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8/13/2019 Forecasting Accuracy of Commodity Futures Against Random Walk
1/22
Can Futures Outperform a Random Walk in Commodity
Markets?
Andrew Krog*
University of Michigan
April 17, 2013
Abstract
The idea that commodity futures are efficient forecasters of future spot prices has never
been proven definitively. Nonetheless, it is a widely held belief among market participants. This
paper explores the theory and tests the forecasting ability of the futures price against a simple
random walk model, that has proven difficult to beat in many forecasting applications. Six highly
traded commodities (corn, crude oil, gasoline, gold, natural gas, silver) from three commodity
classes are included to identify differences in forecasting ability across classes. Recursive
forecasts are constructed for 1-, 3-, 6-, 9-, and 12-month horizons using a spread regression to
obtain mean square prediction errors, which are compared between the regression and random
walk model. The modified Diebold-Mariano statistic is used to test the null hypothesis of equal
predictive power. This statistic is then bootstrapped using a nave block bootstrap to find 95%
confidence intervals. The results show no statistical difference between the random walk and
spread regression, although the MSPE is lower for the random walk for the majority of cases.
Keywords: Recursive Forecasts, Diebold-Mariano Statistic, Block Bootstrap
* Department of Economics, University of Michigan, Ann Arbor, MI 48109. Email: akrog@umich.edu
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1 Introduction
The question of whether or not commodity futures prices are good forecasters of future
spot prices is one that has been frequently addressed yet never answered definitively.
Commodity price forecasts are widely used by policymakers as well as businesses, so
understanding the reliability of these forecasts is crucial to making optimal policy and
investment decisions. Intuitively, a forecasting method should outperform a random walk
benchmark in order to be considered for policy of investment decision making. A frequently
used method, which is the one addressed in this paper, is the futures price.
Futures prices are often looked to as a measure of expected future spot price due to the
assumption that all information available to the market is reflected in the current future price.
However, futures prices have shown differing levels of reliability depending on commodity class
and forecast horizon. This paper will test the forecasting ability of futures prices against a
random walk at 1-, 3-, 6-, 9-, and 12-month horizons for three classes of commodities: softs,
metals, and energy. The six commodities analyzed are corn, crude oil, gasoline, gold, natural gas,
and silver.
Forecasting ability will be determined by constructing recursive forecasts regressing the
change in spot price on the basis (difference between futures price and spot) and an intercept
using least squares. The Diebold-Mariano test statistic will then be used to determine whether the
difference in the predicted mean squared errors for random walk and spread regression is
significantly greater than zero (one-sided test). Since we are considering nested models with
parameter uncertainty we will use the variance of the spread regression predicted mean square
errors to determine the standard error. Furthermore, it is necessary to bootstrap this statistic since
the distribution is not asymptotically N(0,1), a mistake frequently made in past research using the
Diebold-Mariano statistic.
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2 Data
The data begins in December 1993 and is collected monthly through December 2012
except for gasoline which only has futures data starting in 2006. Prices are taken in a three day
window between the 14thand 16thof every month depending on which are trading days.
Furthermore, the month used for each horizon depends on the last trading day on the contract.
Specifications vary by commodity, and the last trading day for contracts differs widely. Trading
for gold and silver ends on the 3rdlast business day of the month; corn ends on the business day
prior to the 15thof delivery month; natural gas terminates trading three days prior to the first day
of delivery month, gasoline on the last day of month before delivery, and crude oil on the 3rd
business day prior to the 25thof the month preceding delivery.
Table 1. Commodity Contract Details
Commodity Currency/Units Spot Futures Exchange
Corn /Bu Corn No. 2 Yellow CBOT
Crude Oil $/bbl WTI Spot Cushing NYMEX
Gasoline /gal Gasoline RBOB NYMEX
Gold $/Troy oz. Gold Bullion London Bullion Market COMEX
Natural Gas $/MMBtu Natural Gas-Henry Hub NYMEX
Silver $/Troy oz. Silver Fix LBM Cash COMEX
The futures price data is collected from Bloomberg using all available contracts for each
commodity. The energy and metals commodities have contracts expiring in each of the twelve
months while corn only offers five. The spot prices were collected from Datastream over the
same period. The future spot prices are taken for the expiration month of the futures contract and
the day is chosen to match the horizon.
