COMMODITY FUTURES IN ASSET ALLOCATION

Transcription

1 The Pennsylvania State University The Graduate School Department of Agricultural Economics and Rural Sociology COMMODITY FUTURES IN ASSET ALLOCATION A Thesis in Agricultural, Environmental and Regional Economics and Operation Research by Shengwu Du 2005 Shengwu Du Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2005

2 The thesis of Shengwu Du has been reviewed and approved* by the following: Spiro E. Stefanou Professor of Agricultural Economics Thesis Adviser Chair of Committee Jeffrey R. Stokes Associate Professor of Agricultural Economics James W. Dunn Professor of Agricultural Economics Jingzhi Huang Associate Professor of Finance Stephen M. Smith Professor of Agricultural and Regional Economics Head of the Department of Agricultural Economics and Rural Sociology *Signatures are on file in the Graduate School.

3 iii ABSTRACT Commodity futures can supply fair investment return and unique diversification benefits to a portfolio manager. The relative performance between commodity futures and traditional financial assets is varied with business and market conditions. In general, commodity futures are negatively correlated with stocks and bonds. This negative correlation comes from the nature of the commodity market. Commodity futures can supply efficient diversification benefit to stocks and bonds investors when it is needed, i.e. when the stock or bond market displays poor performance or has a significant downside movement, commodity futures market usually shows strong performance and provides a good return. Both commodity futures and traditional financial assets display the structure break behavior and their risk/return characters change under different market and economic conditions. Markov regime-switching model can be applied to address this asset return dynamic and corresponding portfolio selection issue. The estimation results of a simple two-regime model support that for three risky asset classes there is a normal regime characterized by relatively high return and low risk and a bad regime characterized by relatively low return and high risk. An investment strategy ignoring the regime switching effect leads to inefficient asset allocation and poorer portfolio performance. When the market is in the bad regime, commodity futures perform relatively stronger than stocks and can provide more significant diversification benefit. Meanwhile, the regime-switching model can predict the break change of financial market and send signal to investors to increase commodity futures investment when financial market switches to the bad regime. A general regime-switching model extends the simple regime-switching model by using economic variables as instruments to predict asset returns. This model helps investigate asset return dynamics and optimal asset allocation under alternative economic environments. The first general regime switching model uses the short rate as the

4 iv instrumental variable to predict expected asset returns and regime switching probability. This model shows that the short rate plays an important role in asset return regime switching and optimal asset allocation. The optimal asset weights on risky assets negatively relate to the T-bill rate. This is not surprising as the T-bill rate determines the risk-free asset. When the return of risk-free asset is increasing, investors expect a greater return from the risky assets and have a greater chance to switch investment to the risk-free asset if the expectation is not met. However, the fraction of commodity futures in the risky asset portfolio increases with the short rate level. The second general regime switching model is the IPI model which uses the monthly change of industrial production index (IPI) as the instrumental variable to model risky asset returns. The estimation results of this model support that the IPI growth rate plays an important role for efficient asset allocation since it determines both the regime transition probability and the expected returns of risky assets. Commodity futures return is always positively correlated with the IPI growth rate and the optimal weights on commodity futures increase with the IPI growth rate. To further investigate the diversification benefit of commodity futures, an alternative approach is provided. Commodity futures are included into the initial portfolio with various stocks and bonds components and a new risk measure, VaR, is adopted to measure the risk of these portfolios. Three different VaR estimations are proposed: parametric VaR with regime-switching model, historical simulation VaR, and the EVT VaR. These estimation results again strongly support that commodity futures can reduce portfolio risk and improve portfolio performance. However, for different initial traditional portfolios, the optimal weight on commodity futures is not identical. Allocating 5-10% capital on commodity futures investment, as most intuitional investors have done, is not the optimal investment strategy, especially for equity investors. Meanwhile, extreme negative market movement for the traditional asset does not occur synchronously with that for commodity futures; hence, commodity futures do provide good downside protection.

5 v TABLE OF CONTENTS LIST OF TABLES... VII LIST OF FIGURES...IX ACKNOWLEDGMENTS...XI CHAPTER 1 INTRODUCTION AND LITERATURE REVIEW COMMODITY FUTURES: AN ALTERNATIVE ASSET RESEARCH ON COMMODITY FUTURES INVESTMENT STATEMENT OF PURPOSE AND PROSPECTS CHAPTER 2 QUANTITATIVE PORTFOLIO SELECTION MODEL OVERVIEW STATIC PORTFOLIO SELECTION THEORY DYNAMIC PORTFOLIO SELECTION THEORY MARKOV REGIME SWITCHING PORTFOLIO SELECTION MODEL PORTFOLIO ANALYSIS WITH VALUE AT RISK CHAPTER 3 COMMODITY FUTURES INDEX INVESTMENT INVESTING IN COMMODITY FUTURES INDEX STAND-ALONE INVESTMENT PERFORMANCE INVESTMENT PERFORMANCE COMPARED WITH STOCKS AND BONDS COMMODITY FUTURES INVESTMENT AND INFLATION COMMODITY FUTURES INVESTMENT AND BUSINESS CYCLE COMMODITY FUTURES INVESTMENT AND EVENT RISK CONCLUSION CHAPTER 4 STATIC ASSET ALLOCATION WITH REGIME-SWITCHING INTRODUCTION OF REGIME SWITCHING MODEL REGIME SWITCHING IN COMMODITY FUTURES SINGLE REGIME SWITCHING MODEL FOR THREE ASSET CLASSES OPTIMAL PORTFOLIO ALLOCATION WITHOUT RISK-FREE ASSET OPTIMAL PORTFOLIO ALLOCATION WITH RISK-FREE ASSET CONCLUSION CHAPTER 5 DYNAMIC PORTFOLIO SELECTION WITH COMMODITY FUTURES GENERAL REGIME-SWITCHING MODEL PORTFOLIO SELECTION WITH T-BILL MODEL PORTFOLIO SELECTION WITH IPI MODEL

