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Technical University Munich

Department of Financial

Mathematics

Commodities as an Asset Class

Diploma Thesis

by

Maria Katharina Heiden

Referent: Prof. Dr. Rudi Zagst

Co-Referent: Dr. Reinhold Hafner

Closing Date: 06.12.2006

This work is dedicated to my parents who always supported me: On my long way

during the study of Financial Mathematics they remained steadfastly at my side

and gave good advice. I’m very lucky to have parents like them.

I declare that I wrote this diploma thesis by my own and that I only used the men-

tioned sources.

Munich, 06.12.2006

ii

Acknowledgements

First and foremost, I would like to thank my academic teacher Prof. Dr. Rudi

Zagst from whom I have learned what I know about financial mathematics. His

great finance lectures have boosted my interest in economics and finance during my

studies. Moreover, his engagement to improve teaching and fit it to students needs,

have forced myself to work hard and show him my thanks with good results.

I would like to express my sincere thanks for his advice and guidance to Dr. Rein-

hold Hafner. He agreed to be my thesis advisor at risklab Germany GmbH and gave

me the chance to write about this great topic. His way to show me the link between

theory and praxis was a key ingredient for my success and joy during my diploma

time.

Moreover, thanks to my colleague Dr. Wolfgang Mader who opened my horizon for

statistics in challenging discussions.

Last but not least, I would like to thank Mr. Nicholas Drude, the best fellow

student someone can imagine. I appreciate his patience in endless mathematical,

philosophical and sometimes unsubstantial discussions.

iii

Contents

1 Introduction 1

2 Overview of Commodity Markets 4

2.1 The different Commodity Types . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1.1 Crude Oil . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1.2 Natural Gas . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.2 Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.2.1 Precious Metals exemplified by Gold . . . . . . . . . 16

2.1.2.2 Industrial Metals . . . . . . . . . . . . . . . . . . . . 19

2.1.3 Agricultures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.3.1 Softs exemplified by Cocoa . . . . . . . . . . . . . . 23

2.1.3.2 Grains exemplified by Corn . . . . . . . . . . . . . . 25

2.1.3.3 Livestock exemplified by Live and Feeder Cattle . . . 26

2.2 Characteristics of Commodity Markets . . . . . . . . . . . . . . . . . 28

2.3 Trading Commodities . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.1 Commodity Derivatives . . . . . . . . . . . . . . . . . . . . . . 32

2.3.1.1 Forwards and Futures . . . . . . . . . . . . . . . . . 33

2.3.1.2 Options . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3.1.3 Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3.1.4 Commodity Linked Structured Notes . . . . . . . . . 38

2.3.1.5 Certificates . . . . . . . . . . . . . . . . . . . . . . . 39

2.3.2 Managed Futures Funds . . . . . . . . . . . . . . . . . . . . . 39

2.3.3 Stocks of Commodity Producing Companies . . . . . . . . . . 41

3 Pricing of Commodity Futures 44

3.1 The Risk Premium Model . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 The Convenience Yield Model . . . . . . . . . . . . . . . . . . . . . . 50

3.3 Relationship of the Risk Premium and Convenience Yield Model . . . 53

3.4 Stochastic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4.1 One Factor Models . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4.2 Two Factor Models . . . . . . . . . . . . . . . . . . . . . . . . 63

3.4.3 Three Factor Models . . . . . . . . . . . . . . . . . . . . . . . 70

4 Commodity Indices 73

4.1 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

v

Contents

4.1.1 Index Composition . . . . . . . . . . . . . . . . . . . . . . . . 74

4.1.2 Index Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.1.3 Rebalancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1.4 Return Calculation . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1.5 Leveraged versus unleveraged Returns . . . . . . . . . . . . . 75

4.2 The Major Market Indices . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2.1 CRB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2.2 GSCI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2.3 DJ-AIGCI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.2.4 DBLCI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.2.5 DBLCI-MR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.2.6 RICI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2.7 Comparison of the Major Market Indices . . . . . . . . . . . . 81

4.3 Index Linked Products . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3.1 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3.2 Mutual Funds . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.3.3 Exchange Traded Funds . . . . . . . . . . . . . . . . . . . . . 86

4.4 Decomposition of Index Returns . . . . . . . . . . . . . . . . . . . . . 88

5 Properties of Commodity Returns 94

5.1 Characteristics of Single Commodities . . . . . . . . . . . . . . . . . . 95

5.1.1 Risk and Return Profile . . . . . . . . . . . . . . . . . . . . . 96

5.1.2 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.1.3 Diversification . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.2 Properties of the DJ-AIGCI Return Components . . . . . . . . . . . 116

5.2.1 Key Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.2.2 Roll Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.2.3 Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.2.4 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.2.5 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6 Asset Allocation with Commodity Derivatives 140

6.1 Mean Variance Spanning . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.2 Dependence to Stocks, Bonds and Inflation . . . . . . . . . . . . . . . 144

6.3 Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 148

7 Conclusions 157

A Data Description 160

vi

Contents

B Characteristics of Selected Commodities 164

B.1 Heating Oil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

B.2 Gasoline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

B.3 Gold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

B.4 Aluminium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

B.5 Copper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

B.6 Lead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

B.7 Nickel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

B.8 Zinc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

B.9 Sugar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

B.10 Coffee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

B.11 Soybean Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

B.12 Lean Hogs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

C Mathematical Preliminaries 183

C.1 Statistical Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

C.2 Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

C.3 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . 193

C.4 Equivalent Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

C.5 Feynman-Kac Representation . . . . . . . . . . . . . . . . . . . . . . 197

D Program Codes 199

D.1 Portfolio Allocation with Commodities . . . . . . . . . . . . . . . . . 199

D.2 Hurdle Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

D.3 Help Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

References 208

vii

List of Figures

2.1 Overview of the different Commodity Types . . . . . . . . . . . . . . 6

2.2 Crude Oil Historical Price Development . . . . . . . . . . . . . . . . . 8

2.3 Crude Oil Reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Net Crude Oil Consumption . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Natural Gas Price and Net Consumption . . . . . . . . . . . . . . . . 13

2.6 Natural Gas and Crude Oil Prices . . . . . . . . . . . . . . . . . . . . 14

2.7 Gold Price Movements between 1960-2006 . . . . . . . . . . . . . . . 17

2.8 Today’s Gold Price Dependence of the US dollar . . . . . . . . . . . . 18

2.9 The London Metals Exchange Index . . . . . . . . . . . . . . . . . . . 20

2.10 Cocoa Bean Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.11 Cocoa Bean Price and Net Consumption Change . . . . . . . . . . . 24

2.12 Corn Price, Stock of Inventory, Production and Consumption . . . . . 26

2.13 Cattle Price, Stock of Inventory, Production and Consumption . . . . 27

2.14 Dependency of Feeder Cattle and Corn Prices . . . . . . . . . . . . . 28

2.15 Commodity Markets Process Chain . . . . . . . . . . . . . . . . . . . 29

2.16 Overview of Commodity Investment Instruments . . . . . . . . . . . . 32

2.17 Commodity Swap Payment Streams . . . . . . . . . . . . . . . . . . . 37

2.18 Commodity Linked Structured Notes . . . . . . . . . . . . . . . . . . 38

2.19 Comparison of Gold and Gold Mining Companies . . . . . . . . . . . 42

3.1 The Risk Premium Model . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Backwardation and Contango . . . . . . . . . . . . . . . . . . . . . . 47

3.3 The Concept of Expectational Variance . . . . . . . . . . . . . . . . . 50

4.1 The CRB Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2 The GSCI Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3 The DJ-AIGCI Index . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4 The DBLCI Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.5 The DBLCI-MR Index . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.6 The RICI Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.7 Index Component Distribution . . . . . . . . . . . . . . . . . . . . . . 83

4.8 Decomposition of Commodity Index Return . . . . . . . . . . . . . . 88

4.9 Term Structure of NYMEX Crude Oil as per July 2006 . . . . . . . . 93

5.1 Relationship between Backwardation and annualized Return . . . . . 98

5.2 Diversification between single commodity groups . . . . . . . . . . . . 104

5.3 Linear Correlation within and between Commodity Groups (1998-2006)107

5.4 Dependence of Market Index (1998-2006) . . . . . . . . . . . . . . . . 110

5.5 Diversification among commodity groups . . . . . . . . . . . . . . . . 112

ix

List of Figures

5.6 Factor Analysis (1991-2006) . . . . . . . . . . . . . . . . . . . . . . . 116

5.7 Performance of DJ-AIGCI Components . . . . . . . . . . . . . . . . . 118

5.8 Return Behavior of DJ-AIGCI Components . . . . . . . . . . . . . . 119

5.9 Performance of DJ-AIGCI Roll Returns . . . . . . . . . . . . . . . . . 121

5.10 Time the DJ-AIGCI spent in Contango or in Backwardation . . . . . 121

5.11 Distribution Change . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.12 Histogram with Norm-Fit of DJ-AIGCI Return Components . . . . . 124

5.13 Kernel Distribution with Norm-Fit of DJ-AIGCI Return Components 131

5.14 Lagged Plot of DJ-AIGCI Return Components . . . . . . . . . . . . . 136

5.15 Autocorrelation and Partial Autocorrelation Function of DJ-AIGCI

Return Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.1 Factor Analysis with other Asset Classes (1991-2006) . . . . . . . . . 143

6.2 Performance of different Asset Classes . . . . . . . . . . . . . . . . . 144

6.3 Efficient Frontiers with and without Commodities (1991-2006) . . . . 154

6.4 Comparison of Portfolio Allocation . . . . . . . . . . . . . . . . . . . 155

6.5 Efficient Frontier and the Hurdle Rate . . . . . . . . . . . . . . . . . 156

B.1 Dependence of Heating Oil Prices to Crude Oil Prices . . . . . . . . . 164

B.2 Heating Oil Prices for Future Delivery . . . . . . . . . . . . . . . . . 165

B.3 Dependence of Gasoline Prices on Crude Oil Prices . . . . . . . . . . 166

B.4 Gold Inventories and Prices . . . . . . . . . . . . . . . . . . . . . . . 168

B.5 Aluminium Inventories and Prices . . . . . . . . . . . . . . . . . . . . 169

B.6 Copper Inventories and Prices . . . . . . . . . . . . . . . . . . . . . . 171

B.7 Lead Inventories and Prices . . . . . . . . . . . . . . . . . . . . . . . 172

B.8 Nickel Inventories and Prices . . . . . . . . . . . . . . . . . . . . . . . 173

B.9 Zinc Inventories and Prices . . . . . . . . . . . . . . . . . . . . . . . . 175

B.10 Sugar Price, Stock of Inventory, Production and Consumption . . . . 176

B.11 Coffee Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

B.12 Coffee Price and Stock of Inventory . . . . . . . . . . . . . . . . . . . 178

B.13 Soybean Price, Stock of Inventory, Production and Consumption . . . 181

B.14 Lean Hogs Price, Stock of Inventory, Production and Consumption . 182

x

List of Tables

2.1 Oil Reserves and Production . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 Equivalent Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.1 Comparison of Commodity Index Characteristics . . . . . . . . . . . 82

4.2 Commodity Index linked Mutual Funds . . . . . . . . . . . . . . . . . 85

4.3 Construction of a Futures Return Series for Crude Oil . . . . . . . . . 89

4.4 Spot, Future and Roll Return Time Series for Crude Oil . . . . . . . 91

4.5 Construction of a Futures Return Series for Copper . . . . . . . . . . 92

4.6 Spot, Future and Roll Return Time Series for Copper . . . . . . . . . 92

5.1 Return Components of different Commodity Indices (1998-2006) . . . 97

5.2 Volatility Components of different Commodity Indices (1998-2006) . . 101

5.3 Pearson Correlation (1998-2006) . . . . . . . . . . . . . . . . . . . . . 104

5.4 Kendall Correlation (1998-2006) . . . . . . . . . . . . . . . . . . . . . 106

5.5 Key Statistics of DJ-AIGCI Components . . . . . . . . . . . . . . . . 119

5.6 Key Statistics of DJ-AIGCI Roll Return . . . . . . . . . . . . . . . . 123

5.7 Distribution Statistics of DJ-AIGCI Return Components . . . . . . . 126

5.8 Significance Tests for Normality of DJ-AIGCI Total Return . . . . . . 130

5.9 Dickey Fuller Test for Stationarity . . . . . . . . . . . . . . . . . . . . 135

5.10 Significance Tests for Autocorrelation . . . . . . . . . . . . . . . . . . 138

6.1 Mean Variance Spanning Coefficients (1991-2006) . . . . . . . . . . . 142

6.2 Mean Variance Spanning Coefficients (2002-2006) . . . . . . . . . . . 142

6.3 Key Statistics of different Asset Classes’ Returns (1991-2006) . . . . . 145

6.4 Kendall/Pearson Correlation between Different Asset Classes and In-

flation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.5 Pearson Correlation between average Return and Volatility . . . . . . 147

xi

1 Introduction

Investing in financial assets was for a long time an exclusive and covered business

which was reserved to institutional and selected wealthy investors. The introduc-

tion of the internet and direct brokerage made transaction costs falling and hence

investment became available to the broad public. While the investment volume in

stock and bond markets grew steadily in America over the last 20 and in Europe

over the last 10 years,, commodities did not play a major role in financial investment

owing their low prices and in-transparency. But the situation has changed over the

past 5 years. Industry has grown to close to 9 billion traded contracts in 2005.1

Due to low performance of stocks and bonds, the time period from 2000 to 2003

boosted the growth of financial commodity markets in particular. The total amount

of traded contracts has nearly tripled during this time. There are two dimensions

which foster the development: hedging and investing. The former is mainly driven

by structural changes in the global economy and the remarkable demand during the

1980s and 1990s: Certain industries lowered their resource intensity noticeably or

became more efficient. This applies for instance to the automobile industry which

has steadily reduced the proportion of metal in their cars. Something similar can

also be noted for farm outputs which could be increased considerably by means of

technological improvements. For this reason, producing commodities became a less

attractive and low profit business. This development led, however, to a decrease in

production and falling inventories. After the collapse of the communist system, ap-

proximately three billion people, accounting for almost 70% of world’s population,

entered into global trade. Their cheap labor costs attracted low educational work of

the producing business or IT-services of international acting companies. This way

of proceeding boosted their economic growth and hence the need for more energy,

new infrastructure, diversified food and thus, for commodities. Caused by the reces-

sion in the commodity producing industry during the 1980s, the business was not

prepared for the sudden demand and prices went through the roof. This forced com-

panies to focus on commodity price risk management what fuelled the demand for

financial risk hedging products. On the other hand, the low performance of stocks

and bonds between 2000 and 2003 moved investors to look around for new, more

attractive return sources. Rising prices of commodities since 2001, have stimulated

their interest and thereupon, the demand for products to enable investment in this

market.

On the following pages, we introduce commodity markets from the investment’s

1See Futures Industry Magazine Jan/Feb 2006, Figure includes contracts on financial assets.

1

1 Introduction

point of view i.e. commodities are highlighted as an asset class. The starting shoot

is in Section 2 an ”Overview of Commodity Markets”. First, we look closer into the

three commodity groups as there are: energy, metals and agriculture. Energy is one

of the most important goods in our daily life. It has a significant economic impact

on all other commodity groups. Afterwards, we give a brief overview of market

participants, their motivation and relevant commodity market characteristics. The

section concludes with a rough outline on commodity trading vehicles

In Section 3 ”Pricing of Commodity Futures” we embody the characteristics of

commodity prices into mathematical forms. Although the underlying trading vehicle

for commodities are futures contracts, i.e. financial derivatives, their price cannot be

valued following general arbitrage arguments. Commodities are assumption goods

and therefore, it is possible that inventories are eaten up in times of scarcity yielding

into the impossibility to create a hedging portfolio that gives the fair price of the

financial derivative. Moreover, the price of commodities is driven by supply and

demand. This has to be taken into consideration.

Lectures2 about stock markets show that investors are attracted to broad diver-

sified exposure in the selected asset class. This is represented by indices which

we introduce in Section 4 ”Commodity Indices”. Moreover, we discuss their com-

mon properties and single characteristics. It might be surprising that the market is

strongly dominated by two indices offered by Goldman Sachs and Dow Jones and

that the different commodity indices generally are not older than five to ten years.

This is reasoned by the adolescence of the market. In the last part of the section

we decompose the index return into its single elements: spot, roll and interest rate

return earned on collateral.

In Section 5 the statistical ”Properties of Commodity Returns” is addressed. The

first part aims to show that the returns of the different single commodities interact

homogenously among each other within the three commodity groups energy, metals

and agricultures but heterogeneously between them. Therefore, an investor can gain

extra profit by diversifying its exposure among the different commodity groups.

Nevertheless, it comes up that the economic significance of oil and oil products

at first seen in Section 2 can also be seen statistically in index returns that are

composed out of different commodity groups. We identify the Dow Jones Index

as balanced commodity exposure and the second part of Section 5 gives a deeper

insight into distribution properties of the single elements of the index. It comes up,

that roll returns had a huge impact on the total value gain of the index.

2For instance [Campbell 2000].

2

Finally, we bring commodity returns into the portfolio context. In Section 6 ”Strate-

gic Asset Allocation with Commodity Derivatives” we show that commodity invest-

ments, represented by the Dow Jones AIG Commodity Index, are indeed an asset

class of its own. Furthermore, we present the cross relations of commodity returns

to stock and bond returns and show that slightly negative correlation characteris-

tics yield to a better risk and return profile of portfolios including stocks, bonds

and commodities in comparison to traditional portfolios including stocks and bonds

only. It is a major conclusion that the better risk and return profile is independent

of the assumption of extraordinary high commodity returns like the ones realized

over the last years.

3

2 Overview of Commodity Markets

After the wall came down and the communist system broke down, emerging coun-

tries have economically grown much faster than industrial countries. The integration

of global financial markets, the expansion of international companies by outsourc-

ing their production into low labor cost countries and the liberalization of trade

restrictions had a positive impact to the economies of the emerging markets. For

instance, China’s share of global Growth Domestic Product (GDP) increased from

6% in 1990 to 15% in 2005 drawing level with the Euro zone. India’s share of global

GDP increased from 4.25% in 1990 after all to 6% in 2005 drawing level with Japan.3

Additionally, the commodity import and export rates increased by two reasons. On

the one hand, the commodity demand of the emerging countries is a function of

the final good demand of the industrial countries. On the other hand, commodities

are needed to push industrialization and electrification, urbanization and expansion

of the road networks. Because global commodity supply was regressive during the

1980s and 1990s, the business didn’t invest into new production areas and reduced

production to stable prices. As, at the beginning of the 21rst century, the internal

demand of emerging countries for commodities increased sharply, the business was

not prepared resulting into short term supply shortages and a strong commodity

price increase.

At the moment we are in the middle of the fourth price rally of the last 100 years.

The first two were caused by the two world wars which were followed by global

economic break downs and the third occurred during the 1970th as the US started

to print money to finance its Vietnam War yielding into a hyper inflation period.

Inflation is measured as the change of a product basket’s value. But products are

made of commodities explaining the brotherhood of inflation and commodity prices.

The price surge was supported by oil supply bottle necks caused by a loss of Iran

as a major oil supplier. The current high price environment in commodity markets

are not driven by high inflation. The major reason is that the supply and demand

equilibrium collapsed.

An additional side effect causing commodity prices to rise has been the depreciating

value of the US dollar, i.e. the value of the dollar in terms of another currency, e.g.

the Euro or the Yen, decreased over the last years. The exchange rate is defined

as the price of one currency in terms of another currency or more plastic spoken:

The exchange rate adjusts the price of a good in one country to be the same in

3See [UBS research 2006].

4

2.1 The different Commodity Types

terms of another countries currency.4 Exchange rates are determined by incomes

and relative prices. Commodities, as real assets, typically rise in price when the

currency in which they are quoted depreciates in value and although commodity

markets grow globally, the main trading currency is still the US dollar. The US

government has built up a massive account deficit which is caused by imports out-

weighting exports. Higher imports than exports cause an over supply in US dollar

and an over demand of foreign currency what weakens the value of the US dollar

and rises commodity prices.

The introductory words uncover the major characteristics in commodity markets.

The prices are driven by current and expected future supply and demand equi-

libriums. In Section 2.1 we will give a more detailed inside into the three major

commodity groups energy, metals and agricultures. It will become clear that the

single elements of the groups follow their own risk factors although technological

improvements enable substitutions. Because our main purpose is the analysis of in-

vestable commodities we only include commodities that are traded at an exchange.

When it comes to commodity investment, we need to understand the interaction of

the different market participants and their trading motivations. This is discussed in

Section 2.2. We close the overview of commodity markets with a brief summary of

the different financial vehicles enabling commodity investments in Section 2.3.


Commodity markets are divided into three major groups: energy, metals and agri-

cultures as it can be seen in Figure 2.1. Into the energy group account all products

which can be used to produce electricity, heat and fuel. While coal drove the indus-

trialization through the 18th and 19th century, the commodity was substituted by

oil since the beginning of the last century. Today’s high price environment drives a

new substitution wave into alternative energy sources. Because crude oil and nat-

ural gas account for around 60% of word’s energy usage we will discuss them in

Section 2.1.1 more detailed.

Appendix B.1 and B.2 will further give a detailed analysis of crude oil downstream

products heating oil and gasoline. It is provided to get a deeper understanding of

the use impact, oil has to our daily life.

4See [Sawyer Sprinkle 2003].

5


Figure 2.1: Overview of the different Commodity Types

The second big commodity group are the metals which again can be divided into the

precious and the industrial metals. Gold as a representative of the first mentioned

sub group cannot be seen as a simple commodity used in jewelery and as electric

conductor but also include the role as a world currency. It is discussed in detail

in the first paragraph of Section 2.1.2 and we will see that although the historical

backing of currencies with gold doesn’t exist anymore, investors still see gold as a

currency and wealth storage yielding into a high negative correlation to US dollar

value changes. The second paragraph of Section 2.1.2 gives a brief introduction to

the industrial metals business. The industry line went through a fundamental change

yielding into a monopoly of some selected producers. In Appendix B.4 to B.8 can

further be found some fundamental analysis of the major metals used in construction

and building, including aluminium, copper, lead, nickel and zinc. The sections show

that a major demand went inventories down and generally current investments in

new mines and refineries will yield fruits five to ten years later. Implicating, that

prices will stay high over the near future.

The third group as shown in Figure 2.1 are the agricultures. This group is by

far the most heterogenous one. It is divided into the softs, the grains and the

livestock products. The major characteristic of these commodities is the seasonality

and the sensitivity to weather conditions and epidemics. While metals and energy

already increased in price sharply, agricultures are lacking behind. The analysis in

Section 2.1.3 will show that inventories are low and new fields of application, e.g.

alternative fuels, cause further demand.

6


2.1.1 Energy

Everybody’s daily life is unimaginable without energy. The first energy sources

of men kind were wood, wind and water. Especially wood was used as a heating

medium. Over the centuries it became scarce around cities and people were forced to

change to another heating source. Great Britain was the first nation that switched

to coal in the 17th century. With ongoing industrialization its usage increased but

with it environmental problems. Since the middle of the 19th century the next

huge change started, the change from coal to oil. Especially the invention and

industrial integration of the combustion engine pushed world’s need for oil as major

energy source. Nowadays, energy became besides fresh water and clean air the

most important element in human life and nobody can imagine to live without

electrical light, heating and automatic transportation. No wonder, that energy

prices have a huge impact to our life and therewith, to industry. During the last

years, there has been a huge oil price surge. The reasons for this are manifold and

shall be explained in Section 2.1.1.1. Moreover, crude oil is the input for two also

exchange traded downstream products: heating oil and gasoline. Their dependence

structure and market characteristics are explained in Appendix B.1 and B.2. The

high price environment provoked by scarcity and political instability put men kind

under evolutionary pressure to change to another major energy source. Because coal

and nuclear energy are, caused by their unattractive environmental impacts, only

short term alternatives, natural gas and so-called alternative energy sources as listed

in Figure 2.1 are getting more popular. Natural gas is today the most promising

alternative and therefore, discussed in the Section 2.1.1.2.

2.1.1.1 Crude Oil

Crude oil is petroleum that is acquired directly from the ground. It is formed millions

of years ago from the remains of tiny aquatic plants and animals that lived in ancient

seas. Around 4,000 BC in Mesopotamia a tarry crude oil, called bitumen, was used

as caulking for ships, as a setting for jewels and mosaics. The walls of Babylon and

the Egyptian Pyramids are hold together with bitumen. During the 19th century

in America, an oil finding was often met with sadness, water was more attractive.

It wasn’t until 1854, with the invention of the oil lamp, that the first large-scale

demand for petroleum emerged. Rockefeller became the first billionaire by giving

away these lamps for free and earning money by selling the kerosine. Today, crude

oil is as important for the economy as air is important for men. It became world’s

first trillion dollar industry and accounts for the single largest product in world

7


trade. The different kinds range from light colorless liquids to black oily sledges and

are named by their origin. The most important kinds of crude oil are:

Brent Blend: a mix of 15 different crude oils from the the North Sea

West Texas Intermediate (WTI) from the USA

Dubai from the Middle East

Tapis from Malaysia

Minas from Indonesia

Crude oil was one of the hottest topics in the last years and the market has be-

come the biggest and most developed of all commodity markets. The development

began in the 1970s as world’s industry realized its dependency of oil and the need

for hedging. In 2001 a new price surge started. In 1999 a barrel crude oil costed

around 10 US dollar and now in 2006 it is quoted around 70 US dollar per barrel.

The historical oil price development is shown in Figure 2.2. Two prices are given:

first, the price of a barrel in dollars of the day and second, the price of a barrel in

2005 US dollars to project the price development in nowadays scale.5

Figure 2.2: Crude Oil Historical Price Development

After the first euphoria about oil’s usability at the end of the 19th century and

the establishment of refineries and production infrastructure, oil prices went into a

long period of stability for around 100 years. The major oil reserves are located

in the Middle East which account for around two thirds of world’s total reserves.6

5See [BP Report 2006].6See Table 2.1.

8


Unfortunately, this region is characterized by political instability. The strong oil

dependence of world’s industry and its fear for supply interruptions causes oil prices

to react heavily to queries in this countries. The first huge oil crises in 1973 was

activated by the announcement of the Organization of the Petroleum Exporting

Countries (OPEC) to stop oil exports to Western countries which supported Israel

in the Yom-Kippur-War against Egypt. The second price surge was caused by a

revolution in the Iran ending with a regime fall. With the new government ruled by

Ayatollah Khomeini oil production was noticeable decreased: In 1978 Iran produced

8.5% of world’s total production, in 1979 it was only 5% of world’s total production

and in 1980 its output was fallen to 2.4% of world’s total production.7 The shocks

during the last century generated a worldwide sensitivity to the dependence on oil

resulting in different arrangements to cut off this chain and therefore, decreasing de-

mand followed by collapsing prices at the end of the century. The OPEC and USA

stabled prices with different engagements until 1990 as the first Golf War started.

An interesting observation is that spot prices jumped up nearby 50% to over 35 US

dollar per barrel but the market saw this increase as a short term movement: the

twelve month later futures contract kept calm in price with 10 US dollar behind the

first month contract.8 Nowadays, this picture has changed. Since the 11th Septem-

ber of 2001 oil prices are rising and it prices suspect that they will stay at a high

level over a long period: the December 2012 futures contract is quoted only 10 US

dollar behind the current spot price. This is indeed an indicator that we entered

a long run high oil price period: The market is willing to pay over 65 US dollar

per barrel crude oil that will be delivered in 6 years. Reasons for this are manifold.

Figure 2.3 presents the statistic about world’s oil reserves.

Since the two major findings in 19869 and worlds biggest finding in 198810, no major

oil field has been discovered during the last 20 years.

Analyzing the data of the [BP Report 2006] including all relevant information re-

garding oil reserves, production and consumption uncovers interesting insides: Mid-

dle East’s reserves remained nearly constant and the US reserves are decreasing.

Table 2.1 shows the distribution of the main oil sources of the world in percentage

share of total world resources in 2005 ranked by its size. All other countries have

reserves below 3.5% of total world reserves in 2005.

7See [BP Report 2006].8See Bloomberg, NYMEX crude oil futures contracts.9over 40 billion barrels in Iran and over 60 billion barrels in the United Arab Emirates

10over 100 billion barrels in Saudi Arabia

9


Figure 2.3: Crude Oil Reserves

From this point of view someone can understand Iran’s and Iraq’s importance for

world oil markets and the huge political efforts around these countries. Under the

dictatorship of Saddam Hussein Iraq’s pumping output had reached scales equal to

them of the United Arab Emirates. But since the war in 2003 it has decreased around

20%. Similarly, Iran couldn’t catch up with former pumping quotes. This is pointed

out in Table 2.1. It shows the production of the biggest resource owners worldwide:

the ownership of oil does not go hand in hand with production. Although countries

like the USA, Mexico, China, Canada and Norway have just a quarter of the reserves

of Kuwait or the United Arab Emirates they support world’s economy more with

higher production quotes as these countries do. However, the proved reserves still

last for more than 40 years but market participants always gets nervous when things

come to an end and the change to substitutes is cost intensive.

A contemporary issue is that the main part of the oil reserves are located in the

political instable regions of the Middle East. Moreover, the USA as the world’s

largest consumer by far had good relations to the Middle East and became aware

Reserves ProductionCountry% of total rank % of total rank

Saudi Arabia 22.0% 1 13.5% 1Iran 11.5% 2 5.1% 4Iraq 9.6% 3 2.3% 3Kuwait 8.5% 4 3.3% 10United Arab Emirates 8.1% 5 3.3% 11Venezuela 6.6% 6 4.0% 7Russia 6.2% 7 12.1% 2

Table 2.1: Oil Reserves and Production

10


after the 11th September of 2001 that huge problems bubble under the earth.

The main problem of world’s dependence of the pumping quotes is the steady rising

consumption. Figure 2.4 shows the daily net consumption (= production minus

consumption) over the last centuries.11

Figure 2.4: Net Crude Oil Consumption

Since 1981 consumption has risen much faster than production because of the pro-

duction arrangements between the USA and the OPEC to stable prices during the

1980th and 1990th and world’s growing industrialization, technological improve-

ments and electrification. Furthermore, the US output and reserves have decreased

since 2000 and so did Norway’s. Indonesia reached the break even point last year

where production and consumption netted off each other and the country is ex-

pected to change from a net exporter to a net importer. Although the OPEC

has increased pumping quotes, reasons mentioned above let to huge negative bars

since 1997 representing the under production and over consumption. Consequently

world’s inventories are low today. Especially the Asia Pacific region has increased

its oil consumption extraordinary. China has on average heightened its consumption

around 8.5% p.a. over the last 10 years. It has grown to the second largest consumer

of crude oil with a yearly share of total world consumption of 8.5%, topped by the

USA with a yearly share of 25% and followed by Japan with a share of 6.5% in

2005.12 But still, China’s barrel per capita usage of oil is very low: it uses 0.005

barrel per day per capita. In comparison, the USA uses 0.07 barrel per day per

capita and Japan uses 0.04 barrel per day per capita.13

11See [BP Report 2006].12See [BP Report 2006].13Population data source : Census Bureau of the U.S. Department of Commerce (www.census.gov).

11


The major problem for the future will be that China is a giant emerging market

and emerging markets have according to experience a higher demand elasticity than

industrial countries. While industrial growth in Western countries is followed by an

oil demand growth of one third, industrial growth in emerging countries is followed

by an oil demand growth of two thirds.14 For instance, if the USA industry grows

around 3% per annum this is followed by an oil demand increase of 1%. If the

Chinese industry growths around 3% per annum this comes in line with an oil

demand increase with 2%. During the last 5 years the Chinese economy grew around

9.5% per annum while the USA economy grew around 6% per annum. This data

forecast a further demand increase over the next years.

Nevertheless, there are movements to switch to alternative energy sources but sub-

stituting long term grown established structures will take a while but will move on

in a high oil price environment. Because crude oil is the input factor for many retail

products like heating oil or gasoline, companies pass through high crude oil prices

to consumers. High costs for heating and transport cut off net salaries like higher

taxes. If no change is expected over a longer period, people are likely to switch

to alternative energy sources although the change is connected to an one time in-

vestment. This movement can already be discovered in Europe and the USA where

biofuels, solar cells or natural gas are getting more popular.

2.1.1.2 Natural Gas

Natural Gas is a fossil fuel that is colorless, shapeless, and odorless in its pure form.

It is combusting, clean burning, and gives off a great deal of energy. Around 500 BC,

Chinese were the first who discovered that the energy in natural gas can be used.

They passed it through bamboo-shoot pipes and then burned it to boil sea water to

get potable fresh water. In the 18th and 19th century natural gas was introduced to

Europe and the USA. There is a vast amount of natural gas estimated still to be in

the ground. Today, the Russian Federation is the major producer worldwide with

a share of total world production of around 22%, followed by the USA with a share

of total world production of around 19% and by far third Canada with a share of

total world production of around 7%.

Natural gas is getting more popular and has the potential to grow to a real alterna-

tive to crude oil. First, only 40% of its reserves are located in the political instable

Middle East and second, natural gas burns clean and with little air pollution impli-

cating an environmental advantage to crude oil and coal. Natural gas was formed

14See[UBS research 2005].

12


in the same way like petroleum and occurs to 30% in combination with its liquid

brother and to 70% in independent fields. Figure 2.5 shows its price and net con-

sumption development over the last 35 years in the scale of oil equivalent for better

comparison.

Figure 2.5: Natural Gas Price and Net Consumption

Natural gas prices are by far not that volatile as crude oil prices are and are by

far not that reactive to political queries because it does not have the importance

in world’s industry, yet. It can be seen as a substitute to crude oil what is getting

specially in an high oil cost environment more popular. Its major drawback to

petroleum is that is comes as a gas. Therefore, its volume is thousand times that of

its liquet brother. Since the 1960s there exist procedures which cool down natural

gas until it goes over into the liquid state of aggregation. In this form it can be

stored and shipped or driven to its place of destination. This procedure is more

complicated and cost intensive as the storage and transportation of oil. Therefore,

the industrial use of natural gas becomes economically justifiable only in a high oil

cost environment.

As we’ve already mentioned, there are different movements to switch from oil based

products to alternative energy sources what explains the higher consumption growth

rates of 2.5% on average over the last 10 years in comparison to crude oil which usage

has been grown on average 1.7% over the last 10 years.15 Natural gas can be used

to substitute traditional heating and electricity sources. Many households already

started to switch their heating fuel from oil to gas what increased its demand. Over

this, car manufacturers are working to get fuel cell based motors ready for commer-

cial use. First test series have been successfully but bulk production stills needs

15See [BP Report 2006].

13


some time because the technology is not stable yet, especially in cold temperature

regions. Furthermore, a gas station infrastructure is not established, yet.

Summing up, natural gas has the potential to become one of the leading energy

sources during the next years. Especially China will be a major consumer. Growing

concerns about pollution from coal burning, China’s major energy source at the

moment, have forced the government to turn to cleaner burning fuels. Especially in

parts where access to coal resources is limited, a number of regasification plants are

currently under construction or planned.16

Closing this section, we will show the structural change in market dependencies

when one product is used as substitute for another. Therefore, Figure 2.6 shows the

historical price movements of natural gas and crude oil.

Figure 2.6: Natural Gas and Crude Oil Prices

Taking a closer look we realize that the price movements became more similar over

the years. Indeed, the correlation between the two energy products increased over

time. While there does not exist a statistically significant correlation between rel-

ative changes in the price series during the periods 1980-1989 and 1990-1999 a

correlation of 0.23 is significant at the 5% alpha level during the period 2000-2006.17

The increasing dependence between the two commodities reflects the substitution

of crude oil with natural gas: when crude oil prices rise market participants switch

to natural gas. This movement increases the demand for natural gas and prices

are rising. Moreover, this movement increases the dependence on steady supply of

16See [EIA Outlook 2006].17For this analysis we took monthly cash data of the [The CRB Commodity Yearbook 2005]

completed with Bloomberg data for 2005 and 2006. We used monthly log returns as ofDefinition C.2. For the mathematical definition of Pearson correlation and the related statisticaltest see Section 5.1.2.

14


natural gas. Therefore, nervous market reactions to possible supply interruptions

can occur. Hurricane Katrina destroyed major pipelines in Oklahoma. The reaction

of the market can clearly be seen in the huge amplitude of natural gas prices in

August 2005.

2.1.2 Metals

Figure 2.1 showed that metals are divided into two big groups: the precious and the

industrial metals which will be introduced in this section.

Precious metals have attracted people with living memory. In former times mainly

used in jewelery, they moved into industrial usage, today. Their representatives

include gold, platinum, palladium and silver. They are mainly used in electric and

computer circuit. The strong market expansion in this area resulted in a high de-

mand for this metals. But for investors gold is still the most attractive representative

of this group. The reason for this can be found in golds role as international currency

and with it as store of value. Therefore, its characteristics and price influencing fac-

tors are presented in Section 2.1.2.1 and shall exemplify an investment in precious

metals. The history of gold is affected by times of the so-called gold standard as

different world currencies were backed up by gold’s value. The last period ended in

1971 as world’s leading currency, the US dollar, was disconnected from gold’s value.

The purpose of the analysis is to find out whether the gold price movements of last

years are still connected to world currencies or not.

Industrial metals are mainly used in construction and building of infrastructure,

transportation and housing. During the 1980s metals industry went through a re-

gression. Technological improvements enabled savings in materials, e.g. cars became

lighter or aluminium and steel were substituted by strong but lighter carbon com-

pounds. Falling prices made the industry line unattractive. Production was driven

down and investments in new production areas disappeared. With the boom of

emerging country’s economies, the demand for industrial metals increased sharply

because new production plants and infrastructure were needed and followed by a

housing boom as standards of living increased. Everybody knows the headlines

about ”China - works ahead!”. The activities pulled of huge amounts of the metals

for what the industry was not prepared. The analysis’ of Appendix B.4 to B.8 show

the inventory destroying effects of these movements followed by rising prices. The

main problem in metals production is the tediousness of the industrial development

of new mines and refineries. In general, many years pass by from the exploration of

a new ore deposit until the first hammer blow. Apply for an official digging approval

15


is time consuming, investors need to be found and equipment and machinery need

to be delivered. Moreover, production costs increased sharply: the cost for an open

pit quintupled over the last ten years and reached 500 million US dollar in 2006. All

these factors resulted into a fundamental change of the industry explained in the

Section 2.1.2.2.

2.1.2.1 Precious Metals exemplified by Gold

The dense, bright yellow metallic element called gold put a spell on people since its

first discovery. Egyptians mined gold since 2,000 BC and worked it up to jewelery

for beauty and religious purposes. Over this, gold is the oldest international cur-

rency and has played a role in most countries’ currency systems for well over 2,000

years. Gold’s scarcity, the fact that it does not corrode or tarnish, coupled with its

malleability so that coins can easily be shaped and the way in which it has been

prized in all civilizations, have made it eminently suitable as a form of money. The

first pure gold coin appeared on the orders of King Croesus of Lydia around 550 BC.

During the Middle Ages in Europe, gold and silver formed the basis of the currency

systems. Although, gold was too valuable for most day-to-day transactions, it was

used as backup system. The so-called gold standard defines a monetary system that

has linked its currency’s value to gold prices at a fixed rate. The only standard-

ized international gold standard existed for a comprehensive short period from the

1870th until the outbreak of the First World War in 1914. A crucial advantage

of the gold standard was the certainty of foreign investors, that the value of their

investment was unlikely to be hurt by the depreciation of the recipient country’s

currency relative to their own. This facilitated large flows of international direct in-

vestments. The capital enabled the fast development of the United States, Canada,

Australia and other emerging markets of that days. Relative to the size of world’s

economy, these flows were as large or even larger than today’s and they were far less

volatile.18

During the worldwide queries of the period between the two world wars it was not

possible to establish a new gold standard. First in 1944 a new gold standard was

introduced by passing the Bretton Woods Convention. It fixed the US dollar to 35

US dollar per ounce while other currencies were defined in terms of the dollar with

fixed but with authorization of the International Monetary Funds adjustable rates.

The US dollar was chosen because the USA were the one country worldwide what

18See World Gold Council.

16


hold credible gold reserves to back their currency.19 The system finally failed caused

by two major drawbacks:

1. The dollar was on the one hand the international reference currency but on the

other hand the currency of the USA. It therefore could change its monetary

policy without bearing any consequences. Indeed all countries involved in the

Bretton Woods Convention financed a part of the huge budged deficit the USA

had cumulated during the Vietnam War what was payed with printing new

money.

2. The creditability of the USA decreased drastic because Germany and France

changed their dollars into gold. At the end of the 1960th the gold reserves of

the USA had fallen around one third.20

Finally US-President Richard Nixon abandoned the system in 1971. The last fixing

price before the ”gold window” was closed was 42.22 US dollar per troy ounce, and

to this price the United States officially valued its gold holdings. Figure 2.7 shows

the long term movements of the gold price and its free flow since 1971.

Figure 2.7: Gold Price Movements between 1960-2006

The newest study about the behavior of gold prices in an international environ-

ment by [Levin Wright 2006] has pointed out the major drivers of gold prices in the

short and long run. For the first case, the authors named changes of the follow-

ing economic indicators to be statistically positively correlated with changes in the

gold price: US inflation, US inflation volatility and credit risk. On the other hand,

changes in the US dollar trade-weighted exchange rate, what reflects the value of

19At this point in history the USA hold 70% of worldwide gold reserves. See [UBS research 2005].20See [UBS research 2005].

17


the US dollar in terms of a basket of the major world currencies, is statistically

significant negatively correlated to changes in the gold price. The second finding

is mentioned in [UBS research 2005] as well and can be seen in Figure 2.8.21 The

correlation between changes in the gold price and changes in the US dollar trade-

weighted index is significant with -0.44. Although, the gold standard has dropped

many years ago, changes in the value of the US dollar and gold exhibit still a strong

dependence structure. Moreover, the high negative correlation suspects, that in-

vestors view gold as a storage of wealth.

Figure 2.8: Today’s Gold Price Dependence of the US dollar

In the long run [Levin Wright 2006] could proof the statistical significance of a pos-

itive dependence between the gold price and the US price level.

Summing up, the research indeed showed that the connection between the gold price

and the US dollar never disengaged because many investors still trust gold to be

a wealth carrier and as a currency hedge. In the period between 1996 and 2000

gold lost more than one quarter of its value. This was caused by a strong American

stock market attracting huge amounts of capital and a sold out of gold reserves by

European Central Banks to standardize their gold reserves to prepare themselves for

the introduction of the Euro. Nowadays, the gold price increased strongly caused

by the problems of the USA resulting in a dollar watering place and a run out of

dollar investments. Specially countries out of the Middle and Far East are backing

up their wealth with gold.

The historical relevance of gold as an international currency provides this commodity

with special features. A simple analysis of supply and demand is not enough to

21Data source: Bloomberg. We used monthly log returns as of Definition C.2. For the mathemat-ical definition of Pearson correlation and the related statistical test see Section 5.1.2.

18


show its value. In fact, geopolitical circumstances outweigh fundamental commodity

market analysis. Nevertheless, this part of the valuation of gold should not be

forgotten and is done in Appendix B.3. Because gold still is a commodity, supply

interruptions or surpluses and abrupt surges or collapses in demand can cause prices

to rise or to fall.

2.1.2.2 Industrial Metals

Metals became one of the most substantial elements in our daily life. But who thinks

about that our mobile phones, ballpoint pens and the aluminium foil in the kitchen

are made of stone?

Mining has been performed since prehistoric times. The people of the Stone Age

used different kinds of mineral quartz for weapons and tools. The first metal that

humans learned to mine and shape was copper. This was the beginning of the

period historically called the age of metal. The oldest known metal mines are the

copper mines at Sinai dating back to 5,000 BC. The oldest known war aiming up

to conquer natural resources was around 2,600 BC under the Egyptian pharaoh

Sechemchet who annexed Sinai with the only purpose to annex its copper mines.

When people discovered that alloying copper and tin produced a stronger and more

durable metal, the so-called Bronze Age started in the Caucasus around 4,000 BC.

From there, the technology spread rapidly all over the Near East. Iron began to be

worked already in Late Bronze Age but was hardly manageable. Traditions tell that

mystery maritime people brought war and destruction and the fluctuating trade of

this times broke down. While people run out of copper, iron ores could be found

and extracted nearly everywhere. As people finally were able to handle iron, the

transition into the Iron Age around 1,200 BC was more of a political change rather

than of new developments in metalworking. The advantages of iron in comparison

to bronze were hardness, durability and cheapness.