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3 Methods
3.1 Forecasting Methods
The model that will be tested for forecasting efficiency of the futures price is a spread
regression used to predict the change in spot price. This model will be tested against a random
walk benchmark.
The future spot price in the random walk model at the h month horizon is simply the
current spot price plus the sum of errors from next period through period h. Since the expected
value of the future period errors is zero, the random walk model implies a no change forecast and
the current spot price is the expected future spot price.
(1) t+h | t= StFor the spread regression, least squares is used to regress the log difference of future and
current spot price on the log difference of current futures price at horizon h and current spot
price.
(2) st+h- st= + *(fthst) + t+h(3) t+h | t=fth- st
The expectation is that =1 and =0 (equation 3) under the null hypothesis of forecast efficiency,
but the intercept term is allowed to be nonzero in the regression.
A key assumption in this least squares construction of the regression is that the policy
makers or investors using the forecasts have a quadratic loss function which may not be the case.
An absolute loss function does not penalize large deviations from the estimated price
disproportionately whereas the least squares method does. Furthermore, the loss function may
not even be symmetric as both the quadratic and absolute loss functions are. One example would
be policy makers concerned with gas prices rising above a certain level but not concerned with
price drops. However, since it is impossible to know the loss function of all commodity price
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forecasters, the quadratic loss function is a reasonable assumption. Symmetry of the loss function
implies risk neutrality and the quadratic nature is used mainly for convenience.
3.2 Recursive Pseudo Forecasts
The two models are used to generate pseudo out-of-sample forecasts to simulate
forecasting with the data in real-time. The initial forecasting period of 24 months is large enough
to give reliable estimates of the regression parameters and small enough to give reliable
estimates of the PMSE with the remaining sample. For each period from May 1996 (the 25th
month of data) until November 2012, the spread regression is re-estimated using the least squares
method described in section 3.1 on all past data. The regression at each period is used to generate
forecasts of the change in spot price at the corresponding horizon, which be compared with the
actual future spot price to construct the PMSE for the model. For the random walk, the PMSE is
simply obtained by using the spot price and h-month horizon future spot.
3.3 Diebold-Mariano Pseudo Out-of-Sample Test for Predictive Accuracy
In order to test for differences in predictive accuracy between the spread regression and
random walk model, the Diebold-Mariano test statistic is employed. The original statistic
supplied by Diebold and Mariano is:
DM =
1
-u0t
-u1t ar 1
-u0t
-u1t ~ N(0,1)
The notation is as follows: and are the recursive pseudo out-of-sampleforecast errors from the null model and alternative model respectively, T is the total sample size,
and R is the initial recursive window.
The spread regression is a nested model of the random walk since the models are
equialent when the parameters are zero. herefore, under the null hypothesis of =0, the
8/13/2019 Forecasting Accuracy of Commodity Futures Against Random Walk
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ariance of the difference in PMSEs will conerge to zero and the statistic will go to infinity.
The solution proposed by McCracken uses the variance of the PMSE of the alternative model,
which must be non-zero (Clark & McCracken, 1999). The new statistic is therefore:
DM =1
- u0t -u1t ar u1t
~ F
However, the distribution of test statistic is no longer asymptotically N(0,1) but rather an
unknown distribution F that must be bootstrapped.
3.4 Bootstrapping the distribution of the modified Diebold-Mariano statistic
Since the distribution F is unknown, nonparametric bootstrapping methods are used. A
nave block bootstrap is chosen due to autocorrelation in the prediction errors that can be seen in
the autocorrelation plots in the appendix. Nave block bootstraps allow serial correlation to be
retained in the resampling by selecting blocks whose length is increased as sample size increases.
