Modelling and Measuring Price
Discovery in
Commodity Markets
Isabel Figuerola-Ferretti
Jesús Gonzalo
Universidad Carlos III de Madrid
Business Department and Economics Department
December 2007
1
Trading Places Movie
2
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.
3
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).
4
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
5
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.
6
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)
*
Pt  Pt  et
7
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.
8
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, -b2), different from the standard and
always present b2 =1. When b2 >1 (<1) the market is under backwardation (contango).

Independent of b2 , 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.
9
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.
10
Literature Review 2:
Price discovery metrics
 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.
11
Literature Review 3
Comparing the two metrics of price discovery:
Special Issue Journal of Financial Markets 2002
 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.
12
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.
13
Equilibrium with infinitely elastic
supply of arbitrage
Standard Assumptions:
_
r
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 market
_
5) Interest
_ rate rt + storage cost ct = rc + I(0),
with rc the mean of (rt + ct)
6) St is I(1).
14
Equilibrium with infinitely elastic
supply of arbitrage
 Let T1=1
 Non-arbitrage equilibrium conditions imply
Ft  St  (rt  c t )
(1)
 Given the above assumptions, equation (1)
implies that St and Ft are cointegrated with
the always present cointegrating vector (1, -1).
15
A bit of more realism:
Convenience Yields
In consumption commodities is very likely that
Ft  St  (rt  c t )
with
Ft  yt  St  (rt  c t )
where
yt
(2)
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.
16
Convenience Yields
One more Assumption:
7) The convenience yield is modeled as
yt   1St   2 Ft
with
(3)
  (0, 1)i 1,2 .
17
 Backwardation refers to futures prices that
decline with time to maturity
 Contango refers to futures prices that rise with
time to maturity
Crude Oil
Gold
$41,50
$405
$41,00
$404
$40,50
Oil price ($/barrel)
$402
$39,50
$39,00
$401
$38,50
$400
$38,00
$399
$37,50
Gold price ($/Troy ounce)
$403
$40,00
$398
$37,00
$397
$36,50
$36,00
April-04
$396
June-04
August-04
September- November-04 December-04 February-05
04
April-05
May-05
July-05
18
Equilibrium with convenience
yields

Substituting (3) into (2) + (a.5)
S t   2 Ft   3  I (0)
with
2 
12
1 1
(4)

and
3 
 rc
1 1
.
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)
19
Equilibrium with convenience
yields
Some remarks:
 The parameters 1 and 2 are not_identified in the
equilibrium equation (4) unless rc 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).
20
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.
21
Equilibrium with finitely elastic supply of
arbitrage services
 Demand schedule of ith participant in spot market
Ei ,t  A  St  Ri ,t  ,
A  0, i  1,..., N s
(5)
where A is the elasticity of demand
 Aggregate cash market demand schedule of arbitrageurs in
period t
H  (  2 F   3 )  St  , H  0
(6)
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.
22
 The cash market will clear at the value of St
that solves
Ns
E
i 1
  Ei ,t  A( St  Ri ,t ) H (  2 Ft   3 )  St  (7)
Ns
i ,t
i 1
 The future market will clear at the value of Ft
such that
NF
E
j 1
NF
j ,t
 E j ,t  A( Ft  R j ,t )  H  (  2 Ft  3 )  St  (8)
j 1
23
Equilibrium with finitely elastic supply of
arbitrage services
 Solving the clearing market conditions as a function of the
N
NS





1
F

1
S
mean reservation prices  R  N
and


S  Ri ,t 
 Rt  N F  R j , t 
 t
j 1


i 1


F
( AN F  H 2 ) N S RtS  HN F  2 RtF  HN F  3
St 
(9)
( H  AN s ) N F  HN S  2
HN S Rt  ( H  AN s ) N F R  HN s  3
Ft 
( H  AN s ) N F  HN S  2
S
F
t
24
Dynamic price relationships
 To derive dynamic price relationships, we
need a description of the evolution of
reservation prices.
Ri ,t  St 1  vt  wi ,t ,
i  1,..., N S
R j ,t  Ft 1  vt  w j ,t , j  1,..., N F
(10)
cov(vt , wi ,t )  0, i
cov( wi ,t , w j ,t )  0, i  j
25
And the mean reservation prices
R S t  St 1  vt  wS t , i  1,..., N S
R
F
 Ft 1  vt  w t , j  1,..., N F
F
t
with
NS
wtS 
w
i 1
NS
S
i ,t
(11)
NF
, wtF 
F
w
 j ,t
j 1
NF
26
Dynamic price relationships: VAR model
 St  H 3  N F 
 St 1   utS 
 F   d  N    M   F    F 
 t
 S
 t 1   ut 
where
and
 utS
 F
u
 t
(12)

 vt  wtS 
  M


 v  wF 
t 

 t
 2 HN F 
1  N S (  2 H  AN F )
M 
HN S
( H  AN S ) N F 
d
d  ( H  AN S ) N F   2 HN S
Garbade and Silver (with b2=1,
b3=0)
their analysis at this point stating that
NF
NS  NF
stop
27
VECM Representation
 St  H  3
 F   d
t 

where
1
M I 
d
 NF 
 St 1   utS 
 N    M  I   F    F 
S 

 t 1   ut 
(13)
  HN F HN F  2 
 HN


HN

S
S
2 

 St  H  N F 
1   2
F   d 
 t
 N S 
 St 1 
S

u


t 
  3  Ft 1   F 

 u
 1   t 
(14)
28
The GG permanent component
is…
NS
NF
(
) St  (
) Ft
NS  NF
NS  NF
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.
29
Two extreme cases:
1.
H=0
2.
H=∞
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
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.
30
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
k
X t   ' X t 1    j X t  j  ut
j 1