6 vi CHAPTER 6 PORTFOLIO SELECTION WITH THE VALUE AT RISK WHY USE THE VALUE AT RISK TRADITIONAL VAR ANALYSIS PORTFOLIO VAR WITH THE EXTREME VALUE THEORY CONCLUSION CHAPTER 7 CONCLUSIONS AND IMPLICATIONS REFERENCES

7 vii LIST OF TABLES Table 3-1: The component breakdown of GSCI main index Table 3-2: Spot returns, collateral yield, and roll yield of GSCI...69 Table 3-3: Summary statistics of historical total returns on GSCI indexes Table 3-4: Correlation structure of GSCI sub-indexes Table 3-5: Historical total returns on GSCI, Stocks, and Bonds Table 3-6: Correlation of GSCI indexes with stocks and T-bond Table 3-7: Portfolio performance of downside risk protection...71 Table 3-8: GSCI correlation with different inflation components...72 Table 3-9: The NBER business cycle chronology Table 3-10: Average asset returns in four phases...72 Table 3-11: Year returns with events Table 4-1: Hansen s LR tests for regime-switching GSCI indexes Table 4-2: Estimates of regime switching model for commodity futures index Table 4-3: Hamilton s LM test for model misspecification Table 4-4: Regime switching model estimates for GSCI, Stocks and Bonds Table 4-5: Correlation of transition probability among three asset classes Table 4-6: Parameters estimation of simple MRS model for three assets Table 4-7: Sharpe ratio of three asset classes Table 4-8: Correlation among three asset classes Table 4-9: Sharpe ratio comparing under different regimes Table 4-10: Sharpe ratio comparing for optimal portfolio without risk-free asset Table 4-11: Sharpe ratio comparing for simulated one period returns over 10 years (without risk-free asset) Table 4-12: Accumulated end wealth over 10 years simulation without risk-free asset Table 4-13: Average weight on GSCI over 10 years investment simulation without risk-free asset Table 4-14: Optimal risky asset allocation when a risk-free asset existing Table 4-15: Sharpe ratio comparing for optimal portfolio with risk-free asset Table 4-16: Sharpe ratio comparing for simulated one period returns over 10 years (with risk-free asset) Table 4-17: Cumulated wealth at the end of 10 years investment simulation (with risk-free asset) Table 4-18: Average optimal weight on GSCI over 10 years investment simulation (with risk-free asset) Table 5-1: T-bill model parameters estimation Table 5-2: Variance-covariance matrix estimation from T-bill model Table 5-3: Correlation matrix estimation from T-bill model Table 5-4: Parameters estimations of Industrial production index model...158

8 Table 5-5: Variance-covariance matrix estimation for IPI model Table 6-1: Percentage of Reward-to-VaR increase in different regime Table 6-2: Historical simulation result of portfolio VaR decrease Table 6-3: Historical simulation result of portfolio Reward-to-VaR increase Table 6-4: Descriptive statistics of three assets daily returns Table 6-5: Maximum estimation results of left tail on three risky asset returns Table 6-6: Percentage of portfolio EVT VaR decrease Table 6-7: Percentage of portfolio EVT Reward-to-VaR increase viii

9 ix LIST OF FIGURES Figure 2-1: Efficient frontier Figure 3-1: Risks and returns tradeoff for GSCI indexes Figure 3-2: The risks and returns characteristics of GSCI, Stocks, and Bonds Figure 3-3: Time series plot graph of return on GSCI, stocks, and bonds Figure 3-4: Correlation under different investment horizon Figure 3-5: Efficient frontier with GSCI, Stocks, and Bonds Figure 3-6: Downside risk protection Figure 3-7: Standardized year change of GSCI total return index and CPI Figure 3-8: Correlation with inflation Figure 3-9: Correlation with different inflation components Figure 3-10: Average returns of commodity futures, stocks and bonds with business cycle Figure 3-11: Average returns of commodity futures, stocks and bonds under different phase Figure 4-1: Smooth probability of state 1 for commodity future indexes Figure 4-2: Smooth probability of state 1 for three asset classes Figure 4-3: Smooth probability of state 1 for three asset classes MRS Figure 4-4: Mean-variance efficient frontier with three asset classes Figure 4-5: Optimal portfolio weight on three assets Figure 4-6: risk and return comparing (no risk-free asset) Figure 4-7: Mean and standard deviation of simulated one-period return (no risk-free asset) Figure 4-8: Accumulated wealth over 10 years investment simulation (no risk-free asset) Figure 4-9: Optimal weight on GSCI over 10 years investment simulation (no risk-free asset) Figure 4-10: Optimal capital allocation with three risky assets and a risk-free asset Figure 4-11: Optimal weight on 4 asset classes Figure 4-12: Risk and return comparing (with risk-free asset) Figure 4-13: Mean and standard deviation of simulated one-period return (with risk-free asset) Figure 4-14: Accumulate wealth over 10 years investment simulation (with risk-free asset) Figure 4-15: Optimal weight on GSCI over the 10 years investment simulation (with risk-free asset) Figure 5-1: Smoother probability of regime 1 at period t for T-bill model Figure 5-2: Transition probability for T-bill model Figure 5-3: One-period ahead optimal asset allocation (T-bill model) Figure 5-4: Multi-period ahead optimal asset allocation (T-bill model)...166

