Has the Role of Commodity Investing Changed Over Time? A longitudinal study on the attractiveness of commodity investing based on trends in the Sharpe ratio and time-varying correlations between commodity futures indices and alternative asset classes.

Master Thesis Faculty of Economics and Business Administration Department of Finance T.N. Boot, MSc.

Author: Administration number: Study program: Graduation date: Supervisor: Second reader:

T.N. Boot, MSc. 626574 Finance 11-22-2012 Dr. J.C. Rodriguez Dr. -

Has the Role of Commodity Investing Changed Over Time? A longitudinal study on the attractiveness of commodity investing based on trends in the Sharpe ratio and time-varying correlations between commodity futures indices and alternative asset classes.

Master Thesis Faculty of Economics and Business Administration Department of Finance

Author: Administration number: Study program: Graduation date: Supervisor: Second reader:

T.N. Boot, MSc. 626574 Finance 11-22-2012 Dr. J.C. Rodriguez Dr. O.G. Spalt

I. Abstract In this study the diversification gains that commodity investing may entail are examined, where analyses are conducted from multiple perspectives. Several investment portfolios are evaluated on their reward-to-volatility ratios, mean-variance frontiers are discussed and regressions for trend testing are run. Next to that, time-varying correlations between commodities and other asset classes are calculated. The returns on several commodity futures indices are used, where the other asset classes of interest are stocks, corporate and government bonds and T-Bills. The techniques used for modeling the correlations are: unconditional correlation, rolling historical correlation, exponentially weighted moving average correlation and multivariate GARCH dynamical conditional correlation. Where the former two techniques provide basic insights, the latter two elaborate on a more advanced financial econometric level. The considered time frame for this research is the past 25 years, ranging from May 1st 1987 to May 1st 2012. Based on analyses conducted over more than 100,000 data points, this study concludes that diversification gains of commodity investing are still present, although they have decreased over time. The results from a portfolio exercise show that augmenting an investment portfolio with commodity futures has become less attractive, where the correlation modeling reports increased inter-commodity and commodity-equity correlations. Increased correlations between several commodity indices and inflation indicate that hedging potential for investors persists. Keywords: Commodities, Futures, Equities, Volatility, Markowitz Portfolio Theory, Sharpe Ratio, Mean-Variance Spanning, Efficient Frontier, Dynamical Conditional Correlation, Rolling Historical Correlation, Exponentially Weighted Moving Average Correlation, Multivariate GARCH.

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II. Preface and acknowledgements Before you lies the final report that is the result of the Master of Science program in Finance of Tilburg University. The process of writing this thesis significantly increased my knowledge of investing in commodity futures and the benefits it may entail. Where most finance courses focus on mainstream investment securities, this thesis especially focuses on one of the alternatives. This is what I found most interesting about the topic, particularly because of the fact that investing in commodity futures has gradually gained more attention. Next to that, I was personally challenged in applying more advanced mathematical techniques for effectively modeling the time-varying correlations. Understanding both the underlying financial econometric concepts and the extensive mathematics is what I found challenging and interesting at the same time. Finally, the programming required for this thesis’ analyses further extended my knowledge of statistical software packages as STATA and SPSS. This thesis would not have been possible without the valuable contributions of many people. First and foremost, I would like to express my appreciation towards my supervisor dr. J.C. Rodriguez. His extensive knowledge of mathematical modeling and clear scope provided useful insights throughout the process. Especially his suggestions regarding the multivariate GARCH modeling proved particularly valuable. Furthermore, I thank my second reader dr. O.G. Spalt for his time and role in the exam committee. Finally, I acknowledge the contributions of the people not directly involved in this thesis. I want to thank my loved ones for providing me the support and confidence for successfully finishing my Master Thesis at Tilburg University. Tom Boot, MSc. Tilburg, November 2012

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III. Management summary At 1:45AM on September 15, 2008 Lehman Brothers Holdings Inc. defaulted. This bankruptcy, the largest in US history, brought global financial markets into panic and sent credit markets worldwide into despair. The consequences of this ‘Lehman moment’ are broadly discussed. Traditionally, investors relied on commodity futures for diversifying investment portfolios because of their low (or even negative) correlation with equities. The Lehman moment however raised some interesting discussion. Following the studies of Jensen and Mercer (2001) and Büyükşahin et al. (2008) this study investigates whether or not the traditional role of commodities as portfolio diversifier or inflation hedge has changed over time. Data on several asset classes is provided by Thomson Reuters Datastream and covers a time period of 25 years. Markowitz portfolio optimization shows that diversification gains of commodities still exist, because they continue to improve portfolio performance, as is shown in figure III.1. However, the shift of the frontier shows that these gains have decreased as of the Lehman moment. Correlation modeling, using four different mathematical techniques, shows that equity-commodity correlations show no clear trend up to the Lehman moment. However, after that moment a structural increase in correlations has been identified, where no clear signs of mean-reversion are observed (figure III.2). Correlations between commodities and bonds or T-Bills show no clear trends. Mean-variance frontier of risky assets

Dynamical conditional correlations

With and without using the GSCI sub-indices

GSCI and S&P 500

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Figure III.1: Mean-variance frontier of risky assets

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Pre-Lehman efficient frontier without commodities Full sample efficient frontier without commodities Tangency portfolio

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σ Pre-Lehman efficient frontier with commodities Full sample efficient frontier with commodities Optimal complete portfolio Global minimum variance portfolio

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Figure III.2: Dynamical conditional correlations of the GSCI and the S&P 500

Commodity index investing has become increasingly popular over the last few years. The consecutive increased ease of investing was fueled by the publication of several influential academic papers that reported negative correlations between commodities and equities; hereby raising investor’s attention. When an increasing number of (non-) financial investors decide to broaden their investment portfolios (containing equities) with commodities, cross-market linkages increase and volatility spill-overs can be identified. When these investors ‘en masse’ decide to liquidate risky positions in case of a crash in the global market (i.e. equities) and flee into safe havens, volatility spill-overs into the satellite market (i.e. commodities) significantly increase correlations. Due to the increasing financialization of the commodity market, equity-commodity correlations are not likely to become as low as they have been traditionally, resulting in consistently lower diversification gains of this asset class. However, the traditional role of commodities as an inflation hedge does seem to persist.

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“As more money has chased (…) risky assets, correlations have risen. By the same logic, at moments when investors become risk-averse and want to cut their positions, these asset classes tend to fall together. The effect can be particularly dramatic if the asset classes are small – as in commodities. (…) This marching-in-step has been described (…) as a ‘market of one’.” The Economist, March 8, 2007.

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IV. List of abbreviations ARCH CCC CDS CPI DCC DJ UBS DJIA EAFE EWMA GARCH GMV GSCI MGARCH MLM MSCI MPT MVS OLS OTC S&P SR TR JCRB USCL VIX

Autoregressive Conditional Heteroscedasticity Constant Conditional Correlation Credit Default Swap Consumer Price Index Dynamical Conditional Correlation Dow Jones UBS commodity index Dow Jones Industrial Average index Europe, Australasia and the Far East Exponentially Weighted Moving Average Generalized Autoregressive Conditional Heteroscedasticity Global Minimum Variance Goldman Sachs Commodity Index Multivariate Generalized Autoregressive Conditional Heteroscedasticity Mount Lucas Management index Morgan Stanley Capital International Markowitz Portfolio Theory Mean-Variance Spanning Ordinary Least Squares Over The Counter Standard and Poor’s Sharpe Ratio Thomson Reuters / Jefferies Commodity Research Bureau index US Corporate Long index Volatility Index of the S&P 500

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V. Table of contents I.

Abstract ........................................................................................................................................... i

II. Preface and acknowledgements ................................................................................................... ii III. Management summary ................................................................................................................ iii IV. List of abbreviations ..................................................................................................................... v V. Table of contents .......................................................................................................................... vi 1.

Introduction ................................................................................................................................... 1 1.1 Motivation for this thesis ........................................................................................................ 1 1.2 Contribution of this thesis....................................................................................................... 2 1.3 Thesis outline .......................................................................................................................... 2

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Literature review and empirical background............................................................................. 3 2.1 Introduction to commodity futures ......................................................................................... 3 2.1.1 The futures contract ........................................................................................................ 3 2.1.2 The mechanics of futures pricing ................................................................................... 5 2.1.3 Why trade in commodities? ............................................................................................ 5 2.1.4 Commodity index investing............................................................................................ 6 2.2 Pre-Lehman literature review ................................................................................................. 7 2.2.1 Historical (portfolio) performance of commodities ........................................................ 7 2.2.2 (Inter-) Commodity correlations ..................................................................................... 9 2.2.3 Commodities as an inflation hedge .............................................................................. 10 2.3 Post-Lehman literature review .............................................................................................. 11 2.3.1 Historical (portfolio) performance of commodities ...................................................... 11 2.3.2 (Inter-) Commodity correlations ................................................................................... 12 2.3.3 Commodities as an inflation hedge .............................................................................. 15

3.

Research questions ...................................................................................................................... 16 3.1 Main research question ......................................................................................................... 16 3.2 Sub-questions........................................................................................................................ 16

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Data............................................................................................................................................... 18 4.1 Commodity indices ............................................................................................................... 18 4.2 Bond indices ......................................................................................................................... 19 4.3 Equity indices ....................................................................................................................... 19 4.4 Inflation indices .................................................................................................................... 20 4.5 T-Bill index........................................................................................................................... 20

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Methodological framework ........................................................................................................ 21 5.1 Volatility ............................................................................................................................... 21 5.2 Sharpe ratio ........................................................................................................................... 22 5.3 Diversification ...................................................................................................................... 23 5.4 Markowitz portfolio theory................................................................................................... 24 5.5 Mean-variance spanning ....................................................................................................... 25 5.6 Unconditional correlation ..................................................................................................... 26 5.7 Conditional correlation ......................................................................................................... 26 5.8 Rolling historical correlation ................................................................................................ 27 vi

5.9 Exponentially weighted moving average correlation ........................................................... 27 5.10 Autoregressive conditional heteroskedasticity ................................................................... 28 5.10.1 ARCH models ............................................................................................................ 29 5.10.2 GARCH models.......................................................................................................... 29 5.10.3 DCC and multivariate GARCH models ..................................................................... 30 5.10.4 Maximum likelihood optimization ............................................................................. 31 5.11 OLS and trend testing ......................................................................................................... 32 6.

Analysis of commodity performance ......................................................................................... 33 6.1 Volatility clustering .............................................................................................................. 33 6.2 Historical stand-alone commodity performance ................................................................... 34 6.2.1 Descriptive statistics full sample .................................................................................. 34 6.2.2 Sub-period definition .................................................................................................... 35 6.2.3 Descriptive statistics sub-periods ................................................................................. 36 6.3 Full sample commodity portfolio performance .................................................................... 37 6.3.1 Markowitz optimization ............................................................................................... 37 6.3.2 Mean-variance spanning ............................................................................................... 39 6.4 Impact of the Lehman moment ............................................................................................. 39 6.4.1 Efficient frontier ........................................................................................................... 39 6.4.2 Sub-period Sharpe ratios and mean-variance spanning ................................................ 40 6.4.3 Optimal yearly portfolio weights.................................................................................. 41

7.

