Theory of Storage and the Dynamics of Metals Forward Curves

Helyette Geman Director, Commodity Finance Centre University of London and ESCP Europe Scientific Advisor to the European Commission

To be presented at the Vale Conference on Commodities

Getulio Vargas Foundation - Rio de Janeiro

August 16 & 17, 2012

Metals, Energy, Agriculturals : A Multi or Unique Asset Class?

→ Commodities have displayed over the last 30 years

. a period of low prices in the 1980s and 1990s, strictly declining if adjusted for inflation, and low volatility

. low correlations between metals, energy, agriculturals

→ Then much higher prices prevailed as of 2002 for crude oil, as of 2004 for copper, 2005 for agriculturals. In the case of iron ore, long-term contracts imposed by steelmakers broke down after the concerted action of Vale, Rio Tinto and BHP Billiton

Æ High correlations appeared, created . by the massive arrival of financial actors buying at the same time several commodities such as copper, gold, crude oil in the form of a commodity index

. by the effects of substitution between commodities and competition for the same rare resources, called electricity , water, land CRB Commodity Index – 1988 to 2011 Brazil Equity versus UK Equity

Data from Yahoo Finance via Matlab Datafeed toolbox

S&P data from Yahoo (ticker ^GSPC) via Matlab Data Source Toolkit Agriculture Index from World Bank http://data.worldbank.org/data-catalog/commodity-price-data

S&P data from Yahoo (ticker ^GSPC) via Matlab Data Source Toolkit Metals & Minerals Index from World Bank http://data.worldbank.org/data-catalog/commodity-price- data Commodity Notes – Correlation 2 - Real World Correlations ©2012 William Smith US Equity versus Energy Index

S&P data from Yahoo (ticker ^GSPC) via Matlab Data Source Toolkit Energy Index from World Bank http://data.worldbank.org/data-catalog/commodity-price-data

ÆNew mining techniques, deeper drilling and mining in untapped places such as Greenland, Mongolia and the Arctic should lead to years in mineral reserves, at current production rates, estimated at 590 for iron ore, 136 for copper, 610 for potash; versus 18.9 for gold, 46.2 for crude oil and 82 for metallurgical coal

Æ Hotelling in his (1931) paper on exhaustible commodities had established that the shadow price of the resource, which is an economic measure of its scarcity, should grow at least at the rate of interest

→ Young (1992) applies Hotelling model to Canadian copper mining firms and finds it poorly depicts the database he analyzes; but the period of analysis ended in 1990 and was the period of price mean- reversion ( G. 2005 : Is Mean Reversion in Commodity Prices Dead? )

→ It is useful to recognize that the possible decline in the quality of the BDI, Copper and the world economy growth Dislocation between BDI and Copper Prices Copper versus Crude Oil

Commodiity Monthly Prices from World Bank http://data.worldbank.org/data-catalog/commodity-price- data

Implied vol

Strike Price Time to Maturity Merrill Lynch, “Modelling the Implied Volatility Surface”, http://finmath.stanford.edu/seminars/docs/ml2004win.pdf Gold 1st Contract (22 Jun 2012) 0.29

0.27

0.25

0.23

0.21 Implied Volatility Implied

0.19

0.17

0.15 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 K/ S Theory of Storage Keynes (1936), Kaldor (1939), Working (1949)

Three fundamentals results:

→ The holder of the physical commodity receives an implicit dividend called convenience yield

→ The volatility of the commodity spot price is high when inventory is low

→ Traditionally, forward curves used to be mostly declining with the maturity ( ‘normal backwardation’) and sometimes in contango. Today, we even get humps

→ The dynamics of the global forward curve matters, in hedging activities in particular, since one never hedges with the prompt- month The Forward Curve

→ The set {FT (t) , T > t} is the forward curve prevailing at date t for a given commodity in a given location

→ It is the fundamental tool when trading commodities, as spot prices may be unabservable and options not always liquid

→ It allows to identify the prices forecasted by the market at future dates since real trades did take place at these prices

→ The shape of the forward curve is a crucial piece of financial information to be compared to all the other sources! Crude Oil in Backwardation in September 2007

68 M1 M5 M9 M13 M17 M21 M25 M29 M33 M37 M41 M45 M49 M53 M57 M61 WTI Oil Prices Jan 2002 - Oct 2007

120

100

0 9/6/2005 1/1/2002 8/8/2006 4/23/2002 8/13/2002 12/3/2002 3/25/2003 7/15/2003 11/4/2003 2/24/2004 6/15/2004 10/5/2004 1/25/2005 5/17/2005 4/18/2006 3/20/2007 7/10/2007 12/27/2005 10/30/2007 11/28/2006 Spot-Forward Relationship for a Storable Commodity

Under no arbitrage ⎡ ⎤ T ⎢ ⎥ f ()t = S t ()1+ r T − (t + )c T −t (− y1 )T −t ( ) ⎢ 123 123 14243 ⎥ ⎢⎣ cos t of financing cos t of storage implicit dividend ⎦⎥ If we define a convenience yield net of cost of storage f T()t = S t ()[1+ ()r − y ()T − t ]