Furthermore, we do not need to use a blocks of blocks bootstrap because the statistic of interest
is symmetric since it involves only a mean and variance. The bootstrap replication is therefore:
{yt*
t=1
= (1*,
*, , r*) where s/l
The length of the original sample is T, the number of blocks used to resample is s, and block
length is l. Also, yt*denotes the difference in squared prediction error between random walk and
spread regression at time t ( 0t -u1t ). For gasoline, l is chosen to be 5 since the sample size isvery short; for all other commodities, 15 is used. To construct the Diebold-Mariano statistic for
each of the 20,000 trials, first a simple mean function is used:
y *= yt=1 t*
Then this mean is divided by the variance of u1t using Newey-West standard errors:
8/13/2019 Forecasting Accuracy of Commodity Futures Against Random Walk
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y *
aru1t
Once the vector of 20,000 simulated statistics is constructed, 95% confidence intervals are found
by calculating the 2.5 and 97.5 percentiles. The null hypothesis is that there is no difference in
forecasting efficiency between the random walk and spread regression. The alternative is that
they do not have the same forecasting efficiency. If the calculated modified Diebold-Mariano test
statistic is outside of the 95% confidence interval then the null hypothesis is rejected at the 5%
level.
3.5 Possible Issues with Proposed Methods
It should be noted that there are multiple ambiguities in the above methods. First of all,
the confidence interval is sensitive to the choice of block length, and there is no general rule for
choosing it. The goal is to preserve any serial correlation in the data, so as sample size increases,
bootstrap block length will also increase. However, there is no obvious reason for choosing
specific numbers in this case. Another issue is the choice of lag for HAC standard errors. The
autocorrelation plots were observed to help with the decision, but these plots are by no means
definitive. This is less of a problem since the same standard errors are included in the calculation
of the bootstrapped distribution.
4 Results
The results show that the spread regression using the futures price had a lower MSPE in
only five of the thirty total cases for six commodities at five different horizons. The only cases
where the random walk has a larger MSPE are corn 3- and 6-month horizons as well as gasoline
3-, 6-, and 12-month horizons. None of these five are significant at the 5% level (Diebold
Mariano statistics and 95% confidence intervals are included in Appendix 1). However, even
though random walk has the lower MSPE in each of the other 25 cases, none of those are
statistically significant at the 5% level. Although there are no statistically significant results, the
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fact that the MSPE is lower for the random walk in the majority of cases across three categories
of highly traded commodities shows that the futures price used in the spread regression is not an
ideal measure of the expected spot price.
Table 2. Mean Square Prediction Errors
Commodity Model 1-month 3-month 6-month 9-month 12-month
Corn RW 1486.5 3112.6 7669.5 11187 12775
Spread 1500.3 3030.9 7643 11594 13498
Crude Oil RW 9.1 80.1 257.8 348.2 377.8
Spread 29.6 126.2 288 360 404.7
Gasoline RW 541.8 1162 2210.7 2761 2969.1
Spread 588.5 1157.8 2126.1 2825 2964
Gold RW 1694.7 3763.9 8579.3 13705 20405
Spread 1703.2 3826.7 9049.9 14984 21840
Natural Gas RW .6476 1.9426 3.5552 4.5541 5.0215
Spread .6584 2.2415 4.1123 5.4413 5.6826
Silver RW 2.3 5.7 12.4 19.6 25.4
Spread 2.4 5.8 12.6 20.2 25.7
One explanation for this, as noted in Alquist & Kilian (2010), is that the difference
between the futures price and spot price is highly variable and may tend to fluctuate around the
current spot price. This is found to also be the case here although to a lesser extent. The
variability is not so large that it made the random walk a statistically significant more efficient
forecaster, although it does.
Another problem with the spread regression is that the bias and MAPE are very close to
those of the random walk for most horizons and commodities. Since the spread is higher variancethan the spot price, it needs to make up for this by being less biased, but that does not appear to
be the case here. One notable exception was corn for which the spread was significantly less
biased than the spot price. This led to lower mean square prediction errors for two horizons.
Furthermore, forecasts for all commodities were biased toward overestimating the future spot
price except for corn and the spread regression for natural gas.
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5 Conclusion
As the results show, using the futures price in a spread regression does not provide a
better forecast for future spot price than a simple random walk model that uses a no-change
forecast. This has important implications for policy makers and investors using futures markets
to forecast or hedge. The assumption of futures prices as an efficient forecasting measure does
not seem to hold across any of the three commodity classes. There was no significant evidence
that the current spot price is a preferred method although it did have lower MSPE for nearly all
commodities and horizons. Further research might explore alternative uses of the futures price in
forecasting, particularly for the commodities in which it was continually overestimating the
future spot.