10 Figure 5-5: Transition probability for IPI model Figure 5-6: Smoother probability of regime 1 at period t for IPI model Figure 5-7: One-period ahead optimal asset allocation (IPI model) Figure 5-8: Multi-period ahead optimal asset allocation (IPI model) Figure 6-1: VaR in regime 1 for different portfolios Figure 6-2: Reward-to-VaR in regime 1 for different portfolios Figure 6-3: VaR in regime 2 for different portfolios Figure 6-4: Reward-to-VaR in regime 2 for different portfolios Figure 6-5: Historical simulation VaR for different portfolios Figure 6-6: Historical simulation Reward-to-VaR for different portfolios Figure 6-7: VaR estimates and the number of exceedances Figure 6-8: the EVT VaR for different portfolios Figure 6-9: the EVT ES for different portfolios Figure 6-10: the EVT Reward-to-VaR for different portfolios x

11 xi ACKNOWLEDGMENTS This dissertation was influenced and improved by advice, knowledge, and assistance of several people. First and foremost, my thanks go to my dissertation advisor Dr. Darren Frechette. His help is invaluable for completing and improving this project. I am grateful to my committee chairman Dr. Spiro Stefanou for his guidance, support, and encouragement. I am proud that his guidance will be foundation of my future scholarly work. I would like to express my deep gratitude to my other committee member: Dr. Stokes, Dr. Huang, and Dr. Dunn for their critical, yet constructive guidance and encouragement. The financial support of the Department of Agricultural Economics and Rural Sociology is acknowledged.

12 1 Chapter 1 INTRODUCTION AND LITERATURE REVIEW Commodities are becoming mainstream in the investment world. There is a growing number of institutional investors increasing their asset allocation to commodity futures to achieve better investment performance. My primary interest in this dissertation lies in studying the return and risk of commodity futures in relation to strategic asset allocation of institutional investors, i.e. the allocation to broad asset classes such as stocks, bonds, and commodity futures. To professional financial investment institutions such as mutual funds, pension funds, and school endowments, the strategic asset allocation is the single most important determinant of the risk and return characteristics of their portfolio. 1.1 Commodity Futures: An Alternative Asset During the past decade, the investment management industry has undergone numerous changes. The great volume of international capital flow makes the equity markets a single global asset class and distinctions between international and domestic stocks are beginning to fade. International equity investments can not provide suitable diversification for a U.S. stock portfolio. Institutional investors are increasingly integrating alternative investment assets into their strategic asset allocation in order to achieve smooth and consistent performance by reducing their overall portfolio risk without sacrificing expected return. Those alternative assets include commodities, hedge funds, managed futures, private equity, and so on. For a long time, commodities were considered as an inappropriate investment category because of their substantial price variation. Recently, a number of studies have confirmed that commodities show a unique risk/return character and can provide

13 2 efficient diversification benefit to traditional portfolio assets compared to other real assets (Irwin and Landa [1987], Arkrim and Hensel [1993], Froot [1995]). In addition, returns for commodities show positive correlation with unexpected inflation and exhibit better performance under high inflation economic regimes (Georgiev [2001] and Gorton and Rouwenhorst [2004]). Therefore, commodities can provide a good inflation hedge for portfolio managers. For many investors, the question is no longer whether commodities are an asset class, but what is the best exposure to this asset class. There are four ways to obtain economic exposure to commodity assets: purchasing the underlying commodity, owning the securities of commodity-based companies, trading commodity futures and options contracts, and buying commodity-linked notes. A commodity futures contract is an agreement to deliver or accept a specified quantity of a commodity at a predetermined price at a designated time within a specified time period. It is the first derivative traded in the world. Although commodity futures account for a relatively small fraction of global futures and options trading at this stage, they have been attracting the attention of some major hedge funds and other institutional investors who sought to diversity their portfolios by investing in nontraditional assets during the past decade. Commodity futures and option are increasing in trading activity, especially metals and agricultural commodities. Georgiev (2001) and Gorton and Rouwenhorst (2004) have shown that direct commodity futures investment is the principal means by which institutional investors can easily obtain exposure to commodity price movements. Commodity futures contracts are traded on exchanges that share the same advantages as stock exchanges: transparent pricing, daily liquidity, clearinghouse security, uniform contract size and terms, and a central marketplace. Moreover, futures contracts do not need physical commodity storage and delivery and can be very flexible to take either a long or short position with just small margin deposits. The advantages of commodity futures make them attractive for the institutional investors who are seeking an easy way to gain exposure to commodities. However, commodities futures prices are subjective to high volatility,