Or in continuous time, at a fixed date t (today), for a given maturity T

fT ()t = S t ()e(r−y)()T−t Copper Forward Curve - 27 May 2008 : Backwardation

Copper COMEX 380

375

370

365

360

355

350

345 M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 M13 M14 M15 M16 M17 M18 M19 M20 M21 M22 M23 M24 Crude Oil Future curve (17/11/2008)

50 M1 M4 M7 M10 M13 M16 M19 M22 M25 M28 M31 M34 M37 M40 M43 M46 M49 M52 M55 M58 M61 A hump in the Oil Forward Curve (bid/ ask) - March 2006

Forward Curve

68,37

67,37

66,37

65,37

Forward Price 64,37

63,37

4 4 3 13 -1 1 - -1 il 62,37 1 1 u 0 -1 -1 il-12 v-12 rs j 8 9 9 -10 -1 il o juil-13nov a 8 -0 il-10 rs u ju n ars-1 m 6 6 -0 -0 v-09 j nov m 6 -07 o ju nov a mars-12 -0 -0 il-07 rs juil n ars-1 m -0 juil-08nov a m rs ju nov m juil nov mars-0 a mars-07 m Maturity

WTI Forward Bid WTI Forward Offer Copper Forward Curve, Oct 2009 Copper 14 Aug 2012 3.45

3.4

3.35

3.3

Copper 3.25

3.2

3.15

3.1 1-Jun-13 1-Jun-14 1-Jun-15 1-Jun-16 1-Jun-17 1-Oct-12 1-Oct-13 1-Oct-14 1-Oct-15 1-Oct-16 1-Apr-13 1-Apr-14 1-Apr-15 1-Apr-16 1-Apr-17 1-Feb-13 1-Feb-14 1-Feb-15 1-Feb-16 1-Feb-17 1-Dec-12 1-Dec-13 1-Dec-14 1-Dec-15 1-Dec-16 1-Aug-12 1-Aug-13 1-Aug-14 1-Aug-15 1-Aug-16 Coal 14 Aug 2012 90

Co al 40

0 NG 14 Aug 2012 7

NG 3

0 1-Jun-20 1-Jun-19 1-Jun-18 1-Jun-17 1-Jun-16 1-Jun-15 1-Jun-14 1-Jun-13 1-Sep-20 1-Sep-19 1-Sep-18 1-Sep-17 1-Sep-16 1-Sep-15 1-Sep-14 1-Sep-13 1-Sep-12 1-Dec-20 1-Dec-19 1-Dec-18 1-Dec-17 1-Dec-16 1-Dec-15 1-Dec-14 1-Dec-13 1-Dec-12 1-Mar-20 1-Mar-19 1-Mar-18 1-Mar-17 1-Mar-16 1-Mar-15 1-Mar-14 1-Mar-13 Iron Ore Futures Curve

From CME Group, http://www.cmegroup.com/trading/metals/ferrous/iron-ore-62pct-fe-cfr-china-tsi-swap-futures_quotes_settlements_futures.html Iron Ore – Evolution of Spot Price

China import Iron Ore Fines 62% FE spot (CFR Tianjin port), US Dollars per Metric Ton http://www.indexmundi.com/commodities/?commodity=iron-ore&months=60 Inventory, Volatility and Shape of the Forward Curve

→ Working (1949) proposed to use the spread of the forward curve (long term forward – short term forward) as a proxy for inventory : when the spread is negative, inventory is low → Fama and French (1988) use LME Future prices over the period 1972 to 1983 to test five base metals (copper, aluminium, copper, lead, tin and zinc) and find that the variance of spot prices declines with high inventories. In the case of gold, forward curve spreads provided little forecast for price volatility. → Ng and Pirrong (1994) analyze four base metals over the period 1986 to 1992 and find persistence of the property that both spot and forward variance declines with inventory in the case of metals

→ G - Nguyen ( 2005) reconstruct a world inventory of soybeans over several years and directly exhibit a quasi- perfect inverse relationship between inventory and spot price volatility → G- Ohana (2009) . Examine at US crude oil and natural gas markets . Show that indeed the spread of the forward curve is a good proxy for inventory . Exhibit that the correlation between the spread of the forward curve and low inventory is particular significant during periods of scarcity

Æ G – Smith (2012) . Reconstruct inventory for copper, lead, iron, tin from the addition of the LME and SHFE data . Validate the use of the spread of the forward curveas a measure of inventory . Display directly an affine relationship between inverse inventory and spot price volatility Copper Inventory – 1985 to 2011

Why Should Different Commodities Be Correlated? Economic Effects

Correlation (A,B) > 0

Resource Competition

Water for Agriculture or Mining or Oil & Gas “Fracking”

