Technical University Munich Commodities as an Asset Class - risklab

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 mentioned
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. Reinhold
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 Inflation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 introduction
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, approximately
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 recession
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 companies
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
1 See 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.
Lectures 2 about stock markets show that investors are attracted to broad diversified
exposure in the selected asset class. This is represented by indices which
we introduce in Section 4 ”Commodity Indices”. Moreover, we discuss their common
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.
2 For instance [Campbell 2000].
2

Finally, we bring commodity returns into the portfolio context. In Section 6 ”Strategic
Asset Allocation with Commodity Derivatives” we show that commodity investments,
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 characteristics
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 countries
have economically grown much faster than industrial countries. The integration
of global financial markets, the expansion of international companies by outsourcing
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
3 See [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 outweighting
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 equilibriums.
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 investable
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.
2.1 The different Commodity Types
Commodity markets are divided into three major groups: energy, metals and agricultures
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 industrialization
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 natural
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.
4 See [Sawyer Sprinkle 2003].
5

2 Overview of Commodity Markets
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 The different Commodity Types
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

2 Overview of Commodity Markets
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 become
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
5 See [BP Report 2006].
6 See Table 2.1.
8

2.1 The different Commodity Types
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 demand
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 September
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 1986 9 and worlds biggest finding in 1988 10 , no major
oil field has been discovered during the last 20 years.
Analyzing the data of the [BP Report 2006] including all relevant information regarding
oil reserves, production and consumption uncovers interesting insides: Middle
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.
7 See [BP Report 2006].
8 See Bloomberg, NYMEX crude oil futures contracts.
9 over 40 billion barrels in Iran and over 60 billion barrels in the United Arab Emirates
10 over 100 billion barrels in Saudi Arabia
9

2 Overview of Commodity Markets
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
Country
Reserves Production
% of total rank % of total rank
Saudi Arabia 22.0% 1 13.5% 1
Iran 11.5% 2 5.1% 4
Iraq 9.6% 3 2.3% 3
Kuwait 8.5% 4 3.3% 10
United Arab Emirates 8.1% 5 3.3% 11
Venezuela 6.6% 6 4.0% 7
Russia 6.2% 7 12.1% 2
Table 2.1: Oil Reserves and Production
10

2.1 The different Commodity Types
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 production
arrangements between the USA and the OPEC to stable prices during the
1980th and 1990th and world’s growing industrialization, technological improvements
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 expected
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
11 See [BP Report 2006].
12 See [BP Report 2006].
13 Population data source : Census Bureau of the U.S. Department of Commerce (www.census.gov).
11

2 Overview of Commodity Markets
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 substituting
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 investment.
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 alternative
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 implicating
an environmental advantage to crude oil and coal. Natural gas was formed
14 See[UBS research 2005].
12

2.1 The different Commodity Types
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 consumption
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 commercial
use. First test series have been successfully but bulk production stills needs
15 See [BP Report 2006].
13

2 Overview of Commodity Markets
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 relative
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
16 See [EIA Outlook 2006].
17 For 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 of
Definition C.2. For the mathematical definition of Pearson correlation and the related statistical
test see Section 5.1.2.
14

2.1 The different Commodity Types
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 demand
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 factors
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 regression.
Technological improvements enabled savings in materials, e.g. cars became
lighter or aluminium and steel were substituted by strong but lighter carbon compounds.
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

2 Overview of Commodity Markets
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 currency
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 standardized
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 investments.
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
18 See World Gold Council.
16

2.1 The different Commodity Types
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 environment
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 following
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
19 At this point in history the USA hold 70% of worldwide gold reserves. See [UBS research 2005].
20 See [UBS research 2005].
17

2 Overview of Commodity Markets
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 tradeweighted
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 investors
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 positive
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
21 Data source: Bloomberg. We used monthly log returns as of Definition C.2. For the mathematical
definition of Pearson correlation and the related statistical test see Section 5.1.2.
18

2.1 The different Commodity Types
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

2 Overview of Commodity Markets
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. Merchants
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
22 For further details see [LME 2006].
23 Data source: Bloomberg.
20

2.1 The different Commodity Types
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 commodity
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
24 See [Commodities 2006].
25 See [Commodities 2006].
21

2 Overview of Commodity Markets
the exception of sugar, are lacking behind.
The softs group includes cocoa, coffee and sugar. It is internally the most heterogenous
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. Therefore,
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.
26 See [Rogers 2005].
22

2.1 The different Commodity Types
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 agricultural
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 commodities
is indirectly driven by weather conditions having direct impact on the input
product feed. Second, the major direct risk factor in livestock markets are epidemics
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

2 Overview of Commodity Markets
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
27 Data source: [The CRB Commodity Yearbook 2005] and [ICCO].
24

2.1 The different Commodity Types
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
28 Thus, it can be assumed that a decline in prices of 10% results in an increase in demand of 2%.
29 A single tree can produce 20 fruits but it needs 400 to get one pound of chocolate.
30 See [The CRB Commodity Yearbook 2005].
31 See [USDA Grain 2006].
32 Moreover, 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.
33 See [USDA Grain 2006] and [The CRB Commodity Yearbook 2005].
25

2 Overview of Commodity Markets
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 operation”
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

2.1 The different Commodity Types
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 premium
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 negative
correlation of -0.15 could be found.
34 See [The CRB Commodity Yearbook 2005].
35 For 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 Pearson
correlation and the related statistical test see Section 5.1.2.
27

2 Overview of Commodity Markets
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
36 See 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.
environment or technological change
weather, political
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 meal 37 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
37 See Appendix B.11.
38 Inspired by [Structured Products 2006]
29

2 Overview of Commodity Markets
The different participants have - caused by their different business purposes - different
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 articles 40 , 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.
2.3 Trading Commodities
To cover all the different needs of the market participants, a range of different
financial products enabling both, hedging and speculation, were developed. Unfortunately,
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 products
without physical delivery. The basic products are described in Section 2.3.1
39 We will examine the theory of normal backwardation in Section 3.1.
40 See e.g. [Edwards Liew 1999] or [Fung Hiesh 1997].
31

2 Overview of Commodity Markets
Figure 2.16: Overview of Commodity Investment Instruments
and called derivatives including futures, the elementary vehicle to trade raw commodities,
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 investment
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 different
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

2.3 Trading Commodities
markets are futures contracts. They are the original vehicle to trade commodities.
Although the price of a futures contract depends on the current spot (cash)
price 41 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 environmental
exposure use financial markets to hedge or manage their price risk. For
instance, at the Chicago Mercantile Exchange (CME) more than one billion contracts
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
41 For the feature of spot prices in commodity markets see the discussion in Section 5.1.1.
42 William 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 the
effect of a firm’s capital structure and dividend policy on market price.
43 Data source: Futures Industry Magazine Mar/Apr 2006, numbers include financial derivatives
(i.e. derivatives on interest rates or equities)
33

2 Overview of Commodity Markets
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 additional
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 respective
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 commodity
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 heating
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.
44 Because forward contracts are over the counter (OTC) bilateral agreements they embody counterparty
default risks.
45 For the formal mathematical proof that forward and futures prices are equal under the assumption
of deterministic interest rates see [Zagst 2002].
46 For further information see [Cox Ross Ingersoll 1981]
34

2.3 Trading Commodities
Historically, the expiration months of futures contracts were without some exceptions
March, June, September and December reflecting the seasonality in commodities
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 commodity
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
47 Data source: Futures Industry Magazine Jan/Feb 2006
35

2 Overview of Commodity Markets
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 registered
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.
48 Data source: Futures Industry Magazine Jan/Feb 2006
36

2.3 Trading Commodities
2.3.1.3 Swaps
Commodity swaps are generally the same like interest rate swaps 49 with the difference
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 t 1 , . . . , t n 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
49 For a general introduction to interest rate swaps see [Zagst 2002]
37

2 Overview of Commodity Markets
types of cash payments are enabled, i.e. someone could think about a price-forinterest
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 exposure
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: Commodity
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

2.3 Trading Commodities
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 demand
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 unattractive
investment vehicle for institutional investors but they are very famous in retail
business. The drawbacks of covered overpricing and credit risk are little communicated.
But their major advantage is high liquidity. Over this, certificates are generally
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 investment
medium. This is the main difference between a CTA and an ordinary trader.
Former have research based investment strategies, including diversification over different
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
50 See [Gong Huber Lanzinner 2006].
51 See Managed Account Research, Inc.;
http : \ \ www.ma − research.com \ managed account vs self − directed.html
39

2 Overview of Commodity Markets
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 demand
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 manage
his funds based on the trading strategy presented in the advisors disclosure
document. The CTA opens an individual account on behalf of the investor, enables
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. Because
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 substantial
net worth. Second, an investor can place his assets at a commodity pool
operator (CTO), who pools funds of different individual investors together and employs
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 enable
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] investigated
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
52 See Managed Account Research, Inc.;
http : \ \ www.ma − research.com \ growth of managed futures.html
40

2.3 Trading Commodities
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 portfolios
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 competing
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 commodities
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 international
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 codes 53 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 exchanges.
54
This methodology was taken by [Gorton Rouwenhorst 2004] to create
an index which replicates their artificially constructed equally weighted commodity
index 55 with commodity producing companies. They showed that the cumulated
stock performance was less than the cumulated performance of the commodities.
53 The Standard Industrial Classification Code (SIC) indicates the company’s type of business.
54 See and further information: [Gorton Rouwenhorst 2004].
55 For further information about different index weighting procedures see Section 4.1.
41

2 Overview of Commodity Markets
Over this there was a higher correlation (0.57) between the commodity producing
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 constructed
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. Therefore,
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
56 There were daily Bloomberg data taken and returns were calculated following Definition C.2 and
the plotted price series starting with 100 in 1999 following Definition C.3.
42

2.3 Trading Commodities
the return correlations 57 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 investor 58 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.
57 See Definition C.2, Definition 5.1 and Equation (5.5).
58 For 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 exchanges
and, as such, translate the underlying supply and demand or, rather, their
expected values at various points in future into absolute figures.
Reflecting expectations
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 pricing 59 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 contract.
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 specifications
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-tonearest
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 Convenience
Yield Model which we will present in Section 3.1 and Section 3.2.
relationship between the two models were first derived in [Markert 2005] and will be
59 For a general introduction to the concept of no-arbitrage pricing and the related definitions
see [Zagst 2002]. The pioneer work summarizing the different concepts of commodity futures
pricing was done by [Markert 2005].
The
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 scenarios.
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 Convenience
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 further
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.
3.1 The Risk Premium Model
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.
of tomorrow over futures contracts. 61
A convenient way to do so is selling today his production
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
60 See [Keynes 1930].
61 Following 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 avoid
counterparty default risk and wants to deal with fair market not with bilateral bargained
prices. Furthermore he does not chose a swap contract because he is not interested in a series
of transactions.
45

3 Pricing of Commodity Futures
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 conditions
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 futures
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 situation
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
62 See also [Geer 2000].
63 See [Merrill Lynch 2006].
46

3.1 The Risk Premium Model
the shape of the forward curve 64 of crude oil 65 and copper 66 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 contango.
Therefore, [Cootner 1990] and [Deaves Kinsky 1995] extended the theory
and formulated the hedging pressure hypothesis. They suggested that both ”backwardated”
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. producers
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 backwardated
commodity futures and for going short contangoed commodity futures.
Putting both theories into mathematical forms we end up with the Risk Premium
Model:
64 For an introduction to forward curves see [Zagst 2002].
65 The values come from the NYMEX light sweet crude oil futures contract with a trading size of
1.000 barrel as per 31. January 2006.
66 The values come from the LME copper futures contract with a trading size of 25 tones as per
31. January 2006.
67 See e.g. [Anderson 2000], [Bessembinder 1992] or [DeRiin Nijman Veld 2000].
47