As civilization developed, the need and the search for minerals accelerated and with

this the trade of metals. The Phoenicians crossed the Mediterranean Sea to work the

copper mines of southern Spain, and their ships sailed to the British Isles to trade

for tin. The Romans improved the mining practices in the lands and first mined

on a large scale, including amongst others the copper and tin ores in Cornwall and

Wales. In 1571, the first place was founded in London where traders of metal and a

range of other commodities began to meet on a regular basis. Because Britain soon

became a major exporter of metals, European merchants arrived to join in these

activities. Later, the coal mines and nation’s production of iron and steel provided

19


the basis for the Industrial Revolution and almost overnight, Great Britain became

the most technologically advanced country in the world, importing large tonnage’s

from all over the world. The major problem was the price uncertainty. Metal traders

having bought ores from as far as Chile and Malaya had no way of knowing what

price would predominate at the time of the ships arrival some month later. Mer-

chants and consumers had to face serious price risks. Technology came to their aid

with the invention of the telegraph. Inter continental lines of communication were

established between the countries of the world and the change from sail to steam

ships made arrival dates more predictable. In 1869 the Suez Canal was opened and

therewith delivery times reduced to three month. The unique three month forward

contract was established and is still alive at the London Metals Exchange (LME)

what was founded in 1877.22 Copper and tin were the first metals that were traded.

In 1920, lead and tin joint and at the end of the 1970th aluminium and nickel were

introduced as well. Finally, in 1999 a silver contract was launched. All contracts

are still traded open outcry circling in a five minute period known as the ”ring”.

The London Metals Exchange Index (LMEX) reflecting the price movements of all

six metals gives market participants since its inception in 2000 an overview of the

industrial metals market development. Figure 2.9 plots the index value evolvement

since its inception.23

Figure 2.9: The London Metals Exchange Index

We clearly observe a strong upward trend during the last years. The major reason

for this is a strong demand from China and India since 2002. Over 20% of copper,

aluminium and zinc world production and nearby every second cargo of iron ore

22For further details see [LME 2006].23Data source: Bloomberg.

20


went to China in 2005.24 Rising prices are highly correlated with a falling stock of

inventory. Pushing production in metals markets is neither easy nor quick. New

mines have to be cultivated and infrastructure in processing and transportation

has to be built. This is mostly combined with environmental sanctions which to

get can last for years, especially in industrial countries. Because of the low margins

during the 1980th and 1990th the industry didn’t invest yielding into capacity bottle

necks at the beginning of the 21rst century followed by the biggest price surge in

metals markets ever. Industry’s huge gains of the last five years were invested in an

acquisition relay, so that today only a couple of mining giants, including Rio Tinto,

BHP Billiton and Anglo American, rule the market. No other industry went through

such a fundamental structural change. Buying other companies means buying their

mines and knowing what you get for your money. Self exploration projects are by far

more risky. But now where only the big survivors are left and demand still exceeds

production, the industry need to go down the traditional road: Rio Tinto will invest

three billion US dollar in both years 2006 and 2007. BHB Billion will invest 12

billion in the exploration of new mines. But this can last until five to ten years and

even more caused by the reasons mentioned introductory to this section.25

Moreover, the industry faces further problems with machinery and educated staff.

While many geologists needed to drive taxi during the 1990th they account to the

most questioned employees, today. The waiting times of Caterpillar, Komatsu and

Liebherr for the giant trucks, costing around three million US dollar and needed to

transport the raw ores in the mining sites, are around 18 month and were supplied

partly without wheels because its producers Michelin and Bridgestone are still afraid

to extend their production.

Although, there is no end in sight in metals reserves because not all mines are

explored yet, putting all these factors together the agers for a long period of high

metal prices is given. Appendix B.4 to B.8 will further give a small inside into the

production and consumption structure of some selected metals including aluminium,

copper, lead, nickel and zinc.

2.1.3 Agricultures

The agricultural market is the most heterogenous market of the three main commod-

ity groups. Figure 2.1 introduced its three sub groups: softs, grains and livestock.

While energy and metals already went through a huge price surge, agricultures, with

24See [Commodities 2006].25See [Commodities 2006].

21


the exception of sugar, are lacking behind.

The softs group includes cocoa, coffee and sugar. It is internally the most het-

erogenous group of the agricultures because it constituencies are no substitutes or

competitors among each other and are no input or output factors for each other.

They only have two things in common: First, they are agricultural products and

therewith, their prices are highly dependent to weather conditions and field pests.

Second, they are used to produce luxury products like candies and pastries. There-

fore, their scale of usage depends on the standard of living in a country. Cocoa will

be discussed in Section 2.1.3.1 and shall serve us as the representative of the softs

group. As a tropical plant it’s mainly grown in Africa. Overproduction pitched the

industry into a regression at the end of the last century. But political queries yielded

to production interruptions and prices became stable over the last years. Coffee will

be discussed in Appendix B.10. It is interesting to know that its world consumption

has grown 10% with the introduction of starbucks.26 When we look around seeing

new coffee shops from different companies, including starbucks, coffee bean and san

francisco coffee company, mushrooming everywhere, price potential can be assumed.

During the last two years sugar went through a renaissance as described detailed in

Appendix B.9. Traditionally, only used to produce candy and pastries, it became

on vogue for ethanol production. Ethanol is used as intermixture to gasoline as

alternative fuel. Brasilia is the heaviest user worldwide and the sudden demand

pushed sugar prices through the roof. This is one of the best examples how high

energy prices and the search for alternatives to oil and oil products beam on other

markets.

The group of grains is one of the biggest commodity groups including corn, different

kinds of wheat, barley, soybeans and rice. Actually soybeans are a member of the

oilseed family but they are generally mentioned under this commodity group for

convenience. Grains are used as animal feed or human food. As an agriculture

their prices are weather dependent as all agricultural prices are. When it comes to

agricultural investment corn, wheat and soybeans are the most traded constituencies.

Because corn and wheat are substitutes to each other we will just introduce corn

to get an idea of the market in Section 2.1.3.2. Historically, corn is mostly used

as animal feed but with the search for new alternative bio fuels it found a new

application. The same happened to soybeans. The soybean complex including

soybeans, soybean meal and soybean oil and its market characteristics are described

in Appendix B.11. Traditionally, soybeans were just used for animal and human

food but this changed in a high energy price environment.

26See [Rogers 2005].

22


Closing this section in 2.1.3.3, we will introduce live and feeder cattle as the major

exchange traded representatives of livestock, the last big sub group of the agri-

cultural commodities. We will see that the major influencing factors of a potential

investment in this commodity group are price stability in grains markets and animal

epidemics. First, livestock prices are influenced by their production costs including

costs for corn, wheat and soybeans. Therefore, this group of agricultural commodi-

ties is indirectly driven by weather conditions having direct impact on the input

product feed. Second, the major direct risk factor in livestock markets are epi-

demics that require to destroy huge herd amounts. Nevertheless, trading lean hogs

is getting more popular as well and new products enable investors to ”feed” hogs

”on paper”. This is described in Appendix B.12.

2.1.3.1 Softs exemplified by Cocoa

500 years ago, Spanish discoverer found a plant in South America and called it

cocoa ”the food of the gods”. Today, it remains a valued commodity used to produce

chocolate and cocoa powder for direct sale or bakery articles and cocoa butter mainly

used for bakery articles, soap and cosmetics. The scope of application is not driven

by possible sudden demand shocks. The sensitive factor is production that might

be interrupted by bad weather conditions. Figure 2.10 shows the historical cocoa

price development since 1960.

Figure 2.10: Cocoa Bean Price

The huge price increase in the 1970s was caused by the US hyper inflation inducing

a general commodity price increase. In 2006, cocoa costs on average 71 cents per

pound that is around 5 cents above its average price since 1960 of 65 cents. Two

thirds of world production comes from Africa whereby Cote d’Ivories with 39% of

23


world supply in 2005 is with far distance to Ghana with 18% of world supply in 2005

the biggest producer. Europe is with 43% of world consumption the biggest user.

Amazing, that the small Netherlands with its 16 million people spent 14% of world

consumption followed by the USA with its 300 million people and 13% of world

consumption. Figure 2.11 gives an overview of world consumption and production

in comparison to price movements.27

Figure 2.11: Cocoa Bean Price and Net Consumption Change

The dark bar series shows the cumulated net consumption starting in 1995, i.e.

assuming that production and consumption started in 1995 the series shows the

development of inventories over the last 10 years. The light bar series shows the

change in yearly net consumption, i.e. the bar is positive when there was more

production than consumption in the year and vise versa. The orange line shows

the realized average price per year and indicates consequently the market reaction

to inventory levels. In 1999, there was a huge over bid caused by an extraordinary

harvest in Africa. Inventories increased heavily what caused prices to fall. Cocoa

prices dropped to their 25 year low with 40 cents per pound. The reaction was

that Brazil and Malaysia noticeable reduced their production and changed to other

crops. Over the last year production and consumption nearby netted off what was

caused by a strong demand for cocoa butter mainly for cosmetics.

It can be expected that the demand for cocoa will rise over the next years because

cocoa products are luxury products which consume naturally will increase with

growing standard of living. [ICCO05] reported that income elasticities of demand

for cocoa are around 0.85 at world scale, i.e. an increase in worlds GDP by 10% cause

an increase in worlds demand for cocoa by 8.5%. Moreover, the study shows that

27Data source: [The CRB Commodity Yearbook 2005] and [ICCO].

24


world’s demand for cocoa is negatively correlated to inflation by -0.2.28 Production

depends on the number and characteristics of the ever green tropic cocoa trees

planted 5 to 6 years ago what results in an inability to react to unexpected short

term demand.29 An other unstable factor in this market is the political situation

of the major producing countries. Cote d’Ivories has just ceased its civil war but

still, there are political queries what cause a production decrease. In 2007 the cocoa

market is likely be nervous because Africa has to fight against a cocoa moth plague.

2.1.3.2 Grains exemplified by Corn

Corn is a member of the grass family of plants and is a native grain of the American

continent. About 5,000 BC, it was first cultivated in Central America to use it as

human food. The cereal was brought to Europe and North America, but remained

poorly grown until the 19th century. Corn is a resistant plant only vulnerable to early

frosts in fall that can be grown in different climates ranging from arid to tropical

and in different regions ranging from flat country to mountain side. Today, corn

accounts for about 70% of the world coarse grain trade and about 75% of its yearly

production are used for animal feed.30 A small amount of around 15% is still used

in human food mainly for oil and vegetarian food and the remaining percentages are

used for alcohol distilleries and the production of ethanol for engines. Latter part

of usage is expected to increase considerably caused by the high oil prices. Today

around 95% of North America’s ethanol is made from corn. Programs on ethanol

production from corn will therefore have a constant influence on corn prices in the

future.31

By far the biggest producer worldwide is the USA with around 40% of share of total

world production in 2006.32 It reached its production high so far in 2004 with a

total output of 300 million tonnes, i.e. 80 million tonnes (= 25%) more than in

2006. Figure 2.12 shows the worldwide development of inventory, production and

consumption and the equivalent year ending prices.33

Low prices since the end of the 90s have made corn business unattractive and

Figure 2.12 clearly shows that inventories have fallen since then. Calculating the

28Thus, it can be assumed that a decline in prices of 10% results in an increase in demand of 2%.29A single tree can produce 20 fruits but it needs 400 to get one pound of chocolate.30See [The CRB Commodity Yearbook 2005].31See [USDA Grain 2006].32Moreover, the USA is the biggest exporter worldwide with around 70% of share of world corn

trade showing world’s corn supply dependence of US production output.33See [USDA Grain 2006] and [The CRB Commodity Yearbook 2005].

25


Figure 2.12: Corn Price, Stock of Inventory, Production and Consumption

stocks as percentage of consumption ratio highlights the situation: while at the end

of the 70th, stocks made up around 25% of consumption, they made up around 35%

at the end of the 80th, 30% at the end of the 90th and finally, only 12% in 2006.

Falling production caused by low prices came hand in hand with a growing demand.

Two major reasons for growing demand can be identified: the use of corn as part of

fuels pulls off a growing part of production and 2006 was the first year, were China

became a net importer of corn and it cannot be expected that this situation will turn

around in the next years. While the country’s stock of inventory were around 123

million tonnes at the end of the 90th, they are around 28 million tonnes today. This

environment of low inventories make prices quite vulnerable for supply interruption.

Figure 2.12 also shows that production is more volatile than consumption is. Corn

output depends on stable weather conditions. Consequently, bad weather will lead

to a bad harvest. This will be followed by exploding prices because inventories are

low and can hardly stand a production collapse.

2.1.3.3 Livestock exemplified by Live and Feeder Cattle

The beef cycle begins with the cow-calf operation, which breeds the new calves.

Because the gestation period is about nine month, most rangers breed their herds of

cows in summer, thus processing the new crop of calves in spring. This allows the

calves to be born during mild weather and they can graze through the summer and

early autumn in the open countryside. After 6 - 8 months the calves can be taken

away from their mothers and most of them are then moved into the ”stocker oper-

ation” where they spend 6 - 10 months. When the cattle reached 600 - 800 pounds,

they are typically sent to a feedlot and become the so-called ”feeder cattle”. In the

26


feedlot, the cattle are fed a special food mix including grain, e.g. corn or wheat, a

protein, e.g. soybeans, and roughage to encourage rapid weight gain. The animal is

considered ”finished” when it reaches full weight typically 1200 pounds. Then it is

sold for slaughter to a meat packing plant.34

Because feeder cattle is a downstream product of live cattle it is traded at a pre-

mium and price movements between the two commodities are highly correlated with

a significant correlation coefficient of 0.53.35 Figure 2.13 shows live cattle statistics

including prices, inventory, production and consumption.

Figure 2.13: Cattle Price, Stock of Inventory, Production and Consumption

The huge price increase in 2002 was caused by a decreasing inventory of 30 million

heads worldwide from 2002 to 2003 caused by high feeding costs. But, the total

number of calves (and their age) is not enough to describe supply: the prices of

feed, i.e. of corn, wheat and soybeans, make a big difference since animals are fed

longer if corn is cheap. This pattern can be seen in Figure 2.14 what shows the price

development of feeder cattle and corn.

Unfortunately, over the long run significant anti - correlation could not be proofed.But

analyzing weekly futures data during the period of 1994 and 2006 a significant neg-

ative correlation of -0.15 could be found.

34See [The CRB Commodity Yearbook 2005].35For the analyzes of this section we took monthly cash data since 1970 of the

[The CRB Commodity Yearbook 2005] completed with Bloomberg data for 2005 and 2006.We used monthly log returns as of Definition C.2. For the mathematical definition of Pearsoncorrelation and the related statistical test see Section 5.1.2.

27


Figure 2.14: Dependency of Feeder Cattle and Corn Prices

2.2 Characteristics of Commodity Markets

Commodity markets are in their origin quite different to traditional financial markets

for stocks, bonds and currencies. Their major difference is that commodities are

real assets that are produced and consumed in industrial processes and prices are

therefore mainly driven by industrial supply and demand. But the interaction of

supply, demand and commodity prices interfere with each other, i.e. production and

consumption are price driving but also price driven. If prices are high so-called high

price producers enter the market. A famous example for this phenomena are the

oil sands in Canada. Current oil prices have reached price levels which enable an

economic extraction of oil there. On the other hand if prices are high consumers try

to substitute commodities against each other as described on Section 2.1.1 the change

from oil to natural gas and other alternative energy sources. Still, commodities are

primarily consumption goods. Demand is therefore not purely price dependent.

Commodities are heterogenous in terms of quality and grade which is reflected in

market prices. This contrasts to traditional asset: a dollar is a dollar or a Siemens

stock is a Siemens stock but coffee or cocoa beans are likely to differ in size and

quality. Another major difference to traditional asset classes is the seasonal pattern

in consumption and production which is manifested in recurring behavior of prices

and volatility, e.g. prices for agricultural products are influenced by crop times or oil

products prices fluctuate with heating or driving seasons.36 Putting all differences

together we realize that there are new value dimensions which have to be considered

when it comes to commodity investments:

Prices are supply and demand driven

36See Appendix B.1 and B.2.

28

2.2 Characteristics of Commodity Markets

Supply and demand limits are not purely price dependent

Timing uncertainly in production and supply

Direct exposure to a variety of exogenous functions, e.g. weather, political

environment or technological change

Some commodities can be substituted by others

Physical accessability introduces transportation and location issues

There are additional costs like storage, insurance and wastage costs

Difference between storable, e.g. metals, and non storable commodities, e.g. electricity

Complex processing chains, some commodities are downstream commodities of

others, e.g. soybeans are needed to get soybean meal37 what on its side is needed

for live cattle breeding

Commodities market enable global trade. Nevertheless, local constraints have to be

kept in mind including the often high transportation costs, costs and risks between

markets (e.g. piracy, ship in distress), industry regulations and currency factors.

Figure 2.15 gives an example of a typical processing chain in commodity markets.38

It covers the major interaction of commodity market members:

Figure 2.15: Commodity Markets Process Chain

37See Appendix B.11.38Inspired by [Structured Products 2006]

29


The different participants have - caused by their different business purposes - dif-

ferent risk and return profiles. The actual real asset traders are the producers,

processors and consumers. The first goes short the commodity because he wants to

sell the raw commodity, e.g. crude oil, soybeans or live cattle. The application of

financial instruments is frequently driven by the pattern of cash flows. Generally

producers have to make significant financing in advance, e.g. a cattleman has to

pay for animal feed, accommodation, medical care etc. to undertake his production.

The live cattle sale some day in the future is exposed to price fluctuations what

makes planing uncertainly. If prices decline sharply revenues may fall short to cover

the costs of serving production’s financing costs. Hence, there is a natural tendency

for producers to hedge their future sale at price levels that ensure adequate returns

without seeking to optimize the potential returns from higher prices.

Processors have a spread exposure in commodities markets. They have to take care

about the price difference between the cost of input and the cost of the output.

Generally processors try to ensure delivery and enlarge the spread gain. Therefore,

they have to make sure that the inventories are filled up with input commodities

properly to balance output demand and input supply fluctuations.

At the end of the chain is the consumer. He goes long the commodity, i.e. he

wants to buy it. His hedging behavior is more complex. His desire to undertake

hedges is influenced by the availability of substitute products and the ability to

pass on higher input costs in its own product market. In many cases there exist

direct bilateral long term supply or purchase contracts between the consumer and

the producer which may include fixed price arrangements to reduce price risk for

both parties. Nevertheless, these agreements include a number of difficulties, e.g.

the lack of transparency, lower liquidity and exposure to counterparty credit risk.

Traders and financial institutions are the lube of commodities markets. They are

responsible to ensure price formations and enable the actual transaction. Therefore,

they act as an agent or principal to secure the sale or purchase of the commodity and

add value to a pure trading relationship by providing risk and portfolio management

expertise. Traders have complex hedging requirements depending on their customer

specific role: acting as an agent a trader generally will have no price exposure but

acting as a principal he will generally have outright price risk which need to be

hedged. Globally he has to handle his client specific risks and hedging requirements

as portfolio to enable diversification effects.

To provide commodities markets with financing and liquidity, financial institutions

are essential. Their role in commodities markets is similar to that in the deriva-

30

2.3 Trading Commodities

tives markets in other asset classes. They provide liquidity improvement, speed of

execution and structural flexibility. Nowadays, their role is becoming more complex

and interdisciplinary. They understand themselves as all finance service provider

to feed their clients with structured products. Their interaction with the last big

participant group of commodities markets, the investors, is growing strongly.

Investors trade commodities as a separate asset class and care about their portfolio

risk and return profile to maximize their revenues. For producers they are essential

as risk takers whereupon the oldest theory of how commodity prices are build up,

the so-called theory of normal backwardation,39 is based on. Investors never will

hold commodities until delivery. Cash settlement or selling before maturity is done.

Investors are necessary for fair price formation, i.e. financial markets can only

be efficient when its members reach a critical mass. Different articles40, which

analyze the return development of actively managed commodity portfolios, mention,

that their pure alpha decreased over time. This pattern indicates that financial

markets become more efficient and arbitrage opportunities are disabled just because

of liquidity and trading volume.


To cover all the different needs of the market participants, a range of different

financial products enabling both, hedging and speculation, were developed. Un-

fortunately, some investors might feel lost in the forest of opportunities. Therefore,

they should first develop a set of requirements that meet their individual investment

needs. Then, they should search and screen and just finally select what will meet

their individual objectives. In the following section the most common commodity

investment vehicles and related products are introduced. For it, Figure 2.16 shall

give a first overview. There are two major ways of investing into commodities: the

direct and the indirect over stocks.

Our main focus lies on the direct way to get commodity exposure. To be very

precise, it is divided into the direct commodity investment over financial products

and the direct commodity investment into the commodity product. But we assume

that no investor is actually interested in camping oil barrels or corn bags in his

basement, we will focus on the direct commodity investment over financial prod-

ucts without physical delivery. The basic products are described in Section 2.3.1

39We will examine the theory of normal backwardation in Section 3.1.40See e.g. [Edwards Liew 1999] or [Fung Hiesh 1997].

31


Figure 2.16: Overview of Commodity Investment Instruments

and called derivatives including futures, the elementary vehicle to trade raw com-

modities, swaps, options linked bonds and certificates. Following, the investment in

commodity portfolios, both actively and passively managed ones, are highlighted.

To get actively managed commodity exposure, an investor has to hire a so-called

Commodity Trading Advisor (CTA). The different ways to do so are described in

Section 2.3.2. Passively managed ones are represented by indices. Exposure can be

taken through index linked products such as index tracking investment or exchange

traded funds. Because the main part of this work will later focus on this investment

vehicle we will dedicate it the whole Section 4. Finally, in Section 2.3.3 we will bring

in mind stocks of commodity producing companies, i.e. the indirect commodity in-

vestment way. For many investors this represents the traditional and familiar way

of taking exposure in commodities markets. Buying stocks or stock funds are an

uncomplicated long term orientated investment methodology that is not connected

with maturities. But our focus lies on direct commodity investment and therefore,

we will keep this topic short.

2.3.1 Commodity Derivatives

As we have already pointed out, that there are different factors which increased the

demand for commodity related products. The following section describes the differ-

ent types of derivatives, i.e. financial products which payoff structure depends on the

price process of another financial instrument that is commonly used to get and/or

hedge commodity exposure. The main structure in exchange traded commodity

32


markets are futures contracts. They are the original vehicle to trade commodi-

ties. Although the price of a futures contract depends on the current spot (cash)

price41 of the underlying commodity, it represents on its side the underlying of other

derivatives like options, swaps and commodity linked bonds.

Many economists, including Alan Greenspan, stated that the financial derivatives

markets have significantly decreased the cost of doing business and thus have risen

the standard of living for everybody. A major step to this development was done in

the work of Merton Miller, Harry Markowitz and William Sharpe who won the 1990s

Novel Price in economics for recognizing and illustrating the value of derivatives in

business application.42 Their theoretical conclusions found their way into practical

applications created by Fischer Black, Myron Scholes, Robert Shiller, Rudi Zagst

and many others.

Today many financial intermediaries, including domestic and international banks,

public and private pension funds, investment companies, mutual funds, hedge funds,

energy providers, asset and liability managers, mortgage companies, swap dealers,

and insurance companies, that face foreign exchange, energy, agricultural or envi-

ronmental exposure use financial markets to hedge or manage their price risk. For

instance, at the Chicago Mercantile Exchange (CME) more than one billion con-

tracts representing an underlying notional value of 640 trillion US dollar were traded

and cleared in 2005.43

2.3.1.1 Forwards and Futures

A forward contract is a bilateral agreement where one party is going to buy an asset

at a today predefined time in the future for a fixed price. Hereby someone can be

long or short the contract depending on the fact whether he took the asset buyer

or seller position. Forwards were originally developed to hedge commodity price

risk and are useful vehicles to look at future prices. As described in Section 2.2

commodity producers need to ensure future cash flows to be cost covering. First

applications of forwards go back in the 18th and 19th centuries. Potato growers in

the state of Maine (USA) started selling their crops at the time of planting in order to

finance the production process. Such arrangements became particularly important

41For the feature of spot prices in commodity markets see the discussion in Section 5.1.1.42William Sharpe was rewarded for the Capital Asset Pricing Model, beta and relative risks, Harry

Markowitz for his theory of efficient portfolio selection and Merton Miller for his work on theeffect of a firm’s capital structure and dividend policy on market price.

43Data source: Futures Industry Magazine Mar/Apr 2006, numbers include financial derivatives(i.e. derivatives on interest rates or equities)

33


in industries where non influenceable external factors like weather conditions are of

high importance and the production process is cost intensive.

The first established emporiums where in terms of quantity, quality and delivery

date standardized forward contracts, so-called future contracts, were traded, are

the New York Cotton Exchange (NYCE), founded 1842, and the Chicago Board of

Trade (CBOT), founded 1848.

Although forwards and futures on the same underlying with the same time to expiry

have the same original spirit ”sell an asset today but deliver it tomorrow” they are

different in many counts, including transaction costs, credit risk,44 customization

and stochastic interest rate. The most noticeable difference between futures and

forwards is that futures are marked-to-market daily and their participants have to

adjust their positions on the so-called margin account which introduces an addi-

tional re-investment risk: while the profit or loss of a forward contract occurs at the

maturity date, the profits and losses of futures contracts are spread over the live of

the contract and occur on a daily basis. For instance, if a participant has a long

position in a futures contract which price went down from one day to the next, then

he gets the so-called margin call from the clearing house standing behind the respec-

tive exchange what requests him to cash settle the difference on the so-called margin

account. From this point of view a future is a series of daily settled forwards and

its value over the whole period is the net present value (NPV) of the single margin

calls. If interest rates are not stochastic the NPV equals the NPV of a forward over

the whole period.45 If interest rates are stochastic the futures price is greater or less

than the forward price depending on the correlation of interest rates and the com-

modity spot price. If they are positively correlated (what in theory should be the

case because commodities are real assets) daily payments from price increases will

on average be more heavily discounted than payments from price decreases, so the

initial futures price must exceed the forward price.46 However, studies have shown,

that the difference is typically small. [Pindyck 1994] compared one-month heat-

ing oil contracts and estimated the difference being less than 0.01%. [French 1983]

compared the futures prices of three month silver and copper contracts with their

equivalent forward prices and found that the difference is about 0.1%. Therefore, it

is common not to differentiate between forward and futures prices. We will do so,

too.

44Because forward contracts are over the counter (OTC) bilateral agreements they embody coun-terparty default risks.

45For the formal mathematical proof that forward and futures prices are equal under the assump-tion of deterministic interest rates see [Zagst 2002].

46For further information see [Cox Ross Ingersoll 1981]

34


Historically, the expiration months of futures contracts were without some excep-

tions March, June, September and December reflecting the seasonality in commodi-

ties markets. This has changed with growing markets. Today there are contracts

for every delivery month and long term maturities until 5 years available.

As we mentioned introductory, futures trading has grown rapidly. For instance, the

New York Mercantile Exchange (NYMEX) Crude Oil Future is meanwhile listed

under the 20th most often traded future contracts worldwide with a trading volume

of over 50 million contracts in 2005. This was an increase of around 15%. Its main

competitor Brent crude oil futures contract, follows with clear distance. Although

its trading volume went up 17% in 2005 it only could reach a volume of 25 million

traded contracts. Over this electronic trading is coming up what will additionally

boost trading volume in the next years.47

The metals markets showed the same picture. The trading of gold at the NYMEX

went up over 6% in 2005 to approximately 13 million traded contracts. The London

Metal Exchange (LME) surpassed Shanghai with the most copper futures traded

worldwide, with volume up approximately 4% to over 16 million.

The agricultural trading has grown as well. Surprisingly, putting future and option

contracts in volume together it has the highest trading volume of all commodity

groups. Although the Asian trading volume went down in 2005 the US exchanges

registered strong increases: the Chicago Board of Trade (CBOT) corn future went

up 14% to over 23 million traded contracts, the CBOT wheat future went up 25% to

over 8 million traded contracts and the New York Board of Trade sugar #1 future

went up 20% to over 12 million traded contracts.

2.3.1.2 Options

Commodity options are options where the underlying asset is a commodity or com-

modity index. In contrast to futures contracts they certify the right but not the

duty to buy or sell an asset at some future point.

Commodity options are identical to options on traditional assets such as stocks and

are primary used to manage risk or to generate premium income through asymmetric

risk exposure. Nowadays, stock options are more common than commodity options.

Nevertheless the options concept was originally developed in commodity markets.

First historical traditions go back to the mathematician, philosopher and astronomer

47Data source: Futures Industry Magazine Jan/Feb 2006

35


Tales. In expectation of a good olive harvest he bought the right to use olive

squeezing machines. In the 17th century options were introduced in the Netherlands

to trade tulips, but the first standardized options exchange, the Chicago Board

Option Exchange, was founded not until 1973. Together with the in the same year

published fundamental Black/Scholes option pricing model this was the starting

shoot for professional financial option trading.

The available standard forms are call and put options. The former are buy options:

the holder of the option has the right to buy the underlying at a predefined price and

time in future. A put option is a sell option: the holder of the option has the right

to sell the underlying at a predefined price and time in future. In addition to the

standard forms there are cap and floor options over the counter (OTC) available. A

cap is a series of call options and a floor is a series of put options on the commodity

itself. They are commonly used to manage ongoing price exposure to the underlying

commodity. Exchange traded options are exercised into a position of the underlying

commodity future contract which is either cash or physically settled. OTC options

are mainly cash settled directly.

Option trading has grown as futures trading did. The LME copper future regis-

tered a trading volume increase of approximately 13% to nearby 2 million contracts

in 2005. Only precious metals options trading is an exception. The New York

Mercantile Exchange reported a decrease of gold option trading of over 40%.

However, heavy trading is reported about the NYMEX crude oil option. Its trading

volume went up over 30% to over 12 million contracts. Together with the NYMEX

crude oil futures trading volume this counts for approximately one quarter of global

energy futures and options trading.48

Putting commodity futures and options trading together it counts for a trading

volume of over 620 million contracts in 2005. Comparing this number with other

market’s trading volumes expansion potential can be suspected: the equity indices

futures and options trading volume counts for over 3.4 billion contracts and the

derivatives trading of individual equities for another 2 billion, followed by the interest

rate market with over 2.1 billion traded futures and options contracts in 2005.

48Data source: Futures Industry Magazine Jan/Feb 2006

36


2.3.1.3 Swaps

Commodity swaps are generally the same like interest rate swaps49 with the dif-

ference that the underlying payment streams are linked to the price movement of

a commodity. A swap is an agreement between two parties to regularly exchange

payments. The most common type is the fixed-for-floating commodity swap. The

buyer of the swap pays at predefined usually equally spaced dates t1, . . . , tn a fixed

price for a commodity times the notional and receives from the seller of the swap

the market value of the commodity times the notional. Hereby the notional is given

in commodity units, e.g. tones of grain or barrels of oil. Figure 2.17 illustrates the

exchange of payments at the oil market.

Figure 2.17: Commodity Swap Payment Streams

In order to hedge his cost structure a crude oil consumer such as an heating oil

refiner enters into the described swap as the fixed leg, e.g. he is going to pay a fixed

price for crude oil times the notional at predefined dates. Generally, he will expect

oil prices to rise. On the other hand of the swap stands the producer of oil, e.g. the

oil extraction company. It can be expected that its financial management forecasts a

price decrease and wants to sell its product to a price fixed on the current high level.

In vocabularies of cash settlement e.g. he is going to pay the floating (respective

market) price times the notional. Usually, just the net positions are cash settled.

Generally, as described under Section 2.2, producer and consumer do not act directly

with each other but traders manage to bring the adequate parties together. We

have seen that the side of the swap entered by a party depends on its expectation of

ongoing price developments. Because many commodity swaps are cash settled today,

investors can speculate on their expectation through entering into the respective side

of a swap instead of entering into a series of the respective futures contracts. Out of

the investors point of view an advantage of swaps is the long term orientation and

the absence of rolling maturing futures contracts.

Swaps can easily be used in structured products where the exchange of different

49For a general introduction to interest rate swaps see [Zagst 2002]

37


types of cash payments are enabled, i.e. someone could think about a price-for-

interest swap. Insurance companies or other institutional investors that wish to

carry a commodity exposure, without being allowed by its regulatory body, may

do so by entering i.e. a price-for-interest swap with a party that is allowed to take

direct commodity exposure, i.e. a bank.

2.3.1.4 Commodity Linked Structured Notes

Commodity linked structured notes are engineered to give investors commodity ex-

posure through an interest rate security where a commodity derivative is embedded.

The issuer of the structured note has no commodity exposure itself. In fact, he is

connected to a commodity desk or dealer which provides the relevant commodity

return cash flows as shown in Figure 2.18.

Figure 2.18: Commodity Linked Structured Notes

Basically, there are three different types of commodity linked structured notes: Com-

modity forward linked notes, commodity option based notes and commodity index

based notes. These instruments generally are designed in two ways: either the final

payment or the coupon payments for the loan are commodity linked. The former

one is constructed as a zero coupon bond with a notional linked to a commodity, i.e.

the notional is calculated as 100% plus/minus a return realized through the linked

commodity. The latter one is constructed as a coupon bond where a fixed coupon is

negotiated and it is up or down graded depending on the realized commodity return.

Because the trader has to ensure a fully collateralized commodity investment, the

structured note still includes an interest component.

Commodity linked structured notes become more and more popular because many

investors already know structured notes from equity markets and do not need to care

about rolling futures and credit risk. Investors seeking exposure to commodities are

generally not comfortable with the credit risk of commodity producers. Linked notes

demerge the wanted commodity price risk from the unfavored credit risk aspects of

such transactions because they are usually offered by high credit grade issuers. Over

38


this, they are designed to meet investors needs and separate them from commodity

producer and consumer requirements.

Finally a regulatory change in commodity mutual fund markets will force the de-

mand for commodity linked interest rate securities.

2.3.1.5 Certificates

A common vehicle to get commodity index exposure in Europe are certificates.

Formally they securitize an obligation of the issuer with a regularly claim for interest

coupon payments. That means that the investor does not purchase stocks or shares

of a mutual fund, he simply lends his money to the issuer. Certificates generally

replicate the price evolvement of an underlying stock or index and therefore count

into the group of derivatives. A major characteristic of derivatives is to have a

maturity: so do certificates. Nowadays, there are open-end versions, which include

an internal rolling mechanism.50

Certificates emerge the whole credit risk of the issuer what makes them an unattrac-

tive investment vehicle for institutional investors but they are very famous in retail

business. The drawbacks of covered overpricing and credit risk are little communi-

cated. But their major advantage is high liquidity. Over this, certificates are gener-

ally available in many customized versions including refunding conditions equipped

with guarantees, bonuses, caps and/or currency risk hedging facilities. Following

[Zagst e.a. 2006] they can be a performance increasing addition to traditional stock

and bond retail portfolios.

2.3.2 Managed Futures Funds

Managed futures funds are managed by commodity trading advisors (CTAs). These

trading advisors manage client’s assets by using global futures markets as an invest-

ment medium. This is the main difference between a CTA and an ordinary trader.

Former have research based investment strategies, including diversification over dif-

ferent markets, risk managing and loss limiting systems whereby ordinary traders

generally are generally just experts in one market. In contrast to traders, who are

usually 100% in the market, CTA’s mainly just invest 10-25% of the assets under

management to absorb losses while waiting for profitable trades.51

50See [Gong Huber Lanzinner 2006].51See Managed Account Research, Inc.;

http : \ \ www.ma− research.com \managed account vs self − directed.html

39


Investment management professionals have been working with managed futures

funds for more than 30 years. But not until 2000 a broad range of institutional

and retail investors seek to invest into managed futures accounts. The steady de-

mand forced industry to grow from about 40 billion US dollar under management

in 2001 to about 130 billion US dollar under management in 2005.52 The growing

use of managed futures by investors may be due to the increased institutional use

of the futures markets. Portfolio managers have become more familiar with futures

contracts. Additionally, investors want greater diversity in their portfolios. They

seek to increase portfolio exposure to international investments and non-financial

sectors, an objective that is easily accomplished through the use of global futures

markets.

There are three types of managed commodity funds available: First, an investor

can directly open an individually managed futures account and hire a CTA to man-

age his funds based on the trading strategy presented in the advisors disclosure

document. The CTA opens an individual account on behalf of the investor, en-

ables him to monitor the activities at any time and his trading authorization can

be revoked whenever the investor does not see his interests represented. Therefore,

this type of participation allows investors the most transparency and liquidity. Be-

cause most advisors have minimum required investments that range from 25,000 to

10 million US dollar this financial investment is open only for investors with sub-

stantial net worth. Second, an investor can place his assets at a commodity pool

operator (CTO), who pools funds of different individual investors together and em-

ploys one or more CTAs to manage the pooled funds. Obtaining information about

this private pools is difficult because they are short in advertising to the public.

Their minimum investment requirements range from 25,000 to 250,000 US dollar.

Third, an investor can purchase the shares of public commodity funds or pools what

is similar to buying shares in a stock or bond mutual fund, except that mutual funds

buy and sell securities rather than commodity futures. Therefore, public funds en-

able small retail investors to participate in commodity markets. Within these groups

there is a wide variety of choices among available managed programs differing from

each other by style, strategy and market focus. In contrast to general advertisement

of the business, research has shown, that many CTA’s practise a trend following

or opportunistic dynamic trading strategy. For instance, [Fung Hiesh 1997] inves-

tigated in over 300 CTA’s during the period 1987 and 1995. [Schneeweiss 2000]

crossed a Rubicon in analyzing CTA portfolios. He reported that ”in general the

52See Managed Account Research, Inc.;http : \ \ www.ma− research.com \ growth of managed futures.html

40


correlation of CTA strategies with other CTA strategies is dependent on the degree

to which the strategies are trend following or discretionary.” Nevertheless, he showed

that adding managed futures to a stock and bond portfolio influences the portfolios

risk and return profile positively. We can find a similar statement in [CBOT 2003]

and [Edwards Liew 1999]. Latter investigated in individual CTAs accounts, private

and public commodity funds, and equally and dollar weighted portfolios created out

of individual CTA accounts over the period 1982 and 1996. They found that port-

folios as a stand alone investment are much better of than individual accounts and

private or public funds. An interesting observation is that the returns of managed

futures went down over the last decade. With more capital and traders compet-

ing for trading profits, commodity markets have become more efficient resulting

in lower returns. The latest Monthly Ranking Report of [ma-research 052006] has

shown dramatic developments. The average yearly returns after fee went down from

over 25% in 1996 to approximately 5% in 2006 so far, what implicates the absence

of arbitrage opportunities occurring in inefficient illiquid markets.

2.3.3 Stocks of Commodity Producing Companies

A traditional stock and bond investor can go the indirect way to invest in com-

modities by taking exposure in commodity producing companies. There exists a

large number of sector indices which are used as benchmarks for a vast amount of

sector funds, e.g. the MSCI World Index series offers amongst others the MSCI

World Energy Index which represents the performance of a broad basket of interna-

tional acting companies in the oil sector, the MSCI World Metals and Mining Index

which represents the performance of international companies which do business in

the metals mining sector or the MSCI World Food Products which represents the

performance of international companies active in the food producing business. If

an investor wants to go indirectly into a single commodity or commodity group

over stocks, he can compare SIC codes53 to find the optimal fitting company, e.g.

there are around 300 stocks of energy producing companies with the SIC code 1310

or 1311 ”crude petroleum and gas extraction” listed at the American stock ex-

changes.54 This methodology was taken by [Gorton Rouwenhorst 2004] to create

an index which replicates their artificially constructed equally weighted commodity

index55 with commodity producing companies. They showed that the cumulated

stock performance was less than the cumulated performance of the commodities.

53The Standard Industrial Classification Code (SIC) indicates the company’s type of business.54See and further information: [Gorton Rouwenhorst 2004].55For further information about different index weighting procedures see Section 4.1.

41


Over this there was a higher correlation (0.57) between the commodity produc-

ing companies stock index and the S&P500 than the correlation (0.4) between the

commodity producing companies stock index and their equivalent raw commodity

index.

Below we are going to take our own view on commodity producing companies. As

an example we picked the gold market. Figure 2.19 shows the performance and the

correlations between the Goldman Sachs Futures Gold Index, the HUI Index and

the S&P 500.56

Figure 2.19: Comparison of Gold and Gold Mining Companies

The Goldman Sachs Futures Gold Index represents a futures index which is con-

structed by rolling long gold futures contracts from the maturing to the next nearby

futures contract in January, March, May July and November of each year. There-

fore, it represents a long only investment in short term gold exposure. The Amex

Gold BUGS (Basket of Unhedged Gold Stocks) Index, known as HUI Index, is a

modified equal dollar weighted index of companies involved in gold mining. The

HUI Index was designed to provide significant exposure to near term movements in

gold prices by including companies that do not hedge their gold production beyond

1.5 years. We found this index to be representative for a portfolio of gold producing

companies which cash flows are highly correlated to nearby gold price movements.

The performance chart of Figure 2.19 shows that there is no consistent truth whether

raw commodities or commodity producing companies were better off in the last

years. Gold producing companies performed better than gold itself. Over this

56There were daily Bloomberg data taken and returns were calculated following Definition C.2 andthe plotted price series starting with 100 in 1999 following Definition C.3.

42


the return correlations57 behaved quite different to the observations reported by

[Gorton Rouwenhorst 2004]. The correlation of daily log returns between the GS

Gold Futures Index and the HUI Index is significant with 0.94 and the correlation

of daily log returns between the HUI Index and the S&P 500 Total Return Index is

significant with -0.25. Over this, the HUI Index is much more volatile than the GS

Gold Futures Index. A HUI Index investor58 had to accept a yearly average standard

deviation of 42.7% in comparison to a GS Gold Futures Index investor who merely

had to accept 16.2% standard deviation. Therefore, the main question to answer is

what kind of exposure an investor wants: one in raw commodities, an asset class

which is driven by its own risk factors, or one in stocks.

57See Definition C.2, Definition 5.1 and Equation (5.5).58For further information which products are available to invest in an index see Section 4.3.

43

3 Pricing of Commodity Futures

Futures prices are the result of open and competitive trading on the floors of ex-

changes and, as such, translate the underlying supply and demand or, rather, their

expected values at various points in future into absolute figures. Reflecting ex-

pectations about future supply and demand, futures prices trigger decisions about

storage, production and consumption that reallocate the supply and demand for

commodities over time. Social welfare is increased by the avoidance of disruption

in the flow of goods and services. In the case of storable commodities, these prices

determine the storage decisions of market participants: higher futures prices signal

the need for greater storage and lower futures prices point to a reduction in current

inventory. Therefore, commodity futures do not represent a pure financial asset and

traditional no-arbitrage asset pricing59 cannot be used to value commodity futures:

Since consumption and processing of the commodity can drive down inventories to

zero, it is not always possible to construct a replicating portfolio for the futures con-

tract. The second factor why commodity futures cannot be valued like pure financial

assets is the non existence of pure spot prices. Although there do exist cash prices

which are actual transaction prices, cash prices often do not pertain to the same

specification of the commodity compared to a respective futures contract’s specifica-

tions in terms of location, grade and quality. In addition cash prices usually include

discounts and premiums that result from longstanding relationships between buyer

and seller. Therefore, cash prices cannot be used as a spot price what is directly

comparable to the futures price. A common technique to estimate spot prices out

of futures prices is an extrapolation of the spread between the nearest and next-to-

nearest active futures contract on a daily basis as described in [Pindyck 1994]. The

use of the nearest-to-maturity future price as a proxy for the spot price is common

as well and described in [Markert 2005] or [Gorton Rouwenhorst 2004]. Because this

technique is used in the construction of the broad indices introduced in Section 4.2

we will follow this procedure as well.

Summing up, commodity prices are a mixture of the prices of an asset, reflecting

expectations of future spot prices and the expected risk premium and consumption

good’s prices, reflecting the current scarcity of a good. Depending on either view,

two general futures pricing models were derived: the Risk Premium and the Con-

venience Yield Model which we will present in Section 3.1 and Section 3.2. The

relationship between the two models were first derived in [Markert 2005] and will be

59For a general introduction to the concept of no-arbitrage pricing and the related definitionssee [Zagst 2002]. The pioneer work summarizing the different concepts of commodity futurespricing was done by [Markert 2005].

