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HEDGING AGAINST COMMODITY PRICE INFLATION: A SECURITY MARKET APPROACH BY GERALD DOUGLAS GAY A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1980 ACKNOWLEDGEMENTS The writer is indebted to Dr. Richard H. Pettway, chairman of nis supervisory committee, and to Dr. Steven Manaster, Dr. Robert W. Kolb, Dr. David A. Denslow, Jr., and Dr. H. Russell Fogler, members of the committee, for their counsel and assistance in the preparation of this dissertation. The writer is especially obligated to Dr. Steven Manaster for his guidance and many helpful suggestions. TABLE OF CONTENTS PAGE ACKNOWLEDGEMENTS................................................. ii ABSTRACT.... ...... ..................................... iv CHAPTER I. INTRODUCTION... ............... .......................... 1 A. Overview.................. ..................... 1 B. Previous Work in Commodity Pricing.................. 2 C. General Outline..................................... 6 II. A LITERATURE REVIEW OF MULTIPERIOD PRICING MODELS....... 9 A. Long's MultiPeriod Pricing Model................... 10 B. A Test of the Long MultiPeriod CAPM.................... 17 III. METHODOLOGY FOR CREATING "QUASIFUTURES" CONTRACTS....... 20 IV. THE FORMATION OF "QUASIFUTURES" CONTRACTS .............. 28 A. The Data............................. .............. 28 B. Empirical Estimation of the Regression Coefficients.. 38 C. Computation and Analysis of the Portfolio Weights.... 42 V. TESTS OF THE PORTFOLIOS AS HEDGING INSTRUMENTS........... 52 A. Testing Price and Risk Characteristics .............. 52 B. Effect of the Nixon Wage and Price Controls.......... 69 C. Quarterly and Semiannual Rebalancing................. 85 VI. SUMMARY AND CONCLUSION................................. 90 A. A Summary of this Research.......................... 90 B. Suggestions for Further Research.................... 95 APPENDICES A. CREATING A "QUASIFUTURES" CONTRACT: AN EXAMPLE........ 98 B. A GENERAL SOLUTION FOR THE PORTFOLIO WEIGHTS............. 100 BIBLIOGRAPHY.................................................. 102 BIOGRAPHICAL SKETCH.............................................. 105 iii Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy HEDGING AGAINST COMMODITY PRICE INFLATION: A SECURITY MARKET APPROACH By Gerald Douglas Gay June, 1980 Chairman: Richard H. Pettway Major Department: Finance, Insurance, and Real Estate Investors often hedge against price inflation in a particular commodity by taking a position in a futures contract, an action that protects the investor from any price fluctuations in the commodity during the period before delivery. This dissertation develops portfolios com posed of defaultfree bills of various maturities and shares of common stock which allow an investor to hedge against commodity price inflation, without actually entering the futures markets. This hedging alternative can be of use to investors who wish to hedge against price inflation in commodities for which organized futures trading does not exist. This study develops hedges by purchasing existing security market instruments, using a previously suggested technique. These hedge port folios, or "quasifutures" contracts, for a particular commodity are constructed such that they are not only highly correlated with the commodity, but also have the same price and risk properties. The port folios and the commodities are shown to have the same expected price at delivery, and the same covariances with the stock market, all other commodities, and longterm bills. iv The hedging portfolios developed for ten various commodities over the period 19651976 are shown to be unbiased substitutes for the commodities themselves. The techniques explored in this dissertation allow investors to hedge against price inflation in any commodity. Consequently, the existence of these "quasifutures" contracts has important implications for the necessity of futures markets and for the development of a multiperiod capital asset pricing model. If markets are perfect and securities are traded costlessly, then futures markets do not provide an investor with a service that is not already available in the existing markets for common stocks and Treasury bills. Additionally, since these "quasifutures" contracts can be created with existing securities, a pricing model, based on a multiperiod economy which allows for shifts in commodity prices and the term structure of interest rates, may well lead to a more complete and realistic view of capital market equilibrium. Finally, the "quasifutures" contracts are shown to require frequent rebalancing. Thus, futures markets may be economically valuable since they allow investors to hedge more efficiently in terms of trans action costs. CHAPTER I INTRODUCTION I. A Overview Investors often hedge against price inflation in a particular commodity by taking a position in a futures contract, an action that pro 'ects the investor from any price fluctuations in the commodity during the period before delivery. This dissertation investigates whether portfolios composed of defaultfree bills of various maturities and shares of common stock allow an investor to hedge against commodity price inflation, with out actually entering the futures markets. This hedging alternative will be especially appealing to investors who wish to hedge against price inflation in commodities for which organized futures trading does not exist. This study develops hedges by purchasing existing security market instruments, using a technique suggested by Long (1974). These hedge portfolios, or "quasifutures" contracts,1 for a particular commodity will not only be highly correlated with the commodity, but will have all the same price and risk properties. The portfolios and the commodities will have the same expected price at delivery, and the same covariances with the stock market, all other commodities, and longterm bills. These portfolios are not true futures contracts for two reasons; (1) they require a positive net investment, and (2) they involve taking a current position in the markets for common stocks and Treasury bills. However, these portfolios will be called "quasifutures" contracts to be consistent with the terminology of Long. 2 The hedging portfolios developed for ten various commodities over the period 19651976 will be shown to be unbiased substitutes for the commodities themselves. Consequently, their existence has important impli cations for the necessity of futures markets and for the development of a multiperiod capital asset pricing model. If markets are perfect and securities are traded costlessly, then futures markets will not provide an investor with a service that is not already available in the existing markets for common stocks and Treasury bills. Additionally, if these "quasifutures" contracts can be created with existing securities, Long's pricing model, based on a multiperiod economy which allows for shifts in commodity prices and the term structure of interest rates, may well lead to a more complete and realistic view of capital market equilibrium. I. B Previous Work in Commodity Pricing Futures markets integrate the actions of two types of investors who are typically described as being either hedgers or speculators. In order to avoid the risks of price fluctuations in the spot commodity, a hedger will initiate a futures market transaction. Thus an investor who desires a particular commodity for future consumption or production purposes will purchase a claim for future delivery of the specified commodity. This is a riskless claim for consumption of the commodity even though the future spot price of the commodity is highly uncertain. On the other side of the market, the speculators underwrite the risks of price fluctuation in the spot commodity in hopes of receiving some compensation. Keynes (1930) first analyzed futures markets as constituting an insurance mechanism. According to Keynes, hedgers pay a significant premium to the speculators for underwriting the risks of price fluctuation in a commodity. Hardy (1940) has argued to the contrary, that futures markets are a socially acceptable form of gambling whereby speculators may actually be willing to pay for this opportunity to gamble. Thus the premium they receive should be zero or possibly negative. Attempts have been made to analyze the returns to holders of futures contracts for the purpose of interpreting the actual need for futures markets. Dusak (1973) presents a portfolio approach and argues that futures markets are no different in principle from the markets for any other risky portfolio assets. Since all assets are candidates for inclusion in an investor's portfolio, the return on any risky asset should be governed by the asset's contribution to the risk of a large and welldiversified port folio of assets. In her paper Dusak investigates the riskreturn relation ship to holders of futures contracts in a capital asset pricing framework. The purchase of a futures contract is like buying a capital asset on credit since the buyer has no capital of his own invested. The margin paid by an investor is merely a goodfaith deposit to acknowledge a later commitment to the contract and thus cannot be treated as a capital invest ment. Thus, in order to approximate the risk premium earned on the spot commodity, Dusak uses the percentage change in the futures price over a given interval. Her results indicate that the returns and systematic portfolio risk are both close to zero for each of the commodity futures studied (wheat, corn, and soybeans) despite the fact that each commodity had a large price variability during the sample period of May 1952 through November 1967. These findings contradict the Keynesian theory which says an investor in commodity contracts should earn a substantial positive return. In addition, the findings only partially support the Hardy gambling casino theory (which predicts a mean return of zero) since the systematic risk was close to zero. Black (1976) also discusses the behavior of futures prices in a model of capital market equilibrium and states that the returns on commodity holdings should obey the capital asset pricing model like any other asset. He develops an expression which states that the expected change in the futures price is proportional to the "dollar beta" of the futures price. If the change in the futures price is uncorrelated with the return on the market portfolio, thus producing a zero beta for the futures price, then one would expect a zero change in the futures price. Had Black empirically tested his equations, he would have expected the same zero returns to holders of futures contracts that Dusak found, since in Dusak's paper covariances with the stock market were close to zero for wheat, corn, and soybean futures. Black also states that since commodity holdings appear to be priced like other assets, then investors who own commodities should be able to diversify away any unsystematic risk. One way this may be done is through futures markets. However, since the majority of commodity holdings is by corporations, the risk may be passed on to shareholders who should hold welldiversified portfolios. This implies that futures markets do not have a unique role in the allocation of commodity price risk since corpor ations can do a more efficient job, especially in the cases where organized futures markets are nonexistent for many commodities. Finally, on the need for futures markets, Black refers to the infor mation content of futures markets. Black (1976, p. 176) states "I believe that futures markets exist because in some situations they provide an inexpensive way to transfer risk, and because many people both in the business and out like to gamble on commodity prices. Neither of these count as a major benefit to society. The big benefit from futures markets 5 is the side effect: the fact that participants in the futures markets can make production, storage, and processing decisions by looking at the pattern of futures prices even if they don't take positions in that market." Stoll (1978) also maintains that traditional explanations of hedging fail to take proper account of the available risk spreading opportunities in the capital market as a whole as opposed to futures markets alone. His rationale for hedging is the inability or reluctance of individuals such as farmers or privately held firms to trade ownership claims on cer tain assets or production techniques with which they are endowed. Stoll describes two types of risk associated with a commodity: (1) price risk arising from future supply and demand uncertainties, and (2) the risk due to uncertainties in storage costs. Futures markets only allow the price risk to be passed on, but both the price and storage risks could be passed on in the stock market if the process has shares traded. Stoll develops a model of futures prices in a capital market equilibrium framework in which there are nontradeable assets. He relates the expected dollar return on a futures contract to the market price of risk and the risk of the futures contract which is measured by the co variance of the commodity return with the return of all other assets. However, even if this systematic market risk is zero as Dusak found, the expected return on futures contracts may not be zero depending on the size of the market value of commodities relative to the market value of all shares of stock. He concludes that in a world of perfect capital markets there is no need for hedging if futures contracts and shares in the production or storage process are traded at no cost. It appears there is beginning to be some success in developing a pricing framework to analyze futures markets. However, there is a lack of any empirical testing of the models in the literature, and even less to substantiate the different authors' explanations for the actual need of futures markets. In their papers, neither Black nor Stoll empirically tests his model. Moreover, Stoll's pricing model for futures contracts, to be properly tested, requires pricing information on nontradeable assets. Dusak did construct and perform some tests, but limited her investigation to three commodities. Correct development of an equilibrium pricing model for futures contracts and subsequent empirical analysis of the returns to holders of futures contracts would not be adequate criteria for justifying the need for futures markets as other authors have claimed. The necessity of futures markets depends on the contribution of futures markets to the completeness of markets. That is, are the services and investment oppor tunities provided by futures markets unique in that they are not available elsewhere? I. C General Outline This dissertation investigates whether there are assets in existing stock and bill markets which can be combined in such a way that the resulting portfolio will have the same price and risk characteristics as does a particular commodity. A stockbill portfolio of this type which serves as a substitute to holding a good itself will be referred to as a "quasi futures" contract. These "quasifutures" contracts for a particular commodity should have (1) the same "withdividend" value at delivery as the spot commodity and (2) the same covariance with the stock market, all other goods, and all longterm bills as does the gooa itself. The concept of a "quasifutures" contract was first presented by Long (1974) in his development of a multiperiod capital asset pricing model. The next chapter presents Long's theoretical discussion for