1 modelling and measuring price discovery in commodity markets isabel figuerola-ferretti jesús...
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Modelling and Measuring Price Discovery in
Commodity Markets
Isabel Figuerola-FerrettiJesús Gonzalo
Universidad Carlos III de MadridBusiness Department and Economics
Department
December 2007
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Trading Places Movie
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Two Whys
There are two standard ways of measuring the contribution of financial markets to the price discovery process:
(i) Hasbrouck (1995) Information Shares (ii) Gonzalo and Granger (1995) P-T
decomposition, suggested by Harris et al. (1997)
We want to find a THEORETICAL JUSTIFICATION for the USE of GG P-T decomposition for price
discovery.
Can the cointegrating vector be different from (1, -1)? Empirically Yes; but theoretically?
YES, TOO.
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Other minor Whys
Why Price Discovery? Markets have two important functions:
Liquidity and Price Discovery, and these functions are important for asset pricing.
Why Commodities? Commodities, in sharp contrast to more
traditional financial assets, are more tied to current economic conditions.
Why Metals? The chief market place is the London Metal
Exchange (LME).
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Road Map
Introduction
Equilibrium Model of Commodity St and Ft Prices with Finite
Elasticity of Arbitrage Services + Convenience Yields (built on Garbade and Silver (1983))
Econometric Implementation : Theoretical model and the GG P-T decomposition
Data (London Metal Exchange : Al, Cu, Ni, Pb, Zn)
Results (Backwardation; and dominant markets in the price discovery process)
Conclusions
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Introduction
Future markets contribute in three important ways to the organization of economic activity:
1. they facilitate price discovery
2. they provide an arena for speculation
3. they offer means of transferring risk or hedging.
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Introduction
Price discovery is the process by which security or commodity markets attempt to identify permanent changes in equilibrium transaction prices.
The unobservable permanent price reflects the fundamental value of the stock or commodity.
It is distinct from the observable price, which can be decomposed into its fundamental value and its transitory effects (due to the bid-ask bounce, temporary order imbalances, inventory adjustments, etc) *
t t tP P e
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Introduction
For producers as well as consumers it is important to determine where the price information and price discovery are being produced.
More on Price Discovery: The process by which future and cash markets
attempt to identify permanent changes in equilibrium transaction prices.
If we assume that the spot and future prices measure a common efficient price with some error, price discovery quantifies the contribution of spot and future prices to the revelation of the common efficient price.
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Introduction
Specific Contributions: We extend the equilibrium model of the term structure of commodity prices
developed by Garbade and Silver (1983) (GS) by incorporating endogenously
convenience yields. This allows us to capture the existence of Backwardation
and Contango. This is reflected on a cointegrating vector (1, -), different from
the standard and always present =1. When >1 (<1) the market is under
backwardation (contango).
Independent of , we prove that the equilibrium model can be written as an error
correction model, where the permanent component of the GG P-T decomposition
coincides with the price discovery process of GS. This justifies theoretically the
use of this type of decomposition.
All the results in the paper are testable, as it can be seen in the application to non-ferrous
metal markets:
(i) All the markets are in Backwardation but Copper
(ii) For those metals with highly liquid future markets, future prices are the
dominant factor in the price discovery process.
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Literature Review 1:Literature on price discovery
Garbade, K. D. & Silver W. L. (1983). Price movements and price discovery in futures and cash markets. Review of Economics and Statistics. 65, 289-297.
Hasbrouck, J. (1995). One security, many markets: Determining the contributions to price discovery. Journal of Finance 50, 1175-1199.
Harris F. H., McInish T. H., Wood R. A. (1997).”Common Long-Memory Components of Intraday Stock Prices: A Measure of Price Discovery.” Wake Forest University Working Paper.
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Hasbrouck, J. (1995). One security, many markets: Determining the contributions to price discovery. Journal of Finance 50, 1175-1199.
Gonzalo, J. Granger C. W. J (1995). Estimation of common long-memory components in cointegrated systems. Journal of Business and Economic Statistics 13, 27-36.
Literature Review 2:Price discovery metrics
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Baillie R., Goffrey G., Tse Y., Zabobina T. (2002). Price discovery and common factor models.
Harris F. H., McInish T. H., Wood R. A. (2002). Security price adjustment across exchanges: an investigation of common factor components for Dow stocks.
Hasbrouck, J. (2002). Stalking the “efficient price” in market microstructure specifications: an overview.
Leathan Bruce N. (2002). Some desiredata for the
measurement of price discovery across markets.