44

3.1 The Risk Premium Model

shown in Section 3.3. The presented models are deterministic and used by traders to

calculate a fair futures price depending on their individual market observations and

the resulting valuations of the input variables. The actual price at which trading

takes place is then generated when seller and buyer prices coincide. But especially

for risk management purposes exogenous stochastic models are needed to simulate

market prices what implicates the ability of portfolio modeling in different market

situations and the observability of the portfolio value in different economic scenar-

ios. Therefore, mathematicians pick one or more input factors of the pricing formula

and assign a stochastic process following a certain distribution to them. The actual

market prices are further fitted over special error minimization procedures, e.g. the

Kalman filter, to choose the model parameters properly. Only stochastic Conve-

nience Yield Models became widely accepted. We will present the most known ones

in Section 3.4. Starting with simple one factor models in Section 3.4.1, we will fur-

ther discuss two factor models in Section 3.4.2, and closing this section with a brief

introduction of three factor models in Section 3.4.3.


The Risk Premium Model values commodity futures contracts with respect to the

expected commodity spot price discounted by an appropriate risk premium. The

idea behind this approach goes back to Keynes’ theory of normal backwardation.60

We have already seen in Section 2.2 that there are different market participants with

different purposes. To get a deeper insight of their motivations and interactions the

famous example of the cattleman is told: Imagine it is February and there is a

cattleman who wants to hedge the value of his live cattle in September when the

herd is ready to sell. A convenient way to do so is selling today his production

of tomorrow over futures contracts.61 Since futures markets are assumed to be

efficient all market participants are assumed to have the same expectation of the

cattle price in September, say 72 cents per pound. However, this price is uncertain

and a variety of events could occur, e.g. heavy barbecuing season, fear of mad cow

disease etc., that might drive the September price up to e.g. 90 cents per pound or

down to e.g. 60 cents per pound to today’s expectation of 72 cents. Producers are

rather interested in covering their production costs with certainty than maximizing

60See [Keynes 1930].61Following Section 2.3.1 there are different financial vehicles the cattleman can use. He picks

exchange traded futures contracts not OTC forward agreements because he wants to avoidcounterparty default risk and wants to deal with fair market not with bilateral bargainedprices. Furthermore he does not chose a swap contract because he is not interested in a seriesof transactions.

45


their final gain. Lets assume the cattleman’s production costs to be 65 cents per

pound, i.e. if the cattleman has to sell his production for less than 65 cents per

pound he will run out of business. To hedge future prices the cattleman goes to

the futures market and sells today his production of tomorrow. To compensate

investors for taking future price risks he needs to sell his production for a discount

say 2 cents per pound and the observable futures price becomes 70 cents per pound.

The mechanism is illustrated in Figure 3.1.62

Figure 3.1: The Risk Premium Model

[Keynes 1930] argues that ”the spot price must exceed the forward price by the

amount which the producer is ready to sacrifice in order to hedge himself, i.e. to

avoid risk of price fluctuations during his production period. Thus, in normal condi-

tions the spot price exceeds the forward price,” i.e. futures prices are set backwards

to expected future spot prices. In the situation of normal backwardation nearby fu-

tures have higher values than long term ones because the insurance premium payed

for price fixity should naturally be higher for longer time distances. The reverse sit-

uation is called contango. A famous example for the two phenomena is the NYMEX

WTI crude oil market. During the past two decades the market was approximately

60% in backwardation. This trend reversed during the last two years. The WTI

crude oil market has spent 81% of the time in contango.63 In this environment, oil

consumers are willing to pay today a higher price for products delivered tomorrow.

Usually this yields to an increase of inventories to guard against expected supply

bottle necks, interruptions or even unavailability of the product. Figure 3.2 shows

62See also [Geer 2000].63See [Merrill Lynch 2006].

46


the shape of the forward curve64 of crude oil65 and copper66 futures traded on the

NYMEX per 31. January 2006. The crude oil market is in contango and the copper

market is in backwardation.

Figure 3.2: Backwardation and Contango

The theory of normal backwardation does not cover the practical phenomena con-

tango. Therefore, [Cootner 1990] and [Deaves Kinsky 1995] extended the theory

and formulated the hedging pressure hypothesis. They suggested that both ”back-

wardated” commodities, where today’s futures price is set below the expected future

spot price, and ”contangoed” commodities, where today’s futures price is set above

the expected future spot price, might have risk premiums. Backwardation occurs

when hedgers are net short and contango occurs when hedgers are net long in the

respective futures market. Different statistical researches report evidence to proof

this hypothesis.67 Backwardated markets provide a hedge for producers, i.e. pro-

ducers are willing to sell their products at an expected loss, and contangoed markets

provide a hedge for consumers, i.e. consumers are willing to purchase products at

an expected loss. As a result, investors receive a risk premium for going long back-

wardated commodity futures and for going short contangoed commodity futures.

Putting both theories into mathematical forms we end up with the Risk Premium

Model:

64For an introduction to forward curves see [Zagst 2002].65The values come from the NYMEX light sweet crude oil futures contract with a trading size of

1.000 barrel as per 31. January 2006.66The values come from the LME copper futures contract with a trading size of 25 tones as per

31. January 2006.67See e.g. [Anderson 2000], [Bessembinder 1992] or [DeRiin Nijman Veld 2000].

47


Theorem 3.1 Risk Premium Model

Let PC(t) be the spot price of a commodity at time t ∈ [0, T ], let Ft denote the

σ-Algebra68 as of Definition C.5 at time t and let rp be the constant asset spe-

cific risk premium. Moreover, define Q as the equivalent martingale measure as of

Definition C.32. Then the price of a commodity future FC(t, T ) at time t ∈ [0, T ] in

the Risk Premium Model is given by:

FC(t, T ) = e−rp(T−t)EQ[PC(T )|Ft] (3.1)

Proof: To distinguish between a traditional financial asset and commodities as

an asset what embodies their consumption good function we are going to use the

following notation:

PA(t) denotes the spot price of a pure financial asset at time t ∈ [0, T ]

PC(t) denotes the spot price of a commodity at time t ∈ [0, T ]

FA(t, T ) denotes the futures price of a pure financial asset at time t ∈ [0, T ]

FC(t, T ) denotes the futures price of a commodity at time t ∈ [0, T ]

Furthermore let rf be the constant risk free interest rate, rp be the constant asset

specific risk premium and u the proportional cost of carry for an asset what can

be seen as a negative dividend yield of a stock. The above assumptions are made

to simplify the model to give an easy introduction of the concept of Risk Premium

Models. Sure, they can be modified what yields into the development of differ-

ent customized applications, i.e. the risk free rate is in practice not constant but

stochastic and the cost of carry may change over time as well.

To exclude arbitrage opportunities in financial markets, FA(t, T ) is the futures price

for which the present value of the expected future payoff equals zero:69

0 = EQ[e−(rf+u)(T−t)(PA(T )− FA(t, T ))|Ft]

FA(t, T ) = e(rf+u)(T−t)EQ[e−(rf+u)(T−t)PA(T )|Ft]

FA(t, T ) = EQ[PA(T )|Ft] (3.2)

68The σ-Algebra Ft embodies all available information until t. For a more detailed mathematicalintroduction see [Zagst 2002] or [Ito 2004].

69The idea behind this approach is that to avoid arbitrage opportunities, the prices of two financialassets producing the same payoff at maturity, have to be equal at each other time beforematurity. For an illustrative introduction to risk neutral derivatives pricing see [Zagst 2002].

48


Risk Premium Models assume that future prices of a consumption good have to

include a risk premium. Therefore, the difference between the price of a commodity

as a financial asset and as a consumption good has to be adjusted:

PC(t) = erp(T−t)PA(t) (3.3)

Putting (3.3) into (3.2), the commodity futures price in the general Risk Premium

Model is given as in Equation 3.1.

2

The general Risk Premium Model represents the point of view that commodity

futures prices equal the expected commodity spot price, discounted by a risk pre-

mium to compensate investors for holding the price risk of a commodity. Based on

Equation (3.1) the return of a futures contract in the interval [s, t] with 0 ≤ s < t ≤T is given by:

rFC(t,T )(s, t) ≡ ln

(FC(t, T )

FC(s, T )

) (3.1)︷︸︸︷= rp(t− s)︸︷︷︸

risk premium

+ ln

(EQ[PC(T )|Ft]

EQ[PC(T )|Fs]

)︸︷︷︸change of price expectation

(3.4)

As we have seen above, according to the Risk Premium Model, the return an in-

vestor has to look forward to is the sum of a risk premium and the change in spot

price expectations. To close the frame lets go back to the introductory example of

the meatpacker. The change in spot price expectation is called the ”expectational

variance” and illustrated in Figure 3.3.70

Recall, the meatpacker has to cover his production costs. Therefore, he needs a fixed

price and is willing to enter into a futures contract to set the September price for his

meat to 70 cents per pound although the expected future spot price is 72 cent. He

pays a risk premium of 2 cents. Depending on possible events such as fear of mad cow

disease, heavy barbecuing season etc., the spot price will run out somewhere between

60 and 90 cents what is either return boosting (positive expectational variance) or

destroying (negative expectational variance).

70See [Geer 2000].

49


Figure 3.3: The Concept of Expectational Variance

3.2 The Convenience Yield Model

The Convenience Yield Model is a no-arbitrage based valuation concept. It val-

ues commodity futures with respect to the current commodity spot price and an

appropriate convenience yield. The fundamental behind this approach goes back

to the theory of storage, first mentioned in [Kaldor 1939] and further analyzed in

[Working 1948] and [Working 1949]. The theory of storage aims to explain the dif-

ferences between spot and futures prices in dependency of the level of inventory and

the resulting benefits: inventories have a productive value since they allow to meet

unexpected demand, avoid the cost of frequent revisions in the production sched-

ule and eliminate manufacturing disruption. In order to represent the advantages

attached to the ownership of the physical good, [Kaldor 1939], [Working 1948] and

[Working 1949] defined the notion of the ”convenience yield”. It describes the ben-

efit that ”accrues to the owner of the physical commodity but not to the holder of

a forward contract.” In the same spirit, the dividend yield is paid to the owner of a

stock but not to the owner of a derivative on the stock. The convenience yield is high

when desired inventories are low and vice versa. Consequently, the concept suggests

on the one hand that inventories might be low for commodities which are difficult to

store. Therefore, they have a high convenience yield. On the other hand inventories

should be high for easy to store commodities and they should have low convenience

yields. [Till 2000] did some related research. She reported that commodities with a

difficult storage situation (storage is impossible, storage is prohibitively expensive,

or producers decide that it is much cheaper to leave the commodity in the ground

50

3.2 The Convenience Yield Model

than store above ground) produced a statistically significant positive return over

the last 40 years. She mentioned amongst others livestock, copper and crude oil.

Implicating, the difficulty for a long term commodity investor is to determine future

stocks of inventories.

Putting the theory into mathematical forms we end up with the Convenience Yield

Model:

Theorem 3.2 Convenience-Yield Model

Let PC(t) be the spot price of a commodity at time t ∈ [0, T ], u be the constant

cost of carry for an asset, c : R 7→ R be the deterministic convenience yield and let

rf denote the constant risk free interest rate, then the price of a commodity future

FC(t, T ) at time t ∈ [0, T ] in the Convenience Yield Model is given by:

FC(t, T ) = PC(t)e(rf+u−c(t))(T−t) (3.5)

Proof: With the notation of Theorem 3.1 the price of a commodity future is

given by:

FC(t, T ) = e−rp(T−t)EQ[PC(T )|Ft] (3.6)

Furthermore, according to the most general form of a financial asset pricing model

the current hypothetical ”asset price” of a physical commodity is the net present

value of its expected future payoff PC(T ):

PA(t) = e−(rf+rp+u)(T−t)EQ[PC(T )|Ft] (3.7)

Setting equal the expectations in Equation 3.6 and 3.7 yields to:

FC(t, T ) = e(rf+u)(T−t)PA(t) (3.8)

Because of the additional consumption value of the physical commodity it is assumed

that the spot price of a commodity differs from the spot price of a pure financial

asset. Therefore, the commodity spot price needs to be adjusted:

PC(t) = (1 + C(t))PA(t), with C(t) ≥ 0 (3.9)

This equation states that the commodity spot price PC(t) exceeds the value of

the spot price of a pure financial asset by the factor (1 + C(t)) that embodies

the consumption good facility of the commodity. Because the convenience yield is

defined as the benefit that accrues from holding the commodity it can be seen as

a dividend which is payed to the holder of the commodity and this yields to the

51


common approximation for the convenience yield:

(1 + C(t)) ∼= ec(t)(T−t) (3.10)

Putting this into Equation 3.9 yields to:

PC(t) = ec(t)(T−t)PA(t) (3.11)

Furthermore, putting this into Equation 3.8 yields to the result as in Equation (3.5).

2

Remark 3.1 Sometimes the convenience yield is defined as net position of benefit

from holding the commodity minus the storage costs.71 The merged equation would

become:

FC(t, T ) = PC(t)e(rf−y(t))(T−t), with y(t) = c(t)− u

Remark 3.2 The convenience yield enters the futures price with a minus: the

holder of the future does not benefit from the physical commodity over the time

interval interval [t, T ]. Therefore, he is not be payed with the yield it provides.

Remark 3.3 As mentioned introductory, the Convenience Yield Model is origi-

nally a no-arbitrage based valuation concepts. Therefore, Theorem 3.2 is in lit-

erature mainly proofed with the following arbitrage argument: If the current fu-

tures price FC(t, T ) was greater than the right hand side of Equation 3.5 namely

PC(t)e(rf+u−c(t))(T−t), one would sell the futures contract, buy the commodity through

a loan, pay the cost of carry, benefit from holding the physical commodity over the

time interval (t, T ) and realize at maturity T a cash and carry arbitrage. Conse-

quently, if FC(t, T ) was strictly smaller than the right hand side PC(t)e(rf+u−c(t))(T−t),

a reverse cash and carry arbitrage would be possible. Therefore, equality must hold.72

In contrast to Remark 3.3 the proof to Theorem 3.2 shows that the Risk Premium

and the Convenience Yield Model are directly connected to each other and therefore,

the two valuation approaches are mutually consistent: backwardation occurs when

the convenience yield is high and contango occurs when the convenience yield is low.

This bridge was first built in [Markert 2005] and is unique in literature so far.

From Equation 3.5 we can take a closer look into the return structure of commodity

71See e.g. [German 2005].72See e.g. [German 2005].

52

3.3 Relationship of the Risk Premium and Convenience Yield Model

futures under the convenience yield model with 0 ≤ s < t ≤ T :

rFC(t,T )(s, t) ≡ ln

(FC(t, T )

FC(s, T )

)(3.5)︷︸︸︷= ln

(PC(t)

PC(s)

)+ (rf + u− c(t))(T − t)

− (rf + u− c(s))(T − s)

= ln

(PC(t)

PC(s)

)︸︷︷︸

change in spot price

+ (c(t)− rf − u)(t− s)︸︷︷︸cost of carry and convenience yield

+ (c(s)− c(t))(T − s)︸︷︷︸change in convenience yield

(3.12)

In the Convenience Yield Model the return provided by a futures contract is the

sum of the change in commodity spot prices, the convenience yield minus the cost

of carry and the change of the convenience yield.

The convenience yield has neither to be constant nor deterministic. In fact, an

assumption of constancy would be very unrealistic because the benefit of holding

a commodity is reverse proportional to the stock of inventory and fluctuates over

time depending on the level of inventory. Therefore, researchers suggest stochastic

models for the convenience yield which allow to explain the different shapes of the

term structure, i.e. the different futures prices as a function of maturity.73 We will

address ourselves to this topic in Section 3.4.

3.3 Relationship of the Risk Premium and Convenience Yield

Model

The convenience yield conceptually links together desired inventories and commodity

futures prices. The benefit from holding the commodity is high when inventories are

low. As a result, the convenience yield can be thought of as a risk premium linked to

inventory levels. Mathematically we can see the connection by comparing Equation

(3.12) and Equation (3.4), i.e. by comparing the return structures according to the

respective model.

73See [Gibson Schwartz 1990], [Schwartz 1997] or [Cassasus Collin-Dufresne 2005].

53


Before we start we need to do the following pre calculation: With the thoughts of

Equation (3.7) and Equation (3.11) the spot price return with 0 ≤ s < t ≤ T can

be calculated as:

ln

(PC(t)

PC(s)

)= ln

(EQ[PC(T )|Ft]

EQ[PC(T )|F∫ ]

)+ (c(t)− rf − rp − u)(T − t)

− (c(s)− rf − rp − u)(T − s)

= ln

(EQ[PC(T )|Ft]

EQ[PC(T )|F∫ ]

)+ (rf + rp + u− c(t))(t− s)

+ (c(t)− c(s))(T − s) (3.13)

Now, we can derive the return according to the Risk Premium Model (R) out of the

return according to the Convenience Yield Model (C):

rFC(t,T ),C(s, t)

(3.12)︷︸︸︷= ln

(PC(t)

PC(s)

)+ (c(t)− rf − u)(t− s)

+ (c(s)− c(t))(T − s)(3.13)︷︸︸︷= ln

(EQ[PC(T )|Ft]

EQ[PC(T )|Fs]

)+ (rf + rp + u− c(t))(t− s)

+ (c(t)− c(s))(T − s) + (c(t)− rf − u)(t− s)

+ (c(s)− c(t))(T − s)

= rp(t− s) + ln

(EQ[PC(T )|Ft]

EQ[PC(T )|Fs]

)(3.4)︷︸︸︷= rFC(t,T ),R(s, t) (3.14)

Therefore, depending on either view the futures price of a commodity is given by:

FC(t, T ) = e−rp(T−t)EQ[PC(T )|Ft]

= PC(t)e(rf+u−c(t))(T−t) (3.15)

Rearranging the right hand side of Equation (3.15) shows the influencing factors of

the expected change in commodity spot prices:

ln(EQ[PC(T )|Ft]

)− ln (PC(t)) = ln

(EQ[PC(T )|Ft]

PC(t)

)= (rf + rp + u− c(t))(T − t) (3.16)

For financial assets, with c(t) = 0 and u equaling a negative dividend yield, above

54

3.4 Stochastic Models

Equation (3.16) states that in an asset pricing equilibrium the spot price is expected

to grow by the risk free rate plus the risk premium minus the dividend yield. In

stochastic stock price models this is captured in the drift rate of the stochastic

process modeling the stock price. Thus, the grass roots needed to understand the

origins of the different stochastic models for commodity prices, e.g. introduced

in [Gibson Schwartz 1990], [Schwartz 1997] or [Cassasus Collin-Dufresne 2005] are

disclosed and the following section shall give a brief introduction of the different

approaches.74


The term structure gives the relationship between the futures prices and the re-

spective time to maturity. It provides useful information for hedging or investment

decisions because it synthesizes the information available in the market and the

operators’ expectations concerning the future. The information is very useful for

management purposes: it can be used to hedge exposure on the physical market

and to adjust the stock level or the production rate. It can also be used to un-

dertake arbitrage transactions, to evaluate derivatives instruments based on futures

contracts, and so on. Therefore, stochastic term structure models aim to reproduce

the futures prices observed in the market as accurately as possible aiming e.g. to

discover futures prices for horizons exceeding exchange traded maturities, to forecast

futures price developments under different economic scenarios, to price structured

products based on futures contracts with minimized errors or to see the interactions

of futures prices with other asset’s price movements.

Over time, different models were introduced ranging from the simplest one factor

models to more sophisticated versions of three factor models. Depending on the

amount of factors, the following factors are modeled stochastically: the spot price,

the convenience yield and the interest rate. Starting in Section 3.4.1 we will intro-

duce two examples of one factor models. The first, called Brownian Motion Model,

will generate the spot price with a stochastic dynamic coming from a Brownian

Motion and a deterministic convenience yield. The second, called Mean Rever-

sion Model, will model the spot price over a mean reverting dynamic structure.

In Section 3.4.2 we will introduce the two most accepted two factor models: the

Convenience Yield and the Long - Short Term Model. Although, the two models

were developed based on different fundamental ideas, the two models are equivalent.

74We will further denote the futures price of a commodity with F (t, T ) = FC(t, T ) and its spotprice with P (t) = PC(t) at t ∈ [0, T ].

55


Closing this section we will give a brief example of a three factor model. It is similar

to the Convenience Yield Model of Section 3.4.2, but extended with a stochastic

interest rate component.

To evaluate futures prices based on the three input factors commodity spot price,

convenience yield and interest rate, given either deterministic or stochastic, the

models borrow from the contingent claim analysis developed for stock and interest

rate models.75 Therefore, the different models of commodity futures pricing share

the following general assumptions: the market for assets is free of frictions, taxes or

transaction costs, trading takes place continuously and lending and borrowing rates

are equal and there are no short sale constrains.

3.4.1 One Factor Models

One factor models are based on the concept that futures prices are determined as

the expectation of the future spot price, conditionally to the available information

at time t. Therefore, the spot price is the main determinant of futures prices.

Thus, following [Lautier 2005], most one factor models rely on the spot price. Two

general approaches are chosen: either to model the stochastic dynamic of the spot

price with a Brownian Motion or with a Mean Reversion Process. The Brownian

Motion Model is more excepted in practice than the Mean Reversion Model because

it allows for a deterministic convenience yield and therefore, covers the consumption

good characteristic of commodities. On the other hand, the Brownian Motion Model

does not cover observed mean reversion pattern in commodity futures prices. We

will introduce both approaches to give a general overview of common market models,

starting in Definition 3.1 with the Brownian Motion Model.

Definition 3.1 Brownian Motion Model

Let P (t) be the spot price of a commodity at time t ∈ [0, T ], µ ∈ R the drift of the

spot price, σP > 0 the spot price volatility and WP (t) a standard Brownian Motion

as defined in Definition C.28. Then the dynamic of the spot price in the Geometric

Brownian Motion Model is

dP (t) = µP (t)dt + σP P (t)dWP (t), t ∈ [0, T ]. (3.17)

Equation (3.17) stats that the commodity spot price is driven by a stochastic that

can be modeled with a simple Brownian Motion. Based on this stochastic process,

75An illustrative introduction can be found in [Zagst 2002].

56


we will further derive the futures price represented by F (P, t) at time t for delivery

of one unit of the commodity at time T . Furthermore, denote:

∂F (P,t)∂t

= Ft(P, t)

∂F (P,t)∂P

= FP (P, t)

∂2F (P,t)∂2P

= FPP (P, t)2.

Using Ito’s lemma as of Definition C.1, the instantaneous change in the futures price

is given as:

dF (P, t) =

[Ft(P, t) + µP (t)FP (P, t) +

1

2σ2

P P (t)2FPP (P, t)

]dt

+ σP P (t)FP (P, t)dWP (t)(3.17)︷︸︸︷=

[Ft(P, t) +

1

2σ2

P P (t)2FPP (P, t)

]dt

+ FP (P, t)

µP (t)dt + σP P (t)dWP (t)︸︷︷︸dP (t)

=

[Ft(P, t) +

1

2σ2

P P (t)2FPP (P, t)

]dt + FP (P, t)dP (t) (3.18)

The difference between commodities as simple financial asset and consumption good

is captured in the net convenience yield76 assumed to be a proportional to the

spot price: c(P, t) = CP (t), with C ∈ R. Following [Brennan Schwartz 1985] the

convenience yield is the flow of services that accrues to an owner of the physical

commodity. He is able to choose where the commodity will be stored and when

to liquidate the inventory. Recognizing the costs for transportation, storage and

insurance, the convenience yield ”may be thought of as the value of being able to

profit from temporary local shortages of the commodity through ownership of the

physical commodity. The profit may arise either from local price variations or from

the ability to maintain a production process as a result of ownership of an inventory

of raw material.” Therefore, the financial spot price process dP (t) has to be amended

with the convenience yield process CP (t)dt yielding to the actual commodity spot

76Compare Remark 3.1 of Section 3.2.

57


price process:77

dP (t) + CP (t)dt = µP (t)dt + σP P (t)dWP (t), t ∈ [0, T ]

⇒ dP (t) = (µ− C)P (t)dt + σP P (t)dWP (t) (3.19)

Following the no arbitrage pricing methodology, the futures contract delivering one

unit of the underlying in T has the same value as the commodity in T . To avoid

arbitrage, the two assets must have the same value before T , as well. Therefore, we

can construct the following portfolio, called risk free hedge portfolio:

V (t) = P (t) + c(P, t)− δF (P, t)

with:

dV (t) = dP (t) + CP (t)dt− δdF (P, t)(3.18)︷︸︸︷= dP (t) + CP (t)dt− δ(

[Ft(P, t) +

1

2σ2

P P (t)2FPP (P, t)

]dt

+ FP (P, t)dP (t))

= [CP (t)− δ(Ft(P, t) +1

2σ2

P P (t)2FPP (P, t))]dt

+ (dP (t)− δFP (P, t)dP (t))︸︷︷︸risk free⇔=0⇔δ= 1

FP (P,t)

δ= 1FP (P,t)︷︸︸︷=

1

FP (P, t)

[CP (t)FP (P, t)− Ft(P, t)− 1

2σ2

P P (t)2FPP (P, t)

]dt

risk free︷︸︸︷≡ rfP (t)dt

Thus, the futures price in the Brownian Motion Model is given as the solution of

the following partial differential equation with the boundary condition F (P (t), T ) =

P (T ):

Ft(P, t) +1

2σ2

P P (t)2FPP (P, t) + FP (P, t)P (t)(rf − C) = 0 (3.20)

77Compare Equation (3.9).

58


Theorem 3.3 Futures Price in the Brownian Motion Model

Let the notations be as in Definition 3.1, c(P, t) = CP (t) with C ∈ R be the deter-

ministic convenience yield and rf be the constant risk free interest rate. Then the

futures price F (P, t) is a function of the spot price and the time to maturity:

F (P, t) = P (t)e(rf−C)(T−t). (3.21)

Proof: It has been shown in [Zagst 2002], that the futures price F (t) of an asset

is the conditional expectation as of Definition C.24, whereby conditional is regarding

the available information of today embodied in σ-Algebra Ft as of Definition C.5,

of its future spot price P (T ) under the equivalent martingale measure Q as of

Definition C.32:78

F (P, t) = EQ[P (T )|Ft], t ∈ [0, T ] (3.22)

Using the Feynman-Kac representation of Theorem C.5, there is an indirect way to

get (3.22). Someone can solve the Cauchy-Problem as given in Definition C.36 to

get the solution of the stochastic differential equation underlying the futures price.

The Feynman-Kac representation then stats that if there exists a solution that it is

equal to conditional expectation of (3.22).79 To get F (P, t) = v(P, t) as requested

in Equation (C.25), we have to define: x := P , r(P, t) ≡ 0 and D(P ) := P (T ).

Therewith, we have to show that F solves the Cauchy-Problem as defined in C.36.

For it, we first have to transfer the spot price P in the world of the equivalent mar-

tingale measure Q which exists because of the Girsanov-Theorem as of Theorem C.3.

Denote with dW the increments of the Brownian motion as of Definition C.28 under

Q. Using the Girsanov-Theorem as of Theorem C.3, we have:

dW (t) = λ(t)dt + dW (t), t ∈ [0, T ] (3.23)

where λ : R 7→ R is called the market price of risk. It results

dP (t) = [µ− σP λ(t)]P (t)dt + σP P (t)dWP (t), t ∈ [0, T ] (3.24)

where it has to be µ−σP λ(t) = rf because the discounted spot price process has to

be a martingale as of Definition C.29. It is

dP (t) = rfP (t)dt + σP P (t)dWP (t), t ∈ [0, T ]. (3.25)

78Also compare Equation (3.2).79Attention: The opposite direction is not always true. See [Zagst 2002].

59


Again, we have to amend the financial spot price of commodities with the conve-

nience yield process as in Equation (3.19). It follows

dP (t) = (rf − C)P (t)dt + σP P (t)dWP (t), t ∈ [0, T ] (3.26)

Then, the adapted Cauchy-Problem as of Definition C.36 is given as:

Ft(P, t) +1

2σ2

P P (t)2FPP (P, t) + FP (P, t)P (t)(rf − C) = 0, t ∈ [0, T ] (3.27)

with the terminal boundary condition F (P, T ) = P (T ). Recall, this is equal to

Equation (3.20) and therefore shows, that solving the Cauchy Problem is in line

with solving the differential equation developed over the no arbitrage approach.

Under the assumption of Equation (3.21), F (P, t) = P (t)e(rf−C)(T−t), with t ∈ [0, T ],

we get:

FP (P, t) = e(rf−C)(T−t),

FP,P (P, t) = 0,

Ft(P, t) = −(rf − C)F (P, t), t ∈ [0, T ].

Putting this into the Cauchy-Problem, Equation (3.27), it follows

0 + (rf − C)F (P, t)− (rf − C)F (P, t) = 0, ∀t ∈ [0, T ]

which shows, that the futures price is indeed F (P, t) = P (t)e(rf−C)(T−t).

2

Although, the Brownian Motion Model is probably the most simple and therewith

the most known one, it has the drawback of not covering mean reversion occurring in

commodity spot prices caused by the consumption good characteristic of commodi-

ties reflecting producers and consumers actions in the physical market.80 When the

spot price is low, industrials expect prices to rise and fill their inventories. Producers

react with a reduction of output providing only low benefits. The increased demand

and the simultaneous reduction of supply have a rising influence on the spot price.

Conversely, when the spot price is higher than its long run average, industrials will

serve their demand with inventories that were build up at low commodity price times

and producers increase their production rate expecting higher margins for the same

80See the latest work [Markert 2005].

60


output. Both movements will push the spot price to lower levels. [Schwartz 1997]

published a one factor model that directly incorporates the mean reversion effect

into the spot price.

Definition 3.2 Mean Reversion Model

Let P (t) be the spot price of a commodity at time t ∈ [0, T ], µ ∈ R the long-

run mean, κ > 0 the speed of adjustment of the spot price, σP > 0 the spot price

volatility and WP (t) a standard Brownian motion as defined in Definition C.28.

Then the dynamic of the spot price in the Mean-Reverting Model is

dP (t) = P (t)κ[µ− ln P (t)]dt + σP P (t)dWP (t), t ∈ [0, T ]. (3.28)

The model covers two characteristics of mean reversion: the spot price has the

prosperity to return to its long-term mean, but simultaneously, random shocks can

move it away in the short-run allowing for sudden price peaks.

Based on the spot price movements we can calculate the futures price in the Mean

Reversion Model:

Theorem 3.4 Futures Price in the Mean Reversion Model

Let the notations be as in Definition 3.2, λ : R 7→ R be the market price of risk

introduced in the Girsanov-Theorem as of Theorem C.3 and rf be the constant risk

free interest rate. Then the futures price F (P, t) is a function of the spot price and

the time to maturity and is expressed by

F (P, t) = exp[e−κ(T−t) ln P (t) + (1− e−κ(T−t))(µ− σ2P /2κ− λ)

+σ2

P

4κ(1− e−2κ(T−t))], (3.29)

with t ∈ [0, T ].

Proof: As shown in Proof 3.4.1, the futures price F must solve the Cauchy-

Problem as defined in Definition C.36 with x := P , r(P, t) ≡ 0 and D(P ) := P (T )

for all P (T ) ∈ R and t ∈ [0, T ]. Following the methodology of Proof 3.4.1 we have

to transfer the stochastic process for the factors into the world of the equivalent

martingale measure Q. Using the Girsanov-Theorem as of Theorem C.3, we have:

dP (t) = P (t)κ[µ− ln P (t)− λ]dt + σP P (t)dWP (t), t ∈ [0, T ].

dWP (t) is the increment of a Brownian motions under the equivalent martingale

measure. Based on this equations the adapted Cauchy-Problem as of Definition C.36

61


is given as:

1

2σ2

P P (t)2FP,P (P, t) + P (t)κ[µ− ln P (t)− λ]FP (P, t) + Ft(P, t) = 0 (3.30)

with t ∈ [0, T ] and the terminal boundary condition F (P, T ) = P (T ).

Under the assumption of Equation (3.34), that

F (P, t) = exp

[e−κ(T−t) ln P (t) + (1− e−κ(T−t))(µ− σ2

P /2κ− λ) +σ2

P

4κ(1− e−2κ(T−t))

]with t ∈ [0, T ], we can calculate the respective derivatives:

FP (P, t) = e−κ(T−t) F (P,t)P (t)

,

FP,P (P, t) = e−2κ(T−t) F (P,t)P (t)2

− e−κ(T−t) F (P,t)P (t)2

,

Ft(P, t) =[κe−κ(T−t)(ln P (t) + λ− µ + σ2

P /2κ)− σ2P

2e−2κ(T−t)

]F (P, t), t ∈ [0, T ].

Putting this into the Cauchy-Problem of Equation (3.30) it follows:

1

2σ2

P F (P, t)(e−2κ(T−t) − e−κ(T−t)

)+ κe−κ(T−t)[µ− ln P (t)− λ]F (P, t)

+

[κe−κ(T−t)(ln P (t) + λ− µ + σ2

P /2κ)− σ2P

2e−2κ(T−t)

]F (P, t)

= κe−κ(T−t)[µ− ln P (t)− λ]F (P, t) + κe−κ(T−t) [ln P (t) + λ− µ] F (P, t)

= 0, t ∈ [0, T ]

which shows that F (P, t) solves the Cauchy-Problem as of Equation (3.30).

Finally, we have to prove that our assumption solves the terminal boundary condi-

tion F (P, T ) = P (T ). Under our assumption it holds

F (P, T ) = exp

[e−κ(T−T )︸︷︷︸

=1

ln P (T ) + (1− e−κ(T−T )︸︷︷︸=0

)(µ− σ2P /2κ− λ)

+σ2

P

4κ(1− e−2κ(T−T )︸︷︷︸

=0

)

]= P (T ).

This proves our assumption.

2

62


The model has the major drawback that it treats positive and negative mean rever-

sion in the same way. Following [Lautier 2005] contango is limited to the storage

costs until a certain maturity resulting in an upper boundary for price spreads be-

tween two maturity following futures contracts, while backwardation is not. To

cover this phenomena, more complex models are needed.

3.4.2 Two Factor Models

Two factor models determine the uncertainty in the commodity spot price over two

random processes. Two approaches are excepted in literature: the convenience yield

and the long-short term approach. Although the models look different on the first

view, they are equivalent what we will show later. Starting this section we introduce

the Convenience Yield Model allowing for a stochastic spot price implicitly driven

by a stochastic convenience yield. Recall, convenience yield determines why and

how commodity spot prices deviate from classical asset prices. The following sto-

chastic model specifies the spot price implicitly driven by the convenience yield that

is modeled exogenously as a mean revering process and determine the futures price

as the risk neutral expectation of future spot prices. The model was first introduced

in [Schwartz 1997] and is given in Definition 3.3.

Definition 3.3 Convenience Yield Model

Let P (t) be the spot price of a commodity at time t ∈ [0, T ], c(t) the convenience yield

at time t, µ ∈ R the drift of the spot price, α ∈ R is the long-run level to which the

convenience yield reverts, κ > 0 is the speed of adjustment of the convenience yield,

σP > 0 the spot price volatility, σc > 0 the convenience yield volatility and dWP (t)

and dWc(t) the increments of two Brownian Motions as defined in Definition C.28

with a correlation

dWP (t)dWc(t) = ρdt, t ∈ [0, T ], ρ ∈ [−1, 1]. (3.31)

The spot price and the instantaneous convenience yield process are assumed to have

the following form:

dP (t) = P (t)[µ− c(t)]dt + σP P (t)dWP (t), (3.32)

dc(t) = κ[α− c(t)]dt + σcdWc(t), t ∈ [0, T ] (3.33)

The spot price P (t) of Equation (3.32) follows a geometric Brownian Motion as

of Definition C.28 with a stochastic convenience yield defined in Equation (3.33).

63


The stochastic convenience yield c(t) as of Equation (3.33) is assumed to be mean

reverting and follows a Mean Reversion process. The inclusion of this process into

Equation (3.32) introduces an implicit mean reversion effect on the commodity spot

price process, when the respective Brownian Motions are positively correlated: An

increase in P (t) from a positive dWP (t) is typically associated with a positive dWc(t)

and an increase of c(t) entering negative the drift rate of P (t) and decreasing the

spot price.81 Based on the spot price movements we can calculate the futures price

in the Convenience Yield Model:

Theorem 3.5 Futures Price in the Convenience Yield Model

Let the notations be as in Definition (3.3), λ : R 7→ R be the market price of risk

introduced in the Girsanov-Theorem as of Theorem C.3 and rf be the constant risk

free interest rate. Then the futures price F (P, c, t) is a function of the spot price,

the convenience yield and the time to maturity and is expressed by

F (P, c, t) = P (t) exp

[−c(t)

(1− e−κ(T−t)

κ

)+ A(T, t)

], t ∈ [0, T ] (3.34)

with

A(T, t) =

(rf − α +

λ

κ+

1

2

σ2c

κ2− σP σcρ

κ

)(T − t) +

1

4σ2

c

1− e−2κ(T−t)

κ3

+

([α− λ

κ

]κ + σP σcρ−

σ2c

κ

)1− e−κ(T−t)

κ2(3.35)

Proof: As shown in Proof 3.4.1, the futures price F must solve the Cauchy-

Problem as defined in Definition C.36 with x := P , r(P, t) ≡ 0 and D(P ) := P (T )

for all P (T ) ∈ R and t ∈ [0, T ]. Following the methodology of Proof 3.4.1, we have

to transfer the stochastic process for the factors into the world of the equivalent

martingale measure Q as defined in Definition C.32. Using the Girsanov-Theorem

as of Theorem C.3, we have:

dP (t) = P (t)[rf − c(t)]dt + σP P (t)dWP (t),

dc(t) = (κ[α− c(t)]− λ)dt + σcdWc(t),

dWP (t)dWc(t) = ρdt, t ∈ [0, T ]

dWP (t) and dWc(t) are the increments of two Brownian motions under the equiva-

lent martingale measure. Based on these equations we can formulated the specific

81See [Markert 2005] for empirical evidence.

64


Cauchy-Problem:

0 =1

2σ2

P P (t)2FP,P (P, c, t) + ρσP σcP (t)FP,c(P, c, t)

+1

2σ2

cFc,c(P, c, t) + [rf − c(t)]P (t)FP (P, c, t)

+ (κ[α− c(t)]− λ)Fc(P, c, t) + Ft(P, c, t), t ∈ [0, T ] (3.36)

with the terminal boundary condition F (P, c, T ) = P (T ).

Under the assumption of Equation (3.34) that

F (P, c, t) = P (t) exp

[−c(t)

(1− e−κ(T−t)

κ

)+ A(T, t)

], t ∈ [0, T ]

with

A(T, t) =

(rf − α +

λ

κ+

1

2

σ2c

κ2− σP σcρ

κ

)(T − t) +

1

4σ2

c

1− e−2κ(T−t)

κ3

+

([α− λ

κ

]κ + σP σcρ−

σ2c

κ

)1− e−κ(T−t)

κ2, t ∈ [0, T ]

we can calculate the respective derivatives:

FP (P, c, t) = F (P,c,t)P (t)

,

FP,P (P, c, t) = 0,

Fc(P, c, t) = −(

1−e−κ(T−t)

κ

)F (P, c, t),

Fc,c(P, c, t) =(

1−e−κ(T−t)

κ

)2

F (P, c, t),

FP,c(P, c, t) = −(

1−e−κ(T−t)

P (t)κ

)F (P, c, t),

and finally,

Ft(P, c, t) =

[c(t)e−κ(T−t) −

(rf − α +

λ

κ+

1

2

σ2c

κ2− σP σcρ

κ

)− σ2

c

2κ2e−2κ(T−t)

−1

κ

([α− λ

κ

]κ + σP σcρ−

σ2c

κ

)e−κ(T−t)

]F (P, c, t)

with t ∈ [0, T ].

65


Putting this into the Cauchy-Problem Equation (3.36) it follows

0 +1

2σ2

c

(1− e−κ(T−t)

κ

)2

F (P, c, t) + [rf − c(t)]F (P, c, t)

− (κ[α− c(t)]− λ + ρσP σc)

(1− e−κ(T−t)

κ

)F (P, c, t)

+


(rf − α +

λ

κ+

1

2

σ2c

κ2− σP σcρ

κ

)− σ2

c

2κ2e−2κ(T−t)

−1

κ

([α− λ

κ

]κ + σP σcρ−

σ2c

κ

)e−κ(T−t)

]F (P, c, t)

= [rf − c(t)]F (P, c, t)− 1

κ(κ[α− c(t)]− λ + ρσP σc)

(1− e−κ(T−t)

)F (P, c, t)

+


(rf − α +

λ

κ− σP σcρ

κ

)−1

κ

([α− λ

κ

]κ + σP σcρ

)e−κ(T−t)

]F (P, c, t)

=1

κ(κ[α− c(t)]− λ + ρσP σc)e

−κ(T−t)F (P, c, t)

+

[c(t)e−κ(T−t) − 1

κ

([α− λ

κ

]κ + σP σcρ

)e−κ(T−t)

]F (P, c, t)

=1


−κ(T−t)F (P, c, t)

− 1


−κ(T−t)F (P, c, t)

= 0, t ∈ [0, T ]

which shows that indeed F (P, c, t) as of Equation (3.34) solves the Cauchy-Problem

Equation (3.36). Still we have to prove that our assumption solves the terminal

boundary condition F (P, c, T ) = P (T ). Under our assumption it holds

F (P, c, T ) = P (T ) exp

−c(T )

(1− e−κ(T−T )

κ

)+ A(T, T )︸︷︷︸

=0

= P (T ).

This proves Theorem 3.5.

2

Thinking about real options under the purpose to find the optimal exercise moment

for exploration ventures, brought up the thought of long term trends and short term

fluctuations in commodity markets. [Schwartz Smith 2000] used the idea and pub-

66


lished their Long - Short Term Model that models mean reversion in short term

prices and uncertainty in the equilibrium level to which prices revert. Although,

these variables are not directly observable in the market, the authors used the fol-

lowing intuition to estimate the parameters of the model from market data: move-

ments in prices for long maturing futures contracts provide information about the

equilibrium price level, and differences between the prices for the short and long

term contracts provide information about short term variations. The mathematical

formulation of the model is given in Definition 3.4.

Definition 3.4 Long - Short Term Model

Let P (t) be the spot price of a commodity at time t ∈ [0, T ], χ(t) the short-term

deviation in prices at time t, ξ(t) the equilibrium price level at time t, µ ∈ R the

drift of the equilibrium price level, κ > 0 the speed of adjustment of the short-term

deviation, σχ > 0 the short-term prices volatility, σξ > 0 the equilibrium price level

volatility and dWχ and dWξ the increments of two standard Brownian Motions as

defined in Definition C.28 with a correlation

dWχ(t)dWξ(t) = ρdt, t ∈ [0, T ], ρ ∈ [−1, 1]. (3.37)

Then the dynamic of this model is

ln P (t) = χ(t) + ξ(t) (3.38)

dχ(t) = −κχ(t)dt + σχdWχ(t) (3.39)

dξ(t) = µξdt + σξdWξ(t), t ∈ [0, T ] (3.40)

Temporary price changes, caused e.g. by abrupt weather alteration or supply inter-

ruptions, are embodied in the short term component χ(t). They are not expected

to persist because market participants will switch to inventories to adjust changing

market conditions. Following [Gabillon 1995], production, consumption, stock level

and the fear of inventory disruptions are the most important explanatory factors in

the short run. Information of these factors are mainly needed for hedging purposes.

Changes in the long term level represent fundamental modifications of the market

conditions and are therefore, are expected to persist. Latter can be caused e.g. by a

change in the number of producers in the industry or the availability of a commodity.

It is also determined by expectations of exhausting supply, improving technology for

the production and macroeconomic influences like inflation, politics and regulatory

effects. Following [Gabillon 1995], the information is used for investment purposes.

