1

Modelling and Measuring Price Discovery in

Commodity Markets

Isabel Figuerola-FerrettiJesús Gonzalo

Universidad Carlos III de MadridBusiness Department and Economics

Department

December 2007

2

Trading Places Movie

3

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.

4

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).

5

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

6

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.

7

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

8

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.

9

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.

10

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.

11

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

12

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

13

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.

14

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

15

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

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

One more Assumption:

7) The convenience yield is modeled as

with .

Convenience Yields

1,2(0, 1)i

17

(3) 21 ttt FSy

18

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

19

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

20

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

21

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.

22

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

23

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

24

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

25

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

26

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

27

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

28

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

29

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.

30

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.

31

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

32

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

33

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???

34

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.

35

The GG Permanent-Transitory Decomposition

36

37

Easy Estimation and Testing

38

39

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.

40

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.

41

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.

42

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).

43

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

44

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

45

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).

46

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

47

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.

48

Graphical AppendixFigure1: Aluminium spot ask settlement prices, 15-month ask forward prices and backwardation

0

500

1000

1500

2000

2500

3000

3500

03/0

1/19

89

03/0

1/19

90

03/0

1/19

91

03/0

1/19

92

03/0

1/19

93

03/0

1/19

94

03/0

1/19

95

03/0

1/19

96

03/0

1/19

97

03/0

1/19

98

03/0

1/19

99

03/0

1/20

00

03/0

1/20

01

03/0

1/20

02

03/0

1/20

03

03/0

1/20

04

03/0

1/20

05

03/0

1/20

06

date

pri

ces

(in

$) a

nd

bac

kwar

dat

ion

-300

-200

-100

0

100

200

300

400

500

600

700

als

al15

backwardation

49

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

89

03/0

1/19

90

03/0

1/19

91

03/0

1/19

92

03/0

1/19

93

03/0

1/19

94

03/0

1/19

95

03/0

1/19

96

03/0

1/19

97

03/0

1/19

98

03/0

1/19

99

03/0

1/20

00

03/0

1/20

01

03/0

1/20

02

03/0

1/20

03

03/0

1/20

04

03/0

1/20

05

03/0

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

50

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

1/19

91

03/0

1/19

92

03/0

1/19

93

03/0

1/19

94

03/0

1/19

95

03/0

1/19

96

03/0

1/19

97

03/0

1/19

98

03/0

1/19

99

03/0

1/20

00

03/0

1/20

01

03/0

1/20

02

03/0

1/20

03

03/0

1/20

04

03/0

1/20

05

03/0

1/20

06

date

pric

es a

nd b

ackw

arda

tion

(in $

)

-2000

0

2000

4000

6000

8000

10000

12000

14000

nis

ni15

backwardation

51

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

90

03/0

1/19

91

03/0

1/19

92

03/0

1/19

93

03/0

1/19

94

03/0

1/19

95

03/0

1/19

96

03/0

1/19

97

03/0

1/19

98

03/0

1/19

99

03/0

1/20

00

03/0

1/20

01

03/0

1/20

02

03/0

1/20

03

03/0

1/20

04

03/0

1/20

05

03/0

1/20

06

dates

Pri

ces

and

bac

kwar

dat

ion

(in

$)

-200

-100

0

100

200

300

400

500

600

pbs

pb15

backwardation

52

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

92

03/0

1/19

93

03/0

1/19

94

03/0

1/19

95

03/0

1/19

96

03/0

1/19

97

03/0

1/19

98

03/0

1/19

99

03/0

1/20

00

03/0

1/20

01

03/0

1/20

02

03/0

1/20

03

03/0

1/20

04

03/0

1/20

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

53

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

54

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

55

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

56

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

57

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

58

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

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)

60

Market Share by Commodity Type in the US, 2003