05 0.05 1.08
k=7 1.04 1.04 0.03 1.23 1.04 1.04 0.05 0.76 1.04 1.04 0.05 0.87
k=8 1.03 1.03 0.03 0.94 1.05 1.05 0.05 0.99 1.01 1.01 0.05 0.24
k=9 1.03 1.03 0.03 0.89 1.06 1.06 0.05 1.15 0.98 0.99 0.04 -0.33
k=10 1.02 1.02 0.03 0.73 1.06 1.07 0.06 1.21 0.97 0.97 0.04 -0.68
k=11 1.02 1.02 0.03 0.69 1.07 1.07 0.05 1.36 0.96 0.96 0.04 -0.95
k=12 1.02 1.03 0.03 0.83 1.07 1.08 0.05 1.49 0.95 0.95 0.04 -1.26
k=13 1.02 1.03 0.03 0.79 1.08 1.08 0.05 1.55 0.94 0.94 0.04 -1.47
k=14 1.03 1.03 0.03 0.90 1.09 1.09 0.05 1.73 0.93 0.93 0.04 -1.90
k=15 1.03 1.03 0.03 0.94 1.09 1.10 0.05 1.77 0.93 0.93 0.04 -1.94
k=16 1.04 1.04 0.03 1.22 1.10 1.10 0.05 2.00 0.94 0.94 0.04 -1.64
k=17 1.04 1.04 0.03 1.39 1.10 1.11 0.05 2.25 0.95 0.95 0.04 -1.33
k=18 1.04 1.04 0.03 1.31 1.10 1.10 0.05 2.01 0.97 0.97 0.04 -0.93
k=19 1.04 1.04 0.03 1.48 1.10 1.10 0.05 1.99 0.98 0.98 0.04 -0.45
k=20 1.04 1.04 0.03 1.51 1.09 1.09 0.05 1.82 1.00 1.00 0.04 -0.13
20
5. Capturing the Excess Volatility in the observed data
Excess volatility in gold prices becomes larger as we move to multiple-days‘ time
windows. The increasing excess volatility in gold prices is captured using Binomial Markov
Random Walk model in the case of path dependency.
5.1. The Random Walk Effect
The Random walk model can be used when intra-day price movements are not continuous.
Suppose we consider a Random Walk model with N steps (RW_N), with each step having
variance 1/N. With a large value of N, the Random Walk model approximates Brownian motion.
In this case, the VRatio < 1.0 for a RW_N and will increase with N, converging to 1.0 as for
Brownian motion. In real observed data, the VRatio could be less or more than 1 on a single day,
but as per the Functional Central Limit Theorem (FCLT), it should converge to 1 if we take
multiple-days‘ time windows. In the case of silver, the data approximately follows Brownian
motion, and the VRatio over k-days converges to 1. Specifically for gold, the VRatio drifts away
from 1 and steadily increases with multiple-days‘ time horizons. Hence, the values observed in
gold prices could not have possibly come from a RW_N model.
In our observed data, the observed VRatio is ~ 1 or > 1 with k-days‘ time windows. The
VRatio increases as we move from a single day to k-days with respect to the length of the time
window. This is captured with the help of the Binomial Markov Random Walk model.
21
5.2. Binomial Markov Random Walk Model (BMRW)
Let us take a simple binomial model where stock prices move up or down by one monetary
unit in each step. Now, we change this model by assigning probabilities. Suppose the continuation
probability is p that the price will move ‗up‘ in the next step when the current step is an ‗up‘, or
that the price will move ‗down‘ in the next step when the current step is a ‗down‘. The reversal
probability, q = (1- p), will mean that the price will move ‗down‘ in the next step when the current
step is ‗up‘ or that the price will move ‗up‘ when the current step is ‗down‘. Hence, p and q can
be regarded respectively as the continuation probability and the reversal probability from one step
to the next. This is a binomial model with state dependent transition probabilities similar to a
Markov Eandom Walk and is defined as the Binomial Markov Random Walk (Fuh, 1997).
In the classic model of efficient markets, price changes at each step are independent, where
p = 1/2 corresponds to the simple binomial model. If p = 1/2, the model is without path
dependence of any kind (Dothan, 2008) and connects market efficiency with the Markov nature of
asset prices.
According to Functional Central Limit Theory (FCLT), the VRatio converges to 1 as N
gets large. This happens because p (0,1) is fixed for all N. To overcome this, we let p to get
smaller as N gets bigger in the following way (Equation 311).
(3.11)
For large N, approximates to a ratio of and as shown in equation 3.12.
22
(3.12)
Figure 4: VRatio in the Binomial Markov Random Walk
(The Continuation Probability p approaches to 0 as N gets Large)
Here we can see that for every fixed N, the VRatio decreases monotonically with respect to
p and is asymptotic to infinity as p approaches 0. In other words, for a given level of λ (here, λ is
a constant), p converges to zero as we increase the number of steps (N) and the VRatio gets
larger. This suggests a possible way to capture the gold spot price data if we allow for path
dependence7 in the model. Different versions of path dependence exist weak form
8, semi-strong
9
form, and strong form10
.