the development of his model from which the concept of a "quasifutures" contract is taken. The economy in Long's model is very realistic in that there is a stock market, a market for defaultfree bills of different maturities, and many consumption goods whose future prices are uncertain. Long relates the price of an asset to not only the systematic market risk, but also to the risk due to changing consumption opportunities (inflation risk) and changing investment opportunities (interest rate risk). Chapter III presents the methodology for creating "quasifutures" contracts. The steps needed to attain the correct expected price and covariance properties of the hedge portfolios are described in detail. Chapter IV describes the actual formation of "quasifutures" con tracts for ten different commodities using asset data from the period January 1961 through March 1976. Selection of the various commodity, bill, and stock data to be used, as well as the formation of a market index, will be discussed. The econometric difficulties encountered in creating the asset weights for the hedging porfolios are also presented. Finally, the composition of the "quasifutures" contracts is examined with regards to the relative weights of the various stocks and bills con tained in each period's hedge. These findings are then compared with those of Fama and Schwert (1977a), who investigate the success of various assets as hedging devices against the different components of the in flation rate. Chapter V outlines the various tests that are conducted in order to verify that the "quasifutures" contracts constructed are indeed true substitutes for the goods themselves. Hypotheses are tested to determine whether each hedging portfolio and good had the same subsequently observed prices and covariances with the stock market, all other goods, and all longterm bills. Interpretation of these results and the implications for the necessity of futures markets is discussed. In addition, the sensi tivity of the results to the Nixon wage and price controls is tested. Since the "quasifutures" contracts are one month hedges which require extensive and costly rebalancing, the tests of hypotheses are repeated for the cases where the portfolios are rebalanced only at quarterly and semiannual intervals. The results from these tests indicate that futures markets provide a less expensive means for hedging against commodity price inflation versus the use of "quasifutures" contracts. The final chapter presents the conclusions of the dissertation and draws the practical implications of the research for hedging. In addition, potential extensions of the dissertation are discussed. CHAPTER II A LITERATURE REVIEW OF MULTIPERIOD PRICING MODELS The capital asset pricing model as developed by Sharpe (1964) and Lintner (1965a,b) abstracts from reality by assuming consumers act to maximize the expected value of a utility function whose only arguments are consumption at time 0 and nominal wealth at time 1. To make this assumption consistent with consumer maximization of the expected utility of a lifetime consumption stream,it becomes necessary to assume future prices of consumption goods are known with certainty and no unpredictable changes occur in the investment opportunities over time. Uncertainty in the rate of commodity price inflation is thereby eliminated. In addition, the capital market cannot contain assets whose future rates of return over a time interval may depend on unanticipated events in the interim. Thus, an economy with a bill market is not allowed. However, commodity price inflation and the term structure of interest rates may play an important role in the determination of equilibrium asset prices. Roll (1973) recognizes that inflation of commodity prices has been almost completely neglected in the literature of asset pricing. He develops a capital asset pricing model that includes the risk of currency inflation and attempts to show how assets and commodities acquire equilibrium prices in competitive markets. Roll points out that the simple Fisherian equation will not correctly specify the relation between nominal interest rates and expected rates of commodity price inflation. His analysis indicates that the nominal expected rate of return on assets will depend on the covariance between nominal asset returns and the rate of inflation. Roll's economy, however, has only one future date and thus the term structure issue is ignored. Merton (1972) presents a continuous time analysis of the demand for assets in an economy where both consumption good prices and the investment opportunity set are allowed to vary randomly over time. He derives a formula that expresses the consumer's demand for risky assets in term of parameters describing his consumption preferences and parameters of the stochastic process governing commodity and asset prices. Like Roll, however, Merton's paper fails to provide testable propositions in the form of precise relations between well defined and readily measurable variables. II. A Long's MultiPeriod Pricing Model Long (1974) provides a very realistic view of capital market equilibrium, by presenting a multiperiod discrete time analysis which deals directly with uncertainty in future commodity prices and future investment opportunities from which he develops empirically testable price formulas for common stock and longterm bills. These formulas are developed such that the relation between equilibrium prices and parameters describing consumer preferences can be directly analyzed. The price of an individual asset can be interpreted in terms of its marginal contribu tions to portfolio characteristics that concern investors such as mean, variance, and covariance. The economy in Long's model contains three markets: a stock market where the shares of N firms are traded, a consumption goods market containing K nonstorable commodities available to consumers, and a market containing defaultfree bills for any maturity date up to and including time T. For each of these markets, the following assumptions are made: (1) markets are perfect in the sense that all items are infinitely divisible, there are no transaction costs, and all traders acts as price takers; (2) on any date all traders have free and equal access to all information which is relevant to assessing the subjective joint proba bility distributions of prices which prevail on subsequent trading dates and, furthermore, all consumers form identical expectations regarding prices to be realized on subsequent dates; (3) markets are only open on trading dates which are equally spaced in time; (4) all production in the economy is accomplished by firms; and (5) the only inputs to produc tion supplied directly by consumers are capital funds supplied by the purchase of the firm's shares and bills. Next, a brief development of the pricing formulas is presented. The notation that will be used is given as follows: 1kt = the price per unit of good k at time t; Cikt = the quantity of good k purchased by consumer i at time t; P, = the exdividend price per share of stock in firm j at time t; Djt = the dividend per share paid at time t to shareholders in firm j; Vjt = the "withdividend" price per share of stock in firm j at time t; X.j = the number of shares in firm j that consumer i chooses at time t to hold during the (t+l)st period; 3tm = the price at time t of a bill which matures and pays one dollar at time m, (O Yitm = the number of bills maturing at time m that consumer i chooses at time t to hold during the (t+l)st period. A vector of prices or quantities will be referred to by omitting the subscript which indexes the elements of the vector. For example, the bundle of goods consumed at time t by consumer i is denoted by Cit = (Cilt, Ci2t, ..., CiKt)' Consumer i is assumed to act at time 0 as if he is maximizing the expected value of a utility function of the form U (Ci0, Ci, ..., CiT) where U is monotonically increasing, strictly concave, continuous, and twice differentiable with respect to its argument. The maximization is done subject to a set of budget constraints which will be given later. Fama (1970b) shows how this problem can be recharacterized as one in which consumer i acts at time 0 as if he is maximizing the expected value of a semiindirect utility function of the form Fi(CiO, wil' 1)' F is defined to be the maximum attainable value of E(Ui) given that Ci0 is the bundle of consumption goods chosen at time 0, wil is the realized value of time 1 nominal wealth, and that I1 is the realized value of the vector of time 1 prices and dividends I (n, P1, D1, B1) The variables wil and jl serve to summarize the opportunity set the consumer will face at time 1. However, it is not necessary to have all of the data specified by wil and 1l in addition to those items known to the consumer at time 0 in order to fully specify the opportunity set seen at time 1. Empirical evidence (see Fama, 1970a) suggests that observed stock prices and dividends do not convey new information about the distribution of rates of return to be earned over subsequent periods that is not already provided by observation of good and bill prices alone. P1 and D1 may then be eliminated as arguments of the function Fi and, therefore, we are left to maximize E [Fi (C 10 l' wi B1)] with respect to (Ci0, Xio, Y i). 13 In order to simplify the problem further, the consumer's subjective joint probability distribution on nl, B1, and w1 is assumed to be multi variate normal at time 0. The expected value of Fi then becomes a function of Cio and the parameters which identify the particular multivariate normal distribution of (wil, nl' 31). The only parameters of the distribution that are affected by the consumer's portfolio choice are 1) the expected value of wil, 2) the variance of wil, 3) the covariance of wil with each. of the time 1 consumption good prices {nkl' k=l, ..., K}, and 4) the covariance of wil with each of the time 1 bill prices {Blm, m=2, ..., T}. Thus there exists a function Gi that can now be used in place of the expected value of Fi as the objective function such that Gi(Cio ei' vi Hi, Ji) = E[F1(Ci0 Wil, n B1)] where ei E(wil); vi = var(wil); H= (Hil, ..., H, ... HiK)' with Hik = cov(wl, kl); and im= (0 .' with J o i i2' im' iT) with Jim = cov(il Blm) Consumer i's current consumptioninvestment decision problem at time 0 will then be to maximize Gi subject to the constraints: K N T C C kO kO + X + Yi0m Om io; k=l j=1 m=l CikO > 0, k = 1, ., K; iN T 11 E ojl V iO m:2 iOm Blm ii i ovil + Yi0i + M i0m lm j=l m=2 By solving the above maximization problem it can be shown that at equilibrium the resulting pricing equation for stocks and bonds is1 P =B [V + 2v ev M + Vb)+ eH + \0o (II1) 0 01 1 ev(aM +eH + eJ e1 B = B [B + 2v e (S + b) + YeH + e] (II2) 0 01 1 ev i + B eH e where v = the number of consumers, e = a weighted average of individual consumers' marginal rates of substitution (MRSs) of expected nominal wealth for variance in nominal wealth, GeH = a vector of weighted averages of individual consumers' MRSs of expected nominal wealth for covariance between nominal wealth and the price at time 1 of each good k, eeJ = a vector of weighted averages of individual consumers' MRSs of expected nominal wealth for covariance between nominal wealth and the price at time 1 of each bill m, b = a vector of the supplies of longterm bills, and m', 4 ', S, ', and z are vectors of covariances whose elements are listed in Figure III1 on page 21. See Long, Appendix A, for the derivation of the firstorder conditions and the resulting price formulas. It is possible to give economic interpretations to each of the above terms. B01, the price of a riskfree one period bill, equals the MRS of current nominal wealth (Wi0) for expected nominal wealth at time 1 (ei). V1 is the marginal contribution per share of stock to the consumer's expected nominal wealth at time 1. The term 2vle ,( + Vb) ev m measures the "expected wealth equivalent" to the average consumer of the marginal nominal risk of a share of stock. The term eOeH measures the "expected wealth equivalent" of the marginal contribution per share of stock to covariance between consumer's time 1 wealth (wil) and the price of each consumption good k, (Ikl). Similarly, 'eJ measures the "expected wealth equivalent" of the marginal contribution per share of stock to covariance between wil and the price at time 1 of bills maturing at time m, (Bm). The price formula for long term bills (112) is interpreted in the same way as the stock price formula (II1) with the words "bills maturing at time m" substituted for "share of stock." Using historic data, estimations can be formed of the expected values and covariances in equations (II1) and (112). To solve for the unobservable e's, {eev, eHk' 0eJm; k = 1, ..., K; m = 2, ..., T} in the formulae, there is a restriction for a nontrivial solution that the number of stocks and longterm bills for which there are pricing formulae must equal or exceed the number of unobservable O's. In other words, it is required tnat N + (T 1) > 1 + K + (T 1), or equivalently, that N > K + 1. If not, a solution will exist for any set of current prices and the model would not have any empirical content. By solving for the unobservable e's, it is possible to derive the following reduced form pricing model2 which not only prices the systematic market risk of a share of stock, but also the inflation risk from changing good prices and the interest rate risk from changes in the term structure: 1 K 1 Pj0 = B1[Vj MVM B 0 MO jk kl 01 FkO T jm (Blm B 01 Bm)], j = 1, ..., N. (113) m=2 m 1 In the above equation VM1 is the mean of the "withdividend" value of the stock market portfolio at time 1 and PMO is its current exdividend price. The assumption that (VM1l, n1' B1) is multivariate normal guarantees the existence of linear regressions within this set of random variables. The symbols 8jM', {k' k = ..., K}, and {6jm, m = 2, ..., T} are, respectively, the coefficients of V M, {1kl, k = 1, ..., K} and {Blm, m = 2, ..., T} in the following multiple regression: K T Vj = tj + j VM + E Ejk kl + EZ jmB + ., j = L, ..., N. k=l m=2 (114) N . In the above regression, VMi1 ` Vjl, the "withdividend" value at time j=1 l 1 of the stock market portfolio. Also, E(ej) = 0 and ej is independent of (VM 1, B1). Finally, FkO is the current exdividend price of any stockbill portfolio whose "withdividend" value at time 1 has a mean equal to Ikl and has the same covariance with the elements of (VM1' I,' B1) as does Hkl* This stockbill portfolio is referred to as a "quasifutures" contract for good k. See Long, Appendix B. II. B A Test of the Long MultiPeriod CAPM Gouldey (1977) presents a test of the Long model. The purpose of Gouldey's study was to test the implications of the Long model, which suggested that investors are concerned with three types of risk when making investment decisions: the traditional systematic market risk, the risk due to a stochastic consumption opportunities set, and the risk due to stochastically changing investment opportunities. Whereas the traditional meanvariance CAPM relates the return on a security to only its systematic market risk, the Long CAPM relates the return on a security to 1) the systematic risk of the security in the stock market, 2) the risk of the security due to expected price changes in each commodity k (k = 1,..., K), and 3) the risk of the security with respect to a bill of maturity m (m = 2, ..., T), due to anticipated shifts in the yield curve. Since there are thus K + T types of risk, each security can be