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References
Alquist, R., & Kilian, L. (2010). What Do We Learn from the Price of Crude Oil Futures?Journal of
Applied Econometrics, 25(4), 539-573.
Bloomberg L.P. (2013, April 10). Commodity Futures Historical Contracts.
Clark, T. E., & McCracken, M. W. (1999). Tests of equal forecast accuracy and encompassing for nested
models.Journal of Econometrics, 105(1), 85-110.
Datastream International Commodity Spot Price Database (2013, April 10). Datastream International.
Diebold, F. X., & Mariano, R. S. (1995). Comparing Predictive Accuracy.Journal of Business &
Economic Statistics, 13(3), 253-263.
CME Group (2013, April 10). Corn Futures Contract Specifications.Retrieved from CME Group:
http://www.cmegroup.com/trading/agricultural/grain-and-oilseed/corn_contract_specifications.html
CME Group (2013, April 10). Gold Futures Contract Specifications.Retrieved from CME Group:
http://www.cmegroup.com/trading/metals/precious/gold_contract_specifications.html
CME Group (2013, April 10).Henry Hub Natural Gas Futures Contract Specifications.Retrieved from
CME Group: http://www.cmegroup.com/trading/energy/natural-gas/natural-
gas_contract_specifications.html
CME Group (2013, April 10).Light Sweet Crude Oil (WTI) Futures Contract Specifications.Retrieved
from CME Group: http://www.cmegroup.com/trading/energy/crude-oil/light-sweet-
crude_contract_specifications.html
CME Group (2013, April 10).RBOB Gasoline Futures Contract Specifications.Retrieved from CME
Group: http://www.cmegroup.com/trading/energy/refined-products/rbob-
gasoline_contract_specifications.html
CME Group (2013, April 10). Silver Futures Contract Specifications.Retrieved from CME Group:
http://www.cmegroup.com/trading/metals/precious/silver_contract_specifications.html
Kilian, L. (1999). Exchange Rates and Monetary Fundamentals: What Do We Learn from Long-Horizon
Regressions?Journal of Applied Econometrics, 14(5), 491-510.
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Appendix 1Tables
Bias
Commodity Model 1-month 3-month 6-month 9-month 12-month
Corn RW -3.37 3.51 20.25 23.48 34.50
Spread -3.87 -6.60 0.40 2.07 10.66
Crude Oil RW 0.43 1.26 2.58 3.61 4.69
Spread 0.43 1.03 2.32 3.56 4.92
Gasoline RW 3.46 13.66 27.99 35.83 43.25
Spread 6.13 21.36 34.28 39.21 45.56
Gold RW 6.14 19.56 38.52 57.52 78.64
Spread 6.38 19.69 37.54 55.05 81.09
Natural Gas RW 0.01 0.01 0.02 0.02 0.04Spread -0.01 -0.08 -0.08 -0.11 -0.07
Silver RW 0.12 0.40 0.78 1.13 1.57
Spread 0.09 0.35 0.67 0.99 1.58
Mean Absolute Prediction Error
Commodity Model 1-month 3-month 6-month 9-month 12-month
Corn RW 24.57 38.39 59.11 70.35 79.39
Spread 24.55 38.67 59.18 72.36 82.56
Crude Oil RW 3.68 6.72 10.20 11.95 13.62
Spread 2.02 5.64 9.48 11.62 13.08
Gasoline RW 19.79 24.63 34.20 44.74 46.43
Spread 18.45 25.64 37.31 44.35 47.62
Gold RW 23.35 38.55 63.17 84.16 100.68
Spread 23.18 38.01 61.24 80.59 97.16
Natural Gas RW 0.54 1.03 1.33 1.55 1.62
Spread 0.54 0.97 1.26 1.48 1.66
Silver RW 0.80 1.28 1.97 2.54 2.87
Spread 0.77 1.27 1.93 2.47 2.82
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Diebold-Mariano Statistics and Bootstrapped 95% Confidence Intervals
Corn
Horizon D-M Statistic Bootstrapped 95% Confidence Interval
1-month 4.20 [ 0.67, 11.44 ]3-month -12.68 [ -55.27, 26.86 ]