14 3 which leads to one of the big disadvantages of investment in commodities futures, the high investment risk. Price movements of commodity futures contracts are influenced by supply and demand relationships, weather, government, agricultural trade, fiscal, monetary and exchange control programs and policies, national and international political and economic events, and changes in interest rates. Most of these factors are unpredictable and subject to high variation. In addition, governments intervene directly and by regulation in certain commodity markets. Such intervention is often intended to influence prices directly. Meanwhile, trading commodity futures on margin involves a high degree of risk. The liability of the account holder is not limited to the initial investment or the equity in the program account. Two methods can be used to hedge against the substantial risk of investing in an individual futures contract. One way is to invest in a diversified basket of commodity futures contracts. There are now a number of investable passive commodity futures indexes which earn return from holding only long positions in unleveraged physical commodity futures. The other way to hedge risk is through managed futures accounts. Instead of passively holding long positions, managed futures accounts, which are discretionarily controlled by commodity futures fund managers or commodity trade advisors, actively use a technical trading system and professional skills to gain a competitive return. Commodity futures indexes are significantly different from managed futures accounts in several ways. First, commodity futures indexes are designed to provide passive exposure to the commodity futures markets while managed futures accounts are actively traded with a skill-based strategy. Second, commodity futures indexes include only physical commodity futures contracts, but managed futures accounts tend to invest across the spectrum of futures markets, and financial futures contracts actually are typically a large part of the portfolio (Irwin and Yoshimaru 1999). Third, commodity futures indexes only take long positions; in contrast, managed futures accounts may invest both long and short. Finally, commodity futures indexes are unleveraged while

15 4 managed futures accounts tend to apply leverage in the purchase and sale of commodity futures contracts. Empirical studies have concluded that passive commodity futures indexes show not only good stand-alone performance but also substantial portfolio diversification benefits (Abanomey and Mathur [2001], Ankrim and Hensel [1993], Anson [1999], Becker and Finnerty [1994], Georgiev [2001], Gorton and Rouwenhorst [2004], Kaplan and Lummer [1998], Johnson and Jensen [2001], and Jensen, Johnson and Mercer [2002]). Adding commodity futures to a portfolio of stocks and bonds has the ability to reduce the risk of the portfolio for a given level of return. However, prior empirical research regarding managed futures is unsettled (Edwards and Ma [1988], Edwards and Park [1996], Elton, Gruber, and Rentzler [1987], Irwin, Krukemyer, and Zulauf [1993], Irwin and Brorsen [1985], Irwin, Zulauf, and Ward [1994], Schneeweis, Savanayanna, and McCarthy [1991], Schneeweis, Spurgin, and McCarthy [1996]). There is no evidence that public commodity pools provide any benefits either as a stand-alone investment or as part of a diversified portfolio, although some studies have indicated that private commodity pools and commodity trade advisors managed accounts can be valuable additions to a diversified portfolio. In addition, the performance of managed futures industry is inconsistent and unpredictable, which makes selecting properly managed futures accounts impossible based on their historical performance. This dissertation discusses portfolio performance of passive commodity futures indexes instead of managed futures based on the following considerations. First, commodity futures index consist only of physical commodities futures contracts, while managed futures invest largely in finance futures (Irwin and Yoshimaru 1999). Finance futures price movement is decided by the underlying assets such as stocks, bonds, and interests rate. Adding financial futures into portfolio is a hedging issue instead of a diversification issue. Second, managed futures, like hedge-funds, are actively trading funds using skill-based strategy. Their performance is mostly decided by their managers experience, trading skill, and inherent luck (Hartzmark 1991). This

16 5 dissertation plans to examine the diversification benefits achieved from the passive addition of a new asset class such as commodity futures. Third, the data for commodity futures indexes are much longer and more easily obtained. Most managed futures accounts are not public, and their performance data are not available. 1.2 Research on Commodity Futures Investment Greer (1978) studied the conservative use of commodity futures and found that diversified, buy-and-hold, and unleveraged commodities were not as risky as stocks. An unleveraged commodities index can serve as an inflation hedge for a stock portfolio. He demonstrated that a combination of a commodity futures index and a stocks index provided better performance with a high mean return for a given risk level. The optimal relative position between these two asset classes was decided by the predicted inflation rate. Bodie (1983) examined how a diverse basket of commodity futures contracts can supplement a portfolio of common stocks, bonds, and bills to improve the risk and return trade-off based on the mean-variance optimization framework. Using historical returns data from 1953 to 1981, he found that a broad-based position in commodity futures tended to perform well when there was unanticipated inflation and provided a substantial diversification benefit. Irwin and Landa (1987) made use of the mean-variance optimization method to compare the portfolio diversification effects of three alternative investment assets: real estate, futures, and gold. They derived the optimal mean-variance portfolio allocation for different risk levels and found that real estate should be held in substantial proportions at lower portfolio risk levels. Both buy-and-hold commodity futures and futures funds were held at lower risk levels. But buy-and-hold commodity futures holdings diminished rapidly as portfolio risk increased, while futures funds holdings increased steadily. Gold was not held in any efficient portfolios.