represented by a point in (K + T + 1) space. Thus, a natural generalization of the security market line (SML) associated with the singleperiod CAPM is the security market hyperplane (SMH) on which each security in equilibrium must lie. The SMH will intersect the expected return axis at the riskless rate just as the SML does in the twodimensional case. Gouldey first tests the Long model using crosssectional methods across individual securities over the period January 1953 through July 1971. He selected 1953 as a starting date since earlier price indices compiled by the Bureau of Labor Statistics (BLS) are inaccurate and misleading due to poor sampling techniques. Also, before 1951, the Fed pegged the interest rates on Treasury bills, thus not allowing Treasury bill rates to adjust to anticipated variation in inflation rates. The tests ended in July 1971 since wage and price controls were imposed in August 1971 which caused the various price indices not to reflect the true cost of consumption goods. Gouldey's data consisted of monthly returns on common stock taken from the monthly returns file of the Center for Research in Security Prices (CRSP) at the University of Chicago. The Standard and Poors Combined Index was used to compute a proxy for the return on the market portfolio. Price indices of the different groupings used by the Bureau of Labor Statistics in compiling the Consumer Price Index (CPI) were used for computing monthly returns for various consumption goods. The consumption bundles selected were food, housing, apparel and upkeep, health and recreation, transportation, and other goods and services. Finally, one month holding rates of returns for U.S. government debt obligations ranging in maturity from one month to twenty years were used. Results of Gouldey's tests using individual stocks indicate the following: 1) the intercept of the SMH is significantly positive, 2) the estimated premium for market risk is less than predicted by the model, and 3) the risk premiums due to unpredictable commodity price inflation and changing interest rates are as predicted by the model. As Gouldey points out, however, the results are suspect due to nonstationarity of many of the parameters. In addition, the results are not conclusive because of the presence of some specification error and also because the residuals may not be independent and identically distributed due to the presence of an industry factor. Therefore, in order to examine the effects of this possible dependence in the residuals, Gouldey repeats the tests using industry portfolios. When portfolio data are used the risk parameters tend to be more stable than those of individual stocks and much of the risk peculiar to individual stocks is diversified away in portfolios. Results of these tests indicate that the implications of the Long CAPM cannot be rejected using industry portfolios. However, the premium earned for market risk is less than predicted by the model. This may be due to other types of risk associated with omitted commodities or bills that investors consider relevant which would cause the estimated market risk premium to be biased downward. Although the Long model did not fare well in his tests, Gouldey states thatthe evidence does not necessarily refute the model, but instead may cast doubt upon the abilities of investors to correctly predict future inflation and interest rates. Again, however, the results of the tests using portfolio industry data indicate that average rates of returns on capital assets do contain risk premiums for systematic market risk and risk due to stochastic consumption and investment opportunities. Those results imply that in principle investors can use stocks and bills to create a hedge against commodity price inflation and rising interest rates. Even though Long suggests that these "quasifutures" contracts exist in principle, there needs to be further development and testing of these hedge portfolios. Gouldey attempted to obtain estimates of the costs of forming "quasifutures" for food, housing, and transportation price indices, but he failed to provide any information concerning the composition of the contracts nor did he test the success of the portfolios as hedging instruments using subsequently observed data. This disserta tion intensively examines the construction and composition of "quasi futures" contracts for ten consumption goods over the period 19651976. The "quasifutures" contracts are subsequently tested for their adequacy as hedging instruments against commodity price inflation. CHAPTER III METHODOLOGY FOR CREATING "QUASIFUTURES" CONTRACTS This chapter presents a general methodology for the construction of "quasifutures" contracts for various consumption goods. In order that a stockbill portfolio can be properly called a "quasifutures" contract for a particular good k, it must have the following two properties: (1) it should have the same expected "withdividend" value at time 1 as the good, k1, and (2) its covariance with ('l' ~1' $1) should be equal to the covariance of ?kl with (M1, 1' i, 1). The first step will be to select the stocks and longterm bills in such a way that the covariance property is attained. Let C be a [(N+T1)X(K+T)] matrix of the covariances appearing in the equilibrium pricing equations (II1) and (112). The elements of C are given in Figure IIII. Let z be the [(K+T)X(K+T)] variancecovariance matrix of (~M1' 1 1) whose elements are presented in Figure 1112. Finally, let P be a [(N+T1)X(K+T)] matrix whose elements are given in Figure 1113. Note that the first N rows of n contain the coefficients taken from the multiple regression of equation (114) for each of the N stocks. The matrices are C, z, and n are related such that C = nz. (III1) For example, this implies that the first element of C is M (Nxl) C ( x j(TI)xl (T1)xK (T1)x(T1) (N+T1)x(K+T) (N+TI)x(K+T) COI(V ,v ) vcov , I Oi V11 ,..l S N.,xl1) I:COV(V ? k NxK COV(V .V I ) COV(Vt ) COV(V ; ) i Scov(3, ,V) cOv(B ,i ) covV(,,Ia ) COV(S3.,V I) COV(,83,,11 . ( V:(T1)xl) :COV(B ,, 1 I (Tl)xK CO T,VM) COV( COV ) I V(312,:1) CCV(Vi!, ) COV(V.,B!T) :COV(Vjl,3 1m, Nx(T; COV(V ,3.; COV(CNv,D .5 COV(B12,312 CO 1COV(BV3, 12), C:COV(V I' m B COV(B,,S2) Co 17, ' Figure III1 Elements of the Covariance Matrix C (NxK) S(1 Nx(T ) . ov(v ,r i K (1x1) (1xK) (KxK) Kx(T1) T1)xK (T1)x(T1) COV(VM, Vh ) I I coVAR(V ) COV( 1 COV(E x ,V !a i COV( V I 2' COV(E.3 VMK ( )x (T1 )x1 iCOv(v ) I I .COV(vi ,r.KI viiC (BV ,M12 ) . (lx) (IxK) COV 2T' 1, ) . SCDV(1 :2141.)CO~~:c2 i covr ) COV(B ' I; : COV(B (T )x * OV IT'BV) 17' (1x(T1)) COV( ,'k ,l)1 COV(E,12'.: )COV(B. COV(BE2,21) I 21 i i COV(l ) COV(E1, ) I I COV(,13, 12) Im'"kl I " K ! W':COV( Kx( :: COV (T1 COV(B T,rK1) COV(BIT,B1) 2 KI 1 3%11 OV( )j, B ' m' r.kl 1,T '"KI i i COV(E,K); (1m' n 1 * )x{71) S COV(BT1,B7T) Figure III2 Elements of the Covariance Matrix E *'i (Kxl) (TI)xl (1x(TI)) OV(BI j ,'V ) icov (B IT'rl i I 23 (Nxl) (NxK) (NxT1) (Tl)xi (TI)xK (T1)x(T.)j S1K * 2K i N i .0I * 0 i &12 022 6N2 0 0 0 o 13 'i 'NT L Nx(T1) 0N3 0 0 . 1 0 1MI i 0 ()xl o 0 Figure I13 Elements of the Augmented Coefficient Matrix n !2 * 'Z2 (NxK) CNi 0 0 ,N2 0 0 . 0 o . 1 (T1)x(1; I 0 (T1)xK 0 . i ^ 0 i K cov (~ = 1Mvar(,, ) + jk cov( Ml k) k=l T + E jmcov(im' 'Ml (1112) m=2 which is equivalent to the covariance of equation (114) with V., If the value of the stock market portfolio is unrelated to the price of goods or bills, then equation (1112) will reduce to the more traditional expression as given in the singleperiod model for the covariance of the value of a stock with the market portfolio. To demonstrate this point let Yi = a + 1Xli + B2X2i + i Our estimate for the vector of regression coefficients, 8, is 1XIx XIX2 1 7, IT 'X X 'X X Y 1 1 1 2 X1 a X2X1 X2X2 X2 , or equivalently  = TX1 2' )1 (X1M 2X)lX1X2(X2'X2)1 X1 'Y I(X2'2)X'X1(X1 2'rX1)1 (X21X2)1+(X2'X2) 1A(X22') X2'Y where A = X2'X1(X1I 2X1)1X'X2; and M2 = IX2(2'X2)2' Now, if 1) X2'X2=0 (a trivial case), or 2) X1'X2=O, which would imply for this paper that the market portfolio (X1) was orthogonal to the vector of good and bill prices (X2), then our estimate of a1 would reduce to S6 = (XI'X1)1x'Y, the traditional measure of the risk of a security relative to the market portfolio. It can be shown that the sum of the first N rows of C is equal to the first row of z by applying the following weighting scheme: 1':.0'] C = [ '11 '] n = [0 0 ... 0] = el (1113) where 1' is a (1XN) vector of ones and 0' is a [1X(T1)] vector of zeros. The above relationship implies the following: N E 1jm (1114) j=l N j Sjk = 0, k=1,...,K, (III5) j=1 N E 6S = 0, m=2,...,T. (III6) j=l The validity of these equations can be demonstrated by summing equation (114) across all N securities and by recalling the definition of N  the value of the market portfolio, VM1 E V.. These conditions, (1114), j=1 J (1115), and (116), must hold during the empirical analysis in the next chapter in order to verify that the estimation procedure used is correct. Let e'l+k be a [1X(K+T)] vector whose (l+k)th element is 1, with all other elements equal to zero. The index k on el+k, k=1,...,K, indicates that e' +k is being associated with the kth consumption good. For example, when k=0, vector el' will refer to the "market" as was previously shown in (1113). The next step is to find the k weighted combinations of the rows of n that will solve for the K row vectors, el+k, k=1,...,K. These weights can be found by solving the following system of equations for bk where bk is a [IX(N+T1)] vector: b' := e' k l+k b'k = e l+k " b'k = e'l+k 1 k=1,...,K. (III7) Each of the K row vectors, b'k can be interpreted as the unit quantities of the respective shares of common stock and longterm bills which correspond to a portfolio whose "withdividend" value at time 1 has a covariance with the vector (VM1, ill B1) equal to the covariance of 1nkl with (VMl, l1, B ). The first N elements of bk refer to the unit quantities of shares of each stock and the last (Tl) elements of b' k refer to the number of each of the longterm bills. Now that the covariance property has been attained, the next step adjusts the expected value of each of the K stockbill portfolios to equal E(nkl), the expected next period's value of the kth good. To set the expected value of the portfolio equal to E(Irkl), a quantity of shortterm bills (m=l) can be added to or subtracted from the portfolio without af fecting its covariance with (V1 1,' Bl). This is possible since the es timated regression coefficients from (114) that are used in solving for bk are independent of the price of a oneperiod bill. Therefore, the quantity of shortterm bills, Yolk' to add to the hedging portfolio is Y0lk = B01 (Ikl bk [P1 Bl]'), k=l,...,K. (III8) A negative value of Y0lk refers to a short position in oneperiod bills or a borrowing of funds at the shortterm bill rate. Thus, a meanadjusted portfolio with a current value of FkO can be created and is referred to as a "quasifutures" contract for good k. A simple example to help clarify the methodology used in solving for the vector of portfolio weights, bk, is presented in Appendix A for an economy containing the shares of two firms, one consumption good, and one longterm bill. It will be the convention during the empirical analysis in this paper to select N, the number of different stocks in the market portfolio, to be equal to one plus the number of consumption goods in the economy. That is, N=K+1. By doing so, there will exist only one "quasifutures" contract per good each period. If N>K+1, then there will exist an infinite number of port folios which could serve as "quasifutures" contracts for a particular good.2 However, of all such portfolios, only that portfolio which had the minimum variance about its next period's "withdividend" expected value would be chosen as the "quasifutures" contract for a particular good. When constructing the hedging portfolios, it is interesting that the number of longterm bills in the economy assumed does not impose any empirical restrictions on the model as does the number of shares and consumption goods assumed. Also, it is shown in Appendix B that the number of shares of each firm held in the hedge portfolio does not depend directly on the regression coefficients, A, associated with the long term bills. However, one can not conclude that bills are unimportant in calculating the share weights. Exclusion of bills in the multiple regression (114) would change the values of B and = and thus would have an effect on the share weights. 2There will exist NK linear independent solutions. Any linear combination of these solutions will also be a solution. CHAPTER IV THE FORMATION OF "QUASIFUTURES" CONTRACTS IV. A The Data In order to select the proper quantities of stocks and longterm bills such that the covariance property of each "quasifutures" contract is attained, the first step is to run the multiple regression of equation (114). The following time series data are required to perform the regression: (1) monthly prices for N=K+1 common stocks, (2) monthly prices for the value of the stock market portfolio, (3) monthly prices for K consumption goods, and (4) monthly prices for T1 defaultfree long term bills. The price data for the above assets were collected for the period January 1961 through March 1976. An explanation for the choice of this time period will be discussed later in this section. Previous studies concerning commodity price inflation (Dusak (1973) and Gouldey (1977)) were somewhat limited in the number of different commodities used. If several goods are used in the analysis, and all the goods were to experience the same relative price changes or inflation rates, then one would expect the same hedging portfolio to be derived for each good except for the amount of shortterm bills needed to adjust the expected value of each good. However, it appears that all goods do not experience the same relative price changes (see Fama and Schwert, 1977b) and thus the composition of each "quasifutures" contract should be different for each good. Therefore, to test the success of "quasifutures" contracts properly, the study should include a wide variety of consumption goods in the analysis. The data concerning the consumption goods to be used presented two main problems. First, to be theoretically consistent with the Long model, all consumption goods should be included in the analysis. Including all these variables as independent variables in the regression (114) may present severe multicollinearity problems. However, failure to include enough of these variables may result in misspecification of the model and missingvariable problems. The second problem concerns finding accurate and available commodity price data. For this purpose prices of seven various groupings containing closely related bundles of consumption goods used by the Bureau of Labor Statistics in compiling the Consumer Price Index (CPI) will be used to proxy for commodity prices. Price indices for these various groupings are readily available. In addition, due to problems to be discussed in some of the CPI components, price indices for three components of the Ulholesale Price Index (WPI) will be used in order to have a larger crosssection of commodity data.1 The various component indices selected to proxy for commodity prices are given in Table IV1. In addition to the reasons previously discussed for not using a single good for the analysis, one would not try to use the overall CPI as a proxy to relect the price changes for a particular good. These reasons are: (1) periodic revisions in