Hasbrouck transforms the VECM into a VMA
X t  ( L)ut
t
X t   (1) ui   * ( L)ut
i 1
 t

X t     ui  l   * ( L ) u t
 i 1 
with Y denoting the common row vector of Y(1) and l a column unit
vector.
31
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)=W=QQ’
and Q a lower triangular matrix (Choleski decomposition of W )

The market-share of the innovation variance attributable to ej is

Q  

2
Sj
j
´
where [YQ]j is the j-th element of the row matrix YQ.
32
Some Comments on the IS metric
(1) Non-uniqueness. There are many square roots of W 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???
33
PT of GG
 St 
Xt   
 Ft 
k
X t   ' X t 1    j X t  j  ut
j 1
P-T decomposition
X t  A1 Wt  A2 zt
wt    ´ X t
where
It exists if det(b’a) different from
zero.
zt   ´ X t
A1      ´  
1
A2   (  ´ ) 1
34
The GG Permanent-Transitory
Decomposition
35
36
Easy Estimation and Testing
37
38
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.
39
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.
40
Empirical Price Discovery in non ferrous
metal markets
Six Simple Steps :
1) Perform unit root test on price levels
2) Determine the rank of cointegration
3) Estimation of the VECM
4) Hypothesis testing on beta
5) Estimation of α and hypothesis testing
on it (e.g. α ´=(0, 1))
6) Set up the PT decomposition.
41
Step 2. Determination of the
cointegration rank
Table 1: Trace Cointegration rank test
Al
Cu
Ni
Pb
Zn
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
Trace test
* Significant at the 20% significance level (80% c.v=15.56).
42
Step 3. Estimation of the VECM (14)
Al
 0.010 
 S t  ( 2.438)
 zˆt 1   k lags of
F   
 t  0.001 
 (0.312) 
 St 1   uˆ 
F    ˆ 
 t 1  u 
S
t
F
t
with zˆt  St  1.20Ft  1.48 and k(AIC)  17
Cu
  0.002 

S
 t  ( 0.871)
 zˆt 1   k lags of
 F   
 t  0.003 
 (1.541) 
 S t 1   uˆtS 
 F    F 
 t 1  uˆt 
With zˆt  St  1.01Ft  0.06 and k(AIC)  14
Ni
Pb
 0.001 
 S t  ( 0.206)
zˆt 1   k lags of
F   
 t  0.013 
 (3.793) 
 S t 1   uˆtS 
F    ˆ F 
 t 1  ut 
with zˆt  St  1.19 Ft  1.25 and k(AIC)  15
Zn
 0.009 


 St  ( 2.709) 
 F   0.001 
 t


 (0.319) 
 zˆt 1   k lags of
 St 1   uˆtS 
 F   uˆ F 
 t 1   t 
with zˆt  St  1.25Ft  1.78 and k(AIC)  16
 0.009 
 St  ( 2.211)
 St 1   uˆtS 





z

k
lags
of
ˆ
t 1
F 
F    F 
 t  0.005 
 t 1  uˆt 
 (1.267) 
with zˆt  St  1.19 Ft  1.69 and k(AIC)  15
43
Step 4. Hypothesis testing on beta
Table 3: Hypothesis Testing on the Cointegrating Vector and Long Run Backwardation
Al
Cu
Ni
Pb
Zn
1
1.00
1.00
1.00
1.00
1.00
2
1.20
1.01
1.19
1.19
1.25
(0.06)
(0.12)
(0.04)
(0.05)
(0.07)
-1.48
-0.06
-1.69
-1.25
-1.78
(0.47)
(0.89)
(0.34)
(0.30)
(0.50)
(0.468) (0.000) (0.000)
(0.000)
Cointegrating vector
(1, -2,- 3)
SE (2)
3 (constant term)
SE (3)
Hypothesis testing
H0:2=1 vs H1:2>1
(p-value)
Long Run
Backwardation
(0.001)
yes
no
yes
yes
yes
44
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
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)
Hypothesis testing (p-values)
Note: is the vector orthogonal to the adjustment vector : `=0. For estimation of and inference on it, see Gonzalo-Granger (1995).
45
Step 6. Set-up the PT decomposition
Al
Cu
Ni
 St  1.177
 0.901 