De Jong, Frank (2002). Measures and contributions to price discovery: a comparison.
Literature Review 3Comparing the two metrics of price
discovery:Special Issue Journal of Financial Markets 2002
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Theoretical Model: Extension of Garbade and Silber (1983)
Equilibrium with infinitely elastic supply of arbitrage
St = Log of the spot market price at time “t”
Ft = Log of the contemporaneous price on a futures contract for a commodity for settlement after a time interval T1= T-t (e.g. 15 months)
rt interest rate applicable to the interval from t to T.
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Equilibrium with infinitely elastic supply of arbitrage
Standard Assumptions:
1) No taxes or transaction cost 2) No limitations on borrowing 3) No costs other than financing + storage a
(short or long) future position 4) No limitations on short sale of the
commodity in the spot market5) Interest rate rt + storage cost ct = +
I(0), with the mean of (rt + ct)
6) St is I(1).
_
r
_
rc_
rc
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Equilibrium with infinitely elastic supply of arbitrage
Let T1=1
Non-arbitrage equilibrium conditions imply
Given the above assumptions, equation (1) implies that St and Ft are cointegrated with the always present cointegrating vector (1, -1).
(1) )c( t ttt rSF
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In consumption commodities is very likely that
with
where is the convenience yield.
Convenience yield is the flow of services that accrues to an owner of the physical commodity but not to an owner of a contract for future delivery of the commodity (Brennan Schwartz (1985) ). The existence of convenience yields can produce two situations very common in commodity markets: BACKWARDATION and CONTANGO.
A bit of more realism: Convenience Yields
)c( t ttt rSF
(2) )c( t tttt rSyF
ty
16
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One more Assumption:
7) The convenience yield is modeled as
with .
Convenience Yields
1,2(0, 1)i
17
(3) 21 ttt FSy
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Backwardation refers to futures prices that decline with time to maturity
Contango refers to futures prices that rise with time to maturity
$36,00
$36,50
$37,00
$37,50
$38,00
$38,50
$39,00
$39,50
$40,00
$40,50
$41,00
$41,50
April-04 June-04 August-04 September-04
November-04 December-04 February-05 April-05 May-05 July-05
Oil
pri
ce (
$/ba
rrel
)
$396
$397
$398
$399
$400
$401
$402
$403
$404
$405
Gol
d pr
ice
($/T
roy
ounc
e)
Crude Oil Gold
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Equilibrium with convenience yields
Substituting (3) into (2) + (a.5)
with and .
It is important to notice the different values that 2 can take
1) 2 >1 then 1>2 . In this case we are under the process of long-run backwardation (“St>Ft” in the long-run)
2) 2=1 then 1=2. In this case we do not observe long-run backwardation or contango
3) 2<1 then 1<2 . In this case we are under the process of long-run contango
(“St<Ft” in the long-run)
(4) )0(32 IFS tt
1
22 1
1
1
3 1
rc
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Equilibrium with convenience yields
Some remarks: The parameters 1 and 2 are not identified in the
equilibrium equation (4) unless is known, or for instance we impose 1 + 2 =1. In the fomer case:
1 = 1+rc/ β3 and 2 = 1- β2 (1- 1 ).
Convenience yields are stationary when β2 =1. When β2 1 it contains a small random walk component. The size depends on the difference (2 -1).
_
rc
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Equilibrium with finitely elastic supply of arbitrage
services
In realistic cases we expect the arbitrage transactions of buying in the cash market and selling the futures contracts or vice versa not to be riskless: unknown transaction costs, unknown convenience yields, constraints on warehouse space, basis risk, etc. These are the cases of finite elasticity of arbitrage services.
To describe the interaction between cash and future prices we must first specify the behaviour of agents in the marketplace.
There are Ns participants in spot market. There are Nf participants in futures market. Ei,t is the endowment of the ith participant immediately prior
to period t. Rit is the reservation price at which that participant is willing
to hold the endowment Ei,t. Elasticity of demand, the same for all participants.
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Demand schedule of ith participant in spot market
where A is the elasticity of demand
Aggregate cash market demand schedule of arbitrageurs in period t
where H is the elasticity of cash market demand by arbitrageurs. It is finite when the arbitrage transactions of buying in the cash market and selling the futures contract or vice versa are not riskless.