Shipping for Metal oreor Agriculturals

Land for Residential Land or Agriculture or Mining

Offshore Experts for Windfarms or Offshore Oil

Correlation (A,B) > 0

Commodiity Montly Prices from World Bank http://data.worldbank.org/data-catalog/commodity-price- data

A is needed to produce B

Aluminium Ore Aluminium (Bauxite)

Lots of Electricity

Correlation (A,B) > 0

Substitution : A or B

Oil

Electricity

Natural Gas

Correlation (A,B) > 0

Alternative Output : A can produce B or C

Sugar

Sugar Cane or

Ethanol as Biofuel Correlation (B,C) > 0 due to competition

Commodiity Monthly Prices from World Bank http://data.worldbank.org/data-catalog/commodity-price- data

Co-Production : A and B produced together

Zinc

together with

Lead

Correlation (A,B) > 0 since supplies of both occur at the same time.

850

800

750

COMEX Gold Price

700

650

600 oct-08 oct-07 avr-08 juin-08 juin-09 juin-10 juin-11 avr-07 mai-07 juin-07 déc-08 déc-09 déc-10 déc-11 déc-07 févr-08 août-08 août-07 mars-07 Contract months Gold Forward Curve - 27 May 2008

Gold 1100

1050

1000

950

900

850

800 M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 M13 M14 M15 M16 M17 M18 Gold Forward Curve – Feb 2010 Gold 4/ 9/ 2012 1800

1750

1700

Gol d

1650

1600

1550 Ju n - 12 Ju n - 13 Ju n - 14 Ju n - 15 Ju n - 16 Ju n - 17 Oct -12 Oct -13 Oct -14 Oct -15 Oct -16 Apr-12 Apr-13 Apr-14 Apr-15 Apr-16 Apr-17 Feb-13 Feb-14 Feb-15 Feb-16 Feb-17 Dec-12 Dec-13 Dec-14 Dec-15 Dec-16 Aug-12 Aug-13 Aug-14 Aug-15 Aug-16 Gold 14 Aug 2012 1740

1720

1700

1680

1660

1640 Gol d

1620

1600

1580

1560

1540 Ju n - 13 Ju n - 14 Ju n - 15 Ju n - 16 Ju n - 17 Oct -12 Oct -13 Oct -14 Oct -15 Oct -16 Oct -17 Apr-13 Apr-14 Apr-15 Apr-16 Apr-17 Feb-13 Feb-14 Feb-15 Feb-16 Feb-17 Dec-12 Dec-13 Dec-14 Dec-15 Dec-16 Dec-17 Aug-12 Aug-13 Aug-14 Aug-15 Aug-16 Aug-17

Gold 1st Contract (22 Jun 2012) 0.29

0.27

0.25

0.23

0.21 Implied Volatility Implied

0.19

0.17

0.15 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 K/ S References

H. Geman and W. Smith (2012) “Inventories and Base Metals Forward Curves”, Resources Policy H.Geman and S. Sarfo (2012) “Seasonality in Cocoa Spot and Forward Markets: Empirical Evidence”, Journal of Agricultural Expansion and Rural Development H.Geman ( 2011) “ Volatility in Commodity Spot Markets: Speculation or Scarcity?”, Swiss Derivatives Review H.Geman (2010) “Commodities and Numéraire”, Encyclopedia of Quantitative Finance H. Geman and Yfong Shi (2009) “ The CEV model for Commodity Prices”, Journal of Alternative Investments H. Geman and S. Kourouvakalis (2008) "A Lattice-Based Method for Pricing Electricity Derivatives under the Geman- Roncoroni Model", Applied Mathematical Finance H. Geman and C. Kharoubi( 2008) “Diversification with Crude Oil Futures : the Time-to- Maturity Effect, Journal of Banking and Finance S. Borovkova and H. Geman (2006) "Seasonal and Stochastic Effects in Commodity Forward Curves", Review of Derivatives Research H. Geman and A. Roncoroni (2006) "Understanding the Fine Structure of Electricity Prices", Journal of Business H. Geman (2005) "Energy Commodity Prices: Is Mean Reversion Dead?", Journal of Alternative Investments H. Geman and S. Ohana (2009) "Inventory, Reserves and Price volatility in Oil and Natural Gas Markets“,Energy Economics H. Geman (2005) "Commodities and Commodity Prices: Pricing and Modeling for Agriculturals, Metals and Energy", Wiley Finance H. Geman and V. Nguyen (2005) "Soybean inventory and forward curves dynamics", Management Science H.Geman (2004) “Water as the Next Commodity”, Journal of Alternative Investments H. Geman and M. Yor (1993) "An Exact Valuation for Asian Option", Mathematical Finance A. Eydeland and H. Geman (1999) "Fundamentals of Electricity options" in Energy Price Modellng, Risk Books H. Geman and O. Vasicek (2001) "Forwards and Futures on Non Storable Commodities", RISK H. Geman (2003) "DCF versus Real Option for Pricing Energy Physical Assets" Conference of the International Energy Agency - Paris [email protected]