67


The derivation of the price of a futures contract with the underlying stochastic

processes as of Definition 3.4 can be found in [Schwartz Smith 2000]. Conceptually,

its derivation runs as the methodology of Proof 3.4.1 and Proof 3.4.2. To avoid

redundance, we will focus on another interesting fact. Although, the model does

not explicitly consider changes in the convenience yield, it is equivalent to the Con-

venience Yield Model of Definition 3.3. The following theorem gives the explanation

how the variables of the one model can be expressed as linear combination of the

variables of the other model:

Theorem 3.6 Equivalence of the Convenience Yield and the Long - Short

Term Model

The Convenience Yield Model as of Definition 3.3 and the Long - Short Model as of

Definition 3.4 are equivalent with the following parameters:

Long - Short Model Convenience Yield Model

κ κσχ

σP

κ

dWχ(t) dWc(t)µξ µ− α− 1

2σ2

P

σξ (σP + σ2c

κ2 − 2ρσP σc

κ)

12

dWξ(t) (σP dWP (t)− σc

κdWc(t))(σP + σ2

c

κ2 − 2ρσP σc

κ)−

12

ρξχ (ρσP − σc

κ)(σP + σ2

c

κ2 − 2ρσP σc

κ)−

12

Table 3.1: Equivalent Parameters

Proof: Following Definition 3.3, the price dynamics in the two factor convenience

yield model are given as of Equation (3.32) and Equation (3.33):

dP (t) = P (t)[µ− c(t)]dt + σP P (t)dWP (t),

dc(t) = κ[α− c(t)]dt + σcdWc(t), t ∈ [0, T ]

With Ito as of Lemma C.1 the log spot price dynamic of (3.32) are given as:

dln(P (t)) =

((1

P (t)

)(P (t)[µ− c(t)])

)dt + 0 +

1

2

(σP P (t)2

(−1

P 2(t)

))dt

+

(1

P (t)

)σP P (t)dWP (t)

= [µ− c(t)− 1

2σ2

P ]dt + σP dWP (t) (3.41)

68


Then, the variables in the long - short model can be written in terms of the variables

of the stochastic convenience yield model as follows:

χ(t) = short term deviation =1

κ(c(t)− α) (3.42)

Therewith, it follows

dχ(t) =1

κdc(t)

(3.33)︷︸︸︷=

1

κ(κ[α− c(t)]dt + σcdWc(t))

= [α− c(t)]dt +σc

κdWc(t)

(3.42)︷︸︸︷= −κχ(t)dt +

σc

κ︸︷︷︸≡σχ

dWc(t)︸︷︷︸≡dWχ(t)

Moreover

ξ(t) = equilibrium price level

= ln(P (t))− χ(t)

= ln(P (t))− 1

κ(c(t)− α) (3.43)

Therewith, it follows

dξ(t) = dln(P (t))− 1

κdc(t)

= [µ− c(t)− 1

2σ2

P ]dt + σP dWP (t)− 1

κ(κ[α− c(t)]dt + σcdWc(t))

= [µ− α− 1

2σ2

P︸︷︷︸≡µξ

]dt + σP dWP (t)− σc

κdWc(t)︸︷︷︸

≡σξ≡dWξ(t)

69


Finally,

ρξχdt = dWχ(t)dWξ(t)

= dWc(σP dWP (t)− σc

κdWc(t))(σP +

σ2c

κ2− 2ρσP σc

κ)−

12

= (σP dWP (t)dWc︸︷︷︸=ρPcdt

−σc

κdWc(t)dWc︸︷︷︸

=dt

)(σP +σ2

c

κ2− 2ρσP σc

κ)−

12

= (ρPcσP −σc

κ)dt(σP +

σ2c

κ2− 2ρσP σc

κ)−

12

(3.44)

showing the last equation of Table 3.1.

2

[Schwartz Smith 2000] showed that the model works best for mid term maturities.

Moreover, the model includes the two one factor models Brownian Motion and Mean

Reversion. The first one is generated by setting σχ equal to zero, i.e. assuming that

there is uncertainty in equilibrium prices, only. A Mean Reversion Model is given

by assuming a constant equilibrium price, i.e. setting σξ equal to zero. Statistical

comparison of the three models by the authors showed significant advantages in cap-

turing the characteristics of commodity futures prices through the two factor model.

But as [Lautier 2005] stats, there is still one question remaining: is it interesting to

represent a stable equilibrium with a stochastic variable? On the other hand, some

pricing perspectives, especially in the real options environment, focus on long term

prices and do not care about short term fluctuation.82

3.4.3 Three Factor Models

Not until 1997, the first three factor model was introduced: [Schwartz 1997] pro-

posed his three factor model with the extension of stochastic interest rates because

the hypothesis of constant interest rates as in the one and two factor models amounts

to saying that the term structure of interest rates is flat, which is far from reality.

Moreover, under this assumption forward and futures prices are equivalent, which

is not the case.83 With a stochastic interest rate, it is possible to determine two

distinct payoff structures for forwards and futures, i.e. to take into account the

margin call mechanism of the futures market. Finally, following [Lautier 2005], the

82Compare [Schwartz 1998].83See [Pindyck 1994] and [French 1983]. Compare Section 2.3.1.1 Paragraph ”Forwards and Fu-

tures”.

70


presence of the interest rate as a third explicative factor is consistent with the theory

of storage. When interest rates are high, storage is more expensive resulting into a

reduction of inventory and therewith, increasing the convenience yield.

Definition 3.5 Convenience Yield Model

Let P (t) be the spot price of a commodity at time t ∈ [0, T ], c(t) the convenience

yield at time t, r(t) the interest rate at time t, µ ∈ R the drift of the spot price,

α ∈ R is the long-run level to which the convenience yield reverts, m ∈ R is the

long-run level to which the interest rate reverts, κ > 0 is the speed of adjustment

of the convenience yield, β > 0 is the speed of adjustment of the interest rate,

σP > 0 the spot price volatility, σc > 0 the convenience yield volatility, σr > 0

the interest rate volatility and dWP (t), dWc(t) and dWr(t) the increments of three

Brownian Motions as defined in Definition C.28 with the following correlations:

dWP (t)dWc(t) = ρPcdt, dWc(t)dWr(t) = ρcrdt and dWr(t)dWP (t) = ρrP dt, with

t ∈ [0, T ] and ρ ∈ [−1, 1]. The spot price, the instantaneous convenience yield and

the the instantaneous interest rate process are assumed to have the following form:

dP (t) = P (t)[µ− c(t)]dt + σP P (t)dWP (t), (3.45)

dc(t) = κ[α− c(t)]dt + σcdWc(t), (3.46)

dr(t) = β[m− r(t)]dt + σrdWr(t), t ∈ [0, T ] (3.47)

The stochastic factors in the models are the commodity spot price, the convenience

yield and the interest rate. By assuming a simple mean reverting process for the

interest rate, it is possible to obtain a closed form solution for futures prices. Their

derivation can be found in [Schwartz 1997].

A new approach comes from [Cortazar Schwartz 2003] as introduced in Definition 3.6.

Again, the spot price and the convenience yield are the first two risk factors but as

third they consider the long term spot price return, allowing it to be stochastic and

to return to a long term average. The temporary price variations are assumed to be

activated by changes in inventory, whereas the long term return is due to changes

in technologies, inflation or demand pattern. The dynamics are modeled as follows:

Definition 3.6 The Long Term - Convenience Yield Model

Let P (t) be the spot price of a commodity at time t ∈ [0, T ], y(t) the demeaned

convenience yield at time t, with y := c − α, where α is the long run mean of the

convenience yield c, v(t) the expected long-term spot price return at time t, with v :=

µ−α, where µ ∈ R is the drift of P , κ, a > 0 the speed of adjustments of the demeaned

convenience yield of v; v ∈ R the long-run mean of the expected long-term spot price

71


return, σP , σy, σv > 0 the corresponding volatilities and dW (t)P , dW (t)y, dW (t)v the

increments of three standard Brownian Motions as defined in Definition C.28 with

correlations ρPy, ρPv, ρyv ∈ [−1, 1]. Then the dynamic of this model is

dP (t) = P (t)[v(t)− y(t)]dt + σP P (t)dWP (t), (3.48)

dy(t) = −κy(t)dt + σydWy(t), (3.49)

dv(t) = a[v − v(t)]dt + σvdWv(t), t ∈ [0, T ]. (3.50)

In practice, the development of three factor models deposit the question of sense

and usage. Although an empirical comparison of three factor to two factor models

show that the introduction of a third factor improves the performance of the models

in terms of their ability to describe the evolution of futures prices, this improvement

is too small to justify for higher computational costs. Especially, for the evaluation

of more complex derivatives parsimony is needed. [Schwartz 1997] concludes, that

the two factor Convenience Yield Model has the best return on investment.

72

4 Commodity Indices

In the following section we will introduce the commodity trading vehicle our main

focus is put on over the next sections: commodity indices. In traditional financial

markets, indices are assumed to produce attractive risk and return profiles. But

what about commodity markets? Indeed, commodity indices represent diversified

portfolios participating from the different facilities of their elements.84 Recall, a

commodity investment over a CTA also provides diversified commodity exposure.

But the main difference between commodity indices and managed futures accounts

or funds is, that the indices introduced in this section represent long only, buy

and hold strategies whereby CTAs actively trade commodity derivatives, i.e. they

are allowed to trade short positions for instance. This yields to different risk and

return structures. [Schneeweiss Spurgin 1996] analyzed various commodity indices

and indices which are used to track managed futures performance. Results indicate

that a buy and hold commodity investment strategy provides a poor forecast of CTA

returns. Therefore, commodity indices have to be treated differently.

Because commodity investment is still adolescent, there is only a very little amount

of commodity indices of less than 20 available. They differ among each other by

e.g. number of commodities involved, their weighting and rebalancing procedures.

The different characteristics are described in Section 4.1 and shall serve us as a

first warming up. The following Section 4.2 provides information about the major

commodity indices. Most of them are not older than ten to 15 years and there are

partly huge creation differences among them.

Investors are accustomed and attracted to the ability of entering a market via cheap

diversified exposure yielding into an increasing demand for commodity linked prod-

ucts that will be introduced in Section 4.3. Products like mutual or exchange traded

funds tracking an index are known from stock and bond markets and famous. Es-

pecially the fees are much lower than managed futures fund fees that can yield up

to 25%.

We already know from Section 3 the source of commodity futures return evolving in

changes of the current supply and demand equilibriums. We will close this section

in 4.4 by decomposing commodity index returns and filtering their origins.

84In Section 5.1.3 we will give the mathematical explanation for this phenomena.

73

4 Commodity Indices

4.1 Characteristics

A commodity index is designed to represent and track price changes in a basket of

commodity futures contracts. The concept of an investable commodity index that is

treated as a separate asset class was first introduced in [Greer 1978]. The underlying

logic is that the returns on the index approximate the returns to an investor holding

a position in the assets underlying basket. Following [Structured Products 2006] the

difference between the different indexes is based on a variety of design factors:

4.1.1 Index Composition

Commodity indices either include a narrow or a broad range of single commodities.

Narrow based indices typically cover major commodities, primarily energy and met-

als. They aim to be sector specific and focus on liquid commodities with a direct link

to industrial production and GDP. Over this they seek to get exposure to factors

such as weather conditions. In contrast, broad based indices cover a large variety of

commodities that are economically significant including energy, metals and agricul-

tures. Although they are more difficult to replicate, they provide the investor with

a diversified exposure because they take advantage of the low correlation between

the different commodity groups.85

4.1.2 Index Weights

The index weights determine the amount of a single commodity with which it enters

the index. The determination methodology is unique and based on different factors.

To get economic weights fundamental economic data such as world production are

taken. Production of commodities can be seen as the equivalent of market capi-

talization in stock markets that is taken to get the index weights of major stock

indices. The easiest way to create an index is to take fixed and equal weights.

There are some indices which weights calculation takes into account market factors.

These include trading volume and open interest of the respective commodity future

contract. In the past some optimized weight schemes based on econometric models

were introduced. They seek to optimize criteria such as level of returns, volatility

of returns or correlation to inflation.

85Further details see Section 5.1.2 and 5.1.3.

74

4.2 The Major Market Indices

4.1.3 Rebalancing

Index rebalancing includes two separate factors: the mechanism of rolling futures

contracts and rebalancing the portfolio weights. As described above futures are de-

signed to mature after a predefined period. To enable long term investments in a

single commodity the futures exposure has to be transferred, i.e. rolled over, from

the maturing futures contract into a fairly long-term future contract. The chosen

time lags (typically 1, 2 or three month) are different. The other element of index

rebalancing is the adjustment of the actual amount of futures contracts per commod-

ity. The different indices have different rebalancing and roll over periods depending

on the structure of the market they reflect. [Erb Harvey 2006] investigated the effect

of rebalancing and show the importance of rebalancing as a return driver.

4.1.4 Return Calculation

Returns may be calculated on an arithmetic or geometric basis. The geometric av-

erage return calculation is a methodology which considers the compounded interest

effect. A geometric average return is always smaller than or equal to the arithmetic

average return depending on the frequency of negative returns.

4.1.5 Leveraged versus unleveraged Returns

A leveraged commodity index also known as excess return index is based on futures

contracts. The terminology ”leveraged” reflects the fact that trading in futures

requires minimal commitment of capital. Capital is just needed for margin require-

ments. To create indices which do not reflect a leverage effect the total amount

invested in commodities have to be invested in collateral, typically in T-Bills. They

are called total return indices and provide the investor with unleveraged return.86


After structuring the different creation characteristics, we want to see what dif-

ferences the most common market indices have among each other. The different

creation characteristics such as index weighting or rebalancing are driven by the

two features of commodity markets: one being the trading platform for consump-

tion goods and one being the platform for financial investments. The indices are

86Attention: The terminology ”leverage effect” describes in stock markets the correlation betweenfalling prices and rising volatility.

75

4 Commodity Indices

generally published in three categories: spot, excess and total return. The spot

return index is just a price index that replicates the underlying commodity spot

price changes. The excess return index replicates the underlying commodity futures

price changes so it includes gains and losses from rolling maturing futures forward.

Finally, the total return index represents a fully collateralized investment in the un-

derlying commodity basket. Only the two last versions are investable because there

does not exist a futures adequate spot market.87 The range of commodity indices is

growing proportionally with the fast growing demand for commodity linked invest-

ment possibilities but is still in size not comparable to the huge investment offers

in stock and bond markets. While in latter markets the available indices are nearly

not countable, there are less than 20 commodity indices launched. In the following

paragraphs we will introduce the most market known ones and closing this section

with a small comparison of them.

4.2.1 CRB

The Commodity Research Bureau Index (CRM) index is the oldest of all commodity

indices. It was introduced in 1957 and is reported in the CRB Yearbook. The index

consists of 17 equally weighted commodities that are shown in Figure 4.1. It repre-

sents a broad basket of common commodity products. As the father of commodity

indices and because of its equal diversification it is widely viewed as a very good

measure of macroeconomic trends.

Figure 4.1: The CRB Index

The index value is calculated with a double-averaging procedure which calculates

87For further details recall introduction to Section 3.

76


first the geometric average return of the single relevant commodities and second, the

geometric average return of the commodity basket. This makes the index robust

against discontinuities associated with temporary supply and demand imbalances in

a given month or commodity.88

Rebalancing only takes place when the index can’t ensure an accurate representation

of the broad commodity market. There have been nine revisions to the component

list, the last in 1995.

Out of the investment point of view the index is a passive, ”buy-and-hold” com-

modity futures basket. Significantly positive returns on this index typically occur

when commodities are short in supply which is followed by rising commodity prices.

This originates its reputation of being a macroeconomic trend measure.

4.2.2 GSCI

The Goldman Sachs Commodity Index (GSCI) consists of 24 commodities and is

a measure of the performance of actively traded, dollar-denominated nearby com-

modity futures. It was first published in 1991, but the rule based methodology was

coupled with historical price data to create a history that begins at January 1970

with five commodities included. Therewith, the index has the longest history of the

commercially available indices.

The weights of the index components are chosen as the 5 year moving average of

their world production volume. Common equity indexes like the S&P 500 or the

Eurostoxx 50 use the market capitalization of the companies entering the index for

the calculation of their component weights. This factor is seen as the equivalent to

world production volume in commodity markets.

Critics say that this puts to much weight to energy which is represented with over

71% as shown in Figure 4.2. However, the choice of its components and the weight-

ing procedure allow the GSCI to reflect world economic growth.

The GSCI weights are reviewed generally once a year. Since they are recalculated

based on world production which in turn is a function of produced quantities and

prices, changes can be heavy. For example, weights to reflect the Energy sector have

varied in the past between 44% and 73% of the index.89

88See [German 2005].89See [Wilshire Research 2005].

77

4 Commodity Indices

Figure 4.2: The GSCI Index

The Index has a major drawback: If prices of a commodity rise, world production

will rise as well because high cost producers are enabled to enter the market. If world

production rises, the GSCI puts more weight to this commodities. But commodity

prices tend to mean revert to a long term price level.90 Implicating the GSCI takes

to much weight into commodities which are expected to fall in price over the coming

periods.

4.2.3 DJ-AIGCI

The Dow Jones - American International Group Commodity Index (DJ-AIGCI) was

designed in 1998 with a backfilled history until 1991 to be a liquid benchmark for

commodity investments. To calculate the weights of its 19 components shown in

Figure 4.3 it takes two measurements into consideration: production and liquidity,

whereby liquidity is the dominant factor. From the financial point of view, fur-

ther is an exogenous quantity of futures markets reflecting the consumption good

character of commodities and latter is an endogenous quantity of futures markets

reflecting the investment character of commodities. Like explained for the GSCI,

production is a useful measure of economic importance but may underestimate the

economic significance of storable commodities (e.g. gold) in comparison to non-

storable commodities (e.g. live cattle). To compensate this, liquidity comes up. It

is an important indicator for the current value placed on a commodity by financial

and physical market participants.

The index weights are reviewed and recalculated once a year by the DJ-AIGCI’s

90See [Bessembinder e.a. 1995].

78


Figure 4.3: The DJ-AIGCI Index

Oversight Committee. Changes are less drastic than in the GSCI because there

exists minimum (2%) and maximum (33%) limits to restrict the weights for single

commodities and sectors.

4.2.4 DBLCI

The Deutsche Bank Liquid Commodity Index (DBLCI) is relatively different to the

other indices. As shown in Figure 4.4 it just includes six of the most liquid commod-

ity futures in terms of trading volume and open interest whereas the other indices

include between 17 to 34 constituents. In 2003, Deutsche Bank research analyzed

the GSCI. They showed that the volatility of a commodity basket depends of the

number of its constituents and decided that there is no need for more than a cou-

ple of single commodities to get the volatility converging against a constant fixed

value. Therefore, they decided that six commodities which are chosen out of the big

commodity groups energy, metal and agriculture are enough to present commodity

markets optimally.91 Figure 4.4 shows the composition of the index. One big ad-

vantage of the index is that it is easy to track because the component weights are

not volatile.

4.2.5 DBLCI-MR

The Deutsche Bank Liquid Commodity Index - Mean Reversion (DBLCI-MR) in-

cludes the same components as the DBLCI. The difference is that the weights of its

91For further details see Section 5.1.3.

79

4 Commodity Indices

Figure 4.4: The DBLCI Index

components are updated relatively to its five year price moving average. Because

commodity prices tend to mean revert over time against a long term price these

averaging methodology has the feature of being an early signal. If a commodity is

cheap relative to its five year moving average price, the index weight gets increased.

If it is relatively expensive, it gets reduced. As shown in [Bessembinder e.a. 1995]

this methodology is not very useful. The drawback is that the mean reversion effect

of the commodity prices is slow. The index reacts directly. Thus, there is no gain

taking over long periods. It can be seen in Figure 4.5 that although energy futures

are still running extraordinary, the DBLCI-MR reduced the energy weight already

and profit taking’s capacity is not fully used.

Figure 4.5: The DBLCI-MR Index

Please note that the weights of the DBLCI-MR are very volatile. At the moment

the main weight is put to agricultures because they are cheap in comparison to their

80


five year moving average price. In other periods crude and heating oil had weights

around 30%. This makes it expensive to track the index.

4.2.6 RICI

The Rogers International Commodity Index (RICI) was launched in 1998 by the

Wall Street legend Jim Rogers who entered the guiness book of records in the 1970s

with his Quantum Fund as best performing fund ever. The investment approach

Jim Rogers always takes is a macroeconomic one. He tries to reflect global eco-

nomic developments. Therefore, the 34 components and its weights are chosen to

reflect their importance in international commerce. Jim Rogers mentioned in his

book: ”The RICI represents my version of the world, it reflects the costs of life and

survival.”92

Figure 4.6: The RICI Index

The index is rebalanced monthly. Research has shown that monthly rebalancing

provides an annualized return advantage of 1.5% to 2% in comparison to annually

rebalancing.93

4.2.7 Comparison of the Major Market Indices

Closing this section we will give a small comparison of the introduced indices.

Table 4.1 summarizes their major characteristics clearly arranged, including index

comparison, major groups as of Figure 2.1, number of integrated commodities, index

92See [Rogers 2005].93See [Erb Harvey 2006] or [Seamans 2003].

81

4 Commodity Indices

weights, its determination, the rebalancing period and the return calculation.

CRB GSCI DJ-AIGCI DBLCI DBLCI-MR RICI

Indexbroad broad broad broad broad broad

Composition

Major Groups94 all all allno Softs, noLivestock

no Softs, noLivestock

all

Number of 17 24 19 5 5 34Commodities

Index Weights equal economiceconomic,market

fix optimized economic

Determinationof Weights

if required annually annually if required monthly annually

Rebalancing monthly annually annually annually monthly monthly

Returngeometric arithmetic arithmetic arithmetic arithmetic arithmetic

Calculation

Table 4.1: Comparison of Commodity Index Characteristics

We only introduced broad diversified indices to reach better comparability. Eco-

nomic index weights like production and consumption are most mentioned and

reflect commodity’s role as consumption good. Moreover, some indices are more

dynamic than other, e.g. rebalancing takes place monthly instead of annually and

the determination of weights ranges from if required to monthly. This fact has to

be considered in index tracking purposes when e.g. transaction costs come up.

Moreover, it becomes clear that there are differences of the amount of different single

commodities integrated in the different indices ranging from five in the DBLCI and

DBLCI-MR to 34 integrated into the RICI. To reach an even better comparison of

the index ingredients we aggregate the single constituencies into their major groups

following Figure 2.1. The result can be seen in Figure 4.7.

We realize that not only the number of commodities differ between the indices but

also their weighting regarding the three major commodity groups. The GSCI and

the DBLCI have over weight in the energy sector while the CRB has over weight

in the agricultural group indicating that the characteristics of these sub markets

will dominate the whole risk and return profile of the respective index. The best

diversified index with nearly one third of its weights in each commodity group is the

DJ-AIGCI. Therefore, we picked it for the further analyzes.

93As of Figure 2.1.

82

4.3 Index Linked Products

Figure 4.7: Index Component Distribution


Today, commodity indices represent the easiest way to get diversified commodity

exposure. The direct way is to use linked derivatives like futures and options. But

this includes caring about maturities and many investors are long term orientated

and therefore, do not want to go down this street. Mutual funds tracking the

performance of major commodity indices are en vogue because it is a simple way

to get a broad commodity portfolio in a convenient way. Furthermore, the product

range was extended by introducing certificates and exchange traded funds recently.

In the following sections we want to take a look at the fast growing commodity index

linked investment opportunities including derivatives in Section 4.3.1, mutual funds

in Section 4.3.2 and exchange traded funds in Section 4.3.3.

4.3.1 Derivatives

Generally, you can get everything over-the-counter (OTC) as long as you find an

adequate dealer. Especially swaps and index linked notes are preferred more and

more to get commodity index exposure. In contrast, exchange traded products are

rare. There are future contracts maturing every January, February, April, June,

August, October, and December listed at the Chicago Board of Trade (CBOT) to

trade the DJ-AIGCI. Each individual futures contract has a fixed ratio to the index

value of DJ-AIGCI, and investors can easily estimate the fair value for each DJ-

AIGCI futures contract on a live basis, based on the prices of the underlying futures

contracts which are used to calculate the DJ-AIGCI.

The GSCI has futures contracts listed on the Chicago Mercantile Exchange (CME)

that have been traded by numerous market makers for over 12 years. The GSCI

83

4 Commodity Indices

is the most liquid commodity index because the maturities are monthly. Over this

there can be traded a long term futures contract maturing in May 2011.

Since 2005 a so-called Rogers TRAKRS future can be traded at the CME. The

exchange in collaboration with Merrill Lynch created a RICI tracking portfolio what

can be accessed by investors through a futures contract with long run maturity until

2010.

For the Deutsche Bank Indices all possible OTC products are available. More-

over, Deutsche Bank offers OTC swaps, forwards, linked notes and options to get

DJ-AIGCI and GSCI exposure.

In 2005 UBS was the first issuer who offered different types of certificates with the

RICI as underlying.

4.3.2 Mutual Funds

Comparing the number of available commodity linked mutual funds with the total

amount of available equity linked funds one would be very surprised. On the com-

modity linked side we have less than 20 funds available but on the equity side the

fund horizon seems to be endless. Not until 1997 as the Oppenheimer Real Asset

Fund was launched, people started doing business in this investment field. Today,

there are approximately 15 billion US dollar of assets under management, this is

more than 10% of the money invested in the managed futures business but less than

0.1% of the total money invested worldwide in mutual funds.94

Generally commodity linked mutual funds do not build up a futures portfolio to track

a commodity index. They get commodity exposure over commodity linked notes or

swaps. The latter ones were very common until an announcement of the Internal

Revenue Service95 that income from commodity-linked swaps is not a ”quantifying

income” because the underlying instruments were not securities. As a result, a

mutual fund that is invested in commodity swaps with more than 10% of its gross

income would lose its status as a registered investment company, and would become

taxable for income and capital gains, rather than passing taxes through to their

investors. From July 2006 on, there will be a huge move out of swaps into structured

notes.

Whatever linked derivative are taken to get a commodity investment, the funds ex-

posure is usually indirect. But at the end of the day their buying power shows up

94See [ICI 2006]95The Internal Revenue Service (IRS) is the US government agency responsible for tax collection

and tax law enforcement.

84


in commodity futures markets: When the major derivatives issuers, e.g. Goldman

Sachs or AIG Financial Products, sell OTC commodity linked derivatives to the

mutual funds, they end up short. To hedge their books, issuers turn around and

buy through own trading desks or external traders equivalent futures to reset their

positions as shown in Figure 2.18. The resulting construct is complex but every-

body is satisfied: investors get commodity exposure through a familiar investment

vehicle, mutual fund companies increase their assets under management, derivatives

dealers earn fees by selling commodity linked derivatives and replicating the index

and futures industry benefits from higher trading volume. To make thinks more

plastic Table 4.2 summarizes the major mutual funds available in July 2006.

Fund Name Index NAV Mil.US dollar

Pimco Commodity Real Return Strategy DJ-AIGCI 11,823Oppenheimer Real Asset Fund GSCI 1,900Fidelity Strategic Real Return Fund DJ-AIGCI 1,934Credit Suisse Commodity Return Strategy DJ-AIGCI 277DWS Scudder Commodity Securities Fund GSCI 183

Table 4.2: Commodity Index linked Mutual Funds

Combining the Net Asset Values96 (NAVs) of the above mentioned mutual funds

the PIMCO Commodity Real Return Fund is by far the biggest flagship. It was

introduced in 2002 and is consequently the second oldest behind Oppenheimer’s

Real Asset Fund. It combines a position in commodity futures backed primarily by

a portfolio of inflation linked interest products. Hereby, the commodity exposure is

passively managed to track the DJ-AIGCI and the fixed income collateral portfolio

is managed actively. At the beginning of July 2006 PIMCO published that they

shift there commodity exposure out of swaps into linked notes as an reaction to the

above mentioned change in regulatory requirements.97

An example of an actively managed commodity fund is the Oppenheimer real asset

fund which was launched in 1997 and thus was the first Real Asset Fund ever. Its

purpose is to outperform the GSCI. The fund holds approximately one third of its

assets in structured notes linked to the GSCI. In addition, the portfolio includes

substantial direct holdings of futures contracts, at the moment 8.2% of its total

assets in energy futures, 3.4% in metals and 2.1% in agricultures.

The Fidelity Strategic Real Return Fund was launched in September 2005 to provide

96The Net Asset Value is defined as the difference between total assets minus total liabilities.97See http : \ \ www.allianzinvestors.com \ commentary \ edu education02072006.jsp

85

4 Commodity Indices

its investors ”with an inflation linked security” what is backed out of 17.3% real

estate investments, 30.5% inflation protected investments, 24.7% floating rate high

yield, 25.2% commodity linked notes and 2.3% cash. The commodity exposure is

build up with structured notes currently to the DJ-AIGCI.

In January 2005 Credit Suisse launched its Commodity Return Fund which primarily

invests in commodity linked swaps which make it receiving a total return rate based

on the DJ-AIGCI and make it paying the 1 month U.S. Treasury Bill rate plus

a spread.98 The portfolio is backed by investment-grade fixed income securities

normally having an average duration of one year or less.

The DWS Scubber Commodity Securities Fund was launched in February 2005. The

fund’s benchmark comprises 50% of the GSCI, 25% of the MSCI World Energy and

25% of the MSCI World Materials Index. Therefore, it’s composed out of 50% com-

modity related common stocks and 50% commodity related structured notes. The

fund uses both top-down analysis to decide which sectors to over- or underweight

based on the supply and demand picture and other fundamental trends in com-

modity markets and bottom-up research to pick promising individual companies. It

invests into the GSCI through linked notes, swap agreements and futures contracts.

It is eye-catching that the market is strongly dominated by the GSCI and the DJ-

AIGCI. These two indices are the oldest investable commodity indices and have

therefore not only a long tradition but are well known in the financial investment

sector. Because investing in commodities as a retail process is quite young and new

products have to be set up. Nevertheless, this market is very active and specially

the RICI is getting more popular. Uhlmann Price Security was the first provider

which enabled investors to get RICI exposure over a mutual fund. In Europe UBS

Investments offers a mutual fund with RICI as benchmark.

4.3.3 Exchange Traded Funds

Exchange Traded Funds (ETFs) are a relatively recent innovative investment con-

cept and were first introduced in 1993. They represent exchange traded investment

funds which in the case of commodities invest long in fully collateralized futures

positions. In comparison to traditional mutual funds ETS’ are permanently traded

like stocks. Therefore, ETFs combine the flexibility of stocks with the risk control

over diversification of traditional mutual funds: any investor can buy or sell shares

98Regarding the change in regulatory guidelines a redeployment into commodity linked notes canbe expected.

86


in ETFs, at a price that is a close approximation of the net asset value per share,99

from virtually any broker, and need not wait for end-of-day pricing or worry about

trading discounts to NAV. Over this, the management fees are generally much lower

then the fees of mutual funds and commodity pools. In the 13 years since their

introduction, the number of ETF’s has grown to 200 listed at American stock ex-

changes with an amount of 300 billion US dollar under management at the end of

2005.100

The first ETF with commodity focus was listed in February 2006 on the American

Stock Exchange under the symbol DBC standing for DB Commodity Index Tracking

Fund. The fund’s objective is to track the DBLCI Excess Return. Because the

fund is not actively managed there is a very low fee of 1.3% annually, including

management and brokerage fees. In comparison, mutual fund management fees are

much higher: there is generally a 5% purchasing fee plus an annual management

fee.

The DBC utilizes a two-tier structure, i.e it invests its assets in a master fund

which is fully owned by Deutsche Bank AG. The master fund, in turn, invests its

assets in exchange traded futures on the commodity respectively its weights in the

DBLCI and a small amount into U.S. Treasury securities to serve margin payments.

This operation method is totally different from that of mutual funds as they get

commodity exposure indirect through swaps and linked notes. Although the fund

has been available for only a few months, it already attracted substantial interest

from retail investors. Net assets at the end of April 2006, were approximately 400

million US dollar what is almost twice the size of the one year old Credit Suisse

Commodity Return Strategy Fund.

The DBC has set a milestone in commodity investments and it can be expected

that other commodity pools will replicate this concept and list their shares via an

ETF type structure. One follower was ABN AMRO in May 2006. It listed the

shares of an ETF which tracks the performance of the RICI on Deutsche Borse.

The big advantage is that investors will have the opportunity to obtain exposure

to the commodities markets in a format that provides unprecedented transparency,

liquidity and cost-effectiveness.

99The NAV per share is defined as:

NAV per share =total assets - total liabilities

total number of shares outstanding

100See [ICI 2006]

87

4 Commodity Indices

4.4 Decomposition of Index Returns

In this section we will get a deeper insight into the return structure of commodity

indices. As already mentioned introductory to Section 4.1 there are three different

index types calculated by the index issuer: the total return index, representing the

return development of a fully collateralized commodity investment, the excess return

index, representing the return development of a leveraged commodity investment,

and the spot return index, representing the simple commodity price changes over

time. But how are the three types connected to each other? Figure 4.8 shall give a

first overview of their team play.101

Total Return = Excess Return + Interest Rate Return

XXXXXXXX

Roll Return+Spot Return

Figure 4.8: Decomposition of Commodity Index Return

A single asset’s index is nothing else but a time series of the prices realized by the

underlying asset. In stock markets this equals a buy and hold trading strategy. In

commodity markets it is not that easy because commodities are traded with futures

contracts, i.e. the underlying has a maturity and therefore, investments have to

be rolled over different positions by and by resulting in the so-called futures or

excess return as the pure return produced by commodity investments. It depends

of the actual price changes of the underlying commodity covered in the spot return

and the roll return realized by rolling futures positions forward under the current

term structure. Later in this section we will see the mathematical derivation of this

dependence structure in Theorem 4.1.

The most common way to construct a single commodity index is to roll someone’s

position from the first to the nearest longer term contract because the nearby con-

tracts have generally the highest liquidity. Futures investments need minimal cash

requirements that are only used to serve margin calls. But to actually add com-

modities as part of an investment portfolio someone has actually to invest a certain

amount reserved for commodity investment. Because this is not possible with fu-

101Figure 4.8 and the following calculations are based on log returns as of Definition C.2. Com-pare [Kat Oomen 2006]. For a commodity return decomposition based on simple returns see[Geer 2000].

88


tures contracts, someone has to invest the reserved amount into a reference asset

called collateral. The issuers of the main indices usually use T-Bills producing his-

torically an annualized return of 3-4%. Because log returns are additive,102 the first

decomposition of Figure 4.8 of total return into excess and interest rate return is

quite intuitive. But what about the second decomposition of excess return into spot

and roll return?

To answer this question we will first give an example calculation by constructing the

futures return time series by rolling the maturing contract into the next nearby con-

tract for the crude oil and copper futures contract already known from in Figure 3.2

in Section 3.1 and second derive the mathematical illustration in Theorem 4.1. For

it, Table 4.3 and Table 4.5 summarize the price movements of the respective con-

tracts. The column header give the maturity T of the respective contract and the

raw header the respective date t at which the price of the contract is measured.

The respective spot return time series is constructed by using the price of the front

month futures contract as a proxy.103 The respective values are highlighted by bold

letters.

Crude Oil (US dollar) Jan 06 Feb 06 Mar 06 Apr 06 May 06 Jun 06 Jul 06

30. Dec 2005 57.98→61.04 ↓ 31.09 62.35 62.70 63.00 63.2531. Jan 2006 68.35→67.92 ↓ 68.74 69.28 69.70 70.0128. Feb 2006 61.10→61.41 ↓ 63.01 64.06 64.8331. Mar 2006 60.57→66.63 ↓ 67.93 68.6728. Apr 2006 71.95→ 71.88 ↓ 73.5031. May 2006 69.23→71.29 ↓30. Jun 2006 68.94

Table 4.3: Construction of a Futures Return Series for Crude Oil

First, we will examine the construction of a futures return series exemplified by

the crude oil price series. The construction follows the arrows in Table 4.3 and

is based on the following thought: From the end of November 2005 to the end of

December 2005 the investor holds the January 2006 contract. Before the contract

expires in January 2006 he closes his position and at the same time he opens a new

position in the February 2006 contract which he holds until the end of January 2006.

Following Definition C.2 the futures return is given as:

rF (t) ≡ ln

(F (t, T )

F (s, T )

), 0 ≤ s < t ≤ T

102See Theorem 4.1.103The procedure is inspired by [Markert 2005] and [Gorton Rouwenhorst 2004].

89

4 Commodity Indices

Implicating, the investor realizes a crude oil futures return of:

rF (Jan) = ln

(F (Jan, Feb)

F (Dec, Feb)

)= ln

(68.35

61.04

)= 11.3%

Again, before the contract expires he closes his February 2006 position and opens

a position in the March 2006 contract. The crude oil futures return time series is

continued with the following value:

rF (Feb) = ln

(F (Feb, Mar)

F (Jan, Mar)

)= ln

(61.10

67.92

)= −10.6%

Running the described construction methodology over the reported times a whole

futures return time series evolves. The results are reported in the first column of

Table 4.4 and also known as excess return as of Figure 4.8.

The next step to encode the different futures return elements is to construct the spot

return. We use the bold highlighted prices in Table 4.3 because the front month

futures contract serves as proxy. Following Definition C.2 the spot return is given

as:

rP (t) ≡ ln

(P (t)

P (s)

), 0 ≤ s < t ≤ T

Implicating, the first crude oil spot return value is given by:

rP (Jan) = ln

(P (Jan)

P (Dec)

)= ln

(68.35

57.98

)= 16.5%

The second value is gives by:

rP (Feb) = ln

(P (Feb)

P (Jan)

)= ln

(61.10

68.35

)= −11.2%

Again, running the described calculation rule over the reported times a whole spot

return time series evolves. All values are listed in the second column of Table 4.4.

Although crude oil went up in price over the last months and could realize a high spot

return the positive slope of the term structure as shown in Figure 3.2 disembogue

into a negative difference between futures and spot returns over the whole period as

documented in the last column of Table 4.4. This gap is caused by rolling a maturing

futures contract into the next nearby month futures contract. Because the market

is in contango the next nearby month futures contract is more expensive than the

maturing futures contract and the investor realizes a loss amounting to -17.4% by

90


Future Return Spot Return Difference = Roll Return

Jan 2006 11.3% 16.5% -5.1%Feb 2006 -10.6% -11.2% 0.6%Mar 2006 -1.4% -0.9% -0.5%Apr 2006 7.7% 17.2% -9.5%May 2006 -3.8% -3.9% 0.1%Jun 2006 -3.4% -0.4% -2.9%

Total -0.1% 17.3% -17.4%

Table 4.4: Spot, Future and Roll Return Time Series for Crude Oil

rolling his position forward. The so-called roll return first introduced in Figure 4.8

is mathematically derived in Theorem 4.1:

Theorem 4.1 Roll Return

Let F (t, T ) denote the commodity futures price at time t ∈ [0, T ] and let P (t) be the

commodity spot price at time t ∈ [0, T ]. Moreover, we have 0 ≤ s < t ≤ T . Then

the roll return is given by:

rr(t) = ln

(F (t, T )

F (s, T )

)︸︷︷︸

futures return

− ln

(P (t)

P (s)

)︸︷︷︸

spot return

(4.1)

Proof: Recall, the spot price, denoted by P (t), is approximated by the front

month futures price, denoted by F (t, T ), i.e. we have: P (t) = F (t, T ). Therewith,

we can calculate:

rF (t) ≡ ln

(F (t, T )

F (s, T )

)= ln(F (t, T )︸︷︷︸

=P (t)

)− ln (F (s, T )) + ln (P (s))− ln (P (s))

= ln

(P (t)

P (s)

)︸︷︷︸

spot return

+ ln

(P (t− 1)

F (s, T )

)︸︷︷︸

roll return

, 0 ≤ s < t ≤ T (4.2)

Rearranging yields to the result.

2

As shown in Figure 3.2 the copper market is in backwardation, e.g. the negative

slope of the term structure disembogues into a positive roll return what we will show

in the following example. The price data of the respective futures contract are given

in Table 4.5.

Calculating the return series with the same methodology described for the crude oil

91

4 Commodity Indices

Copper (US dollar) Jan 06 Feb 06 Mar 06 Apr 06 May 06 Jun 06 Jul 06

30. Dec 2005 4,538→4,489 ↓ 4,431 4,359 4,291 4,231 4,17331. Jan 2006 4,912→4,886 ↓ 4,853 4,815 4,768 4,72128. Feb 2006 4,881→4,842 ↓ 4,812 4,778 4,74231. Mar 2006 5,440→5,423 ↓ 5,400 5,37528. Apr 2006 7,118→7,066 ↓ 7,00831. May 2006 8,001→7,968 ↓30. Jun 2006 7,425

Table 4.5: Construction of a Futures Return Series for Copper

example we end up with the values given in Table 4.6. Recall, the futures return

series is calculated by following the arrows and the spot return series by following

the bold letters. The ”backwarded” term structure produced a positive roll return

amounting to 3.9% as shown in the last column of Table 4.6.

Future Return Spot Return Difference = Roll Return

Jan 2006 9.0% 7.9% 1.1%Feb 2006 -0.1% -0.7% 0.5%Mar 2006 11.7% 10.8% 0.8%Apr 2006 27.2% 26.9% 0.3%May 2006 12.4% 11.7% 0.7%Jun 2006 -7.1% -7.5% 0.4%

Total 53.0% 49.2% 3.9%

Table 4.6: Spot, Future and Roll Return Time Series for Copper

The examples above have shown the impact of the term structure to the investors

return. If a market is in contango the negative roll return will diminish the final

return in spite of price increases yielding to positive spot returns. To push back

the negative rolling impact in contangoed markets, someone could think about ex-

tending the rolling periods. For instance, if the investor of the crude oil example

had avoided rolling forward the positions monthly, and instead would have invested

in January 2006 directly into the July 2006 contract he would have realized a fu-

tures return of ln(

68.9463.25

)= 8.6% because the roll return would have decreased to

ln(

63.2557.98

)= −8.7%. This conclusion is used by Merrill Lynch. In May 2006 they

introduced the ML Oil Return and Income Index that rolls forward its oil futures

positions every third month.104 Backtesting has shown that in fact they could realize

an excess return in comparison to one month rolling, long only oil futures indices

104See [Merrill Lynch 2006]. To be precise, Merrill Lynch employ a short option trading facility aswell to minimize the negative influence of contango to the roll return.

92


over the last 2 years. Given the current term structure as of July 2006 of NYMEX

crude oil shown in Figure 4.9 this strategy is expected to work over the next nine

months namely until April 2007 properly. From this point on, the market is ex-

pected to be in backwardation again yielding to positive roll returns. Implicating,

monthly rolling will be more attractive again.

Figure 4.9: Term Structure of NYMEX Crude Oil as per July 2006

Generally, the big public commodity indices described in Section 4.2 roll every month

over a five day period each with 20% of the total futures investment caused by

liquidity reasons. Trading volume is clustered around the front month contracts.

For instance, the most traded commodity futures contract worldwide, the NYMEX

crude oil future, has in July 2006 approximately 230.000 open interests in the con-

tract maturing in August 2006 less than half of this about 130.000 open interests in

the contract maturing one month later namely in September 2006 and the contract

maturing one year later namely in July 2007 has just 10.000 open interests. The

example is supported by different issuer’s studies proofing that liquidity is clustered

around the nearby contracts. For instance, following [Merrill Lynch 2006] the second

nearby futures contract has only a trading volume of two thirds of the trading vol-

ume of the first month futures contract. Nevertheless, Deutsche Bank has changed

its trading strategy. They implemented the so-called optimum yield rolling strategy.

Depending on the shape of the forward curve, they roll the contracts forward into

contracts that under liquidity requirements maximize the roll return.

93

5 Properties of Commodity Returns

Investor’s attention is generally attracted by asset classes that are on an upwards

move. Legends like Jim Rogers have helped to establish commodities as an asset

class and to convince many investors that the only way for commodities is up. The

economic boom of emerging market countries and the long lasting expansion of pro-

duction capacities will push prices further over the next years. But investors first

have to understand that there is not the ”average commodity”. Therefore, the first

part of this section, i.e. Section 5.1, will concentrate on single commodity returns

and their interactions. Introductory, Section 5.1.1 shall give a first inside into their

different risk and return profiles. We will use the conclusions from Section 3 and

Section 4.4 to decompose excess returns, i.e. the pure commodity return, aiming

to identify whether commodities offer a risk premium or not and how much risk

an investor has to bear when investing into selected commodities. An interesting

observation will be, that in contrast to traditional asset classes, the risk measure

volatility goes up in bullish markets. Commodity price surges come in line with low

inventories and the fear of supply interruptions yields into nervous market move-

ments.

Although, the different types of commodities are influenced by their own specific

risk factors, technological progress allows new substitution possibilities. So, com-

modities that are on the first view totaly different among each other, might be more

and more driven by the same risk factors and demand sources. But in which ex-

tent can similar price movements be observed? Section 5.1.2 will show that only

commodities of the same group show high overlapping among their price movement

characteristics while combining different commodity groups will yield into balanced

risk and return profiles. Section 5.1.3 will finally give the mathematical explanation

of diversification and therewith will state, why commodity indices are suitable to

get balanced commodity exposure.

The second part of this section, i.e. Section 5.2, will further focus on the statis-

tical properties of such a balanced commodity exposure’s return. While different

research focused on the construction and analysis of artificial commodity indices

including e.g. [Gorton Rouwenhorst 2004] and [Erb Harvey 2006], little is done in

analyzing actual market indices, e.g. [Kat Oomen 2006]. We will close the gap by

analyzing the DJ-AIGCI total return index and its pure commodity return compo-

nents.105 We will uncover roll returns and show their impact on total returns in

105The switch from excess return in Section 5.1 to total return in the Section 5.2 is motivated asfollows: The first part of Section 5 concentrates on the characteristics of single commodity

94

5.1 Characteristics of Single Commodities

Section 5.2.1 and 5.2.2. Our findings in Section 5.2.3 stand in contrast to findings of

[Gorton Rouwenhorst 2004] and [PIMCO 2006]. While we report negative skewness,

they published positive. We reason this with two facts: First, they both construct

artificial indices that are not investable and second, they consider a period from

1970 until 2005. Therewith, the value development considers the two major price

surges over the last 100 years.