Equilibrium with finitely elastic supply of arbitrage services
, , , 0, 1,..., (5)i t t i t sE A S R A i N
2 3( ) , 0 (6)tH F S H
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The cash market will clear at the value of St that solves
The future market will clear at the value of Ft such that
(7) )()( 321
,,1
, tt
N
ititti
N
iti SFHRSAEE
ss
, , , 2 31 1
( ) ( ) (8)F FN N
j t j t t j t t tj j
E E A F R H F S
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Equilibrium with finitely elastic supply of arbitrage services
Solving the clearing market conditions as a function of
the mean reservation prices and
2
3
2
322
)(
)(
(9) )(
)(
SFs
sFtFs
StS
t
SFs
FFtF
StSF
t
HNNANH
HNRNANHRHNF
HNNANH
HNRHNRNHANS
SN
itiS
St RNR
1,
1
FN
jtjF
Ft RNR
1,
1
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Dynamic price relationships
To derive dynamic price relationships, we need a description of the evolution of reservation prices.
, 1 ,
, 1 ,
,
, ,
, 1,...,
, 1,..., (10)
cov( , ) 0,
cov( , ) 0,
i t t t i t S
j t t t j t F
t i t i
i t j t
R S v w i N
R F v w j N
v w
w w i j
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And the mean reservation prices
with
1
1
, 1,...,
, 1,..., (11)
S St t t t S
F Ft t t t F
R S v w i N
R F v w j N
,,11 ,
FSNN
FSj ti t
jS Fit t
S F
www w
N N
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Dynamic price relationships: VAR model
where
and
Garbade and Silver (with stop their analysis at this point stating that
FS
F
NN
N
Measures the importance of future
markets relative to cash markets
13
1
(12)S
t F t tF
t S t t
S N S uHM
F N Fd u
Ftt
Stt
Ft
St
wv
wvM
u
u
2 2
2
( )1( )
( )
S F F
S S F
S F S
N H AN HNM
HN H AN Nd
d H AN N HN
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VECM Representation
13
1
2
2
(13)
1
St F t t
Ft S t t
F F
S S
S N S uHM I
F N Fd u
where
HN HNM I
HN HNd
(14)
1
1 1
1
32
Ft
St
t
t
S
F
t
t
u
uF
S
N
N
d
HF
S
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The GG permanent component is…
tFS
Ft
FS
S FNN
NS
NN
N)()(
This is our price discovery metric, which coincides with the one proposed by GS. Our metric does not depend on the existence of backwardation or contango.
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Two extreme cases:
1. H = 0 No VECM, no cointegration. Spot and Future prices will
follow independent randon walks. This eliminates both the risk transfer and the price discovery functions of future markets
2. H = ∞ In VAR (12) the matrix M has reduced rank
(1, -2)M =0 , and the errors are perfectly correlated. Therefore the long run equilibrium relationship (4), St= 2 Ft + 3, becomes an exact relationship. Future contracts are in this situation perfect substitutes for spot market positions and prices will be “discovered” in both markets simultaneously.
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Two Metrics for Price Discovery:IS of Hasbrouck (1995) and
PT of Gonzalo and Granger (1995) See Special Issue of the Journal of Financial Markets, 2002, 5
Both approaches start from the estimation of the VECM
Hasbrouck transforms the VECM into a VMA
with denoting the common row vector of and l a column unit vector.
k
jtjtjtt uXXX
11'
tt uLX )(
t
t
iit uLuX )(*)1(
1
t
t
iit uLluX )(*
1
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Two Metrics for Price Discovery:IS of Hasbrouck (1995) and
PT of Gonzalo and Granger (1995)
The information share (IS) measure is a calculation that attributes the source of variation in the random walk component to the innovations in the various markets. To calculate it we need to have uncorrelated innovations:
ut=Qet, with Var(ut)=QQ’
and Q a lower triangular matrix (Choleski decomposition of )
The market-share of the innovation variance attributable to ej is
where [Q]j is the j-th element of the row matrix Q.
´
2
jj
QS
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Some Comments on the IS metric
(1) Non-uniqueness. There are many square roots of and not even the Cholesky square root is unique.
Solution: To calculate all the Choleskys, and form upper and lower bounds of the IS.
Problem: Theses bounds can be very distant.
(2) It is not clear how to proceed when the cointegrating vector is different from (1, -1).
(3) It presents some difficulties for testing
(4) Economic Theory behind it???
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PT of GG
1
1
12
´
´
´
( ´ )
t t
t t
w X
z X
A
A
k
jtjtjtt uXXX
11'
1 2 t t tX A W A z
t
tt F
SX
P-T decomposition
where
It exists if det(’) different from zero.