7 Different degrees of path dependence exist with the different forms of the efficient markets hypothesis.
8 Case of diffusion scaling or weak or a mild form of path dependence: p(N) approaches 1/2 at the rate of 1/√N,
the VRatio converges to 1 as N becomes large. 9 Semi-strong form of path dependence: In this case, p is fixed to be the same for all N. Here also, the VRatio does
converge to 1 as N becomes large, thanks to the Functional Central Limit Theorem (FCLT). This means excess
volatility disappears if the holding period is longer. 10 Strong form of path dependence: we get when p(N) converges to 0 as N becomes large. In this situation, price
reversals become the rule and the VRatio does not converge to 1.0 but increases with N.
23
In the case of gold, the excess volatility becomes larger and it keeps increasing as we move
to multiple-days‘ time windows. The only way for the VRatio to increase continuously with
respect to the length of each multiple-days‘ time windows is if the continuation probability p
approaches 0 as N becomes large. Hence, our empirical findings suggest that volatility in gold
arises from excessive path dependence (strong-form) in prices. Thus, the nature of volatility in
gold is quite different from that of silver. Such a distinction can be useful to have during
discussions of portfolio management because the characteristics of price movements in gold and
silver are different.
In Appendix 3.A, we use a method of risk named Expected Lifetime Shortfall (ELS) where
the risk depends on the continuation probability p. We have shown that as we increase the number
of steps in price changes the continuation probability tends to zero. With the help of a simple two-
step model, we show that the ELS is lowest when p = 0. Intuitively, we can understand that this
risk is minimum irrespective of the number of steps (N) when p converges to 0. A simulation
study with random numbers for each different step of N has shown the same result graphically.
Using the same model, we can infer that gold is less risky an asset than silver in terms of
investments. Strong path dependence and high negative correlation in gold prices mean that if
gold prices fall in the medium-term then it is very likely that they will go up again and vice versa.
In contrast, silver is more likely to stay in the same price range for longer periods. Hence, gold is
a better investment choice when compared with silver.
24
6. Conclusion
This study has examined the volatility of gold and silver, using the extreme value estimator
RS proposed by Rogers and Satchell (1991) and the VRatio proposed by Maheswaran et al (2011).
The multiple-days‘ time horizons (k-days, with k = 1 to 20 days) used here for analyzing the 15
years‘ time series data allow examination the volatility over the different time frame. The
Brownian motion model provides a steadily improving fit to the data with an increase in ‗k‘, thus
supporting the use of the k-days‘ horizon. Bootstrap simulations are employed for computing
standard error and also to check for the statistical validity of our analysis. Unlike the initial
observations, there is excess volatility in gold prices than those of silver over the multiple-days‘
time horizons. The excess volatility increases for gold with an increase in ‗k‘. This helps in
capturing the gold data effectively with the Binomial Markov Random Walk model in the case of
a strong form of path dependence. In this study, a new measure of risk called the Expected
Lifetime Shortfall (ELS) ratio is proposed to check for the presence of mean reversion in asset
prices. Using the same model, we can be more confident that gold is a less risky asset than silver
for making investments in the medium-term. For gold, the VRatio increases and this suggests
extensive mean-reverting characteristics in its price movements, viz., when prices are low, the
tendency to move higher exists. In the case of silver, the VRatio doesn‘t change much over time
and is found to hover around 1, which suggests the Random walk or Brownian motion model. The
mean-reverting characteristic is not very strong in silver, viz., if prices move up then they are more
likely to stay up for a relatively long time. Similarly, if prices for silver go down then they are
more likely to stay low for a longer time as compared to gold. Gold is less volatile when we hold
it over the medium term because of a negative correlation between successive price changes.
25
Hence, from an investor‘ point of view, gold should be more preferred in their portfolio than
silver.
Appendix A.
Expected Lifetime Shortfall (ELS) as a measure of long-term risk
Here, we first describe a simple N-step binomial Markov Random Walk model (see
Maheswaran et al, 2011 for details) to calculate ELS. We set normalized logarithm of asset prices
to move up or down each step. In the first step, the probabilities of moving up and down are the
same at 0.5. Now, let the continuation probability, i.e., the probability that price will move ‗up‘ in
the next step when the current step is an ‗up‘, or the probability that price will move ‗down‘ in the
next step when the current step is a ‗down‘, be p. The reversal probability is q = (1 – p), which
means that price will move ‗down‘ in the next step when the current step is ‗up‘ or the price will
move ‗up‘ when the current step is ‗down‘. In other words, the transition probabilities of this
binomial model are state dependent. In this model,
Here,
;
for
26
Where { } and s are generated from the distribution mentioned above.
Simulations are carried out to generate ELS for N-steps and the results are shown in Table 5 and
Figures 6 and 7. This is best understood with a simple two-step model. First, we start with S0 = 0.