6-month -2.13 [ -46.50, 90.80 ]
9-month 22.61 [ -39.28, 127.92 ]
12-month 32.13 [ -41.35, 142.20 ]
Crude Oil
Horizon D-M Statistic Bootstrapped 95% Confidence Interval
1-month 41.78 [ 12.45, 87.21 ]3-month 33.46 [ 2.13, 87.70 ]
6-month 12.87 [ .36, 30.59 ]
9-month 4.60 [ -.56, 10.99 ]
12-month 8.61 [ -.96, 21.71 ]
Gasoline
Horizon D-M Statistic Bootstrapped 95% Confidence Interval
1-month 17.26 [ -4.0, 25.10 ]
3-month -0.69 [ -73.82, 43.75 ]6-month -9.25 [ -77.80, 11.61 ]
9-month 10.65 [ -42,14, 22.90 ]
12-month -0.67 [ -26.82, 19.50 ]
Natural Gas
Horizon D-M Statistic Bootstrapped 95% Confidence Interval
1-month 0.18 [ -0.06, 0.38 ]
3-month 1.83 [ -0.52, 5.36]
6-month 2.00 [ -1.33, 6.99 ]
9-month 2.69 [ -0.17, 7.26 ]
12-month 1.64 [ -1.13, 5.67 ]
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Gold
Horizon D-M Statistic Bootstrapped 95% Confidence Interval
1-month 3.29 [ -.98, 10.50 ]
3-month 9.15 [ -0.10, 29.77 ]
6-month 37.71 [ 7.71, 85.73 ]9-month 57.09 [ 20.17, 159.45 ]
12-month 49.34 [ 18.26, 147.51 ]
Silver
Horizon D-M Statistic Bootstrapped 95% Confidence Interval
1-month 1.34 [ -0.08, 2.11 ]
3-month 0.17 [ -0.19, 0.48 ]
6-month 0.40 [ -0.35, 1.38 ]
9-month 0.77 [ -0.23, 2.27]
12-month 0.36 [ -1.22, 1.48 ]
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Appendix 2Figures
Autocorrelation Function Plots
Corn
Crude Oil
0 5 10 15 20 25 30-0.5
0
0.5
1
1-Month Horizon
0 5 10 15 20 25 30-0.5
0
0.5
1
3-Month Horizon
0 5 10 15 20 25 30-0.5
0
0.5
1
6-Month Horizon
0 5 10 15 20 25 30-0.5
0
0.5
1
9-Month Horizon
0 5 10 15 20 25 30-0.5
0
0.5
1
12-Month Horizon
0 10 20 30-1
0
1
1-Month Horizon
0 10 20 30-1
0
1
3-Month Horizon
0 10 20 30-1
0
1
6-Month Horizon
0 10 20 30-1
0
1
9-Month Horizon
0 10 20 30-1
0
1
12-Month Horizon
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Gasoline
Gold
0 10 20 30-1
0
1
1-Month Horizon
0 10 20 30-1
0
1
3-Month Horizon
0 10 20 30-1
0
1
6-Month Horizon
0 10 20 30-1
0
1
9-Month Horizon
0 10 20 30-1
0
1
12-Month Horizon
0 10 20 30-1
0
1
1-Month Horizon
0 10 20 30-1
0
1
3-Month Horizon
0 10 20 30
-1
0
1
6-Month Horizon
0 10 20 30
-1
0
1
9-Month Horizon
0 10 20 300
0.5
1
12-Month Horizon
8/13/2019 Forecasting Accuracy of Commodity Futures Against Random Walk
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Natural Gas
Silver
0 10 20 30-1
0
1
1-Month Horizon
0 10 20 30-1
0
1
3-Month Horizon
0 10 20 30-1
0
1
6-Month Horizon
0 10 20 30-1
0
1
9-Month Horizon
0 10 20 30-1
0
1
12-Month Horizon
0 10 20 30-1
0
1
1-Month Horizon
0 10 20 30-1
0
1
3-Month Horizon
0 10 20 30-1
0
1
6-Month Horizon
0 10 20 30-1
0
1
9-Month Horizon
0 10 20 30-1
0
1
12-Month Horizon
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Futures Prices
1994 1996 1998 2000 2002 2004 2006 2008 2010 2012
20
40
60
80
100
120
140
Crude Oil Futures Price for all Horizons
Date
Price($/bbl)
1-month
3-month
6-month
9-month
12-month
2007 2007.5 2008 2008.5 2009 2009.5 2010 2010.5 2011 2011.5 2012
150
200
250
300
350
Gasoline Futures Price for all Horizons
Date
Price(cents/gal)
1-month
3-month
6-month
9-month
12-month
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1994 1996 1998 2000 2002 2004 2006 2008 2010 2012
400
600
800
1000
1200
1400
1600
1800
Gold Futures Price for all Horizons
Year
Price($/Troyoz.)