To close this section, we will report two major time series characteristics: stationarity

in Section 5.2.4 and autocorrelation in Section 5.2.5. Our findings are in line with

[Kat Oomen 2006]. They’ve already reported that the facility of autocorrelation in

selected commodity returns, including among others corn, soybeans, live cattle, oil

and gold, got lost in index returns.106


As we introduced the different commodity types in Section 2.1 it became clear that

the single members of the commodity market differ among each other. Neverthe-

less, we identified dependencies resulting from substitutions or production hierar-

chical structures. The question to answer in this section is consequently, how these

macroeconomic dependencies can be seen in statistical characteristics of return se-

ries’ calculable from futures price time series following Definition C.2 and how the

interaction of different commodities can yield to diversification effects.

For it, we first analyze the risk and return profile of different commodities in

Section 5.1.1. Caused by the consumption good facility of commodities, different

pattern to traditional asset classes occur. Moreover, it will come up that broad

indices as of Section 4.2 have the most attractive risk and return profile in com-

parison to single and group commodity indices. This might indicate diversification

effects. The mathematical basic for diversification is imperfect correlation between

different assets. Therefore, we will analyze possible co-movements of selected com-

modity returns in Section 5.1.2. Finally, we will collect all results in Section 5.1.3

and will come up with the conclusion that commodity investment is most attractive

in products linked to broad commodity indices that are balanced weighted over the

three commodity groups, e.g. like the DJ-AIGCI. Closing, we show that the energy

returns and their interactions among each other, i.e. we focus on the pure commodity return.The second part of Section 5 aims to show distributional behavior of commodity exposure’sreturn. Pure futures return can be seen as an overlay to a portfolio but if an investor actuallywants to invest a part of his wealth into commodities, he has to do so over the collateralizedversion as described in Section 4.4, i.e. he had to consider total returns.

106A detailed data description of our sample can be found in Appendix A.

95


market’s price movements have a huge impact to broad indices although they are

balanced weighted.

5.1.1 Risk and Return Profile

When it comes to financial investing the first two regarded measurements are risk and

return. When an investor puts his money into an asset he is interested in the profit

he will earn, i.e. the expected return of the investment, and the entered risk, e.g.

measured by the volatility of the expected return. Recall, in Section 3 we identified

the two drivers of commodity futures prices to be the spot price respectively the

expectation of the future spot price and a risk premium respectively a risk premium

on inventories called convenience yield. Because our investment focus is long term

orientated and commodities are traded with futures having a maturity we have to

roll over the investment by and by. As we have seen in Section 4.4 the calculable

excess return representing the pure commodity return can be divided into the spot

return, i.e. a return that is generated by the value change of a commodity, and the

roll return, i.e. a return that is generated by the change of risk premiums. At the

end of the day expected future returns are based on the experiences of the past.

Therefore, this section shall give an empirical overview of the risk and return profile

of historical commodity returns.

For it, we identified a small peer group including respectively a single commodity

from each commodity group as of Figure 2.1, a sub index representing each com-

modity group and the two market dominating broad indices, the DJ-AIGCI and the

GSCI.

To examine the value development of the different commodity indices we use the

annualized sample mean for a small peer group.107 Continuous returns are time ad-

ditive108 and so annualized values are reached by linear scaling of the sample mean

by the average number of observations per year. Table 5.1 shows the results for the

return components109 of the different commodity indices of our small peer group.

107To be precise: Let r1, . . . , rT be a discrete random sample of returns as of Definition C.2 attimes t ∈ 1, . . . , T. The sample mean is defined as:

r =1T

T∑t=1

rt (5.1)

108See Equation (C.5).109The single return components were separated as described in Section 4.4. Excess and spot return

series are published by the index issuers and the roll returns were calculated as of Theorem 4.1.

96


Excess Return Spot Return Roll Return

Gasoline 33.6% 29.3% 4.3%Natural Gas -16.0% 26.5% -42.5%Nickel 35.5% 32.1% 3.4%Zinc 11.9% 20.5% -8.6%Gold 9.4% 14.2% -4.8%Corn -25.7% 1.7% -27.4%Lean Hogs -13.5% 6.6% -20.1%Sugar 7.5% 9.4% -1.9%Energy Index 25.5% 29.7% -4.3%Industrial Metals Index 17.7% 20.1% -2.4%Precious Metals Index 10.3% 14.5% -4.2%Agricultural Index -14.9% 3.3% -18.2%DJ-AIGCI 12.0% 19.9% -7.9%GSCI 15.3% 22.0% -6.7%

Table 5.1: Return Components of different Commodity Indices (1998-2006)

Over the 8 year period starting in August 1998 all commodities have produced on

average a positive spot return. But because commodity investments include rolling

futures positions forward we need to take the roll returns into consideration. All

group and broad indices including more than one participant produced on average

negative roll returns. Implicating, most commodities have been in contango. Hi-

lary Till, co-founder of Premia Capital Management LLC, has investigated into the

source of steady commodity returns. In [Till 2000] she identified commodities with

statistically significant returns as these, whose underlying commodity have difficult

storage situations. For these commodities, either storage is impossible, prohibitively

expensive, or producers decide, it is much cheaper to leave the commodity in the

ground than to store it. Her findings are in line with earlier research by [Kolb 1996]

who examined 45 commodity futures contracts between 1982 and 2004. Both men-

tion soybean meal, live cattle, live hogs, crude oil, gasoline and copper to be difficult

to store and to have significant positive returns. Storage can act as a buffer. If too

little of a commodity is produced, one can draw on storage and price does not need

to ration demand. But for commodities with a difficult storage situation, ”... price

has to do a lot (or all) of the work of equilibrating supply and demand ...”.110

[Kolb 1996] showed that the average geometric excess return of the difficult to store

commodities was 3.5% over the period of 1982 to 2004. In contrast, the average

geometric excess return of the not difficult to store commodities was -4.3% over the

same period.

110See [Till 2000].

97


Morgan Stanley investigated into the relationship between excess returns and the

time a commodity spent in contango respectively in backwardation. In the presenta-

tion [Nash Shrayer 2004] they show findings regarding the existence of a weak linear

relationship between the average annualized return produced by a commodity and

the time it spend in backwardation. They examined 18 commodities over the period

1983 to 2004 and identified heating oil, live cattle, copper, crude oil and gasoline as

commodities with positive return and positive time the commodity spend on average

in backwardation.

Figure 5.1 shows the percentage time a commodity spend in backwardation plotted

against the annualized mean of its excess return as of Table 5.1. Indeed, we can also

identify a linear relationship between this two components.

Figure 5.1: Relationship between Backwardation and annualized Return

More recently, [Till Feldman 2006] extended the framework originated in the work

of [Nash Shrayer 2004]. They found that the power of backwardation to explain

commodity futures return is indeed valid, but requires the investor to have a very

long investment horizon when relying on this indicator. Specifically, they examined

soybean, corn and wheat futures over the period of 1950 to 2004. They found

that a contracts average level of backwardation only explains 25% of the variation

in futures returns over one year time frames, 42% of variation over two year time

frames, 63% of variation over five year time frames and robust 77% of variation over

eight year time frames.

All these research aims to answer the question whether commodities offer a signifi-

cant risk premium or not. This depends on how futures prices deviate from expected

future spot prices or equivalent on how high their convenience yield is. This is very

different from equities. Since the main reason to buy stocks is investment, for stocks

98


it is plausible that prices are set such that the expected return exceeds the inter-

est rate and is higher for more risky stocks. For commodity futures to offer a risk

premium, we need hedging demand to pull futures prices away from the respective

expected future spot price. For the identified difficult to store commodities there is

plausible tendency for hedgers to be predominantly on the sell side. As a result, the

expected futures return is more likely to be positive than negative.

In general, no uniform conclusion about significant excess returns can be made.

But we came to the conviction that commodity’s risk premium vary over time de-

pendent on the current and expected supply and demand situation. Moreover, the

price of commodities and therewith the realized returns move through cycles over

time caused by commodity’s consumption good facility. In periods of scarcity and

high hedging demand with high risk premiums new supply will enter the market

yielding, according to experience, into over supply periods with falling prices, low

or negative risk premiums and negative industry growth with falling supply. New

demand thrusts are firstly buffered by inventories to a certain degree but yielding

again, according to experience, in a new period of scarcity and the circle starts anew.

The current price surge came in line with high price movements over short periods,

such as recently seen in crude oil and copper markets, for instance. Therefore,

commodities are often thought to be extremely volatile. Indeed, in response to

weather related events, supply shocks, e.g. caused by news about existing reserves,

and speculative trading some commodity prices may exhibit large swings over short

periods. First research regarding this phenomena goes back to the theory of storage.

Following [Kaldor 1939] volatility is inversly related to the level of inventories. When

there are little or no inventories to buffer supply and demand disequilibriums, prices

may rise dramatically. As a consequence, rising prices and rising volatility come in

line and both are negatively correlated to the level of inventory.

Today, there exists a vast amount of literature what investigates the volatility of com-

modity futures. A statistical study performed by [Fama French 1987] on a number

of commodity futures including metals, wood and animals shows that the variance

of prices increases adversely to inventory levels. [German Nguyen 2002] investigated

worldwide soybean inventories over a 10 year period and showed that volatility can

be written as an exact inverse function of inventory. Regarding energy markets, the

property is the same and widely discussed in actuality: whenever there is a down-

ward adjustment of the estimated oil reserves in the US or another region, oil prices

and their volatility increase sharply.

99


To investigate the variability of different commodity indices we calculate the an-

nualized sample standard deviation, the minimum and the maximum of daily log

returns separately for the respective excess return (ER), spot return (SP) and roll

return (RR). The annualized sample standard deviation, denoted by σ, is calculated

as the root of the the annualized sample variance111 and gives an absolute measure

of the variability of returns to either the negative or positive side of the mean. In

Table 5.2 we represent our findings for the small peer group already known from

Table 5.1.

The first observation is that spot volatility explains the main part of excess return

volatility. The dispersion of roll returns are generally quite small in comparison

to spot volatility. To understand this we have to recover that the spot price of a

commodity is approximated by the price of the first nearby futures contract. The roll

return is made by rolling the investment from the first into the second month futures

contract and therefore, the difference between these two prices relative to the price

of the first nearby futures contract, e.g. the spot price. This difference depends

of the shape of the forward curve. As we have already seen in Section 5.1.1, the

shape of the forward curve is an expression of the current and expected supply and

demand equilibrium. The rolling periods of our sample are monthly and therewith

short term orientated. Caused by the small time difference between the first and

the second nearby futures contract, sudden extreme events will effect both prices

and rolling over the investment will create only small roll returns. The described

phenomena can be seen in the copper example of the previous section. Going back

to Table 4.5 we see a huge sudden spot price surge during March and April from

5,440 US dollar per contract to 7,118 US dollar per contract. But the price of the

second month contract was influenced in the same way. From Table 4.6 we take

a small roll return of 0.3% in this month. This observation goes in line with the

Samuelson effect well known and often analyzed in commodity related research,

e.g. [Samuelson 1965] and [Anderson Danthine 1983]. The Samuelson effect is called

the property of commodity price volatility to decrease with increasing maturity. It

111To be more precise: Let r1, . . . , rT be a discrete random sample of returns as of Definition C.2at times t ∈ 1, . . . , T and r be the sample mean as in Equation (5.1). The sample variance isdefined as:

σ2 =1

T − 1

T∑t=1

(rt − r)2 (5.2)

Annualized values are calculated by scaling linear with the average number of observations peryear because continuous returns are time additive. The square root of the sample variance:

σ =√

σ2 (5.3)

is called the sample standard deviation.

100


Return Std. Deviation Minimum Maximum

ER 38.2% -12.8% 11.2%Gasoline SP 38.3% -12.8% 11.2%

RR 5.4% -2.5 1.9ER 55.0% -16.7% 18.8%

Natural Gas SP 55.5% -16.7% 18.8%RR 7.3% -7.7% 2.0%ER 36.0% -18.3% 13.6%

Nickel SP 35.5% -18.2% 12.4%RR 4.9% -4.7% 10.8%ER 23.3% -8.9% 8.9%

Zinc SP 23.1% -9.0% 8.9%RR 2.0% -2.1% 4.1%ER 16.4% -7.6% 8.8%

Gold SP 16.3% -7.6% 8.8%RR 1.1% -1.4% 1.4%ER 22.2% -5.3% 6.5%

Corn SP 22.7% -5.3% 6.5%RR 4.3% -3.3% 3.0%ER 27.5% -7.4% 6.9%

Lean Hogs SP 30.4% -12.2% 11.8%RR 11.3% -5.4% 7.9%ER 32.9% -9.3% 8.4%

Sugar SP 33.3% -9.3% 8.4%RR 4.5% -2.5% 2.5%

ER 33.3% -14.4% 8.0%Energy Index SP 33.3% -14.4% 8.0%

RR 3.5% -3.7% 2.8%ER 19.5% -9.0% 7.6%

Industrial Metals Index SP 19.1% -9.1% 7.6%RR 2.5% -2.3% 4.4%ER 16.3% -8.3% 8.5%

Precious Metals Index SP 16.3% -8.2% 8.5%RR 1.2% -1.5% 1.5%ER 17.0% -10.5% 8.6%

Agricultural Index SP 17.4% -12.5% 9.8%RR 4.3% -2.9% 5.0%

ER 15.0% -4.3% 4.8%DJ-AIGCI SP 15.2% -4.3% 4.8%

RR 1.9% -2.1% 0.8%ER 22.7% -9.2% 6.5%

GSCI SP 22.7% -4.3% 4.8%RR 2.2% -2.3% 1.6%

Table 5.2: Volatility Components of different Commodity Indices (1998-2006)

is explained by the fact that the arrival of news (e.g. on inventories) will have an

immediate impact on short-term futures prices, while long-term contract prices tend

to remain unchanged since production adjustments are likely to take place before

the contracts come to delivery at maturity.

The second observation is regarding the dispersion of the different commodities.

They differ not only among each other, but also among each commodity group as

the sub indices tell. Moreover, the annualized standard deviations range from as

much as 55.0% for natural gas to as low as 16.4% for gold. Therefore, general

statements about the ”high volatility” implicating high risks for investors cannot

be supported. [Kat Oomen 2006] examined the development of commodity return

101


volatility during different periods of the business cycle over a period of 1965 to 2005

and conclude, that changes in the dispersion level can be observed. Especially oil’s

and oil product’s prices react differently in different business cycle periods. During

recessions they tend to be high volatile and at the beginning of an expansion phase

their variability tend to decrease. Moreover, they report that most commodities

including i.e. oil and oil products, silver, platinum, copper, soybeans, cocoa and

corn tend to be more volatile when the forward curve is in backwardation. This

is not surprising when interpreting backwardation as an indication of scarcity that

usually is followed by price surges and as described above this is positively related

to volatility increases.112

Third, sub indices exhibit in general smaller standard deviations as their partici-

pants. This might be an indication for diversification effects and our guess is under-

lined by the small standard deviation of the broad indices.

5.1.2 Correlation

Correlation113 is the normalized covariance114 of two variables and measures their

co-movements in a range of plus to minus one. Strong positive correlation indicates

that upward movements in one returns series tend to come in line with upward move-

ments in the other, and similarly, strong negative correlation indicate that downward

movements of the two series tend to go together. To measure the strongness of the

relation someone can calculate the so-called correlation coefficients including Pear-

son’s, Kendall’s and Spearman’s. They identify links between two variables in a

range of plus to minus one. A positive value indicates a positive relationship and

vice versa, a negative value identifies a negative connection. Absolute higher values

indicate stronger co-movements.

Looking at historical correlations is aiming to answer the question whether the con-

sidered investment universe is homogeneous or heterogeneous. Or, alternatively as

[Erb Harvey 2006] state, ”is the commodity market a collection of securities that

behave in a similar way, or is the market a collection of dissimilar securities?” While

introducing the different commodity types in Section 2.1 we already uncovered some

dependence structures between the single commodities. In Appendix B more exam-

ples can be found, including a strong connection between the row commodity and

its downstream products, e.g. oil and heating oil in Appendix B.1 or the inter-

dependencies in the soybean complex in Appendix B.11. We saw a link between

112Compare Table 6.5.113For the formal mathematical definition see Definition C.15.114For the formal mathematical definition see Definition C.14.

102


macroeconomic influences and commodities, e.g. gold in Section 2.1.2.1, the be-

havior of prices when commodities can serve as a complement or substitute for each

other, e.g oil and natural gas and finally, we saw adverse price movements in livestock

markets when grains markets move in a respective way in Section 2.1.3.3.

To investigate the co-movement of futures returns over the last years in detail we

will analyze Person’s and Kendall’s correlation coefficient. The first one is generally

the most well known and most used one. Unfortunately, it is restricted to the

discovery of linear relationships between variables and therefore, situations with

non-linear dependence structures keep covered. Nevertheless, to get a first idea of

the interactions between commodity returns the Pearson correlation coefficient is

helpful and for sample returns calculated as follows:

Definition 5.1 Pearson Correlation

Let r1, . . . , rT and l1, . . . , lT be two discrete random samples of different returns

at times t ∈ 1, . . . , T and r respectively l be the sample mean as of Equation 5.1.

The sample Pearson correlation coefficient is defined as:

ρ =1

T−1

∑Tt=1(rt − r)(lt − l)√

1T−1

∑Tt=1(rt − r)2

√1

T−1

∑Tt=1(lt − l)2

(5.4)

Our findings regarding the spot and the roll returns are documented in Table 5.3.115

A correlation matrix is always symmetric because it makes no difference whether to

take the correlation between e.g. nickel and zinc price movements or zinc and nickel

price movements. If two commodity prices move statistically independent then a

good estimate of their correlation should be insignificantly different from zero high-

lighted with a brown color in Table 5.3.

Examining our findings we realize higher similar price co-movements among com-

modity groups than between them. But still, the price movements of commodities

among a group are imperfectly correlated and an investor can improve investment

characteristics by spreading his wealth by investing in sub instead of single com-

modity indices. Figure 5.2 illustrates the phenomena in the risk return space. The

115Yellow values are significant at the 1% alpha level, blue values are significant at the 5% alphalevel and brown values are insignificant. We tested the null hypothesis of zero correlation andused the t-test with the following test statistic:

t =ρ√

1− ρ2∗√

T − 2 (5.5)

It follows a Student’s t-distribution with T − 2 degrees of freedom. For further details ofhypothesis testing see Section 5.2.3, [Bamberg Baur 2002] or [Kanji 1999].

103


Table 5.3: Pearson Correlation (1998-2006)

commodity group indices are denoted by a circle and some of its constituents by a

quadrat or rectangle. The dependence of the single commodities to the respective

sub index is highlighted with a black oval.

Figure 5.2: Diversification between single commodity groups

Someone realizes that the investment in sub indices is more attractive than in single

commodities. Caused by the imperfect correlation between commodity group mem-

bers, strong price movements of one commodity are balanced by moderate price

104


movements of another. Implicating, the volatility of the whole group decreases and

an investment that may not produce more revenues is still more attractive because

of its lower risk structure.

However, Definition 5.1 uncovers the problem with the Pearson correlation. Only

the first two moments of a distribution are involved to measure the degree of depen-

dence. Only the normal distribution family is wholly explainable with its first two

moments. Thus, two variables could have zero correlation and still be related over

higher moments. Therefore, we take Kendall’s correlation coefficient also known as

Kendall’s tau additionally into consideration. It measures the degree of an arbi-

trary monotone relationship between two variables without any assumptions such

as linearity or distribution. Moreover, it is robust against outliers.

To calculate Kendall’s tau for two return series with observations at the same dates,

the value of each observed return per index denoted with rt respectively lt is tacked

with its rank relative to the other return observations in the sample, i.e. with

1, 2, . . . T . Now all values come from the uniform distribution of numbers 1, 2, . . . T .

If all rt respectively lt have different values, each number is found exactly one time.

If some observations of rt respectively lt have the same value, they get an average

rank. In either case, the sum of all assigned ranks is equal to the sum of numbers

1, 2, . . . T namely(

12T (T+1)

). Kendall’s tau uses the relative order of the ranks to

identify correlations: the rank is higher, lower or equal if the values are higher, lower

of equal. Therefore, it checks for all t = 1, . . . , T whether

1. a pair is concordant at time t, i.e. rt > rt+1 and lt > lt+1 and the number of

concordant states over the whole period is denoted with nc

2. a pair is discordant at time t, i.e. rt < rt+1 and lt < lt+1 and the number of

discordant states over the whole period is denoted with nd

3. there is a tied observation, i.e. rt = rt+1 and lt = lt+1 and the number of

tied r-observations is denoted by ntr and the number of tied l-observations is

denoted by ntl

Definition 5.2 Kendall’s Correlation


at times t ∈ 1, . . . , T and recall the definitions of concordant, discordant and tied

observations above. The total number of possible pairings of rt and lt is(

T (T−1)2

).

If there are no tied observation the Kendall correlation coefficient is defined as:

τ =nc − nd

T (T − 1)/2(5.6)

105


If there are tied observation the Kendall correlation coefficient is defined as:

τ =nc − nd√(

T (T−1)2

−∑ntr

i=1ntr,i(ntr,i−1)

2

)(T (T−1)

2−∑ntl

i=1ntl,i(ntl,i−1)

2

) (5.7)

Our findings are documented in Figure 5.4.116 As someone can see, Kendall’s tau is

also standardized to a range minus to plus one.

Table 5.4: Kendall Correlation (1998-2006)

Table 5.3 and Table 5.4 and show the respective correlation coefficients for the two

sources of excess return, i.e. for the spot return (SP) and the roll return (RR). We

have already seen in Section 5.1.1 that the main part of investor’s return is driven by

the spot price movements and therefore, it explains the main part of co-movements

between different commodities.117 Confirming statements in literature including

116Yellow values are significant at the 1% alpha level, blue values are significant at the 5% alphalevel and brown values are insignificant. We tested the null hypothesis of zero correlation andused the Kendall rank correlation test with the following test statistic:

t =τ√

T (T − 1)(2T + 5)/18(5.8)

It follows a standard normal distribution. For further details of hypothesis testing seeSection 5.2.3, [Bamberg Baur 2002] or [Kanji 1999].

117Indeed, the correlations among the excess returns do not differ relevantly from the correlationsamong the spot returns. To avoid redundance, they are not explicitly reported.

106


[Kat Oomen 2006a] and [Erb Harvey 2006], there exists a high dependence within

commodity groups, e.g. between gasoline and natural gas or between nickel and

zinc, but a small dependency between the groups what is confirmed by the small

correlation coefficients not only between the single commodities of different groups

but also between the different group commodity indices. Especially agricultural

products exhibit price movements that are not in line with price movements of the

other commodity groups. To visualize this phenomena we plotted in Figure 5.3 in

the left diagram the return observations of nickel against the return observations of

zinc and in the right diagram the return observations of the industrial metals index

against the observations of the agricultural index. The red line gives an idea of how

100% correlation of two variables would look like.

Figure 5.3: Linear Correlation within and between Commodity Groups (1998-2006)

Comparing Pearson’s and Kendall’s correlation coefficient we realize partly high dif-

ferences. Especially the high linear correlations over 0.2 including the correlation

between gasoline and natural gas, nickel and zinc and industrial and precious metals

are only half that high in Kendall’s scale indicating that the first two moments are

not enough to describe the relationship between commodity price movements.

In financial markets, the existence of non-linear dependence structures between re-

turns is well known and implicating the fact, that correlations may not be an appro-

priate measure of co-dependence. Nevertheless, correlation is related to the slope

parameter of a linear regression model.

107


Definition 5.3 Linear Regression Model


at times t ∈ 1, . . . , T, α and β two constant inR and εt be an error term of identical

and independent distributed random variables with mean zero and variance σ2ε . Let

rt denote the dependent variable and the lt the independent. A linear regression

model is given by the equation of a line:

rt = α + βlt + εt (5.9)

To fit such line through actual observed values we need an estimated line, denoted

by rt = α + βlt where α and β denote the estimate of the line intercept α and the

slope β. The residuals are defined as et = rt − rt. Then, the actual data points are

the fitted model plus the residuals: rt = α + βlt + et.

It is logical to choose a method of estimating these parameters that in some way

minimizes the residuals, since then the predicted values of the dependent variable

will be closer to the observed values. Choosing estimates to minimize the sum of

the residuals will not work, because large positive residuals would cancel out large

negative residuals. The sum of the absolute residuals could be minimized, but the

mathematical properties of the estimators are much nicer if we minimize the sum of

the squared residuals. This is called the ordinary least squares (OLS) criteria.

Theorem 5.1 Estimates for the Linear Regression Model

The ordinary least squares estimates for the linear regression model of Definition 5.3

α and β are given by:

β =

∑Tt=1(rt − r)(lt − l)∑T

t=1(rt − r)2(5.10)

or β = 0 if rt = r ∀t ∈ 1, . . . , T and

α = r − bl (5.11)

Whereby r and l denote the sample means as of Equation (5.1).

Proof: Following the idea of OLS estimation we need to minimize the sum of

squared residuals, i.e. the following equation:

L(α, β) =T∑

t=1

[rt − (α + βlt)]2

To find an optimum we need to set the first derivation regarding α and β zero and

108


solving for the respective unknown.

∂

∂αL(α, β) =

T∑t=1

−2rt + 2(α + βlt)

=T∑

t=1

−rt + nα + β

T∑t=1

lt ≡ 0 (5.12)

∂

∂βL(α, β) =

T∑t=1

−2rtlt + 2(α + βlt)lt

=T∑

t=1

−rtlt + αT∑

t=1

lt + βT∑

t=1

l2t ≡ 0 (5.13)

With (5.12) = (5.13) there exists a unique solution given in 5.10 if and only if not

all values of lt are equal. Putting the solutions into the respective second partial

derivations shows that the solutions yield indeed to a minimum.118

2

Comparing Equation 5.10 and Equation 5.4 someone can see the connection between

the correlation coefficient and the slope parameter of linear regression model:

β =

∑Tt=1(rt − r)(lt − l)∑T

t=1(rt − r)2

=1

T−1

∑Tt=1(rt − r)(lt − l)

1T−1

∑Tt=1(rt − r)2

=1

T−1

∑Tt=1(rt − r)(lt − l)√

1T−1

∑Tt=1(rt − r)2

√1

T−1

∑Tt=1(lt − l)2

∗

√1

T−1

∑Tt=1(lt − l)2√

1T−1

∑Tt=1(rt − r)2

= ρ ∗

√1

T−1

∑Tt=1(lt − l)2√

1T−1

∑Tt=1(rt − r)2

(5.14)

This concept is well known in equity markets and used in the Capital Asset Pricing

Model (CAPM) to measure the variability of stock returns in dependence of the

variability of the market index. Figure 5.4 shall give a visual impression of the de-

pendency of the GSCI as market index to the energy market.119

118For further details see [Bamberg Baur 2002]119The goodness of fit parameter R2 =

PTt=1(rt−r)2PT

t=1(rt−r)2= 0.93 and therewith relatively high. The

beta coefficient is significant at the 1% alpha level. For an introduction of significance testsregarding regression coefficients see e.g. [Bamberg Baur 2002].

109


Figure 5.4: Dependence of Market Index (1998-2006)

It became clear how high the GSCI returns depend on returns of the energy mar-

ket. This is not astonishing because recall Figure 4.7 that uncovers the high over-

weighting of the GSCI in favor of energy products.

Summing up, the above results show that the commodity market is not a homoge-

neous but a heterogeneous universe. As Section 2.1 let us already guess the mar-

kets consists of dissimilar assets that depend on their own price influencing factors.

Therefore, spreading its wealth among more than one commodity can increase in-

vestors performance. Especially the low correlation between the different commodity

groups is attractive and provides diversification benefits.

5.1.3 Diversification

As [Campbell 2000] stats, many economists pronounce that ”there is no such thing

as a free lunch”, but finance theory does offer a free lunch: the reduction in risk

that is obtainable through diversification. An investor who spreads his wealth among

different investments can reduce the volatility of his portfolio, provided only that the

underlying investments are imperfectly correlated. Because this has no impact on

the return of a portfolio, there is ”no bill for a lunch”. Still, many investors ignore

diversification possibilities and are overinvested in favor of selected asset such as the

stocks of their own country instead of diversifying internationally. They might feel

110


that the volatility reduction is small: ”the free lich is so meagre that it is nor even

worth lining up at the buffet table”.

The mathematical explanation for the phenomena diversification given in Theorem

5.2 in comparison to the results from Section 5.1.2 highlight the substantial benefit:

Theorem 5.2 Diversification

Consider a portfolio with n equally weighted assets and denote its weights with

x∗(n) = ( 1n· · · 1

n)T ∈ Rn. Let C = (cij)i=1...n,j=1...n be the n× n symmetric co-

variance matrix of the n assets. Then the portfolio variance σ2(x∗(n)) converges

forn −→∞ against the average covariance σc ∈ R:

σ2(x∗(n)) = σc +1

n(σ2 − σc) −→ σc for n −→∞ (5.15)

Proof: Let x define the asset weights of a portfolio. The portfolio variance is

than defined as:

σ2(x) = xT Cx

=n∑

i,j=1

xixjcij

=n∑

i=1

x2i σ

2i +

n∑i=1

n∑j=1,i6=j

xixjcij

Now put x = x∗(n) = ( 1n· · · 1

n)T ∈ Rn and it follows:

σ2(x∗(n)) =n∑

i=1

x∗i (n)2︸︷︷︸1

n2

σ2i +

n∑i=1

n∑j=1,i6=j

x∗i (n)︸︷︷︸1n

x∗j(n)︸︷︷︸1n

cij

=1

n(1

n

n∑i=1

σ2i )︸︷︷︸

≡ σ2 average variance

+n− 1

n(

1

n(n− 1)

n∑i=1

n∑j=1,i6=j

cij)︸︷︷︸≡ σc average covariance

limn→∞= σc +1

n(σ2 − σn)︸︷︷︸

0

= σc

what is a constant independent of n

2

111


Recall, correlation is defined as the normalized covariance. Following, the Theorem

shows that creating a portfolio of assets that are imperfectly correlated among each

other provides diversification because the portfolio variance is balanced. The average

covariance represents a lower bound, portfolio variance goes against with increasing

constituents.

[Campbell 2000] examined the stock market to investigate the phenomena. He

showed that individual stocks are getting more volatile over time but their cor-

relation to other stocks is fallen. Therefore, an investor needs to avoid concentrated

portfolios more than ever and identified around 25 to 30 stocks to be sufficient for

optimal diversification. In commodity markets Deutsche Bank first investigated di-

versification effects in 2003.120 They examined the 24 commodities of the GSCI and

found 5 constituencies to be sufficient for optimal diversification yielding into the

composition of the DBLCI. Viewing Figure 5.5 we identify the DJ-AIGCI as the

optimal investment vehicle to get broad diversified commodity exposure.

Figure 5.5: Diversification among commodity groups

An investor who needs to limit its downside as many institutional investors, is more

attractive to get a low risk portfolio producing a predefined excess return over cash

return than to maximize the upside. As we have already seen in Section 5.1.2

diversification effects are higher between commodities of different commodity groups

than between commodities of the same group. In Figure 5.5 this is visualized through

the left position of the broad diversified indices relative to the sub indices labeled

again with a circle already known from Figure 5.2. In Section 4.2 we investigated

the differences of the major commodity indices and identified differences in their

construction. They differ from the amount of constituents until their weights. Recall,

120See [Wilshire Research 2005].

112


in Figure 4.7 we summed up the weights of the single commodities and sorted

them into the buckets energy, industrials121 and agricultures to identify the degree

of diversification among the three bid commodity groups. Independently of the

weighting procedure or the number of constituencies of an index we realized an

over-weighting of the GSCI, the DBLCI and the RICI to the energy sector and an

over-weighting of the CRB index to the agricultural sector. Only the DJ-AIGCI has

a balanced distribution of its weights among the three commodity groups. Recalling,

its weighting procedure is linked to world’s production of a commodity but also

bounders the single weights to be higher than 2% and lower than 33%. Examining

Figure 5.5 we realize that this procedure pays off and the index participates best

from the different statistical properties of its single constituencies.

Summing up, portfolio’s mean is the average of its underlying investment’s returns

and portfolio’s variance is the average of its underlying covariances. This explains

why it is better to invest in broad than in single group commodity indices. With

the exception of the agricultural group, we have seen that commodity groups are

generally homogeneous yielding into low diversification benefits. Only the softs

and therewith the agricultural group behaves heterogenous among its constituen-

cies. The disadvantage of this group are lower returns and high dependencies to

unpredictable and uncertain weather conditions. Broad indices bring the homoge-

nous groups that are heterogenous among each other together yielding into strong

diversification effects. But only if the index has no excess weights in favor of one

commodity group maximal diversification benefits can be gathered.

In Section 5.1.2 we have already seen that the energy group is dominant not only

in producing returns but also in having high correlations to all other commodities.

This dominance together with the excess weights in favor to energy of the GSCI gave

birth to very strong co-movements of the two return series’. As we have seen, the DJ-

AIGCI does not have this excess weights but still, Table 5.3 and Table 5.2 report

high correlation between this return series’. To get a deeper inside into the risk

factor structure underlying the returns of the different commodity groups included

into the DJ-AIGCI we performed a so-called factor analysis. The main purpose of

the factor analysis is to decompose a data matrix R consisting of different return

time series into specific (U) and common factors (F ). It is very similar to a more

dimensional linear regression model. The difference is that in a regression model

the regressors are explicitly given, in a factor model the regressors are implicitly

extracted from the data matrix. The degrees, known as loadings, the common or

specific factors influence the return series’ can simultaneously be used to decompose

121This bucket includes metals but also commodities like rubber and cotton.

113


the correlation structure existing among the time series’. Therewith, identifying

common risk factors and their degree of influence comes in line with decomposing

the underlying risk structure. Theorem 5.3 gives the mathematical formulation of

the model.

Theorem 5.3 Model of Factor Analysis

Let (R)t=1,...,T ;j=1,...,n ∈ R(T×n) be the data matrix which includes column-wise the

discrete random samples r1j, . . . , rTj for j = 1, . . . , n different assets at times

t ∈ 1, . . . , T. Denote with rj, j = 1, . . . , n, the assets’ specific sample mean as

of Equation (5.1) and with σj , j = 1, . . . , n, the assets’ specific sample standard

deviation as of Equation (5.3).

Define the normalized variables ztj =rtj−rj

σjand put them into the normalized data

matrix (Z)t=1,...,T ;j=1,...,n ∈ R(T×n). Furthermore, define the matrix (F )t=1,...,T ;l=1,...,k ∈R(T×k) coding the normalized common factors and put its weights aj,l with j =

1, . . . , n and l = 1, . . . , k into the matrix A ∈ R(n×k). Additionally, define the matrix

(U)t=1,...,T ;j=1,...,n coding the normalized specific factors and put its weights dij 6= 0 if

i 6= j and zero otherwise for i, j = 1, . . . , n into the matrix D ∈ R(n×n).

The Factor Analysis decomposes the standardized data matrix in the following form:

Z = FAT + UD (5.16)

Denote the correlation matrix with (P )i=1,...,n;j=1,...,n ∈ R(n×n) including the Pearson

correlation coefficients between the n assets as of Definition C.15. With the repre-

sentation of Z in Equation (5.16), the correlation matrix has the following form:

R = AAT + DD (5.17)

Proof: Denote with I the identity matrix, i.e. a symmetric matrix with ones at

the main diagonal and zeros otherwise. With the notions of Theorem 5.3 the single

elements of the standardized data matrix shall have the following representation for

t = 1, . . . , T and j = 1, . . . , n:

ztj =k∑

l=1

ajkFl + djUij

In matrix notion:

Z = FAT + UD

114


We assume that:

the normalized specific factors are uncorrelated among each other, i.e.

1

nUT U = I (5.18)

the normalized specific factors are uncorrelated to the normalized common fac-

tors, i.e.

UT F = 0 (5.19)

the normalized specific factors are uncorrelated among each other, i.e.

1

nF T F = I (5.20)

With Definition C.15 follows:

R =1

nZT Z

(5.16)︷︸︸︷= (FAT + UD)T (FAT + UD)

= F T F︸︷︷︸= I︸︷︷︸(5.20)

AAT + (UT F︸︷︷︸= 0︸︷︷︸(5.18)

)T (AT DT )T + UT F︸︷︷︸= 0︸︷︷︸(5.19)

AT DT + UT U︸︷︷︸= I︸︷︷︸(5.18)

DT D

= AAT︸︷︷︸common normalized variance

+ DD︸︷︷︸specific normalized variance

Statistical estimates are generated using, e.g. maximum likelihood methods. Further

explanations are out of the scope at this point and can be found among others e.g.

in [Bamberg Baur 2002].

2

The visualization of our results are shown in Figure 5.6 where we plotted the com-

mon factor loadings that influence the respective return series and therewith its

normalized variance most.

We extracted three common risk factors implicitly from the data. The major chal-

lenge is the interpretation of these factors. In Figure 5.6 we see that the grains,

agricultural, softs and non-energy sub index stick out in one direction resulting in

our conclusion that this might indicate a common risk factor symbolizing the agri-

115


Figure 5.6: Factor Analysis (1991-2006)

cultural risk. Furthermore, the precious metals, non-energy and industrial metals

sub index protrude in the same direction. We deduce that this indicates a common

risk factor embodying the risk related to metal’s investments. Finally, the energy

and the DJ-AIGCI index show in the one direction no other sub index stick out to.

We infer that energy is the last separate risk factor driving mainly the DJ-AIGCI

returns, although the index has no extra weight in favor to this commodity group.

Therefore, the analysis underlies the impact of the energy market to the whole com-

modity market.

Summing up, this section has shown that putting different commodities into a port-

folio disembogues into diversification effects providing the investor with attractive

risk and return profiles. Nevertheless, the energy market has the strongest influ-

ence to the whole commodity market in comparison to the metals and agricultural

markets. To show this, we used two different types of analysis: an explicit and an

implicit one. Both produced the same result. Therefore, it is proofed how impor-

tant broad diversified commodity exposure is and that excess weights in favor of one

commodity group can cause unidirectional risk profiles.

5.2 Properties of the DJ-AIGCI Return Components

The last section has shown that commodity investing cannot be seen as a homoge-

nous one but is driven by different risk and influencing factors. Finally, we deduced

116


that broad diversified commodity indices represent the optimal vehicle to get com-

modity exposure. Diversification effects cause better risk and return profiles in

comparison to single commodity’s or commodity group’s. We identified the DJ-

AIGCI as the index that weights are best balanced among the commodity groups

under consideration.

The following lines will introduce the specifications of this index’ returns. Starting,

we will first give a brief overview of the performance and return characteristics in

Section 5.2.1. Therefore, we divided the DJ-AIGCI total return into its elements as

of Figure 4.8 and show the value and risk propositions of the single elements. Recall

from Section 4.4, the elements included interest rate, spot and roll return. Second,

we will analyze the roll returns embodied in a DJ-AIGCI investment’s return in

Section 5.2.2.

The main purpose of analyzing returns is to identify their distributional properties.

This is needed to assume their future behavior and especially, their behavior in the

portfolio context. Therefore, Section 5.2.3 will give a brief overview of this analysis.

A more and more popular getting research field in statistics is time series analysis

and based on its results time series modeling of returns. Generally, these models are

constructed as regression models which take historical values as regressor. To do so,

someone needs two major characteristics, stationarity and autocorrelation. The first

one examines the change of the distribution characteristics over time. Our findings

are presented in Section 5.2.4. Autocorrelation investigates the linear dependence of

returns following on each other. This analysis is reported in Section 5.2.5.

5.2.1 Key Statistics

The DJ-AIGCI was introduced at 01.01.1991 with a starting value of 100.122 The

DJ-AIGCI manual [DJAIGCI 2006] gives inside into the calculation methodology

that is in line with our explanations in Section 5.1.1 about the construction of a

commodity index. As already mentioned in Section 4.2 the DJ-AIGCI is available

for investment in two different types, the excess and the total return index, and it

is additionally calculated as spot return. Therewith, it is possible to analyze the

single return elements of the index as of Figure 4.8. The calculation followed the

simple scheme: first, we calculated the log returns of the time series published in

Bloomberg. Subtracting the excess from the total return identified the interest rate

component and subtracting the spot from the excess return identified the roll return

122From this point on we use data series starting at 01.01.1991 to cover the whole history of theindex.

117


component. To show the development of the single elements and its value propo-

sition we calculated price series following Definition C.3 and plotted the results in

Figure 5.7.

Figure 5.7: Performance of DJ-AIGCI Components

Comparing the price series it becomes clear that although commodity spot prices

have performed extraordinary over the last years, an investor could not participate

fully because he had to bear the negative roll returns caused by shifting the in-

vestment forward over time, i.e. rolling futures contracts. But if an investor had

gone into the collateralized version of the index, he would have been better off. The

interest rate return was that high that it over compensated the negative roll returns

and an investor could have picked up the extraordinary spot return development of

the last years. The question to answer is, which type of return either total or excess

has to be considered when it comes to commodity investment. If an investor wants

to get commodity exposure as actual part of his portfolio, he needs to consider the

total return. He takes a certain amount of his wealth and directly invests it. Be-

cause this is not possible in futures contracts, he puts the money as collateral to his

futures engagement in a risk free interest earning instrument. Recall, this is done

by all mutual funds tracking a certain commodity index. The other alternative is,

that an investor wants to deal with the leverage effect, futures investments provide

by the minimal cash requirements only used to serve possible margin calls. Then he

has to consider the excess return as the actual pure commodity return. Following

Theorem 4.1 and because of the publishing of the DJ-AIGCI spot return index it is

quite easy to divide the excess return into its elements: excess, spot and roll return

as shown in Figure 5.8.

118


Figure 5.8: Return Behavior of DJ-AIGCI Components

As mentioned in [DJAIGCI 2006], rolling takes place monthly around the eighths

business day. The futures are rolled forward in 20% portions over a five day period.

This procedure can be observed in the resulting roll return series noticeable by

clearly different from zero returns over the five day rolling period. Because roll

returns are a seasonal phenomena once a month they show different patterns than

the spot or excess returns as it can be seen in Figure 5.8.

Closing this section, the key figures of the commodity return elements are summa-

rized in Table 5.5 and the above mentioned became clearer in real values: although

spot commodity prices produced on average an annual return of 6.84%, an investor

realized only 54% of total namely 3.72% per annum. The additional 3.12% per an-

num were eaten up by negative roll returns. But taking DJ-AIGCI exposure over a

collateralized investment would have produced on average 7.63% per annum.

Total

Return

Excess

Return

Spot

Return

Roll

ReturnAnnualized arithmetic mean 7.63% 3.70% 6.62% -2,85%Total value gain 228.88% 78.03% 174,23% -35,85%Annualized standard deviation 12.66 12.66% 12.74% 1.71%Minimum (daily) -9.15% -9.17% -9.17% -0.58%Maximum (daily) 4.85% 4.82% 4.82% 2.16%Mean (daily) 0.03% 0.01% 0.03% -0.01%Median (daily) 0.04% 0.03% 0.04% 0.00%99% VaR -2.03% -2.04% -2.04% -0.41%95% VaR -1.24% -1.26% -1.26% -0.18%

Table 5.5: Key Statistics of DJ-AIGCI Components

The minimum and maximum values show the total dispersion of the returns. In

119


comparison to excess and spot returns, roll returns are small. This can also be

seen in Figure 5.8 by comparing the scales123 and the huge difference in annualized

standard deviations underlies the statement. Because the mean and the standard

deviation are sensitive against outliers we calculated the median and the Value at

Risk (99% and 95%). The median is defined as the middle value of a series, i.e.

sorting the members of series ascending, the median is the value that lies in the

middle so that 50% of the series’ members are smaller and 50% are bigger than

the median. Its formal definition is given in Definition C.17. The Value at Risk is

actually the quantile of the return distribution as defined in Definition C.18.124 The

99% VaR this value that only 1% of the return series is smaller than the 99% VaR

and 99% of the return series are bigger than the 99% VaR. In the same way, the

95% VaR is this value that only 5% of the return series are smaller than the 95% VaR

and 95% of the return series are bigger than the 95% VaR. Taking the minimum

daily log return of -9.17% in comparison to the VaR values we realize that this was

really an unusual outlier and that in 99% of the days over the last 15 years the

negative returns didn’t fall below -2.04%.