Now, S1 = +1 (up) with probability 0.5 and S1 = 1 (down) with probability 0.5. Hence, in the
second step, we have S2 = +2 (up-up) with joint probability p/2, and S2 = –2 (down-down) with
joint probability p/2 and S2 = 0 (up-down and down-up) with joint probability (1 – p) respectively
(see Figure 5 and Table 4). ELSs are calculated below for the two-step model.
Figure 5: A two-step tree representation of the Binomial Markov Random Walk.
27
Table 4
Two step Binomial Markov Random Walk Model
This table shows a two-steps step binomial Markov Random Walk model. S¬0 is the initial stage, S¬1 is
step one and S¬2 is step two. ― ‖ is the absolute minimum value of prices in all three stages.
0 +1 +2 0
0 +1 0 0
0 -1 0 1
0 -1 -2 2
The first moment of Expected Lifetime Shortfall (ELS) is denoted here by ELS1 in
equation 3.13.
ELS1 = 1 * (1-p)/2 + 2 * p/2 = (1+p)/2 (3.13)
The second moment of Expected Lifetime Shortfall (ELS) is denoted here by ELS2 in
equation 3.14.
ELS2 = 1^2 * (1-p)/2 + 2^2 * p/2 = (1+3p)/ 2 (3.14)
ELS is a measure of risk. The lower the ELS, the lower the risk. ELS1 attains its lowest
value of 0.5 when p = 0 and highest value 1 when p = 1. ELS2 attains its lowest value of 0.5 when
p = 0 and highest value 2 when p = 1. Although the mathematical calculations become complex
as we increase the number of steps, ELS1 and ELS2 attain their lowest values of 0.5 when p = 0
irrespective of the step numbers.
28
Table 5
The Expected lifetime shortfalls
This table reports Expected lifetime shortfall measures (ELS1 and ELS2) for different values of
continuation probabilities (p) and the number of steps (N) in the path,
N=2 N=4 N=8 N=16 N=32 N=64 N=128 N=256 N=∞
ELS 1xpected Lifetime Shortfall 1
p=0.05 0.373 0.288 0.236 0.208 0.193 0.187 0.183 0.182 0.183
p=0.10 0.390 0.324 0.288 0.272 0.266 0.264 0.261 0.262 0.266
p=0.20 0.426 0.393 0.383 0.381 0.384 0.387 0.389 0.391 0.399
p=0.30 0.460 0.459 0.469 0.480 0.490 0.499 0.504 0.509 0.522
p=0.40 0.496 0.528 0.556 0.580 0.599 0.615 0.623 0.630 0.651
p=0.50 0.532 0.595 0.646 0.686 0.717 0.740 0.755 0.766 0.798
p=0.60 0.567 0.666 0.745 0.806 0.853 0.887 0.911 0.927 0.977
p=0.70 0.602 0.740 0.859 0.954 1.028 1.082 1.122 1.145 1.219
p=0.80 0.638 0.820 1.001 1.153 1.276 1.361 1.424 1.469 1.596
p=0.90 0.673 0.907 1.178 1.453 1.694 1.879 2.019 2.126 2.394
p=0.95 0.691 0.953 1.288 1.675 2.081 2.424 2.705 2.912 3.478
Expected Lifetime Shortfall 2
p=0.05 0.289 0.170 0.110 0.082 0.066 0.059 0.055 0.054 0.053
p=0.10 0.326 0.215 0.160 0.134 0.121 0.115 0.111 0.111 0.111
p=0.20 0.402 0.313 0.272 0.255 0.248 0.246 0.245 0.245 0.250
p=0.30 0.475 0.423 0.405 0.402 0.403 0.409 0.412 0.416 0.429
p=0.40 0.552 0.554 0.568 0.590 0.607 0.623 0.630 0.638 0.667
p=0.50 0.627 0.706 0.775 0.831 0.876 0.910 0.931 0.947 1.000
p=0.60 0.702 0.889 1.042 1.165 1.256 1.318 1.367 1.395 1.500
p=0.70 0.777 1.102 1.409 1.665 1.855 1.993 2.097 2.148 2.333
p=0.80 0.852 1.357 1.951 2.493 2.922 3.226 3.439 3.589 4.000
p=0.90 0.927 1.657 2.755 4.093 5.380 6.392 7.140 7.689 9.000
p=0.95 0.964 1.822 3.310 5.554 8.393 11.084 13.305 14.904 19.000
29
Figure 6: Expected lifetime short-fall 1
(For different values of continuation probabilities (p) and the number of steps (N) in the path)
Figure 7: Expected lifetime short-fall 2
(For different values of continuation probabilities (p) and the number of steps (N) in the path)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0.0
1
0.0
6
0.1
1
0.1
6
0.2
1
0.2
6
0.3
1
0.3
6
0.4
1
0.4
6
0.5
1
0.5
6
0.6
1
0.6
6
0.7
1
0.7
6
0.8
1
0.8
6
0.9
1
0.9
6
ELS
1
Continuation Probability p
ELS1 in the classic BMRW Model (As a function of p for different values of N)
n=2
n=4
n=8
n=16
n=32
n=64
n=128
n=256
Limit
30
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