17 6 Arkrim and Hensel (1993) studied the inflation hedging effect and portfolio improvement role of fully collateralized commodities futures using Goldman Sachs Commodity Index (GSCI) and Investable Commodity Index (ICI) as proxy. Based on a long term monthly return series from 1972 to 1990, commodity spot returns were correlated positively with unexpected inflation, which negatively impact returns of traditional financial assets. A mean-variance optimizer was used to calculate the efficient portfolios for two risk-aversion levels and the optimization results showed that adding commodity futures can generally reduce portfolio volatility without weakening returns. They also examined the case for commodity futures using the reverse optimization method which derives the required return for commodity futures to be included at the efficient portfolio. The implied expected return necessary to hold 2.5% in commodity futures is only 12 basis points over the risk-free asset, which is much lower than average returns of commodity futures. Becker and Finnerty (1994) examined the risk and return properties of equity/bond portfolios before and after the inclusion of a diversified portfolio of long commodity futures. Inclusion of commodities (using Commodity Research Bureau Index (CRB) and GSCI as proxy) improved portfolio performance by enhancing the risk and return characteristics. The improvement of the risk/return characteristic is more substantial during the high inflation periods such as the decade of the 1970s than during the low inflation period such as the decade of the 1980s. This result confirms that commodity futures can serve as an inflation hedge. Froot (1995) compared three classes of real assets: real estate, commodity futures, and the stocks of companies that are commodity producers. He found that commodity futures were a more effective hedging tool against unexpected inflation than real estate and the stock of commodity producing companies. Commodity futures rendered the other real assets ineffective when they were the initial hedge in a portfolio. However, when they were added to the portfolio as the secondary hedge after other real assets had already been added, commodity futures still received a significant portfolio

18 7 weight as a diversifier. The same conclusion did not hold for real estate and commodity-based equity. Satyanarayan and Varangis (1996) examined the diversification benefits of commodity futures in a global portfolio context. They found that commodity futures returns were correlated negatively with the returns to all developed markets and with three of six emerging markets. The reason is that emerging markets tend to be net suppliers of commodity inputs. Kaplan and Lummer (1998) found that, in the long run, fully collateralized long positions in GSCI futures contract not only provided good diversification for stock and bond portfolio but also hedged against inflation risk; however, their portfolio benefits could not be achieved over short periods of time. Over the long run, commodity investment was correlated negatively with stocks and bonds investments and had better performance during a rising inflation period than during a falling inflation period. But over a short period such as March 1992 to February 1997, commodity futures showed positive correlation with traditional assets. This result is in sharp contrast with preceding empirical studies and leads to more careful thinking on commodity futures investment. Anson (1999) examined the portfolio diversification role of four commodity futures indexes: GSCI, Chase Physical Commodity Index (CPCI), ICI and JPMorgan Commodity Index (JPMCI). Using quarterly returns from 1947 to 1997, he maximized mean-variance utility over four asset classes: large-capitalization stocks (S&P500), small-capitalization stocks (NASDAQ), long-term bonds, and commodity futures. The results showed that how the addition of commodity futures to a diversified portfolio was determined by the investor s level of relative risk aversion. Despite the popular perception that commodity futures are too risky for the typical investment, he demonstrated that the more risk averse the investor is, the higher his utility will be by investing in commodity futures. Gibson (1999) examined the rewards of multiple-asset-class investing in a

19 8 broader equity context. He found that a commodity futures index (GSCI) investment has a pattern of returns that is the most dissimilar to the other asset classes (stocks, bonds, and real estate) and accordingly produces the strongest diversification effect when combined with other asset classes. Georgiev (2001) investigated the relative risk and return advantages of direct commodity futures investment using GSCI as a proxy. Investment in GSCI was shown to result in a significant diversification benefit based on monthly returns data for a sample period from January 1990 through December This benefit stemmed from the unique exposure of commodities to market forces such as unexpected inflation as well as the potential of positive roll return in periods of high spot price volatility. This diversification benefit was beyond that achievable from commodity-based stock and bond investment (indirect investment on commodities). Using monthly returns of GSCI from 1973 to 1997, Jensen, Johnson and Mercer (2000, 2002) examined the investment performance of commodity futures as a stand-alone investment and as a portfolio component. In the overall sample period, the stand-alone performance was poor for commodity futures since they had lower returns and higher risk. In the portfolio context, the diversification benefit was significant for commodity futures and mean-variance optimization yielded substantial capital allocation to them. By dichotomizing the sample period into expansive-versus-restrictive monetary environments using the most recent changes in the Federal Reserve discount rate as the classifying criterion, the authors studied the investment performance of commodity futures under different economic conditions. The results were dramatically different in the two economic conditions. During restrictive monetary period, commodity futures tended to display strong return/risk performance as a stand-alone asset and significantly enhanced the portfolio performance. In contrast, during the expansive period, the return/risk performance was very poor and no portfolio improvement was founded. The authors also hypothesized that the sensitivity to Federal Reserve monetary policy is likely to vary across commodities and expanded their investigation by examining the