the weighting of subcomponents, (2) insen sitivity of the CPI to price changes during periods of low mean rates of inflation due to rounding or truncation to one digit after the decimal, (3) differences in the accuracy of measurement across components, and 1The following discussion concerning the CPI components is taken heavily from Fama and Schwert (1977b). Table IV1 Selected Consumption Bundles Component Source Symbol 1. Food CPI FD 2. Gas and Electricity CPI GE 3. Apparel and Upkeep CPI AU 4. Private Transportation CPI PT 5. Reading and Recreation CPI RR 6. Medical Care CPI MC 7. Household Furnishings CPI HFO 8. Metal and Metal Products WPI MMP 9. Lumber and Wood Products WPI LWP 10. Chemicals and Allied Products WPI CAP (4) problems in defining the overall inflation rate if relative price changes induce substitution effects causing consumers to consume goods in different proportions at different times. To retain the uniqueness of each price series, monthly price data for the selected consumption bundles were used. Fama and Schwert (1979) show that inflation rates of different goods can be broken into a part common to all goods and a part peculiar to each good itself. An example of a part peculiar to a particular good is one that may be due to seasonals. If production for a good is seasonal whereas demand occurs smoothly throughout the year, then seasons in the price of such a good may be observed to offset the cost of storing the output. However, as one goes from using monthlyprices to prices of longer intervals, the variability of the price series of a component becomes more and more like that of the CPI. Thus, the variability of price indices of various goods will become more and more similar as they are measured over longer intervals. Final selection of the consumption bundles used in this study depends in many ways on the construction of the CPI. An overall break down of the major groupings composing the CPI is given in Table IV2. Since the data to be used in the study are on a monthly basis, it will be desirable to have the selected CPI components sampled and priced monthly. In addition, it is desirable for each component chosen to be a fairly homogeneous grouping of goods. Approximately 50 percent of the items and locations used to construct the CPI are sampled every month. The rest of the component prices are collected on a quarterly basis, but on a rotating basis so that there is some monthly revision of prices. How ever, components which are sampled on quarterly intervals will reflect changes in prices which actually occurred during the preceding two months. Therefore, some components at times will be incorrectly calculated as being unchanged. These lags in the updating of prices may introduce autocorrelation in the price series. The Food component which represents 22.4 percent of the CPI is priced monthly in all locations and is probably the cleanest series used in the study. The Gas and Electricity index which is a subcomponent of the Fuel and Utilities index represents about 3 percent of the CPI and is also priced monthly in all locations. However, since most of this component depends on utility rates determined by governmental agencies, the behavior of its price series may be different had the prices been determined in a free market as the other components. Despite this shortcoming, this series was retained in the study due to its importance as an everyday consumption item. to a i k eI C Ln n I u c i a I j i L io co o n > W V, c io L; 4 ea a1(A 3Si QIn V v 0 0 C 0  S6 >n . 0ga c D r I '* > u I c >d i. 3 C >< 1 C C. 00 i = C U0.5. 0 C C OL ci o c 4 I' o a a a 3 c  va c0 ci r r 0 C  n 0 C e i ; O ) x aa in = Q a i c ? oa u=o i ,I = o u 1 3 e a o u o  S su 1 ) 0 0 ,' a ac c o U =i w 0 (v 0 I L, Q I 0 6 w ia c CM E I o 1 3 C I ci2 'V 0 0 0 0 L. C X C 1 0 C S M C C c c sO ci . c i = S. SQ C 0 ( m C W0 C 4.'. C C. ci = ; i 1 r 0n "^ L O C QC O 0e1 c\iomcM < c me c xQ Se t =i t V C 00 Ei 0 0 0 01 0 0 C c O l 2 u a 0 1 ) J \ It seemed desirable to include the Homeownership series in the list of components chosen for the study. However, there are many problems with this series due to its subcomponents such as Mortgage Interest Expenses and Real Estate Taxes. The expense incurred in financing a good is not a factor of its price. In addition, government agencies, and not a free market, determine real estate taxes. A third subcomponent of the Homeownership series is the Home Purchase Price Index. This index is a three month moving average of newly insured FHA housing prices and thus suffers from the problem of builtin lags. Finally, another shelter related index, Rent, was also not included in this study, since this index is based on contract rents for a fixed sample of apartments, where the rent for a given apartment is sampled every six months with all unsampled months assumed to be unchanged in price. In addition, since most rental contracts are negotiated on an annual basis, this component does not truly reflect the actual price changes in the rental market. Household Furnishings (about 7.8% of the CPI) is a fairly homogeneous grouping and was selected for the study since it did not suffer from the shortcomings previously discussed. Apparel and Upkeep which comprises about 10.6% of the CPI is a fairly clean and homogenous grouping and was selected, as was Private Transportation. Private transportation (about 12.6% of the CPI) reflects the cost of new and used cars, and the price of gasoline, motor oil, and auto parts. The Health and Recreation index is a very heterogeneous mix and contains many builtin lags. Two of its subcomponents, Medical Care (5.7%) and Reading and Recreation (5.9%) are much cleaner indices and were thus selected. Due to the lack of correctly measured CPI components, three price indices from the Wholesale Price Index (WPI) were selected to enlarge the sample of goods used. The three selected series are Metal and Metal Products, Lumber and Wood Products, and Chemicals and Allied Products. Despite the fact that these series are not truly consumer prices, they are homogeneous groupings and appear to accurately reflect price movements in the particular goods. In summary, monthly price data for the period January 1961 through arch 1976 weregathered for K=lO consumption bundles taken from the Consumer Price Index and the Wholesale Price Index to be used as proxies for various commodity prices. One reason for selecting January 1961 as a starting point for the study was due to an important change in the way the CPI was constructed beginning in 1964. At this time the coverage of the index was extended and the weights assigned to the different items are updated in line with the 19601961 Consumer Expenditure Survey. However, it was still possible to gather revised data for the selected price series back to January 1961 according to the 1964 updating scheme. One final point related to the commodity price indices concerns the Nixon wage and price controls. Beginning in August 1971, the Nixon administration instituted price controls which were not lifted until 19731974. Since the controls led to queues and shortages, reported inflation rates probably understated true changes in purchasing costs to consumers. After the controls were lifted, the reported inflation rates probably overstated the true inflation rates. Tests will be performed in Section V. B to see if the controls had any significant effect on the success of the "quasifutures" contracts as hedging devices. Based on the decision to use K=10 commodities in the study, monthly price data on N=ll (K+1) common stocks are required. This empirical restriction that the number of stocks to be used equals one plus the number of goods was discussed in Section II. A. Recalling that if N>K+1, then there will exist an infinite number of hedging portfolios which have all the properties of a "quasifutures" contract for a particular good. However, of all such portfolios, only that portfolio which had the minimum variance about its next period's "withdividend" expected value would be chosen as the "quasifutures" contract for a particular good. Despite not being the optimal approach for creating a "quasifutures" contract, the problem of searching for a minimum variance portfolio will be avoided by setting N=K+1, whereby, only one solution is produced. As Gouldey (1977) points out, the risk parameters using portfolio data tend to be more stable than those of individual stocks and much of the risk peculiar to individual stocks is diversified away in portfolios. Instead of using individual securities in the analysis, industry portfolios are formed.2 Various industry classifications as provided by the Compustat files will be used to create the industry portfolios. The eleven industry groupings to be used are listed in Table IV3. The industries selected are widely varied and an attempt was made to have all the industries correspond to each of the consumption bundles selected. For example, the "Motor Vehicles and Auto Trucks" portfolio corresponds to the Private Transportation component. The purpose of this is to see if the best hedging strategy against inflation in a particular good consists of buying the stock of a firm in a related industry. 2 It would be desirable to use individual stocks. However, due to the instability of the regression estimates when individual stocks are used, industry portfolios are created. It is realized that industry portfolios may not be fully diversified. Table IV3 Industry Portfolios Name 1. ChemicalsMajor ChemicalsMinor 2. Textile Apparel MGF 3. Forest Products 4. Home Furnishings 5. DrugsEthical DrugsProprietary DrugsMedical Supply 6. SteelMajor SteelMinor 7. Motor Vehicles Auto Trucks 8. Elec UtilitiesNormalized Elec UtilitiesFlow Thru 9. RetailFood Chains 10. Leisure Time Products Amusement & Recreation Prof. Sports & Arenas 11. Natural Gas Companies Compustat Industry Code 2801 2802 2300 2400 2510 2835 2836 2837 3310 3311 3711 3713 4912 4911 5411 3948 7949 7941 4924 Symbol CH TA FP HF D ST A price index was created to represent a set of prices for each of the industry portfolios. The following procedure was used to create each price index. First, each company within an industry grouping was checked to see if it had a monthly return (RET1) listed in the monthly returns file of the Center for Research in Security Prices (CRSP) at the Univer sity of Chicago. For all companies that had a monthly return listed in a given month, the returns were summed and an average monthly return was computed for that industry. This procedure was repeated for each month from January 1961 through March 1976. Firms were frequently observed to enter and leave the sample when computing each industry average monthly return. This method allowed the maximum number of companies to be used each month in the calculation of the industry average monthly return. The regression in equation (114) requires time series data in the form of prices rather than returns. To convert the eleven industry average return series into price indices, the following steps were taken for each industry. First, each industry was assigned a base value of 100 at the beginning of the time period. Then, the value of an industry portfolio in any subsequent month t was calculated by multiplying the price in month tl by one plus the portfolio return in month t. By per forming these calculations for all months, eleven industry price indices were created. To be consistent with the theory underlying the model, the value of the stock market portfolio in month t is defined to equal the sum of the N values of all stocks in month t. That is VMt = E Vjt, for all months t. j=1 Therefore, a stock market price index was created by summing each industry price index across industries. Finally, Long's model requires data for defaultfree couponless bills of all maturities. The prices of U.S. Government Treasury bills were used because of their availability and reliability. The interest rate on Treasury bills was pegged by the Fed before 1951, thus not allowing Treasury bill rates to adjust to anticipated variation in inflation rates. Price data were available on bills with maturities from one to three months until March 1959, at which time they became available for maturi ties up to six months. Price data on Treasury bills with maturities up to twelve months became available beginning in August 1964. Since the data on the consumption bundles werecollected back to 1961, monthly prices for Treasury bills with maturities ranging from one to six months were collected for the period January 1961 through March 1976.3 It will be the convention to treat the one month bill as a shortterm bill. The Treasury bills ranging in maturity from two to six months are treated as longterm bills. Thus, there are five (Tl) longterm bills and one shortterm bill used in the study. IV. 3 Empirical Estimation of the Regression Coefficients Following the collection of the various monthly price data, the next step is to estimate the regression coefficients in the following multiple regression whose parameters were discussed in Chapter II: S K T V1 = *j + 'Ml + Z jk + = m B + j j=,... N. (II4) jk=l m=2 The above equation requires using realizations of time 1 values of the variables. Since we do not have perfect foresight, the empirical counter part to equation (114) will be the following: See Fama (1975) for a description of the Treasury bill data used in this study. ^K ^ T E(Vjl) = + jME(VMI) + z jk E(nkl) + m E(Bl ) + Wj, k= M=2 jm j = 1,..., N. (II4a) A naive expectations model is used to estimate each variable's expected value. Thus, the convention in this study will be: (1) to use today's stock and commodity prices as estimates of next period's value; (2) to use today's value of the stock market portfolio as an estimate of next period's value; and (3) to use today's forward prices as implied by the term structure as the best estimates of next period's bill prices. Conventions (1) and (2) assume that stock and commodity prices follow a martingale. Convention (3) assumes the expectations theory of the term structure. Today's forward price for each bill is then computed according to 1 E(Bm) = B1 Bm, m = 2,..., T. (IV1) The number of observations used in each of the N=11 regressions performed over each time interval must exceed the number of coefficients (K+T+1 = 17) to be estimated. Five years of monthly observations will be used in each time series regression. The first set of N regressions will be performed on sixty months of observations extending from January 1961 through December 1965. The estimated coefficients from this time interval are then used to compute the one period or one month "quasi futures" contracts for each of the goods according to the method dis cussed in Chapter III. Thus, the first sixty months of observations are used to create portfolios which serve as hedging devices against price inflation in the various selected goods during January 1966. Since the "quasifutures" contracts are oneperiod contracts, new estimates of the regression coefficients must be computed each month in order to rebalance the portfolios. This is done by repeating the regression after first dropping the observations from the oldest month and adding observations from the latest month. By using this updating scheme, observations from the most recent sixty months are used in each regression. For example, the portfolio weights of "quasifutures" contracts for goods to be delivered in February 1966 are computed using the estimated regression coefficients from the period February 1961 through January 1966. During the initial runs of the regressions, serial correlation in the residuals was found. In order to get unbiased estimates of the regression coefficients, the data were transformed into first differences. When the method of first differences is used, it is believed that rho, the first order serial coefficient of the regression disturbances, is close to unity. In order to justify the assumption that the value of rho is sufficiently close to unity, the HildrethLu4 maximum likelihood scanning procedure was used in regressions over various selected time intervals. This technique transforms the data by rho (e.g.: Xt PXt1) and selects the value of rho which results in the lowest transformed error sum of squares. A sample of the estimated values of p for each of the eleven regressions over the interval January 1961 through December 1965 is presented in Table IV4. Since p was sufficiently close to unity in the sample regressions, the method of first differences was used in all regressions.5 In addition, the constant term was omitted from the regressions since there is no reason to include a trend variable. 