Analysis of commodity correlations........................................................................................... 43 7.1 Unconditional full sample correlation .................................................................................. 43 7.1.1 Traditional asset classes................................................................................................ 43 7.1.2 Inter-commodity ........................................................................................................... 44 7.1.3 Inflation ........................................................................................................................ 45 7.2 Unconditional sub-period correlation ................................................................................... 45 7.2.1 Traditional asset classes................................................................................................ 45 7.2.2 Inter-commodity ........................................................................................................... 46 7.2.3 Inflation ........................................................................................................................ 46 7.3 Conditional correlation ......................................................................................................... 46 7.3.1 Stocks ........................................................................................................................... 46 7.3.2 Bonds ............................................................................................................................ 48 7.3.3 T-Bills ........................................................................................................................... 49 7.3.4 Inter-commodity ........................................................................................................... 50 7.3.5 Inflation ........................................................................................................................ 50

8.

Discussion ..................................................................................................................................... 52

9.

Conclusion and recommendations ............................................................................................. 54 9.1 Answering the sub-questions ................................................................................................ 54 9.1.1 Commodities as a diversification tool .......................................................................... 54 9.1.2 Commodity portfolio performance ............................................................................... 54 9.1.3 Inflation hedging .......................................................................................................... 55 9.2 Answering the main research question ................................................................................. 55 9.3 Implications for investors ..................................................................................................... 55

10. Limitations and future research directions .............................................................................. 56

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11. References .................................................................................................................................... 58 11.1 Articles................................................................................................................................ 58 11.2 Books and readers ............................................................................................................... 62 11.3 Websites.............................................................................................................................. 63 12. Appendices ................................................................................................................................... 64 12.1 Appendix 1: List of figures ................................................................................................. 64 12.2 Appendix 2: List of tables................................................................................................... 65 12.3 Appendix 3: Pre-Lehman literature review ........................................................................ 67 12.4 Appendix 4: Post-Lehman literature review ....................................................................... 70 12.5 Appendix 5: Additional index information ......................................................................... 73 12.6 Appendix 6: Commodity indices weights ........................................................................... 74 12.7 Appendix 7: Descriptive statistics full sample ................................................................... 75 12.8 Appendix 8: Historical performance benchmark indices.................................................... 77 12.9 Appendix 9: Sub-period descriptive statistics .................................................................... 78 12.10 Appendix 10: Optimal portfolio weights across different risk levels ............................... 81 12.11 Appendix 11: Mean-variance frontiers of risky assets ..................................................... 83 12.12 Appendix 12: Sub-period mean-variance spanning results .............................................. 84 12.13 Appendix 13: Optimal portfolio weights per year ............................................................ 85 12.14 Appendix 14: Unconditional correlation matrices full sample ......................................... 86 12.15 Appendix 15: Sub-period unconditional correlation matrices .......................................... 88 12.16 Appendix 16: Dynamical conditional correlation graphs benchmark indices .................. 91 12.17 Appendix 17: Time-varying correlations graphs commodity sub-indices ........................ 92 12.18 Appendix 18: OLS regression results matrix for conditional correlations ....................... 95

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1. Introduction This chapter will start off with explaining why this study has been conducted. What was the event that potentially changed the traditional insights and what was this event exactly about? The second paragraph explains what this study comprehends and how it contributes to the prevailing body of academic research. Finally, the last paragraph will address the thesis’ outline and thoroughly explain the order in which all relevant issues are discussed.

1.1 Motivation for this thesis At 1:45 AM on September 15, 2008 Lehman Brothers Holdings Inc. filed for chapter 111 bankruptcy. Before this date, Lehman was the fourth largest investment bank in the United States after Goldman Sachs, Morgan Stanley and Merrill Lynch respectively. Traditionally, Lehman Brothers was a bank specialized in fixed income products, and therefore had significant mortgage businesses in the US and the UK. Next to that, it was a leading dealer in the over the counter (OTC) derivatives market. In the OTC market we speak of bilateral trade, i.e. deals directly emerge between two parties instead of over an exchange. These derivatives included credit default swaps (CDS), which offer an insurance against the default of (for instance) bonds. A CDS thus allows parties to diversify a company’s credit risk away. When the housing crises began, Lehman held large amounts of mortgages and leveraged loan assets. Consequently, the principal causes of its default were market concerns about the capital adequacy of the firm rather than its liquidity position. This was because of its ongoing exposure to illiquid assets. When the results of Q3 2008 appeared much worse than expected and Lehman was not able to raise new capital, the market’s and counterparty’s confidence collapsed. This mass loss of confidence ultimately led to the firm’s liquidation and with this many of its overseas subsidiaries. This sudden failure of Lehman Brothers on September 15, 2008 is widely viewed as the crucial moment in the global financial crisis (PricewaterhouseCoopers, 2012). With over $639 billion in assets and $613 billion in liabilities, Lehman Brothers’ default is the largest in US history, almost twice the size of Washington Mutual’s bankruptcy (a savings bank) which defaulted two weeks later. Lehman’s collapse brought global financial markets into a panic and sent credit markets worldwide into despair (New York Times, 2012). In the years that followed, the term ‘Lehman moment’ became known as a synonym for the financial event with explosive and unpredictable consequences. This Lehman moment raised some interesting questions in numerous security markets. People state that the financial world has changed as of that point in time. Where investors used to rely on textbook examples for effectively diversifying their investment portfolios, the Lehman moment possibly changed the rules of the game. With most research being done on mainstream securities like stocks and bonds, it’s interesting to look for possible changes in the commodity market and its relation with those mainstream investment assets. This is precisely what can be considered as the main motivation for this study. 1

One of the six different types of bankruptcy in the US (also known as corporate bankruptcy), which is a corporate financial reorganization that allows companies to continue operations while being obliged to follow certain debt repayment plans.

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1.2 Contribution of this thesis As the title of the thesis indicates, the main goal of this study is to determine whether or not the role of commodity futures investing has changed over time, and (if so) to what extent and in which direction. This is particularly interesting in light of the 2008 (and ongoing) economic downturn as was discussed in the previous paragraph. Where research on commodity investing has gained more attention throughout the years, studies on the period after Lehman’s default (i.e. Post-Lehman period from here onwards) are relatively scarce, which consequently poses an interesting field of study. This thesis aims at investigating whether the prevailing viewpoints on commodity investing do indeed still hold, with a specific focus on correlations between commodities and other more traditional asset classes (e.g. stocks). This study’s analyses elaborate on the commodity market from two different viewpoints. First by elaborating on the added value, measured in reward-to-volatility figures (i.e. Sharpe ratio, which will be algebraically introduced in section 5.2), commodity futures may provide to diversified investment portfolios. Second, by using a mathematically challenging financial econometric estimation model to obtain time-varying correlations, which will subsequently be analyzed for trends. The main contribution of this thesis can be found in the use of a very recent dataset, a portfolio composition exercise using this recent data and a non-straightforward correlation estimation model.

1.3 Thesis outline This report will subsequently address the steps necessary for conducting an academically sound analysis. Basically, this report can be subdivided into two separate parts. Chapter 2 summarizes the prevailing academic literature on commodity investing, which will be done for further clarification of the subject and precisely listing past and recent findings. Next, chapter 3 will elaborate on the specific research questions on which this thesis’ methodological framework is built. Not only the research questions themselves are discussed, but also the reasoning towards obtaining these key questions is explained. With the exact goals formulated, it is necessary to define the data that is going to be used for the analyses. This will be done in chapter 4, where the data sources and the different indices will be thoroughly explained and analyzed. How the research questions, formulated in chapter 3, will be answered is discussed in chapter 5. Here the empirical background is provided that is needed for understanding the techniques that will be used in this study. The goal of this specific chapter will thus mainly be to explain the different (mathematical) modeling techniques and why their use is justified. The second part of this study will be about the actual analyses themselves. After the framework has been drawn in the first five chapters, the following ones will address and analyze the collected data. Chapter 6 discusses the results on commodity performance, where chapter 7 elaborates on the correlation modeling. In chapter 8 the analyses are further discussed in the context of the literature review of chapter 2. Chapter 9 concludes this thesis by formally answering the research questions. Finally, the limitations of this study and possible future research directions are addressed in chapter 10. Where chapter 11 holds the academic references, chapter 12 provides the appendices, including lists of figures and tables, for supporting the main text.

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2. Literature review and empirical background This chapter will provide an academic background to the thesis. The main body of this literature review will be concentrated around three issues: commodity returns, commodity correlations and the role of commodity futures as an inflation hedge. First, a brief introduction to commodities as an investment asset will be provided, together with an overview of the basics of futures investing. After these concepts are clarified, the literature review looks at the research that has been conducted in the past. A clear distinction regarding the Lehman moment will be made to thoroughly distinguish between papers that were written before and after this starting point of the financial crisis. This subdivision will prove beneficial in deducting trends in the academic findings. It will also be used later on in this report in conjunction with this study’s findings. After this thesis’ results are obtained, the insights gained from the literature study and the analyses of the empirical findings will further be discussed in chapter 8. In appendices 12.3 and 12.4 an overview of the main literature that is used in this study is presented for reference purposes. It is interesting to read these tables top-down as they are chronologically ordered; preliminary analysis shows that some trends appear to emerge in the academic findings.

2.1 Introduction to commodity futures In this section an introduction to the derivative under study will be provided. First the concepts of the futures contract are discussed, where consecutively the pricing, the typical commodity investors and their corresponding investing methodologies are elaborated. 2.1.1 The futures contract Risk is defined as the variability in the value of an asset and can be split into price risk, counterparty risk, operations risk and business risk (Léautier, 2007). More specifically, price risk entails the variability that is due to changes in for instance exchange rates, interest rates and commodity prices. Businesses may choose to control these risks. They might choose to do so by the use of derivative instruments, which are types of contracts that transfer the risk of changing prices of the underlying asset from one party to another (Levontuiev, 2011). One particular derivative is the futures contract. Investing in commodities is typically done through the futures market (Kat and Oomen, 2006). A commodity future is an agreement to buy (or sell) a specified quantity of a commodity at a future date at the so called futures price (Gorton and Rouwenhorst, 2005). Next to that, additional information is specified in the futures contract. One might for instance think of quality specifications or the place (or means) of delivery. A futures contract is different from a forward contract in the sense that it is a standardized contract that is traded on a derivatives exchange, where a forward contract is a custom contract between a buyer and seller. Since centralized trading makes futures contracts more liquid than actual physical commodities, the main body of commodity research is focused on futures (Nguyen and Sercu, 2010). Consequently, this study will be concerned with futures prices of commodities. Note that the terms commodity investing and commodity futures investing will be used interchangeably.