5.2.2 Roll Returns

As we have seen in the previous section roll returns are a major reason why in-

vestors couldn’t participate wholly on the commodity price surge of the last years.

Figure 5.9 shows the negative performance of the DJ-AIGCI roll return in larger

scale than in Figure 5.7. In this large scale the rolling periods can better be seen

and we clearly observe that roll returns follow a jump process caused by their sea-

sonal occurrence monthly.

Negative roll returns come in line with a time when the majority of the underlying

commodities are in contango and positive roll returns occur when the majority of the

underlying commodities are in backwardation. Figure 5.9 shows that longer periods

of backwardation are followed by longer periods of contango and that the negative

returns in contango periods are higher than the positive returns in backwardation

periods. This explains the wavelike downwards move of the performance line. At

the moment discussions are coming up that speculate about a synthetic created

contango caused by a similar rolling procedure of the major commodity indices. To

investigate this problem we show in Figure 5.10 the percent of time the DJ-AIGCI

123Because the minimum return of -9.17% occurred as a stand alone outlier at the inception of theDJ-AIGCI, we cut of the value for better observability of the general return development.

124For a detailed discussion of the VaR see [Zagst 2002].

120


Figure 5.9: Performance of DJ-AIGCI Roll Returns

spent in contango versus the time it spend in backwardation. Indeed, there is a

small trend that the time of contango is increasing. But this is not a phenomena

created during the last years it seems to be a steady process of 4% growth over a

five year period.

Figure 5.10: Time the DJ-AIGCI spent in Contango or in Backwardation

Matt Schwab, a managing director in the investor coverage group at AIG Financial

Products, mentioned in an interview, the people who are involved in the creation and

maintenance of the DJ-AIGCI are aware of the fact that commodities spent histor-

ically more time in contango than in backwardation. Moreover, ”when institutions

ask me if passive flows are causing the contango and hurting index performance,

we highlight the fact that at the end of 2005, passive money was just 3% of the

size of the overall over-the-counter commodity derivatives market.”125 In contrast to

125See [Risknet 2006].

121


Deutsche Bank who changed their rolling procedure into a dynamic optimum yield

one, many investors are not interested in these kind of trading strategy. John Bryn-

jolfsson, head of the Pimco Real Return Commodity Strategy Fund126 stated in an

interview: ”Aside from missing the liquidity that is present in the front-end month,

having an index that can make or lose money by extending to different calendars is

a relatively speculative process that certainly should not be part of a passive index

definition strategy.”127

Because roll returns are zero during the non rolling periods and this is the main time

during a month, the zero return is the dominant one as shown in the left diagram

of Figure 5.11.

Figure 5.11: Distribution Change

We plotted on the left side a histogram128 of the real roll returns and on the right

side we plotted the pure roll returns, i.e. the roll return that actually occurred dur-

ing the rolling periods. Of the original 3902 daily observations, only 908 data points

are left taking only the pure roll returns into consideration. This has the advantage

that we can separately analyze the contango and the backwardation times of the

market, i.e. how are positive and negative roll returns distributed. As the right

diagram in Figure 5.11 clearly shows negative returns occurred historically more of-

ten than positive ones, i.e. the bars on the negative side of the diagram are higher

than the bars on the positive side. But high irregular outliers can be found more

126Recall, the fund was introduced in Section 4.3 and is by far the biggest commodity mutual fundon the market. It tracks the DJ-AIGCI.

127See [Risknet 2006].128Further details about the contraction of a histogram see Section 5.2.3.

122


often on the positive side of the distribution, i.e. the distribution has a long right tail.

Finally, Table 5.6 shows the key statistics for both, the actual and the pure roll

returns.

Roll Return Pure Roll Return

Annualized arithmetic mean -2.85% -2.85%Total value gain -35.85% -35.85%Annualized standard deviation 1.71% 1.68%Minimum (daily) -0.58% -0.58%Maximum (daily) 2.16% 2.16%Mean (daily) -0.01% -0.05%Median (daily) 0.00% -0.04%99% VaR -0.41% -0.48%95% VaR -0.18% -0.40%

Table 5.6: Key Statistics of DJ-AIGCI Roll Return

It can clearly be seen that mean and median are negative in pure roll returns what

additionally underlies the statement that commodities where historically more of-

ten in contango than in backwardation. Because the data population decreased, the

VaR values have more explanatory power and are not biased to zero.

In the following section we will switch between the actual and the real pure returns

depending on the analysis. It makes no sense to investigate in Section 5.2.3 the

distribution of the actual roll returns because as it can be seen in the left diagram

of Figure 5.11 the distribution is too much biased to zero. But on the other hand,

it makes no sense to analyze pure roll returns in Section 5.2.4 and 5.2.5 from the

time series point of view.

5.2.3 Distribution

To study commodity returns and their effects to other asset classes, it is best to

study their distributional properties, i.e. we want to understand the behavior of

historical returns across assets to implicate later, how we can model their distrib-

ution129 for forecasting and/or portfolio allocation purposes. Because this analysis

aims to analyze a real commodity investment in the portfolio allocation space, total

returns are considered.

129For a general introduction or review of statistical distribution and their moments see any intro-ductory statistics or time series analysis book, e.g. [Bamberg Baur 2002] or [Tsay 2002].

123


All major theories in finance are based on the assumption of normal distributed

log returns respectively the multivariate normal distribution of multiple assets in

a portfolio. Therefore, it is important to examine whether the data sample under

consideration satisfies the assumption of normality or not. A first introduction to

the distribution gives a so-called histogram that shows the frequency a single return

occurs. Because commodity log returns are real numbers infinite many values can

occur.130 Therefore, in dependence of the dispersion of the data small buckets are

defined and every historical return is put into one and the number of returns in the

bucket are counted and printed as bars.131

Figure 5.12 shows the histograms for the DJ-AIGCI total return and for the pure

commodity return components, the spot and the roll return, for the period 1991-2006.

Additionally a plot of a normal distribution with the sample mean and sample stan-

dard deviation is plotted for better comparability.

Figure 5.12: Histogram with Norm-Fit of DJ-AIGCI Return Components

Unfortunately, viewing the data in this way already tells that the historically oc-

curred log returns did not follow a perfect normal distribution. Especially, the roll

returns are far away of being normal distributed. However, to get a deeper inside

into the distribution properties of the returns we need to examine their defining

moments. We have already introduced the sample mean and the sample variance

of a return series. But the first two moments uniquely just determine the normal

distribution. To check whether the data sample under consideration satisfies the

130For a brief introduction to number theory see [Broecker 1995]131To be more precise: Let r1, . . . , rT be a discrete random sample of returns at times t ∈ [1, T ]

and r be the sample mean as in Equation (5.1). Given the origin r and a bin width h, thenthe bins of the histogram are defined as the intervals [r + mh, r + (m + 1)h] for m ∈ Z. Thedensity estimator f in a histogram is defined as:

f(r) =1

Th(no. of ri in the same bin as r) (5.21)

124


assumption of normality or not, as a first indicator we need to investigate the sym-

metry and tail behavior of the sample. The two characteristics are described by the

third centered and with respect to the second moment, i.e. the standard deviation,

normalized moment called skewness and the fourth centered and with respect to the

second moment, i.e. the standard deviation, normalized moment called kurtosis.

The conventional sample coefficients are given by:132

Definition 5.4 Sample Skewness

Let r1, . . . , rT be a discrete random sample of returns at times t ∈ 1, . . . , T.Denote with r the sample mean as of Equation (5.1) and with σ the sample standard

deviation as of Equation (5.3). The sample skewness is defined as:

S =T

(T − 1)(T − 2)

T∑t=1

(rt − r

σ

)3

(5.22)

Skewness is a measure of the symmetry of the probability distribution of a real-

valued random variable. Roughly speaking, a distribution has positive skew or is

right-skewed if the right tail, i.e. the tail with the higher returns, is longer and nega-

tive skew or is left-skewed if the left tail, i.e. the tail with the lower returns, is longer.

Definition 5.5 Sample Kurtosis

Let r1, . . . , rT be a discrete random sample of returns at times t ∈ 1, . . . , T.Denote with r the sample mean as of Equation (5.1) and with σ the sample standard

deviation as of Equation (5.3). The sample kurtosis is defined as:

K =T (T + 1)

(T − 1)(T − 2)(T − 3)

T∑t=1

(rt − r

σ

)4

− 3(T − 1)2

(T − 2)(T − 3)(5.23)

Kurtosis is a measure of the ”peakedness” and the tail behavior of the probability

distribution of a real-valued random variable. Higher kurtosis means the variance is

influenced by infrequent extreme deviations. For a standard normal distribution the

skewness is zero and the kurtosis is three. Therefore, the definition of the kurtosis

is mainly given relative to the standard normal distribution, i.e. adjusted by three

132The given sample estimators are adjusted to be unbiased and used in many statistical programs.For further details of their derivation see [Groeneveld Meeden 1984] or [Bamberg Baur 2002].

125


as in Definition 5.5.

Unfortunately, both measures are not robust against outliers because both, the

mean and the standard deviation are influenced by them. [Kim White 2004] study

different alternatives and we decided to introduce a robust skewness measure first

mentioned in [Bowley 1920] based on quantiles. The α% - quantile value qα is chosen

that α% of all realized returns in the sample are smaller than the α% - quantile value

qα.133

Definition 5.6 Bowley Skewness

Let r1, . . . , rT be a discrete random sample of returns at times t ∈ 1, . . . , T and

let qα be as defined in C.18 with α ∈ [0, 1]. The Bowley skewness is defined as:

SB =q0.75 + q0.25 − 2q0.5

q0.75 − q0.25

(5.24)

It can easily be seen that for a symmetric distribution the Bowley skewness is zero.

Both values, the sample and the Bowley skewness for the different types of the DJ-

AIGCI return components are documented in Table 5.7. The robust estimation of

Bowley is much closer to zero. Implicating, the return distribution is influenced by

extreme events.

Total Return Spot Return Pure Roll Return

Mean 0.03% 0.03% -0.05%Median 0.04% 0.04% -0.04%Standard deviation 0.08% 0.81% 0.22%Sample Skewness -0.32 -0.31 1.81Bowley Skewness -0.03 -0.05 -0.32Sample Kurtosis 5.98 5.83 13.08Moors Kurtosis 0.12 0.10 0.55

Table 5.7: Distribution Statistics of DJ-AIGCI Return Components

[Moors 1988] showed that the conventional measure of kurtosis K can be interpreted

as a measure of dispersion of a distribution around the two values µ± σ.134 Hence,

K can be large when probability mass is concentrated either near the mean µ or in

the tails of the distribution. Based on this interpretation he proposed the following

robust kurtosis measure:

133The formal definition can be found in C.18.134The formal definitions of µ and σ can be found in C.10 and C.11.

126


Definition 5.7 Moors Kurtosis


let qα be as defined in C.18 with α ∈ [0, 1]. The Moors kurtosis is defined as:

KM =(q0.875 − q0.625) + (q0.375 − q0.125)

q0.75 − q0.25

− 1.23 (5.25)

Moors created this estimator based on the idea that the two terms, (q0.875 − q0.625)

and (q0.375 − q0.125), are large respectively small if relatively small respectively high

probability mass is concentrated in the neighborhood of q0.75 and q0.25 corresponding

to large respectively small dispersion around the two values µ + σ and µ − σ. The

denominator is a scaling factor, ensuring that the statistic is invariant under linear

transformation. It is easy to calculate that the Moors kurtosis has the value 1.23

for a standard normal distribution.135 For comparability, Definition 5.7 gives the

adjusted value relative to the normal distribution.

Both values, the sample and the Moors kurtosis for the different types of the

DJ-AIGCI return components are documented in Table 5.7. On the first view it

might be surprising that both values fall much apart from each other. But recall

the difference in the definitions. The sample kurtosis measures both, the peakedness

around the mean and fat tails. In contrast, the Moors kurtosis just measures the

dispersion around µ ± σ. Viewing again the histograms in Figure 5.12 uncovers

that indeed the dispersion around µ ± σ is quite modest but the peak around the

mean is very high. This explains why the two values fall much apart from each other.

Summing up, the return distributions of the DJ-AIGCI components seem to be dif-

ferent from the normal distribution. To check whether our observations are randomly

for our special sample or whether we can generalize the statement of non-normality

to different observations, we performed some hypothesis tests for normality. The

major idea of hypothesis testing is to compare distribution’s characteristics of a

sample with the distribution’s characteristics of a reference distribution, in our case,

to compare the sample distribution of DJ-AIGCI log returns with the normal dis-

tribution.

How does it work in practise? First we formulate the null hypothesis of normality

that we will test against the alternative of non normality, aiming to reject the null

hypothesis with a manageable alpha error to reach significant statements, i.e. to

make sure that the observation is not randomly for the specific data sample but

135q0.875 = −q0.125 = −1.15, q0.25 = −q0.75 = −0.68, q0.375 = −q0.125 = −0.32

127


can be generalized. Second, we choose a normal distribution facility and derive a

measure function called test statistic that compares the value of the sample facility

with the value of the normal distribution facility. Third, we judge based on the out-

come of the test statistic whether to reject the null hypothesis or not. Over time,

different test statistics based on different facilities were derived. We will present

the three mainly used ones, the Lilliefors Kolmogorov-Smirnov, the Shapiro and the

Jaque-Bera Test.

The Lilliefors Test is an adaption of the Kolmogorov-Smirnov Test. It is named

after Hubert Lilliefors, professor of statistics at George Washington University but

actually the test was developed independently by Lilliefors and by Van Soest. The

basic idea behind the test is the same as the idea behind the Kolmogorov-Smirnov

Test of finding the biggest distance between the empirical and reference cumulative

probability density function of a sample observation:

Definition 5.8 Lilliefors Kolmogorov Smirnov Test

Let r1, . . . , rT be a discrete random sample of returns at times t ∈ 1, . . . , T.The total number of observation be T . Let r denote the sample mean as defined in

Equation (5.1), σ2 denote the sample variance as defined in Equation (5.21) and

σ =√

σ2 be the sample standard deviation as of Equation (5.3). Moreover, define

the transformation of the sample observations Zt = rt−rσ

, count in Ft the number of

sample observations that are smaller or equal to Zt and define the sample probability

as: L(Zt) = Ft

T. Finally, define the probability under the assumption of standard

normality as: N (Zt) =∫ Zt

−∞1√2π

e(−12x2)dx. The Lilliefors Test Statistic is defined as:

L = maxt|L(Zt)−N (Zt)|, |L(Zt)−N (Zt−1)| (5.26)

So L measures the absolute value of the biggest difference between the probability

associated to Zt when Zt is normally distributed, and the empirical probability of

Zt, e.g. the frequencies actually observed. Because the empirical distribution is

discrete, the term |L(Zt)−N (Zt−1)| is needed to cover that the maximum absolute

difference can occur at either endpoints of the empirical distribution.

For different alpha errors the critical values can be found in amongst others, e.g.

[Lilliefors 1967], [Van Soest 1967] and [Molin Abdi 1998]. The outcome of the test

for our sample data of DJ-AIGCI excess returns can be found in Table 5.8.

The next test brings us back in the world of regression introduced in Section 5.1.2.

128


The idea is to obtain a test statistic for normality by dividing the square of an

appropriate linear combination of the sample order statistic by the usual estimator

of the sample variance:

Definition 5.9 Shapiro Test

Let r1, . . . , rT be a discrete random sample of returns at times t ∈ 1, . . . , T.The Shapiro Test statistic tests the null hypothesis that the sample r1, . . . , rT comes

from a normal distribution. First, create the order statistic r(1), . . . , r(T ) as in C.21.

Second, define (a1, . . . , aT ) = mT V −1

mT V −1V −1mwith m = (m1, . . . ,mT )T denoting the

expectations of an order statistic of an i.i.d. sample from the standard normal dis-

tribution and V is the covariance matrix of those order statistic. The Shapiro Test

statistic is defined as:

W =(∑T

i=1 air(i))2

(∑T

i=1(ri − r)2(5.27)

The basic idea of the test is to think about a regression between an order statistic of

the observation sample and an order statistic of a sample generated from the stan-

dard normal distribution: Let x(1), x(2), . . . , x(T ) denote an ordered random sample

of size T from a standard normal distribution as defined in C.21 and define with

Definitions C.21, C.10 and C.14 for i, j = 1, 2, . . . , T :

E(xi) = mi

cov(xi, xj) = vij

Let r(1), . . . , r(T ) denote the order statistic of the sample as defined in C.21. The

objective is to derive a test for the hypothesis that this is a sample from a normal

distribution with unknown mean µ and variance σ2. If the r(i) are a normal sample

then r(i) can be expressed as:

r(i) = µ + σx(i) (5.28)

for i = 1, 2, . . . , T . Then, the derivation of W is based on the Aitken’s general-

ized least squares estimation of regression coefficients136 and the results of Lloyd to

derive least square estimates based on order statistics.137 The test rejects the null

hypothesis of normality if W is too small because the numerator of the statistic is

the normalized least square regression coefficient of the regression between the order

statistic of the observation sample and an order statistic of a sample generated from

the normal distribution and the denominator of the statistic is the sample variance.

136See [Aitken 1935] or [Powell 2006].137See [Lloydes 1952].

129


If the sample comes from a normal distribution both values should have the same

value. Following Lemma 2 and 3 in [Shapiro Wilk 1965] W is caped by 1 and floored

by(

Tm21

T−1

).

Finally, the Jarque Bera Test uses a test statistic based on skewness and kurtosis to

test for normality.

Definition 5.10 Jarque Bera Test


let S and K be defined in Definition 5.4 and Definition 5.5. The Jaque Bera Test

statistic is defined as:

JB =(T − k)

6

(S2 +

(K − 3)2

4

)(5.29)

k denotes the number of estimated coefficients that were needed to create the series.

The Jaque Bera Test statistic has an asymptotic chi-squared distribution with two

degrees of freedom. Since samples from a normal distribution have an expected

skewness of 0 and an expected kurtosis of 3, any deviation from this increases the

Jarque Bera test statistic what yield to finally with crossing the critical value to a

rejection of the null hypothesis.

Table 5.8 summarizes the empirical results for our data sample of DJ-AIGCI excess

returns.

Statistic p-Value Action

Lilliefors Test 0.05 0.0% reject H0

Shapiro Test 0.97 0.0% reject H0

Jarque Bera Test 5853 0.0% reject H0

Table 5.8: Significance Tests for Normality of DJ-AIGCI Total Return

All tests reject the null hypothesis of normality. Implicating, we could not proof

that historical log returns of the DJ-AIGCI components are normally distributed.

However, we can construct an empirical distribution, the so-called Kernel distribu-

tion as seen in Figure 5.13. The first step of viewing empirical data was the construc-

tion of a histogram. Intuitively, the basic idea of getting more accurate estimates

for the empirical distribution would be to make the bins smaller and to define the

130


empirical density function as the sum of the bins. Actually, this method is called the

naive estimation of the density and more accurate explained in [Silverman 1992]. A

more sophisticated idea is to define dumps around the observations and to add them

together as the empirical density function. This method is called kernel estimation

and defined as follows:

Definition 5.11 Kernel Estimation


let h define the window width, also called smoothing parameter or bandwidth. The

empirical density function is defined as:

f(r) =1

Th

T∑t=1

K

(r − rt

h

)(5.30)

whereby the so-called kernel function satisfies:∫ ∞

−∞K(r)dr = 1 (5.31)

In statistical computation the smoothing parameter h = (4/3)1/5 σT 1/5 derived in

[Silverman 1992] and the Gaussian kernel function

K(r) =1√2π

e

− r2

2

(5.32)

because of its continuity and differentiability properties became excepted and there-

fore, we will use them, too. The result of this estimation technique are shown in

Figure 5.13. Additionally, we plotted a normal distribution for comparability.

Figure 5.13: Kernel Distribution with Norm-Fit of DJ-AIGCI Return Components

Summing up, commodity returns exhibit negative skewness and excess kurtosis when

131


considering the traditional estimates as of Definition 5.4 and as of Definition 5.5.

Their against outliers robust estimators, the Bowley Skewness and the Moors Kur-

tosis, have shown that the distribution is influenced by extreme events and peaked

around the mean. Considering the results of the robust estimators they are much

nearer to the reference values for a normal distribution, i.e. cleaning the data of

outliers might result into returns that are nearly normally distributed. Nevertheless,

using the original data sample all tests for normal distribution failed and finally, the

construction of an empirical density showed visually that indeed commodity returns

are not normally distributed.

5.2.4 Stationarity

In time series analysis the first step is to investigate stationarity, i.e. changes of

the distribution characteristics over time. As we already known, the first interesting

characteristic is the location parameter mean. Taking again a look at Figure 5.7 we

observe changes in the mean of the DJ-AIGCI price series, i.e. there are increasing

and decreasing periods. But viewing again Figure 5.8 namely the time series of the

log returns we realize that they seem to shake around a constant mean. But being

mathematically precise, stationarity is defined as follows.

Definition 5.12 Strictly Stationarity

Let r1, . . . , rT be a discrete random sample of returns at times t ∈ 1, . . . , T. The

time series rt is said to be strictly stationary if the joint distribution of (rt1 , . . . , rtk)

is identical to that of (rt1+t, . . . , rtk+t) for all t, where k is an arbitrary positive in-

teger and (t1, . . . , tk) is a collection of k positive integers.

In other words, strict stationarity requires that the joint distribution of (rt1 , . . . , rtk)

is invariant under time shifts. This is empirically hard to find and therefore, a weaker

form of stationarity shall help to handle small distribution changes over time.

Definition 5.13 Weakly Stationarity

Let r1, . . . , rT be a discrete random sample of returns at times t ∈ 1, . . . , T.The time series rt is said to be weakly stationary if the mean of rt and the covariance

between rt and rt+k is time invariant for an arbitrary integer k.

132


Weak stationarity exactly embodies what was said introductory. It implies that the

data fluctuate with constant variation around a constant level. To test for station-

arity, different tests were established over time. Because we are already familiar

with regressions and the OLS estimation methodology from Section 5.1.2, we will

introduce the Dickey Fuller Test that is based on the same principles. For it, we

first set up a simple autoregressive time series model:

rt = βrt−1 + εt (5.33)

As in Definition 5.3 εt is an error term of identical and independent distributed ran-

dom variables with mean zero and variance σ2ε . Furthermore, assume for simplicity

that r0 = 0 and that εt’s come from a normal distribution. If β = 1 the time series

model of (5.33) reduces to:

rt = εt + εt−1 + . . . + ε1 (5.34)

This describes a random walk with rt ∼ N (0, σ2εt), i.e the rt’s have a time dependent

distribution in contradiction to Definition 5.13. Therefore, to proof the hypothesis of

stationarity [Dickey Fuller 1979] suggested the null hypothesis of β = 1. Rejecting

the null hypothesis implicates, that a series can be seen as stationary, statistically

significant.

Definition 5.14 Dickey Fuller Test

Let r1, . . . , rT be a discrete random sample of returns at times t ∈ 1, . . . , T.With the derivation methodology as of the proof to Theorem 5.1, the OLS estimate

for β in (5.33) is given as:

β =

∑Tt=1 rt−1rt∑Tt=1 r2

t−1

(5.35)

Furthermore, the usual OLS standard error for the estimation coefficient is given

as:

σ2β

=1

T−1

∑Tt=1(rt − βrt−1)

2∑Tt=1 r2

t−1

=σ∑T

t=1 r2t−1

(5.36)

133


Then, the Dickey and Fuller test statistic under the null hypothesis of β = 1 is

defined as:

DF =β − 1

σβ

(5.37)

To decide whether to reject the test or not, we need to know the distribution of DF

test statistic to identify the critical regions.

Theorem 5.4 Distribution of the Dickey Fuller Test Statistic

Let r1, . . . , rT be a discrete random sample of returns at times t ∈ 1, . . . , Tand let W (t) denote a Wiener process as of Definition xy. The symbol

d−→ denotes

convergence in distribution as of Definition xy. With the notion of Definition 5.14

and T −→∞ it follows that the Dickey Fuller test statistic DF as of Definition 5.14

is under the null hypothesis of β = 1 asymptotically distributed as:

DFd−→

12(W (1)2 − 1)(∫ 1

0W (r)dr

)1/2(5.38)

Proof: With the notion of Definition 5.14 and under the null hypothesis of β = 1,

it follows:

DF =β − 1

σβ

H0:β=1︷︸︸︷=

β − β

σβ

With (5.33) and (5.35):

DF =

(PTt=1 rt−1rtPTt=1 rt−1

)−(PT

t=1(rt−εt)PTt=1 rt−1

)σβ

Furthermore, taking (5.36) yields to:

DF =

(PTt=1 rt−1εtPTt=1 r2

t−1

)(

σ2PTt=1 r2

t−1

)1/2

134


Expanding with 1 = TT:

DF =

(1T

PTt=1 rt−1εt

1T2

PTt=1 r2

t−1

)(σ2 T 2PT

t=1 r2t−1

)1/2

=

(1T

∑Tt=1 rt−1εt

1T 2

∑Tt=1 r2

t−1

)∗

1(σ2 T 2PT

t=1 r2t−1

)1/2

=

(1T

∑Tt=1 rt−1εt

)(

σ2

T 2

∑Tt=1 r2

t−1

) d−→12(W (1)2 − 1)(∫ 1

0W (r)dr

)1/2

The convergence follows with Proposition 9 in [Hamilton 1994] page 486. Its multi-

lateral proof is out of the scope of this thesis.

2

The respective values of DF for different critical values can be found in amongst

others [Hamilton 1994], for instance.

Our findings for the DJ-AIGCI total return and its pure commodity return compo-

nents can be found in Table 5.9.

Statistic p-Value Action

Total Return -15.00 0.00% reject H0

Spot Return -15.23 0.00% reject H0

Roll Return -11.32 0.00% reject H0

Pure Roll Return -4.01 0.01% reject H0

Table 5.9: Dickey Fuller Test for Stationarity

The test clearly shows that commodity returns can assumed to be stationary.

5.2.5 Autocorrelation

In Section 5.1.2 we already introduced the measure for linear dependency of two

variables, the correlation, and Figure 5.3 visualized the degree of dependence. In

this section we are interested in analyzing the linear dependence between returns

following each other. Therefore, the equivalent to Figure 5.3 is done in Figure 5.14.

But there we didn’t plot two different sample realizations at the same time against

each other, we plotted the realizations of the same sample at different points in time

135


against each other. Plotting the sample against the time shifted sample is done

with the so-called time lag. In Figure 5.14 the DJ-AIGCI total return and its pure

commodity return components are plotted until the fourth time lag. The diagonal

line is plotted for better comparability and represents 100% correlation.

Figure 5.14: Lagged Plot of DJ-AIGCI Return Components

We realize that both, total and spot returns don’t show any autocorrelation pattern,

but roll returns do. The zero cross pattern occurs through the non rolling periods.

After visual examining the data we need to check mathematically whether auto-

correlation is significant or not. Therefore, we first define the so-called sample

autocorrelation function (ACF):

Definition 5.15 Sample Autocorrelation Function

Let r1, . . . , rT be a discrete random sample of returns at times t ∈ 1, . . . , T.The time series rt is weakly stationary. The k-lag sample autocorrelation is defined

as the correlation between rt and its past value rt−k:

ρk =

∑Tt=k+1(rt − r)(rt−k − r)∑T

t=1(rt − r)2(5.39)

The autocorrelation function is therefore a function in k.

The sample autocorrelation is a biased estimator for Definition C.16 of order (1/T ),

i.e. for our sample of order 10−4 and therewith relatively small. The autocorrelation

function for the DJ-AIGCI total return and its pure commodity return components

are plotted in the left diagrams of Figure 5.15 and reported until the sixth lag in

Table 5.10. The horizontal lines in Figure 5.15 give the 5% alpha levels, indicating

significant autocorrelation.

136


Figure 5.15: Autocorrelation and Partial Autocorrelation Function of DJ-AIGCI ReturnComponents

The plot proofs our first impression gotten by Figure 5.14 that total and spot returns

don’t show autocorrelation pattern. In contrast, roll return exhibit strong positive

autocorrelation until the fourth lag, decreasing with increasing lag. It is interesting

that the correlation decreases with 0.2 steps. Recall, the DJ-AIGCI rolling proce-

dure rolls forward 20% of the futures per day over a five day period. Because the

randomness of price changes are captured in spot returns and the slow changing,

generally long term lasting term structure is reflected by roll returns, this pattern

can be explained.

Moreover, we calculated the so-called partial autocorrelation function (PACF). It

removes the effect of shorter lag autocorrelation from the correlation estimate at

longer lags and is defined as follows:

137


Definition 5.16 Partial Autocorrelation Function


as in Definition 5.15. The partial autocorrelation function is then defined as:

pρkk =rk −

∑k−1i=1 pρk−1,iρk−1

1−∑k−1

i=1 pρk−1,iρi

(5.40)

In the right diagrams of Figure 5.15 we plotted the adequate PACFs for the three

DJ-AIGCI return components. The 20% rolling effect is removed but still, roll

returns exhibit significant autocorrelation.

To be precise, we need to test jointly for significant autocorrelation. Therefore, the

Box and Pierce Test is excepted. It test the null hypothesis that the autocorrelation

function as of Definition 5.15 of a time series until the m-th lag for all i ∈ 1, . . . ,m

is zero against the alternative that there is at least one i ∈ 1, . . . ,m for that the

autocorrelation function is not zero. The test statistic is defined as follows:

Definition 5.17 Box and Pierce Test


as in Definition 5.15. The Box and Pierce test statistic is then defined as:

Q(k) = Tk∑

i=1

ρ2i (5.41)

Under the assumption that r1, . . . , rT is independently and identical distributed

Q(k) is chi-squared distribution with m degrees of freedom.

We performed the test for the DJ-AIGCI total return and its pure commodity return

components. Our results are documented in Table 5.10.

DJ-AIGCI (TR) DJ-AIGCI (SP) DJ-AIGCI (RR)

Lag i ρi Statistic p Value ρi Statistic p Value ρi Statistic p Value

1 1.00 0.16 69.04% 1.00 0.00 97.71% 1.00 1949.48 0.00%2 -0.01 1.02 60.19% 0.00 0.57 75.22% 0.71 3011.15 0.00%3 -0.01 1.13 76.87% -0.01 0.78 85.48% 0.52 3448.09 0.00%4 0.01 1.64 80.19% 0.01 1.35 85.35% 0.33 3546.22 0.00%5 -0.01 1.91 86.15% -0.01 1.39 92.55% 0.16 3546.87 0.00%6 0.01 8.05 23.46% 0.00 8.30 21.66% -0.01 3547.52 0.00%

Table 5.10: Significance Tests for Autocorrelation

138


Not surprisingly, the test cannot be rejected for the total and the spot return but for

the roll return the test of no autocorrelation has to be rejected. The results are in

line with former findings of [Kat Oomen 2006]. Additionally, they found significant

autocorrelation on single commodity futures returns including among others corn,

soybeans, live cattle, oil and gold. Implicating, the property of autocorrelation gets

lost in commodity index returns.

139

6 Asset Allocation with Commodity Derivatives

One of the most important decisions many people face is the choice where to invest

their earned money for saving purposes. Generally, their individual needs are quite

different: some may have relative short-term objectives, others may be saving to

make college tuition payments in the medium term, yet others may be saving for

retirement or to ensure the well-being of their heirs. Nevertheless, they all have one

common decision to make: which asset classes shall be allocated in my portfolio and

how much weight shall they get? Depending on the individual preferences, the de-

gree of allocation in the single asset classes vary. The process is even more complex

when institutional investors come into consideration. They have generally to take

care about legal restrictions on the one hand but the pressure to reach return tar-

gets on the other hand. We will investigate these questions in the following sections.

Starting with Section 6.1, we will analyze the risk premium of commodities with the

purpose to categorize commodities as a separate asset class. Only if there exists a

part of the risk premium embodied in commodity returns that cannot be explained

by an other asset class’ returns, commodities can be seen as independent investment

opportunity. Second, we will analyze the behavior of commodity returns in compar-

ison to stock and bond returns. As we have seen in Section 5.1.3 the most attractive

risk and return profiles can only be reached by combining portfolio participants with

different return characteristics. The preceding sections have already uncovered that

commodity returns have different risk and return sources than stock and bond re-

turns have, suspecting different return facilities. This shall be addressed in Section

6.2 compactly. Finally, we will view commodities in the portfolio allocation process

with stocks and bonds in Section 6.3. We will stretch an efficient frontier that allows

investors to pick the optimal asset allocation depending on their individual risk and

return preferences. Generally, the analysis will show that allocating commodities to

a traditional stock and bond portfolio improves the characteristics of the investment.

This is not reasoned by the extraordinary commodity returns of the last years but in

the attractive risk and correlation characteristics commodity returns have to stock

and bond returns.

6.1 Mean Variance Spanning

The first question to answer in investment practice is whether the investment medium

can be seen as a separate investment class or not. If it can be seen as a separate

asset class the second step is to categorize it as traditional or alternative asset class.

140


Following [Greer 1997] commodities account into the group of alternative assets.

Furthermore, he distinguishes between three super asset classes, including ”capital

assets”, ”asset that can be used as economic inputs” and ”assets that are a store of

value”. The first group consists of all financial assets whose value is determined by

their future cash flows. They provide a source of ongoing value. As a result, this

assets are valued based on the net present value of their expected returns. Equities

and bonds are the main representatives of this group. But also hedge funds, private

equity funds and credit derivatives are included because their value is determined by

the present value of the expected future cash flows from the securities in which they

invest. The second group called ”assets that can be used as economic inputs” can be

consumed as part of the production cycle. Deducting form Section 2, commodities

count into this group. Additionally, we have already seen in Section 3 that these

assets cannot be valued using a simple net present value approach. Finally, com-

modities like gold and silver count, amongst others e.g. art, into the third group.

Owning jewelery or gold bars don’t produce future cash flows. But especially in

the emerging parts of the world these assets are a medium of maintaining wealth.

In these countries, residents don’t have access to the same range of financial prod-

ucts that are available to residents of more developed nations. Consequently, they

accumulate their wealth through a tangible asset as opposed to a capital asset.

The example of gold shows that the lines between the three super asset classes can

become blurred. However, what is an asset class anyway in the correct statistical

interpretation? Following [DeRoon Nijman 2001] any suspected asset class ri that

actually earns a risk premium above cash c that cannot be explained by other

already existing asset classes rj is actually an asset class in its own right. The

following definition shall give the correct mathematical explanation:138

Definition 6.1 Mean Variance Spanning

Let ri denote the return of a representative of the suspected asset class, let c denote

the return of cash, e.g. Treasury Bills, and let rj denote the return of other asset

classes. The suspected asset class can be seen as a separate asset class if the α

coefficient of the following regression is statistically significant:

ri − c = α +n∑

j=1

βj(rj − c) + ε (6.1)

Definition 6.1 shows that an asset class can be seen as separate if there is a part

of the risk premium that cannot be explained by other asset classes. Following

138For details about solving such regression models see Definition 5.3 and Theorem 5.1.

141


Equation (5.14) the higher the correlation between two assets, the more systematic

common risk exposure and hence common risk premium exists.

For our analysis we used the DJ-AIGCI total return index, the S&P 500 total return

index and the J.P. Morgan USA Government Bond Index. As shown in Section 4.4

the total return of a commodity index is summed up by the excess return and the

interest rate return earned on collateral. Following [DJAIGCI 2006] Dow Jones uses

US Treasury Bill’s returns to calculate the total return index. To be consistent we

used this return as cash in the regression. Table 6.1 summarizes our results for the

period (1991-2006). A statistically significant regression coefficient is marked with

a ”∗”.

Value t-value p-value

α 0.000166 1.298 0.19453βStocks -0.00865 -0.68 0.49585βBonds -0.15454* -3.47 0.0005

Table 6.1: Mean Variance Spanning Coefficients (1991-2006)

Unfortunately, we cannot identify a statistically significant risk premium. Similar

research done by Deutsche Bank using the same traditional reference indices as we

did, identified over the period from January 1989 to January 2005 no statistically

significant risk premium for the GSCI but for the DBLCI and the DBLCI-MR. As

already known, commodities went through a regression through the 1990th and first

became attractive for investing at the beginning of the 21rst century. Therefore, we

performed the same analysis for the period (2002-2006). The results are documented

in Table 6.2.

Value t-value p-value

α 0.000592* 2.022 0.044βStocks 0.049133 1.674 0.0944βBonds 0.040089 0.410 0.6819

Table 6.2: Mean Variance Spanning Coefficients (2002-2006)

Over the smaller period we identified a small139 but statistically significant risk

premium.

139But recall, the analysis is in daily scale.

142


Furthermore, this analysis show the time dependent character of commodity invest-

ment. Because commodities don’t provide future cash flows and are not valued by

discounted cash flow methods their performance is highly dependent to scarcity em-

bodied in the convenience yield or risk premium on inventories. Moreover, we have

seen that commodity markets have been in contango more often than in backwarda-

tion producing negative roll returns with a major impact on the total excess return

performance.

Finally, we performed again a factor analysis following Definition 5.3 with different

representatives of the three asset classes under consideration. We used respectively

three indices of the traditional asset classes including a USA only, a Europe only and

a global index. As representatives for the bond market we took the J. P. Morgan

US Government Bond Index, the J. P. Morgan Europe Government Bond Index and

the the J. P. Morgan Global Government Bond Index. The S&P 500 Total Return,

the MSCI World and the MSCI Europe serve as examples for the stock market.

Again, the DJ-AIGCI total return represents the commodity exposure. Figure 6.1

summarizes our findings:

Figure 6.1: Factor Analysis with other Asset Classes (1991-2006)

We identified three common risk factors. The bond indices stick out in one direction

what we therefore identify as the risk factor mainly influencing a bond investment.

The second dimension is uniquely stretched by the stock indices resulting to our con-

clusion that this is the risk factor driving the stock returns. Finally, the DJ-AIGCI

protrudes in his own direction. We deduce that this might symbolize a separate

143


commodity risk factor, although the DJ-AIGCI additionally sticks out in the bond

direction. This finding is in line with the mean variance spanning result as reported

in Table 6.1.

Summing up, the section showed that commodities tend to be a separate asset class

that produced a statistically significant risk premium over selected periods and is

driven by its own risk factors. This promises diversification effects in the portfolio

context with the traditional asset classes stocks and bonds.

6.2 Dependence to Stocks, Bonds and Inflation

After we identified commodities as a separate asset class we need to take a deeper

look into their performance over the period under consideration. In Figure 6.2 we

plotted the development of an 100 US dollar investment into the three asset classes

stocks, bonds and commodities at the 01.01.1991.140 Additionally, inflation is drawn

in to show that all asset classes have outperformed the natural value loss of money

over time.

Figure 6.2: Performance of different Asset Classes

On the first view we realize that all three asset classes have outperformed inflation,

implicating that investment in general payed off over the last years. Stocks produced

the highest return with a total value gain of around 433% followed by a fully collat-

eralized commodity investment with a total value gain of around 228% and latest

bonds with a total value gain of around 176%. Moreover, Table 6.3 summarized the

140The calculation followed Definition C.3.

144


key statistics of the investment opportunities.

DJ-AIGCI S&P 500 JPM Bond USA

Annualized arithmetic mean 7.63% 10.73% 6.50%Total value gain 228.18% 433.09% 175.57%Annualized standard deviation 12.64% 15.91% 4.54%Minimum (daily) -9.15% -7.11% -1.61%Maximum (daily) 4.85 % 5.58% 1.59%Mean (daily) 0.03 % 0.04% 0.03%Median (daily) 0.04 % 0.04% 0.03%99% VaR -2.03 % -2.62% -0.78%95% VaR -1.24 % -1.58% -0.45%

Table 6.3: Key Statistics of different Asset Classes’ Returns (1991-2006)

Recall from Table 5.5, the average excess return of commodities was on average

3.70% per annum, i.e. the 7.63% average commodity total return per annum con-

sists of 48.5% commodity excess return and 51.5% interest rate return earned on

collateral. Therewith, only a fully collateralized commodity investment produced a

return situated in the middle of stock and bond returns. There are different research

papers, including [Gorton Rouwenhorst 2004] and [PIMCO 2006], identifying com-

modities as the best performing asset class over the long run. Both studies start

at the beginning of the 1970th as commodities run into their first huge price surge

and end 2004 in the middle of the second huge commodity price surge. During the

whole period stock performance is characterized by a steady growth interrupted by

the regression period at the end of the 1990th and the bull period at the beginning

of the 21st century. For instance, [PIMCO 2006] reported an annualized arithmetic

return of 14.06% for commodities and an annualized arithmetic return of 12.60% for

stocks. Additionally, literature and our analysis shows that commodity investment

shall be long term orientated conforming findings of [Till 2000] as reported in Sec-

tion 5.1.1.

In Section 5.1.3 we introduced the free lunch in finance: diversification. The main

purpose of asset allocation is to improve the risk return profile of an investors port-

folio. Theorem 5.2 uncovered the main driver of diversification: the average covari-

ance or respective its normalized brother the average correlation of the portfolio

constituencies. Following Definition 5.1 and Definition 5.2 we calculated Pearson’s

and Kendall’s correlation coefficient between the three asset classes and reported

145


the results in Table 6.4. Additionally, the correlations to inflation are reported.141

DJ-AIGCI S&P 500 JPM Bond USA Inflation

DJ-AIGCI 1.00 -0.01 -0.06* 0.13S&P 500 -0.00 1.00 -0.03* -0.09JPM Bond USA -0.04* 0.04* 1.00 -0.07Inflation 0.09 -0.08 -0.04 1.00

Table 6.4: Kendall/Pearson Correlation between Different Asset Classes and Inflation

In the left lower rectangle of Table 6.4 the Kendall correlation coefficient is reported

and in the right upper rectangle the Pearson correlation coefficient is entered. We

highlighted statistically significant values with a ”∗”.

Commodity returns don’t show correlation pattern to the traditional asset classes,

i.e. the correlation coefficients are nearby zero. On the contrary, to bond returns

they exhibit a negative dependence structure. In the portfolio context, this adverse

pattern promises diversification effects and we will address ourselves to this problem

in Section 6.3.

Our findings are in line with past research. [Gorton Rouwenhorst 2004] extended the

analysis and showed the Pearson correlation coefficient for average returns taken over

different time frames, including monthly, quarterly, one year and five years. They

showed that the anti correlation rises with increasing time period. This suggests that

the diversification benefits of commodities tend to be larger for longer horizons.

Investors ultimately care about the real purchasing power of their returns, which

means that the threat of inflation is a concern for investors. Table 6.4 shows that

traditional asset classes are a poor hedge against inflation. Only commodities have

a positive correlation to inflation indicating positive price movements coming in line

with increasing inflation. This is not astonishing because commodities are real as-

sets. Inflation is measured as the change of a product basket’s value. But products

are made of commodities explaining the co-movement of inflation and commodity

prices. [Gorton Rouwenhorst 2004] extended the analysis and calculated the Pear-

son correlation coefficient to the above described average returns for different time

periods. Again, they could show that the correlation increases over time. More-

over, they divided inflation in its expected and unexpected part. For it, they used a

141Inflation is measured in monthly scale. Linear interpolating to daily values would destroy theactual dependence structure. Therefore, the correlation coefficients of the three asset classes toinflation are calculated with monthly data.

146


model suggested by [Fama Schwert 1977] and [Schwert 1981]. The short term Trea-

sury Bill’s rate is a proxy for the market’s expectation of inflation, if the expected

real rate of interest is constant over time. Consequently, unexpected inflation can

be measured as the actual inflation rate minus the nominal interest rate which is

known ex ante. They showed that the negative sensitivities of stocks and bonds and

the positive sensitivities of commodities are higher to unexpected inflation than to

inflation itself.

To close the section, the following analysis shall give another interesting inside into

the behavior of commodity returns. Stock returns are getting more volatile in falling

markets, i.e. stock returns and their volatility are negatively correlated. This pattern

can be seen in Table 6.5 and is known as ”leverage effect”. Following Definition 5.1

we calculated the Pearson correlation coefficient between the average return and

the average volatility over a time lag of five, 20 and 60 days. Again statistically

significant values are marked with a ”∗”.