20 9 performance of five GSCI sub-indexes. They found that the diversification benefit varied across commodities with metals, energy, and agricultural futures providing quite high returns during the restrictive period and livestock contracts performing much better during the expansive period. Nijman and Swinkels (2003) showed that commodity futures investments helped pension funds in a mean-variance framework by shifting up the efficient frontier. They tested the statistical significance of the shift for multiple investment horizons using the regression analysis developed by Huberman & Kandel (1987). They concluded that commodity investment could significantly reduce the risk of real pension funds, and the timing strategies between commodities and stocks can improve the strategic mean-variance frontier even further. The timing strategies were based on four macro economic variables: the yield on 10-year government bonds, the term spread, the default spread, and the inflation rate. Since there is some evidence that expected returns and covariances change with current economic condition, the authors examined both unconditional and conditional spanning of commodities by the traditional asset classes. The results show that commodities can improve the mean-variance level both conditionally and unconditionally for real pensions. Gorton and Rouwenhorst (2004) studied return/risk properties of commodity futures as an asset class by constructing an equally-weighted and fully-collateralized index. Over a long-term period from July 1959 to March 2004, commodity futures have offered the same average return and Sharpe ratio as equities and showed negative correlation with equities and bonds but positive correlation with inflation, unexpected inflation and changes in expected inflation. Those results strongly support that commodity futures should enter the investment mainstream not only as a stand-alone asset class but also as a portfolio component. In addition, the authors demonstrated that the negative correlation between commodity futures and the other asset classes was largely due to different behavior over the business cycle.

21 10 In conclusion, prior researchers present ample evidence about the benefits of commodity futures investment. Since commodity futures are negatively or non-significantly correlated with traditional assets, such as stocks and bonds, in according with Markowitz s mean-variance theory, adding commodity futures into the portfolio should improve the portfolio performance by reducing the total risk without sacrificing expected return. In addition, commodity futures returns exhibit negative correlation with inflation rates, which tend to reduce stock and bond returns. Commodity futures can serve as a good inflation hedging tool for portfolio construction. However, the portfolio performance improvement role of commodity futures is not consistent at all times. The risk/return property of commodity futures will vary with the underlying economic condition. To study this variation is the primary object of this dissertation. 1.3 Statement of Purpose and Prospects The main purpose of this dissertation is to examine the portfolio performance of commodity futures indexes under a dynamic regime switching economic environment. It is hypothesized that the risk/return characteristics of commodity futures and financial assets will be different under distinct economic regimes and moreover the economic regimes are related to each other rather than totally independent. When portfolio managers make a decision about asset allocations based on time-varying expected returns and variations, they should consider current economic conditions and also take the regime switching probability effects into account. This dissertation will make at least four unique contributions to the existing literature. First, instead of assuming that the distribution of asset return is identical independent distributed (i.i.d.) normal; this paper applies a regime-switching model to examine the structure break behavior existing in asset return dynamics and investigate the relevance with asset allocation. Second, in addition to studying static portfolio selection for a single period, this paper examines the multi-period portfolio allocation problem with commodity futures under different economic conditions. Third, the Value

22 11 at Risk (VaR), a new risk measure is used to examine the risk reduction effect of commodity futures. Traditional approaches use the variance of return distributions as the risk measure. Finally, while traditional analyses examine the diversification benefits of commodity futures under normal market conditions, this work extends the literature by using the Extreme Value Theory (EVT) to examine the extreme market movements and event risk for a portfolio including commodity futures and traditional assets. Prior literature on the strategic benefits of commodity futures adopts the mean-variance optimization framework and measures asset returns performance using historical sample averages. The mean-variance model was developed by Markowitz (1952) and Tobin (1958). The optimal solution of portfolio weights in their model is a function on the first and second moments of the asset return distributions. According to this model, making a portfolio will significantly reduce the total risk without sacrificing expected return when the correlation among individual assets is negative or closes to zero, which is the case for a portfolio of commodity futures and traditional assets. Although the mean-variance model provides an elegant mathematical optimization method to decide the efficient portfolio allocation, there are three important shortcomings of this theory when applied to portfolio analysis. First, the mean-variance model can only handle the static portfolio allocation problem and the optimization procedure is based on historical average returns. The key assumptions of this model are that asset returns are normally distributed and investment risk is measured by the variance. Time-varying aspects of the assets performance are not considered in this model. Those aspects include skewness, kurtosis, serial correlation, and time-varying means and variances, all of which can only be examined using a dynamic model. Samuelson (1969) and Merton (1969, 1971) initiated multi-period portfolio optimization study using stochastic dynamic control theory. Their continuous time models show that if the investor has constant relative risk aversion and returns are