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The futures price is the price one agrees upon when the contract is entered. Commodity futures are quite different from other asset classes like stocks and bonds. Viewed from the firm’s perspective, the main goal of stocks and bonds is to raise resources for the firm, resources that can again be allocated to generate future cash flows. These future cash flows are however uncertain and the risk of this uncertainty is carried by those investors that choose to invest in these stocks and bonds. Commodity futures however, do not raise resources for a firm. They offer firms insurance against their future outputs. Investors in these commodity futures bear the risk of short-term commodity price fluctuations (Gorton and Rouwenhorst, 2005). Where the standard futures contract basically obliges the buyer to deliver the physical commodity at maturity, this rarely occurs. Investors want to avoid the costs of actually holding physical commodities. Futures are mostly contracts with delivery time longer than one month. Interesting note on this is that these contracts often become illiquid in the delivery month, precisely because investors are reluctant to deliver or accept delivery of these physical commodities (Gorton and Rouwenhorst, 2005; Hong and Yogo, 2010). After the maturation of a first-month contract, the second month contract becomes active. This is called a contract ‘roll’, i.e. replacing the current contract in the index with a following one (Kat and Oomen, 2006). In the contract, a trader might choose between two different standpoints. A ‘long position’ obliges the trader to purchase the commodity at delivery date, where a ‘short position’ commits to delivering the commodity. The terms buy and sell are figuratively speaking, because the contract is not bought as is the case with stocks and bonds, but is entered by mutual agreement. Buying or selling a futures contract does not require any payment other than the initial margin. When assuming a 10% margin, this means only $10 is needed to buy exposure to a $100 worth of commodities. Where this is useful for speculation, it is very risky at the same time. Since a 5% price fluctuation in the underlying exposure means a 50% fluctuation in the margin payment. To bring the risk in line with other asset classes like stocks and bonds, investors typically provide 100% initial margin, i.e. they fully collateralize the contracts they hold (Kat and Oomen, 2006). The trader holding the long position benefits from any price increases, where the short seller benefits from a lower price. The profit to the holder of the long position (who has to purchase the commodity at delivery date) is equal to the difference between the spot price at maturity (i.e. the prevailing commodity price at that moment in time) and the original futures price. As an example to this, one may consider oil futures where the holder of the long position is obliged to purchase 1000 barrels of crude oil. If the original futures price is lower than the prevailing spot price at maturity, the long holder has to buy 1000 barrels at the price specified in the original futures contract, which he can subsequently sell at the higher prevailing spot price, thereby taking the spread as a profit. Adversely, the holder of the short position (i.e. the one selling the futures contract) benefits from a price decrease. Namely, in the case of decreasing prices the buyer of the contract has to buy the amount of commodities specified in the contract for a higher price than the seller would have received from the market at that point in time. Furthermore, the futures contract is a zero sum game. This means that gains and losses exactly cancel out. For this reason the existence of the commodity futures market theoretically should not have a major impact on the prices in the spot market for that commodity (Bodie et al., 2011). However, this statement is known to have raised some discussion among academics (among others Krugman, 2008). 4

2.1.2 The mechanics of futures pricing Now the basics of the commodity future are discussed, it is important to understand the underlying concepts of futures pricing. A lot of research has been done to understand the mechanical relationship between the futures price and the expected spot price at maturation date. This section will elaborate on an example of Bodie et al. (2011), beginning with three basic theories that contributed to the development of the Markowitz (or Modern) Portfolio Theory (introduced by Harry Markowitz in 1952). First, consider the simplest theory of futures pricing called the ‘expectations hypothesis’. This theory presumes that the futures price at time zero F0 is equal to the expected future spot price E(PT) at maturity time T, or stated algebraically F0=E(PT). It is easily shown that the expected profit for both parties is zero. This theory assumes that all participants are risk neutral. It however ignores the fact that investors typically are not risk neutral but instead most are risk averse, which thus means risk premia need to be taken into consideration. Normal backwardation is a theory attributed to John Maynard Keynes and John Hicks. This theory basically is about the fact that providers of commodities (e.g. farmers) are natural hedgers and wish to dispose of risk (i.e. they want to take on short positions). Since each contract involves at least two parties, a counterparty that is willing to take on a long position is needed. Normal backwardation assumes that these can be found in the form of speculators. However, speculators are only willing to take on this long position if an expected profit might be realized. This is the case when the expected spot price is higher than the original futures price, so they can take the spread of E(PT)-F0. This spread again is the loss for the commodity provider (i.e. in this case the farmer) who is willing to pay for alleviating the risk of uncertain prices. Where normal backwardation is an improvement over the expectations hypothesis in the sense that it indeed incorporates risk premia, these are based on total variability instead of on systematic risk. Contango states the exact opposite of the backwardation theorem. This theory says that the investors taking a long position (and hereby insuring them against any possible price increases) are willing to pay for shedding this risk. Again, speculators must be paid a premium before they are willing to enter a contract and will thus only fulfill the short position if the expected spot price E(PT) is lower than the futures price F0. Markowitz Portfolio Theory (MPT) effectively generalizes the above mentioned reasoning on risk premia (Markowitz, 1952). It basically says that when commodity prices pose positive systematic risk, the futures price F0 must be lower than the expected spot price E(PT). The actual pricing is based on the risk-free rate and the required rate of return of the investor, which is again determined by its risk-aversion level. Concluding, the MPT thus states that speculators with well diversified portfolios are only willing to take on long positions if they receive enough compensation for bearing the risk. This profit is generated when the futures price is lower than the expected spot price, as it thus generates positive returns. The inverse holds for investors in a short position. More on Markowitz’ theory will be discussed in chapter 5 with an elaboration of the methodological framework. 2.1.3 Why trade in commodities? Commodity futures are a relatively unknown investment class although they have been traded in the US for more than 100 years. This might be explained by the fact that commodities themselves are 5

strikingly different from more mainstream asset classes like stocks and bonds. According to Basu and Gavin (2011) the gross market value of commodity derivatives rose by a factor of 25 between June 2003 and June 2008 to a total of $2.13 trillion. As argued, commodities are derivative securities instead of actual claims on organizations. They are real assets that show signs of seasonality in price levels and volatility opposed to financial assets (Gorton and Rouwenhorst, 2005). However, investing in commodities might be interesting for several reasons. Traditionally, one may classify several types of traders who are interested in investing in commodities (Gold, 1961; Teweles et al., 1969). First, there are people who use the commodity market for hedging (e.g. of physical inventories). In general, hedging refers to techniques that offset specific sources of risk, rather than the search for an optimal risk-return profile for a complete portfolio (Bodie et al., 2011). In a specific commodity context we define hedging as a protective procedure designed to minimize losses that are due to price fluctuations (Bache & Co., 1958). If we consider the example of hedging commodity inventories we refer to the use of futures contracts, which (as argued before) determine the price of a commodity for future delivery. A company might choose to sell short an amount of commodity futures equal to the amount of physical commodities in its inventory. If its physical stock drops in value, it should be offset by the gains on its short sale. In this way the company thus hedges away specific price fluctuations. A long hedge entails the purchase of a commodity future to offset (or lessen) from a possible advance in the value of the physical commodity that is not yet owned, but is needed for business operations at prices that are currently set (Bache & Co., 1958). Nowadays, as far as hedging goes, commodity futures are used for instance for hedging inflation, which will be discussed in greater detail later on in this study. Speculators on the other hand are primarily interested in the gains resulting from a price change in a commodity futures contract (Bach & Co., 1958). Again, the speculator often is not interested in the actual delivery of the commodity. Speculators however are different from each other in many ways. For instance, in the length of time during which they hold positions or the type of position (long vs. short) they take. A third and final reason for investing in commodities is because of the diversification benefits it might entail (note that section 5.3 will elaborate on the mathematical explanation of the concept). Next to hedging, diversification is the other well-known technique for reducing investment risk. Diversification is about spreading your risk between multiple investment assets. A portfolio comprised of only one type of stock is said to be completely undiversified, with all risk being tied up in one asset. Risk can be split into diversifiable (i.e. firm specific or idiosyncratic risk) and nondiversifiable (i.e. market or systematic) risk. By including multiple stocks (with each its own characteristics) in an investment portfolio the idiosyncratic risk can be diversified away. The same holds when including other asset classes like bonds or commodities. Key with this is that a low correlation between the different assets helps drive down the portfolio risk (as it decreases the idiosyncratic risk), but generally all correlations less than one are beneficial. 2.1.4 Commodity index investing An increasingly popular investing strategy is not to invest in stand-alone commodity futures, but in a basket instead (Stoll and Whaley, 2009; Tang and Xiong, 2011). These baskets of commodities are 6

called commodity indices and function like for instance an equity index (e.g. the S&P 500). Each commodity in the index is assigned a particular predetermined weight, where the total value of the specific basket explains the index value. In this way, commodity indices provide returns that are comparable to long positions in listed commodity futures contracts (Tang and Xiong, 2011). There are several commodity indices constructed, of which the Standard and Poor’s (S&P) Goldman Sachs Commodity index (GSCI) and the Dow Jones UBS Commodity Index (DJ UBS) are most commonly used. These commodity indices will be part of the data used in this study when evaluating the commodity performance over time. In chapter 4 a specific overview of the indices is provided, where appendix 12.6 elaborates on the exact composition of the indices that will be used in this study.

2.2 Pre-Lehman literature review In this paragraph the first part of the literature review will be addressed. Here, the prevailing academic literature on commodity performance and correlation of the Pre-Lehman period (i.e. the period before September 15, 2008) is discussed. Next to that, we elaborate on the role of commodities as an inflation hedge. Where the following sub-sections provide a thorough overview, appendix 12.3 provides additional tables containing summarized findings of the most influential papers of this PreLehman era. 2.2.1 Historical (portfolio) performance of commodities Numerous studies have been conducted on determining whether or not commodities should be added to a well-diversified portfolio. Among others, Irwin and Lande (1987), Bjornson and Carter (1997) and Jensen and Mercer (2001) show that commodity futures indeed provide diversification benefits and improve performance of portfolios. More specifically, Laws and Thompson (2007) and Kat and Oomen (2006) agree that two papers about commodity investing may be considered as being the most influential (up till that point in time). First, Gorton and Rouwenhorst (2005) construct an equally-weighted index of commodity futures using data from the Commodity Research Bureau (CRB) for the 1959-2004 period. The commodities are studied in a portfolio context in order to reduce any noise that might hamper the detection of risk premia. They find that historically commodity futures have significantly outperformed the return on spot commodities, where both have outpaced inflation. During this timeframe stocks and commodity futures have experienced higher volatility than bonds, where stocks have offered a slightly higher return and standard deviation than commodities. Consecutively, Sharpe ratios of commodities and equities are basically the same. The return distribution of equities is found to be negatively skewed, where commodities show positive skewness. This means that stocks carry relatively more weight in the left tail of the distribution, where the opposite is true for commodities. Interesting about the paper is that Gorton and Rouwenhorst (2005) investigate the benefits that so called ‘pure plays’ might entail as well. Traditionally, investors have also gained exposure to commodity prices by investing in stocks of companies involved in producing the actual commodities (i.e. ‘pure plays’). The analysis of Gorton and Rouwenhorst (2005), which is based on the company’s SIC-code, shows significant differences 7