DJ-AIGCI (ER) DJ-AIGCI (TR) S&P 500 JPM Bond USA

5 days 0.00 0.00 -0.10* -0.17*20 days 0.04* 0.03* -0.21* -0.26*60 days 0.15* 0.14* -0.33* -0.27*

Table 6.5: Pearson Correlation between average Return and Volatility

While stock and bond returns are getting more nervous in falling markets, com-

modity returns exhibit the adverse pattern: they are getting more nervous in ris-

ing markets. This phenomena was already discussed in Section 5.1 and named by

[German 2005] as the ”negative leverage effect”. Recall, price surges in commodity

markets come in line with falling inventories and the fear of possible supply inter-

ruptions. This makes the market nervous and volatility rising. In contrast, stocks

and bonds are valued by future cash flows. Falling prices indicate company and

issuer problems exciting sells that increases the volatility.

This anti-cyclical pattern gives again evidence to suggest that the combination of

stocks, bonds and commodities yield to attractive risk and return profiles of a port-

folio.

147


6.3 Portfolio Optimization

The last two sections introduced the historical performance and return character-

istics of commodities and their dependence structure to traditional asset class’ re-

turns, i.e. stock and bond returns. We will close this section with a brief analysis of

the interaction of all three asset classes in the portfolio context. Our findings from

Section 6.2 already suspected positive diversification effects, commodities could gen-

erate in a traditional stock and bond portfolio. The basic tool for calculating as-

set allocation is Harry Markowitz’s mean variance optimization first published in

[Markowitz 1952]. The idea is to construct a portfolio that has maximum return

based on the constrain of a predefined risk boundary depending on the risk aversion

of the investor. Imagine n assets under consideration. Denote an asset’s weight in the

portfolio with xi, i = 1, . . . , n, its average return with µ as defined in Definition C.10.

The covariance matrix including the asset’s variances as of Definition C.11 at the

main diagonal and the covariances among the assets as of Definition C.14 on the

non diagonal entries be denoted by (C)i,j=1,...,n. The mathematical formulation of

the Markowitz’s mean variance optimization is given as:

Pµ =

µT x −→ max

xT Cx = σ2

1T x = 1 with 1 = (1, . . . , 1)T

(6.2)

Alternatively, the problem can be re-written to construct a portfolio that has mini-

mum risk, i.e. variance, under the constrain that a predefined return is still gener-

ated:

Pσ2 =

xT Cx −→ min

µT x = µ

1T x = 1 with 1 = (1, . . . , 1)T

(6.3)

The problem’s solution requires three input factor: means, standard deviations re-

spectively volatilities and correlations respectively covariances. Based on these three

inputs, en efficient frontier is constructed in which each point maximizes the return

per unit risk. This provides investors with individual risk and return guidelines with

the adequate portfolio composition x. The mathematical solution of Pσ2 and the

representation of the efficient frontier is given in Theorem 6.1.

Theorem 6.1 Mean Variance Optimization

Denote with C the covariance matrix which is assumed to be positive definite. More-

over, define µ as is Definition C.10 and denote:

148


a = 1T C−1µ, b = µT C−1µ, c = 1T C−11, d = bc− a2

The optimal solution of Pσ2 is given as:

x =1

d

((cµ− a)C−1µ + (b− aµ)C−11

)(6.4)

with

σ2(µ) = xT Cx =cµ2 − 2aµ + b

d(6.5)

The Minimum Variance Portfolio denoted with xMV P is given as:

xMV P =1

cC−11 (6.6)

It is located in the risk return space at:

(µMV P , σMV P ) = (a

c,

√1

c) (6.7)

The efficient frontier is given as:

µ = µMV P ±√

d

c(σ2 − σ2

MV P ) (6.8)

Note, the negative case of the efficient frontier of Equation (6.8) is dominated by

the positive case and therefore, can be categorized as not efficient. Anticipating

Figure 6.3, compare the two portfolios with risk equal to 4.9% annualized standard

deviations. In the negative case it produces an annualized return of around 6.6%,

but in the positive case it produces an annualized return of around 7.5%. Following,

the positive case portfolio will be chosen because although taking the same risk as

in the negative case, more return is generated.

Proof: Because C−1 is positive definite, it follows:

b = µT C−1µ > 0 (6.9)

and furthermore:

c = 1T C−11 > 0 (6.10)

With the scalar product < x, y >≡ xT C−1y, the Chauchy-Schwarz inequation and

149


x ≡ 1, y ≡ µ follows:

< x, y >2 = (1T C−1µ)2 = a2

< < x, x >< y, y >= (1T C−11)(µT C−1µ) = bc

⇔

d = bc− a2 > 0 (6.11)

Furthermore, the Lagrange function is given as:

L(x, u) =1

2xT Cx + u1(µ− µT x) + u2(1− 1T x) (6.12)

x is optimal for P if there exists an u = (u1, u2)T ∈ R that satisfies the so-called

Kuhn-Tucker conditions:

∂L

∂xi

(x, u) =n∑

j=1

ci,jxj − u1µi − u2 = 0 (6.13)

∂L

∂u1

(x, u) = µ− µT x = 0 (6.14)

∂L

∂u2

(x, u) = 1− 1T x = 0 (6.15)

(6.13) ⇔ Cx = u1µ + u21

⇔ x = u1C−1µ + u2C

−11(6.16)

(6.15)&(6.16)︷︸︸︷⇒ 1T x = u1 1T C−1µ︸︷︷︸≡a

+ u2 1T C−11︸︷︷︸≡c

= au1 + cu2

(6.15)︷︸︸︷= 1

(6.17)

(6.14)&(6.16)︷︸︸︷⇒ µT x = u1 µT C−1µ︸︷︷︸≡b

+ u2 µT C−11︸︷︷︸≡a

= bu1 + au2

(6.14)︷︸︸︷= µ

(6.18)

(6.17)&(6.18)︷︸︸︷⇔ (a c

b a

)︸︷︷︸

≡A

(u1

u2

)︸︷︷︸

≡u

=

(1

µ

)(6.19)

150


Calculate the inverse of A as:

A−1 = 1det(A)

(a −c

−b a

)

= 1

bc− a2︸︷︷︸≡−d

(a −c

−b a

)

= 1d

(−a c

b −a

) (6.20)

With (6.19) and (6.20) follows:

u = A−1

(1

µ

)

=1

d

(cµ − a

b − aµ

)(6.21)

Putting (6.21) into (6.16) yields to Equation (6.4), the optimal solution x of Pσ2 :

x

(6.16)︷︸︸︷= u1C

−1µ + u2C−11

(6.21)︷︸︸︷= 1

d((cµ− a)C−1µ + (b− aµ)C−11)

With it, Equation (6.5) follows with:

σ2(µ) = xT Cx(6.13)︷︸︸︷= u1µ

T x + u21T x

(6.17)&(6.18)︷︸︸︷= u1µ + u2

(6.21)︷︸︸︷=

1

d((cµ− a)µ + (b− aµ))

=cµ2 − 2aµ + b

d

Furthermore, the minimum of Equation (6.5) yields to the Minimum Variance Port-

folio denoted with xMV P , i.e. Equation (6.6), and its points in the risk return space,

151


i.e. Equation (6.7):

∂σ2(µ)

∂µ=

1

d(2cµ− 2a) ≡ 0

⇒ µMV P =a

c(6.22)

To check for a minimum, the second partial derivative is positive:

∂2σ2(µ)

∂2µ=

2c

d(6.10)&(6.11)︷︸︸︷

> 0

Putting this into Equation (6.5) yields to:

σMV P =√

σ2(µMV P )(6.5)︷︸︸︷=

√cµ2

MV P − 2aµMV P + b

d

(6.22)︷︸︸︷=

√c(

ac

)2 − 2a(

ac

)+ b

d

=

√1

c

Together we have shown Equation (6.7), the location of the Minimum Variance

Portfolio in the risk return space:

(µMV P , σMV P ) = (a

c,

√1

c)

To find the Minimum Variance Portfolio xMV P respectively the weights of the Mini-

mum Variance Portfolio coded in xMV P , we put Equation (6.7) into Equation (6.4):

xMV P

(6.4)︷︸︸︷=

1

d

((cµMV P − a)C−1µ + (b− aµMV P )C−11

)(6.22)︷︸︸︷=

1

d

((c(a

c

)− a)C−1µ + (b− a

(a

c

))C−11

)=

1

cC−11

152


Finally, the efficient frontier is calculated from Equation (6.5), whereby we have to

define σ ≡ σ(µ):

σ2

(6.5)︷︸︸︷=

cµ2 − 2aµ + b

d⇔

d

cσ2 = µ2 − 2

a

cµ +

b

c

=(µ− a

c

)2−a2

c2+

b

c︸︷︷︸bc−a2

c2

(6.22)&(6.11)︷︸︸︷= (µ− µMV P )2 +

d

c

1c

= (µ− µMV P )2 +d

cσ2

MV P

⇔

(µ− µMV P )2 =d

c(σ2 − σ2

MV P )

⇔

µ = µMV P ±√

d

c(σ2 − σ2

MV P )

2

Following Theorem 6.1 we calculated the efficient frontier with and without com-

modities. For it, we took the annualized means over the period 1991 until 2006, i.e.

caused by the time additivity of log returns this is equal to taking rolling averages

over one year periods, and linear correlations of stock, bond and commodity returns

as reported in Table 6.4. Our results are shown in Figure 6.3. The red line repre-

sents the efficient frontier with commodities and the brown line draws the efficient

frontier without commodities.

The efficient frontier with commodities is superior to the efficient frontier without

commodities, i.e. including commodities in the opportunity set improved the risk

and return tradeoff over the entire risk levels under consideration. If an investor had

invested into a portfolio including commodities, he would have realized the different

return expectations with lower risk amounting on average to 0.77% annualized stan-

dard deviation. The Minimum Variance Portfolio decreased in risk around 0.8%

annualized standard deviation. This is shown in Figure 6.3: the Minimum Vari-

ance Portfolio (MV P ) without commodities has an annualized standard deviation

of around 4.5% whereby the Minimum Variance Portfolio (MV Pc) with commodi-

ties has an annualized standard deviation of around 3.8%. Implicating, including

153


Figure 6.3: Efficient Frontiers with and without Commodities (1991-2006)

commodities into the asset allocation improves the risk structure in comparison to

a portfolio only including traditional asset classes.

An other interesting inside into the team play of the three asset classes in a port-

folio gives an efficient frontier area graph as in Figure 6.4. It display the change

of the asset weights x of the efficient frontier across the entire risk spectrum. Con-

sequently, the efficient frontier area graph is similar to a standard asset allocation

pie chart that shows the asset allocation that corresponds to a particular spot on

the efficient frontier, except the efficient frontier area graph displays all of the asset

allocations on the efficient frontier. It is helpful to identify the substitution of the

different asset classes over the different risk levels. The black bar in the respective

diagram of Figure 6.4 represents the Minimum Variance Portfolio’s asset allocation.

All weighting combinations on its right side yield to efficient risk and return profiles

of the resulting portfolio, i.e. the respective asset allocation yield to risk and return

profiles of the positive case in Equation 6.8.

The left diagram in Figure 6.4 shows the asset allocation of traditional portfolios

only including stocks and bonds. It is not astonishing that with increasing stock

allocation the portfolios’ risk rises. The Minimum Variance Portfolio consists of

11.8% stocks and 88.4% bonds. The right diagram in Figure 6.4 shows the alloca-

tion of a portfolio including stocks, bonds and commodities. Over the entire risk

range commodities are allocated positively ranging from 14.9% in the Minimum

Variance Portfolio until 27.7% in a high risk portfolio. Not until a standard de-

viation of around 12% per annum, commodities are substituted by stocks as well.

This is not astonishing when comparing the input parameters of Table 6.3. At the

154


Figure 6.4: Comparison of Portfolio Allocation

right end of the efficient frontier a complete stock portfolio as riskiest opportunity

is located.

The historical mean variance analysis has shown that commodities are an essential

part of the asset allocation if portfolios shall be generated that have superior risk

and return profiles in comparison to stock and bond only portfolios. But we have

to take the ever-present disclaimer into consideration that ”past performance is no

guarantee of future performance”. Moreover, mean variance optimization is very

sensitive to the estimates of returns, standard deviations and correlations.142 But

of the three inputs required to create an efficient frontier, returns are by far the

most important, and unfortunately, the least stable. [Chopra Ziemba 1993] esti-

mated that at a moderate risk level, mean variance optimization is 11 times more

sensitive to small changes in returns relative to small changes in the risk measure

standard deviation. Furthermore, mean variance optimization is two times more

sensitive to small changes in risk relative to small changes in correlations. We can

assume that the underlying historical correlation structure between the traditional

asset classes and commodities will not fundamentally change because it is not ex-

pected that the underlying economic dependence structure between the traditional

asset classes and commodities will change. Expected returns are not that stable.

Therefore, we calculated the minimum annual return, commodities should produce

to be allocated with stocks and bonds in a portfolio, the so-called Hurdle Rate. As

reference portfolio we used a 25% stock and 75% bond allocation. We identified

a hurdle of 4.5%, i.e. when commodities produce an annualized return of 4.5% or

more, they are allocated in the mean variance framework. The modification of the

efficient frontier when allocating commodities under the described assumptions, is

142See [Best Grauer 1991] and [Michaud 1998].

155


plotted in Figure 6.5.

Figure 6.5: Efficient Frontier and the Hurdle Rate

The Minimum Variance Portfolio of a traditional stock and bond only portfolio has

as reported in Figure 6.4 a stock allocation of 11.6% and a bond allocation of 88.4%.

Adding commodities yield to an improvement in two directions: choosing portfolio

PC2 at the efficient frontier with commodities would decrease the taken risk around

0.3% per annum while producing the same annualized return of 4.48%. In the sec-

ond possible case, choosing portfolio PC1 at the efficient frontier with commodities

would increase the return around 0.2% per annum while taking the same risk of

7.08%. This shows that especially the attractive standard deviation and correlation

structure commodity returns have to stock and bond returns cause the need for

allocating this asset class and not the extraordinary returns generated during the

last years. Nevertheless, 64% of the historical annualized returns were bigger than

the hurdle rate.

Closing this section we can state, that allocating commodities is essential for attrac-

tive risk and return profiles especially in the low risk space.

156

7 Conclusions

As globalization and computerization enabled multinational companies to spread

out their production plants, the economic renaissance of Asian countries has begun.

To satisfy the growing need for infrastructure, new buildings and electrification,

huge amounts of metals and energy were pulled of global trade into the emerging

markets. But the commodity producing industry was not prepared and scarcity

let prices rise. This forced company’s financial managers to maintain their risk

management systems and brought them to mind the need for financial risk hedging

products, investors were attracted by the extraordinary returns. The goal of this

thesis has been to highlight commodities as an asset class. Introductory, we gave an

overview of commodity markets which consists of three sub markets: energy, metals

and agriculture. Different characteristics and fields of usage rose our awareness that

there does not exist the ”average” commodity. The macroeconomic and statistical

facilities of the commodities under consideration differ essentially among each other

resulting in diversification benefits when considering commodity baskets.

Nevertheless, commodities embody commonly a special facility: they are consump-

tion goods. Therefore, the elementary financial products to trade commodities can-

not be valued following the same arbitrage arguments as they are used in traditional

financial derivatives pricing. Depending on the view that is taken of commodities

as consumption goods or financial assets, two different commodity futures pricing

concepts were developed. If commodity futures are seen as derivatives written on a

non-tradable reference figure, i.e. on the price of a consumption good, they should

be valued based on equilibrium asset pricing concepts. But if commodity futures

are seen as derivatives written on an asset-like underlying, they should be valued

based on arbitrage related concepts. Risk Premium Models are equilibrium asset

pricing concepts and Convenience Yield Models are arbitrage related valuation con-

cept. The convenience yield captures the additional value of the commodity as

consumption good on top of its value as tradable asset. Following [Markert 2005],

we have shown that both valuation concepts are mutually consistent and can be

derived from each other if the convenience yield is interpreted as the deviation of

the commodity spot price from the value of the commodity as a pure financial asset.

Especially for risk management purposes, stochastic models were discussed to clone

observed market prices. We introduced the most common one, two and three factor

models. Although latter ones fit best the term structure of commodity futures, two

factor models are still more accepted in practice because they have the best trade

off between fitting advantage and computational costs.

157

7 Conclusions

The main focus of this thesis was put on diversified commodity exposure represented

by the DJ-AIGCI. The commodity index analyzes of Section 4.2.7 and Section 5.1.3

identified it as balanced commodity investment basket using the attractive diversifi-

cation effects offered by the heterogeneous commodity market. If commodities shall

actually be allocated in a portfolio, total returns have to be considered. Their de-

composition has shown that they consist of spot, roll and interest rate returns. For-

mer represent the simple price changes of the included commodities over time. But

these are only tradable with futures contracts having fixed maturities. Therefore,

long term orientation includes rolling the futures investment forward and creating

therewith the so-called roll returns. To trade futures contracts, only minimal cash

is required to serve margin calls. Therefore, fully collateralized futures portfolios

are considered to produce additional interest rate return. Recall Figure 5.7 where

the performance of single return elements are shown and the huge negative impact

of roll returns on the total return.

The ensuing statistical analysis of the return components aimed to identify the

behavior of commodity returns in the portfolio context. Normality of the DJ-AIGCI

total return could not be proofed. The distribution is influenced by outliers and

peaked around the mean. Nevertheless, we found that correlations to the returns

of the traditional asset classes, stocks and bonds, were slightly negative. Moreover,

while stock and bond returns show the well known leverage effect, i.e. returns

and their volatility are positively correlated, commodity returns exhibit a ”negative

leverage effect”, i.e. returns and their volatility are positively correlated. Price

surges in commodity markets are caused by low inventories and the fear of possible

supply interruptions. This makes market participants nervous and it is expressed

by higher volatility.

To answer the question if commodities indeed represent an asset class of its own,

the mean variance spanning has to identify a statistical significant risk premium

that cannot be explained by other already established asset classes. We found a

selected period that satisfies this condition. Moreover, a factor analysis identified

a single implicit risk factor driving commodity returns independently to stock and

bond returns. We deduce that commodities tend to be indeed a separate asset

class with its own risk and return facilities that are different to the characteristics

stock and bond returns exhibit. Mean variance and hurdle rate analyzes have shown

that allocating commodities to a traditional stock and bond portfolio yield to more

attractive risk and return profiles. This is reasoned by the risk and correlation profile

commodity returns have to stock and bond returns and not by the extraordinary

commodity returns of the last years.

158

Appendix

A Data Description

Availability of data in commodity markets is rather scarce since the broad market

development started only a couple of years ago and different sources are spread out.

Because commodities are traded in US dollar, all analyzes are in US dollar.

The first analysis in Section 2.1 and Appendix B yield to show the macroeconomic

embedding of commodities as consumption good. The main data source was the

[The CRB Commodity Yearbook 2005] which describes/publishes production, con-

sumption and price data. Over the long run, only spot price data were available.

Therefore, the analyzes are based on this type of commodity return.

Since the [The CRB Commodity Yearbook 2005] was not appropriate as data source,

we used Bloomberg publishing ending stocks, production and consumption data for

selected commodities including different agricultures and metals. Sometimes, the

specific organizations also provided these information, i.e. [USDA Livestock 2006].

In Section 4.4 we used different futures prices to show how commodity time series

are created. Data came from Bloomberg and the ticker were composed following the

lower methodology:

NYMEX crude oil futures contract with ticker CLxy Comdty, whereby

x ∈ (F (=Jan), G (=Feb), H (=Mar), J (=Apr), K (=May), M (=Jun), N

(=Jul), Q (=Aug), U (=Sep), V (=Oct), X (=Nov), Z (=Dec))

representing the respective month starting with and y ∈ (5, 6) representing the

respective year

LME copper futures contract with ticker LPxy Comdty, whereby

x ∈ (F (=Jan), G (=Feb), H (=Mar), J (=Apr), K (=May), M (=Jun), N

(=Jul), Q (=Aug), U (=Sep), V (=Oct), X (=Nov), Z (=Dec))

representing the respective month starting with and y ∈ (5, 6) representing the

respective year

While there exists a huge amount of literature that examines time series constructed

from futures prices, little is said about market indices and their components. There-

fore, our analysis is focused on the indices in Section 5. In the first part, in

Section 5.1 we aim to give a market overview and compare single commodities,

different commodity groups and major market indices. The RICI is the youngest

index with its introduction in August 1998. We set this time as a start to compare

different indices over the same period of time. Taking into consideration that Gol-

160

man Sachs (GS) publishes as single provider a spot, excess and total return for all

its single and sub indices, we used in this analysis the GS single and sub indices.

The broad indices were calculated by their respective issuer. The Bloomberg tickers

are as follows:

GS Gasoline Spot Return: GSCCHUSP Comdty

GS Gasoline Excess Return: GSCCHUER Comdty

GS Natural Gas Spot Return: GasGSCCNGSP Comdty

GS Natural Gas Excess Return: GSCCNGER Comdty

GS Nickel Spot Return: GSCCIKSP Comdty

GS Nickel Excess Return: GSCCIKER Comdty

GS Zinc Spot Return: GSCCIZSP Comdty

GS Zinc Excess Return: GSCCIZER Comdty

GS Gold Spot Return: GSCCGCSP Comdty

GS Gold Excess Return: GSCCGCER Comdty

GS Corn Spot Return: GSCCCNSP Comdty

GS Corn Excess Return: GSCCCNER Comdty

GS Lean Hogs Spot Return: GSCCLHSP Comdty

GS Lean Hogs Excess Return: GSCCLHER Comdty

GS Sugar Spot Return: GSCCSBSP Comdty

GS Sugar Excess Return: GSCCSBER Comdty

GS Energy Spot Return: GSENSPOT Comdty

GS Energy Excess Return: GSENER Comdty

GS Industrial Metals Spot Return: GSINSPOT Comdty

GS Industrial Metals Excess Return: GSINER Comdty

GS Precious Metals Spot Return: GSPMSPOT Comdty

GS Precious Metals Excess Return: GSPMER Comdty

GS Agricultures Spot Return: GSCAGSPT Comdty

GS Agricultures Excess Return: GSCAGER Comdty

DJ-AIGCI Spot Return: AIGDJAIGSP Comdty

DJ-AIGCI Excess Return: DJAIG Comdty

161

A Data Description

GSCI Spot Return: GSCISPOT Comdty

GSCI Excess Return: GSCIER Comdty

DBLCI Mean Reversion Excess Return: DBLCMMCL Comdty

DBLCI Excess Return: DBLCIX Comdty

RICI Excess Return: RICIGLER Comdty

In Section 5.1.3 we decided to direct any further analyzes to the DJ-AIGCI. To

cover the whole index history, we took data available as from 01.01.1991. Dow

Jones (DJ) publishes excess return time series for its single and sub indices. Only

for the sub indices it publishes the spot return series, too. Therefore, the analysis of

the DJ-AIGCI return components was confined. Nevertheless, the data for the factor

analysis in Section 5.1.3 and the other analyzes in Section 5.2 were downloaded from

Bloomberg. The tickers are as follows:

DJ-AIGCI Total Return: DJAIGTR Comdty

DJ-AIGCI Spot Return: DJAIGSP Comdty

DJ-AIGCI Excess Return: DJAIG Comdty

DJ Energy Spot Return: DJAIGENSP Comdty

DJ Energy Excess Return: DJAIGEN Comdty

DJ Non - Energy Spot Return: DJAIGXESP Comdty

DJ Non - Energy Excess Return: DJAIGXE Comdty

DJ Industrial Metals Spot Return: DJAIGINSP Comdty

DJ Industrial Metals Excess Return: DJAIGIN Comdty

DJ Precious Metals Spot Return: DJAIGPRSP Comdty

DJ Precious Metals Excess Return: DJAIGPR Comdty

DJ Agricultures Spot Return: DJAIGAGSP Comdty

DJ Agricultures Excess Return: DJAIGAG Comdty

DJ Softs Spot Return: DJAIGSOSP Comdty

DJ Softs Excess Return: DJAIGSO Comdty

DJ Grains Spot Return: DJAIGGRSP Comdty

DJ Grains Excess Return: DJAIGGR Comdty

162

DJ Livestock Spot Return: DJAIGLISP Comdty

DJ Livestock Excess Return: DJAIGLI Comdty

Finally, in Section 6 we did some analysis to embody commodity returns into the in-

vestment environment of the traditional asset classes. For this purpose, the following

data were used:

DJ-AIGCI Total Return: DJAIGTR Comdty

MSCI World USD: MSDUWI Index

S&P500 Total Return: SPTR Index

MSCI Europe: MSDUE15 Index

J.P. Morgan Global Government Bond: JPMGGLBL Index

J.P. Morgan Government Bond Index USA: JPMTUS Index

J.P. Morgan Government Bond Index Europe: JPMGEURO Index

OECD US Consumer Price Index, All Items: OEUSC009 Index

All data were available in daily frequency, excluding the OECD US Consumer Price

Index. It is given in a monthly frequency and its changing serves as measure of

inflation.

163

B Characteristics of Selected Commodities

B.1 Heating Oil

About 25% of a barrel crude oil is used to produce heating oil. It is the second

largest downstream product after gasoline refined from crude oil. Therefore, its

price is highly correlated to crude oil prices. The consumer’s price for home heating

oil generally includes up to 50% for crude oil, 11% refining costs, and 39% marketing

and distribution costs.143 Hence, there exists a trade off between heating and crude

oil prices. Demand and supply shifts caused by changes in weather or refinery shut-

downs result in higher heating oil prices and effect the simultaneous rise of crude oil

prices. Figure B.1 shows the similar price pattern over the last 40 years. The sig-

nificant correlation coefficient between price changes in the two price series is 0.66.144

Figure B.1: Dependence of Heating Oil Prices to Crude Oil Prices

The processing margin which is earned when refiners buy crude oil and refine it into

heating oil and gasoline is called ”crack spread”. It is common industry practice to

react to the crack-spread ration 3-2-1, which involves selling 1 heating oil contract

and 2 gasoline futures contracts and buying 3 crude oil contracts. As long as the

crack spread is positive, it is profitable for refiners to buy crude oil and refine it into

the downstream products.

Heating oil is mainly used for residential heating. In the US, there are still over

7 million households which use it as primary heating fuel. The peak in demand

143See Energy Information Administration: stand 2002.144For this analysis we took monthly cash data of the [The CRB Commodity Yearbook 2005]


164

B.1 Heating Oil

occurs during the winter months from October through March. Therefore, heating

oil prices fluctuate seasonally. Prices increase during the filling months from March

through October. Moreover, unexpected long and hard winters can cause further

price rises due to pumping, pipeline or refinery bottlenecks within the period from

December to March. In the last years demand in heating oil decreased in industrial

countries. Many households began to switch over to more convenient heating sources

such as natural gas. Furthermore, there is a trend to mix traditional heating oil

with natural sources. One idea to do so comes from the Purdue University (USA).

They found a combination of 20% of soybean oil and 80% of conventional oil to be

sufficient: The mix can be used in conventional furnaces without altering existing

equipment, is relatively easy to produce and produces no sulphur emissions.

Thus, it appears that the high cost environment which we currently have in crude oil

markets, applies to heating oil markets, too. Figure B.2 shows the prices of heating

oil at the New York Mercantile Exchange (NYMEX) for different future delivery

dates as of July 2006.145

Figure B.2: Heating Oil Prices for Future Delivery

First, we clearly discover the seasonality: prices are expected to fall towards the end

of winter in March until the beginning of the filling season in July. In August 2007

prices are expected to rise again. Second, the market expects heating oil to be

more expensive in 2007 than in 2006. Market participants are willing to pay over

5 US dollar more for heating oil that will be delivered in one and a half year than

for heating oil that will be delivered this winter.

145Data source: Bloomberg.

165


B.2 Gasoline

About 50% of a barrel crude oil is used to refine gasoline. In the USA, it accounts

for about 17 % of the energy consumed yearly.146 The primary use of gasoline is in

automobiles and light trucks. Gasoline also fuels boats, recreational vehicles, var-

ious farm machines, and other equipment. While gasoline is produced year-round,

extra volumes are made for the summer driving season. There are three main grades

of gasoline: regular, mid-grade, and premium. Each grade has a different octane

level.147 Octane is a measure of a gasoline’s ability to resist the pinging and knock-

ing noise of the engine. Additional refining steps are required to increase the octane

which increase the retail price. Figure B.3 shows the price movements of gasoline

compared to price movements with its major input factor crude oil. Changes in both

price series have a significant correlation coefficient of 0.67.148 Both, heating oil and

gasoline, are priced with a refinery margin on top of the crude oil price. But in

contrast to heating oil, the gasoline peak time is in summer throughout the driving

season. That is why these prices were more strongly submitted to the impacts of

hurricane Katrina in August 2005 than the heating oil prices were.

Figure B.3: Dependence of Gasoline Prices on Crude Oil Prices

In 2004, US retail prices of gasoline were summed up of 44% for crude oil, 27% for

federal and state taxes, 14% for distribution and marketing and 15% for refining

costs and profits. But in 2005, the US retail price of gasoline was summed up

146See [The CRB Commodity Yearbook 2005].14787 (R+M)/2, 89 (R+M)/2 and 93 (R+M)/2148For this analysis we took monthly cash data of the [The CRB Commodity Yearbook 2005]


166

B.3 Gold

of 47% for crude oil, 23% for federal and state taxes, 12% for distribution and

marketing and 18% for refining costs and profits.149 Both, crude oil prices and

margins rose. In real terms, the margin for refinery increased around 10 US cents

per gallon. This is typically for market environments with rising prices and costs

being passed on to customers.

High costs and environmental considerations have brought about reduced gasoline

consumption over the course of years. In an attempt to improve air quality and

reduce harmful emissions from internal combustion engines, the US Congress passed

the Clean Air Act to mandate the addition of ethanol to gasoline in 1990. The

most common blend is E10, which contains 10% ethanol and 90% gasoline. Auto

manufacturers have approved a fuel mixture that is produced by fermenting and

distilling crops such as corn, barley, wheat and sugar. In Brazil, this proportion

is much higher. Ethanol accounts for 25% to 35% of the fuel. However, diesel

and bio fuels are getting more and more popular in general. South America has

by far the highest usage rates, but Europe and North America are increasing their

consumption, too. Nevertheless, we should also consider the growing demand of fuel

and gas from emerging Asian markets. Taking into account that Asian countries pay

little attention to environmental issues, we may expect that these demands continue

to grow.

B.3 Gold

Since 1886 South Africa has been the gold producing mecca of the world. At its

peak production in 1970 South Africa contributed 79% of the world’s annual supply.

Its dominant position has waned in the last 30 years. Although it is still world’s

largest producer, it just accounts for 20% of world production followed by the USA

that accounts for around 10% of world production and China, Russia and Australia

which all account for nearby 10% of world production.150 South Africa’s diminishing

production dominance is primarily due to vast gold discoveries in North America

and Australia. Nevertheless, it continues to be unchallenged in the more important

category of gold reserves. Moreover, recent manpower cuttings and rationalizations

resulted in major cost reductions throughout the South African industry. Average

production costs are now lower than North American mine ones. Especially during

the years 1996 and 2001 as the gold price was very low South Africa’s currency, the

Rand, depreciated to the US dollar over 50%. This dependency of the gold price

149See US Energy Information Administration (www.eia.doe.gov)150See [UBS research 2005].

167


cried for economical diversification. Therefore, South African producers shifted their

focus into other precious metals like platinum which provided lucrative profits over

the last years and together with an increasing gold price the Rand re-appreciated.

However, production extended in other parts of the world like Indonesia, Peru, Ar-

gentina and the USA netting off South Africa’s production downturn. After a weak

supply year 2004 that left a negative production consumption balance of -135.8

tonnes, the supply increased in 2005 to 3.9 million tonnes and could fill the bal-

ance gab of the last year, although consumption increased by around 7%. The gold

demand is influenced by three independent factors: the investment, the industrial

and the hedging demand. First, geopolitical pressure and wealth insurance caused

by a weak dollar are the main drivers of the building up of gold reserves. Second,

gold’s major industrial use is jewelery, dentistry and electricity. The first two are

mainly driven by standards of life. Gold jewelery has its major use in the Middle

East and India, not only to dress women but also for religious purposes. With an

increase of wealth a demand surge can be expected. Third, the hedging activities of

gold producing companies influence the gold price. In the first quarter of 2006 they

held back 18% of mine production to sell it at higher prices. Figure B.4 shows the

development of gold prices and inventories between 1992 and 2006.151

Figure B.4: Gold Inventories and Prices

Although inventories went up over the last years the gold price did as well. This

reflects the currency facility of gold in international systems and the negative US

economy influences. The left scale is also used to reflect the values of the US dollar

trade-weighted index to show the depreciation of the US dollar over time in contrast

to the appreciation of gold over the same time horizon.

151Data source: Bloomberg

168

B.4 Aluminium

B.4 Aluminium

The silvery, lightweight metal called aluminium is extracted from an aluminium ore,

also known as bauxite. The primary method is the electrolytic reduction. It was

simultaneously discovered in 1886 by Charles Martin, USA, and Paul L.T. Heroult,

France.152 Bauxite can be found in the tropical and sub-tropical areas of Africa,

India, South America and Australia. By volume, aluminium weighs a third as steel

and has therefore a high strength-to-weight ratio which makes it ideal for building

and construction what accounts for 22% of its total usage. Due to its resistance to

corrosion in salt water, it is used in boat hulls and various marine devices. Generally,

26% of the aluminium consumption comes from transportation business including

auto mobiles and air plants. Another 22% of the total consumption are used in

packing industry and the rest is split to cooking materials, low-temperature nuclear

reactors, machinery and electricity.

As mentioned by way of introduction, China is the heaviest user worldwide. Al-

though 24% of world production is done in Asia, Chinese have massive problems

with old and inefficient smelting plants that hardly can serve the internal demand.

In 2005, Beijing introduced a 5% export tax to focus local producer on home mar-

kets. While other global operating companies could buffer the trade drop from Far

East over the short run, the situation became critical in 2006 as Figure B.5 shows.

Inventories hit its 7 years low.

Figure B.5: Aluminium Inventories and Prices

Huge requirements of energy for the electrolytic reduction process present the main

problem in aluminium production. The production of one ton of aluminium requires

152See [The CRB Commodity Yearbook 2005].

169


the same amount of energy as necessary to ensure the power supply of a detached

family house throughout two years. Implicating, aluminium production costs are

highly correlated to energy prices and many aluminium smelting plants are married

to coal power plants that are nowadays replaced by water or nuclear power plants

because of the high environmental pollution caused by burning coal to energy. The

world’s biggest aluminium producer, Alcoa, is investing approximately one billion to

build a smelting plant on Island where volcanic natural heat of the earth is cheap.

The costs for the transportation of raw material from Brazilian ore mines to Iceland

by sea and the backhaul of pure aluminium to consumer markets in Europe, North

America and Asia do not net off the high energy costs somewhere else in the world

making this complex logistic solution to the most cost efficient solution.

Looking at today’s heaviest Aluminium consumer China, it has still massive energy

problems which limit the aluminium production of the country. In view of these

problems, it cannot be assumed that significant production expansion will decrease

prices for the short term.

B.5 Copper

The red coloured metal copper is the oldest metal in the history of mankind. It is

extracted and worked up since 5,000 BC. Copper is, however, not only the oldest

metal used by humans, but also one of the most widely used industrial metals.

It is an excellent conductor, highly corrosion-resistant and ductile. Employment

within electrical industry accounts about 41% of total copper usage with increasing

tendency, owing to its better electric conductivity in comparison with aluminium.

Therefore, it serves as a high-class substitute for aluminium. Building construction

is the single largest market accounting for 48% of total usage.153 For better clearness:

the average US home contains 200 kilograms of copper. Moreover, copper is biostatic

which means that bacteria cannot grow on its surface. Therefore, it is used in air-

conditioning systems, on worktops and doorknobs to prevent a spread of diseases.154

The biggest copper output comes from Chile with 37% of world production. Alone

8% of the world supply comes from the biggest BHB Billiton owned copper mine

Escondido in the Atacama Desert. Last year, the company invested around 400

million US dollar in a northerly located newly opened pit to meet the world copper

demand. A range of smaller suppliers are Australia, the USA, Russia, Peru and In-

donesia, all accounting between 6% and 8% of world production. Figure B.6 shows

153See LME154See [The CRB Commodity Yearbook 2005].

170

B.6 Lead

the price and inventory development over the last years.

Figure B.6: Copper Inventories and Prices

The huge drop in inventories came in line with a huge price increase in 2003 which

was caused by an accident in the world’s third largest mine, the Indonesian Grasberg

mine. A wall came down and production was discontinued for several month. But

as it can be seen, inventories grew again. Although, China can only meet 4% of

its demand, production is stable again and higher than consumption yielding into a

refill of inventories. Vast copper reserves give reason to assume that copper prices

will soon come down as it is the most over-priced metal.

B.6 Lead

Lead is a dense, toxic, gray metallic element. It is known for its stability and is

one of the oldest mined and worked up metals. In the past, the metal was used in

decorative elements, windows, roofs and pipelines routed in castles and churches.

Water pipelines in Roman Aqueducts were made of lead. As scientists found out

about the poisoning factor of lead, they speculated that lead poisoning could yield

into the fall of the Roman Empire. But these theories were later disproved.

Today, lead is mainly used in electronics, that accounts for about 75% of its total

usage. Batteries of nearly all transportation vehicles contain lead. Moreover, lead

is extensively used as a radiation shielding material owing to its high density and

nuclear properties. The steadily worldwide increasing demand for electricity and

the rediscovery of nuclear energy have caused a heavy demand for lead over the past

years. Furthermore, lead is used as protection against radiation from computers,

171


television and telecommunication.155 Because production did not increase with de-

mand, the stock of inventory fell over 75% between 2002 and 2004. As a natural

reaction, lead prices went up as Figure B.7 clearly shows.

Figure B.7: Lead Inventories and Prices

The tremendous drop was mainly driven by a production decline in Western coun-

tries. Caused by low lead prices during the 1990s companies went out of busi-

ness. But demand increased steadily and in 2005, there was a world deficit in

the production-consumption balance sheet of 86,000 tonnes. The major producer

is China with nearly one third of total world production and high growth rates of

10.7% in 2005. Since 2001, refined lead metal output in China has doubled. The ma-

jor lead producer is the Yugang Lead Group which produced nearly 230,000 tonnes

of refined lead in 2005, but on the other hand, China also caused the tremendous

demand increase for lead. Last year, its demand increased around 40%. Therewith,

China overtook the USA as the world’s largest consumer of lead metal.

Rising prices attracted new producers to jump into the market. For instance, in

Australia the Magellan mine was opened in 2005, promising 70,000 tonnes output

per year. But also Kazakhstan and India, where Hindustan Zinc commissioned a

new plant earlier this year, pushes production. The International Lead and Zinc

Study Group forecasts that supply will exceed demand in 2006. This will give rising

prises a rest.

155See [Rogers 2005].

172

B.7 Nickel

B.7 Nickel

Nickel is a hard and ductile metal that has a silvery tinge. In 2005, 30% of

world supply came from Canada followed by Russia (23%), Australia (15%) and

Indonesia (10%).156 It is a good conductor of heat and electricity. It is used in

rechargeable batteries and in electric circuitry but only accounting for 8% of total

consumption. Primary nickel can resist corrosion and maintains its physical and

mechanical properties even if exposed to extreme temperatures. When these prop-

erties were recognized, the development of primary nickel began. It was found that

by combining primary nickel with steel, even in small quantities, the durability and

strength of steel increased significantly as did its resistance to corrosion. Today,

the production of stainless steel, a mixture of steel and nickel, is the single largest

consumer of primary nickel accounting for over 75% of total nickel consumption.

Therefore, it is not astonishing, that the run for nickel came in line with the run for

steel mainly caused by China’s building and construction boom. The price surge

and the drop in inventories can be seen in Figure B.8 and is highly dependent on

China’s need for stainless steel and other corrosion-resistant alloys.

Figure B.8: Nickel Inventories and Prices

The huge inventory loss through shrinkage was mainly caused by production prob-

lems in three Australian nickel projects (Murrin Murrin, Cawse and Bulong). But

this is all about to change. Moreover, the demand from China can be expected to

drop due to previous over-ordering. Nevertheless, inventories are low and further

supply interruptions or the inability to meet production plans will result in nervous

price amplitudes. While Australia’s nickel miners are doing well, the nickel smelting

156See [UBS research 2005].

173


industry worldwide is operating at close to capacity. Furthermore, world produc-

tion highly depends on Canada but caused by strong winters the country remains a

seasonal supplier.

B.8 Zinc

Zinc is a bluish-while metallic metal. It is never found in its pure state but rather

in zinc oxide, zinc silicate, zinc carbonate, zinc sulphide, and in different minerals.

China with 22%, Australia with 14%, Peru with 15% and Canada with 11% of

total world production are the major zinc suppliers.157 Primarily zinc is utilized

as a protective coating for other metals, such as iron and steel, in a process called

galvanizing and in copper-zinc alloys. The galvanizing process increases corrosion

resistance and accounts for almost half its modern-day demand. Viewing business

lines, 57% of total consume is pulled by building and construction business. Another

33% of total consumption are needed in transportation and machinery. Moreover,

zinc is used as the negative electrode in dry cell (flashlight) batteries and in round flat

batteries which are normally used in watches, cameras, and other electric devices.

With 22% of world production China is the biggest producer worldwide but with

a demand of 20% this nearly nets off. Chinese net imports of refined zinc metal

totalled 265,000 tonnes in 2005. The primary source of imported material continued

to be Kazakhstan, although substantial quantities were also sourced from Australia.

As is can be seen in Figure B.9 zinc inventories have decreased over the last three

years. Since 2004 zinc metal production has exceeded demand. In the Western

World there was a shortfall of 319,000 tonnes and globally of 317,000 tonnes in

2005.158 Although the USA decreased its demand, the growth in Chinese demand

exceeded the drop and caused the negative ending stocks.resulting in falling inven-

tories as Figure B.9 shows.

Although, mine output is stable and growing with rates around 4%, the International

Lead and Zinc Study Group forecasts an ending stock deficit of 437,000 tonnes in

2006. The global usage of refined zinc metal will increase again, strongest in Asia

where demand is forecast to rise by 7.3% in China, 9.1% in India, 4.5% in Japan and

4.4% in the Republic of Korea. This might put prices furthermore under pressure.

157See [UBS research 2005].158See The International Lead and Zinc Study Group.

174

B.9 Sugar

Figure B.9: Zinc Inventories and Prices

B.9 Sugar

”Sugar makes life sweet”, an advertising slogan that points out the importance of

this commodity for everybody. The brown and white crystalline fabric is a substance

produced from sugar cane or sugar beets. Sugar cane was originally known in tropical

regions of the world. People chew pieces of the stalk to extract the sweet taste. In

India, China and the Middle East the first refinery methodologies were introduced.

Since 1800, sugar is produced industrially and traded all over the world.

Today, sugar cane or sugar beets are planted in over 100 countries worldwide. Sugar

cane counts for 70% of world production and sugar beets count for the remainder.

The trend has been that production of sugar from cane is relatively increasing to

that produced from beets because sugar cane is a perennial, while sugar beet is an

annual plant. Due to the longer production cycle, sugar cane production is generally

more resistant to changes in price than sugar beet production.

The worldwide biggest producer is Brazil with around 20% of the world supply.

In the last 10 years it increased its production around 9% annually. Because of

low sugar prices between 1999 and 2002 many countries like Thailand, Pakistan

and India have reduced their production. The Brazilian production increase could

therefore just net of their reductions what caused inventories to fall and the prices

to rise. Figure B.10 shows the historical development of sugar production, consump-

tion, inventories and cash prices. Higher volatility of production in comparison to

volatility of consumption indicates that supply is the more susceptible component

affected e.g. by unexpected weather conditions.

Taking a look into futures prices of futures maturing in 2008 we notice that prices

will remain at a high level what is due to the growing demand for bio fuels as a

175


Figure B.10: Sugar Price, Stock of Inventory, Production and Consumption

substitute for gasoline. Ethanol refined from sugar is an ingredient to produce these

alternative fuels. Brazil is the biggest user worldwide with an increasing demand

for ethanol fuel caused by an enthusiastic reception by consumers of the flex-fuel

vehicle. Production reached its all time high with 17.4 billion liters. In the USA, 95

ethanol refineries were in production in 2005, 14 began production, 30 were under

construction, 10 were expanded and the industry produced a record of 15.1 billion

litres.159 Implementation of the Renewable Fuel’s standard (RFS) is in process and

will ensure a strong, long term future for bio fuel. Europe is focused to establish

bio fuel standards, too, but many countries are lagging behind. Nevertheless, in

November 2005 the ”biomass action plan” was passed including bio fuel targets and

assessments of how bio fuel incentives fit in with reforms of the Common Agriculture

Policy.