4See Hildreth and Lu (1960) for a description of this procedure. It is assumed that the residuals follow a firstorder autoregressive process. Since the values of rho are sufficiently close to unity, a firstdifference transformation of equation (Il4a) is actually used to estimate the regression coefficients. Table IV4 Estimation of Equation II4a: HildrethLu Scanning Technique January 1961 to December 1965 Dependent Variable RHO R CH .99 .9948 TA .97 .9955 FP .80 .9957 HF .91 .9958 D .97 .9893 ST .82 .9779 MV .99 .9918 EU .95 .9927 F 1.00 .9703 LT .70 .9620 NG .81 .9824 In order to verify that the estimation procedure used to calculate the regression coefficients is correct, three conditions must hold over each time interval. Recalling from Chapter III, these conditions are as follows: N j Bjm 1 (1114) j=l N j Sjk = 0, k = 1,...,K, (III5) j=1 N z 6j = 0, m = 2,...,T. (III6) j=1 Jm These qualities exist due to the definition of the value of the market index. The estimation procedure used proved to be quite accurate as demon strated in Table IV5, which presents a summary of the regression coef ficients and their totals across industries from the eleven regressions over the sample interval March 1971 through February 1976. IV. C Computation and Analysis of the Portfolio Weights Following the estimation of each set of regression coefficients, the next step is to create the matrix n from which the monthly portfolio weights are computed. The matrix 0 is a [(N+T1) X (K+T)] matrix whose elements were given in Figure III3. For each time interval, the regression coefficients from each set of N = 11 regressions are placed in the first N rows of n. The last 5 (T1) rows are completed as given in Figure III3. Recalling from Chapter III, let e'+k be a [Ix(K+T)] vector whose (l+k)th element is 1 with all other elements equal to zero. The index k on el+k, k=1,...,K, indicates that the vector e+k is associated with the kth consumption good. The next step is to find the K = 10 weighted com binations of the rows of Q that will solve for the K row vectors, 43 SO 0 0 0 0 0 0 O 0 N SCO m O O C) o L. LO O r CO O r FL LO .o Lo CO 0 o *C) CO 0 I O S O O LO C 'C CD C\ CA CO O u c C m q C C 0 m I II I I I 0000 00000C o 0C) 0n C C CD o o cc ND N c0 CO N 0 N 000 C O3 COM O 0 ..M COJOCCIO N O .4 , ,.r LA e ( N n r r , I II II I I P LoO O NC r a; cN v O LO 0 c C0OO COO LA OO O 4 C: c1 1 1 : C r " c o 0 4 C C r o O C ) La% CO O C  SCO I C OI t I I I I L SMq) O .C LO c cIj i II aU  SCOO CON N 0 O O LA 0 O o)= C L O t CO r Lt C1) o LO 0 I C I I I I I 0 c  i *r = r 0 *r LA C 0 I000 C) O C E r L N N c0r CO m n C CO 0) r > C A On l0 COC N 0 CD 1 OOO0O(. 0 0 LA Lo cO r r r N co cO i 0o o INo CO p' m oo c NC o A ) o0 I I I I I r ' o0 r' LA o o. 0 N CO CO 0CM '0 N* o LA M (oM COO 0 N L A O rw N CO CO CD C n C O DL O N LA0 0D CM 0 0 0 0 O(U C C C t LL I > CD w C. i L. i" 0 ( S: ui LZ U .. j 0. ( 0 cU >. Q ~ 44 S0 CO cL O l 0 N0 0C0 0 CO 000 0 CO C0 0C o cJ o !. n O; 1 ( Io c OI 0 n O 00 CO N CJ 0 I I I I I I I C Cm C 'c co co O L oC Io CD o o o o) C C C I LO " m Ln CQ C CO r CD O COr D CD CD C I I IW I C I C0 a C JC) C o0 0 0 O O CO O C OO t 0.0 0C 4r Ln CQ LO f o C 0 M o I0 I I I Uw C C O C) C) O') COM C C a o Ln eq o Ln o t o cr o O C O O o P' C a O N O CO CO C O N 0 I I I Il i M el+k, k = ..., 10, from the following system of equations: bh = e+k "1, k = 1, ..., 10. (1117) The vector bk then contains the unit quantities of the respective shares and longterm bills which correspond to a portfolio whose "withdividend" value one month later has a covariance with the vector (VnM1, i1, B) equal to the covariance of nkl with (V,1, 1, B1). After attaining the correct covariance properties for each stock bill portfolio, the expected value for next month is set equal to the next month's expected value of the good for which it is hedging. Adding a quantity of bills with one month to maturity to the portfolio accomplishes this. These shortterm bills will not disturb the covariance of the port folio with (VM1', B1). The quantity of onemonth bills, YOlk' to add to each portfolio is Yolk = 801 (kkl bk [P1 1]'), k = 1, ..., 10.6 (III8) A negative value of YOlk refers to a short position in one month bills. After the quantity of one month bills is computed and added to each port folio, a "quasifutures" contract for each good is created which will have all the same desirable risk and return properties that concern investors as the goods themselves. The "quasifutures" contracts for each good are of one month in length. Their portfolio weights are calculated using the regression coefficients from the sixty most recent months of price observations. 6The bars over the variables indicate time 1 expectations. Therefore, every month starting from December 1965 and extending to February 1976, a new "quasifutures" contract is constructed for each good for delivery one month later.7 The quantities of each of the shares and bills in the hedging portfolios for a particular good changed significantly from month to month, thus indicating a need for rebalancing. Table IV6 presents a typical example of the fluctuations in the portfolio weights of the "quasi futures" contracts for Food (FD) over the interval September 1969 through December 1969. As one might expect, the fluctuations are more apparent in the quantities of the various bills due to their high degree of correlation and substitution. The next chapter investigates the need for rebalancing further. Fama and Schwert (1977a)investigate the question of which assets provided effective hedges against inflation during the 19531971 period. By regressing the returns on various assets against both the anticipated and unanticipated components of the inflation rate, they conclude that U.S. Government bonds and bills were a complete hedge against expected inflation and that common stocks were negatively related to the expected component of the inflation rate, and probably also to the unexpected component. These results imply that bills serve as a better hedge against inflation than do common stocks during the 19531971 inflationary period. In addition, the "quasifutures" contracts which serve as hedging devices should therefore consist mostly of bills. In fact, if returns on common stock were negatively related to the inflation rate, one would expect to observe a short position in common stocks each month. Tables specifying the composition of each "quasifutures" contract are available from the author upon request. Table IV6 Composition of "QuasiFutures" Contracts for Foodl September 1969 through December 1969 Months Industries 9/69 10/69 11/69 12/69  .217 .014  .269  .139  .035 .264 .244  .453 .718 .063 .546 Bills2 2Month 3Month 4Month 5Month 6Month 1Month3 10.085 13.168 40.013 1.261 11.987 29.960  .160 .099  .310  .237  .029 .777  .121  .025 .863 .071  .417 18.990 30.131 39.643 12.041 29.916 33.716 .468 .044 .194  .069  .077 .560  .940 .094 .202 .071  .701 1.880 34.815 9.057 14.245 17.706 22.342 .525 .063 .012  .104  .093 .627  1.053 .083 .279 .086  .752 3.816 40.306 8.498 17.365 20.889 26.443 Numbers in table and bills. represent quantities of the respective shares of stock Bills are assumed to have a face value of $100. 1month bills are used only for adjusting the expected value of the portfolio. Examining the percentage of each of the two types of assets in each monthly portfolio confirms these implications. Table IV7 presents the fraction of the total dollar investment in common stock and in bills in each of the monthly "quasifutures" contracts for the consumption bundle "Lumber and Wood Products" from December 1965 through February 1976. In all but 14 of the 123 months there was a larger dollar invest ment in bills. In fact, in more than half of the months there was a negative dollar investment or a short position in common stock. The results for all of the other consumption goods were similar to those for "Lumber and Wood Products." Thus it appears that bills do serve as a better hedge against commodity price inflation than do common stocks. While bills were a better hedging mechanism than common stocks, no conclusion can be drawn as to which of the bill maturities is better. Each of the various bills were frequently observed to be held short from month to month. Finally, the results imply that purchasing the common stock of a firm that produces a good similar to the hedged commodity would not contribute to the hedging efficiency of the portfolio. 3 u t r m co U) 0) )n a O CM LO k (0 cnbcOI S t, r. 0 3c 0 *_O 0 0 0 (l 0 (0 4) S c B m .0 S . CU 5 I 0 I U. > E U) 0 OT  > 0 o S, S e0a 00S o c C 4= <, 0 4 )1 4C I 0= C U S LL ,0 J I 0 r 0 VC) CD F 4 na (0 0O t C) I i II C) M: J U, (0 0, I i I I * CM 0 3 I i CM i 0 I I I I Ln c3 U) Co Cn 0< 0 0 I 000 a)C I FU CM cr) 0 CM * CMo 0 . CMO VCM (0 NP. on CM 0 CM m0 O Co> CD CM C CM C 0000 0 0 II 0 CD CD C) z m = > = __. . 0> a o ". U. x 7" , ) <. 0 C ro ,I *  i I I I" U) LO 00 CM C0V) 00 !a Co~O ko C) I ca J 1 cn (l ) C) 01 i 0( () J 1 C) I J J I 0 I c 00 r0 C o n c0 co to k0 r% (O ko co C) c . In C o o o0 o co I I I I qd 0 Do mo in 0 c  0 CV) C) I I w qOo m3 oc x LO O m aC *O e C\J C (o * C\j I I . 0 o O 0 LO CD C) r ( V) tD C % cO U LO r%% C 4 cJ I' mMON N MO)~ C~C . CMj 0) (n S I CM in r CMQ o n C) C) O O 0 CO Im0 0 0 0) 0 I I C I 0 ' 0 0 0 CM I I I co "M uD m) 1. C" I C) a r m n M r, o Co o .Do to m. LO mD w0o O O Co r J 0 1 00 o) 1w kC.o "M m m r  c o m o r m O M ri to cw C) k) 1. Co n ^o o CV) m. N. CM 3 0 1 .. C. .0 C. r. I I I I I I I I I I 0 * . CI 0 O 0  O 0 * * I1 LAO I i U) CO) M 0 0 CD M L o 4LA) OM M A LO M M C\1 1.0 LO j r" NLO LA ko ko P k.0 '. 0 0 0n 4 0^ o c', 0) qr 0 C) C r Nc O ir) 4mn LO le :d t o cr L)4*n Ln c m r o m O r C) "N iD 4r M M r0 MO C) CD N r 5i . t I Ln pV 1I I " r I I iI _I cN C M C C V ) r, C\ V c L O I I N cnl cc '1 .o C\ 0 C'd 0e> 0 c) A N 0d0) I i CM ' 0 ' n o N r \ cM LA t  r ( LA CJ Cc 3 '. o F cM CM o V ^ Nn i i I i i i) I I I oo CHAPTER V TESTS OF THE PORTFOLIOS AS HEDGING INSTRUMENTS Following their construction, the monthly "quasifutures" contracts are tested to determine their success as hedging devices against commodity price inflation. As discussed previously, a "quasifutures" contract for a particular good is constructed such that its expected value at the delivery date is equal to nkl. In addition, the portfolio should also have a covariance with (~M1' i) equal to the covariance of tk with V. A Testing Price and Risk Characteristics Two tests were designed to investigate whether each series of "quasi futures" contracts were indeed true substitutes for owning the goods them selves. The first of these tests was to indicate if the subsequently observed value at delivery of each "quasifutures" contract was identical to the observed value of the good at the same delivery date. This was accomplished by testing the significance of the estimated regression coefficients in the following time series regressions for each of the 10 consumption goods: Fkt = IkO + Ykl "kt + Ekt' k = 1,...,10, (Vl) where Fkt is the actual observed value at delivery date t of a "quasi futures" contract which was purchased one month earlier. The value of Fkt at delivery is computed according to Fkt = btl Pt Bt] + 100ttk k = l,...,10. (V2) Let a perfect hedge be defined as one where the hedging portfolio and the good have identical observed values each period. This would require the intercept and slope coefficients in equation (Vl) to be exactly zero and one, respectively, with zero standard errors. In addition, the R2 of each regression should equal one. Let an unbiased hedge be defined as one in which the estimated slope and intercept coef ficients are not statistically different from zero and one, respectively. Thus, the hypothesis of whether the "quasifutures" contracts do indeed have the same subsequently observed values as the goods is tested by examining the regression coefficients in equation (Vl) which should have the following expected values: E(ykO) = 0, k = 1,...,10, and E(ykl) = 1, k = 1,...,10. Estimates of the regression coefficients in equation.(Vl) from the period January 1966 to rlarch 1976 are presented in Table Vl for each of the consumption goods. The estimates of the intercepts coefficients, YkO' k=1,...,10, are all within one standard error from zero, with three exceptions, as indicated by the t statistics. The three exceptions are Gas and Electricity (t = 1.04), Apparel and Upkeep (t = 1.71), and Household Furnishings (t = 1.18). However, none of the t values are significant at a .05 significance level. Similarly, the estimates of the slope coefficients, ykl' k=1,...,10, are all within one standard error from one, with one exception (Apparel and Upkeep: t = 1.62). Again, Table V1 Portfolios as Substitutes for Consumption Goods: Ordinary Least Squares Fkt = YkO + Ykl kt + :kt January 1966 March 1976, T=123 (standard errors in parenthesis) SDurbin Ykl t(Ykl1)a Watson 1.15 .63 2.26 (.24) .83 .81 2.36 (.21) 1.42 1.62 2.00 (.26) 1.01 .04 2.16 (.23) .91 .75 2.32 (.12) 1.07 .39 2.15 (.18) .87 .68 2.35 (.19) 1.37 .59 2.21 (.63) 1.11 .92 2.25 (.12) 1.10 .83 2.07 (.12) R2 .16 .11 .20 .14 .31 .23 .15 .04 .41 .43 F(Yko=0:, Yll)b .19 1.29 1.80 .04 .89 .99 .70 1.18 .43 1.53 aCritical t.05,121 1.980 bCritical F.052,121= 3.07 05;2,121'230 Good FD GE AU PT RR MC MMP LWP CAP HFO YkO 19.03 (31.25) 27.63 (26.33) 53.39 (31.20) .08 (26.86) 8.84 (14.68) 13.96 (22.75) 21.58 (24.67) 80.32 (88.75) 13.25 (14.57) 16.59 (14.12) t(YkO=0)a .61 1.04 1.71 .00 .60 .61 .87 .90 .91 1.18 none of the t values are significant at a .05 significance level. Finally, an F test can be conducted (see Theil, p. 133) which will test simultane ously the joint hypothesis that YkO = 0 and Ykl = 1 for each good. This statistic is determined according to F2,T2(YkO' Ykl) = :[kO YkO kl Ykl] (s2C) O kO kl Ykl /2 where s C is the variancecovariance matrix of the regression coefficients. The values of F with 2 and 121 degrees of freedom for each regression are clearly insignificant at a .05 significance level as indicated in Table V1. The intercept and slope coefficients in equation (V1) are re estimated using seemingly unrelated estimation. This procedure will provide better estimates whenever the residual terms are contemporaneously correlated across regressions. Table V2 presents these estimates of the intercept and slope coefficients along with the t statistics. None of the t values are significant at a .05 significance