(generally underperformance) between these equities and the traditional commodity futures. Furthermore, the correlation between both assets has only been 0.40, where the equities had a correlation of 0.57 with the S&P 500. Since these pure plays are known not to be close substitutes to commodity futures, as their price and volatility incorporate much more sources of risk than commodity futures do, these will not be considered in this study. Second, Erb and Harvey (2006) study data from 1982 to 2004 and find that of the 36 individual commodity futures that Gorton and Rouwenhorst (2005) studied; exactly half had geometric means larger than zero (the other eighteen were lower than zero). They conclude that the average commodity future provides a risk premium of zero and ascribe the fact that Gorton and Rouwenhorst (2005) did find significant returns to the rebalancing of their portfolio. Erb and Harvey (2006) emphasize that it is unusual to infer the long-term performance of any asset class by looking at the returns from an equally-weighted portfolio. In addition, Arnott, Hsu and Moore (2005) point out as well that equallyweighted portfolios lack the liquidity and capacity that are found in traditionally market-weighted portfolios. Because of this, equally-weighting yields characteristics that are not representative of the corresponding asset class. One paper that raised particular interest and that will be used throughout this study has been written by Jensen and Mercer (2001). They examine the benefits of adding managed and unmanaged commodity futures to a portfolio of traditional investment assets; containing US and foreign equity, corporate bonds and T-Bills. Their study is an extension of Jensen et al. (2000). The former broadened the amount of futures contracts that is considered, where the latter only focuses on the GSCI. Jensen et al. (2000) note a link between the added value of commodities to a portfolio and the type of monetary policy the Federal Reserve (Fed) is pursuing. When the Fed is following a restrictive policy, i.e. when interest rates are raised, commodities can provide significant contributions to portfolio performance. If the Fed is following an expansive policy, i.e. interest rates are lowered; other asset classes (e.g. stocks) are preferred over commodity futures. This reasoning implies that commodities act as an inflation hedge; because interest rates are typically raised in periods of heightened inflation concern. Jensen and Mercer (2001) look into the GSCI and its corresponding six sub-indices for approximating the returns with a passive buy-and-hold investment in commodity futures. Next to that, they consider the Mount Lucas Management index (MLM), to serve as a benchmark for actively managed futures. They also chose to extend the paper of Jensen et al. (2000) because of the bias that is present in the GSCI. A common critique of this index is that it is heavily tilted towards energy futures (especially oil), since the GSCI is a value-weighted index based on world production. Jensen and Mercer (2001) conclude that the stand-alone performance of commodities is sub-par to stocks, which is consistent with prior research (among others Edwards and Park, 1996), indicating that commodities are a poor stand-alone investment. The GSCI Energy index is found to be highly volatile, where the Precious Metals index shows poor risk-return tradeoff. This is because precious metals (e.g. gold) are known to be relatively volatile, but have little market risk (Mezger, 2012). Because of the latter reason they are often considered beneficial for portfolio diversification, while they make a poor stand-alone investment. During the timeframe 1970-1999 commodities provided significant performance increases. Hereby the largest gain was obtained by active management (in line with Edwards and Liew, 1999). A simple trading rule based on the type of monetary policy the Fed is pursuing can however significantly enhance stand-alone performance of unmanaged 8

commodity futures as well. As argued, most investors are known to be risk-averse (Bodie et al., 2011). Looking at the role commodities have fulfilled across different risk levels we refer to Anson (1999). In his paper he studied the timeframe of 1974-1997 and found that an investor’s demand for commodity futures rises as its risk aversion level rises. An investor with high risk aversion should invest around 20% of its wealth in commodity futures. Jensen and Mercer (2001) also looked at this topic by examining the cash-collateralized GSCI for the period 1973-1997. They concluded that this percentage is somewhere between 5% and 36% (depending on the investor’s appetite for risk). 2.2.2 (Inter-) Commodity correlations Where one part of research is concerned with returns and reward-to-volatility analyses, correlation modeling is another important field of study. Chance (1994) argues that for the period 1971-1992 the GSCI was negatively correlated with the S&P 500 (-0.42), the MSCI EAFE index (-0.27), Treasury Bonds (-0.32) and T-Bills (-0.20). When considering moving average correlations he states that the relationship between commodity futures returns and the S&P 500 index returns is relatively unstable. The correlations are shown to be large for some time, and then again particularly low. When averaged for the study’s long time period, the overall correlation tends to smooth out; leaving him to conclude that no clear relationship exists between futures returns and stock returns. Erb and Harvey (2006) find similar results and elaborate more specifically on the inter-commodity correlations, which they calculate for the period 1982-2004. All (but one) inter-commodity correlations are found to be relatively low, but positive (which is again confirmed by Stockton, 2007). The correlation between Livestock and Industry Metals is slightly negative as can be seen in the table 2.1 below. Excess Return Correlations Dec. 1982 - May 2004 (monthly data) GSCI Agriculture Agriculture Energy Industry Metals Livestock Non-Energy Precious Metals

0.24 0.91 0.13 0.20 0.36 0.19

0.01 0.17 0.12 0.78 0.08

Energy Industry Metals

0.03 0.01 0.06 0.14

-0.02 0.31 0.20

Livestock

0.63 0.03

NonEnergy

0.20

Source: Erb and Harvey (2006) - The Strategic and Tactical Value of Commodity Futures

Table 2.1: Erb and Harvey (2006) correlation overview

In table 2.1 it is again shown that the GSCI commodity index is heavily influenced by the Energy index; as was initially indicated by Jensen and Mercer (2001). Based on the fact that commodities show little correlation with each other, Erb and Harvey (2006) conclude that commodities should not be viewed as one, but instead as a market of dissimilar assets. Kaplan and Lummer (1997) provide an update to Lummer and Siegel (1993) and find similar results as Chance (1994). For the period 1970-1997, they find a negative yearly correlation between fully collateralized GSCI futures and stocks and bonds. Correlations based on monthly returns over the whole sample period are close to zero. When they consider the last five years of their dataset (using monthly returns) they find the following correlations: bonds (0.12), the S&P 500 (0.28), the MSCI EAFE (0.39), T-Bills (0.25) and inflation (-0.06). All signs are opposite to that of the yearly correlation measures. Greer (2000) studies the yearly returns as well, but by using the Chase Physical 9

Commodity Index for the period 1970-2000. He finds a negative correlation with stocks (-0.14) and bonds (-0.32), which is in line with the yearly findings of Kaplan and Lummer (1997). Büyükşahin, Haigh and Robe (2008) thoroughly elaborate on the correlations and study more frequencies. They report commodity returns on a daily, weekly and monthly level for the period 19912008 and construct both simple (i.e. unconditional) and time-varying correlations. No significant correlation between the GSCI and the S&P 500 is found on either a weekly or monthly level. Using daily data however, a small negative correlation of -0.035 is found. The data is split into three consecutive sub-periods (that will be discussed in chapter 6): June 1992 to May 1997, June 1997 to May 2003 and June 2003 to May 2008. Using weekly data they find no significant unconditional correlations between the GSCI sub-indices and the S&P 500, except for Industry Metals (0.135) and Non-Energy (0.062). Again, this study reports a high correlation between the GSCI and the Energy index (0.967). As a general conclusion of the paper they state that the correlation between equities and commodities has not significantly increased in the past fifteen years. This paper’s detailed correlation modeling will act as a guideline for this study’s methodological framework (see chapter 5). Where Büyükşahin et al. (2008) did not find any significant correlation, Sieczka and Holyst (2009) argue otherwise. They conduct a minimum spanning tree analysis on 35 individual commodity futures for the period 1998-2007 and find an increase in inter-commodity correlations. They argue that a growing demand (e.g. energy, metals and food) from China and India created a commodity boom and corresponding intensive speculations, which subsequently resulted in the increasing comovements. Gorton and Rouwenhorst (2005) however find a negative correlation between commodity futures and traditional asset classes and explain this by the fact that commodities tend to behave differently over the business cycle. They examine the correlation on different intervals: monthly, quarterly, yearly and 5-yearly. This is done to identify any changes in comovements that might be caused by short term price fluctuations. For the period 1959-2004 they find a negative correlation between commodities and stocks (only significant at the 5 year window with a correlation of -0.42). For corporate bonds also a negative correlation is found at all frequencies, where inflation is positively correlated (although non-significant at the monthly and quarterly level). Interesting note is that the correlation of the commodities with stocks, bonds and inflation tends to increase with the holding period. 2.2.3 Commodities as an inflation hedge Inflation may be split into an expected and unexpected part. Expected (or anticipated) inflation is foreseen and therefore also results in high bond yields or equity returns. It is the unexpected inflation that should be of concern to the investor. This might result in negative returns for stocks and bonds, but increases in commodity prices. Intuitively it makes sense to take the correlation between nominal asset returns and inflation rates as a measure for the hedging capacity of the asset (Bodie, 1983; Spierdijk and Umar, 2011). Greer (2000) finds a negative correlation of stocks and bonds with inflation. Next to that, the paper shows that commodities had a correlation of 0.25 with annual inflation, and a correlation of 0.59 with the change in inflation. In addition, Strongin and Petsch (1996) conclude that the return on the GSCI is tied to current economic conditions. When inflation rises, the GSCI offers relatively good performance (compared to stocks and bonds). Therefore analysis between commodity futures prices and the US Consumer Price Index (CPI) makes sense. Nijman and Swinkels (2003) elaborate on these results and find that portfolio efficient frontiers (both 10

nominal and real) can be improved by timing the allocation of wealth to the GSCI. This timing should be based on several macroeconomic variables, including inflation. In line with the findings of Greer (2000), Kaplan and Lummer (1997) find a significant positive correlation between the annual returns of the GSCI and inflation (0.26). The GSCI has performed well in periods of both accelerating and decelerating inflation. However, (as expected) the GSCI performed significantly better with rising inflation. Precisely in these periods it is the GSCI that acts as a decent hedge, since the traditional assets are expected to generate low (or even negative) returns. Gorton and Rouwenhorst (2005) also suggest that commodities provide a better hedge against inflation than stocks and bonds. This because they are a bet on expected commodity prices, which are the components of inflation. Gorton and Rouwenhorst (2005) find a positive correlation between commodities and inflation at each frequency of data (i.e. monthly, quarterly, yearly and 5-yearly) which again confirms the findings of Greer (2000) and Kaplan and Lummer (1997). However, care should be taken in hedging inflation with the GSCI. Namely, the CPI is comprised of a services part of 60% and a commodities part of 40%. Energy commodities account for 4% of the CPI weight, where food commodities take care of 14%. Clearly a commodity index like the GSCI leaves out many important components of the CPI, including the single largest component, i.e. rent. Therefore, Erb and Harvey (2006) conclude that a mismatch might be present for commodities to act as a decent inflation hedge. However, they conclude that the GSCI has a positive (but non-significant) relation with inflation and a positive (and significant) relation with unexpected inflation (which is in line with Bjornson and Carter, 1997). The Precious Metals index (just as the individual commodities gold and silver) shows a (statistically significant) negative inflation beta. The remainder of the commodities is non-significantly correlated with inflation. Finally, Stockton (2007) also found that Energy, Livestock and Industry Metals had large correlations with inflation and unexpected inflation. However, the author stresses that commodities should not be viewed as a proper hedge against inflation. Instead they may be considered as an asset class that is expected to perform well in times of inflation. Namely, Stockton (2007) argues that the GSCI’s inflation hedging potential is dependent on the type of collateral that is used, not on the performance of the actual commodities.