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The GG Permanent-Transitory Decomposition
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Easy Estimation and Testing
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Some Comments on GG PT
Advantages: The linear combination defining Wt is unique
Easy estimation (by LS) Easy testing (chi-squared distribution) Economic Theory behind it (well not always ha ha ha ha).
Problems: It needs to invert a matrix so it may not exist (probability
zero) Wt may not be a random walk; but it can be.
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Empirical Price Discovery in non Ferrous Metal Markets. Data
Daily spot and future (15 months) for Al, Cu, Ni, Pb, Zn, quoted in the LME
Sample January 1989- October 2006 Source Ecowin.The LME data has the advantage that
there are simultaneous spot and forward ask prices, for fixed maturities, every business day.
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Empirical Price Discovery in non ferrous metal markets
Six Simple Steps :1) Perform unit root test on price levels2) Determine the rank of cointegration3) Estimation of the VECM4) Hypothesis testing on beta5) Estimation of α and hypothesis
testing on it (e.g. α ´=(0, 1))
6) Set up the PT decomposition.
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Step 2. Determination of the cointegration rank
Table 1: Trace Cointegration rank test
Al Cu Ni Pb Zn
Trace test
r ≤1 vs r=2 (95% c.v=9.14) 1.02 1.85 0.57 0.84 5.23
r = 0 vs r=2 (95% c.v=20.16) 27.73 15.64* 42.48 43.59 23.51
* Significant at the 20% significance level (80% c.v=15.56).
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Step 3. Estimation of the VECM (14)
Al Pb
Cu Zn
Ni
F
t
St
t
tt
t
t
u
u
F
Soflagskz
F
S
ˆ
ˆ ˆ
)312.0(001.0
)438.2(
010.0
1
11
17k(AIC) and 48.120.1ˆ ttt FSzwith
F
t
St
t
tt
t
t
u
u
F
Soflagskz
F
S
ˆ
ˆ ˆ
)541.1(003.0
)871.0(
002.0
1
11
14k(AIC) and 06.001.1ˆ ttt FSzWith
F
t
St
t
tt
t
t
u
u
F
Soflagskz
F
S
ˆ
ˆ ˆ
)267.1(005.0
)211.2(
009.0
1
11
15k(AIC) and 69.119.1ˆ with ttt FSz
F
t
St
t
tt
t
t
u
u
F
Soflagskz
F
S
ˆ
ˆ ˆ
)793.3(013.0
)206.0(
001.0
1
11
15k(AIC) and 25.119.1ˆ with ttt FSz
11
1
0.009
( 2.709) ˆˆ
0.001 ˆ
(0.319)
St t t
t Ft t t
S S uz k lags of
F F u
16k(AIC) and 78.125.1ˆ ttt FSzwith
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Step 4. Hypothesis testing on beta
Table 3: Hypothesis Testing on the Cointegrating Vector and Long Run Backwardation
Al Cu Ni Pb Zn
Cointegrating vector(1, -2,- 3)
1 1.00 1.00 1.00 1.00 1.00
2 1.20 1.01 1.19 1.19 1.25
SE (2) (0.06) (0.12) (0.04) (0.05) (0.07)
3 (constant term) -1.48 -0.06 -1.69 -1.25 -1.78
SE (3) (0.47) (0.89) (0.34) (0.30) (0.50)
Hypothesis testing
H0:2=1 vs H1:2>1
(p-value) (0.001) (0.468) (0.000) (0.000) (0.000)
Long Run Backwardation yes no yes yes yes
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Step 5. Estimation of and hipothesis testing on it
Table 4: Proportion of spot and future prices in the price discovery function (
Estimation Al Cu Ni Pb Zn
1 0.09 0.58 0.35 0.94 0.09
2 0.91 0.42 0.65 0.06 0.91
Hypothesis testing (p-values)
H0: ´=(0,1) (0.755) (0.123) (0.205) (0.000) (0.749)
H0: ´=(1,0) (0.015) (0.384) (0.027) (0.837) (0.007)
Note: is the vector orthogonal to the adjustment vector : `=0. For estimation of and inference on it, see Gonzalo-Granger (1995).