W

 F  0.983 t  0.083 Z t



 t 
Pb
 St  1.010
 F   0.849 Wt 

 t 
Wt  0.088St  0.912 Ft
Wt  0.937S t  0.062 Ft
Z t  St  1.197 Ft
Z t  S t  1.190Ft
 St  1.004
 F   0.995 Wt 

 t 
 0.409 
 0.585 Z t


Zn
 St  1.223
 F   0.978 Wt 

 t 
Wt  0.582S t  0.418Ft
Wt  0.089 S t  0.911Ft
Z t  S t  1.010Ft
Z t  S t  1.251Ft
 St  1.117
 F   0.938 Wt 

 t 
 0.055 
 0.794 Z t


 0.893 
 0.086 Z t


 0.613 
 0.325 Z t


Wt  0.345S t  0.654 Ft
Z t  S t  1.191Ft
46
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.
47
03/01/2006
03/01/2005
03/01/2004
03/01/2003
03/01/2002
03/01/2001
03/01/2000
03/01/1999
03/01/1998
03/01/1997
03/01/1996
03/01/1995
03/01/1994
03/01/1993
03/01/1992
03/01/1991
03/01/1990
03/01/1989
prices (in$) and backwardation
Graphical Appendix
Figure1: Aluminium spot ask settlement prices, 15-month ask forward prices and backwardation
3500
700
3000
600
500
2500
400
2000
300
200
1500
1000
0
als
al15
backwardation
100
0
-100
500
-200
-300
date
48
03/01/2006
03/01/2005
03/01/2004
03/01/2003
03/01/2002
03/01/2001
03/01/2000
03/01/1999
03/01/1998
03/01/1997
03/01/1996
03/01/1995
03/01/1994
03/01/1993
03/01/1992
03/01/1991
03/01/1990
03/01/1989
Prices and backwardation (in$)
Figure 2: copper spot ask settlement prices, 15 month forward ask prices and backwardation
10000
1600
9000
1400
8000
1200
7000
1000
6000
800
5000
600
4000
400
3000
200
2000
0
1000
-200
0
-400
cus
cu15
backwardation
date
49
03/01/2006
03/01/2005
03/01/2004
03/01/2003
03/01/2002
03/01/2001
03/01/2000
03/01/1999
03/01/1998
03/01/1997
03/01/1996
03/01/1995
03/01/1994
03/01/1993
03/01/1992
03/01/1991
03/01/1990
03/01/1989
prices and backwardation (in $)
Figure 3: Nickel spot ask settlement prices, 15-month ask forward prices and backwardation
40000
14000
35000
12000
30000
10000
25000
8000
20000
6000
15000
4000
10000
2000
5000
0
nis
ni15
backwardation
0
-2000
date
50
03/01/2006
03/01/2005
03/01/2004
03/01/2003
03/01/2002
03/01/2001
03/01/2000
03/01/1999
03/01/1998
03/01/1997
03/01/1996
03/01/1995
03/01/1994
03/01/1993
03/01/1992
03/01/1991
03/01/1990
03/01/1989
Prices and backwardation (in $)
Figure 4: Lead spot ask settlement prices, 15-month forward prices and backwardation
1800
600
1600
500
1400
400
1200
300
1000
200
800
400
200
-100
0
-200
pbs
pb15
backwardation
600
100
0
dates
51
03/01/2006
03/01/2005
03/01/2004
03/01/2003
03/01/2002
03/01/2001
03/01/2000
03/01/1999
03/01/1998
03/01/1997
03/01/1996
03/01/1995
03/01/1994
03/01/1993
03/01/1992
03/01/1991
03/01/1990
03/01/1989
Prices and backwardation (in $)
Figure 5: zinc spot ask settlement prices, 15-month forward prices and backwardation
5000
1000
4500
4000
800
3500
600
3000
2500
400
0
zis
zi15
backwardation
2000
1500
200
1000
0
500
-200
date
52
Figure 6: Range of annual Aluminum convenience yields in %
40
30
20
10
0
-10
1/02/89
11/02/92
9/02/96
7/03/00
5/03/04
53
Figure 7: Range of annual Copper convenience yields in %
30
20
10
0
-10
1/0 3/89
11 /03/9 2
9/0 3/96
7/0 4/00
5/0 4/04
54
Figure 6: Range of annual Nickel
convenience yields in %
80
60
40
20
0
-20
1/03/89
11/03/92
9/03/96
7/04/00
5/04/04
55
Figure 9: Range of annual Lead convenience yields in %
50
40
30
20
10
0
-10
1/03/89
11/03/92
9/03/96
7/04/00
5/04/04
56
Figure 10: Range of annual Zinc convenience
yields in %
40
30
20
10
0
-10
1/03/89
11/03/92
9/03/96
7/04/00
5/04/04
57
Figure 6: Average yearly LME Futures Trading Volumes-Non Ferrous Metals
January 1990- December 2006
90000
80000
70000
60000
50000
40000
30000
20000
10000
0
Al
Cu
Ni
Future contract
Pb
Zn
58
Spot and total future volumes for LME traded contracts
May 2006 – December 2007
Al
Cu
Ni
Pb
Zn
157664
71018
27606
14295
13707
6859
2043
1729
12
10
14
8
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)
33309 Futures
4646 Spot
7 Ratio Vf/Vs
Market Share by Commodity Type
in the US, 2003
60