2.3 Post-Lehman literature review Where section 2.2 elaborates on the influential papers of the Pre-Lehman period, this section will discuss the academic research that has been conducted after 2008. Just as in section 2.2, the findings will be divided into performance, correlation and inflation hedging. Aim of this section is to provide clear insight into the changing academic results on commodity investing, as the papers in this section will be discussed in comparison to those of the Pre-Lehman period. Please refer to appendix 12.4 for the tables containing summarized findings of the most influential papers of this Post-Lehman period. 2.3.1 Historical (portfolio) performance of commodities After more than 40 years of real declines, commodity prices increased dramatically between 2002 and 2008. Following the stock market crash of the early 2000s, investors became to see commodities as important assets for portfolio diversification, rather than assets that were considered incautious and 11

relatively difficult to hedge. Next to that, increases in capital from institutional investors have been noted. Some estimate the amount of passive investment to be around $150-200 billion by 2008 (Silvennoinen and Thorp, 2010). A continuing depreciation of the US dollar and low interest rates created a thriving environment, while a larger demand for food, metals and oil was coming from the emerging economies. Before its major drop in September 2008, the GSCI kept up the pace with the S&P 500. For the period 1990-2009 Silvennoinen and Thorp (2010) found the GSCI to exceed the mean T-Bill rate, where the volatility exceeded that of the S&P 500 (which is attributed to the GSCI’s large weight in Energy). Non-commercial traders (i.e. financial investors or speculators) are considered to be the driver of this returns volatility (Stoll and Whaley, 2009). Similar results are found by Mellon Capital Management (2008). They argue that the GSCI has climbed 41% over the first half of 2008, where it subsequently fell 49% from July to October. They state as well that, with the exception of crude oil, commodity prices are close to where they were around 1900. This should not be surprising, since commodities (as opposed to for instance equity) do not have an earnings expectation. Any supply and demand characteristics that might have influenced commodity prices are said to have cancelled out. Demand has increased from emerging economies, where on the other side economic growth in Europe, Japan and the US has slowed down. Next to that, much of the increasing demand has been offset by advances in production technology. Consequently, they make the statement: ‘Depending on where you stop the clock, it may appear as though performance is quite attractive for one or more commodities. But the longer-term perspective shows more of a path to nowhere with a lot of volatility’. Daskalaki and Skiadopoulos (2011) use a dataset from 1989 to 2009 and find that the S&P 500 outperformed the GSCI both on mean return (10% vs. 8% respectively) and volatility (14.9% vs. 21.4%). This study again confirms that commodities form a relatively poor stand-alone investment (in line with Jensen et al., 2000). To assess the added value commodities may have in a portfolio context they use a regression based technique known as mean-variance spanning (MVS). MVS, originally introduced by Huberman and Kandel (1987), analyzes the effect that the introduction of additional risky assets has on the mean-variance frontier of benchmark assets (DeRoon et al., 2001). Spanning occurs when the mean-variance frontier of the augmented risky asset set coincides with the predefined benchmark frontier (see section 5.5 for the specific testing procedure). Daskalaki and Skiadopoulos (2011) construct a benchmark portfolio containing stocks, bonds and T-Bills. Next to that, they construct the augmented set by adding commodity futures. They find that spanning occurs for both the GSCI and DJ UBS indices, as for individual commodities. Their results thus challenge the belief that commodities should be added to an investor’s portfolio (as argued by Nijman and Swinkels, 2003). However, when they repeat the mean-variance testing over the period August 2007 to December 2009 they find an exception for gold. This finding, that gold does improve portfolio performance in periods of economic downturn, is in line with Baur and McDermott (2010) who provide evidence for the commodity’s role as a safe haven. 2.3.2 (Inter-) Commodity correlations As was indicated in section 2.2.2, the majority of empirical studies of the Pre-Lehman period report a low or non-significant correlation (among others Büyükşahin et al., 2008; Chong and Miffre, 2008), or even a negative correlation between stocks and commodities (Chance, 1994; Kaplan and Lummer, 12

1997; Greer, 2000; Gorton and Rouwenhorst, 2005). As can be viewed from appendix 12.4, numerous studies find increasing correlations in the last decade; the causes and possible trends (i.e. whether or not the increased correlation is going to mean-revert) are however heavily discussed. With respect to the causes of the increase in equity-commodity correlations there basically are two views. One of them attributes the commodity boom-and-bust simply to supply and demand characteristics. Emerging economies like China and India significantly increased commodity demand, causing prices to increase (Krugman, 2008; Hamilton, 2009; Kilian, 2009). The prices consecutively fell sharply when the world-wide economic downturn caused demand to fade. Opposed to this view, Tang and Xiong (2011) state that an increasing demand from China could not have contributed significantly to these increasing correlations. This because in the period 2006-2008 the commodity prices in China remained relatively stable. However, they did find that commodities became increasingly correlated with oil, especially the indexed commodities (both for the GSCI and the DJ UBS index). A consequence of this larger correlation with oil is that the indexed (Non-Energy) commodities have obtained significantly larger price volatility. The increase in correlation starts to emerge as of 2004, which might be related to the second prevailing view. This second view ascribes the large volatility of commodity prices to the large passive investments into commodity indices. The commodity market gradually gained more attention after the equity market collapse of 2000. Investors’ interest was aroused by numerous academic studies (Greer, 2000; Erb and Harvey, 2006; Gorton and Rouwenhorst, 2005) that reported low correlations between commodity futures and the traditional asset classes. The number of open contracts in commodity exchanges world-wide has subsequently, between 2002 and 2008, increased by around 170%. Similar trends have been spotted in the OTC trading (Redrado et al., 2009; Domanski and Heath, 2007). Büyükşahin et al. (2010) also mention the increased investments of hedge funds or commodity index funds as of 2003, where their 2011 paper reports that excess speculation increased from about 11% in 2000 to 40-50% in 2008. As this index investing into the commodity market has increased in recent years (Tang and Xiong, 2011), commodity and equity markets have become more integrated. This process of financialization is extensively discussed by Silvennoinen and Thorp (2010). If the systematic component of commodity prices increasingly dominates returns, correlation with traditional assets will increase. The estimated conditional variance models confirm the significant positive spill-overs from macroeconomic factors, including the equity market volatility (where the VIX index acts as proxy), exchange rates, and financial traders’ positions. Silvennoinen and Thorp (2010) find that correlations between the commodity indices and the S&P 500 (and European stocks) have significantly increased, especially during the recent crisis. These findings of decreasing diversification benefits of commodities are clearly not in line with Pre-Lehman literature as Büyükşahin et al. (2008) or Chong and Miffre (2009). Büyükşahin et al. (2010) revisit their analyses of 2008 by extending the dataset from May to November 2008. They conclude that unconditional correlations increased both economically and significantly in the period from June 2003 to November 2008, which is largely ascribed to the market distress of mid-September 2008. When they model the time-varying correlations, they note that large 13

fluctuations have occurred through time; the DCC can be as high as +0.5 or as low as -0.5. These peaks however, remain lower than in the previous decade. Although they do not find evidence for a clear shift in correlations, they conclude that commodity- and financial markets are becoming more ‘a market of one’ during extreme events, which is in line with Hartmann et al. (2004) and Solnik and Longin (2001). They find that inter-commodity correlations mostly remain low or even insignificant, as was found by Erb and Harvey (2006). However, they do note that this presumption does not hold in periods of extensive financial distress; as for instance in 2008. Concluding, they state: ‘the commodity umbrella leaks during heavy storms’. Interesting fact is that this conclusion relies heavily on the 2008 period. Diversification gains are still said to exist, although in an ever decreasing extent. Büyükşahin and Robe (2011) again elaborate on the findings of Büyükşahin et al. (2010) with a more recent data set ranging from 1991 to 2010, hereby capturing a larger Post-Lehman period. Büyükşahin and Robe (2011) clearly stress the ever increasing financialization of the commodity market. Their paper is unique in the sense that it uses non-public trader data. Traders are becoming more intrigued by the presumable diversification gains of the commodity futures. But suppose that, during times of economic downturn, these investors have to liquidate their risky positions. Their exit from satellite markets (i.e. commodities) after a major shock in the central market (i.e. equities) could cause cross-market volatility spill-overs (among others Kyle and Xiong, 2011; Pavlova and Rigobon, 2008). Büyükşahin and Robe (2011) argue that commodity-equity comovements are related to the extent to which speculators (especially hedge funds) participate in the commodity market. This effect is larger when hedge funds invest in both equity and commodity markets. A 1% increase in the overall commodity futures share of hedge funds is associated with a 4% increase in commodity-equity correlations. This correlation sharply increased after the default of Lehman Brothers and stayed that way through March 2010 (the end of their sample). Before the Lehman moment, the correlations seemed to fluctuate over time; up till August 2008 no upward trend was apparent. After that moment correlations rose to levels never seen in the prior two decades. By using a time dummy, they conclude that the recent crisis is different from what has occurred before as the cross market linkages are persistently high, which is contrary to for instance Büyükşahin et al. (2010), Tang and Xiong (2011), Silvennoinen and Thorp (2010) and Chong and Miffre (2009). However, it should be noted that no specific cause is indicated, where it is shown that hedge funds only had a minor contribution to this increase. Other financial traders (e.g. swap dealers or index traders) do not have much explanatory power here either. They recommend further research into the time dummy, but ascribe possible explanations to the financialization and spill-over hypotheses. Finally, Li et al. (2012) aim to investigate whether or not the diversification gains of commodity futures have vanished, given the recent increase in correlation with stocks. They argue as well that the long-run trends are highly likely to be influenced by industrialization and financialization. Industrialization tends to increase the amount of commercial traders, where financialization increases the number of non-commercial traders. Li et al. (2012) find significant increases in correlations between the Pre- and Post-Lehman period. This paper is one of the first that discusses the possibility that the Post-Lehman correlations might not mean-revert, i.e. a long-run phenomenon might be apparent. Where Büyükşahin and Robe (2011) created a time dummy, they decompose the DCC model into a secular (i.e. long-run) and cyclical (i.e. short-run) component and find that 32 out of the 45 equity markets show an upward long-run trend in the correlation with commodity futures between 14

2000 and 2010. In addition, 43 out of the 45 equity markets had large upswings in their long-run trends due to the 2008 credit crunch. Finally, 39 out of the 45 equity markets tend to move towards or above their long-run trends when equity market volatility increases. 2.3.3 Commodities as an inflation hedge Spierdijk and Umar (2011) discuss the inflation hedging potential of the GSCI during 1982-2010, for investment horizons between one month and ten years. The average monthly inflation rate is 0.24% with a standard deviation of 0.27%, where the GSCI composite provided a monthly return of 0.52%. The GSCI was however highly volatile with a standard deviation of 5.79%. For inflation, negative skewness and large positive excess kurtosis are reported, indicating large departures from normality. When Spierdijk and Umar (2011) split the sample around August 2008, they find considerable differences in commodity returns, kurtosis and skewness. They find that each frequency shows statistically significant hedging potential, that increases with the length of the horizon. When they split the GSCI into sub-indices and individual commodities, they find that Energy and Non-Precious Metals (e.g. copper) commodities provide the best inflation hedging potential, although Agriculture and Livestock show some hedging potential as well. Attié and Roache (2009) find that commodities are the best performing asset class over a short horizon, where the long term effects fall gradually over time. One year after an inflation shock, commodity prices increase by 0.4 percentage point. All three tested commodity indices (the TR JCRB, the GSCI and the gold spot price) experience higher returns when inflation rises. The effects are both statistically and economically significant; a 1% higher inflation results in an increase between 3.8% and 10% among these indices.