3 Pricing of Commodity Futures
Theorem 3.1 Risk Premium Model
Let P C (t) be the spot price of a commodity at time t ∈ [0, T ], let F t denote the
σ-Algebra 68 as of Definition C.5 at time t and let r p be the constant asset specific
risk premium. Moreover, define ˜Q as the equivalent martingale measure as of
Definition C.32. Then the price of a commodity future F C (t, T ) at time t ∈ [0, T ] in
the Risk Premium Model is given by:
F C (t, T ) = e −rp(T −t) E ˜Q[P C (T )|F t ] (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:
P A (t) denotes the spot price of a pure financial asset at time t ∈ [0, T ]
P C (t) denotes the spot price of a commodity at time t ∈ [0, T ]
F A (t, T ) denotes the futures price of a pure financial asset at time t ∈ [0, T ]
F C (t, T ) denotes the futures price of a commodity at time t ∈ [0, T ]
Furthermore let r f be the constant risk free interest rate, r p 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 different
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, F A (t, T ) is the futures price
for which the present value of the expected future payoff equals zero: 69
0 = E ˜Q[e −(r f +u)(T −t) (P A (T ) − F A (t, T ))|F t ]
F A (t, T ) = e (r f +u)(T −t) E ˜Q[e −(r f +u)(T −t) P A (T )|F t ]
F A (t, T ) = E ˜Q[P A (T )|F t ] (3.2)
68 The σ-Algebra F t embodies all available information until t. For a more detailed mathematical
introduction see [Zagst 2002] or [Ito 2004].
69 The idea behind this approach is that to avoid arbitrage opportunities, the prices of two financial
assets producing the same payoff at maturity, have to be equal at each other time before
maturity. For an illustrative introduction to risk neutral derivatives pricing see [Zagst 2002].
48

3.1 The Risk Premium Model
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:
P C (t) = e rp(T −t) P A (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.
✷
The general Risk Premium Model represents the point of view that commodity
futures prices equal the expected commodity spot price, discounted by a risk premium
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:
r FC (t,T )(s, t) ≡ ln
( ) (3.1)
( )
FC (t, T ) {}}{
E ˜Q[P C (T )|F t ]
= r p (t − s) + ln
F C (s, T ) } {{ } E ˜Q[P C (T )|F s ]
risk premium } {{ }
change of price expectation
(3.4)
As we have seen above, according to the Risk Premium Model, the return an investor
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).
70 See [Geer 2000].
49

3 Pricing of Commodity Futures
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 values
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 differences
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 schedule
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 benefit
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 P C (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 ↦→ R be the deterministic convenience yield and let
r f
denote the constant risk free interest rate, then the price of a commodity future
F C (t, T ) at time t ∈ [0, T ] in the Convenience Yield Model is given by:
F C (t, T ) = P C (t)e (r f +u−c(t))(T −t)
(3.5)
Proof:
given by:
With the notation of Theorem 3.1 the price of a commodity future is
F C (t, T ) = e −rp(T −t) E ˜Q[P C (T )|F t ] (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 P C (T ):
P A (t) = e −(r f +r p+u)(T −t) E ˜Q[P C (T )|F t ] (3.7)
Setting equal the expectations in Equation 3.6 and 3.7 yields to:
F C (t, T ) = e (r f +u)(T −t) P A (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:
P C (t) = (1 + C(t))P A (t), with C(t) ≥ 0 (3.9)
This equation states that the commodity spot price P C (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

3 Pricing of Commodity Futures
common approximation for the convenience yield:
(1 + C(t)) ∼ = e c(t)(T −t) (3.10)
Putting this into Equation 3.9 yields to:
P C (t) = e c(t)(T −t) P A (t) (3.11)
Furthermore, putting this into Equation 3.8 yields to the result as in Equation (3.5).
✷
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:
F C (t, T ) = P C (t)e (r f −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 originally
a no-arbitrage based valuation concepts. Therefore, Theorem 3.2 is in literature
mainly proofed with the following arbitrage argument: If the current futures
price F C (t, T ) was greater than the right hand side of Equation 3.5 namely
P C (t)e (r f +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. Consequently,
if F C (t, T ) was strictly smaller than the right hand side P C (t)e (r f +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
71 See e.g. [German 2005].
72 See 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 :
( )
FC (t, T )
r FC (t,T )(s, t) ≡ ln
F C (s, T )
(3.5) ( )
{}}{ PC (t)
= ln + (r f + u − c(t))(T − t)
P C (s)
− (r f + u − c(s))(T − s)
( )
PC (t)
= ln
P C (s)
} {{ }
change in spot price
+ (c(t) − r f − 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.
73 See [Gibson Schwartz 1990], [Schwartz 1997] or [Cassasus Collin-Dufresne 2005].
53

3 Pricing of Commodity Futures
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)
P C (s)
( )
E ˜Q[P C (T )|F t ]
= ln
+ (c(t) − r
E ˜Q[P f − r p − u)(T − t)
C (T )|F ∫ ]
− (c(s) − r f − r p − u)(T − s)
( )
E ˜Q[P C (T )|F t ]
= ln
+ (r
E ˜Q[P f + r p + u − c(t))(t − s)
C (T )|F ∫ ]
+ (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):
r FC (t,T ),C(s, t)
(3.12) ( )
{}}{ PC (t)
= ln + (c(t) − r f − u)(t − s)
P C (s)
+ (c(s) − c(t))(T − s)
( )
(3.13)
{}}{ E ˜Q[P C (T )|F t ]
= ln
+ (r
E ˜Q[P f + r p + u − c(t))(t − s)
C (T )|F s ]
+ (c(t) − c(s))(T − s) + (c(t) − r f − u)(t − s)
+ (c(s) − c(t))(T − s)
( )
E ˜Q[P C (T )|F t ]
= r p (t − s) + ln
E ˜Q[P C (T )|F s ]
(3.4)
{}}{
= r FC (t,T ),R(s, t) (3.14)
Therefore, depending on either view the futures price of a commodity is given by:
F C (t, T ) = e −rp(T −t) E ˜Q[P C (T )|F t ]
= P C (t)e (r f +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 ( E ˜Q[P C (T )|F t ] ) ( )
E ˜Q[P C (T )|F t ]
− ln (P C (t)) = ln
P C (t)
= (r f + r p + 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
3.4 Stochastic Models
The term structure gives the relationship between the futures prices and the respective
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 undertake
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 introduce
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 Reversion
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.
74 We will further denote the futures price of a commodity with F (t, T ) = F C (t, T ) and its spot
price with P (t) = P C (t) at t ∈ [0, T ].
55

3 Pricing of Commodity Futures
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, σ P > 0 the spot price volatility and W P (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 + σ P P (t)dW P (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,
75 An illustrative introduction can be found in [Zagst 2002].
56

3.4 Stochastic Models
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
= F t (P, t)
∂F (P,t)
∂P
= F P (P, t)
∂ 2 F (P,t)
∂ 2 P
= F P P (P, t) 2 .
Using Itô’s lemma as of Definition C.1, the instantaneous change in the futures price
is given as:
dF (P, t) =
(3.17)
{}}{
=
=
[
F t (P, t) + µP (t)F P (P, t) + 1 ]
2 σ2 P P (t) 2 F P P (P, t) dt
+ σ P P (t)F P (P, t)dW P (t)
[
F t (P, t) + 1 ]
2 σ2 P P (t) 2 F P P (P, t) dt
⎡
⎢
⎥
+ F P (P, t) ⎣µP (t)dt + σ P P (t)dW P (t) ⎦
} {{ }
dP (t)
[
F t (P, t) + 1 ]
2 σ2 P P (t) 2 F P P (P, t) dt + F P (P, t)dP (t) (3.18)
⎤
The difference between commodities as simple financial asset and consumption good
is captured in the net convenience yield 76 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
76 Compare Remark 3.1 of Section 3.2.
57

3 Pricing of Commodity Futures
price process: 77
dP (t) + CP (t)dt = µP (t)dt + σ P P (t)dW P (t), t ∈ [0, T ]
⇒ dP (t) = (µ − C)P (t)dt + σ P P (t)dW P (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:
with:
V (t) = P (t) + c(P, t) − δF (P, t)
dV (t) = dP (t) + CP (t)dt − δdF (P, t)
(3.18)
[
{}}{
= dP (t) + CP (t)dt − δ( F t (P, t) + 1 ]
2 σ2 P P (t) 2 F P P (P, t) dt
+ F P (P, t)dP (t))
= [CP (t) − δ(F t (P, t) + 1 2 σ2 P P (t) 2 F P P (P, t))]dt
δ= 1
F P (P,t)
{}}{
=
risk free
{}}{
≡
+ (dP (t) − δF P (P, t)dP (t))
} {{ }
risk free⇔=0⇔δ= 1
F P (P,t)
[
1
CP (t)F P (P, t) − F t (P, t) − 1 ]
F P (P, t)
2 σ2 P P (t) 2 F P P (P, t) dt
r f P (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 ):
F t (P, t) + 1 2 σ2 P P (t) 2 F P P (P, t) + F P (P, t)P (t)(r f − C) = 0 (3.20)
77 Compare Equation (3.9).
58

Definition C.32: 78 F (P, t) = E ˜Q[P (T )|F t ], t ∈ [0, T ] (3.22)
3.4 Stochastic Models
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 deterministic
convenience yield and r f 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 (r f −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 F t as of Definition C.5,
of its future spot price P (T ) under the equivalent martingale measure ˜Q as of
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 martingale
measure ˜Q which exists because of the Girsanov-Theorem as of Theorem C.3.
Denote with d ˜W the increments of the Brownian motion as of Definition C.28 under
˜Q. Using the Girsanov-Theorem as of Theorem C.3, we have:
d ˜W (t) = λ(t)dt + dW (t), t ∈ [0, T ] (3.23)
where λ : R ↦→ R is called the market price of risk. It results
dP (t) = [µ − σ P λ(t)]P (t)dt + σ P P (t)d ˜W P (t), t ∈ [0, T ] (3.24)
where it has to be µ − σ P λ(t) = r f because the discounted spot price process has to
be a martingale as of Definition C.29. It is
dP (t) = r f P (t)dt + σ P P (t)d ˜W P (t), t ∈ [0, T ]. (3.25)
78 Also compare Equation (3.2).
79 Attention: The opposite direction is not always true. See [Zagst 2002].
59

3 Pricing of Commodity Futures
Again, we have to amend the financial spot price of commodities with the convenience
yield process as in Equation (3.19). It follows
dP (t) = (r f − C)P (t)dt + σ P P (t)d ˜W P (t), t ∈ [0, T ] (3.26)
Then, the adapted Cauchy-Problem as of Definition C.36 is given as:
F t (P, t) + 1 2 σ2 P P (t) 2 F P P (P, t) + F P (P, t)P (t)(r f − 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 (r f −C)(T −t) , with t ∈ [0, T ],
we get:
F P (P, t) = e (r f −C)(T −t) ,
F P,P (P, t) = 0,
F t (P, t) = −(r f − C)F (P, t), t ∈ [0, T ].
Putting this into the Cauchy-Problem, Equation (3.27), it follows
0 + (r f − C)F (P, t) − (r f − C)F (P, t) = 0, ∀t ∈ [0, T ]
which shows, that the futures price is indeed F (P, t) = P (t)e (r f −C)(T −t) .
✷
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 commodities
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
80 See the latest work [Markert 2005].
60

3.4 Stochastic Models
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 longrun
mean, κ > 0 the speed of adjustment of the spot price, σ P > 0 the spot price
volatility and W P (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 + σ P P (t)dW P (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 ↦→ R be the market price of risk
introduced in the Girsanov-Theorem as of Theorem C.3 and r f 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
with t ∈ [0, T ].
Proof:
F (P, t) = exp[e −κ(T −t) ln P (t) + (1 − e −κ(T −t) )(µ − σ 2 P /2κ − λ)
+ σ2 P
4κ (1 − e−2κ(T −t) )], (3.29)
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 + σ P P (t)d ˜W P (t), t ∈ [0, T ].
d ˜W P (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

3 Pricing of Commodity Futures
is given as:
1
2 σ2 P P (t) 2 F P,P (P, t) + P (t)κ[µ − ln P (t) − λ]F P (P, t) + F t (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 − σP 2 /2κ − λ) + σ2 P
4κ (1 − e−2κ(T −t) )
with t ∈ [0, T ], we can calculate the respective derivatives:
F P (P, t) = e −κ(T −t) F (P,t)
P (t) ,
F P,P (P, t) = e −2κ(T −t) F (P,t)
− e −κ(T −t) F (P,t)
,
P (t) 2
P (t) 2 ]
F t (P, t) =
[κe −κ(T −t) (ln P (t) + λ − µ + σP 2 /2κ) − σ2 P
2
e −2κ(T −t) F (P, t), t ∈ [0, T ].
Putting this into the Cauchy-Problem of Equation (3.30) it follows:
1
2 σ2 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 + σP 2 /2κ) − σ2 P
2 e−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 condition
F (P, T ) = P (T ). Under our assumption it holds
F (P, T ) = exp
This proves our assumption.
[
−κ(T −T )
−κ(T −T )
e} {{ } ln P (T ) + (1
}
− e
{{ }
)(µ − σP 2 /2κ − λ)
=1
=0
]
+ σ2 P
−T )
(1 − e−2κ(T
4κ } {{ }
)
=0
= P (T ).
✷
62

3.4 Stochastic Models
The model has the major drawback that it treats positive and negative mean reversion
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 between
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 stochastic
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,
σ P > 0 the spot price volatility, σ c > 0 the convenience yield volatility and dW P (t)
and dW c (t) the increments of two Brownian Motions as defined in Definition C.28
with a correlation
dW P (t)dW c (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 + σ P P (t)dW P (t), (3.32)
dc(t) = κ[α − c(t)]dt + σ c dW c (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

3 Pricing of Commodity Futures
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 dW P (t) is typically associated with a positive dW c (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 ↦→ R be the market price of risk
introduced in the Girsanov-Theorem as of Theorem C.3 and r f 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
[ ( )
1 − e
−κ(T −t)
F (P, c, t) = P (t) exp −c(t)
κ
]
+ A(T, t) , t ∈ [0, T ] (3.34)
with
A(T, t) =
(
r f − α + λ κ + 1 σc
2
2
+
([
α − λ ]
κ + σ P σ c ρ − σ2 c
κ
κ
κ − σ )
P σ c ρ
(T − t) + 1 2 κ
4 σ2 c
) 1 − e
−κ(T −t)
−2κ(T −t)
1 − e
κ 3
κ 2 (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)[r f − c(t)]dt + σ P P (t)d ˜W P (t),
dc(t) = (κ[α − c(t)] − λ)dt + σ c d ˜W c (t),
d ˜W P (t)d ˜W c (t) = ρdt, t ∈ [0, T ]
d ˜W P (t) and d ˜W c (t) are the increments of two Brownian motions under the equivalent
martingale measure. Based on these equations we can formulated the specific
81 See [Markert 2005] for empirical evidence.
64

3.4 Stochastic Models
Cauchy-Problem:
0 = 1 2 σ2 P P (t) 2 F P,P (P, c, t) + ρσ P σ c P (t)F P,c (P, c, t)
+ 1 2 σ2 c F c,c (P, c, t) + [r f − c(t)]P (t)F P (P, c, t)
+ (κ[α − c(t)] − λ)F c (P, c, t) + F t (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
[ ( )
1 − e
−κ(T −t)
F (P, c, t) = P (t) exp −c(t)
κ
]
+ A(T, t) , t ∈ [0, T ]
with
A(T, t) =
(
r f − α + λ κ + 1 σc
2
2
+
([
α − λ ]
κ + σ P σ c ρ − σ2 c
κ
κ
κ − σ )
P σ c ρ
(T − t) + 1 2 κ
4 σ2 c
) 1 − e
−κ(T −t)
−2κ(T −t)
1 − e
κ 3
κ 2 , t ∈ [0, T ]
we can calculate the respective derivatives:
F P (P, c, t) = F (P,c,t)
P (t)
,
F P,P (P, c, t) = 0,
F c (P, c, t) = −
F c,c (P, c, t) =
(
1−e
(
F P,c (P, c, t) = −
−κ(T −t)
κ
(
1−e−κ(T −t)
κ
1−e−κ(T −t)
P (t)κ
)
F (P, c, t),
) 2
F (P, c, t),
)
F (P, c, t),
and finally,
F t (P, c, t) =
[
(
c(t)e −κ(T −t) − r f − α + λ κ + 1 σc
2
2 κ − σ )
P σ c ρ
2 κ
− 1 ([
α − λ ]
κ + σ P σ c ρ − σ2 c
κ κ
κ
)
e −κ(T −t) ]
F (P, c, t)
− σ2 c −t)
e−2κ(T
2κ2 with t ∈ [0, T ].
65

3 Pricing of Commodity Futures
Putting this into the Cauchy-Problem Equation (3.36) it follows
0 + 1 ( ) 1 − e
−κ(T −t) 2
2 σ2 c
F (P, c, t) + [r f − c(t)]F (P, c, t)
κ
( )
1 − e
−κ(T −t)
− (κ[α − c(t)] − λ + ρσ P σ c )
F (P, c, t)
κ
[
(
+ c(t)e −κ(T −t) − r f − α + λ κ + 1 σc
2
2 κ − σ )
P σ c ρ
− σ2 c −t)
2 e−2κ(T
κ 2κ2 − 1 ([
α − λ ]
) ]
κ + σ P σ c ρ − σ2 c
e −κ(T −t) F (P, c, t)
κ κ
κ
= [r f − c(t)]F (P, c, t) − 1 κ (κ[α − c(t)] − λ + ρσ P σ c ) ( 1 − e −κ(T −t)) F (P, c, t)
[
(
+ c(t)e −κ(T −t) − r f − α + λ κ − σ )
P σ c ρ
κ
− 1 ([
α − λ ] ) ]
κ + σ P σ c ρ e −κ(T −t) F (P, c, t)
κ κ
= 1 κ (κ[α − c(t)] − λ + ρσ P σ c )e −κ(T −t) F (P, c, t)
[
+ c(t)e −κ(T −t) − 1 ([
α − λ ] ) ]
κ + σ P σ c ρ e −κ(T −t) F (P, c, t)
κ κ
= 1 κ (κ[α − c(t)] − λ + ρσ P σ c )e −κ(T −t) F (P, c, t)
− 1 κ (κ[α − c(t)] − λ + ρσ P σ c )e −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 ⎢ 1 − e
−κ(T −T )
⎣ −c(T ) + A(T, T ) ⎥
κ
⎦
} {{ }
=0
= P (T ).
This proves Theorem 3.5.
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

3.4 Stochastic Models
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 following
intuition to estimate the parameters of the model from market data: movements
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, σ χ > 0 the short-term prices volatility, σ ξ > 0 the equilibrium price level
volatility and dW χ and dW ξ the increments of two standard Brownian Motions as
defined in Definition C.28 with a correlation
dW χ (t)dW ξ (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 + σ χ dW χ (t) (3.39)
dξ(t) = µ ξ dt + σ ξ dW ξ (t), t ∈ [0, T ] (3.40)
Temporary price changes, caused e.g. by abrupt weather alteration or supply interruptions,
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

3 Pricing of Commodity Futures
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 Convenience
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
κ
σ χ
Convenience Yield Model
dW χ (t)
dW c (t)
µ ξ µ − α − 1 2 σ2 P
σ ξ
(σ P + σ2 c
− 2ρσ κ 2 P σ c
) 1 2
κ
dW ξ (t)
ρ ξχ
κ
σ Pκ
(σ P dW P (t) − σc
κ dW c(t))(σ P + σ2 c
κ 2 − 2ρσ P σ c
κ
) − 1 2
(ρσ P − σc
κ )(σ P + σ2 c
κ 2 − 2ρσ P σ c
κ
) − 1 2
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 + σ P P (t)dW P (t),
dc(t) = κ[α − c(t)]dt + σ c dW c (t), t ∈ [0, T ]
With Itô as of Lemma C.1 the log spot price dynamic of (3.32) are given as:
dln(P (t)) =
(( ) 1
P (t)
( 1
+
P (t)
)
(P (t)[µ − c(t)])
)
σ P P (t)dW P (t)
dt + 0 + 1 2
( ( )) −1
σ P P (t) 2 dt
P 2 (t)
= [µ − c(t) − 1 2 σ2 P ]dt + σ P dW P (t) (3.41)
68

3.4 Stochastic Models
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 (c(t) − α)
κ
(3.42)
Therewith, it follows
Moreover
dχ(t) = 1 κ dc(t)
(3.33)
{}}{
= 1 κ (κ[α − c(t)]dt + σ cdW c (t))
= [α − c(t)]dt + σ c
κ dW c(t)
(3.42)
{}}{
= −κχ(t)dt + σ c
dW c (t)
}{{} κ } {{ }
≡dW
≡σ χ(t)
χ
ξ(t) = equilibrium price level
= ln(P (t)) − χ(t)
= ln(P (t)) − 1 (c(t) − α) (3.43)
κ
Therewith, it follows
dξ(t) = dln(P (t)) − 1 κ dc(t)
= [µ − c(t) − 1 2 σ2 P ]dt + σ P dW P (t) − 1 κ (κ[α − c(t)]dt + σ cdW c (t))
= [µ − α − 1 2 σ2 P ]dt + σ P dW P (t) − σ c
} {{ }
κ dW c(t)
} {{ }
≡µ ξ
≡σ ξ ≡dW ξ (t)
69

3 Pricing of Commodity Futures
Finally,
ρ ξχ dt = dW χ (t)dW ξ (t)
= dW c (σ P dW P (t) − σ c
κ dW c(t))(σ P + σ2 c
κ − 2ρσ P σ c
2 κ
) − 1 2
= (σ P dW P (t)dW
} {{ } c − σ c
κ dW c(t)dW c )(σ
} {{ } P + σ2 c
κ − 2ρσ P σ c
2 κ
=ρ P c dt
=dt
) − 1 2
= (ρ P c σ P − σ c
κ )dt(σ P + σ2 c
κ − 2ρσ P σ c
2 κ
) − 1 2
(3.44)
showing the last equation of Table 3.1.
✷
[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 σ χ 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 σ ξ equal to zero. Statistical
comparison of the three models by the authors showed significant advantages in capturing
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] proposed
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
82 Compare [Schwartz 1998].
83 See [Pindyck 1994] and [French 1983]. Compare Section 2.3.1.1 Paragraph ”Forwards and Futures”.
70

3.4 Stochastic Models
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,
σ P > 0 the spot price volatility, σ c > 0 the convenience yield volatility, σ r > 0
the interest rate volatility and dW P (t), dW c (t) and dW r (t) the increments of three
Brownian Motions as defined in Definition C.28 with the following correlations:
dW P (t)dW c (t) = ρ P c dt, dW c (t)dW r (t) = ρ cr dt and dW r (t)dW P (t) = ρ rP 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 + σ P P (t)dW P (t), (3.45)
dc(t) = κ[α − c(t)]dt + σ c dW c (t), (3.46)
dr(t) = β[m − r(t)]dt + σ r dW r (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

3 Pricing of Commodity Futures
return, σ P , σ y , σ v > 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 ρ P y , ρ P v , ρ yv ∈ [−1, 1]. Then the dynamic of this model is
dP (t) = P (t)[v(t) − y(t)]dt + σ P P (t)dW P (t), (3.48)
dy(t) = −κy(t)dt + σ y dW y (t), (3.49)
dv(t) = a[¯v − v(t)]dt + σ v dW v (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 products
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. Especially
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.
84 In 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 metals.
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 agricultures.
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 capitalization
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.
85 Further 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 designed
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 commodity.
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 average
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 requirements.
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
4.2 The Major Market Indices
After structuring the different creation characteristics, we want to see what differences
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 consumption
goods and one being the platform for financial investments. The indices are
86 Attention: The terminology ”leverage effect” describes in stock markets the correlation between
falling 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 underlying
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 investment
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 represents
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
87 For further details recall introduction to Section 3.
76

4.2 The Major Market Indices
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” commodity
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 commodity
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 weighting
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
88 See [German 2005].
89 See [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, further
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 nonstorable
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
90 See [Bessembinder e.a. 1995].
78

4.2 The Major Market Indices
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 commodity
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 couple
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 advantage
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) includes
the same components as the DBLCI. The difference is that the weights of its
91 For 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

4.2 The Major Market Indices
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 economic
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
92 See [Rogers 2005].
93 See [Erb Harvey 2006] or [Seamans 2003].
81

4 Commodity Indices
weights, its determination, the rebalancing period and the return calculation.
Index
Composition
CRB GSCI DJ-AIGCI DBLCI DBLCI-MR RICI
broad broad broad broad broad broad
Major Groups 94 all all all
Number of
Commodities
no Softs, no
Livestock
no Softs, no
Livestock
17 24 19 5 5 34
Index Weights equal economic
Determination
of Weights
economic,
market
all
fix optimized economic
if required annually annually if required monthly annually
Rebalancing monthly annually annually annually monthly monthly
Return
Calculation
geometric arithmetic arithmetic arithmetic arithmetic arithmetic
Table 4.1: Comparison of Commodity Index Characteristics
We only introduced broad diversified indices to reach better comparability. Economic
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.
93 As of Figure 2.1.
82

4.3 Index Linked Products
Figure 4.7: Index Component Distribution
4.3 Index Linked Products
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. Moreover,
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 commodity
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 Service 95 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 exposure
is usually indirect. But at the end of the day their buying power shows up
94 See [ICI 2006]
95 The Internal Revenue Service (IRS) is the US government agency responsible for tax collection
and tax law enforcement.
84