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Step 6. Set-up the PT decomposition
Al Pb
Cu Zn
Ni
tt
t ZWF
S
083.0
901.0
983.0
177.1t
ttt
ttt
FSZ
FSW
197.1
912.0088.0
tt
t ZWF
S
585.0
409.0
995.0
004.1t
ttt
ttt
FSZ
FSW
010.1
418.0582.0
tt
t ZWF
S
325.0
613.0
938.0
117.1t
ttt
ttt
FSZ
FSW
191.1
654.0345.0
tt
t ZWF
S
794.0
055.0
849.0
010.1t
ttt
ttt
FSZ
FSW
190.1
062.0937.0
tt
t ZWF
S
086.0
893.0
978.0
223.1t
ttt
ttt
FSZ
FSW
251.1
911.0089.0
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Conclusions and Extensions
We introduce a way of modelling endogenously convenience yields, such that Backwardation and Contango are captured in the cointegrating vector. Cointegrating vector that is different from the standard and always present (1, -1)
As a by-product we can calculate convenience yields
An Economic Theoret¡cal justification for the GG PT decomposition
For those metals with most liquid future markets the future price is the major contributor to the revelation of the efficient price (price discovery). This means that for those commodities producers and consumers should rely on the LME future price to make their production and consumption decisions
On going extensions : (1) To other commodities (2) Backwardation and contango jointly in
the model. This will imply a non-linear ECM.
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Graphical AppendixFigure1: Aluminium spot ask settlement prices, 15-month ask forward prices and backwardation
0
500
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date
pri
ces
(in
$) a
nd
bac
kwar
dat
ion
-300
-200
-100
0
100
200
300
400
500
600
700
als
al15
backwardation
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Figure 2: copper spot ask settlement prices, 15 month forward ask prices and backwardation
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
03/0
1/19
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05
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1/20
06
date
Pri
ces
and
bac
kwar
dat
ion
(in
$)
-400
-200
0
200
400
600
800
1000
1200
1400
1600
cus
cu15
backwardation
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Figure 3: Nickel spot ask settlement prices, 15-month ask forward prices and backwardation
0
5000
10000
15000
20000
25000
30000
35000
4000003
/01/
1989
03/0
1/19
90
03/0
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92
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06
date
pric
es a
nd b
ackw
arda
tion
(in $
)
-2000
0
2000
4000
6000
8000
10000
12000
14000
nis
ni15
backwardation
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Figure 4: Lead spot ask settlement prices, 15-month forward prices and backwardation
0
200
400
600
800
1000
1200
1400
1600
1800
03/0
1/19
89
03/0
1/19
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05
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1/20
06
dates
Pri
ces
and
bac
kwar
dat
ion
(in
$)
-200
-100
0
100
200
300
400
500
600
pbs
pb15
backwardation
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Figure 5: zinc spot ask settlement prices, 15-month forward prices and backwardation
0
500
1000
1500
2000
2500
3000
3500
4000
4500
500003
/01/
1989
03/0
1/19
90
03/0
1/19
91
03/0
1/19
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03/0
1/19
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03/0
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05
03/0
1/20
06
date
Pri
ces
and
bac
kwar
dat
ion
(in
$)
-200
0
200
400
600
800
1000
zis
zi15
backwardation
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Figure 6: Range of annual Aluminum convenience yields in %
-10
0
10
20
30
40
1/02/89 11/02/92 9/02/96 7/03/00 5/03/04
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Figure 7: Range of annual Copper convenience yields in %
-10
0
10
20
30
1/03/89 11/03/92 9/03/96 7/04/00 5/04/04
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Figure 6: Range of annual Nickel convenience yields in %
-20
0
20
40
60
80
1/03/89 11/03/92 9/03/96 7/04/00 5/04/04
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Figure 9: Range of annual Lead convenience yields in %
-10
0
10
20
30
40
50
1/03/89 11/03/92 9/03/96 7/04/00 5/04/04
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Figure 10: Range of annual Zinc convenience yields in %
-10
0
10
20
30
40
1/03/89 11/03/92 9/03/96 7/04/00 5/04/04
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Figure 6: Average yearly LME Futures Trading Volumes-Non Ferrous Metals January 1990- December 2006
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
Al Cu Ni Pb Zn
Future contract
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Spot and total future volumes for LME traded contractsMay 2006 – December 2007
Al Cu Ni Pb Zn
157664 71018 27606 14295 33309 Futures
13707 6859 2043 1729 4646 Spot
12 10 14 8 7 Ratio Vf/Vs
0.9200156 0.9119252 0.9310938 0.8920994 0.8775919 Vf/(Vf+Vs)
0.07998436 0.08807478 0.0689062 0.1079006 0.1224081 Vs/(Vf+Vs)
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Market Share by Commodity Type in the US, 2003