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3. Research questions As can be seen from the previous chapter a lot of discussion still persists about the developments in the commodity markets. This has been explicitly shown with the split of the literature review into a Pre- and Post-Lehman section. In this chapter the aim of this study will be set forward; translated into the formulation of several relevant research questions. Section 3.1 introduces the main research question that subsequently will be broken down into three sub-questions (section 3.2).

3.1 Main research question As mentioned, based on the literature review that was stated in chapter 2, this chapter will elaborate on the question that is the focal point of this study. In the prevailing academic literature, no strong consent appears to be present about what benefits commodity investing may entail, both on an individual risk / reward basis and the usability for portfolio diversification. It thus strongly appears to be the case that the commodity market is changing as traditional perceptions (e.g. on correlations) might not be taken for granted. As the title of the thesis indicates, the main goal of this study is to determine whether or not the role of commodity futures investing has changed over time, and if so to what extent and in which direction. This study’s main research question is: Has the role of commodity investing changed over time?

3.2 Sub-questions Traditionally, commodities have been used for basically two reasons (Bodie et al., 2011): 1. Diversification. 2. Inflation hedging. Both issues can be assessed by looking into reward-to-volatility ratios and correlations between the different securities (as will be shown algebraically in chapter 5). In order to examine whether these traditional perceptions are still valid, the following sub-questions will be answered: 1.

Have the diversification benefits of commodities changed over time and are there any differences between the commodity futures (sub-)indices? This question relies on the study that Jensen and Mercer (2001) conducted for the period 1970 to 1999 (see section 2.2.1). Using several portfolio exercises, they analyze the commodity subindices in order to assess which commodity class offered the most diversification benefits. As for the correlations part, this question will address two different aspects. First, the correlation of commodities with traditional asset classes (e.g. stocks and bonds) is assessed using the methodology of Büyükşahin et al. (2008). Second, a closer look is given to the inter- commodity correlations as was done by Erb and Harvey (2006).

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2.

Has the Sharpe ratio of commodity futures changed over time? It is important to assess whether or not commodities may prove attractive stand-alone investments, or provide benefits when added to diversified investment portfolios. The answer to this question will be distilled from analyses that are combined with the first question. In answering the first sub-question, a commodity portfolio selection exercise (using Markowitz portfolio optimization) is conducted to detect any possible trends in the added value of commodity futures to a well-diversified investment portfolio. Next to that, the Sharpe ratios for different frequencies of data and time intervals are calculated to allow for several trend analyses.

3.

Is commodity investing still suitable for inflation hedging? This question addresses the return of the different commodity portfolios, and the correlation with the Consumer Price Index (CPI) as was done by Spierdijk and Umar (2011). Where the first sub-question focuses on the correlation between commodities and the traditional asset classes, this question will look specifically at the correlation with inflation and whether or not any trends seem to emerge. As argued, correlations determine the suitableness of an asset class to act as (an inflation) hedge.

17

4. Data As mentioned before, the purpose of this study is to closely examine the diversification benefits of commodity futures investing, particularly in light of the 2008 credit crunch. Therefore a time horizon of 25 years is chosen, based on daily, weekly and monthly information. The indices are chosen based on Jensen and Mercer (2001), Lankhorst (2012) and Erb and Harvey (2006). All data can be accessed by using Thomson Reuters Datastream and is available for all three frequencies, except inflation which is only presented at a monthly basis. All returns are total (log)returns in US dollar and for the period May 1st 1987 to May 1st 2012, unless mentioned otherwise. Please refer to appendix 12.5 for additional index information and a detailed specification of the data sources. For the equity and commodity categories (which will receive the most attention in this study) this thesis will mainly use the S&P 500 and GSCI, the additional ones are used for validation / robustness checks. More precisely, this means that calculations are repeated with the additional indices in order to exclude any biases resulting from the indices’ composition. For each index-category a short explanation and justification will be presented prior to the overview of the exact data sources.

4.1 Commodity indices Commodity indices are relatively unknown to the general public; however among investors they have gradually gained more attention. As argued, the index that is most known is provided by Standard and Poor’s and Goldman Sachs, i.e. the S&P Goldman Sachs Commodity Index (GSCI). This is a composite index of commodity sector returns that represents an unleveraged, fully collateralized, long-only investment in commodity futures that is broadly diversified across available commodity futures (Goldman Sachs, 2012). In appendix 12.6 the weights of the GSCI, which are based on world production, can be found. The quantity of each commodity is set equal to the average quantity of production in the last five years. Currently the GSCI contains 24 commodities from 5 different sectors: Agriculture, Energy, Industry Metals, Livestock and Precious Metals. This diversity aims to minimize any effects of idiosyncratic events. As can be seen in the appendix 12.6, the weights of the energy commodities (e.g. different classifications of oil) are relatively high. This is precisely why S&P also publishes data on the GSCI with Energy excluded, i.e. the Non-Energy commodity subindex. These weighting critiques also lead to two other important commodity indices. The Dow Jones / UBS Commodity Index (DJ UBS) and the Thomson Reuters / Jefferies Commodity Research Bureau Index (TR JCRB). The DJ UBS index is comprised of 20 commodities in 5 categories: Agriculture, Energy, Industry Metals, Precious Metals and Livestock. All commodities carry weights that account for economic significance and market liquidity (Dow Jones, 2012). In order to remain a well-diversified index (instead of the GSCI which is energy focused), DJ UBS makes sure that no related group of commodities constitutes more than 33% of the index and no single commodity represents more than 15% or less than 2% of the index (between rebalancings, weights may fluctuate outside these limits). The TR JCRB index is composed of 19 different commodities which are subdivided into four different groups: petroleum based products, liquid assets, highly liquid assets and diverse 18

commodities. The TR JCRB has fixed weights and is rebalanced every month to prevent weighting shifts that are the result of relative price movements (Thomson Reuters, 2012). Through this rebalancing process the TR JCRB index thus systematically decreases its exposure to commodities that have increased in value, and increases exposure to commodities that may be relatively inexpensive. In appendix 12.6 the TR JCRB weights can be found. Note that these are subdivided into five categories for easy comparison with the other two indices. In addition, the GSCI sub-indices are used, because it could be incorrect to draw general conclusions about commodity futures by only looking at the returns of the overall index as is done in Gorton and Rouwenhorst (2005) and Jensen et al. (2000). Namely, not all commodities have shown to offer similar risk premia (among others Kat and Oomen, 2006; Gorton et al., 2008). In order to be able to make general statements about the commodity market as a whole, these three indices (and the corresponding sub-indices) will be used: • The S&P Goldman Sachs Commodity Index with sub-indices: • Agriculture. • Energy. • Industry Metals. • Livestock. • Non-Energy. • Precious Metals. • Dow Jones / UBS Commodity Index2. • Thomson Reuters / Jefferies CRB Index3.

4.2 Bond indices This study will use three fixed income indices, two bond indices and a risk-free (i.e. T-Bill) index. Corporate bonds are bonds issued by corporations in order to expand their business or finance their operations. Investors may choose to invest in bonds since these, opposed to equities, usually pay (taxable) coupons and are considered safer investment securities. To proxy for corporate bonds, the Barclays US Corporate Long Bond Index (Barclays USCL) will be used. This index includes US investment grade bonds issued by industrial, utility and financial companies with maturities greater than 10 years. Subsequently, for government bonds, which are provided by national governments, the US 10 yr. Government Bond Index will be used. So the bond indices used in this study are: • Barclays US Corporate Long Bond Index. • US Benchmark 10 yr. Government Bond Index.

4.3 Equity indices Equity indices are commonly used for assessing market trends or for passive investing by using index funds, which yield a return equal to that of the corresponding index. One of the oldest equity indices is the Dow Jones Industrial Average Index (DJIA). It consists of 30 large corporations and has been computed since 1896 (Bodie et al., 2011). Because the DJIA holds one share for each of these 30 2 3

The DJ UBS index starts at 02-01-1991. The TR JCRB index starts at 01-01-1994.

19

companies, the weight of an individual company is determined by its share price. The DJIA is therefore called a price-weighted index. Because the DJIA only takes into account 30 stocks, care must be taken in drawing market-wide conclusions based on the index’ movements. Some other wellknown indices as the Wilshire 5000 and the Russell 2000 have drawbacks for this study as well. Because they include large amounts of small caps, the indices’ characteristics may be different from that of equity indices containing only larger corporations. The Standard and Poor’s 500 Composite Index is a capitalization weighted index of 500 stocks and is constructed by weighting the market capitalization of each firm. Firms with a higher value of outstanding equity receive a larger weight in the index. These stocks are top publicly traded stocks of US based firms. When one wants to look at a broader, more international level, the international stock market indices come into play. A leader in constructing these international indices has been Morgan Stanley Capital International (MSCI). They construct (among others) the MSCI World Index (MSCI World), which is a free float weighted index. This means that the weights are determined by the value of the shares that are freely tradable, i.e. not in possession of for instance governments or investment families. The MSCI World consists of 24 developed country market indices and thus is a measure for stock market performance in developed nations. Because both the S&P 500 and the MSCI World include US based companies, a bias towards this nation might be present. When one wants to look at the performance of the developed nations’ stock market index with the US excluded, one may look at the MSCI index for Europe, Australasia and the Far East (MSCI EAFE). This index, constructed by MSCI Barra, measures equity performance of developed nations outside the US and Canada. The equity indices under scope are: • S&P 500 Composite. • MSCI World Index. • MSCI EAFE Index.

4.4 Inflation indices In order to take changes in the prices of goods and services into account the US Consumer Price Index (CPI) is considered. This index represents a time series measure of the US price level and is calculated by observing price changes in a wide array of consumer products. Two measures are used; the CPI-U (or CPI - All urban) represents all products in urban areas, where the CPI (core) excludes the goods that have high price volatility like food and energy. Volatility in both categories often has other causes (e.g. bad harvests or OPEC politics). The inflation indices under consideration thus are: • CPI - All urban: all items. • CPI - All items less food and energy (core).

4.5 T-Bill index In order to consider short-term lending to the US government, T-Bills will be taken into account as well. Where government bonds are of longer maturity and provide (semi-annual) coupon payments, T-Bills provide a short term investment and are sold at discount from face value. The index used is: • Barclays US Treasury Bill 1-3 months4. 4

The US T-Bill index starts at 12-01-1991.