4.3 Index Linked Products
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 everybody
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 11,823
Oppenheimer Real Asset Fund GSCI 1,900
Fidelity Strategic Real Return Fund DJ-AIGCI 1,934
Credit Suisse Commodity Return Strategy DJ-AIGCI 277
DWS Scudder Commodity Securities Fund GSCI 183
Table 4.2: Commodity Index linked Mutual Funds
Combining the Net Asset Values 96 (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
96 The Net Asset Value is defined as the difference between total assets minus total liabilities.
97 See 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% commodity
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 commodity
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 concept
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
98 Regarding the change in regulatory guidelines a redeployment into commodity linked notes can
be expected.
86

4.3 Index Linked Products
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 exchanges
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 Börse.
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.
99 The NAV per share is defined as:
100 See [ICI 2006]
NAV per share =
total assets - total liabilities
total number of shares outstanding
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
✘ ✘✘ ✘ ✘✘ ✘❳❳ ❳❳
❳ ❳❳
Spot Return + Roll 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 contracts
have generally the highest liquidity. Futures investments need minimal cash
requirements that are only used to serve margin calls. But to actually add commodities
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-
101 Figure 4.8 and the following calculations are based on log returns as of Definition C.2. Compare
[Kat Oomen 2006]. For a commodity return decomposition based on simple returns see
[Geer 2000].
88

4.4 Decomposition of Index Returns
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 historically
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 contract
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 contracts.
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 63.25
31. Jan 2006 68.35 →67.92 ↓
68.74 69.28 69.70 70.01
28. Feb 2006 61.10 →61.41 ↓
63.01 64.06 64.83
31. Mar 2006 60.57 →66.63 ↓
67.93 68.67
28. Apr 2006 71.95 → 71.88 ↓
73.50
31. 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:
r F (t) ≡ ln
( ) F (t, T )
, 0 ≤ s < t ≤ T
F (s, T )
102 See Theorem 4.1.
103 The procedure is inspired by [Markert 2005] and [Gorton Rouwenhorst 2004].
89

4 Commodity Indices
Implicating, the investor realizes a crude oil futures return of:
r F (Jan) = ln
( ) ( )
F (Jan, F eb) 68.35
= ln = 11.3%
F (Dec, F eb) 61.04
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:
r F (F eb) = ln
( ) ( )
F (F eb, Mar) 61.10
= ln = −10.6%
F (Jan, Mar) 67.92
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:
r P (t) ≡ ln
( ) P (t)
, 0 ≤ s < t ≤ T
P (s)
Implicating, the first crude oil spot return value is given by:
r P (Jan) = ln
The second value is gives by:
r P (F eb) = ln
( ) ( )
P (Jan) 68.35
= ln = 16.5%
P (Dec) 57.98
( ) ( )
P (F eb) 61.10
= ln = −11.2%
P (Jan) 68.35
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

4.4 Decomposition of Index Returns
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:
Proof:
( ) ( )
F (t, T ) P (t)
rr(t) = ln
− ln
F (s, T ) P (s)
} {{ } } {{ }
futures return spot return
(4.1)
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:
r F (t) ≡
( ) F (t, T )
ln
F (s, T )
= ln(F (t, T )) − ln (F (s, T )) + ln (P (s)) − ln (P (s))
} {{ }
=P (t)
Rearranging yields to the result.
( ) ( )
P (t) P (t − 1)
= ln + ln
, 0 ≤ s < t ≤ T (4.2)
P (s) F (s, T )
} {{ } } {{ }
spot return roll return
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 4,173
31. Jan 2006 4,912 →4,886 ↓
4,853 4,815 4,768 4,721
28. Feb 2006 4,881 →4,842 ↓
4,812 4,778 4,742
31. Mar 2006 5,440 →5,423 ↓
5,400 5,375
28. Apr 2006 7,118 →7,066 ↓
7,008
31. 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 extending
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 futures
return of ln ( 68.94
63.25)
= 8.6% because the roll return would have decreased to
ln ( 63.25
57.98)
= −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
104 See [Merrill Lynch 2006]. To be precise, Merrill Lynch employ a short option trading facility as
well to minimize the negative influence of contango to the roll return.
92

4.4 Decomposition of Index Returns
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 expected
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 contract
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 volume
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 production
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 movements.
Although, the different types of commodities are influenced by their own specific
risk factors, technological progress allows new substitution possibilities. So, commodities
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 extent
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 statistical
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 components.
105
We will uncover roll returns and show their impact on total returns in
105 The switch from excess return in Section 5.1 to total return in the Section 5.2 is motivated as
follows: 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
5.1 Characteristics of Single Commodities
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. Nevertheless,
we identified dependencies resulting from substitutions or production hierarchical
structures. The question to answer in this section is consequently, how these
macroeconomic dependencies can be seen in statistical characteristics of return series’
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 comparison
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 commodity
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’s
return. Pure futures return can be seen as an overlay to a portfolio but if an investor actually
wants to invest a part of his wealth into commodities, he has to do so over the collateralized
version as described in Section 4.4, i.e. he had to consider total returns.
106 A detailed data description of our sample can be found in Appendix A.
95

5 Properties of Commodity Returns
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 commodity
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 additive
108 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 components 109 of the different commodity indices of our small peer group.
107 To be precise: Let {r 1 , . . . , r T } be a discrete random sample of returns as of Definition C.2 at
times t ∈ {1, . . . , T }. The sample mean is defined as:
¯r = 1 T
T∑
r t (5.1)
t=1
108 See Equation (C.5).
109 The 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

5.1 Characteristics of Single Commodities
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. Hilary
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 mention
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.
110 See [Till 2000].
97

5 Properties of Commodity Returns
Morgan Stanley investigated into the relationship between excess returns and the
time a commodity spent in contango respectively in backwardation. In the presentation
[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 significant
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

5.1 Characteristics of Single Commodities
it is plausible that prices are set such that the expected return exceeds the interest
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 dependent
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 commodity
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 downward
adjustment of the estimated oil reserves in the US or another region, oil prices
and their volatility increase sharply.
99

5 Properties of Commodity Returns
To investigate the variability of different commodity indices we calculate the annualized
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 variance 111 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
111 To be more precise: Let {r 1 , . . . , r T } be a discrete random sample of returns as of Definition C.2
at times t ∈ {1, . . . , T } and ¯r be the sample mean as in Equation (5.1). The sample variance is
defined as:
¯σ 2 = 1 T∑
(r t − ¯r) 2 (5.2)
T − 1
Annualized values are calculated by scaling linear with the average number of observations per
year because continuous returns are time additive. The square root of the sample variance:
is called the sample standard deviation.
t=1
¯σ = √¯σ 2 (5.3)
100

5.1 Characteristics of Single Commodities
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 1.9
ER 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

5 Properties of Commodity Returns
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 participants.
This might be an indication for diversification effects and our guess is underlined
by the small standard deviation of the broad indices.
5.1.2 Correlation
Correlation 113 is the normalized covariance 114 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 movements
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 Pearson’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 considered
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 examples
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 interdependencies
in the soybean complex in Appendix B.11. We saw a link between
112 Compare Table 6.5.
113 For the formal mathematical definition see Definition C.15.
114 For the formal mathematical definition see Definition C.14.
102

5.1 Characteristics of Single Commodities
macroeconomic influences and commodities, e.g. gold in Section 2.1.2.1, the behavior
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 {r 1 , . . . , r T } and {l 1 , . . . , l T } 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
∑
1 T
T −1 t=1 (r t − ¯r)(l t − ¯l)
∑ T
t=1 (r t − ¯r) 2 √
1
T −1
∑ T
t=1 (l t − ¯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 highlighted
with a brown color in Table 5.3.
Examining our findings we realize higher similar price co-movements among commodity
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 commodity
indices. Figure 5.2 illustrates the phenomena in the risk return space. The
115 Yellow values are significant at the 1% alpha level, blue values are significant at the 5% alpha
level and brown values are insignificant. We tested the null hypothesis of zero correlation and
used the t-test with the following test statistic:
t =
ρ
√
1 − ρ
2 ∗ √ T − 2 (5.5)
It follows a Student’s t-distribution with T − 2 degrees of freedom. For further details of
hypothesis testing see Section 5.2.3, [Bamberg Baur 2002] or [Kanji 1999].
103

5 Properties of Commodity Returns
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 members,
strong price movements of one commodity are balanced by moderate price
104

5.1 Characteristics of Single Commodities
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 dependence.
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 arbitrary
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 r t respectively l t 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 r t respectively l t have different values, each number is found exactly one time.
If some observations of r t respectively l t 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 . Kendall’s tau uses the relative order of the ranks to
1
2T (T +1)
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. r t > r t+1 and l t > l t+1 and the number of
concordant states over the whole period is denoted with n c
2. a pair is discordant at time t, i.e. r t < r t+1 and l t < l t+1 and the number of
discordant states over the whole period is denoted with n d
3. there is a tied observation, i.e. r t = r t+1 and l t = l t+1 and the number of
tied r-observations is denoted by n tr and the number of tied l-observations is
denoted by n tl
Definition 5.2 Kendall’s Correlation
Let {r 1 , . . . , r T } and {l 1 , . . . , l T } be two discrete random samples of different returns
at times t ∈ {1, . . . , T } and recall the definitions of concordant, discordant ( and tied )
observations above. The total number of possible pairings of r t and l t is T (T −1)
.
If there are no tied observation the Kendall correlation coefficient is defined as:
2
τ =
n c − n d
T (T − 1)/2
(5.6)
105

5 Properties of Commodity Returns
If there are tied observation the Kendall correlation coefficient is defined as:
τ =
√ (
T (T −1)
2
− ∑ n tr
i=1
n tr,i (n tr,i −1)
2
n c − n d
) (
T (T −1)
2
− ∑ n tl
i=1
) (5.7)
n tl,i (n tl,i −1)
2
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
116 Yellow values are significant at the 1% alpha level, blue values are significant at the 5% alpha
level and brown values are insignificant. We tested the null hypothesis of zero correlation and
used 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 see
Section 5.2.3, [Bamberg Baur 2002] or [Kanji 1999].
117 Indeed, the correlations among the excess returns do not differ relevantly from the correlations
among the spot returns. To avoid redundance, they are not explicitly reported.
106

5.1 Characteristics of Single Commodities
[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 differences.
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 returns
is well known and implicating the fact, that correlations may not be an appropriate
measure of co-dependence. Nevertheless, correlation is related to the slope
parameter of a linear regression model.
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5 Properties of Commodity Returns
Definition 5.3 Linear Regression Model
Let {r 1 , . . . , r T } and {l 1 , . . . , l T } be two discrete random samples of different returns
at times t ∈ {1, . . . , T }, α and β two constant inR and ε t be an error term of identical
and independent distributed random variables with mean zero and variance σε. 2 Let
r t denote the dependent variable and the l t the independent. A linear regression
model is given by the equation of a line:
r t = α + βl t + ε t (5.9)
To fit such line through actual observed values we need an estimated line, denoted
by ˆr t = ˆα + ˆβˆl t where ˆα and ˆβ denote the estimate of the line intercept α and the
slope β. The residuals are defined as e t = r t − ˆr t . Then, the actual data points are
the fitted model plus the residuals: r t = ˆα + ˆβl t + e t .
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
ˆα and ˆβ are given by:
ˆβ =
∑ T
t=1 (r t − ¯r)(l t − ¯l)
∑ T
t=1 (r t − ¯r) 2 (5.10)
or ˆβ = 0 if r t = ¯r ∀t ∈ {1, . . . , T } and
ˆα = ¯r − ˆb¯l (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(ˆα, ˆβ) =
T∑
[r t − (ˆα + ˆβl t )] 2
t=1
To find an optimum we need to set the first derivation regarding ˆα and ˆβ zero and
108

5.1 Characteristics of Single Commodities
solving for the respective unknown.
∂
∂ ˆα L(ˆα, ˆβ) =
=
∂
∂ ˆβ L(ˆα, ˆβ) =
=
T∑
−2r t + 2(ˆα + ˆβl t )
t=1
T∑
−r t + nˆα + ˆβ
t=1
T∑
l t ≡ 0 (5.12)
t=1
T∑
−2r t l t + 2(ˆα + ˆβl t )l t
t=1
T∑
−r t l t + ˆα
t=1
t=1 t=1
T∑
l t + ˆβ
T∑
lt 2 ≡ 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 l t are equal. Putting the solutions into the respective second partial
derivations shows that the solutions yield indeed to a minimum. 118
✷
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:
ˆβ =
∑ T
t=1 (r t − ¯r)(l t − ¯l)
∑ T
t=1 (r t − ¯r) 2
=
=
∑
1 T
T −1 t=1 (r t − ¯r)(l t − ¯l)
∑
1 T
T −1 t=1 (r t − ¯r) 2
√
1
T −1
√
= ρ ∗ √
∑
1 T
T −1 t=1 (r t − ¯r)(l t − ¯l)
∑ T
t=1 (r t − ¯r) 2 √
1
T −1
1
T −1
∑ T
t=1 (l t − ¯l) 2
1
T −1
√
∑ T
t=1 (l t − ¯l)
∗ √
2
1
T −1
1
T −1
∑ T
t=1 (l t − ¯l) 2
∑ T
t=1 (r t − ¯r) 2
∑ T
t=1 (r t − ¯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 dependency
of the GSCI as market index to the energy market. 119
118 For further details see [Bamberg Baur 2002]
119 The goodness of fit parameter R 2 T
t=1
= P (ˆrt−¯r)2
T = 0.93 and therewith relatively high. The
t=1 (rt−¯r)2
beta coefficient is significant at the 1% alpha level. For an introduction of significance tests
regarding regression coefficients see e.g. [Bamberg Baur 2002].
109

5 Properties of Commodity Returns
Figure 5.4: Dependence of Market Index (1998-2006)
It became clear how high the GSCI returns depend on returns of the energy market.
This is not astonishing because recall Figure 4.7 that uncovers the high overweighting
of the GSCI in favor of energy products.
Summing up, the above results show that the commodity market is not a homogeneous
but a heterogeneous universe. As Section 2.1 let us already guess the markets
consists of dissimilar assets that depend on their own price influencing factors.
Therefore, spreading its wealth among more than one commodity can increase investors
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

5.1 Characteristics of Single Commodities
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) = ( 1 n · · · 1
n )T ∈ R n . Let C = (c ij ) i=1...n,j=1...n be the n × n symmetric covariance
matrix of the n assets. Then the portfolio variance σ 2 (x ∗ (n)) converges
forn −→ ∞ against the average covariance ¯σ c ∈ R:
σ 2 (x ∗ (n)) = ¯σ c + 1 n (¯σ2 − ¯σ c ) −→ ¯σ c for n −→ ∞ (5.15)
Proof:
than defined as:
Let x define the asset weights of a portfolio. The portfolio variance is
σ 2 (x) = x T Cx
n∑
= x i x j c ij
=
i,j=1
n∑
x 2 i σi 2 +
i=1
n∑ n∑
x i x j c ij
i=1 j=1,i≠j
Now put x = x ∗ (n) = ( 1 n · · · 1
n )T ∈ R n and it follows:
σ 2 (x ∗ (n)) =
=
n∑
n∑
x ∗ i (n) 2 σi 2 +
} {{ }
i=1 1
i=1
n 2
1
( 1 n∑
σi 2 )
n n
i=1
} {{ }
≡ ¯σ 2 average variance
n∑
j=1,i≠j
x ∗ i (n) x ∗
} {{ } }
j(n)
{{ }
1 1
n n
+ n − 1
n ( 1
c ij
n∑
n∑
c ij )
n(n − 1)
i=1 j=1,i≠j
} {{ }
≡ ¯σ c average covariance
lim n→∞
= ¯σc + 1 n (¯σ2 − ¯σ n )
} {{ }
0
= ¯σ c
what is a constant independent of n
✷
111

5 Properties of Commodity Returns
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 correlation
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 diversification
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,
120 See [Wilshire Research 2005].
112

5.1 Characteristics of Single Commodities
in Figure 4.7 we summed up the weights of the single commodities and sorted
them into the buckets energy, industrials 121 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 constituencies.
The disadvantage of this group are lower returns and high dependencies to
unpredictable and uncertain weather conditions. Broad indices bring the homogenous
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
121 This bucket includes metals but also commodities like rubber and cotton.
113

5 Properties of Commodity Returns
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 {r 1j , . . . , r T j } for j = 1, . . . , n different assets at times
t ∈ {1, . . . , T }. Denote with ¯r j , j = 1, . . . , n, the assets’ specific sample mean as
of Equation (5.1) and with ¯σ j , j = 1, . . . , n, the assets’ specific sample standard
deviation as of Equation (5.3).
Define the normalized variables z tj = r tj−¯r j
¯σ j
and 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 a j,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 d ij ≠ 0 if
i ≠ 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 = F A T + 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 representation
of Z in Equation (5.16), the correlation matrix has the following form:
R = AA T + 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:
z tj =
k∑
a jk F l + d j U ij
l=1
In matrix notion:
Z = F A T + UD
114

5.1 Characteristics of Single Commodities
We assume that:
the normalized specific factors are uncorrelated among each other, i.e.
1
n U T U = I (5.18)
the normalized specific factors are uncorrelated to the normalized common factors,
i.e.
U T F = 0 (5.19)
the normalized specific factors are uncorrelated among each other, i.e.
With Definition C.15 follows:
1
n F T F = I (5.20)
R = 1 n ZT Z
(5.16)
{}}{
= (F A T + UD) T (F A T + UD)
= F} {{ T F}
AA T + (U} {{ T F}
) T (A T D T ) T + U} {{ T F}
A T D T + U} {{ T U}
D T D
}{{} = I
(5.20)
}{{} = 0
(5.18)
}{{} = 0
(5.19)
}{{} = I
(5.18)
= AA }{{}
T + }{{} DD
common normalized variance 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].
✷
The visualization of our results are shown in Figure 5.6 where we plotted the common
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 challenge
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

5 Properties of Commodity Returns
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 commodity
market.
Summing up, this section has shown that putting different commodities into a portfolio
disembogues into diversification effects providing the investor with attractive
risk and return profiles. Nevertheless, the energy market has the strongest influence
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 important
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 homogenous
one but is driven by different risk and influencing factors. Finally, we deduced
116

5.2 Properties of the DJ-AIGCI Return Components
that broad diversified commodity indices represent the optimal vehicle to get commodity
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
122 From this point on we use data series starting at 01.01.1991 to cover the whole history of the
index.
117

5 Properties of Commodity Returns
component. To show the development of the single elements and its value proposition
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 investment
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. Because
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

5.2 Properties of the DJ-AIGCI Return Components
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 summarized
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 annum
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
Return
Annualized 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

5 Properties of Commodity Returns
comparison to excess and spot returns, roll returns are small. This can also be
seen in Figure 5.8 by comparing the scales 123 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 investors
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 seasonal
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
123 Because the minimum return of -9.17% occurred as a stand alone outlier at the inception of the
DJ-AIGCI, we cut of the value for better observability of the general return development.
124 For a detailed discussion of the VaR see [Zagst 2002].
120

5.2 Properties of the DJ-AIGCI Return Components
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 historically
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
125 See [Risknet 2006].
121

5 Properties of Commodity Returns
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 Brynjolfsson,
head of the Pimco Real Return Commodity Strategy Fund 126 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 histogram 128 of the real roll returns and on the right
side we plotted the pure roll returns, i.e. the roll return that actually occurred during
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 often
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
126 Recall, the fund was introduced in Section 4.3 and is by far the biggest commodity mutual fund
on the market. It tracks the DJ-AIGCI.
127 See [Risknet 2006].
128 Further details about the contraction of a histogram see Section 5.2.3.
122

5.2 Properties of the DJ-AIGCI Return Components
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 often
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 distribution
129 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.
129 For a general introduction or review of statistical distribution and their moments see any introductory
statistics or time series analysis book, e.g. [Bamberg Baur 2002] or [Tsay 2002].
123

5 Properties of Commodity Returns
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 standard
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 occurred
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
130 For a brief introduction to number theory see [Broecker 1995]
131 To be more precise: Let {r 1 , . . . , r T } 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, then
the bins of the histogram are defined as the intervals [¯r + mh, ¯r + (m + 1)h] for m ∈ Z. The
density estimator ˆf in a histogram is defined as:
ˆf(r) = 1
T h (no. of r i in the same bin as r) (5.21)
124

5.2 Properties of the DJ-AIGCI Return Components
assumption of normality or not, as a first indicator we need to investigate the symmetry
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 {r 1 , . . . , r T } 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∑
( ) 3 rt − ¯r
(5.22)
t=1
¯σ
Skewness is a measure of the symmetry of the probability distribution of a realvalued
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 negative
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 {r 1 , . . . , r T } 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∑
( ) 4 rt − ¯r 3(T − 1)2
−
¯σ (T − 2)(T − 3)
t=1
(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
132 The 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

5 Properties of Commodity Returns
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 q α is chosen
that α% of all realized returns in the sample are smaller than the α% - quantile value
q α . 133
Definition 5.6 Bowley Skewness
Let {r 1 , . . . , r T } be a discrete random sample of returns at times t ∈ {1, . . . , T } and
let q α be as defined in C.18 with α ∈ [0, 1]. The Bowley skewness is defined as:
¯ S B = q 0.75 + q 0.25 − 2q 0.5
q 0.75 − q 0.25
(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 1.81
Bowley Skewness -0.03 -0.05 -0.32
Sample Kurtosis 5.98 5.83 13.08
Moors 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:
133 The formal definition can be found in C.18.
134 The formal definitions of µ and σ can be found in C.10 and C.11.
126

5.2 Properties of the DJ-AIGCI Return Components
Definition 5.7 Moors Kurtosis
Let {r 1 , . . . , r T } be a discrete random sample of returns at times t ∈ {1, . . . , T } and
let q α be as defined in C.18 with α ∈ [0, 1]. The Moors kurtosis is defined as:
K¯
M = (q 0.875 − q 0.625 ) + (q 0.375 − q 0.125 )
− 1.23 (5.25)
q 0.75 − q 0.25
Moors created this estimator based on the idea that the two terms, (q 0.875 − q 0.625 )
and (q 0.375 − q 0.125 ), are large respectively small if relatively small respectively high
probability mass is concentrated in the neighborhood of q 0.75 and q 0.25 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 different
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 distribution.
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
135 q 0.875 = −q 0.125 = −1.15, q 0.25 = −q 0.75 = −0.68, q 0.375 = −q 0.125 = −0.32
127

5 Properties of Commodity Returns
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 outcome
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 {r 1 , . . . , r T } 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), ¯σ 2 denote the sample variance as defined in Equation (5.21) and
¯σ = √¯σ 2 be the sample standard deviation as of Equation (5.3). Moreover, define
the transformation of the sample observations Z t = rt−¯r , count in F ¯σ
t the number of
sample observations that are smaller or equal to Z t and define the sample probability
as: L(Z t ) = Ft
T
normality as: N (Z t ) = ∫ Z t
−∞
. Finally, define the probability under the assumption of standard
√1
2π
e (− 1 2 x2 )dx . The Lilliefors Test Statistic is defined as:
L = max{|L(Z t ) − N (Z t )|, |L(Z t ) − N (Z t−1 )|} (5.26)
t
So L measures the absolute value of the biggest difference between the probability
associated to Z t when Z t is normally distributed, and the empirical probability of
Z t , e.g. the frequencies actually observed. Because the empirical distribution is
discrete, the term |L(Z t ) − N (Z t−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

5.2 Properties of the DJ-AIGCI Return Components
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 {r 1 , . . . , r T } be a discrete random sample of returns at times t ∈ {1, . . . , T }.
The Shapiro Test statistic tests the null hypothesis that the sample {r 1 , . . . , r T } comes
from a normal distribution. First, create the order statistic r (1) , . . . , r (T ) as in C.21.
m
Second, define (a 1 , . . . , a T ) = T V −1
with m = (m m T V −1 V −1 m 1, . . . , m T ) T denoting the
expectations of an order statistic of an i.i.d. sample from the standard normal distribution
and V is the covariance matrix of those order statistic. The Shapiro Test
statistic is defined as:
W = (∑ T
i=1 a ir (i) ) 2
( ∑ T
i=1 (r i − ¯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 standard
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(x i ) = m i
cov(x i , x j ) = v ij
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 σ 2 . 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 generalized
least squares estimation of regression coefficients 136 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.
136 See [Aitken 1935] or [Powell 2006].
137 See [Lloydes 1952].
129

5 Properties of Commodity Returns
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 .
(
T m 2
1
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 {r 1 , . . . , r T } be a discrete random sample of returns at times t ∈ {1, . . . , T } and
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
(
¯S 2 + ( ¯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 H 0
Shapiro Test 0.97 0.0% reject H 0
Jarque Bera Test 5853 0.0% reject H 0
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 distribution
as seen in Figure 5.13. The first step of viewing empirical data was the construction
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