23 12 identical independent distributed, the optimal long-term portfolio is the myopic portfolio, i.e. the portfolio optimal for the investor with a one period investment horizon. If returns are not identical independent distributed, this is in general no longer true and the optimal portfolio for the long-term investor that is allowed to rebalance will differ from the myopic portfolio. This difference is called intertemporal hedging. This dissertation introduces Ang and Bekaert s regime-switching portfolio selection model to examine portfolio improvement role of commodity futures in both one-period and multi-period context. In the one-period context, a simple (without instrument) regime switching model is applied to model asset return dynamics. The asset risk and return are distinguished in different regimes and the regime variable follows a first-order Markov chain. For the one-period ahead optimal portfolio allocation, the optimal asset weights derived from a mean-variance optimization are decided by the regime expectation in the next period. In the multi-period context, asset return dynamics are decided by a regime variable as well as an economic instrument. A dynamic programming approach is adopted to estimate the optimal portfolio weights at each investment horizon and under different economic conditions. Markov regime-switching model is adopted to capture the investment performance of risky assets and the solution of optimal weights is based on this special data generation process. The expected returns and variation of risky assets vary through time and exhibit regime switching characteristics, so the investment opportunity set is not constant over the investment horizon and the optimal capital allocation to commodity futures should be dependent on the underlying regime. A second critical limitation to be addressed is that the mean-variance model measures investment risk based on the variation of the return distribution. A number of studies (see Balzer [1994] for an overview) have shown that the variance is only a suitable measure of risk when the expected returns are normally distributed. Higher moments should be considered when the distribution of asset returns is non-normal. In addition, investors are mainly concerned about downside risk caused by adverse market

24 13 movement, instead of general volatility. Value at Risk (VaR), measuring the left (right) tail risk for holding a long (short) market position at a given confidence level, can be obtained for general distribution and is consistent with investors intuition. While VaR has become a popular risk measurement tool in the investment world, no literature appears to examine the risk reduction performance of commodity futures based on the VaR risk measure. This study will fill in this gap. Empirical evidence shows that the returns of commodity futures and financial assets are not normally distributed and exhibit unusual levels of skewness and kurtosis. (For example: Deaton and Laroque [1992], Yang and Brorsen [1992], Myers [994], Vercammen [1995], Hilliard and Reis [1999], Wei and Leuthold [2000], Roberts [2001], Kraus and Litzenberger [1976], Duffie and Pan [1997], Timmermann [2000]) As a consequence, portfolio analysis based solely on mean and variance may be leading to wrong conclusions and decisions. Amin and Kat (2003), Bacmann and Scholz (2003) and Bacmann and Pache (2003) examined hedge funds diversification effect and showed that while hedge funds combine well with stocks and bonds in the mean-variance framework, this is no longer the case when skewness is considered. Alexander and Baptista (2002) developed a mean-var model for portfolio selection instead of the traditional mean-variance model and concluded that the optimal portfolio solution with a mean-var model converges to the solution of a mean-variance model only when the asset returns distribution is normal and the confidence level for the VaR measure is very high. In 2003, they proposed a new investment performance measure called reward-to-var ratio as a complement to the traditional Sharpe ratio. The reward-to-var measure ranks portfolio performance differently from the Sharpe ratio under non-normality. All of above discussed studies encourage us to examine the risk reduction effects of commodity futures using VaR as risk measurement. The third important weakness of the mean-variance model is that optimal portfolio allocations are based on historical average performance of asset returns. Event risk and market extreme movement are not considered. It is well known that financial

25 14 markets are subject to extreme variations, mostly because of financial turmoil, large credit defaults, war, nature disaster, and political crisis. These market extreme movements will result in substantial negative returns. Fund managers monitoring their portfolio risk only based on historical average information might suffer a serous loss when the market moves extremely in an adverse direction. A question is naturally raised: while commodity futures show wonderful diversification effects under normal market conditions, can they serve to hedge against event risk for institutional investors to achieve smooth performance? Previous studies have showed that commodities, in contrast with traditional financial assets, tend to have positive exposure to event risk (Deaton and Laroque [1992], Georgiev [2001], Anson [2002]). No statistical models have been designed to formally test this issue. This dissertation fills in this gap by taking advantage of the Extreme Value Theory (EVT) to describe the investment risk associated with extreme events for a portfolio among stocks, bonds and commodity futures. The Extreme Value Theory provides a very powerful tool to analyze the extreme tail distribution of a random variable. Recently, more and more research has applied this tool to analyze the extreme variations in financial markets. For example, Diebold, Schuermann, and Stroughair (1998) provided an interesting discussion about how to use the EVT to measure the VaR of equities investment; Bacmanna and Gawron (2004) examined the fat tail risk in portfolios of hedge funds and traditional investments. This empirical study can provide some statistical evidence about the diversification effect of commodity futures under extreme market conditions. To sum up, the conclusion of past study about the benefit of commodity futures investment is questioned and disputable since it is based on a static model and an improper risk measure. This dissertation provides some complementary studies by extending static analysis to a dynamic context and by assessing portfolio risk with the VaR measure. Markov regime-switching model is used to capture the time-varying investment opportunity for stocks, bonds and commodity futures. The issue of the

26 15 diversification benefits and risk reduction of commodity futures is examined under distinct economic conditions. Moreover, the EVT is adopted to examine the diversification benefit under extreme market movement. This dissertation intends to provide a comprehensive examination on commodity futures investment and supplies more persuasive evidence on its diversification benefit to traditional portfolio.