1-month
3-month
6-month
9-month
12-month
1994 1996 1998 2000 2002 2004 2006 2008 2010 2012
5
10
15
20
25
30
35
40
Silver Futures Price at all Horizons
Year
Price($/Troyoz.)
1-month
3-month
6-month
9-month
12-month
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1994 1996 1998 2000 2002 2004 2006 2008 2010 2012
2
4
6
8
10
12
14
Natural Gas Futures Price for all Horizons
Year
Price($/MMBtu)
1-month
3-month
6-month
9-month
12-month
1994 1996 1998 2000 2002 2004 2006 2008 2010 2012
200
300
400
500
600
700
Corn Futures Price for all Horizons
Year
Price(cents/bu)
1-month
3-month
6-month
9-month
12-month
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Spreads
1995 2000 2005 2010-0.05
0
0.05
0.1
1-month1995 2000 2005 2010
-0.2
-0.1
0
0.1
0.2
Corn Spread for All Horizons
3-month1995 2000 2005 2010
-0.3
-0.2
-0.1
0
0.1
0.2
6-month
1995 2000 2005 2010
-0.2
-0.1
0
0.1
0.2
0.3
9-month1995 2000 2005 2010
-0.2
-0.1
0
0.1
0.2
0.3
0.4
12-month
1995 2000 2005 2010
-0.2
-0.1
0
0.1
1-month1995 2000 2005 2010
-0.2
-0.1
0
0.1
Crude Oil Spread for All Horizons
3-month1995 2000 2005 2010
-0.2
-0.1
0
0.1
0.2
6-month
1995 2000 2005 2010-0.3
-0.2
-0.1
0
0.1
0.2
9-month1995 2000 2005 2010
-0.3
-0.2
-0.1
0
0.1
0.2
12-month
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2007 2008 2009 2010 2011 2012-0.3
-0.2
-0.1
0
0.1
1-month2007 2008 2009 2010 2011 2012
-0.2
-0.1
0
0.1
0.2
Gasoline Spread for All Horizons
3-month2007 2008 2009 2010 2011 2012
-0.3
-0.2
-0.1
0
0.1
0.2
6-month
2007 2008 2009 2010 2011 2012
-0.3
-0.2
-0.1
0
0.1
0.2
9-month2007 2008 2009 2010 2011 2012
-0.2
-0.1
0
0.1
0.2
12-month
1995 2000 2005 2010
-0.01
-0.005
0
0.005
0.01
0.015
1-month1995 2000 2005 2010
0
0.01
0.02
0.03
Gold Spread for All Horizons
3-month1995 2000 2005 2010
0
0.01
0.02
0.03
0.04
0.05
0.06
6-month
1995 2000 2005 2010
0
0.02
0.04
0.06
0.08
9-month1995 2000 2005 2010
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
12-month
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1995 2000 2005 2010
-0.5
0
0.5
1-month1995 2000 2005 2010
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Natural Gas Spread for All Horizons
3-month1995 2000 2005 2010
-0.4
-0.2
0
0.2
0.4
6-month
1995 2000 2005 2010-0.5
0
0.5
9-month1995 2000 2005 2010
-0.4
-0.2
0
0.2
0.4
12-month
1995 2000 2005 2010-0.06
-0.04
-0.02
0
0.02
0.04
1-month1995 2000 2005 2010
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Silver Spread for All Horizons
3-month1995 2000 2005 2010
-0.04
-0.02
0
0.02
0.04
0.06
0.08
6-month
1995 2000 2005 2010
0
0.05
0.1
9-month1995 2000 2005 2010
-0.2
0
0.2
0.4
12-month
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