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5. Methodological framework In this chapter the empirical background and methodology of this thesis will be discussed. Different modeling techniques will be explained by looking at their mathematical foundation. Consecutively, it will be made clear why these specific methodologies are relevant for this thesis’ analyses. Correlations based on four different measures are introduced (Büyükşahin et al., 2008): • Unconditional correlation. • Rolling historical correlation. • Exponentially weighted moving average correlation. • Dynamical conditional correlation.

5.1 Volatility All analyses in this study are based on financial returns data. Return series are interesting for financial statistics because they show some attractive statistical properties. When assumed in a continuous fashion they can be obtained as follows (Guida and Matringe, 2004):

 PA ,t rA ,t = ln   PA ,t −1

   

(5.1)

Where: rA,t = the return on asset A at observation t. PA,t = the price (or index value) of asset A at observation t. PA,t-1 = the price (or index value) of asset A at observation t-1. In order to explain the importance of using a proper technique to model an asset’s variability we first provide an introduction based on Bodie et al. (2011). Asset managers, holding portfolios of assets, generally are interested in the extent to which the returns on the portfolio’s assets tend to vary. In order to get a feeling for this, they calculate the portfolio’s variance5. Consider the global variance of an individual asset A:

σ A2 =

1 n 2 rA,t − E (rA ) ∑ n t =1

[

]

(5.2)

Equation (5.2) shows that this measure of variability weights the squared deviations from the mean, based on n (equally-weighted) observations. When considering a portfolio constructed from two assets, respectively A and B, we obtain formula 5.3 for the portfolio’s variance (Bodie et al., 2011).

5

Another possible measure of variability is to use that absolute deviation from the mean. This so called Mean Absolute Deviation (MAD) measure is not commonly used and has several drawbacks. MAD will not be considered here.

21

n

σ P2 = 1 ∑ [wA a(t ) + wB b(t )]2 n

=

t =1

1 n 2 ∑ wAa(t )2 + wB2 b(t ) 2 + 2wA wB a(t )b(t ) n t =1

[

]

1 N 1 N 1 N = w ∑ a(t ) 2 + wB2 ∑ b(t ) 2 + 2wA wB ∑ a(t )b(t ) n t =1 n t =1 n t =1

(5.3)

2 A

= wA2σ A2 + wB2σ B2 + 2wA wB

1 N ∑ a(t )b(t ) n t =1

Where the deviations from the mean a(t) and b(t) are respectively: a ( t ) = r A ,t − E ( r A ) and b ( t ) = rB ,t − E ( rB ) Concluding, the portfolio variance is: N

σ P2 = wA2σ A2 + wB2σ B2 + 2wA wB 1 ∑ [rA,t − E (rA )][rB ,t − E (rB )] n

(5.4)

t =1

= wA2σ A2 + wB2σ B2 + 2σ AB = wA2σ AA + wB2σ BB + 2σ AB When considering equation (5.4) we see that the variance of the portfolio P is constructed using three components, the weighted sum (wA and wB) of the assets’ variances and a third term, the covariance, which is of particular interest. This term quantifies the extent to which the two assets co-vary. Why it is the covariance that matters will be shown in section 5.3.

5.2 Sharpe ratio Now that volatility has been introduced we look at how investors use it to evaluate their investments. Investors are known not to be interested in returns, but in excess returns. Thus returns over a specific risk-free measure. T-Bills are normally considered to be a safe investment and are thus used as a benchmark for the risk-free return. Investments are priced according to the amount of excess return they offer. This amount depends on the riskiness of the investment. Riskiness is usually defined as the standard deviation of this excess return. This suggests to measure the attractiveness of an investment by the ratio of its risk premium and the standard deviation of its excess return, which is exactly what William Sharpe (1994) did. This reward-to-volatility ratio is defined as:

Sp =

E (r p ) − r f

(5.5)

σp

The nominator gives the portfolio’s return over the risk-free rate. This ratio gives an idea of the expected risk premium over one unit of risk.

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5.3 Diversification Consider equation (5.4) for two assets and generalize this for n assets into: n

n

n

n

σ P2 = ∑ ∑ wi w jCov ( ri , rj ) = ∑ ∑ wi w jσ ij i =1 j =1

(5.6)

i =1 j =1

To further examine the importance of respectively the assets’ variances and covariances we elaborate on another example, based on Bodie et al. (2011). Assume an asset manager holds a portfolio of n assets. When considering a naïve diversification strategy we use an equally-weighted portfolio, where each asset receives the weight wi = 1/ n . For i=j, equation (5.6) can be rewritten into the following form:

σ P2 =

1 n 1 2 n n 1 σ ij ∑ σ i + ∑∑ 2 n i =1 n j =1 i =1 n

(5.7)

j ≠i

Notice that Cov ( ri , ri ) = σ ii = σ i2 . Equation (5.6) has thus effectively been broken down into one variance component and one covariance component. Consider an n*n (co)variance matrix, notice that the variance terms are placed on the diagonal (i.e. n terms), thus leaving n(n-1) covariance terms. We can calculate the asset’s average variance (5.8) and covariance (5.9) as:

1 n 2 σ = ∑σ i n i =1 2

Cov =

(5.8)

n n 1 ∑∑σ ij n(n − 1) j =1 i=1

(5.9)

j ≠i

Hereby, the average portfolio variance becomes: 2

nσ n(n − 1) 1 2 n −1 σ = 2 + Cov = σ + Cov 2 n n n n 2 p

(5.10)

From this equation we see that the first term will be driven to zero for a large n, leaving only the covariance term. Thus when firm specific risk (represented by the first term) is diversified away, the systematic risk (determined by the covariance of the returns of the individual securities) only matters; this is precisely why the covariance receives that much attention of investors. Using above definitions of variances and covariances an elaboration to several correlation measures can be found in the following sections. First the algorithm for calculating the optimal portfolio weights will be discussed.

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5.4 Markowitz portfolio theory The goal of a portfolio manager is to achieve the best possible trade-off between risk and return (De Goeij, 2011). In order to achieve this, the reward-to-volatility ratio, i.e. Sharpe ratio as it was introduced in section 5.2, is maximized. When considering a portfolio comprised of one risk-free asset and many risky assets, the optimum for a mean-variance investor is achieved by: 1. Maximizing the Sharpe ratio for the portfolio that is completely comprised of risky assets. 2. Calculating the mean and standard deviation of this portfolio (i.e. tangency portfolio) of risky assets. 3. Maximizing the investor’s utility function by calculating the optimal weight for the tangency portfolio, and thus the weight to be invested in the risk-free asset. These weights are determined by the investor’s risk-aversion level. Where a risk-neutral investor is only concerned with the return on an investment, most investors are known to be risk averse (to varying degrees). Using matrix notation, the above mentioned steps may be completed simultaneously. Define the weight of the risk-free asset to be:

wrf = 1− w'ι = 1−ι' w

(5.11)

Where: wrf = a scalar representing the weight of the risk-free asset. w = a (K*1) vector containing the weights of the risky assets.

ι = a (K*1) vector of 1’s. Now the optimal complete portfolio is defined as:

rc = (1− w'ι )rf + w' rp

(5.12)

= rf + w' (rp − r' f ι) With the corresponding optimization problem:

w* = A−1Σ−1µ e = A−1Σ−1 (E(rp ) − r' f ι )

(5.13)

Where: A = the investor’s risk aversion level.

µ e = the matrix with excess returns.

Σ = the (co)variance matrix of the excess returns6. 6

The variance of the excess returns is assumed to be equal to the variance of the returns.

24

An optimal solution to this mathematical problem can be found by using Excel’s solver function. The target function for the optimal complete portfolio is the maximization of its Sharpe ratio, subject to the constraints that the sum of the weights has to equal one and no individual weight can be smaller than zero. The optimization problem is algebraically defined as:

 E ( rc ) − r f  Max [S c ] = Max   σc  

(5.14)

s.t. N

∑w

=1

(5.15)

0 ≤ wi i = 1,..., N

(5.16)

i

i =1

N

N

∑∑ w w σ i

j

ij

= target portfolio standard deviation

(5.17)

i =1 j =1

Constraint 5.16 is set because the analyses in this thesis will not take short sales into consideration. The final constraint 5.17 is optional, as it is only used for determining the optimal portfolio weights for different levels of risk (as measured by varying degrees of portfolio standard deviation).

5.5 Mean-variance spanning As briefly discussed in chapter 2, mean-variance spanning (MVS), originally introduced by Huberman and Kandel (1987), analyzes the effects that the introduction of additional risky assets has on the mean-variance frontier of benchmark assets (DeRoon et al., 2001). Spanning occurs when the mean-variance frontier of the augmented risky asset set coincides with the benchmark frontier. Consider a benchmark portfolio ‘Bench’ and an augmented commodity portfolio ‘Com’. More specifically in the context of this study, MVS tests whether the addition of commodity futures to the benchmark portfolio improves performance (section 5.11 will elaborate on the concept of OLS).

RtCom = α + β ⋅ RtBench + ε t

(5.18)

Where:

RtBench = the return on the benchmark portfolio. RtCom = the return on the augmented portfolio containing commodities. If the intercept α is zero and the slope β is one, spanning occurs. It means that the returns for the commodity portfolio can completely be explained by the returns on the benchmark portfolio, i.e. commodities do not contribute to portfolio performance as they receive no weight. This state is referred to as the test hypothesis H0. If H0 does not hold, so either α is unequal to zero or β is unequal to one, adding commodities to the portfolio indeed proves beneficial.

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5.6 Unconditional correlation When scaling the covariance by the standard deviations of the variables, we obtain the standardized covariance, better known as the correlation coefficient. The correlation between two random variables A and B with each mean 0 is defined as:

ρ AB =

E ( rA , rB ) 2 A

2 B

=

Cov( rA , rB )

E ( r ) E (r )

σ Aσ B

=

σ AB

(5.19)

σ A2σ B2

Because of standardizing, formula 5.19 yields a value between -1 and +1. A correlation coefficient of +1 indicates that variables A and B are perfectly positively correlated, where -1 yields a perfect negative correlation (also called anticorrelation). A correlation coefficient of 0 means there is no linear relation between the two variables whatsoever. The absolute value of the correlation coefficient describes the proportion of the variance that is in common between variables A and B (Verbeek, 2008). Note that the correlation coefficient is symmetric. If variables A and B are independent, the correlation coefficient is 0, but the converse is not true. The correlation between A and B may be 0 where there still is some dependency present. This is because the coefficient only detects linear dependencies, which is important to keep in mind when evaluating correlation figures later on. One other critical point that needs to be taken into consideration is the relation between correlation and causality. Note that correlation cannot be used to deduce a causal relationship between two variables; it can however indicate a possible causal relationship (Field, 2005).

5.7 Conditional correlation In the preceding section we looked into the overall correlation between two assets (i.e. unconditional correlation). Since investment assets like stocks and bonds are known to be volatile, the correlations may change through time. In order to get a feeling for these time-varying correlations one might calculate a correlation dependent on the time t, i.e. the conditional correlation, which is defined as (Engle, 2000):

ρ AB, t =

Et −1 ( rA,t , rB ,t ) Et −1 ( rA2,t ) Et −1 ( rB2,t )

=

σ AB ,t

(5.20)

σ A2 ,tσ B2 ,t

Measuring the relationship between different variables at multiple points in time, instead of using a single measure for an entire sample-period, provides information on the evolution of the correlations over time (Büyükşahin et al., 2008). There are several ways to express these time-varying (i.e. conditional) correlations; in the following sections these will be discussed in ascending order of mathematical complexity.