Closing this section, we shall take a look at Far East. China is with 8% of world

supply the third biggest producer but with 9% of world usage the second biggest

consumer. India is with 10% of world supply the second biggest producer but with

14% of world usage the biggest consumer. According to an ISO study of May 2006,

China is likely to increase its consumption about one third of today’s usage caused

by growing standard of living. We assume that this will be similar in India. When

it comes to food, we should keep in mind that around 3 billion people are hungry

to enter western standards.

159See International Sugar Organization (ISO) ”Quarterly Reports”

176

B.10 Coffee

B.10 Coffee

The black hot drink made out of coffee powder was first popular in Arabian countries

in the 13th century. Historians ascribe its popularity to the ban of alcohol in these

countries. The secret of planting coffee trees and roasting coffee beans, which are

finally crashed to coffee powder, was strictly kept. English men were the first who

cultivated coffee drinking in Europe in the 17th century. But caused by tree illnesses

in the colonial plants they switched over to tea. Today, the USA is the biggest

consumer with around 21 million bags in 2005, followed by Germany with around 8

million bags in 2005 and Italy and France, both with around 5 million bags in 2005.

The evergreen tropical shrub can grow up to 3.5 meters and it takes around 9

month to ripe the coffee beans. Generally there are two brands of coffee: Arabica

and Robusta.

The most widely produced coffee is Arabica, which makes up about 70% of total

worldwide production with Brazil and Colombia being the major producers. Ro-

busta is a more resistant brand and can be planted in a soil which is not suitable

to grow Arabica. The main producers are Indonesia, Vietnam and West Africa.

South America accounts generally for around two thirds of world production and

Africa and Asia share nearby equally the other third. Figure B.11 shows the price

development of Arabica coffee traded at the New York Board of Trade (NYBT) over

the last 60 years.

Figure B.11: Coffee Price

There were two major coffee crises during the last 25 years but the latter between

1999 and 2003 was the worst one: prices dropped around 50 cents per pound in 2002.

Although retail prices remained stable and industry’s income exceeded 70 billion US

177


dollar in 2005, the tiny fraction of 5.5 billion US dollar went to the producing coun-

tries where 125 million people are dependent on coffee with their livelihood. Prices

fell down that hardly that many producers went out of business and tried either to

change to crop or left the country. For instance, the USA had severe problems with

illegal immigrants from South America in this time, and in Colombia and Guatemala

there was a strong increase in coca planting. The price collapse was caused by over

production: extraordinary harvests in Brazil and a vast extension of production in

Vietnam by nearly doubling its output from around 7 million bags in 1998 to around

12 million bags in 1999 and becoming the world’s major producer of Robusta and

the second largest producer of coffee worldwide behind Brazil. Figure B.12 shows

the increase in inventory and the related price decrease.

Figure B.12: Coffee Price and Stock of Inventory

The years of coffee recession caused a production decline over the last 3 years from

121 million bags in 2002 to 106 million bags in 2005. On the other hand, consump-

tion developed constantly with annual growth rates of 2% over the same period. As

a result, inventories started to decrease and prices increased again. Because negative

price shocks generally resonate for a long time, it cannot be expected that produc-

tion will boom soon and it might be that coffee markets finally run into a period of

stable prices.

B.11 Soybean Complex

The soybean is a member of the oilseed family and is an ancient food crop from

China, Japan and Korea. There it has been known for more than 4,500 years. The

plant was introduced to its currently biggest producer, the USA, in the early 1800s.

178

B.11 Soybean Complex

Today, soybeans are the second largest crop produced in the USA behind corn. The

key value of soybeans lies in the relatively high protein content, its similarity to

corn and its remarkable resistibility which brought it the name ”miracle plant”.

The beans are planted in spring, usually in April and May, but at the latest until

early July. Late crop runs the risk of being caught by an early frost in fall and

may have difficulties flowering and setting pods in August. The seeds are harvested

100 - 150 days later in autumn.

Soybeans are used to produce a wide variety of food products. Its high protein

content makes it an excellent source of protein without many of the negative factors

of animal meat. Popular soy-based food products include whole soybeans roasted

for snacks or used in sauces, soy oil for cooking and baking, soy protein concentrates

which contain up to 92% protein, soy milk, yogurt and cheese, tofu, tofu products

and meat alternatives such as hamburger and sausages.

When it comes to exchange tradable soybean products, the market talks about the

”Soybean Complex” including soybeans, soybean meal and soybean oil whereby lat-

ter both are produced by crushing soybeans. Typically, about 19% of a soybean’s

weight can be extracted as crude soybean oil. The oil content of a bean corre-

lates directly with the temperatures and amount of sunshine during the soybean

pod - filling period. Soy oil is cholesterol-free and high on polyunsaturated fat. Of

the edible vegetable oils, soy oil is the world’s largest at about 32%, followed by palm

oil and rapeseed oil. An important product extracted from soybean oil is lecithin

which is used in many food preparations as an emulsifier. Soybean meal makes

up about 35% of the weight of raw soybeans. If the seeds are of particularly good

quality, then the processor can get more meal by including more hulls in the meal

while still meeting the 48% protein minimum needed to meet exchanges’ quality

requirements. Generally, soybean meal is used for animal feed for poultry, hogs and

cattle. It accounts for about two thirds of the world’s high protein animal feed. Its

main competitor is corn but owing to its higher protein content it exhibits a price

premium.160

Because soybean oil and meal are downstream products of soybeans, they are traded

at a premium called ”crush spread”, and price series are highly correlated with

significant coefficients of 0.82 between soybeans and soybean meal and 0.65 between

soybeans and soybean oil.161 It is a very popular agricultural spread and traded

160See [The CRB Commodity Yearbook 2005].161For the analysis we took monthly cash data since 1970 of the

[The CRB Commodity Yearbook 2005] completed with Bloomberg data for 2005 and

179


by the simultaneous purchase or sale of soybean futures and the sale or purchase of

soybean oil and soybean meal futures. Trading the spread between oil and meal is

also possible: if meal demand is high and oil demand is not, all the same processors

proceed crushing and allow oil stocks to build up in anticipation of future demand.

Bedding on the spread’s changes can therefore pay off.162

Prior to the 1970s, the USA had a monopoly on soybeans. Caused by a feed shortage

in protein, the secret of planting was passed on to Brazil and Argentina. Both coun-

tries boosted their production extraordinarily: while the USA accounted for around

50% of world production and 73% of world exports in 1995, their share reduced to

37% of world production and 42% of world exports in 2006. The lost market shares

went to Argentina and Brazil. Especially Brazil became the major competitor of

the USA: in 2005 they supplied around 39% of world soybean trade in comparison

to the USA what accounted for around 38% of world soybean trade. This year the

picture has changed dramatically: only 60% of this year’s Brazilian soybean crop

has been sold, yet. In comparison to the 5-year average, this is a sales drop of 10%.

It originates from the strong Brazilian Real compared to the US dollar. Whereas in

2004, it was possible to get over three Real for one US dollar, it is today merely 2.1.

This makes Brazilian products expensive in comparison to USA ones yielding to a

falling export rate to 36% for Brazil and a rising export rate of 42% for the USA

in 2006. Figure B.13 shows the global development of soybean inventory, production

and consumption over the last centuries.163

Soybean inventory has grown since the South American countries have pushed for

the market. The biggest importer worldwide is China, followed by Europe. While

Europe imported around 45% of world supply in 1995, the country of Far East

just accounted for 2.5% of world supply. In 2006 China’s imports have grown to

45% of world supply, while Europe’s imports fall back to 20% of world supply. The

strong increase is caused by rising meat consumption in China, a focus to industrial

products and a migration into cities. For the coming years, a further consumption

increase can be expected because the standard of living will increase and therewith

meat consumption.

2006. We used monthly log returns as of Definition C.2. For the mathematical definition ofPearson correlation and the related statistical test see Section 5.1.2.

162Because demand for oil and meal is driven by different factors their price series among eachother are less correlated than their price series to soybean prices are. The significant correlationcoefficient is 0.36.

163See [USDA Oilseeds 2006], Bloomberg and [The CRB Commodity Yearbook 2005].

180

B.12 Lean Hogs

Figure B.13: Soybean Price, Stock of Inventory, Production and Consumption

B.12 Lean Hogs

Hogs are generally bred twice a year in a continuous cycle designed to provide a

steady flow of production. The gestation period for hogs is 3 and a half months

and the average litter size is 9 pigs. After 3 - 4 weeks the pigs are taken away from

their mother and then fed to maximize weight gain. The food consists primarily of

grains such as corn, barley, wheat and soybeans for protein. Hogs typically gain 3.1

pounds per pound feed. The time until slaughter is usually 6 month when the pigs

have reached a weight of around 190 pounds.164

The biggest producer worldwide is China with around 53% of share of total world

production. Thus, small changes in Chinese production and consumption have a

significant impact to the world-hog markets. From 2005 to 2006 China increased its

production by almost 5%. This development may substantiated through manifold

reasons: pork’s’ popularity in Chinese diet, continued higher disposable incomes,

strong profitability in pork business, increased investment in the sectors’ operations

and the substitution of poultry due to the bird flue. The influence of latter one

can clearly be seen in Figure B.14 in a huge price increase of pork in 2003, the

year as bird flue was first mentioned in the media. Nevertheless, China’s per capita

consumption is around 39 kilograms per person. This is 5 kilograms less than in

Europe, the world’s second largest pork consumer in 2006.165 If China wants to

reach European standards they will have to produce 12 million tons more pork per

year i.e. they will have to increase their production around 25% and to cut off their

exports.

164See [The CRB Commodity Yearbook 2005].165Surprisingly, Hong Kong is the world’s biggest pork user. People there consume around

65 kilograms per year. See [USDA Livestock 2006]

181


Figure B.14: Lean Hogs Price, Stock of Inventory, Production and Consumption

Europe is the world’s largest exporter accounting for around 30% of world trade

and second largest producer with 21.5 million kilograms in 2006. No wonder that

the outbreak of swine fever at the end of the 1990s had tremendous impacts on

world pork prices as it can be seen in Figure B.14. Somebody might remember the

headlines as the Netherlands had to kill more than 12 million pigs in one go those

days.

Nowadays, EU business is pushed by the substitution of poultry and the cheap food

costs in the new member countries. Japan was one of the biggest addresses for

European pork. But prices can’t stand USA prices. It increased its exports by

around 80% since 2001. While its exports accounted for 21% of total world supply

in 2001 it accounts know for 25% of total world supply.

Since 2004, the so-called ”hog crush” and ”cattle crush” can be traded what was

enabled by a bilateral engagement of the Chicago Mercantile Exchange (CME) and

the Chicago Board of Trade (CBOT). The hogs spread is constructed by buying

one corn futures contract and selling two lean hog futures contracts because one

corn contract contains 5000 bushels or 2800 pounds, which is almost enough corn

to raise 400 pigs, the equivalent to 2 hog futures contracts. The cattle spread is

constructed by buying one corn and one feeder cattle contract and selling two live

cattle contracts. These new products developed originally out of the producers

hedging point of few enable speculators to ”feed” hogs or cattle ”on paper”.166

166See [Crush Spread 2006].

182

C Mathematical Preliminaries

C.1 Statistical Basics

In this section we will give some basic definitions of used terminology. Generally,

we tried to define the relevant term during the description of the respective analysis

for better connection of theory and praxis.

Our general starting point of all analysis are the price series available in Bloomberg.

Let Pt denote the price of an asset at time t. From this we can calculate two types

of return, a discrete and continuous one:

Definition C.1 Discrete Return

The one period simple or discrete return is defined as:

Rt =Pt − Pt−1

Pt−1

(C.1)

Definition C.2 Continuous Compounding Return

The one period continuous compounding or log return is defined as:

rt = ln

(Pt

Pt−1

)(C.2)

The following equation links both return definitions:

Rt = ert − 1 or rt = ln(Rt + 1) (C.3)

In general, the difference between continuous and simple returns is very small, es-

pecially for short time scales like ticks, days or months. This can be seen with the

Taylor series:

Rt

(C.3)︷︸︸︷= ert − 1 =

∞∑i=0

rit

i!− 1

=∞∑i=1

rit

i!= rt +

∞∑i=2

rit

i!(C.4)

Equation (C.4) shows that the two return definitions just differentiate from each

other over higher order terms. For values around zero, they have little weight causing

183


only small differences between the two return definitions.167

A major facility of log returns is its time additivity. Following Definition C.2, with

0 ≤ s < t ≤ T , we have:

r0→t = ln

(P (t)

P (0)

)= ln

(P (t) ∗ P (s)

P (0) ∗ P (s)

)= ln

(P (s)

P (0)

)+ ln

(P (t)

P (s)

)= r0→s + rs→t

(C.5)

For continuous compounding returns the recalculation to prices is given as:

Definition C.3 Price Series

Let rt denote the log return of an asset at time t. The price of the asset is then

defined as: P (t) = P (t− 1) ∗ ert .

The available information are mathematically embodied in the σ-Algebra.

Definition C.4 σ-Algebra

A system F of subsets of the sample space Ω is called σ-Algebra if it has the following

features:

1. ∅ ∈ F

2. A ∈ F ⇒ Ac ∈ F

3. A1, A2, A3 . . . ∈ F ⇒⋃n

i=1 Ai ∈ F

The σ-Algebra of all open intervals of R is called the Borel-σ-Algebra.

An important example of such a sigma-Algebra is he so-called Borel sigma-Algebra

B(Rk),with R denoting the real numbers, that is the smallest sigma-Algebra con-

taining all open sets in Rk.

Definition C.5 Probability Mass

Let F be a σ-Algebra in Ω. A probability mass is a function Q : F → R with the

following features:

1. Q(A) ≥ 0 for all A ∈ F

2. Q(Ω) = 1

3. for all Ai ∈ F , i ∈ N with Ai

⋂Aj = ∅ for all i, j ∈ N with i 6= j is:

Q

(∞∑i=1

Ai

)≡ Q

(∞⋃i=1

Ai

)=

∞∑i=1

Q(Ai) (C.6)

167Compare [Dorfleitner 2002].

184


The triple (Ω,F , Q) is called probability space.

A random variable is a mathematical function that maps outcomes of random ex-

periments to numbers.

Definition C.6 Random Variable

Let (Ω,F , Q) be a probability space and let B denote the Borel-σ-Algebra. The

function R : Ω 7→ R with R−1(B) ∈ F for all B ∈ B is called random variable.

A reel number R(ω) = r, ω ∈ Ω is called the realization of R.

Definition C.7 Distribution

The probability mass QR on (R, B1) defined by QR(B) ≡ Q(R−1(B)) is called

distribution of R.

A probability density function can be seen as a ”smoothed out” version of a his-

togram, e.g. Figure 5.12: if one empirically measures values of a continuous random

variable repeatedly and produces a histogram depicting relative frequencies of out-

put ranges, then this histogram will resemble the random variable’s probability den-

sity (assuming that the variable is sampled sufficiently often and the output ranges

are sufficiently narrow). Mathematically, the probability density function serves to

represent the probability distribution of a random variable.

Definition C.8 Density Function

A probability density function is any function f(r) that describes the probability

density in terms of the input variable r with the following characteristics:

1. f(r) is greater than or equal to zero for all values of r

2. the total area under the graph is 1:∫ ∞

−∞f(r)dr = 1

The cumulated distribution function describes based on the existence of a density

the probability that the random variable R takes on a value less than or equal to r

and is defined as:

Definition C.9 Cumulated Probability Function

Define a random variable R. The cumulated probability function is defined as:

F (r) = Q(R ≤ r)

∫ r

−∞ f(x)dx : if R is continuous∑rt≤r Q(R = rt) =

∑rt≤r q(rt) : if R is discrete168

(C.7)

185


The expected value of a random variable or its mean is the sum of the probability of

each possible outcome of the experiment multiplied by its payoff (”value”). Thus,

it represents the average amount one ”expects” as the outcome of the random trial

when identical odds are repeated many times. More mathematical spoken, the mean

is like the center of gravity of a density, i.e. the location of the density. It is called

the first moment of the density function and defined as follows:

Definition C.10 Mean

Define the random variable R ∈ R(T×1) describing all possible return realizations

of a commodity index over time. The distribution of R is described by the density

function f(r). Then the mean is defined as:

µ = E(R) =

∫∞−∞ rf(r)dr : if R is continuous∑T

t=1 rtq(rt) : if R is discrete(C.8)

The variance measures the dispersion of the density function about the mean and is

called the second moment. It indicates how possible values are spread around the

expected value, i.e. mean. While the mean shows the location of the distribution,

the variance indicates the scale of the values.

Definition C.11 Variance

Define the random variable R ∈ R(T×1) describing all possible return realizations

of a commodity index over time. The distribution of R is described by the density

function f(r). The variance is defined as:

σ2 = var(R) = E[(R− E[R])2] = E[R2]− E[R]2 (C.9)

A more understandable measure is the square root of the variance, called the stan-

dard deviation: σ =√

σ2. As its name implies it gives in a standard form an

indication of the possible deviations from the mean.

In financial theory the most used distribution is the normal distribution. Actually

it is a family of distributions, differing among their location and scale parameters,

i.e. their mean and variance. The standard normal distribution is the normal

distribution with mean zero and variance one.

Definition C.12 Normal Distribution

A random variable R is called normally distributed with mean µ variance σ2, i.e.

R ∼ N(µ, σ2),

186


if its density function is defined as:

f(x) =1

x√

2πσ2e−

(x−µ)2

2σ2

A random variable is called standard normally distributed if it is normally distrib-

uted with mean zero and variance one.

Definition C.13 Log-normal distributed

A random variable R is called log normally, if ln R is normally distributed with

mean µ variance σ2, i.e.

ln R ∼ N(µ, σ2),

if its density function is defined as:

f(x) =1

x√

2πσ2e−

(ln r−µ)2

2σ2 , r > 0.

The covariance describes a linear dependence between the variance of two random

variables.

Definition C.14 Covariance

Define the random variables L and R ∈ R(n×1) describing all possible return real-

izations of a two commodity indices over time. The covariance is defined as:

cov(R,L) = E[(R− µR)(L− µL)] (C.10)

The correlation is the normalized covariance.

Definition C.15 Correlation

Define the random variables L and R ∈ R(T×1) describing all possible return re-

alizations of a two commodity indices over time. The correlation is a standardized

form of the covariance and defined as:

ρ =cov(R,L)√

var(R)√

var(L)=

E[(R− µR)(L− µL)]√var(R)

√var(L)

(C.11)

The autocorrelation describes a linear dependence between return realization at

different points in time.

187


Definition C.16 Autocorrelation

Consider a weakly stationary return series rt. The correlation coefficient between

rt and rt−l is called the lag-l autocorrelation of rt and defined as:

ρl =cov(rt, rt−l)√

var(rt)√

var(rt−l)=

cov(rt, rt−l)

var(rt)(C.12)

While the mean is the average outcome of an experiment the median is the middle

value of the sample.

Definition C.17 Median

To be precise: Let r1, . . . , rT denote a random sample. The order statistic is defined

as r(1) ≤ r(2) ≤ . . . ≤ r(T ). Then the median is defined as:

rM =

r(T+1

2) : if T is odd

r( T2 )

+r(1+ T

2 )

2: if T is even.

(C.13)

Definition C.18 Quantile

To be precise: Let r1, . . . , rT denote a random sample and R is the random variable

embodying all possible realizations rt. Let quantile qα is defined as the value that α%

of the possible realizations are smaller than qα, i.e.

Q(R ≤ qα) = α (C.14)

Definition C.19 Skewness

Define the random variable R ∈ R(T×1) describing all possible return realizations of

a commodity index over time and let µ and σ be their mean and standard deviation.

The coefficients of skewness S is defined as:

S = E(R− µ

σ)3 (C.15)

Definition C.20 Kurtosis

Define the random variable R ∈ R(T×1) describing all possible return realizations of

a commodity index over time and let µ and σ be their mean and standard deviation.

The coefficients of kurtosis K is defined as:

K = E(R− µ

σ)4 − 3 (C.16)

Definition C.21 Order Statistic

Let r1, . . . , rT be a sample of T independent observations of the random variable

188


R. Arrange the rt in ascending order of magnitude and denote the ordered set by

(r(1), . . . , r(T )), so that: (r(1) ≤ . . . ≤ r(T )). (r(1), . . . , r(T )) is called the order statistic.

Definition C.22 Distribution of Order Statistic

Let f(r) be the probability density function and F(r) be the cumulative distribution

function of R. Then the probability density of the k’th statistic can be found as

follows:

fR(k)(r) =

T !

(k − 1)!(T − k)!F (r)k−1(1− F (r))T−kf(r) (C.17)

Proof:

fR(k)(r) =

d

dxFR(k)

(r) =d

dxQ(R(k) ≤ r)

=d

dxQ(at leats k of the T R’s are ≤ r)

=d

dxQ(≥ ksuccesses in T trials)

=d

dx

T∑i=k

(T

j

)Q(R1 ≤ r)j(1−Q(R1 ≤ r))T−j

=d

dx

T∑i=k

(T

j

)F (r)j(1− F (r))T−j

=T∑

i=k

(T

j

)(jF (r)j−1f(r)(1− F (r))T−j

+ F (r)j(T − j)(1− F (r))T−j−1(−f(r)))

=T∑

i=k

(T(

T − 1j − 1

)F (r)j−1(1− F (r))T−j

− T

(T − 1

j

)F (r)j(1− F (r))T−j−1)f(r)

= Tf(r)(T−1∑

i=k−1

(T − 1

j

)F (r)j(1− F (r))(T−1)−j

−T∑

i=k

(T − 1

j

)F (r)j(1− F (r))(T−1)−j)

= Tf(r)((

T − 1k − 1

)F (r)k−1(1− F (r))(T−1)−(k−1)

−(

T − 1T

)F (r)T (1− F (r))(T−1)−T︸︷︷︸

0

)

=T !

(k − 1)!(T − k)!F (r)k−1(1− F (r))T−kf(r)

2

189


C.2 Probability Theory

The following definitions are inspired by [Zagst 2002]. For a detailed introduction

to financial market theory, please refer to it.

Definition C.23 Measurable

A k-dimensional function f : Ω 7→ Rk is called (F −B(Rk)-) measurable or simply

(F-) measurable if

f−1(B) = ω ∈ Ω : f(ω) ∈ B ∈ F ∀B ∈ B

From this point on we assume that we are working on a complete probability space

(Ω,F , Q). In this case a k dimensional measurable function X : Ω 7→ Rk, k ∈ N,

is called a random vector. For k = 1 we call R a random variable. The smallest

sigma-Algebra containing all sets X−1(B) = ω ∈ Ω : X(ω) ∈ B, where B runs

through the Borel sigma-Algebra B(Rk), is called the sigma-Algebra generated by

X, and will be denoted by F(R).

Definition C.24 Conditional Expectation

Let X be an integrable random variable on the probability space (Ω,F , Q) and G ⊂ Fbe a sub-sigma-Algebra of F . The conditional expectation of X given G is implicitly

defined to be the G-measurable function EQ[X|G] with∫A

XdQ =

∫A

EQ[X|G]dQ ∀A ∈ G Q− a.s.

The function

Q : F 7→ [0, 1]

is a probability mass.

For the main properties of the conditional expectation see [Zagst 2002].

Definition C.25 Filtration

A filtration F is a non-decreasing family of sub-sigma-algebras (Ft)t≥0 with Ft ⊂ Fand Fs ⊂ Ft for all 0 ≤ s < t < ∞. We call (Ω,F , Q, F) a filtered probability space,

and require that

1. F0 contains all subsets of the (Q-) null sets of F ,

2. F is right-continuous, i.e. Ft = Ft+ := ∩s>tFs

190

C.2 Probability Theory

Ft represents the information available at time t, and F = (Ft)t≥0 describes the flow

of information over time, where we suppose that we don’t lose information as time

passes by.

The price behavior of financial products over time is usually described by a so-called

stochastic process.

Definition C.26 Stochastic Process

A stochastic process (vector process) is a family X = (X(t))t≥0 of random variables

(vectors) defined on the filtered probability space (Ω,F , Q, F). We say that:

1. X is adapted (to the filtration F) if Xt = X(t) is (Ft-) measurable for all

t ≥ 0

2. X is measurable of the mapping X : [0,∞]×Ω → Rk, k ∈ N is B([0,∞))⊗F-

B(Rk)−measurable

3. X is progressively measurable if the mapping X : [0, t] × Ω → Rk, k ∈ Nis B([0, t])⊗Ft-B(Rk)−measurable for each t ≥ 0.

Note that we either write Xt or X(t), whichever is more comfortable. Also note

that a stochastic process is a function in t for each fixed or realized ωinΩ. If the

stochastic process X is measurable, the mapping X(·, ω) : [0,∞) 7→ Rk, k ∈ N,

B([0,∞)) ⊗ B(Rk) is measurable for each fixed ωinΩ. For each fixed ωinΩ we call

X(ω) = (Xt(ω))t≥0 = (X(t, ω))t≥0 a path or realization of the stochastic process.

Definition C.27 L2[0,T]-Prozess

Let (Ω,F , Q, F) bw a filtered probability space and X be a stochastic process adapted

to F. We call a stochastic process L2[0,T]-process, if X is progressively measurable

and

‖X‖2T := EQ[

∫ T

0

X2(t)dt] < ∞.

One of the atoms of modern finance is the following special stochastic process called

Wiener process, sometimes also known as Brownian motion.169

Definition C.28 Wiener process Let (Ω,F , Q, F) be a filtered probability space.

The stochastic process W = (Wt)t≥0 = (W (t))t≥0 is called a Q- Brownian motion or

Q- Wiener process if

1. W (0) = 0 Q-a.s.

169For further details to the naming conventions see [Zagst 2002].

191


2. W has independent increments, i.e. W (t) −W (s) is independent of W (t′) −W (s′) for all 0 ≤ s′ ≤ t′ ≤ s ≤ t < ∞

3. W has stationary increments, i.e. the distribution of W (t + u) − W (t) only

depends on u for u ≥ 0

4. Under Q, W has Gaussian increments, i.e. for 0 ≤ s ≤ t:

W (t)−W (s) ∼ N(0, t− s).

with the definitions of C.12.

5. W has continuous path Q-a.s.

We call W with W T = (W1, . . . ,Wm) = (W1(t), . . . ,Wm(t))t≥0 a m-dimensional

Wiener process, m ∈ N, if its components Wj, j = 1, . . . ,m, m ∈ N, are independent

Wiener processes.

Definition C.29 Martingale Let (Ω,F , Q, F) be a filtered probability space. A

stochastic process X = X(t); t ≥ 0, that is adapted with EQ[|X(t)|] < ∞, ∀t ≥ 0,

is called:

martingale relative to (Q, F), if EQ[X(t)|Fs] = X(s) Q-a.s. ∀0 ≤ s ≤ t < ∞

super-martingale relative to (Q, F), if EQ[X(t)|Fs] ≤ X(s) Q-a.s. ∀0 ≤ s ≤t < ∞

sub-martingale relative to (Q, F), if EQ[X(t)|Fs] ≥ X(s) Q-a.s. ∀0 ≤ s ≤t < ∞

192

C.3 Stochastic Differential Equations

C.3 Stochastic Differential Equations

A major tool to describe the price behavior of financial assets and derivatives is the

Ito process.

Definition C.30 Ito process

Let W = (W1, . . . ,Wm), m ∈ N, be a m-dimensional Wiener process. A stochastic

process X = (X(t))t≥0 is called an Ito process if ∀t ≥ 0 we have:

X(t) = X0 +

∫ t

0

µ(s)ds +

∫ t

0

σ(s)dW (s)

= X0 +

∫ t

0

µ(s)ds +m∑

j=1

∫ t

0

σj(s)dWj(s), (C.18)

where X0 is (F0-)measurable and µ = (µ(t))t≥0 and σ(t) = (σ1(t), . . . , σm(t))t≥0 are

(m-dimensional) progressively measurable stochastic processes with∫ t

0

|µ(s)| ds < ∞ Q− a.s. , (C.19)

and ∫ t

0

σ2j (s)ds < ∞ Q− a.s. (C.20)

∀t ≥ 0, j = 1, . . . ,m.

A n-dimensional Ito process is given by a vector X = (X1, . . . , Xn), n ∈ N, with

each Xi being an Ito process, i = 1, . . . , n.

For convenience we write symbolically instead of (C.18)

dX(t) = µ(t)dt + σ(t)dW (t) = µ(t)dt +m∑

j=1

σj(t)dWj(t);

and call this a stochastic differential equation (SDE).

Definition C.31 Quadratic Covariance Process

Let m ∈ N and W = (W1, . . . ,Wm) be a m-dimensional Wiener process. Further-

more, let (X1(t))t≥0 and (X2(t))t≥0 be two Ito processes with

dXi(t) = µi(t)dt + σi(t)dW (t) = µi(t)dt +m∑

j=1

σijdWj(t), i = 1, 2. (C.21)

193


Then we call the stochastic process < X1, X2 >= (< X1(t), X2(t) >)t≥0 defined by

< X1, X2 >:=m∑

j=1

∫ t

0

σ1j(s)σ2j(s)ds (C.22)

the quadratic covariance (process) of X1 and X2. If X1 = X2 =: X we call the

stochastic process < X >:=< X, X > the quadratic variation (process) of X.170

Theorem C.1 (Ito´s Lemma) Let W = (W1, . . . ,Wm) be a m-dimensional Wiener

process, m ∈ N,and X = (X(t))t≥0 be an Ito process with

dX(t) = µ(t)dt + σ(t)dW (t) = µ(t)dt +m∑

j=1

σj(t)dWj(t)

Furthermore, let G : R× [0,∞) 7→ R be twice continuously differentiable in the first

variable, with derivatives denoted by GX and GXX , and once continuously differen-

tiable in the second, with derivative denoted by Gt. Then we have ∀t ∈ [0,∞)

dG(X(t), t) = Gt(X(t), t)dt + GX(X(t), t)dX(t) +1

2GXX(X(t), t)d < X(t), X(t) >

=(Gt(X(t), t) + GX(X(t), t)µ(t) +

1

2GXX(X(t), t)

m∑j=1

σ2j (t))dt

+ GX(X(t), t)m∑

j=1

σj(t)dWj(t)

Whereby, the Wj, j = 1, . . . ,m, are assumed to be independent.

170For a detailed definition see [Zagst 2002].

194

C.4 Equivalent Measure

C.4 Equivalent Measure

Definition C.32 Equivalent Measure

Let Q and Q be two measures defined on the same measurable space (Ω,F). We say

Q is absolutely continuous with respect to Q, written Q Q,if Q(A) = 0 whenever

Q(A) = 0, A ∈ F . If both Q Q and Q Q, we call Q and Q equivalent measures

and denote this by Q ∼ Q.

The definition of equivalent measures states that two measures are equivalent if and

only if they have same null sets.

Definition C.33 Radon Nikodym Derivative

Let Q be a sigma-finite measure and Q be a measure on the measurable space (Ω,F)

with Q < ∞. Then Q Q if and only if there exists an integrable function f ≥ 0

Q-a.s. such that

Q(A) =

∫A

fdQ ∀A ∈ F

f is called the Radon-Nikodym derivative of Q with respect to Q and is also written

as f = dQdQ

.

Let γ = (γ(t))t≥0 be a m-dimensional progressively measurable stochastic process,

m ∈ N, with ∫ t

0

γ2j (s)ds < ∞ Q− a.s. ∀t ≥ 0, j = 1, . . . ,m

Let the stochastic process L(γ) = (L(γ, t))t≥0 = (L(γ(t), t))t≥0, ∀t ≥ 0 be defined

by

L(γ, t) = e−R t0 γ(s)′dW (s)− 1

2

R t0 ||gamma(s)||ds

Note that the stochastic process X(γ) = (X(γ, t))t≥0 = (X(γ(t), t))t≥0 with

X(γ, t) :=

∫ t

0

γ(s)′dW (s) +1

2

∫ t

0

||γ(s)||2ds

or

dX(γ, t) :=1

2||γ(t)||2dt + γ(t)′dW (t)

is ∀t ∈ [0,∞) an Ito process with with µ(γ(t), t) = 12||γ(t)||2 = 1

2

∑mj=1 γ2

j (t) and

σ(γ(t), t) = γ(t)′. Thus, using the transformation G : R×[0,∞) 7→ R with G(x, t) =

e−x and Ito’ lemma as of C.1 with G(X(γ, t), t) = e−X(γ,t) = L(γ, t) we get:171

dL(γ, t) = −L(γ, t)γ(t)′dW (t)

171For a detailed calculation see [Zagst 2002].

195


Theorem C.2 Novikov Condition

Let γ and L(γ) be as defined above. Then L(γ) = (L(γ, t))t∈[0,T ] os a continuous

(Q -) martingale if

EQ

[e

12

R t0 ||γ(s)||2ds

]< ∞

For each T ≥ 0 we define the measure Q = QL(γ,T ) on the measure space (Ω,FT ) by

Q(A) := EQ[1A ∗ L(γ, T )] =

∫A

L(γ, T )dQ ∀A ∈ FT

which is the probability measure if L(γ, T ) is a (Q -) martingale. In this case,

L(γ, T ) is a (Q -) density of Q, i.e. L(γ, T ) = dQdQ

on (Ω,FT ). The following Girsanov

theorem shows how the adequate (tildeQ -) Wiener process W =(W (t)

)t∈[0,T ]

is

constructed, starting with a (Q -) Wiener process W = (W (t))t∈[0,T ].

Theorem C.3 Girsanov Theorem

Let W = (W1, . . . ,Wm) = (W (t)1, . . . ,W (t)m)t∈[0,T ] be a m-dimensional (Q -)

Wiener Process, m ∈ N, γ, L(γ), Q and T ∈ [0,∞) be defined as above, and the

m-dimensional stochastic process W = (W1, . . . , Wm) =(W (t)1, . . . , W (t)m

)t∈[0,T ]

be defined by

dW (t) := γ(t)dt + dW (t), t ∈ [0, T ]

If the stochastic process L(γ) is a (Q -) martingale, then the stochastic process W

is a m-dimensional (Q -) Wiener Process on the measure space (Ω,FT ).

196

C.5 Feynman-Kac Representation

C.5 Feynman-Kac Representation

We now will move on to solve stochastic differential equations. For it, we consider

a special form of stochastic differential equation. To do this, let µ : R× [0,∞) → Rand σ : R×[0,∞) → Rm be measurable functions (with respect to the corresponding

Borel sigma-Algebras) with∫ t

0

|µi(s)|ds < ∞ and

∫ t

0

σ2ij(s)ds < ∞ Q− a.s., (C.23)

and j = 1, . . . ,m, i = 1, . . . , n, n and m ∈ N and ∀t ≥ 0.

Definition C.34 Strong Solution of the SDE

If there exists a n-dimensional stochastic process X = (X(t))t≥0 on the probability

space (Ω,F , Q, F) satisfying (C.23), i.e. an Ito process, such that ∀t ≥ 0

X(t) = x +

∫ t

0

µ(X(s), s)ds +

∫ t

0

σ(X(s), s)dW (s) Q− a.s.,

X(0) = x, x ∈ Rn, fixed,

we call X the strong solution of the stochastic differential equation (SDE)

dX(t) = µ(X(t), t)dt + σ(X(t), t)dW (t) ∀t ≥ 0 and X(0) = x. (C.24)

Theorem C.4 Existence and Uniqueness

Let µ and σ of the stochastic differential equation (SDE) be continuous functions

such that for ∀t ≥ 0, x, y ∈ Rn and for K > 0 the following conditions hold:

‖µ(x, t)− µ(y, t)‖+ ‖σ(x, t)− σ(y, t)‖ ≤ K ‖x− y‖ , (Lipschitz − Condition)

‖µ(x, t)‖2 + ‖σ(x, t)‖2 ≤ K2(1 + ‖x‖2), (Growth− Condition)

Then there exists a unique, continuous strong solution X = (X(t))t≥0 von (C.24) of

(SDE) and a constant C, depending on K and T > 0, such that

EQ[‖X(t)‖2] ≤ C(1 + ‖x‖2)eCt ∀t ∈ [0, T ].

To move on to the so-called Feynman-Kac Representation the following definition

is very helpful.

197


Definition C.35 characteristic Operator

Let X = (X(t))t≥0 be the unique solution of the stochastic differential equation

( (C.24)) under the assumptions of Theorem C.4. Then the operator D defined by

(Dυ)(x, t) = υt(x, t) + µ(x, t)υx(x, t) +1

2σ2(x, t)υxx(x, t)

with υ : R × [0,∞) → R twice continuously differentiable in x, once continuously

differentiable in t and

σ2(x, t) =m∑

j=1

σ2j (x, t)

is called the characteristic operator for X(t).

The operator is used to define the so-called Cauchy problem.

Definition C.36 Cauchy Problem

Let D : R → R and r : R×[0, T ] → R be continuous and T > 0 be arbitrary but fixed.

Then, the Cauchy problem is stated as follows: Find a function υ : R× [0, T ] → R,

which is continuously differentiable in t and twice continuously differentiable in x

and solves the partial differential equation (sometimes called Kolmogorov equation)

Dυ(x, t) = r(x, t)υ(x, t) ∀(x, t) ∈ R× [0, T ], (C.25)

υ(x, T ) = D(x) ∀x ∈ R. (C.26)

Theorem C.5 Feynman-Kac Representation

Under the assumption, that the function µ, σ, r, v und D as defined above, satisfy

sufficient regulatory conditions, the solution v of the Cauchy problem os given by the

following conditional expectation called Feynman-Kac representation:

v(x, t) = Ex,tQ

[e−

R Tt r(X(s),s)dsD(X(T ))

](C.27)

with X(0) = x.

198

D Program Codes

The following section show the program codes which were used to generate the

results of Section 6. I produced the programs in collaboration with my colleague

at risklab germany Dr. Wolfgang Mader. I appreciate his expertise and thank him

very much for his support.

D.1 Portfolio Allocation with Commodities

% Function calculates the Efficient Frontier with Commodites based on bootstrapped

%data

% Author’s Information

%———————————————————————

% risklab germany GmbH

% Nypmhenburger Strasse 112 - 116

% D-80636 Muenchen

% Germany

% Internet: www.risklab.de

% email: [email protected]

% Implementation Date: 2006 - 09 - 27

% Author: Dr. Wolfgang Mader, Maria Heiden

%———————————————————————

% Calculate Efficient Frontier

returns = RollingAverage;

Steps = 500;

ub = [1;1;1]; i = 1;

% Portfolio with Commodities

for mu = min(mean(returns)):(max(mean(returns))-min(mean(returns))) ...

/Steps:max(mean(returns))

199

D Program Codes

[weights,value] = szenoptiRM(returns, mu, ub);

weightsPF(i,:)=weights’;

muePF(i)=mu;

riskPF(i)=value;

i=i+1;

end

muePF=muePF’; riskPF=riskPF’;

ub = [1;1];

i = 1;

returnsOld = returns(:,2:3);

% Portfolio without Commodities

for mu = min(mean(returnsOld)):(max(mean(returnsOld))-min(mean(returnsOld)))

... /Steps:max(mean(returnsOld))

[weights,value] = szenoptiRM(returnsOld, mu, ub);

weightsPFNo(i,:)=weights’;

muePFNo(i)=mu;

riskPFNo(i)=value;

i=i+1;

end

muePFNo = muePFNo’; riskPFNo =riskPFNo’;

% plotting efficient frontiers

figure(1) plot(riskPF,muePF,’-.r’) hold on

plot(riskPFNo,muePFNo,’-.k’)

legend hurdle noCom grid on

200

D.2 Hurdle Rate

D.2 Hurdle Rate

% Hurdle Rate is the return an alternative has to produce to be allocated in a

% Stock-Bond-Portfolio


%———————————————————————



% D-80636 Muenchen

% Germany





%———————————————————————

% Start Values

returns = RollingAverage;

allocationBonds = .75;

startHurdle = .0385;

stepsImplied = .0001;

outperformanceTrigger = .0001;

meanBonds = mean(returns(:,3));

meanStocks = mean(returns(:,2));

meanAlternative = mean(returns(:,1));

sdBonds = std(returns(:,3));

sdStocks = std(returns(:,2));

sdAlternative = std(returns(:,1));

201

D Program Codes

warning off;

% Reference Values of Start Allocation (25% Stocks, 75% Bonds)

% Reference Portfolio Standard Deviation

stdPFTest=std(returns(:,2:3)*[(1-allocationBonds);allocationBonds]);

% Reference Portfolio Mean

muPFTest = (1-allocationBonds)*meanStocks+allocationBonds*meanBonds;

% Check Optimization

ub = [1;1];

[weights,mue] = szenoptiValue(returns(:,2:3), stdPFTest, ub);

stdev = std(returns(:,2:3)*weights);

% Loop for Hurdle Rate

impliedHurdleRate = startHurdle;

ub=[1;1;1];

% Move Returndistribution of Commodities to Start Value

returns(:,1) = returns(:,1)-mean(returns(:,1))+impliedHurdleRate;

while (impliedHurdleRate ¡ meanAlternative)

disp(’————————————————————’)

returns(:,1) = returns(:,1) + stepsImplied;

disp(’impliedHurdleRate’)

mean(returns(:,1))

[weights,value] = szenoptiValue(returns, stdPFTest, ub);

202

D.2 Hurdle Rate

disp(’current weights’)

weights

disp(’current return’)

value

disp(’target return’)

muPFTest

disp(’STDEV’)

std(returns*weights)

% Hurdle Rate is found if new portfolio return is bigger than reference portfolio

% return plus outperformance-trigger

returnTest = value;

if and(returnTest¿muPFTest+outperformanceTrigger,weights(1)¿0)

break

end

impliedHurdleRate = impliedHurdleRate + stepsImplied;

end

% Calculating Efficient Frontiers

Steps = 500;

i = 1;

for mu = min(mean(returns)):(max(mean(returns))-min(mean(returns))) ...

/Steps:max(mean(returns))

[weights,value] = szenoptiRM(returns, mu, ub);

weightsPFHurd(i,:)=weights’;

muPFHurd(i)=mu;

riskPFHurd(i)=value;

i=i+1;

end

203

D Program Codes

muPFHurd=muPFHurd’; riskPFHurd=riskPFHurd’;

ub = [1;1];

i = 1;

returnsOld = returns(:,2:3);

for mu = min(mean(returnsOld)):(max(mean(returnsOld))-min(mean(returnsOld)))

... /Steps:max(mean(returnsOld))

[weights,value] = szenoptiRM(returnsOld, mu, ub);

weightsPFNo(i,:)=weights’;

muPFNo(i)=mu;

riskPFNo(i)=value;

i=i+1;

end

muPFNo = muPFNo’;

riskPFNo =riskPFNo’;

% plotting efficient frontiers

figure(1)

plot(riskPFHurd,muPFHurd,’-.r’) hold on

plot(riskPFNo,muPFNo,’-.k’)

legend hurdle noCom

204

D.3 Help Function

D.3 Help Function

% Function calculates the efficient frontier

function [weights,value] = szenoptiValue(returns, riskTarget, boundAlternatives)


%———————————————————————



% D-80636 Muenchen

% Germany





%———————————————————————

T=size(returns,1);

N=size(returns,2);

x0=repmat(1/N,1,N)’;

A=[];

b = [];

Aeq(1,:)=ones(1,N);

beq =[1];

lb=zeros(N,1);

ub=[1;1;boundAlternatives];

optimizationOptions = optimset(’Display’, ’off’, ’LargeScale’, ’off’);

[weights,value]=fmincon(@(x) optiMean(x,returns),x0,A,b,Aeq,beq,lb,ub, ...

@(x) mycon(x,returns,riskTarget),optimizationOptions);

205

D Program Codes

value = -value;

function [c,ceq] = mycon(w,returns,riskTarget) objective=optiStd(w,returns);

c = [];

% Compute nonlinear inequalities at x

ceq = objective - riskTarget;

function objective = optiMean(w,returns)

objective=-mean(returns*w);

% Function calculates the efficient frontier

function [weights,value] = szenoptiRM(returns, muTarget, ub)


%———————————————————————



% D-80636 Muenchen

% Germany





%———————————————————————

T=size(returns,1);

N=size(returns,2);

x0=repmat(1/N,1,N)’;

A=[];

b = [];

Aeq(1,:)=ones(1,N);

Aeq(2,:)=mean(returns);

beq=[1;muTarget];

lb=zeros(N,1);

206

D.3 Help Function

optimizationOptions = optimset(’Display’, ’off’, ’LargeScale’, ’off’);

[weights,value]=fmincon(@(x) optiStd(x,returns),x0,A,b,Aeq,beq,lb,ub, ...

[],optimizationOptions);

function objective = optiStd(w,returns)

objective=std(returns*w);

207

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