5.2 Properties of the DJ-AIGCI Return Components
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 {r 1 , . . . , r T } be a discrete random sample of returns at times t ∈ {1, . . . , T } and
let h define the window width, also called smoothing parameter or bandwidth. The
empirical density function is defined as:
ˆf(r) = 1
T h
T∑
( ) r − rt
K
h
t=1
(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
−
e
r2 2
2π
(5.32)
because of its continuity and differentiability properties became excepted and therefore,
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

5 Properties of Commodity Returns
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 Kurtosis,
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 {r 1 , . . . , r T } be a discrete random sample of returns at times t ∈ {1, . . . , T }. The
time series r t is said to be strictly stationary if the joint distribution of (r t1 , . . . , r tk )
is identical to that of (r t1 +t, . . . , r tk +t) for all t, where k is an arbitrary positive integer
and (t 1 , . . . , t k ) is a collection of k positive integers.
In other words, strict stationarity requires that the joint distribution of (r t1 , . . . , r tk )
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 {r 1 , . . . , r T } be a discrete random sample of returns at times t ∈ {1, . . . , T }.
The time series r t is said to be weakly stationary if the mean of r t and the covariance
between r t and r t+k is time invariant for an arbitrary integer k.
132

5.2 Properties of the DJ-AIGCI Return Components
Weak stationarity exactly embodies what was said introductory. It implies that the
data fluctuate with constant variation around a constant level. To test for stationarity,
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:
r t = βr t−1 + ε t (5.33)
As in Definition 5.3 ε t is an error term of identical and independent distributed random
variables with mean zero and variance σε. 2 Furthermore, assume for simplicity
that r 0 = 0 and that ε t ’s come from a normal distribution. If β = 1 the time series
model of (5.33) reduces to:
r t = ε t + ε t−1 + . . . + ε 1 (5.34)
This describes a random walk with r t ∼ N (0, σεt), 2 i.e the r t ’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 {r 1 , . . . , r T } 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:
ˆβ =
∑ T
t=1 r t−1r t
∑ T
t=1 r2 t−1
(5.35)
Furthermore, the usual OLS standard error for the estimation coefficient is given
as:
ˆσ 2ˆβ =
=
∑
1 T
T −1 t=1 (r t − ˆβr t−1 ) 2
∑ T
t=1 r2 t−1
ˆσ
∑ T
t=1 r2 t−1
(5.36)
133

5 Properties of Commodity Returns
Then, the Dickey and Fuller test statistic under the null hypothesis of β = 1 is
defined as:
DF = ˆβ − 1
ˆσ ˆβ
(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 {r 1 , . . . , r T } be a discrete random sample of returns at times t ∈ {1, . . . , T }
d
and let W (t) denote a Wiener process as of Definition xy. The symbol −→ 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:
DF
d
−→
1
(W 2 (1)2 − 1)
( ∫ ) 1/2
(5.38)
1
W (r)dr 0
Proof: With the notion of Definition 5.14 and under the null hypothesis of β = 1,
it follows:
DF = ˆβ − 1
ˆσ ˆβ
H 0 :β=1
{}}{
= ˆβ − β
ˆσ ˆβ
With (5.33) and (5.35):
DF =
( P T
)
t=1 r t−1r t
P T −
t=1 r t−1
ˆσ ˆβ
( P T
)
t=1
P (rt−εt)
T
t=1 r t−1
Furthermore, taking (5.36) yields to:
DF =
( P T
)
t=1 r t−1ε t
P T
t=1 r2 t−1
(
) 1/2
ˆσ
P 2
T
t=1 r2 t−1
134

5.2 Properties of the DJ-AIGCI Return Components
Expanding with 1 = T : T
DF =
=
=
(
1
T
P T
t=1 r t−1ε t
1
T 2 P T
t=1 r2 t−1
(
ˆσ 2 T 2
P T
t=1 r2 t−1
)
) 1/2
(
1
∑ T
T t=1 r t−1ε t
∑
1 T
T 2 t=1 r2 t−1
(
1
T
∑ T
t=1 r t−1ε t
)
(
ˆσ 2
T 2 ∑ T
t=1 r2 t−1
)
⎛
)
⎜
∗ ⎝
d
−→
1
(
ˆσ 2 T
P 2
T
t=1 r2 t−1
⎞
⎟
) 1/2
⎠
1
(W 2 (1)2 − 1)
( ∫ ) 1/2
1
W (r)dr 0
The convergence follows with Proposition 9 in [Hamilton 1994] page 486. Its multilateral
proof is out of the scope of this thesis.
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 components
can be found in Table 5.9.
Statistic p-Value Action
Total Return -15.00 0.00% reject H 0
Spot Return -15.23 0.00% reject H 0
Roll Return -11.32 0.00% reject H 0
Pure Roll Return -4.01 0.01% reject H 0
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

5 Properties of Commodity Returns
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 autocorrelation
is significant or not. Therefore, we first define the so-called sample
autocorrelation function (ACF):
Definition 5.15 Sample Autocorrelation Function
Let {r 1 , . . . , r T } be a discrete random sample of returns at times t ∈ {1, . . . , T }.
The time series r t is weakly stationary. The k-lag sample autocorrelation is defined
as the correlation between r t and its past value r t−k :
ˆρ k =
∑ T
t=k+1 (r t − ¯r)(r t−k − ¯r)
∑ T
t=1 (r t − ¯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

5.2 Properties of the DJ-AIGCI Return Components
Figure 5.15: Autocorrelation and Partial Autocorrelation Function of DJ-AIGCI Return
Components
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 procedure
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

5 Properties of Commodity Returns
Definition 5.16 Partial Autocorrelation Function
Let {r 1 , . . . , r T } be a discrete random sample of returns at times t ∈ {1, . . . , T }.
The time series r t is weakly stationary. The k-lag sample autocorrelation is defined
as in Definition 5.15. The partial autocorrelation function is then defined as:
pρ ˆ kk = r k − ∑ k−1
i=1 pρ ˆ k−1,i ˆρ k−1
1 − ∑ k−1
i=1 pρ ˆ
(5.40)
k−1,i ˆρ i
In the right diagrams of Figure 5.15 we plotted the adequate PACFs for the three
DJ-AIGCI return components.
returns exhibit significant autocorrelation.
The 20% rolling effect is removed but still, roll
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
Let {r 1 , . . . , r T } be a discrete random sample of returns at times t ∈ {1, . . . , T }.
The time series r t is weakly stationary. The k-lag sample autocorrelation is defined
as in Definition 5.15. The Box and Pierce test statistic is then defined as:
Q(k) = T
k∑
ˆρ 2 i (5.41)
i=1
Under the assumption that {r 1 , . . . , r T } 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 ˆρ i Statistic p Value ˆρ i Statistic p Value ˆρ i 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

5.2 Properties of the DJ-AIGCI Return Components
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 degree
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 targets
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 comparison
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 returns
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

6.1 Mean Variance Spanning
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, commodities
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 products
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 r i that
actually earns a risk premium above cash c that cannot be explained by other
already existing asset classes r j 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 r i denote the return of a representative of the suspected asset class, let c denote
the return of cash, e.g. Treasury Bills, and let r j 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:
n∑
r i − c = α + β j (r j − c) + ε (6.1)
j=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
138 For details about solving such regression models see Definition 5.3 and Theorem 5.1.
141

6 Asset Allocation with Commodity Derivatives
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 0.19453
β Stocks -0.00865 -0.68 0.49585
β Bonds -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 0.044
β Stocks 0.049133 1.674 0.0944
β Bonds 0.040089 0.410 0.6819
Table 6.2: Mean Variance Spanning Coefficients (2002-2006)
Over the smaller period we identified a small 139
premium.
but statistically significant risk
139 But recall, the analysis is in daily scale.
142

6.1 Mean Variance Spanning
Furthermore, this analysis show the time dependent character of commodity investment.
Because commodities don’t provide future cash flows and are not valued by
discounted cash flow methods their performance is highly dependent to scarcity embodied
in the convenience yield or risk premium on inventories. Moreover, we have
seen that commodity markets have been in contango more often than in backwardation
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 conclusion
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

6 Asset Allocation with Commodity Derivatives
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 collateralized
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
140 The calculation followed Definition C.3.
144

6.2 Dependence to Stocks, Bonds and Inflation
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 consists
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 commodities
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 Section
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 portfolio.
Theorem 5.2 uncovered the main driver of diversification: the average covariance
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

6 Asset Allocation with Commodity Derivatives
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* 0.13
S&P 500 -0.00 1.00 -0.03* -0.09
JPM Bond USA -0.04* 0.04* 1.00 -0.07
Inflation 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 assets.
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 Pearson
correlation coefficient to the above described average returns for different time
periods. Again, they could show that the correlation increases over time. Moreover,
they divided inflation in its expected and unexpected part. For it, they used a
141 Inflation is measured in monthly scale. Linear interpolating to daily values would destroy the
actual dependence structure. Therefore, the correlation coefficients of the three asset classes to
inflation are calculated with monthly data.
146

6.2 Dependence to Stocks, Bonds and Inflation
model suggested by [Fama Schwert 1977] and [Schwert 1981]. The short term Treasury
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, commodity
returns exhibit the adverse pattern: they are getting more nervous in rising
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 interruptions.
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 portfolio.
147

6 Asset Allocation with Commodity Derivatives
6.3 Portfolio Optimization
The last two sections introduced the historical performance and return characteristics
of commodities and their dependence structure to traditional asset class’ returns,
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 generate
in a traditional stock and bond portfolio. The basic tool for calculating asset
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 x i , 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:
⎧
⎪⎨ µ T x −→ max
P µ = x
⎪⎩
T Cx = ¯σ 2
(6.2)
1 T x = 1 with 1 = (1, . . . , 1) T
Alternatively, the problem can be re-written to construct a portfolio that has minimum
risk, i.e. variance, under the constrain that a predefined return is still generated:
⎧
⎪⎨ x T Cx −→ min
P σ 2 = µ
⎪⎩
T x = ¯µ
(6.3)
1 T x = 1 with 1 = (1, . . . , 1) T
The problem’s solution requires three input factor: means, standard deviations respectively
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 P σ 2 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. Moreover,
define µ as is Definition C.10 and denote:
148

6.3 Portfolio Optimization
a = 1 T C −1 µ, b = µ T C −1 µ, c = 1 T C −1 1, d = bc − a 2
The optimal solution of P σ 2
is given as:
˜x = 1 d
(
(c¯µ − a)C −1 µ + (b − a¯µ)C −1 1 ) (6.4)
with
σ 2 (¯µ) = ˜x T C ˜x = c¯µ2 − 2a¯µ + b
d
The Minimum Variance Portfolio denoted with x MV P is given as:
(6.5)
x MV P = 1 c C−1 1 (6.6)
It is located in the risk return space at:
(µ MV P , σ MV P ) = ( a c , √
1
c ) (6.7)
The efficient frontier is given as:
¯µ = µ MV P ±
√
d
c (σ2 − σ 2 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 = µ T C −1 µ > 0 (6.9)
and furthermore:
c = 1 T C −1 1 > 0 (6.10)
With the scalar product < x, y >≡ x T C −1 y, the Chauchy-Schwarz inequation and
149

6 Asset Allocation with Commodity Derivatives
x ≡ 1, y ≡ µ follows:
< x, y > 2 = (1 T C −1 µ) 2 = a 2
< < x, x >< y, y >= (1 T C −1 1)(µ T C −1 µ) = bc
⇔
d = bc − a 2 > 0 (6.11)
Furthermore, the Lagrange function is given as:
L(x, u) = 1 2 xT Cx + u 1 (¯µ − µ T x) + u 2 (1 − 1 T x) (6.12)
˜x is optimal for P if there exists an u = (u 1 , u 2 ) T
Kuhn-Tucker conditions:
∈ R that satisfies the so-called
∂L
∂x i
(˜x, u) =
n∑
c i,j ˜x j − u 1 µ i − u 2 = 0 (6.13)
j=1
∂L
(˜x, u)
∂u 1
= ¯µ − µ T ˜x = 0 (6.14)
∂L
(˜x, u)
∂u 2
= 1 − 1 T x = 0 (6.15)
(6.13) ⇔ C ˜x = u 1 µ + u 2 1
⇔ ˜x = u 1 C −1 µ + u 2 C −1 1
(6.16)
(6.15)&(6.16)
{}}{
⇒ 1 T ˜x = u 1 1 T C −1 µ
} {{ }
≡a
+ u 2 1 T C −1 1 } {{ }
≡c
= au 1 + cu 2
(6.15)
{}}{
= 1
(6.17)
(6.14)&(6.16)
{}}{
⇒ µ T ˜x = u 1 µ T C −1 µ
} {{ }
≡b
+ u 2 µ T C −1 1
} {{ }
≡a
= bu 1 + au 2
(6.14)
{}}{
= ¯µ
(6.18)
(6.17)&(6.18)
{}}{
⇔
(
a
c
) (
}
b a
{{ }
≡A
)
u 1
u 2
} {{ }
≡u
=
(
1
¯µ
)
(6.19)
150

6.3 Portfolio Optimization
Calculate the inverse of A as:
A −1 =
1
det(A)
=
1
bc − a 2
} {{ }
≡−d
=
1
d
(
(
(
a
−b
a
−b
−a
b
)
−c
a
)
−c
a
)
c
−a
(6.20)
With (6.19) and (6.20) follows:
( )
u = A −1 1
¯µ
(
= 1 c¯µ − a
d b − a¯µ
)
(6.21)
Putting (6.21) into (6.16) yields to Equation (6.4), the optimal solution ˜x of P σ 2:
˜x
(6.16)
{}}{
= u 1 C −1 µ + u 2 C −1 1
(6.21)
{}}{
= 1 d ((c¯µ − a)C−1 µ + (b − a¯µ)C −1 1)
With it, Equation (6.5) follows with:
σ 2 (¯µ) = ˜x T C ˜x
(6.13)
{}}{
= u 1 µ T ˜x + u 2 1 T ˜x
(6.17)&(6.18)
{}}{
= u 1¯µ + u 2
(6.21)
{}}{
=
=
1
((c¯µ − a)¯µ + (b − a¯µ))
d
c¯µ 2 − 2a¯µ + b
d
Furthermore, the minimum of Equation (6.5) yields to the Minimum Variance Portfolio
denoted with x MV P , i.e. Equation (6.6), and its points in the risk return space,
151

6 Asset Allocation with Commodity Derivatives
i.e. Equation (6.7):
∂σ 2 (¯µ)
∂ ¯µ
= 1 (2c¯µ − 2a) ≡ 0
d
⇒ µ MV P = a c
(6.22)
To check for a minimum, the second partial derivative is positive:
∂ 2 σ 2 (¯µ)
∂ 2¯µ
=
2c
d
(6.10)&(6.11)
{}}{
> 0
Putting this into Equation (6.5) yields to:
σ MV P = √ σ 2 (µ MV P )
√
(6.5)
{}}{ cµ
2
=
MV P
− 2aµ MV P + b
(6.22)
{}}{
=
√
d
c ( )
a 2 (
c − 2a a
)
c + b
d
=
√
1
c
Together we have shown Equation (6.7), the location of the Minimum Variance
Portfolio in the risk return space:
(µ MV P , σ MV P ) = ( a c , √
1
c )
To find the Minimum Variance Portfolio x MV P respectively the weights of the Minimum
Variance Portfolio coded in x MV P , we put Equation (6.7) into Equation (6.4):
(6.4)
{}}{
x MV P = 1 (
(cµMV P − a)C −1 µ + (b − aµ MV P )C −1 1 )
d
(6.22)
{}}{
= 1 ( ( a
)
( a
) )
(c − a)C −1 µ + (b − a )C −1 1
d c c
= 1 c C−1 1
152

6.3 Portfolio Optimization
Finally, the efficient frontier is calculated from Equation (6.5), whereby we have to
define σ ≡ σ(¯µ):
(6.5)
σ 2 {}}{
=
c¯µ 2 − 2a¯µ + b
d
⇔
d
c σ2 = ¯µ 2 − 2 a c ¯µ + b c
(
= ¯µ − a ) 2 a 2
−
c
c 2 + b c } {{ }
bc−a 2
c 2
(6.22)&(6.11)
{}}{
= (¯µ − µ MV P ) 2 + d 1
c c
= (¯µ − µ MV P ) 2 + d c σ2 MV P
⇔
(¯µ − µ MV P ) 2 =
⇔
d
c (σ2 − σ 2 MV P )
¯µ = µ MV P ±
√
d
c (σ2 − σ 2 MV P )
Following Theorem 6.1 we calculated the efficient frontier with and without commodities.
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 represents
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 standard
deviation. The Minimum Variance Portfolio decreased in risk around 0.8%
annualized standard deviation. This is shown in Figure 6.3: the Minimum Variance
Portfolio (MV P ) without commodities has an annualized standard deviation
of around 4.5% whereby the Minimum Variance Portfolio (MV P c ) with commodities
has an annualized standard deviation of around 3.8%. Implicating, including
153

6 Asset Allocation with Commodity Derivatives
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 portfolio
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. Consequently,
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 allocation
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 deviation
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

6.3 Portfolio Optimization
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] estimated
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 expected
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
142 See [Best Grauer 1991] and [Michaud 1998].
155

6 Asset Allocation with Commodity Derivatives
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
P C2
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 second
possible case, choosing portfolio P C1 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 attractive
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 consumption
goods. Therefore, the elementary financial products to trade commodities cannot
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 concept.
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 diversification
effects offered by the heterogeneous commodity market. If commodities shall
actually be allocated in a portfolio, total returns have to be considered. Their decomposition
has shown that they consist of spot, roll and interest rate returns. Former
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, consumption
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. Therefore,
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 investment
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 shutdowns
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 significant
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
143 See Energy Information Administration: stand 2002.
144 For 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 of
Definition C.2. For the mathematical definition of Pearson correlation and the related statistical
test see Section 5.1.2.
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.
145 Data source: Bloomberg.
165

B Characteristics of Selected Commodities
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, various
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 knocking
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
146 See [The CRB Commodity Yearbook 2005].
147 87 (R+M)/2, 89 (R+M)/2 and 93 (R+M)/2
148 For 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 of
Definition C.2. For the mathematical definition of Pearson correlation and the related statistical
test see Section 5.1.2.
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
149 See US Energy Information Administration (www.eia.doe.gov)
150 See [UBS research 2005].
167

B Characteristics of Selected Commodities
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, Argentina
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 balance
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.
151 Data 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. Although
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 markets.
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
152 See [The CRB Commodity Yearbook 2005].
169

B Characteristics of Selected Commodities
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 airconditioning
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 Indonesia,
all accounting between 6% and 8% of world production. Figure B.6 shows
153 See LME
154 See [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

B Characteristics of Selected Commodities
television and telecommunication. 155 Because production did not increase with demand,
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 countries.
Caused by low lead prices during the 1990s companies went out of business.
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 major
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.
155 See [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 properties
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 problems
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
156 See [UBS research 2005].
173

B Characteristics of Selected Commodities
industry worldwide is operating at close to capacity. Furthermore, world production
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 inventories
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.
157 See [UBS research 2005].
158 See 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, consumption,
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

B Characteristics of Selected Commodities
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.
159 See 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. Robusta
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

B Characteristics of Selected Commodities
dollar in 2005, the tiny fraction of 5.5 billion US dollar went to the producing countries
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, consumption
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 production
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 latter
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 correlates
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
160 See [The CRB Commodity Yearbook 2005].
161 For the analysis we took monthly cash data since 1970 of the
[The CRB Commodity Yearbook 2005] completed with Bloomberg data for 2005 and
179

B Characteristics of Selected Commodities
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 countries
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 of
Pearson correlation and the related statistical test see Section 5.1.2.
162 Because demand for oil and meal is driven by different factors their price series among each
other are less correlated than their price series to soybean prices are. The significant correlation
coefficient is 0.36.
163 See [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.
164 See [The CRB Commodity Yearbook 2005].
165 Surprisingly, Hong Kong is the world’s biggest pork user. People there consume around
65 kilograms per year. See [USDA Livestock 2006]
181

B Characteristics of Selected Commodities
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
166 See [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 P t 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:
R t = P t − P t−1
P t−1
Definition C.2 Continuous Compounding Return
The one period continuous compounding or log return is defined as:
( )
Pt
r t = ln
P t−1
(C.1)
(C.2)
The following equation links both return definitions:
R t = e rt − 1 or r t = ln(R t + 1) (C.3)
In general, the difference between continuous and simple returns is very small, especially
for short time scales like ticks, days or months. This can be seen with the
Taylor series:
R t
(C.3)
{}}{
= e rt − 1 =
=
∞∑
i=1
∞∑
i=0
r i t
i! = r t +
r i t
i! − 1
∞∑ rt
i
i!
i=2
(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

C Mathematical Preliminaries
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:
r 0→t = ln
( ) P (t)
= ln
P (0)
( )
P (t) ∗ P (s)
= ln
P (0) ∗ P (s)
( ) P (s)
+ ln
P (0)
( ) P (t)
= r 0→s + r s→t
P (s)
(C.5)
For continuous compounding returns the recalculation to prices is given as:
Definition C.3 Price Series
Let r t denote the log return of an asset at time t. The price of the asset is then
defined as: P (t) = P (t − 1) ∗ e rt .
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 ⇒ A c ∈ F
3. A 1 , A 2 , A 3 . . . ∈ F ⇒ ⋃ n
i=1 A i ∈ 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(R k ),with R denoting the real numbers, that is the smallest sigma-Algebra containing
all open sets in R k .
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 A i ∈ F, i ∈ N with A i
⋂
Aj = ∅ for all i, j ∈ N with i ≠ j is:
( ∞
) (
∑
∞
)
⋃
Q A i ≡ Q A i =
i=1
i=1
167 Compare [Dorfleitner 2002].
∞∑
Q(A i )
i=1
(C.6)
184

C.1 Statistical Basics
The triple (Ω, F, Q) is called probability space.
A random variable is a mathematical function that maps outcomes of random experiments
to numbers.
Definition C.6 Random Variable
Let (Ω, F, Q) be a probability space and let B denote the Borel-σ-Algebra.
function R : Ω ↦→ R with R −1 (B) ∈ F for all B ∈ B is called random variable.
The
A reel number R(ω) = r, ω ∈ Ω is called the realization of R.
Definition C.7 Distribution
The probability mass Q R on (R, B 1 ) defined by Q R (B) ≡ Q(R −1 (B)) is called
distribution of R.
A probability density function can be seen as a ”smoothed out” version of a histogram,
e.g. Figure 5.12: if one empirically measures values of a continuous random
variable repeatedly and produces a histogram depicting relative frequencies of output
ranges, then this histogram will resemble the random variable’s probability density
(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
−∞
∑
r Q(R = r t≤r t) = ∑ r q(r t≤r t) : if R is discrete 168 (C.7)
185

C Mathematical Preliminaries
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 r tq(r t ) : 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:
σ 2 = var(R) = E[(R − E[R]) 2 ] = E[R 2 ] − E[R] 2
(C.9)
A more understandable measure is the square root of the variance, called the standard
deviation: σ = √ σ 2 . 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 σ 2 , i.e.
R ∼ N(µ, σ 2 ),
186

C.1 Statistical Basics
if its density function is defined as:
f(x) =
1
x √ (x−µ)2
e− 2σ 2
2πσ2 A random variable is called standard normally distributed if it is normally distributed
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 σ 2 , i.e.
ln R ∼ N(µ, σ 2 ),
if its density function is defined as:
f(x) =
1
x √ (ln r−µ)2
e− 2σ 2 , r > 0.
2πσ2 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 realizations
of a two commodity indices over time. The covariance is defined as:
cov(R, L) = E[(R − µ R )(L − µ L )]
(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 realizations
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 − µ R)(L − µ L )]
√
var(R)
√
var(L)
(C.11)
The autocorrelation describes a linear dependence between return realization at
different points in time.
187

C Mathematical Preliminaries
Definition C.16 Autocorrelation
Consider a weakly stationary return series r t . The correlation coefficient between
r t and r t−l is called the lag-l autocorrelation of r t and defined as:
ρ l =
cov(r t , r t−l )
√
var(rt ) √ var(r t−l ) = cov(r t, r t−l )
var(r t )
(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 r 1 , . . . , r T denote a random sample. The order statistic is defined
as r (1) ≤ r (2) ≤ . . . ≤ r (T ) . Then the median is defined as:
¯r M =
{
r (
T +1
2 ) : if T is odd
r ( T
2
) +r (1+ T 2 )
2
: if T is even.
(C.13)
Definition C.18 Quantile
To be precise: Let r 1 , . . . , r T denote a random sample and R is the random variable
embodying all possible realizations r t . Let quantile q α is defined as the value that α%
of the possible realizations are smaller than q α , i.e.
Q(R ≤ q α ) = 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:
Definition C.20 Kurtosis
S = E( R − µ ) 3 (C.15)
σ
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:
Definition C.21 Order Statistic
K = E( R − µ ) 4 − 3 (C.16)
σ
Let {r 1 , . . . , r T } be a sample of T independent observations of the random variable
188

C.1 Statistical Basics
R. Arrange the r t 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.
follows:
Proof:
Then the probability density of the k’th statistic can be found as
f R(k) (r) =
T !
(k − 1)!(T − k)! F (r)k−1 (1 − F (r)) T −k f(r) (C.17)
f R(k) (r) = d
dx F R (k)
(r) = d
dx Q(R (k) ≤ r)
= d Q(at leats k of the T R’s are ≤ r)
dx
= d Q(≥ ksuccesses in T trials)
dx
T∑
= d
dx
= d
dx
=
T∑
i=k
i=k
T∑
i=k
( T
j
)
Q(R 1 ≤ r) j (1 − Q(R 1 ≤ r)) T −j
( T
j
)
F (r) j (1 − F (r)) T −j
( T
j
)
(jF (r) j−1 f(r)(1 − F (r)) T −j
+ F (r) j (T − j)(1 − F (r)) T −j−1 (−f(r)))
T∑
( ) T − 1
= (T F (r) j−1 (1 − F (r)) T −j
j − 1
i=k
( ) T − 1
− T F (r) j (1 − F (r)) T −j−1 )f(r)
j
T∑
−1 ( ) T − 1
= T f(r)(
F (r) j (1 − F (r))
j
i=k−1
(T −1)−j
T∑
( ) T − 1
−
F (r) j (1 − F (r)) (T −1)−j )
j
i=k
( ) T − 1
= T f(r)( F (r) k−1 (1 − F (r))
k − 1
( ) T − 1
− F (r) T (T −1)−T
(1 − F (r)) )
T
} {{ }
0
T !
=
(k − 1)!(T − k)! F (r)k−1 (1 − F (r)) T −k f(r)
(T −1)−(k−1)
✷
189

C Mathematical Preliminaries
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 ↦→ R k is called (F − B(R k )-) 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 ↦→ R k , 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(R k ), 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 ⊂ F
be a sub-sigma-Algebra of F. The conditional expectation of X given G is implicitly
defined to be the G-measurable function E Q [X|G] with
∫ ∫
XdQ = E Q [X|G]dQ ∀A ∈ G Q − a.s.
A
A
The function
Q : F ↦→ [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 (F t ) t≥0 with F t ⊂ F
and F s ⊂ F t for all 0 ≤ s < t < ∞. We call (Ω, F, Q, F) a filtered probability space,
and require that
1. F 0 contains all subsets of the (Q-) null sets of F,
2. F is right-continuous, i.e. F t = F t+ := ∩ s>t F s
190

C.2 Probability Theory
F t represents the information available at time t, and F = (F t ) 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 X t = X(t) is (F t -) measurable for all
t ≥ 0
2. X is measurable of the mapping X : [0, ∞]×Ω → R k , k ∈ N is B([0, ∞))⊗F-
B(R k )−measurable
3. X is progressively measurable if the mapping X : [0, t] × Ω → R k , k ∈ N
is B([0, t]) ⊗ F t -B(R k )−measurable for each t ≥ 0.
Note that we either write X t 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, ∞) ↦→ R k , k ∈ N,
B([0, ∞)) ⊗ B(R k ) is measurable for each fixed ωinΩ. For each fixed ωinΩ we call
X(ω) = (X t (ω)) t≥0 = (X(t, ω)) t≥0 a path or realization of the stochastic process.
Definition C.27 L 2 [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 L 2 [0, T]-process, if X is progressively measurable
and
∫ T
‖X‖ 2 T := E Q[ X 2 (t)dt] < ∞.
0
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 = (W t ) t≥0 = (W (t)) t≥0 is called a Q- Brownian motion or
Q- Wiener process if
1. W (0) = 0 Q-a.s.
169 For further details to the naming conventions see [Zagst 2002].
191

C Mathematical Preliminaries
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 = (W 1 , . . . , W m ) = (W 1 (t), . . . , W m (t)) t≥0 a m-dimensional
Wiener process, m ∈ N, if its components W j , 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 E Q [|X(t)|] < ∞, ∀t ≥ 0,
is called:
martingale relative to (Q, F), if E Q [X(t)|F s ] = X(s) Q-a.s. ∀0 ≤ s ≤ t < ∞
super-martingale relative to (Q, F), if E Q [X(t)|F s ] ≤ X(s) Q-a.s. ∀0 ≤ s ≤
t < ∞
sub-martingale relative to (Q, F), if E Q [X(t)|F s ] ≥ 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
Itô process.
Definition C.30 Itô process
Let W = (W 1 , . . . , W m ), m ∈ N, be a m-dimensional Wiener process. A stochastic
process X = (X(t)) t≥0 is called an Itô process if ∀t ≥ 0 we have:
X(t) = X 0 +
= X 0 +
∫ t
0
∫ t
µ(s)ds +
µ(s)ds +
∫ t
0
j=1 0
0
m∑
σ(s)dW (s)
∫ t
σ j (s)dW j (s),
(C.18)
where X 0 is (F 0 -)measurable and µ = (µ(t)) t≥0 and σ(t) = (σ 1 (t), . . . , σ m (t)) t≥0 are
(m-dimensional) progressively measurable stochastic processes with
and
∀t ≥ 0, j = 1, . . . , m.
∫ t
0
∫ t
|µ(s)| ds < ∞ Q − a.s. , (C.19)
0
σ 2 j (s)ds < ∞ Q − a.s. (C.20)
A n-dimensional Itô process is given by a vector X = (X 1 , . . . , X n ), n ∈ N, with
each X i being an Itô process, i = 1, . . . , n.
For convenience we write symbolically instead of (C.18)
m∑
dX(t) = µ(t)dt + σ(t)dW (t) = µ(t)dt + σ j (t)dW j (t);
and call this a stochastic differential equation (SDE).
Definition C.31 Quadratic Covariance Process
Let m ∈ N and W = (W 1 , . . . , W m ) be a m-dimensional Wiener process. Furthermore,
let (X 1 (t)) t≥0 and (X 2 (t)) t≥0 be two Itô processes with
m∑
dX i (t) = µ i (t)dt + σ i (t)dW (t) = µ i (t)dt + σ ij dW j (t), i = 1, 2. (C.21)
j=1
j=1
193

C Mathematical Preliminaries
Then we call the stochastic process < X 1 , X 2 >= (< X 1 (t), X 2 (t) >) t≥0 defined by
< X 1 , X 2 >:=
m∑
∫ t
j=1 0
σ 1j (s)σ 2j (s)ds
(C.22)
the quadratic covariance (process) of X 1 and X 2 . If X 1 = X 2 =: X we call the
stochastic process < X >:=< X, X > the quadratic variation (process) of X. 170
Theorem C.1 (Itô´s Lemma) Let W = (W 1 , . . . , W m ) be a m-dimensional Wiener
process, m ∈ N,and X = (X(t)) t≥0 be an Itô process with
m∑
dX(t) = µ(t)dt + σ(t)dW (t) = µ(t)dt + σ j (t)dW j (t)
Furthermore, let G : R × [0, ∞) ↦→ R be twice continuously differentiable in the first
variable, with derivatives denoted by G X and G XX , and once continuously differentiable
in the second, with derivative denoted by G t . Then we have ∀t ∈ [0, ∞)
j=1
dG(X(t), t) = G t (X(t), t)dt + G X (X(t), t)dX(t) + 1 2 G XX(X(t), t)d < X(t), X(t) >
=
(
G t (X(t), t) + G X (X(t), t)µ(t) + 1 2 G XX(X(t), t)
+ G X (X(t), t)
m∑
σ j (t)dW j (t)
j=1
m∑
j=1
)
σj 2 (t) dt
Whereby, the W j , j = 1, . . . , m, are assumed to be independent.
170 For 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 = d ˜Q
dQ .
Let γ = (γ(t)) t≥0 be a m-dimensional progressively measurable stochastic process,
m ∈ N, with
∫ t
0
γ 2 j (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 t
R
0 γ(s)′ dW (s)− 1 t
2 0 ||gamma(s)||ds
Note that the stochastic process X(γ) = (X(γ, t)) t≥0 = (X(γ(t), t)) t≥0 with
or
X(γ, t) :=
∫ t
0
γ(s) ′ dW (s) + 1 2
∫ t
0
||γ(s)|| 2 ds
dX(γ, t) := 1 2 ||γ(t)||2 dt + γ(t) ′ dW (t)
is ∀t ∈ [0, ∞) an Itô process with with µ(γ(t), t) = 1 2 ||γ(t)||2 = 1 2
∑ m
j=1 γ2 j (t) and
σ(γ(t), t) = γ(t) ′ . Thus, using the transformation G : R×[0, ∞) ↦→ R with G(x, t) =
e −x and Itô’ lemma as of C.1 with G(X(γ, t), t) = e −X(γ,t) = L(γ, t) we get: 171
171 For a detailed calculation see [Zagst 2002].
dL(γ, t) = −L(γ, t)γ(t) ′ dW (t)
195

C Mathematical Preliminaries
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
E Q
[e 1 2
R t
0 ||γ(s)||2 ds]
< ∞
For each T ≥ 0 we define the measure ˜Q = Q L(γ,T ) on the measure space (Ω, F T ) by
∫
˜Q(A) := E Q [1 A ∗ L(γ, T )] = L(γ, T )dQ
A
∀A ∈ F T
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 ) = d ˜Q on (Ω, F dQ T ). The following Girsanov
theorem shows how the adequate (tildeQ -) Wiener process ˜W
( )
= ˜W (t) is
constructed, starting with a (Q -) Wiener process W = (W (t)) t∈[0,T ]
.
Theorem C.3 Girsanov Theorem
t∈[0,T ]
Let W = (W 1 , . . . , W m ) = (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 = ( ˜W 1 , . . . , ˜W
(
m ) = ˜W (t)1 , . . . , ˜W
)
(t) m
be defined by
d˜W (t) := γ(t)dt + dW (t), t ∈ [0, 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, F T ).
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, ∞) → R
and σ : R×[0, ∞) → R m be measurable functions (with respect to the corresponding
Borel sigma-Algebras) with
∫ t
0
|µ i (s)|ds < ∞ and
∫ t
0
σ 2 ij(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 Itô process, such that ∀t ≥ 0
X(t) = x +
∫ t
µ(X(s), s)ds +
∫ t
0
0
σ(X(s), s)dW (s)
Q − a.s.,
X(0) = x, x ∈ R n ,
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 ∈ R n 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 ≤ K 2 (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
E Q [‖X(t)‖ 2 ] ≤ C(1 + ‖x‖ 2 )e Ct ∀t ∈ [0, T ].
To move on to the so-called Feynman-Kac Representation the following definition
is very helpful.
197

C Mathematical Preliminaries
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) = υ t (x, t) + µ(x, t)υ x (x, t) + 1 2 σ2 (x, t)υ xx (x, t)
with υ : R × [0, ∞) → R twice continuously differentiable in x, once continuously
differentiable in t and
m∑
σ 2 (x, t) = σj 2 (x, t)
j=1
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) = E x,t
Q
[e − R ]
T
t r(X(s),s)ds D(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: info@risklab.de
% 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
% Author’s Information
%———————————————————————
% risklab germany GmbH
% Nypmhenburger Strasse 112 - 116
% D-80636 Muenchen
% Germany
% Internet: www.risklab.de
% email: info@risklab.de
% Implementation Date: 2006 - 09 - 27
% Author: Dr. Wolfgang Mader, Maria Heiden
%———————————————————————
% 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)
% Author’s Information
%———————————————————————
% risklab germany GmbH
% Nypmhenburger Strasse 112 - 116
% D-80636 Muenchen
% Germany
% Internet: www.risklab.de
% email: info@risklab.de
% Implementation Date: 2006 - 09 - 27
% Author: Dr. Wolfgang Mader, Maria Heiden
%———————————————————————
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)
% Author’s Information
%———————————————————————
% risklab germany GmbH
% Nypmhenburger Strasse 112 - 116
% D-80636 Muenchen
% Germany
% Internet: www.risklab.de
% email: info@risklab.de
% Implementation Date: 2006 - 09 - 27
% Author: Dr. Wolfgang Mader, Maria Heiden
%———————————————————————
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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