level. Table V2 also presents an F statistic which differs from the previously described F statistic. This F statistic tests the general linear hypothesis that the intercept and slope coefficients across regressions are jointly equal to zero and one, respectively; see Zellner (1962). The F value is clearly insignificant at a .05 significance level. The evidence supports the claim that each hedging portfolio and consumption good had statistically equivalent observed values each month. Although the results indicate that the portfolios were not perfect sub stitutes for the goods, the portfolios did serve as unbiased hedges against commodity price inflation in each good. Table V2 Portfolios as Substitutes for Consumption Goods: Seemingly Unrelated Estimation Fkt = YkO+ Ykl kt+ Ekt January 1966March 1976, T = 123 (standard errors in parenthesis) Yk1 1.15 (.20) .92 (.14) 1.36 (.22) 1.10 (.16) .96 (.09) 1.05 (.13) .96 (.12) 1.30 (.36) 1.08 (.10) 1.09 (.09) t(kl=)a .75 .57 1.64 .63 .44 .38 .33 .83 .80 1.00 aCritical t.05,121 = 1.980 F20,1210 = .69; Critical F.05 = 1.58 Good FD GE AU PT RR MC MMP LWP CAP HFO ^0 YkO 19.60 (25.64) 16.72 (17.74) 46.36 (25.89) 10.28 (18.84) 3.37 (11.24) 11.50 (16.67) 11.24 (16.06) 69.71 (53.25) 9.74 (12.50) 14.09 (11.46) t(YkO=)a .76 .94 1.79 .55 .30 .69 .70 1.31 .78 1.23 The second step to indicate whether the "quasifutures" contracts were true substitutes for the consumption goods is to test whether the relative risk properties of the stockbill portfolios were similar to that of the goods. That is, the "quasifutures" contract for good k should have a covariance with (V' M1' 1 ) equal to the covariance of ,kl with ( s i ). This can be stated as COV *,kl (11'' 1) } = COV {0kl' ( t' ) ,) } k = 1,...,K. (V3) Since the previous tests indicated that the portfolios had equiva lent observed values at each delivery date, then any differences in the actual monthly values should be uncorrelated with (~1' f1' 1S). If equation (V3) is true then the following can be stated: COV {( kt kt) (VI' .11 1 I)} = 0, k = 1,...,K. (V4) Letting Dkt be equal to Fkt Ikt, the difference in the observed monthly prices between the hedging portfolio and good k, the following set of 16 (K+T) regressions is run for each good k, k = 1,...,K. D 1 1 1 Dkt= 0kO + kI Vt + pkt' j+ + j+l + j+l kt =kO + kl jt "kt, j = K~ti K+i K+i k = k+ + "Ki B + Ki i = 2,...,T. (V5) kt kO kl t,t+il kt, If the differences, Dkt, are truly uncorrelated with (V5M1 l' il ), then all the intercept and slope coefficients in equation (V5) should equal zero. Thus, the hypothesis to be tested is 1 2 K+T HO : "kO = "kO : "'= = kO = 0 0 kO kO k 0 1 2 K+T "kl kl ~ = .'" : k1 : O, k = 1,...K. An F test can be conducted (see Maddala, p. 323) which will test the above hypothesis. The F ratio is F = (S2 Sl)/ 2(K+T) S/ K+T z M.r 2(K+T) i=1 where 1) M. is the number of observations in the ith regression, 2) S, = the sum of the unrestricted residual sum of squares from each regression K+T K+T E RSS. with df = S M. 2(K+T), i= 1 i=I 1 and 3) S2 = the restricted residual sum of squares from a pooled regression of the data 2 K+T = (K+T) Z Dt with df = z M.. t i= 1 Estimates of the regression coefficients in equation (V5) from the period January 1966 to March 1976 are presented in Tables V3 through V12 for each of the 10 consumption goods. In addition, the t statistics for the coefficients are given along with the R2 and the residual sum of squares from each regression. When trying to reject a null hypothesis, selecting a larger significance level produces a more stringent test. For all of the consumption goods, with the exception of Apparel and Upkeep, none of the estimated intercept or slope coefficients were greater than Table V3 Covariance Properties of Contracts Fit "it = a0 + all (Ind Var)t January 1966 March 1976, T t(al0=0) all .37 .01 .61 .47 .33 .51 .34 .36 .53 .34 .79 .50 .87 .51 .39 .62 .67 .15 .12 .14 .19 .14 .10 .11 .07 .18 .17 59.01 16.05  7.91  9.43  8.15 t(all=0) .38 .61 .47 .33 .51 .34 .36 .53 .34 .81 .50 .87 .51 .39 .62 .67 Residual Sum of R2 Squares .001 613715.5 .003 .002 .001 .002 .001 .001 .002 .001 .005 .002 .006 .002 .001 .003 .004 612553.1 613356.6 613909.9 613139.1 613878.3 613803.4 613009.0 613870.7 611166.4 613194.2 610615.2 613124.4 613687.4 612501.1 612180.4 S1 = 9807704.7 S2 = 9831382.4 F32,1936 15 Critical F.05 = 1.45 05 df = 1936 df = 1968 for Food + 1It = 123 Ind. Var. MKT FD GE AU PT RR MC MMP LWP CAP HFO B1 B2 B3 B4 B5 a10 12.10 19.03 15.30 17.10 22.75 17.20 12.62 14.74 9.49 21.39 20.12 5878.04 1591.37 779.82 924.96 795.53 Table V4 Covariance Properties of Contracts for Gas and Electricity F2t 2t = 20 + a21(Ind Var)t + "2t January 1966 March 1976, T = 123 Ind. Var. MKT FD GE AU PT RR MC MMP LWP CAP HFO B1 B2 B3 B4 B5 "20 10.05 26.91 27.63 47.07 34.82 42.69 30.79 24.78 24.95 18.61 32.54 1182.28 631.33 628.49 426.53 290.78 t(ci20=0) .38 1.08 1.05 1.14 .98 1.05 1.10 1.11 1.12 .86 1.01 .21 .25 .39 .36 .31 a21 .01 .16 .17 .34 .24 .30 .19 .14 .13 .10 .21 11.25 6.44 6.45 4.42 3.05 t(a21=0) .10 .80 .79 .97 .78 .88 .85 .81 .81 .54 .79 .21 .26 .40 .36 .31 R2 .000 .005 .005 .008 .005 .006 .006 .005 .005 .002 .005 .000 .001 .001 .001 .001 Sl = 6295925.9 S2 = 6424353.6 F32,1936 = 1.23 df = 1936 df = 1968 Critical F05 = 1.45 .05 Residual Sum of Squares 394886.6 392847.4 392914.3 391882.3 393949.7 392406.2 392567.7 392816.6 392784.9 393982.2 392887.3 394782.8 394708.3 394413.2 394493.0 394603.4 Table V5 Covariance Properties of Contracts for Apparel and Upkeep F3t 3t = 930 + a31(Ind Var)t + 3t January 1966 March 1976, T=123 Ind. Var. MKT FD GE AU PT RR MC MMP LWP CAP HFO B1 B2 B3 B4 B5 a30 .16 33.89 28.48 53.40 43.92 47.71 34.78 28.23 27.24 25.32 39.11 8791.19 3993.25 2224.94 1704.25 1509.15 t(a30=0) .01 1.80 1.43 1.71 1.63 1.56 1.64 1.68 1.62 1.55 1.60 2.18 2.15 1.85 1.90 2.12 a31 .01 .24 .20 .42 .34 .37 .24 .19 .17 .18 .29 88.30 40.32 22.59 17.40 15.50 t(a31=0) .21 1.62 1.25 1.59 1.50 1.44 1.48 1.48 1.42 1.34 1.46 2.18 2.15 1.85 1.90 2.13 R2 .000 .021 .013 .021 .018 .017 .018 .018 .016 .015 .017 .038 .037 .028 .029 .036 Residual Sum of Squares 227742.5 222999.0 224913.8 223136.2 223679.6 223986.7 223780.1 223772.9 224116.7 224477.2 223884.7 219232.9 219441.1 221568.3 221169.8 219606.8 = 3567508.3 = 3677036.8 df = 1936 df = 1968 32,1936 1.86 Critical F. 5 = 1.45 Table V6 Covariance Properties of Contracts for Private Transportation F4t 4t = a40 + 41 (Ind Var)t + 4t January 1966 March 1976, T=123 Ind. Var. MKT FD GE AU PT RR MC MMP LWP CAP HFO B1 B2 B3 B4 B5 a40 12.72 2.75 .75 6.08 .08 4.32 2.54 1.44 6.18 1.99 1.11 5749.06 2737.57 1480.92 1025.41 800.91 t(a40=O) .64 .15 .04 .20 .00 .14 .12 .09 .37 .12 .05 1.42 1.47 1.23 1.15 1.12 a41 .01 .03 .01 .04 .01 .03 .01 .01 .05 .03 .01 57.71 27.60 15.00 10.44 8.19 t(a41=0) .60 .20 .09 .16 .04 .11 .07 .03 .44 .19 .00 1.42 1.47 1.23 1.15 1.12 2 R2 .003 .000 .000 .000 .000 .000 .000 .000 .002 .000 .000 .017 .018 .012 .011 .010 Sl = 3548072.8 S2 = 3566366.8 F32,1936 = .31 df = 1936 df = 1968 Critical F.05 = 1.45 Residual Sum of Squares 222115.1 222696.9 222758.7 222724.6 222771.6 222752.0 222763.9 222772.6 222417.0 222707.2 222773.8 219103.4 218843.8 220015.1 220380.1 220477.0 Table V7 Covariance Properties of Contracts for Reading and Recreation F5t "5t = 50 + 51 (Ind Var)t + St January 1966 March 1976, T=123 Ind. Var. MKT FD GE AU PT RR MC MMP LWP CAP HFO B1 B2 B3 B4 B5 "50 5.09 2.02 3.91 8.37 7.14 8.84 5.11 3.53 .32 1.87 5.82 2258.23 1062.91 808.43 483.15 377.80 t(a50=0) .53 .22 .41 .56 .55 .60 .50 .44 .04 .24 .50 1.16 1.18 1.40 1.12 1.10 "51 .01 .03 .05 .09 .08 .09 .05 .04 .01 .03 .06 22.69 10.74 8.21 4.94 3.89 t( 51O0) .73 .42 .60 .68 .69 .73 .69 .67 .26 .47 .65 1.16 1.18 1.40 1.12 1.10 R2 .004 .002 .003 .004 .004 .004 .004 .004 .001 .002 .004 .010 .012 .016 .010 .010 = 825262.6 = 836043.2 df = 1936 df = 1968 F32,1936 = 79 Critical F05 = 1.45 05 Residual Sum of Squares 51656.0 51804.2 51725.1 51684.1 51675.2 51655.3 51679.7 51688.8 51851.5 51784.6 51697.5 51313.2 51285.8 51053.8 51344.3 51363.5 Table V8 Covariance Properties of Contracts for Medical F6t "6t = a60 + a61 (Ind Var)t + 6t January 1966 March 1976, T=123 Ind. Var. MKT FD GE AU PT RR MC MMP LWP CAP HFO B1 B2 B3 B4 B5 060 9.18 18.26 16.64 19.74 19.99 16.83 13.96 15.64 10.06 19.28 18.95 3686.31 1873.08 1323.96 1049.99 785.51 t(a60=0) .43 .90 .78 .59 .69 .51 .61 .87 .55 1.10 .72 .84 .93 1.02 1.09 1.02 "61 .01 .10 .09 .12 .12 .10 .07 .08 .03 .12 .11 37.06 18.95 13.48 10.75 8.10 t(a61=0) .70 .64 .53 .42 .50 .34 .37 .57 .25 .81 .52 .84 .93 1.02 1.09 1.02 = 4131077.2 = 4209499.2 df = 1936 df = 1968 F32,1936 = 1.15 Critical F 05 = 1.45 .05 Care Residual Sum of Squares 258216.2 258395.9 258670.2 258880.0 258724.9 259011.2 258965.0 258565.6 259127.5 257881.1 258693.0 257751.4 257413.4 257038.9 256723.9 257019.0 R2 .004 .003 .002 .002 .002 .001 .001 .003 .001 .005 .002 .006 .007 .009 .010 .009 Table V9 Covariance Properties of Contracts for Metal and Metal Products F7t 7t = a70 + a71 (Ind Var)t + "7t January 1966 March 1976, T=123 Residual Sum of Ind. Var. a70 t(a70=0) a71 t(a71=0) R2 Squares MKT 14.14 .48 .01 .30 .001 483556.9 FD 21.39 .77 .13 .58 .003 482556.2 GE 22.01 .75 .14 .57 .003 482595.6 AU 44.21 .97 .33 .85 .006 481038.2 PT 30.93 .78 .22 .65 .004 482235.4 RR 38.55 .85 .28 .74 .005 481722.8 MC 28.08 .90 .18 .74 .005 481744.7 MMP 21.58 .87 .13 .67 .004 482138.4 LWP 19.46 .79 .10 .58 .003 482584.1 CAP 14.38 .60 .08 .38 .001 483337.7 HFO 28.70 .80 .19 .66 .004 482197.3 B1 1855.61 .31 18.57 .31 .001 483530.1 B2 151.93 .06 1.48 .05 .000 483899.1 B3 14.31 .01 .09 .00 .000 483910.2 B4 44.64 .03 .51 .04 .000 483904.6 B5 69.15 .07 .77 .07 .000 483890.3 SI = 7724841.6 df = 1936 S2 = 7803676.8 df = 1968 F32,1936 = .55 Critical F = 1.45 .05 Table V10 Covariance Properties of Contracts for Lumber and Wood Products F8t X8t = a80 + a81 (Ind Var)t + "8t January 1966 March 1976, T=123 Ind. Var. MKT FD GE AU PT RR MC MMP LWP CAP HFO B1 B2 B3 B4 B5 c80 55.01 110.51 106.49 204.64 144.83 177.49 126.47 111.62 80.32 92.44 142.03 3326.18 3214.50 2723.63 2181.21 2122.31 t(a80=O) .52 1.11 1.02 1.25 1.02 1.10 1.14.. 1.26 .90 1.08 1.11 .15 .32 .43 .46 .56 c81 .01 .65 .64 1.48 .99 1.26 .77 .65 .37 .54 .94 33.68 32.72 27.90 22.52 22.03 t(a81=0) .25 .84 .75 1.08 .83 .93 .89 .96 .59 .76 .89 .16 .33 .43 .47 .57 R2 .001 .006 .005 .010 .006 .007 .007 .008 .003 .005 .007 .000 .001 .002 .002 .003 Residual Sum of Squares 6225374. 6192771. 6199474. 6169336. 6193731. 6184320. 6188088. 6181574. 6210587. 6199034. 6187793. 6227339. 6223068. 6219045. 6217445. 6211973. S1 = 99230952 S2 = 101330400.0 F32,1936 = 1.28 df = 1936 df = 1968 Critical F05 = 1.45 05 Table V11 Covariance Properties of Contracts for Chemicals and Allied Products F9t H9t = a90 + a91 (Ind Var)t + 19t January 1966 March 1976, T=123 ^o "90 17.53 17.69 20.15 28.61 29.12 30.83 22.09 12.73 13.17 13.25 24.13 1146.73 285.07 141.67 34.42 145.54 t(a900) .98 1.05 1.14 1.03 1.21 1.13 1.17 .85 .87 .91 1.11 .31 .17 .13 .04 .23 a91 .01 .14 .17 .24 .25 .26 .17 .10 .09 .11 .20 11.51 2.87 1.43 .35 1.49 t( 91 =0) .99 1.06 1.15 1.03 1.22 1.13 1.18 .86 .89 .92 1.12 .31 .17 .13 .04 .23 = 2862602.3 = 2881004.8 df = 1936 df = 1968 F 1936 = .39 32,1936 j Critical F0 = 1.45 .05 Ind. Var. MKT FD GE AU PT RR MC MMP LWP CAP HFO B1 B2 B3 B4 B5 Residual Sum of Squares 178612.7 178401.5 178119.7 178505.3 177879.3 178171.4 178007.9 178971.8 178900.8 178800.2 178223.7 179914.0 180017.5 180034.9 180057.4 179984.2 R2 .008 .009 .011 .009 .012 .011 .011 .006 .006 .007 .010 .001 .000 .000 .000 .000 Table V12 Covariance Properties of Contracts for Household Furnishings Flot l0t = a10,0 + a10,1 (Ind Var)t + 10tt January 1966 March 1976, T=123 t(al0,o=0) .00 1.40 1.15 1.09 1.18 1.00 1.16 1.22 1.19 1.39 1.18 1.52 1.60 1.56 1.62 1.56 "10 1 .01 .09 .08 .14 .13 .12 .08 .07 .06 .08 .11 35.91 17.53 11.05 8.60 6.65 t(a l01=0) .30 1.11 .87 .90 .96 .81 .89 .89 .89 1.06 .94 1.52 1.61 1.56 1.63 1.57 = 1200515.9 = 1237188.8 df = 1936 df = 1968 F32,1936 = 1.85 Critical F = 1.45 .05 Ind. Var. MKT FD GE AU PT RR MC MMP LWP CAP HFO B1 B2 B3 B4 B5 10,0 .05 15.31 13.28 19.71 18.36 17.72 14.20 11.92 11.62 13.15 16.59 3573.56 1734.89 1086.31 840.49 646.29 Residual Sum of Squares 75796.2 75092.4 75382.1 75344.1 75273.7 75442.9 75358.6 75357.2 75394.0 75158.8 75299.5 74430.6 74266.1 74356.0 74227.3 74336.4 R2 .001 .010 .006 .007 .008 .005 .007 .007 .006 .009 .007 .019 .021 .020 .021 .020 1.63 standard errors from zero. Thus, the hypothesis that each of the regression coefficients is individually equal to zero cannot be rejected as indicated by the critical t vale of 1.658 at a .10 significance level with 121 degrees of freedom. Only three of the regressions pertaining to Apparel and Upkeep contained t values which were slightly greater than 2. The hypothesis that all the intercept and slope coefficients are jointly equal to zero cannot be rejected for eight of the ten consumption goods as indicated by the F statistic at the bottom of each table. All the F values with the exception of Apparel and Upkeep (AU) and Household Furnishings (HFO) are insignificant at a .05 significance level. Finally, the values of R2 in all the regressions are extremely small (the highest in any being .037) indicating almost no relationship between the difference in the value of the hedging portfolios and goods each period and the vector of prices (VMl1' ,I B1). Despite two significant F values, the findings essentially indicate that each series of "quasifutures" contracts had the same covariance with the vector of prices (VM1, nl, B1 ) as did each good. In addition, the neglible values of R2 in each regression indicate that any observed difference each month between the value of the portfolio and the good was nonsystematically related to (VM1' 1 B1). V. B Effect of the Nixon Wage and Price Controls On August 15, 1971, President Nixon announced his New Economic Policy intended to keep wage increases down to a 5.5 percent annual rate and also to limit price increases, and thus inflation, to a 2.5 percent annual rate. Economists thought that the 3 percent difference would result from a rise in productivity and thus curtail the rising inflation rate in addition to stimulating real growth. They also believed at the time that if the inflation was of the costpush variety, then controls could be a success since rising costs of production would be pushing prices higher. Under demandpull inflation, economists thought controls would be a waste of time since the demand itself requires limiting. Although the freeze applied to prices and wages, prices of stocks and bondswere not included. Interest rates were not frozen, but bankers and other lenders were urged to hold the line. The impact of these wage and price controls on the ability of the "quasifutures" contracts to serve as successful hedging instruments against commodity price inflation is examined next. The controls which went into effect beginning on August 16, 1971, led to queues and shortages which caused reported inflation rates through the price indices to understate the true changes in purchasing costs to consumers. August 1971 will be used as the starting date for the controls. As the controls were lifted, the reported inflation rates probably over stated the true inflation rates. For instance, prices of many goods on the commodity exchanges frequently reached their permissible daily price increases within a few minutes of the opening of the exchanges. During the ending months of the controls, the Administration tried to stagger price increases by gradually lifting the controls from various industries at a time. December 1974 will be used as the ending date when the con trols were lifted from almost all industries. The tests described in Section V. A will be used again to examine the success of the portfolios as hedging devices during the period August 1971 through December 1974. Table V13 presents the results of the first test which investigates whether the portfolios had the same observed value each month as each of the goods. The hypothesis is that the intercept coefficient in equation (V1) is equal to zero and the 71 Table V13 Portfolios as Substitutes for Consumption Goods: Ordinary Least Squares Fkt = YkO + klkt + kt August 1971 December 1974, T=41 (standard errors in parenthesis) Good FD YkO  87.19 (102.23) GE 24.45 (53.89) AU 162.16 (122.20) PT 35.59 (122.77) RR 57.58 (111.95) MC 153.63 (141.79) MMP 6.17 (39.21) LWP .50 (217.49) CAP 23.08 (24.26) HFO 53.15 (55.70) t(Yko=O)a .85 Ykl 1.59 (.73) .45 .84 (.42) 1.33 2.24 (.96) .29 1.26 (.99) .51 1.42 (.88) 1.08 2.02 (1.02) .16 .00 .95 (.28) .86 (1.31) .95 .81 (.20) .95 1.37 (.44) t(kl=1)a .81 .38 1.29 .26 .48 1.00 .18 .11 .95 .84 Durbin Watson 2.13 R2 F(YkO=O Ykl=1)b .11 .41 2.16 .09 2.49 .12 2.16 .04 2.46 .06 .27 .39 .22 1.05 2.31 .09 1.17 2.32 .23 2.38 .01 2.57 .29 2.56 .20 .01 .30 .46 2.39 aCritical t.10,39= 1.684 Critical F052 3.23 .05;2,39= slope coefficient is equal to one. The estimates of YkO' k = 1,...,10, are all within one standard error from zero, with two exceptions, as indicated by the t statistics. None of the t values are significant for a critical value of t = 1.68 at a .10 significance level with 39 degrees of freedom. Similarly, the estimates of Ykl' k = 1,...,10, are all within one standard error from one, with one exception. Again, none of the t values are significant at a .10 significance level. The values of F in each regression, which test the joint hypothesis that YkO = 0 and Ykl = 1 for each good k, are all insignificant for a critical value of F = 3.23 at a .05 significance level with 2 and 39 degrees of freedom. Finally, the values of R2 from each regression appear to be slightly lower than those from the January 1966 to March 1976 period. The intercept and slope coefficients are reestimated using seemingly unrelated estimation. Table V14 presents these estimates of the intercept and slope coefficients along with the t statistics. None of the t values are significant at a .10 significance level. The F statistic which tests the general linear hypothesis that the intercept and slope coefficients across regressions are jointly equal to zero and one, respectively, is also presented in Table V14. The F value is insignificant at a .01 significance level. The evidence again supports the claim that each hedging portfolio and consumption good had statistically equivalent values each month. The wage and price controls appear to have had little or no effect on the success of the portfolios to serve as unbiased hedges for the consumption goods. Table V14 Portfolios as Substitutes for Consumption Goods: Seemingly Unrelated Estimation Fkt = Yk + Ykl Ikt + kt August 1971December 1974, T = 41 (standard errors in parenthesis) t(YkO0O)a 1.55 .97 .23 .07 .10 .23 .43 .59 .43 .48 Ykl 1.39 (.26) 1.26 (.24) 1.08 (.51) .95 (.45) 1.00 (.38) .81 (.46) 1.06 (.13) 1.03 (.24) 1.06 (.12) 1.03 (.17) t(kl=1)a 1.50 1.08 .11 .41 YkO 59.06 (38.02) 29.75 (30.80) 14.97 (65.79) 3.80 (56.87) 4.95 (47.84) 14.97 (64.29) 8.34 (19.30) 28.97 (48.93) 6.29 (14.49) 10.82 (22.48) t.0,39= 1.684 1.83; Critical F.01 = 1.92 Good .18 MMP LWP CAP HFO aCritical F20,390 The second set of tests for the wage and price control period will check to see if the covariance properties of the stockbill portfolios were similar to that of the goods. Tables V15 through V24 contain the results of regressing the monthly observed price difference between the hedging portfolios and each good against each of the variables in (VMI, 111 B1). The hypothesis is that each of the intercept and slope coefficients in equation (V5) should jointly equal zero. That is, 1 2 K+T HO: akO akO "'." = kO : 0 1 2 K+T k1 : 2k1 "' aK+T 0 k = 1,...K. ckl kl kl In the tables for each good, none of the estimated intercept or slope coefficients are significantly different from zero as indicated by the critical t value of 2.02 at a .05 significance level with 39 degrees of freedom. The hypothesis that all the intercept and slope coefficients are jointly equal to zero cannot be rejected for nine of the ten con sumption goods. With the exception of Household Furnishings (HFO), all the F values are insignificant at the .05 significance level. The findings essentially indicate that each series of "quasifutures" contracts had the same covariance with the vector of prices (VM' 1 1, B1) as did each good. Any observed difference between the value of the portfolio and the good each month was nonsystematically related to (VM1, 11 1) Some care must be given in the interpretation of these results. The hedge portfolios appear to provide successful hedges during the price control period. However, if the price index numbers for the commodities do not reflect true transaction prices, the portfolios only provide hedges against changes in the government reported price levels and not hedges against actual price changes in the market place. Table V15 Covariance Properties of Contracts for Food Flt l t = a10 + a11(Ind Var)t + "lt August 1971 December 1974, T = 41 Ind. Var. MKT FD GE AU PT RR MC MMP LWP CAP HFO B1 B2 B3 B4 B5 al0 28.07 87.19 86.60 251.02 124.66 263.71 183.16 59.74 70.89 53.86 139.22 3180.74 1617.98 1214.28 1275.07 1244.98 S1 = 3983398.8 S2 = 4048830.0 F32,624 = .32 t(al0=0) .41 .85 .63 1.05 .71 .90 .90 .76 .74 .77 .83 .30 .35 .41 .57 .70 all" .01 .59 .63 1.93 .96 2.04 1.28 .39 .40 .41 1.05 31.99 16.38 12.37 13.06 12.82 t(all =0) .49 .81 .60 1.30 .68 .89 .88 .71 .70 .70 .81 .30 .35 .41 .57 .70 df = 624 df = 656 Critical F.05 = 1.47 .05 R2 .006 .017 .009 .026 .012 .020 .019 .013 .012 .013 .016 .002 .003 .004 .008 .013 Residual Sum of Squares 250464.1 247838.6 249730.8 245385.5 249030.9 247038.8 247134.0 248806.8 248897.8 248837.3 247880.9 251428.9 251220.6 250935.9 249916.0 248851.9 Table V16 Covariance Properties of Contracts for Gas and El F2t 2t = "20 + a21(Ind Var)t + 2t August 1971 December 1974, T = 41 Ind. Var. MKT FD GE AU PT RR MC MMP LWP CAP HFO B1 B2 B3 B4 B5 "20 15.18 17.76 24.45 37.10 23.70 35.52 30.18 11.35 11.42 9.44 21.32 1274.99 733.12 491.35 392.49 150.71 SI = 614349.8 S2 = 623216.6 F32,624 = .28 t(a20=0) .57 .44 .45 .39 .34 .31 .38 .37 .31 .34 .32 .31 .40 .42 .45 .22 "21 .01 .10 .16 .27 .16 .25 .19 .06 .05 .05 .14 12.84 7.43 5.02 4.04 1.58 t(a21=0) .70 .36 .40 .36 .30 .28 .34 .27 .22 .23 .28 .31 .40 .42 .45 .22 ectricity R2 .012 .003 .004 .003 .002 .002 .003 .002 .001 .001 .002 .002 .004 .005 .005 .001 df = 624 df = 656 Critical F.05 = 1.47 Residual Sum of Squares 38052.7 38396.3 38372.9 38401.4 38439.8 38449.6 38415.4 38456.2 38478.7 38475.5 38451.9 38434.7 38366.2 38351.6 38327.7 38479.2 Table V17 Covariance Properties of Contracts for Apparel and Upkeep F3t 3t = a30 + 31 (Ind Var)t + "3t August 1971 December 1974, T = 41 Ind. Var. MKT FD GE AU PT RR MC MMP LWP CAP HFO B1 B2 B3 B4 B5 030 42.28 76.39 79.81 162.16 111.96 199.35 137.93 50.12 76.75 41.03 102.80 9228.92 3753.98 1881.52 1538.71 1425.97 S1 = 1012352.3 S2 = 1072020.0 F32,624 = 1.15 t(a30=0) 1.22 1.48 1.14 1.33 1.26 1.35 1.34 1.26 1.61 1.14 1.21 1.75 1.62 1.25 1.36 1.60 (31 .01 .52 .59 1.24 .87 1.54 .96 .33 .44 .31 .77* 92.72 37.93 19.13 15.74 14.67 t(a 31=0) 1.37 1.41 1.09 1.29 1.21 1.32 1.30 1.17 1.53 1.04 1.16 1.75 1.62 1.25 1.37 1.61 R2 .046 .048 .029 .041 .036 .043 .041 .034 .057 .027 .034 .073 .063 .039 .046 .062 df = 624 df = 656 Critical F 05 = 1.47 Residual Sum of Squares 63210.2 63039.3 64299.5 63522.2 63840.1 63425.3 63507.0 64003.2 62487.3 64454.5 64026.1 61436.8 62059.7 63698.7 63217.9 62124.5 Table V18 Covariance Properties of Contracts for Private Transportation F4t 4t = a40 + 41(Ind Var)t + v4t August 1971 December 1974, T = 41 Residual Sum of Ind. Var. 40 t(a40=0) a41 t(41=0) R Squares MKT 25.20 .53 .01 .60 .009 120573 FD 50.93 .72 .34 .68 .012 120269 GE 35.19 .37 .25 .34 .003 121333 AU 65.44 .39 .49 .37 .004 121260 PT 35.59 .29 .26 .27 .002 121466 RR 90.73 .44 .69 .43 .005 121116 MC 53.56 .38 .36 .36 .003 121294 MMP 12.42 .23 .07 .17 .001 121592 LWP 97.07 1.50 .57 1.46 .052 115348 CAP 4.30 .09 .01 .03 .000 121685 HFO 24.68 .21 .17 .19 .001 121579 81 11964.60 1.67 120.18 1.67 .066 113604 B2 5422.77 1.74 54.76 1.74 .072 112958 B3 3324.03 1.65 33.76 1.65 .065 113752 84 2174.45 1.42 22.21 1.43 .050 115653 85 1781.66 1.47 18.31 1.47 .053 115267 SI = 1898749 df = 624 S2 = 1952864 df = 656 F32624 = .56 Critical F 1.47 .05 Table V19 Covariance Properties of Contracts for Reading and Recreation F5t 5t = "50 + "51 (Ind Var)t + 5t August 1971 December 1974, T = 41 Residual a2 Sum of Ind. Var. a50 t(a50=0) 51 t(a51=0) R Squares MKT 3.12 .12 .01 .31 .002 36312.5 FD 30.62 .79 .19 .67 .001 35988.2 GE 23.42 .45 .14 .36 .003 36281.4 AU 43.01 .47 .30 .41 .004 36240.3 PT 16.58 .25 .09 .18 .001 36271.1 RR 57.58 .51 .42 .47 .006 36193.2 MC 37.05 .47 .23 .41 .004 36240.3 MMP 9.11 .30 .03 .15 .001 36380.0 LWP 51.12 1.43 .28 1.31 .042 34862.4 CAP 5.40 .20 .01 .02 .000 36399.5 HFO 18.16 .28 .10 .21 .001 36359.1 B1 4172.33 1.04 41.95 1.04 .027 35415.3 B2 2000.95 1.15 20.24 1.15 .032 35207.1 B3 1436.35 1.29 14.62 1.29 .041 34911.2 B4 943.25 1.12 9.67 1.12 .031 35256.1 B5 723.24 1.08 7.47 1.09 .029 35331.6 S1 = 573749.3 df = 624 S2 = 597444.3 df = 656 F32,624 = 81 Critical F.05 = 1.47 Table V20 Covariance Properties of Contracts for Medical 6t 6t = 60 + a61(Ind Var)t + 6t August 1971 December 1974, T = 41 Ind. Var. MKT FD GE AU PT RR MC MMP LWP CAP HFO B1 B2 B3 B4 B5 a60 25.63 82.87 94.37 202.41 106.13 210.45 143.62 54.94 79.72 46.43 111.95 3968.58 2411.99 1781.24 1476.00 1054.09 s1 = 1928744.3 S = 2062723 F32,624 = 1.35 t(a60=O) .54 1.17 .99 1.21 .87 1.03 1.08 1.01 1.21 .94 .96 .53 .75 .86 .95 .85 a61 .01 51 .64 1.49 .76 1.57 1.02 .31 .41 .29 .78 39.97 24.47 18.20 15.18 10.94 t(a61 =0) .80 1.00 .86 1.14 .77 .97 1.00 .80 1.03 .71 .86 .53 .75 .86 .96 .86 df = 624 df = 656 Critical F05 = 1.47 Care R2 .016 .025 .019 .032 .015 .024 .025 .016 .027 .013 .019 .007 .014 .019 .023 .019 Residual Sum of Squares 120950.6 119849.0 120629.1 118980.5 121105.4 120023.2 119867.9 120978.8 119665.0 121374.6 120658.9 122048.3 121200.2 120637.2 120124.3 120651.3 Table V21 Covariance Properties of Contracts for Metal and Metal Products F7t "7t = 70 + a71 (Ind Var)t + "7t August 1971 December 1974, T = 41 Ind. Var. MKT FD GE AU PT RR MC MMP LWP CAP HFO B1 B2 B3 B4 B5 a70 7.85 10.00 1.24 5.11 11.24 15.81 1.84 6.17 37.60 7.39 8.87 2773.82 1406.69 1014.10 600.30 513.11 = 995736.0 S2 = 999851.1 32,624 = .08 t(a70=0) .23 .19 .02 .04 .13 .11 .02 .16 .79 .21 .11 .52 .61 .68 .54 .58 df = 624 df = 656 Critical F.05 = 1.47 a71 .01 .07 .01 .04 .09 .12 .01 .05 .23 .06 .07 27.86 14.20 10.29 6.13 5.27 t(a71=O) .23 .19 .02 .04 .13 .11 .02 .17 .80 .22 .11 .52 .61 .68 .54 .58 Residual Sum of Squares 62407.3 62429.6 62487.7 62485.3 62461.0 62470.4 62488.0 62444.5 61488.8 62411.1 62469.4 62054.1 61901.4 61750.8 62029.4 61957.2 R2 .001 .001 .000 .000 .000 .000 .000 .001 .016 .012 .000 .007 .009 .012 .007 .009 Table V22 Covariance Properties of Contracts for Lumber and Wood Products F8t 8t = a80 + "81 (Ind Var)t + "8t August 1971 December 1974, T = 41 Ind. Var. MKT FD GE AU PT RR MC MMP LWP CAP HFO B1 B2 B3 84 B5 a80 4.08 93.97 165.19 367.43 224.60 325.75 267.07 134.36 .50 120.35 241.80 13172.8 5385.3 3631.57 1634.4 1065.81 S1 = 20666073 S2 = 21148416 F32,624 = .46 t(a80=0) .03 .40 .53 .63 .56 .49 .57 .76 .00 .75 .63 .54 .51 .53 .32 .26 881 .03 .50 1.10 2.70 1.62 2.39 1.75 .79 .14 .82 1.71 132.04 54.11 36.61 16.43 10.69 t(a81=0) .13 .30 .45 .63 .50 .45 .52 .63 .10 .61 .57 .54 .51 .53 .32 .26 R2 .000 .002 .005 .010 .007 .005 .007 .010 .000 .010 .008 .008 .007 .007 .003 .002 Residual Sum of Squares 1298446 1295921 1292129 1286048 1290585 1292205 1289881 1285864 1298596 1286537 1288075 1289221 1290454 1289646 1295676 1296789 df = 624 df = 656 Critical F.05 = 1.47 Table V23 Covariance Properties of Contracts for Chemicals and Allied Products F9t 9t = "90 + a91 (Ind Var)t + 9t August 1971 December 1974, T = 41 19.66 41.24 50.84 93.02 62.89 122.67 84.96 29.77 34.94 23.08 61.12 3072.94 1023.48 335.23 439.32 444.68 = 471743.2 = 483588.1 t(90=0) "91 .83 .01 1.18 .29 1.08 .39 1.12 .72 1.04 .50 1.22 .96 1.22 .60 1.11 .20 1.07 .21 .95 .19 1.06 .47 .84 30.87 .64 10.34 .32 3.41 .57 4.49 .72 4.57 t(91=0)) .89 1.15 1.06 1.11 1.03 1.21 1.20 1.08 1.04 .92 1.05 .84 .64 .32 .57 .72 R2 .020 .033 .028 .031 .026 .036 .036 .029 .027 .021 .027 .018 .010 .003 .008 .013 df = 624 df = 656 32,624 = .49 Critical F05 = 1.47 .05 Ind. Var. MKT FD GE AU PT RR MC MMP LWP CAP HFO B1 82 B3 B4 B5 Residual Sum of Squares 29573.7 29180.9 29330.7 29252.1 29383.0 29079.7 29096.8 29297.1 29355.2 29533.4 29351.2 29643.2 29865.2 30095.4 29929.6 29776.0 Table V24 Covariance Properties of Contracts for Household Furnishings F0,t 10,t = "10,0 + al0,1(Ind Var)t + v10,t August 1971 December 1974, T = 41 t(a10,0 =0) .55 1.37 1.07 1.18 .89 1.09 1.12 1.04 1.68 .94 .95 1.07 1.27 1.37 1.34 1.30 "10,1 .01 .29 .33 .69 .37 .79 .50 .15 .28 .13 .37 37.72 19.57 13.64 10.12 7.84 t(al O,1=) .85 1.19 .93 1.10 .79 1.03 1.02 .80 1.49 .68 .84 1.07 1.27 1.37 1.35 1.31 Residual S Sum of R Squares .018 27542.6 .035 .022 .030 .016 .026 .026 .016 .054 .012 .018 .028 .040 .046 .045 .042 27074.4 27443.5 27206.9 27612.1 27310.2 27314.7 27599.0 26546.1 27725.5 27550.4 27253.6 26934.6 26753.8 26797.4 26873.2 S1 = 435538 S2 = 476083.4 F32,624 = 1.82 Critical F.05 = 1.47 .05 Ind. Var. MKT FD GE AU PT RR MC MMP LWP CAP HFO B1 B2 B3 B4 B5 "10,0 12.58 46.20 48.58 94.25 52.36 106.30 75.67 26.99 52.27 22.07 53.15 3749.76 1932.97 1338.18 985.59 757.59 df = 624 df = 656 V. C quarterly and Semiannual Rebalancing From the conclusions reached in the previous hypothesis testing, it appears that one can successfully create an unbiased hedge against commodity price inflation using shares of common stock and Treasury bills of various maturities. These hedging portfolios or "quasifutures" con tracts had the same observed price and risk characteristics each month as did the goods. Theoretically, since the portfolios serve as one month hedges, new portfolios must be created each month in order to maintain the proper hedge. One may speculate that some minor rebalancing each month is necessary due to the realization of events. An example of the fluctuations in the portfolio weights for a typical series of contracts was presented in Table IV5. The portfolio weights appear to be very sensitive to small changes in the data used in the multiple regressions. Multicollinearity, due to the bills, was probably introduced into the regressions, since price movements of bills with close maturities are strongly interrelated. Whenever a high degree of multicollinearity is encountered, the estimates of the regression coefficients will be highly imprecise and nonstationary over successive time intervals. This could possibly explain the large observed changes in the portfolio weights each period. Adjusting the portfolio weights each month by the indicated amounts would involve substantial transaction costs, but it might be possible to reduce these costs and see how important the shifts in the portfolios weights are, by reducing the frequency of the rebalancing. That is, the success of the "quasifutures" contracts as hedging instruments will be examined when the stockbill portfolios are rebalanced: 1) every 3 months 86 or at quarterly intervals; and 2) every 6 months or at semiannual inter vals. The procedure will be to keep the same position in each stock as determined at the beginning of the interval until the interval elapses, at which time the portfolio is rebalanced based on the sixty previous months of price data. As for the bills, the proceeds from each bill when it matures will be reinvested in the bill of the longest maturity. That is, at maturity the funds from the one month bill wil.1 be placed in a six month bill. A comparison of the observed values of the hedging portfolios and the goods will first be made for the quarterly rebalancing case and then for the semiannual case. Table V25 contains the results of the time series regressions of equation (Vl) for the period January 1966 through March 1976. The hypothesis that YkO = 0 for each good is rejected for two of the ten goods (Apparel and Upkeep: t = 4.03, and Household Furnishings: t = 2.26) at a critical t value of 1.658 for a .10 signif icance level with 121 degrees of freedom. The hypothesis that Ykl = 1 for each good is also rejected forthe same two goods. The values of F, which test the joint hypothesis that YkO = 0 and Ykl = 1 for each good are presented in the last column. The hypothesis is rejected for the two goods, Apparel and Upkeep and Household Furnishings, at a critical F value of 3.07 for a .05 significance level with 2 and 121 degrees of freedom. Finally, the values of the R2 of each regression are lower than when the portfolios were rebalanced monthly. Before concluding that the portfolios still serve as an unbiased hedge for eight of the ten goods, the values of the DurbinWatson statis tics should be carefully examined. The critical DurbinWatson value at a .05 significance level is 1.65 for 123 observations. Thus, each of the Table V25 Portfolios as Substitutes for Consumption Goods: Quarterly Rebalancing Fkt = YkO + Yklkt + kt January 1966 March 1976, T=123 (standard errors in parenthesis) Good kO t(kOO)a kl t(kl)a WDurbin 2 F(kO=0, y b Good k0_ tkO) tk1 tkl' Watson R2 (YkO ^^kl= FD 18.58 .37 1.24 .62 1.01 .08 .85 (49.89) (.39) GE 19.91 .49 .93 .20 1.00 .06 1.40 (40.26) (.33) AU 235.43 4.03 2.89 3.85 .96 .22 8.91 (58.48) (.49) PT 16.22 .35 1.19 .47 .83 .07 .55 (46.07) (.39) RR 21.79 .78 1.12 .51 .92 .16 2.42 (28.09) (.24) MC 15.06 .49 1.04 .17 1.21 .14 1.51 (30.74) (.24) MMP 44.92 1.04 .72 .84 .67 .04 .84 (43.37) (.33) LWP 37.55 .37 .89 .15 .84 .01 2.53 (101.42) (.72) CAP 24.06 .98 1.22 1.06 1.15 .23 .59 (24.51) (.20) HFO 56.55 2.26 1.41 2.00 .98 .28 3.52 (25.06) (.21) aCritical t = 1.658 S 10,121 Critical F = 3.07 .05;2,121 regressions exhibit positive autocorrelation of the residuals. This fact alone is sufficient to conclude that the "quasifutures" contracts did not provide an unbiased hedge in the months they were not rebalanced. When the portfolio is not rebalanced in any given month, there is much information contained in the previous residual that is not being used by the investor to adjust the expected value of the portfolio to that of the good. The results for the semiannual case are presented in Table V26, The results indicate a further deterioration of the success of the port folios as hedging instruments. The hypothesis that the estimated inter cept and slope coefficients are equal to zero and one, respectively, can be rejected for nine of the ten goods. In addition, the values of the DurbinWatson statistics are somewhat lower. It appears that rebalancing at quarterly or semiannual intervals in order to avoid frequent brokerage commissions is a poor strategy for an investor who wishes to maintain an unbiased hedge. The realization of events each month appears to signi ficantly influence the portfolio weights. Futures markets allow investors to maintain a perfect hedge against commodity price inflation. This dissertation demonstrates how an investor can create an unbiased hedge, on a monthly basis, using the markets for common stocks and Treasury bills. Thus, the amount contributed by futures markets to the overall completeness of financial markets is questionable. In addition, trading in futures contracts does not exist for many commodi ties. However, the techniques explored in this dissertation allow inves tors to hedge against price inflation for any commodity. In order to maintain an unbiased hedge, frequent rebalancing of the portfolios is required. Thus, futures markets may be economically valuable since they allow investors to more efficiently hedge in terms of transaction costs. Table V26 Portfolios as Substitutes for Consumption Goods: Semiannual Rebalancing Fkt =kO +klukt + kt January 1966 March 1976, T=123 (standard errors in parenthesis) Good YkO FD 165.51 (74.17) GE 33.81 (58.81) AU 443.56 (74.38) PT 92.23 (75.02) RR 15.63 (28.79) MC 35.17 (53.25) MMP 50.16 (66.03) LWP 92.87 (171.18) CAP 162.73 (39.16) HFO 132.94 (33.56) t(ko0)a kl 2.23 2.63 (.58) .57 1.07 (.48) 5.96 4.77 (.62) 1.23 2.03 (.63) .54 1.21 (.24) .66 1.32 (.41) .76 .90 (.50) .54 .44 (1.21) 4.16 2.44 (.33) 3.96 2.17 (.28) t(kl=l )a 2.81 .15 6.08 1.63 .88 .78 .20 1.19 4.36 4.18 Durbin Watson .85 .55 .69 .52 .50 .56 .47 .51 1.30 .87 R2 .15 .04 .33 .08 .17 .08 .03 .01 .32 .34 F(yko=O'Ykl=1 ) 7.52 7.48 18.33 4.81 3.15 .47 3.17 4.50 9.80 9.65 aCritical t1 Critical F121 .05;2,121 1.658 = 3.07 CHAPTER VI SUMMARY AND CONCLUSION VI. A A Summary of This Research This dissertation has investigated whether portfolios composed of defaultfree bills of various maturities and shares of common stocks exist that will allow an investor to hedge against commodity price inflation, without actually entering the futures markets. These hedging portfolios, or "quasifutures" contracts, for a particular commodity were constructed such that they would have the same price and risk proper ties as the commodity itself, such as the expected price at delivery and covariances with the stock market, all other commodities, and longterm bills. The results indicate that an investor can create a stockbill portfolio which will serve as an unbiased hedge against commodity price inflation. Thus, if markets are perfect and securities are traded cost lessly, futures markets may be unnecessary since they do not provide an investor with a service that cannot already be duplicated in the existing markets for common stocks and Treasury bills. The techniques explored in this dissertation allow investors to hedge against price inflation in any commodity. Previously, investors could only obtain perfect hedges against price inflation in commodities for which they could purchase a futures contract. Thus, hedging by means of "quasifutures" contracts will be especially appealing to investors who wish to hedge against price inflation in commodities for which organized futures trading does not exist. Examination of the portfolio weights needed to maintain each month's hedge indicated a need for frequent rebalancing. This extensive rebal ancing, which would be costly to an investor, indicated that futures markets do provide a necessary service once the assumption of perfect markets is relaxed. In addition to the value of their information con tent (see Black (1976)), organized commodity exchanges are economically valuable since they provide a less expensive means for hedging against commodity price inflation versus the use of "quasifutures" contracts. Explanations by various authors for the necessity of futures markets were presented in Section I. B. Included among these were (1) the insurance viewpoint of Keynes; (2) the gambling casino viewpoint of Hardy; (3) the information content derived from the pattern of futures prices discussed in Black; and (4) Stoll's rationale that farmers or privately held firms are reluctant or unable to trade ownership claims on certain assets or production techniques with which they are endowed. The literature, however, contains little empirical analysis of the above viewpoints. Dusak examined the returns to holders of futures contracts and found evidence which was directly contrary to Keynes's viewpoint and only partially supportive of Hardy's. Long (1974), in his development of a multiperiod capital asset pricing model, uses an economy that not only includes a stock market, as do the traditional singleperiod models, but also includes a market for defaultfree bills of different maturities and many consumption goods whose future prices are uncertain. Long relates the price of an asset to not only the systematic market risk, but also to the risk due to changing consumption opportunities (inflation risk) and changing investment oppor tunities (interest rate risk). Implementation of Long's pricing equation (see equation (113)) requires the development of a specific stockbill portfolio. This stockbill portfolio, which has a current exdividend price of FkO, is constructed so as to have a time 1 "withdividend" value equal to the expected time 1 price of a particular good, 1k1, and also to have the same covariances with the elements of (VMI1, nIl Bl) as does 11kl. This hedging portfolio is referred to as "quasifutures" contract for good k. Chapter III presented a detailed methodology of how to create a "quasifutures" contract for a particular commodity. The steps to be taken included: (1) performing time series regressions using historic information on stock, commodity, and Treasury bill price data; (2) in verting a matrix containing the above regression coefficients, and inter preting the kth row as being the unit quantities of a stockbill portfolio that has the same covariances with the vector of prices (VMI1, i' B1) as does the kth good; and (3) setting the expected value of each hedging portfolio equal to the expected value of the kth good by adding a quan tity of shortterm bills. Section IV. A discussed the various assets used in the study for which data were collected. The commodity data consisted of seven consump tion bundles used in constructing the Consumer Price Index and three components from the Wholesale Price Index. The selected commodity indices are listed in Table IV1. Eleven various industry portfolios were created since the risk parameters using portfolio data are more stable than those of individual stocks. Table IV3 lists the industries used which were selected from the Compustat classifications. Finally, Treasury bills, with maturities from one month up to six months, were used to represent prices of defaultfree couponless debt obligations. Estimation of the regression coefficients needed for the calculation of the portfolio weights was discussed in Section IV. B. Each time series regression consisted of observations on stock, good, and bill price data taken from the sixty most recent months. Problems with serial correlation in the residuals were encountered in the regressions. This problem was alleviated by using the method of first differences. The use of this technique was defended by showing that the values of the first order serial coefficients of the regressions were close to unity. The "quasifutures" contracts for each good were constructed to serve as one month hedges. In order to properly maintain each hedge, new portfolio weights were computed each month using the regression coefficients that were estimated from the sixty most recent months of price observations. Thus, every month from December 1965 through February 1976, new "quasifutures" contracts were constructed for each good for delivery one month later. The analysis of the composition of the portfolios indicated that the bills served as a better hedge against commodity price inflation than did common stocks. It was shown in Table IV7 that the portfolios were largely composed of Treasury bills with a short position in stocks frequently observed. These results were similar to those of Fama and Schwert (1977a),who found that bills served as a complete hedge against expected inflation while the returns on common stock were negatively related to the expected component of the inflation rate. Section V. A presented tests of hypotheses concerning the actual success of the "quasifutures" contracts as hedging instruments. The first test was to indicate if the subsequently observed value at delivery of each "quasifutures" contract was identical to the observed value of the good. Although the hedge was not perfect, the portfolios did serve as statistically unbiased hedges against commodity price inflation as seen in Tables V1 and V2. The second test indicated that the risk properties of the stockbill portfolios were similar to that of the goods. If each series of "quasifutures" contracts had the same covari ance with the vector of prices (VM1' "1' B1) as did each good, then any observed difference between the value of the portfolio and the good each month should be nonsystematically related to (VMIl, 1 B1). The results in Tables V3 through V12 showed that these differences were insignifi cant, indicating the covariance properties of the hedging portfolios and the goods were similar. The tests were repeated for the period August 1971 through December 1974 when the Nixon wage and price controls were in effect. The results presented in Tables V13 and V14, and Tables V15 through V24 indicated that the controls had little or no effect on the success of the portfolios as hedging instruments. An attempt was made to reduce the costs of maintaining the proper portfolio weights by reducing the frequency of the rebalancing. The effect on the quality of each hedge was examined when the portfolios were rebalanced at quarterly and semiannual intervals. A comparison of the observed values of the portfolios and the goods for the quarterly rebalancing case was presented in Table V25. There was a significant difference in the two values for two of the goods. More importantly, all of the regressions exhibited positive serial correlation of the residuals as indicated by the low DurbinWatson statistics. The "quasi futures" contracts did not provide an unbiased hedge during the unrebal anced months. In addition, rebalancing would have incorporated any infor mation from the previous month's error. The results for the semiannual case presented in Table V26 indicated a further deterioration in the success of the portfolios as hedging instruments. Thus, rebalancing at quarterly or semiannual intervals in order to avoid costly brokerage commissions is a poor strategy for an investor who wishes to maintain an unbiased hedge. The realization of events each month appears to significantly influence the portfolio weights. V. B Suggestions for Further Research This dissertation constructed portfolios which hedged against various consumption bundles used in compiling the Consumer Price Index. A logical step would next be to construct hedging portfolios using actual commodities for which organized commodity exchanges exist. An investor would then be able to compare the price of a "quasifutures" contract for a particular commodity with the actual price of a futures contract for that commodity. If the price of the "quasifutures" contract was lower, then an investor could arbitrage the difference in prices by selling the futures contract and purchasing the "quasifutures" contract. At the delivery date the investor would use the funds from the "quasi futures" contracts to settle his short position in the futures contract. An alternate approach for hedging against commodity price inflation might be taken from Manaster (1979). Manaster shows how all nominal efficient portfolios can be made real efficient by the simple addition 