26

5.8 Rolling historical correlation Rolling historical correlations do take into account the time-varying nature of financial assets, by calculating the correlation at any point in time for a specified window of length k (where k is smaller than the size of the sample). For each subsequent correlation, this window moves one observation, hence the name rolling correlation. Engle (2000) provides the following formula:

ρ AB,t =

1 t −1 ∑ rA,s rB,s k s =t −k −1  1 t −1 2  1 t −1 2   ∑ rA,s  ∑ rB ,s   k s =t −k −1  k s =t −k −1 

t −1

∑r

=

r

A,s B ,s s =t − k −1

(5.21)

 t −1 2  t −1 2   ∑ rA, s  ∑ rB , s   s =t −k −1  s =t −k −1 

In formula 5.21, rA,s is the deviation from the mean of variable A (analogous for variable B). This estimation technique assigns equal weight to each observation in the estimation window, namely (1/k). However, this method has several drawbacks. First of all, formula 5.21 only takes observations into account that fall in the window, i.e. a zero weight is assigned to older observations. Next to that, equally-weighting each observation might not be desirable because older observations could carry less information than more recent ones. Finally, issues are known to be raised (Büyükşahin et al., 2008) in determining the window length k. For financial data often 30 or 90 period windows are used. The analyses in this study will be conducted by using the 30 period window.

5.9 Exponentially weighted moving average correlation As already mentioned in section 5.8, with rolling historical correlations the past observations are all equally-weighted, which might not be desirable if one wants to attach more weight to recent observations. This is where the exponentially weighted moving average (EWMA) correlation comes into play, as it uses declining weights based on a specific parameter lambda. This parameter emphasizes current data by assigning more weight to recent observations, i.e. assigning exponentially decreasing weights to older ones, hereby never reaching the zero weight. One point of discussion with using this technique is the determination of lambda which is used for all corresponding assets. Usually lambda is set to 0.94 for weekly and monthly data, and 0.97 for daily observations (Engle, 2000; Büyükşahin et al., 2008). t −1

t −1

(1 − λ )∑ λt −1−s rA,s rB ,s

ρ AB,t =

∑λ

t −1− s

s =1 t −1 t −1 (1 − λ ) ∑ λt −1−s rA2,s (1 − λ ) ∑ λt −1−s rB2,s   s =1   s =1 

=

rA,s rB,s

s =1

 t −1 t −1−s 2  t −1 t −1−s 2   ∑ λ rA,s  ∑ λ rB,s   s =1  s =1 

(5.22)

In formula 5.22 (Engle, 2000) we identify the covariance divided by the product of two standard deviations and will rewrite it into recursive form by using the following example.

27

When calculating formula 5.22 for time t=5 (and t=4 for illustrational purposes) we obtain: 4

(1 − λ )∑ λ4−s rA,s rB,s

ρ AB,5 =

s =1

  4 4− s 2  4− s 2  (1 − λ ) ∑ λ rA,s (1 − λ ) ∑ λ rB ,s   s =1   s =1  4

(5.23)

When considering the variances we obtain for variable A: 3

σ A2 , 4 = (1 − λ )∑ λ3− s rA2, s = (1 − λ )(λ2 rA2,1 + λ1rA2, 2 + λ0 rA2,3 )

(5.24)

s =1 4

σ A2 ,5 = (1 − λ )∑ λ4− s rA2, s = (1 − λ )(λ3 rA2,1 + λ2 rA2, 2 + λ1rA2,3 + λ0 rA2, 4 )

(5.25)

s =1

We thus see that each iteration weights the preceding periods’ variance by λ, and the preceding periods’ squared return by (1-λ). Generalize the formula for variable A’s variance into:

σ A2,n = λσ A2,n−1 + (1− λ)rA2,t −1

(5.26)

For the numerator, containing variable A and B’s covariance, we follow analogous reasoning and generalize into: 2 2 σ AB ,n = λσ AB,n−1 + (1 − λ )(rA,t −1rB,t −1 )

(5.27)

Just as with the rolling historical correlations the weights have to sum up to one. Each preceding covariance is weighted with λ, and each preceding product of returns by 1-λ. However, the exponentially weighted moving average has a drawback that it lacks a term for mean-reversion. Since variance terms in practice tend to mean-revert, another improvement to the used methodology can be made by including such a term. Next to that, the EWMA technique, as with the rolling historical correlations, is not suited best for taking changes in volatility into account. Because of the fact that the estimated correlation is highly sensitive to the volatility (which is estimated as well), it may be hard to deduce the true relationship between the two variables (Büyükşahin et al., 2008). Since the correlations are subject to volatility shocks, it may be especially hard to interpret these in periods of high volatility. Here the use of GARCH models comes into play.

5.10 Autoregressive conditional heteroskedasticity In financial time series one often observes what is referred to as volatility clustering (Verbeek, 2008). This refers to the fact that big shocks are often followed by big shocks (in either direction), and small shocks are followed by small shocks. We thus often observe so called autocorrelation in the riskiness of financial assets. When considering stock markets one can often classify periods of ‘high activity’ as 28

well as quieter periods (i.e. lower volatility). This observation is particularly true for high frequent observations (i.e. daily or weekly) opposed to monthly observations, which can be explained by using the theory of time-diversification. When modeling (financial) variables one needs to pay particular attention to the form of the error term. A stochastic variable with a constant variance is called homoscedastic, where heteroscedasticity indicates varying volatility. Where more traditional estimation techniques often assume the homoscedasticity characteristic to hold, this obviously would not suit this thesis’ financial data particularly well. French et al. (1987) show that the daily S&P 500 returns show conditional heteroscedasticity. Assuming that the variance is constant can lead to drawing false conclusions. Engle (1982) proposed the concept of autoregressive conditional heteroskedasticity (ARCH) to deal with just these issues, which was later generalized (GARCH) by Bollerslev (1986). Concluding we may state that the application of ARCH and GARCH models in the field of finance has been particularly successful because of the heteroskedastic nature of financial time series (Engle, 2001). 5.10.1 ARCH models The ARCH(p) model uses past disturbances to model the variance of a time series and is defined as: p

2 t

2 1 t −1

σ =ϖ +α ε

2 2 t −2

+α ε

2 p t− p

+ ... + α ε

= ϖ + ∑ α j ε t2− j

(5.28)

j =1

Where p refers to the number of autoregressive (i.e. backward looking) terms. Basically, it says that the variance at time t depends upon a long run average, plus the squared error terms from previous periods. When considering the model with one autoregressive term we obtain: σ t2 ≡ E {ε t2 | Ι t −1 } = ω + αε

2 t −1

(5.29)

So, the variance at time t is defined as the squared error term at time t, given the information set I available (typically containing all previous error terms), and is equal to the unconditional variance plus the influence of the previous period’s volatility. The second term of the equation thus says that when a big shock happens in period t-1, it is more likely that ε t has a large (absolute) value as well. This appears in line with the volatility clustering theorem discussed above. The effect of this previous shock is determined by the weight α. 5.10.2 GARCH models Because ARCH models are difficult to estimate, the model has been generalized by a student of Engle, named Bollerslev (1986). The GARCH(p,q) is defined as: p

q

j =1

j =1

σ t2 = ϖ + ∑ α j ε t2− j + ∑ β jσ t2− j

(5.30)

29

Where p refers to how many autoregressive lags, or ARCH terms, appear in the equation while the q refers to the amount of moving average (variance) terms. Since the GARCH(1,1) model is known to perform well in practice (Verbeek, 2008; Li et al., 2012; Engle, 2004), we will consider the following: σ t2 = ϖ + αε

2 t −1

+ βσ

2 t −1

(5.31)

This model is more intuitive than Engle’s ARCH model as it basically generalizes the pure autoregressive ARCH model into an autoregressive moving average model. As can be seen from equation 5.31, the GARCH(1,1) model, which calculates the variance at time t, contains three components: the intercept (ω) that acts as the long-run average, a shock from the previous period (with weight α) and the variance forecast from the previous period (weighted with β). The weights α and β determine how fast the variance changes with new information or how fast it reverts to the long-run average. A large β indicates that shocks to the conditional variance take a long time to fade off, i.e. the volatility is said to be persistent (Chong and Miffre, 2009). A large α on the other hand means that the volatility reacts strongly to any recent movement in the market. 5.10.3 DCC and multivariate GARCH models Where the univariate GARCH model is used for the estimation of one specific financial variable’s variance, a multivariate analysis would include additional dependent variables. Multivariate analysis is particularly useful when modeling the comovements of returns on two financial assets (Su and Huang, 2010), in which case we deal with a bivariate model. A multivariate GARCH model (MGARCH) would result in more reliable results than separate univariate models. MGARCH models allow both the conditional mean and conditional covariance to be dynamic (STATA, 2012). Engle (2002b) proposes a new class of multivariate estimators called dynamical conditional correlation models (DCC). This DCC model has been deduced from Bollerslev’s (1990) constant conditional correlation (CCC) model and does not smooth out fluctuations while also preserving trends (Li et al., 2012). The CCC model is a multivariate time series model with time-varying conditional (co)variances, but constant conditional correlations. When the conditional correlation parameters that weight the nonlinear combinations of the conditional variance are not considered constant anymore, but are also left for optimization one obtains the DCC model. The DCC model relies on two separate steps to estimate the time-varying correlation between two variables. In the first step the time-varying (co)variances are estimated using the GARCH methodology. In the second step, a time-varying correlation matrix is estimated using the standardized residuals from the first step (Büyükşahin et al., 2008). To clarify the relation between the conditional correlation and conditional variances we rewrite the returns as the conditional standard deviation times the standardized disturbance. Consider for asset A:

rA,t = σ A,t ε A,t

(5.32)

30

Where: ε A,t = the standardized disturbance with mean zero and variance one. Now, consider the return series r with mean zero and variance-covariance matrix Ht to have the following structure. rt ~ N ( 0 , H t )

(5.33)

H t = D tR tD t

(5.34)

Where: Ht = the conditional covariance matrix at time t. Rt = the time-varying correlation matrix at time t (in the CCC model this matrix is constant). Dt = the diagonal matrix of time-varying standard deviations, i.e. univariate GARCH volatilities. Engle (2002) suggests the following dynamics of the time-varying correlation matrix Rt: R t = Q t* −1 Q t Q t* −1

(5.35)

Qt = (1 − α − β )Q + α (ε A,t −1ε B ,t −1 ) + β Qt −1

(5.36)

Where: Q = the unconditional correlation matrix of standardized residuals. Qt* = diagonal matrix composed of the square roots of the diagonal elements of Qt .

In formula 5.36 one can identify the same principle as in formula 5.31. Namely, it consists of a long run average correlation, the previous period’s shock and the previous period’s correlation matrix. Note that mean-reversion is guaranteed when α+β