27 16 Chapter 2 QUANTITATIVE PORTFOLIO SELECTION MODEL 2.1 Overview Asset managers construct a portfolio of assets with many different single assets. In general, there are two ways to construct a portfolio in the investment world: the discretionary approach and the quantitative approach. The discretionary approach uses fundamental analysis or technical (charts) analysis to predict the forward market movement and asset price variation. Decisions concerning the selection of instruments and asset allocation are made based on personal experience and analytical skills. Personal insights play a large part in this portfolio construction process. The quantitative approach uses statistical models, simulation techniques, and optimization processes to discover the best asset selection. This method holds that the portfolio selection problem can be expressed by a mathematic structure based on assumptions, parameters, input variables, and output variables. Statistical models and the prediction of asset price variations play the key roles for this portfolio approach. In the past two decades, the rapid growing of financial markets, especially, the derivative market has complicated investment analysis and more quantitative-oriented strategies. Meanwhile, the development of new technology makes it possible to apply well-defined portfolio theory and advanced statistic model to portfolio construction practice. The inauguration of modern portfolio theory came with a static portfolio model of Markowitz (1952) and Tobin (1958) that discussed how to construct an optimal portfolio based on a mean-variance (M-V) framework. This normative theory was extended by Sharpe (1964) and Lintner (1965) to a general equilibrium model of asset prices, which is the famous Capital Asset Pricing Models (CAPM). Roll and Ross (1976)

28 17 criticized the usefulness of the CAPM and published the arbitrage pricing theory (APT) as an alternative of CAPM. The M-V, CAPM, and APT were all static portfolio theories, which basically discussed the myopic portfolio selection under one period context. In multi-period case when rebalancing is allowed, Samuelson (1969) and Merton (1969, 1971) apply stochastic dynamic control theory to show that if the investor has constant relative risk aversion and returns are i.i.d., the optimal long-term portfolio is the myopic portfolio, i.e. the portfolio optimal for the investor with a one period investment horizon. If returns are not i.i.d., in general, this is no longer true and the optimal portfolio for the long-term investor that is allowed to rebalance will differ from the myopic portfolio. Recent literature that investigates intertemporal hedging includes Brandt (1999) and Ang and Bekaert (1999) who find the hedging demand to be relatively small whereas Brennan, Schwartz and Lagnado (1997), Barberis (2000) and Campbell and Viceira (1999, 2000) find the intertemporal hedging demand to be quite large. Static portfolio theory is not new to agricultural economists. Johnson (1960) discussed the optimal portfolios involving commodity inventory and short positions in futures contracts using mean-variance theory. Agricultural economists have modified the financial portfolio theory to study agricultural market issues such as optimal hedging with futures and options, optimal crop selection, risk premium and so on. Tomek and Peterson (2001) provided a comprehensive review of this field. This section provides an introduction about the recently developed dynamic portfolio methods. 2.2 Static Portfolio Selection Theory The Mean-Variance Model In the 1950s, there was no specific measure for investment risk. Harry Markowitz (1952) derived the expected rate of return for a portfolio of assets and an

29 18 expected risk measure. He derived the formula for computing the variance of a portfolio and showed that variance was a meaningful measure of portfolio risk under a reasonable assumption. The formula for the variance of a portfolio not only indicated the importance of diversifying your investments to reduce the total risk of a portfolio, but also showed how to effectively diversify. Mathematic programming We assume there are N investment assets available and T time periods. Let r it be the one period return of asset i at time period t (i=1,, N: t=1,, T; the return could be the continuously compounded return or the simple return, they are not very different when one period is not long). Let the means and variance of r it be Er ( ) = μ and Varr ( ) = σ respectively, and let the covariance of r it and r jt be 2 it it it it Cov( r, r ) = σ. In general, it jt ijt μ σ σ are dependent on time t. The Markowitz 2 it, it, and ijt theory is a one-period theory with T=1 in which an investor at time 0 is supposed to make an optimal portfolio among the N assets for time T=1 with respect to his objective function based on predicted returns and variances. Hence in the static Markowitz theory, the variability of μ σ σ over time is irrelevant since T=1. Thus we can leave 2 it, it, and ijt out the t at subscript. Let w=(w 1,, w N ) is the portfolio weight vector with denoting the portfolio value ratio of the i-th asset. Then the portfolio return is: R ( w) = w r = w r + L + w r (2.1) p 1 1 N N The mean and variance of portfolio return are respectively given by: μ ( w) = σ N wμ p i i i= 1 N N 2 p( w) = ww i jσ ij i= 1 j= 1 (2.2) Hence if μ σ σ are known, then an investor at time 0 can optimize his 2 it, it, and ijt objective function basing on expected returns and variances: