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The convenience yield : a model and empirical examination of the relationship between commodity futures… Howe, Maureen E. 1987

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T H E C O N V E N I E N C E Y I E L D : A M O D E L A N D E M P I R I C A L E X A M I N A T I O N OF T H E R E L A T I O N S H I P B E T W E E N C O M M O D I T Y F U T U R E S PRICES A N D C U R R E N T SPOT PRICES By M A U R E E N E. H O W E B . Comm. Honors, The University of Manitoba, 1979 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R OF P H I L O S O P H Y in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department of Finance) We accept this thesis as conforming to the required standard U N I V E R S I T Y OF BRITISH C O L U M B I A September, 1987 ©Maureen E. Howe In p resen t i ng this thesis in part ial fu l f i lment o f the requ i remen ts for an a d v a n c e d d e g r e e at the Un ivers i ty o f Br i t ish C o l u m b i a , I agree that the Library shal l m a k e it f reely avai lable for re fe rence a n d s tudy . I fur ther agree that pe rm iss ion for ex tens i ve c o p y i n g o f th is thesis for scho la r l y p u r p o s e s may be g ran ted by the h e a d o f m y depa r tmen t o r by his o r her represen ta t i ves . It is u n d e r s t o o d that c o p y i n g o r pub l i ca t i on of this thesis fo r f inanc ia l ga in shal l no t b e a l l o w e d w i t hou t m y wr i t ten p e r m i s s i o n . D e p a r t m e n t o f ^ V ^ > ^ > c ^ T h e Un ivers i ty o f Bri t ish C o l u m b i a 1956 M a i n M a l l V a n c o u v e r , C a n a d a V 6 T 1Y3 Da te CfbfP /<?3 ii ABSTRACT This thesis examines the cross-sectional and time series variation between commodities futures prices and current prices. The 'Theory of Storage' states that the difference between the two prices will be a function of two factors: The first is the cost of storing the commodity over the term of the futures contract (carrying costs). The second factor is the value of the convenience yield. The convenience yield is a concept which evolved from the theory of storage and is explained as the benefit which accrues to the individual or firm that holds the commodity in storage but does not accrue to the holder of the futures contract. It is generally assumed that the value of a commodity's convenience yield is decreasing in the aggregate inven-tory available and some indirect empirical support has been generated for this assumption, however, an economic model has not been provided which derives the result. There are two objectives of this thesis: The first is to provide a model of the convenience yield which explains the relationship between the level of inventories and the value of the convenience yield. The second objective is to empirically test the predictions of the model. The model provided shows the convenience yield to be decreasing in the level of ag-gregate inventory. In addition, the value of the convenience is found to be related to the time-series process of shocks to demand. An analogy is drawn between the convenience yield and an option with a stochastic exercise price. Using futures price data and aggregate inventory data, the empirical implications of the model are tested. The results support the hypothesis that a commodity's convenience yield is decreasing in aggregate inventory. Some evidence is also provided that the convenience yield is decreasing in the correlation between shocks to demand. i i i T A B L E OF C O N T E N T S Page A B S T R A C T ii LIST OF T A B L E S v LIST OF F I G U R E S vii A C K N O W L E D G M E N T S viii C H A P T E R I - I N T R O D U C T I O N 1 C H A P T E R II - L I T E R A T U R E R E V I E W 4 2.0 Introduction ' 5 2.1 Theories of Hedging and Returns of Speculators 5 2.2 Theories of Asset Pricing and Compensation For Risk 11 2.3 Theory of Storage 19 2.4 Discussion and Critique of the Existing Literature of the Theory of Storage 26 C H A P T E R III - T H E O R E T I C A L M O D E L 28 3.1 Description of the Economy 29 3.2 Storage and Production Decisions 30 3.3 Characterization of the Decision 32 3.4 Analysis of Convenience Yield 35 3.5 Options with Stochastic Exercise Prices 37 3.6 Summary of Results 38 C H A P T E R IV - E M P I R I C A L A N A L Y S I S 40 4.1 Data 41 4.1.1 Description 41 4.1.2 Time-series Plots 43 4.1.3 Summary Statistics 44 4.2 Econometric Design 46 4.2.1 General Linear Model 46 4.2.2 Heteroscedasticity 47 4.2.3 Serial Correlation 49 4.2.4 Normality 51 4.2.5 Outliers 51 4.2.6 Cross-Dependancies 53 4.2.7 Non-Constant Parameters 54 4.3 Results 55 4.3.1 OLS and GLS Regressions 55 4.3.2 Trimmed Samples 56 4.3.3 Seasonal Parameters 56 4.3.4 Seemingly Unrelated Regressions (SUR) 57 4.3.5 Dummy Variable Regressions 57 4.3.6 Replication of Fama and French 59 4.3.7 Autoregression in Demand Shocks 61 4.4 Summary 62 iv T A B L E O F C O N T E N T S - cont. C H A P T E R V - C O N C L U S I O N 63 A P P E N D I X 68 1 - Proof:The Correlation Between h\ and h2 is p. 69 2 - ProofrThe Convenience Yield is Decreasing in Aggregate Inventory. 70 3 - Proof:The Convenience Yield is Zero When p — 1. 74 4 - Proof:When Marginal Costs of Production Are Constant 76 The Convenience Yield is Decreasing in p. 5 - Adjustment For Heteroscedasticity 77 6 - Derivation of the Univariate F-Statistic 78 7 - The Correlation Between P\ and P2 is Increasing in 2 — j - . 81 dir. 8 - The Comparative Static, < 0. 84 9 - Data Sources 86 10 - Tables 88 11 - Figures 113 B I B L I O G R A P H Y 139 V LIST OF T A B L E S Number Title Page 1 Correlation Coefficients Between Commodity Spot Prices and Futures Prices of Contracts for Immediate Delivery 88 2 Summary Statistics - U.S. Commercial Stocks of Commodities 89 3 Summary Statistics - Quarterly Spot Prices 90 4 Summary Statisitics - Quarterly Futures Prices 91 5 Summary Statistics - Quarterly Convenience Yields 92 6 Summary Statistics - Quarterly Ratios of Convenience Yield to Spot Price 93 7 Summary Statistics - Quarterly Futures Returns 94 8 Tests for Heteroscedasticity in Convenience Yields 95 9 Tests for Heteroscedasticity in Futures Returns 96 10 Goldfeld-Quandt Test Statistics of Within Season Heteroscedasticity 97 11 Tests of Serial Correlation and Normality of the Data 98 12 Indirect Tests of Autoregression in Demand Shocks From Spot Prices 99 13 Regression of the Convenience Yield Projected on Inventory 100 14 Regressions of the Convenience Yield Projected on the Negative of the Reciprocal of Inventory 101 15 Regression of the Convenience Yield Projected on the Natural Log of Inventory 102 16 OLS Regressions of the Convenience Yield Projected on Inventory Omitting Outliers 103 17 Pre-Harvest Quarter Regressions of the Convenience Yield Projected on Inventory 104 18 Post-Harvest Quarter Regressions of the Convenience Yield Projected on Inventory 104 19 Pre-Harvest Quarter Regressions of the Convenience Yield Projected on Inventory Using a GLS Model 105 20 Pre-Harvest Quarter Regressions of the Convenience Yield Projected on the Negative of the Reciprocal of Inventories Using a GLS Framework 105 21 Pre-Harvest Quarter Regressions of the Convenience Yield Projected on the Natural Log of Inventory Using a GLS Model 105 22 Regression of Convenience Yield Projected On Inventory Using a SUR Framework 106 23 F-Statistic For Tests of Cross-Sectional Differences in the Slope Parameter for Inventory 106 VI LIST OF T A B L E S - cont. 24 Regression of the Convenience Yield Projected On The Negative of the Reciprocal of Inventory 107 25 F-Statistic For Tests of Cross-Sectional Differences in the Slope 107 Parameter for the Negative of the Reciprocal of Inventory 107 26 Regression of the Convenience Yield Projected on the Natural Log of Inventory Using a SUR Framework 108 27 F-Statistic For Tests of Cross-Sectional Differences in the Slope Parameter for the Natural Log of Inventory 108 28 GLS Regressions of the Convenience Yield on Quarterly Seasonal Dummies 109 29 F-Statistics for Tests of Seasonaltiy in the Quarterly Convenience Yield 110 30 GLS Regressions of the Futures Return on the Quarterly Nominal Yield and Quarterly Seasonal Dummies 111 31 F-Statistics for Tests of Seasonality in the Quarterly Dummy Variables 112 vii LIST OF F I G U R E S Number Title Page 1 Equilibrium Prices when p = 0. 113 2 Equilibrium Prices When p = — 1. 114 3 Equilibrium Prices When p = 1. 115 4 Time Series of Wheat Inventories 116 5 Time Series of Corn Inventories 117 6 Time Series of Soybean Inventories 118 7 Time Series of Soyoil Inventories 119 8 Time Series of Copper Inventories 120 9 Time Series of Silver Inventories 121 10 Time Series of Gold Inventories 122 11 Time Series of Wheat Prices From 1974 to 1986 123 12 Time Series of Corn Prices From 1974 to 1986 124 13 Time Series of Soybean Prices From 1974 to 1986 125 14 Time Series of Soyoil Prices From 1974 to 1986 126 15 Time Series of Copper Prices From 1974 to 1986 127 16 Time Series of Silver Prices From 1974 to 1986 128 17 Time Series of Gold Prices From 1974 to 1986 129 18 Wheat (Chicago) - Convenience Yield vs Inventory 130 19 Wheat (Kansas) - Convenience Yield vs Inventory 131 20 Wheat (Minneapolis) - Convenience Yield vs Inventory 132 21 Corn (Chicago) - Convenience Yield vs Inventory 133 22 Soybeans (Chicago) - Convenience Yield vs Inventory 134 23 Soyoil (Chicago) - Convenience Yield vs Inventory 135 24 Copper (CMX) - Convenience Yield vs Inventory 136 25 Silver (CMX) - Convenience Yield vs Inventory 137 26 Gold (IMM) - Convenience Yield vs Inventory 138 viii ACKNOWLEDGEMENTS I would like to thank those people who have helped in the completion of this thesis. First are the members of my thesis committee: Rob Heinkel, my advisor, Jim Brander and Jack Hughes. Each member unselfishly contributed his time and effort and I am truly grateful to them. My graditude is also extended to Rex Thompson who made me believe in my ability and provided encouragement and support at every step in the process. Finally, I would like to thank my sister Jan for helping me bear much of the agony of the PhD programme, while reaping none of the benefits. To each of the above, I dedicate this dissertation. CHAPTER ONE I N T R O D U C T I O N 2 1.0 Introduction This thesis examines the cross-sectional and time series behavior of the relationship between commodities futures prices and current spot prices. The theory which describes the relationship between these two prices is termed 'the theory of storage'. In simple terms, the theory of storage states that a commodity's futures price will equal the spot price plus storage costs. This simple explanation of the relationship is unable to address the existence of a commodity's futures price being below its spot price, (inverse carrying charges), a phenomena known to exist at times in commodity markets. In response to the issue of inverse carrying charges the concept of a 'convenience yield' evolved. The convenience yield is an abstract concept. It is described as the benefit which accrues to the individual or firm that holds an inventory of the commodity, but not to the owner of a futures contract written on the same commodity. The convenience yield has been described as the value of the increase in flexibility associated with owning the commodity (ie. deciding where to store the good and when to liquidate the stock). Generally it is assumed that the value of a commodity's convenience yield is decreasing in the level of aggregate inventories. However, no economic model of the convenience yield has been provided that examines the comparative static of the convenience yield with respect to aggregate inventories and identifies other exogenous variables that will be important in determining the value. The first objective of this paper is to provide such a model of the convenience yield. The second objective of the thesis is to test the implications emanating from the proposed model. The major implication is that the value of the convenience yield is decreasing in aggregate inventory. While other authors have generated indirect support for this hypothesis, none have actually calculated commodities convenience yields and compared them directly to aggregate inventories. The remaining four chapters cover the following topics in addressing the above objectives: Chapter Two summarizes the literature on commodity futures pricing. The model of the con-3 venience yield is derived in Chapter Three. Chapter Four describes the data, the methodology and the results from the empirical analysis. Conclusions and the direction of futures research are presented in Chapter Five. CHAPTER TWO L I T E R A T U R E R E V I E W 5 2.0 Introduction This chapter reviews the literature on the pricing of commodity futures. It is divided into three sections, each of which developes a different perspective on the relationship between spot and futures prices. They include: 1) Theories of hedging and returns to speculators. 2) Theories of asset pricing and compensation for risk. 3) Theories of storage and the convenience yield. Inasmuch as the model presented falls into the third classification, more detailed attention is given to studies considered in that section than to studies considered in the other two. 2.1 Theories of Hedging and Returns to Speculators A n early theory on the relationship between futures prices and expected future spot prices is Keynes' (1930) theory of normal backwardation. In Keynes' view, futures contracts are instruments for providing insurance. Traders in those contracts include hedgers, who are net short in the commodity, and speculators, who are willing to provide insurance at a premium in the form of a futures price below the expected future spot price. For example, farmers would hedge by selling their grain forward in the futures market and would, in effect, pay an insurance premium by way of accepting a price for the grain below the spot price expected to prevail at the time of delivery. The size of the premium will be related to the term of the contract. Futures prices in markets that are characterized by normal backwardation will be downward biased predictors of expected future spot prices and the futures price will rise over the maturity of the contract. Hicks (1947) and Houthaker (1957) expand on the theory of normal backwardation by assuming that, in addition to a premium for bearing risk, speculators may also profit from an ability to forecast future spot prices. Hypotheses as to whether speculators receive risk premia and/or systematically profit from their forecasting ability are tested empirically by both Houthaker (1959) and Rockwell (1967). Essentially, they find that large speculators appear to 6 outperform small speculators in forecasting ability, but that neither appear to earn risk premia. In contrast to the theory of normal backwardation, Telser (1958) argues that competition and free entry to futures markets drive speculators gains (i.e. risk premia) to zero. He then tests the implication that futures prices are unbiased esitmates of future spot prices empirically, and finds evidence to support that prediction. Cootner (1960) takes issue with Telser's results, claiming that Telser assumes away the problem by appealing to the 'unbiased expectations ) theory'. 1 Using similar data, Cootner fails to reject the presence of a bias consistent with normal backwardation. Telser (1960) replied to Cootner's criticism, arguing that hedging commitments may change during the year for seasonal commodities. Early in the crop year and during the latter part, short hedging commitments increase. Since the nature of hedging commitments change, he states that bias in the futures price will also change, and backwardation is not sufficient to completely describe the process. Cootner (1967) proposes that trends in futures prices, whether they are increasing or de-creasing, will be related to hedging patterns. If hedging is dominated by firms shorting the commodity, futures prices should increase over the term of the contract. When hedging is dom-inated by firms which are long the commodity, futures prices will decrease over the term of the contract and will be upward biased. This latter relationship between a commodity's futures price and its spot price is termed 'normal contango'. Other papers that incorporate hedging into the analysis of futures pricing and which derive models of futures pricing from an 'insurance' framework include Johnson (1960), Stein (1960), Heifner (1972) and Peck (1975). Stein (1961) derives a geometric technique for simultaneous determination of spot and futures prices in commodity markets and explains the allocation between hedged and unhedged holdings of stocks. A number of papers generalize the role of hedging into an equilibrium pricing framework. 1 The unbiased expectations theory simply states that futures prices are unbiased predictors of future expected spot prices. 7 Stoll (1979) models the simultaneous determination of commodity futures prices and spot prices from a hedging framework and incorporates the possibility of risk-spreading. in the capital market as well as in the futures market. Previous papers demonstrated that in a world of perfect capital markets, where shares in the production or storage process and the futures contract are fully traded at no costs, there is no need for hedging. 2 The rationale offered for hedging is the inability to trade ownership claims on certain assets or production techniques. Stoll's model is developed in a two-date world where there exists certain untradable assets. Given the existence of positive storage costs, futures prices are below expected future spot prices unless the covariance between prices and stock prices is sufficiently negative. With a large negative covariance the futures price can be greater than the expected future spot price. Papers by Cox, Ingersoll and Ross (1981), Jarrow and Oldfield (1981), Richard and Sun-daresan (1981), and French (1983) examine the relationship between forward prices and futures prices. When a futures contract is entered, the profit or loss made on the change in the futures price is settled at the end of each day by the brokerage house with whom the account is held. This daily debiting or crediting of the investors account is refered to as 'marking to market'. Alternatively, with a forward contract, the profit or loss is realized at the maturity of the con-tract. A futures contract can be thought of as a series of one day forward contracts, where every day the profit or loss is realized and a new contract is written at the current futures price, or the price that sets the value of the contract equal to zero. Richard and Sudaresan construct a rational expectations general equilibrium model in a continuous time, multigood, identical consumer economy with constant stochastic returns to scale production. They show that a futures price is actually a forward price for the delivery of a random number of units of a good. The random number is independent of the good and is equal to the return earned from continuous reinvestment in instantaneously riskless bonds until maturity of the futures contract. The model has implications for the theory of normal backwardation and normal contango and how the theories relate to both forward and futures 2 Baron (1976),, Stoll (1979) 8 prices. Richard and Sudaresan show that the ability of investors to hedge their future consumption to protect against welfare losses will determine whether forward prices are characterized by nor-mal backwardation or contango. It is found that when forward contracts are poor hedges, in the sense that profitability is low when marginal utility of consumption is higher than average and high when marginal utility of consumption is lower than average, forward prices are downward biased. When forward contracts are good hedges, the forward price will exceed the expected spot price. The bias in futures contracts is found to have two components. The first depends on the contracts usefulness as a hedge. The second component depends on the risk of the future contract's profit at time T, the maturity of the contract. If the spot price and the instantaneous interest rate are positively correlated, then risk increases and will lead to a higher futures price. The two sources of bias can be offsetting or reinforcing. It is the combined effect of both that will determine whether the futures price characterizes normal backwardation or contango. Anderson and Danthine (1983) recast the theory of hedging in futures markets in a rational expectations, multiperiod world and introduce production uncertainty. Keynes' original theory considered individuals who were subject only to price risk, but not quantity risk. Producers will be concerned with the variances of price and output, and how the two covary. A n implication of the model obtained by Anderson and Danthine is that when demand is known but output is uncertain, futures prices will be upward or downward biased according to demand elasticity.3 Papers by Kawai (1983) and Turnovsky (1983) also approach the problem from a rational expectations perspective. Kawai developes a model of non-storable commodities with no pro-duction uncertainty. His optimization model assumes risk aversion and price uncertainty and he solves for the futures price in terms of the expected spot price. He finds the futures price to be a biased estimate of the next period's spot price, and a priori, predicts the existence of a non-negative risk premium. He also finds that the introduction of a futures market reduces the 3 Similar results are obtained by Conroy and Rendleman (1983). 9 long run variance of serially correlated spot prices. 4 By providing insurance the futures market stimulates production thus increasing the level of expected supply and lowering the level of expected spot prices in the long run. Turnovsky (1983) takes a rational expectations approach to study futures markets equi-librium for storable commodities and allows for production uncertainty as well as risk averse producers and speculators. He investigates how the implementation of a futures market affects the demand and supply functions of the spot commodity. In the absence of transactions costs, he concludes that futures prices will equal expected future spot prices. When these costs are non-zero the extent of speculation in the futures market will depend on the difference between the current futures prices and the expected spot price in the next period. With risk averse behavior the current futures price is found to be a weighted average of the current spot price and the expected future spot price. The futures price is found to be a biased predictor of the future spot price with the bias depending on the parameters of the cost function for trading inventories and the parameters of the cost function for producing the commodity. In general, the futures price will under predict the future spot price. The introduction of a futures market is also found to decrease the variance and the mean of spot prices. Finally, Turnovsky finds that both in the short run and in the long run the futures price is more stable than the spot price. Britto (1984) derives a similar result to that of Anderson and Danthine (1983), that hedging will depend on the price and income elasicity of demand for producer's output as well as on the degree to which consumers are risk-averse. Britto examines the equilibrium properties of a rational expectations model of hedging when there is futures trading. Futures contracts are assumed to provide individuals and firms with insurance against price risk. When faced with both price and output risk (or production risk), the price and income elasticities will determine whether insurance is purchased through the futures markets. 4 Papers by Danthine (1977) and Lucas (1978) show that market efficiency does not necessarily imply a martingale process for commodity prices. 1 0 Anderson and Danthine (1983), Turnovsky (1983), Kawai (1983) and Britto (1984) provide models which involve individuals concerned with wealth rather than ultimate consumption of different commodities. Hirshleifer (1986) changes the approach of studying futures pricing in a world with price and quantity risk in two respects. First, in his model individuals maximize expected utility levels by choosing consumption levels. Second, he allows for sequential arrival of information about crop output. Earlier partial equilibrium models found that demand elasticity would determine whether futures prices were positively or negatively biased predictors of future spot prices. This result is shown not to hold up in a general equilibrium model without transactions costs, and the paper provides conditions under which futures prices are unbiased predictors of future spot prices regardless of demand elasticity. When fixed transactions cost are introduced, small traders and consumers tend to be driven from the futures markets, leaving only large traders. This phenomena will lead to the prediction of a downward bias under inelastic demand, and upward bias for elastic demand. This last result is consistent with those of the partial equilibrium papers. Hirshleifer's paper also looks at optimal hedges. He finds the optimal hedge to be related to the correlation of a producers's output with aggregate output, the sensitivity of output to the environment relative to the sensitivity of other producer's output, and to demand elasticity. In a later paper that integrates the stock market, futures trading opportunities, demand and output shocks and the cost as well as the revenue side of production, Hirshleifer (1986) provides some additional insights into hedging and futures price bias. The paper has similarities to the earlier paper by Stoll (1979) in that it looks at capital market equilibrium where there exists untradable assets. The results, however, are quite different from Stoll's paper. In this study the bias in futures prices is found to have two components, one due to the covariance with market returns, and one due to producer hedging. The first component represents the effect emphasized by the C A P M 5 approach to future pricing, the second represents the effect 5 C A P M is an abreviation for the capital asset pricing model, Sharpe (1964), and Lintner (1965). 11 emphasized by the hedging approach. The interaction of market risk with hedging brings a new prediction concerning the hedging component of bias: if demand shocks arise from variation in income then the risk premium will rise with the absolute value of income elasticity. While the previous literature found that price bias is positive or negative depending on whether demand is inelastic or elastic, by introducing production costs, there is a tendancy towards backwardation. 2.2 Theories of Asset Pricing and Compensation For Risk The papers summarized in the previous section analyzed the bias in futures prices by deriving models of hedging and speculation. This section examines papers in which futures contracts are priced in equilibrium pricing models including other assets. The most common such model is the Sharpe/Lintner capital asset pricing model. The first application of the C A P M to the pricing of commodity futures contracts was Dusak (1973). Dusak argues that futures markets are no different in principle from the markets of any other risky assets and, therefore, will adhere to the same pricing model. The portfolio approach to the pricing of futures contracts predicts that the bias, or risk premium, embodied in the futures price will depend on the contract's contributution to the risk of a large and well diversified portfolio of assets. The standard form of the C A P M is written as E(Ri) = Rf + [E{RW) - Rf}/3i, where E(RW) is the expected return on total wealth, E{Rt) is the expected rate of return on asset it Rf is the pure time return to capital and = — T S T - ^ - T ^ - . In applying the Sharpe/Lintner model of capital asset pricing to the pricing of a futures contract Dusak encounters the problem of defining the appropriate capital asset and defining and calculating its rate of return. Futures contracts are bought on margins that range from 5 to 10 percent of the face value of the contract and no additional investment is made when the contract is entered. The difficulty is in determining what constitutes the capital investment when calculating the rate of return on the asset. The approach followed by Dusak is to use 12 the percentage change in the futures price. While this change cannot represent a return on capital it can represent the risk premium, i£, — Rf, on the spot commodity. The spot holder who chooses to hedge his/her holding owns a riskless asset and will earn Rf. Representing E(RA in terms of period 0 and period 1 prices as ——>Jk — , the equilibrium risk-return relation on asset i is expressed as ( 2 1 ) p''° ~ — F+W) • The price of asset i under the agreement to purchase the asset at time 0 but with payment deferred until time 1 is given by P, )o(l + -ff/)- Multiplying both sides of equation (2.1) by (l + Rf) the equation is transformed to (2.2) Pifi{l + Rf) = E{PiA) - \E[RW) - R}]Piflf3i. Dusak interprets P,;o(l + Rf) as the current futures price, PF,O, for delivery of the spot commodity one period later, and E(Pii\) as the spot price expected to prevail one period later. Setting Pffi = Pi,o(l +• Rf) and rearranging terms, the risk premium in futures prices is represented as (2-3) ^ - P ^ = m R w ) - R f ] . Equation (2.3) expresses the risk premium on the spot commodity as the change in futures price divided by the period 0 spot price. Measurement of the risk premium in equation (2.3) is accomplished through percentage changes in the futures price. Following along the lines of Dusak's own argument, that Ppt = P ^ l + Rf), this method underestimates the risk premium on the spot commodity by a factor of ^ R • Given the length of the intervals over which she computes returns, she assesses the factor is likely to be very small. Dusak tests the risk-return relationship in the futures markets on wheat, corn, and soy-beans. Semi-monthly price quotations are used for the 15-year period from May 15, 1952 through November 15, 1967, giving a sample size of 300 observations per contract. Returns are 13 calculated separately for each contract which results in a discontinuous series since contracts are usually for a 9 or 10 month span. Based on the Keynes measure of risk, which is variability of returns, she finds the risk of futures returns to be similar to that of the Standard and Poor's Index. However, while the semi-monthly excess return to the Index was about .0029 (excluding dividends) the returns to the futures contracts were close to zero. Using the Standard and Poor's Index to represent the market portfolio and the 15-day Treasury Bi l l rate to represent the risk free rate, beta coefficients are calculated for the different contracts of the three commodities. These coefficients are found to be very small, ranging in size from .007 to .119. The results imply that beta is a better measure of risk than variability of returns. Commodity futures returns conform better to the capital asset pricing model than to the Keynsian model. Two problems with Dusak's methodology are identified by Hughes and Holthausen (1976): First, she fails to recognize storage costs in the dynamics of futures prices. Second, she does not not allow for variations in production. This latter point will determine the investment opportunity set and could cause changes in the parameters of the return distribution. These considerations could be quite problematic for agricultural commodities. In the case of metals, investment returns should outweigh the returns to storage. Seasonality of production is also not a problem for metals. Using copper and silver the authors follow a methodology which is similar to that of Dusak. The one difference is that where Dusak calculated the risk premium on the futures contract as the percentage change in the futures price, acknowledging that it was downward biased by a factor of i^jij, Hughes and Holthausen incorporate the factor into the risk premium such that Bimonthly returns are used and each contract is treated as a separate commodity. The sample period is 1970-1974 inclusive. The Standard and Poor's 500 Index is used to proxy the market. 14 The results are quite different for copper and silver than those of Dusak for agricultural commodities. For wheat, corn and soybeans the estimates of and the mean returns were found to be close to zero. The /?,'s for silver and copper contracts are quite large and often significant at the 10 percent level. The mean returns are also found to be larger than those for agricultural commodities. A comparison of results found for metals and those for agricultural commodities are sup-portive of the portfolio approach to valuing risk premiums in futures prices. The results within the metals are not as clear. There is no obvious relationship between risk and return on the different metals contracts. The authors raise the question, "Do speculators consider markets other than commodity futures markets as appropriate for diversifying risk ". If not, than some measure other than the Standard and Poor's 500 Index must be used as a measure of market return. A n additional investigation of the ability of the capital asset pricing model to explain the variabiltiy of risk premium in futures prices is conducted by Bodie and Rosansky (1980). Using the sample period 1950 to 1976 and 23 individual commodities, this study calculates commodity's excess returns and betas. To examine the relationship between the two variables a cross-sectional regression is run. The authors find the estimated slope of the security market line is negative, which is clearly inconsistent with the C A P M . Betas of many individual commodities are negative and close to zero while the mean excess rates of returns are large and positive. The fact that the returns are positive lends support to the normal backwardation hypothesis. Black (1976) also recognizes that the expected changes in the futures price can be modelled according to the C A P M . He believes, however, that because the initial value of the futures contract is zero, the standard C A P M must be modified so that it applies to dollar returns rather than percentage returns. If future prices are independent of the return on the market, it is hypothesized that the systematic risk associated with holding a futures contract is zero, and the futures price is the expected spot price. The problems associated with applying the general 15 C A P M stem from the stringent assumptions under which it is derived. The capital asset pricing model holds in a single-period world but only under the assump-tions of a stationary investment opportunity and consumption set. The implications of such assumptions are constant relative prices and interest rates, conditions which would make a com-modity futures contract riskless. Such problems have lead researchers to apply more general models to the pricing of futures contracts. Grauer and Litzenberger (1979) model the equilibrium expected excess returns of futures contracts in a state preference exchange economy allowing for absolute and relative price un-certainty. Returns are derived in terms of real market betas. The derivation is in a two-period context and individuals are assumed to have identical homothetic preferences. Futures prices are found to equal a ratio of the real return on a nominal bond to the real rate of interest times the sum of the expected future spot price, a premium for inflation and the contracts risk premium. The risk premium is proportional to the covariance between the real future spot price of the commodity and the social marginal utility of real wealth. Breeden (1980) conducts an analysis applying the Breeden (1979) continuous time multi-good form of the C A P M to futures pricing. He calculates 'consumption-betas' for 20 different commodities over the 1960-1978 period. The continuous time multi-good form of the C A P M differs from the standard Sharpe/Lintner model in its measurement of systematic risk. "The in-tertemporal pricing model states that the equilibrium expected excess return on an asset should be proportional to its covariance of returns with changes in aggregate real consumption." 6 Indi-viduals' preference functions are allowed to be general and diverse. In his model the covariance of a commodity's price with real consumption will depend on its elasticity of supply and de-mand with respect to expenditure and commodity prices, and the covariance between the rate of production and aggregate consumption. Breeden finds that commodities with high income elasticities and relatively low supply uncertainties, have high consumption betas. Meats (cat-tle, hogs, broilers, and pork bellies) and industrial materials (copper, platinum, and plywood) 6 Breeden (1980), page 504. 16 fall into this category. For some commodities the effects of a high income elasicity and low or negative covariance between production and consumption tend to offset one another. Breeden concludes that cocoa, cotton orange juice, potatoes and sugar are such commodities. For these foods, consumption betas vary greatly from large and negative for sugar to large and positive for orange juice. The consumption betas for grains were found to be low or negative. Breeden resolves that his estimates of consumption betas are, in general, consistent with his theoretical model, although he provides no statistical test of the model's predictions. If the continuous time consumption based C A P M is an accurate interpretation of the world it is understandable why studies which apply the standard Sharpe/Lintner form of the C A P M to futures pricing have generated ambiguous support of the model. Studies applying the standard C A P M will have measured systematic risk incorrectly. Extending the ideas presented in Breeden (1980), Hazuka (1984) developes a linear rela-tionship between a commodity's 'consumption beta' and its risk premium. The model is tested using 15 commodities, each which is classified as either non-storable, seasonally storable, or non-seasonally storable. Using the consumption betas calculated by Breeden for the fifteen commodities in the sample, Hazuka finds the correlation coefficient between the slope coeffi-cients and consumption betas is .563. This coefficient is significantly different from zero at the 95 percent confidence level and is in the predicted direction. These results provide evidence that consumption betas may be a better estimate of the systematic risk an investor is exposed to when entering into commodity futures contracts than the conventional Sharpe/Lintner betas. A recent paper by Fama and French (1987) compares the risk premium theory with the theory of storage.7 These authors study the "long and continuing controversy" of risk premiums in futures prices using a methodology different from the portfolio approach. The difference between a commodity's futures price and current spot price is divided into a forecast of the change in the spot price, Et[S(T) - S(t)}, and an expected premium, Et[P(t, T)], where S(T) is the future spot price, S(t) is the current spot price, and P(t, T) is the premium in the futures 1 The theory of storage is discussed in the next section of this chapter. 17 price. The expected premium represents the bias in futures prices as forecasts of expected spot prices. The equation representing this relationship appears as (2.5) F(t, T) - S(t) = Et[P[t, T)] + Et[S(T) - S{t)} and the expected premium as (2.6) Et[P(t, T)\ =F(t,T)-Et[S(T)}, where F(t, T) is the current futures price for delivery at time T. Using the regression approach of Fama (1984a, 1984b) the authors look at the coefficients of two regressions, the change in the spot price, Evidence that the futures price has forecast power of the future spot price is provided if b\ is positive. If 62 is positive it is evidence that the observed premium at time t contains information about the premium to be received at T, which in turn is evidence of time-varying expected premiums. These regressions are run for 21 commodities and the results are reported for maturities of 2-, 6- and 10- months. Fama and French find large variation in futures prices and attribute the low significance of their test statistics to it. They conclude there exists a relation between basis variability and the ability of futures prices to forecast future spot prices, however, they have difficulty identifying a time-varying expected premium. To increase the power of univariate tests of the existence of a non-zero expected premium, they combine contracts of a particular commodity to ensure an observation is available each month for every commodity, and then combine commodities into portfolios. This method provides marginal evidence of normal backwardation, but the authors S(T) - S(t) = ax + b![F(t, T) - S(t)\ + u(t, T), and the premium on the basis, F{t, T) - S{T) = a 2 + b2[F{t, T) - S{t)\ + z(t, T). 18 conclude " . . . the evidence is not strong,enough to resolve the long-standing controversy about the existence of nonzero expected premiums." 8 The papers that have been summarized in this section describe the equilibrium relationship between futures and spot prices from a C A P M or extended C A P M approach. The one exception is the paper by Fama and French which does not attempt to model the risk premium in an equilibrium setting, but instead investigates empirically the existence of such a premium. The final paper to be discussed in this section is Gay and Manaster (1986). This paper looks at the quality option embodied in futures contracts. Sellers of commodities in futures markets have some discression over the quality of commodity they deliver. At the time the contract is entered there is some uncertainty regarding which grade will be more valuable at the maturity. This provides an option to the seller of the commodity, which will decrease the futures price. Ignoring the convenience yield, the equilibrium relationship between futures price and the spot price which is necessary to prevent arbitrage is (2-7) Ft(T) = ^-^[Xu + StiT) - Wt(T,Xu,X2t)\, where Ft(T) is the futures price at time t for a contract that matures at time T, Xft is the spot price of the z"1 good in the deliverable set at time t, St(T) is the current value of the cost of storing the deliverable goods from time t to time T, Wt{ T, Xu, X2t) is the value at time t of an option to exchange asset X2 for asset Xi at time T, and r is the risk free rate. A t maturity, the futures price will equal the spot price of the least expensive deliverable asset. Assuming that the spot price follows a Wiener process and trades in frictionless markets, the Margrabe (1978) formula for an exchange option is applied to value an option to exchange number two hard winter wheat for number two soft red wheat. These two grades of wheat are delivered almost exclusively on the Chicago exchange although the C B O T wheat contract allows eleven different qualities of wheat to be delivered. Examining the period from January 8 Fama and French (1987), page 72. 19 1975 to December 1980 the arbitrage condition of equation (2.7) is tested. The evidence suggests that when the quality option is explicitly considered the futures price is more highly correlated with spot prices, however, the evidence deteriorates as time to maturity increases. The value of the option should be increasing with time to maturity. One possible explanation provided for this inconsistency is that the assumptions made are short term in nature and do not hold up over longer time intervals. 2.3 The Theory of Storage The risk premium theories described in the previous section examine the relationship be-tween the expected future spot price of a commodity and its futures price. In contrast, the theory of storage attempts to explain the relationship between a commodity's current spot price and its futures price. This theory was first formalized by Kaldor (1939) and Working (1948,1949). In its simplest form the theory states that the futures price will equal the spot price plus storage costs. Take an economy with two dates, to and t\, and with no transactions costs. Individuals wishing to take a speculative position in the commodity have two choices: 1. Buy a futures contract. 2. Buy the commodity and store it. In either case consider a 100% leveraged position. If a futures contract is purchased at io, at expiration of the contract the net cash flow is (2.8) - * b , i + h = CashFlow(futures), where, -Fo,i = the futures price at to for delivery at t\, and P i = the unknown future spot price in time period one. If the commodity is purchased and stored for speculation on the future spot price, the net cash flow at t\ is (2.9) - P 0 ( l + Ri) + h ~ C = CashFlow(storage), 20 where, C — the storage costs, Po = the spot price at io. and Ri = the risk free rate over the period. Let 7 = -p^. Storage costs can then be expressed as, C- = 7 P 0 , Now equation (2.9) can be written as, (2.9') - P 0 ( l + Rt+.i) + h = CashFlow(storage). For futures contracts to be held in equilibrium, CashFlow(futures) > CashFlow(storage). The relationship between the cash flow from the two alternatives implies, -F0li > -P0{l + Ri+i), or, (2.10) F0:i < P0{l + Rx + i). In a two date economy equation (2.10) should hold with equality. In multiperiod economy a futures contract maturing more than a single period ahead is not necessarily equivalent to the stored commodity. Kaldor and Working used the concept of a convenience yield derived from stocks of the commodity held in storage to explain the phenomena of "inverse carrying charges".9 This yield may arise from situations such as increased flexibility in production runs, meeting unexpected demand and maintaining a given level of output at a lower cost. "Recognizing the time lost and the costs incurred in ordering and transporting a commodity 9 Brennan (1958) defines inverse carrying charges as futures prices below spot prices or prices of deferred futures below that of near futures. 21 from one location to another, the marginal convenience yield includes both the reduction in costs of acquiring inventory and the value of being able to profit from temporary local shortages of the commodity through ownership of a larger inventory." 1 0 The convenience yield must be subtracted from storage costs in calculating net storage costs. In equilibrium the difference between the future price and the spot price of a commodity will equal storage costs (ie. interest charges, insurance, and rent) less the marginal convenience yield. Inserting an additional term to represent the convenience yield, in a multiperiod world equation (2.10) becomes (2.11) F0,t-Po{l + Rt + lY= ~Kt, where Kt is the convenience yield, Fo,t is the futures price for delivery in period t, and, Rt is the. risk free rate over t periods. In a multiperiod world the futures ratio is defined as (2.12) ^ = (i + R t + ^ y - k t ! where kt= p^. The futures ratio can be reinterpreted in return notation as, F0,t - Pi Po °~ = {l + Rt + 1)t-{kt+l), where, —4, is the "futures return". Kaldor and Working hypothesized the marginal convenience yield to be a decreasing func-tion of the level of inventories. The lower the inventory level, the greater will be the convenience yield of an additional unit. In the case where the level of stocks is extremely low, the marginal convenience yield may actually exceed marginal storage costs. In this situation, the spot price would be higher than the future price, resulting in inverse carrying charges. Alternatively, when stocks of the commodity are sufficienty high, the convenience yield may be near or equal zero. 1 0 Brennan (1986), page 2. 22 The assumption that the convenience yield falls to zero at high inventory levels was made explicitly by Brennan (1958) in estimating the risk premium in the supply curve of storage. In an empirical examination of agricultural commodities, Brennan found the difference between the marginal risk premium and the marginal convenience yield is increasing in stocks of the commodity. Brennan's study does not directly support the proposition that the marginal con-venience yield is decreasing in stocks since the marginal risk premium is hypothesized to be positively related to the level of inventories. Telser (1958) investigates the relationship between the basis, (which he defines as the difference between futures prices of successive maturities adjusted for storage costs, the general price level, and the time spanned by the maturity dates of the adjacent futures contracts), and aggregate inventories for wheat and cotton. Plotting the basis against the logarithm of stocks, the spread between succesive futures contracts is greatest when inventories are low. To statistically test this relationship a regression of the form (2.13) W = a + b • log(y) + c • log(q) + d-(g) is estimated where W is the spread between futures prices, y is aggregate stocks, q is consump-tion and g is the fraction of total stocks held by the government. The variables a, b, c, and d are regression coefficients. Telser includes the variable g since the government set a lower limit on their selling price which would thus decrease the convenience yield of a given quantity of stocks. In Telser's study, W equals the marginal cost of storage less the marginal convenience yield of a quantity of stocks y. The estimated coefficients for cotton have signs consistent with the described theory but they are not statistically significant. The coefficients for the regressions on wheat have some signs that are inconsistent with the theory, but they are also not significant. Several regressions for wheat do have statistically significant results that are consistent with the theory proposed in the paper. Telser concludes that his empirical results support the proposition that the spread in futures prices is decreasing in inventories. 23 One of the propositions forwarded by Working (1947) is that the current level of inventory was the only variable that would play a role in determining the value of the convenience yield: "The results from all lines of investigation concur in indicating that prices quoted at one time in a futures market, for two dates of delivery, stand in a relation which in general does not reflect expectations regarding events that may occur between the two delivery dates." 1 1 Simply stated, the point being made by Working is that existing inventory is the only variable responsible for intertemporal price spreads. Events that are expected to occur in the future will not impact on the convenience yield. Weymar (1966) shows that while Working's proposition may be true in general, if the time interval between successive price expectations is long enough to allow for a change in the expected inventory level during that interval, his conclusion can be rejected. Weymar uses Working's definition of the price spread, written solely as a function of current inventories, to show that in a multiperiod economy expected inventories matter. Following along similar reasoning, Weymar also examines the dynamic behavior of spot prices as they relate to inventories. More recent evidence of the Kaldor-Working hypothesis that the convenience yield is de-creasing in aggregate inventories is provided by Brennan (1986). The objective of his paper is to test the assumption made in Brennan and Schwartz (1985) that the convenience yield of a commodity may be written solely as a function of the spot price. The predictive ability of three different Brennan-Schwartz type convenience yield functions are tested against an alternative model in which the convenience yield follows an autonomous stochastic process and is not de-pendent on the spot price. Comparing the standard errors of the model's convenience yield predictions, the results reveal that while the Brennan-Schwartz assumption works reasonably well for precious metals, the autonomous model of the convenience yield performed the best for all commodities. Brennan tests the Kaldor-Working hypothesis that the convenience yield is decreasing in aggregate inventory. Using the autonomous model of the convenience yield specified in the 1 1 Working (1949), page 1255. 24 paper Brennan generates time series estimates of the instantaneous convenience yield. These estimates are then related to the level of inventories through the estimation of the non-linear regression (2.14) C=a+b(±)n, where C is the estimate of the instantaneous rate of convenience, I is the level of inventories, and X is a measure of the sales rate. The variables a, b and n are parameter estimates. In estimating these parameters for gold and silver the period from September, 1979 to May, 1981 is omitted from the sample due to unusual market events. For these commodities the estimated autonomous convenience yield is found to be large and positive at low inventory levels and small and negative at high inventory levels. For platinum, a negative realtionship is observed only when the entire sample period is used. For the remaining commodities, copper, heating oil, lumber and plywood, strong evidence in favour of the Kaldor-Working hypothesis is provided. It is interesting to note that through the non-linear estimation technique constant costs of storage implies that a < 0, b > 0, n < 0. "For six of the commodities the relation between the convenience yield and the level of inventories is consistent with constant marginal costs of storage; for the seventh, heating oil, the relation is consistent with increasing convex storage costs." 1 2 In the previously discussed paper by Fama and French (1987) the authors specifically acknowledge seasonality in the convenience yield in their empirical investigation of the theory of storage, although they interpret seasonality as a proxy for inventory level. Refering to the difference between the futures price and the current spot price as the 'basis', Fama and French gather monthly observations of the difference between the futures price and the current spot price (the basis) for 1-, 3-, 6- and 12-month maturities. The standard deviations of the basis are found to differ systematically across commodity groups. Silver has the lowest standard deviation of 1.5%, followed by 2% for gold, and 4.2% for platinum. Agricultural products, (grains), range 1 2 Brennan (1986), page 24. 25 from 4.6% for corn to 9.7% for oats. The standard deviations for animal products are the largest ranging from 5.6% for cattle to 22.2% for eggs. While no model is provided in the paper, the authors suggest that demand and supply shocks generate variation in the basis. The larger the inventory level of a commodity, the less likely such a shock will be transmitted to price effects. High storage costs deter storage, therefore, variation in the basis should be an increasing function of storage costs. For more direct tests of the theory of storage the authors run regressions of the basis on the interest rate and monthly seasonal dummies. The equation estimated is where dm equals one if the futures contract matures in month m and zero otherwise. When the commodity is continuously stored it is hypothesized that the slope /? should be 1.0, in other words, the basis should vary perfectly with the nominal interest rate. The seasonal dummies should capture variation in the marginal convenience yield. The strongest evidence of seasonals in the basis is found for animal products (broilers, cattle, eggs, and pork bellies). For these products the interest rate explains only a small fraction of the basis variation and much of the basis variation can be attributed to seasonals. Seasonals in the basis are also found for many of the seasonally produced agricultural commodities (corn, oats, orange juice, soybeans and wheat), however, for five other agricultural commodities (cocoa, coffee, cotton, soymeal and soyoil) there is no evidence of seasonals in the basis. Metals produce the strongest evidence that the variation of the basis tracks interest rates with no evidence of seasonality. (2.15) m=l 26 2.4 Discussion and Critique of the Existing Literature of the Theory of Storage While the literature dealing with the theory of storage extends back to Kaldor (1939), the theoretical development as well as the empirical examinations each have fundamental issues that have not been fully addressed. Working (1948,1949) provides an intuitive economic explanation of why the convenience yield should be related to current inventories. No formal model is provided by Working and his concept of a convenience yield is somewhat vague. Weymar (1966) proves that, according to Working's own formulation of the expected price change, expected inventories could be an important determinant in the price change. He agrees with Working's agrument that over short time intervals, (periods of one to two months), expectations may not be important. Weymar goes on to examine the dynamics of spot price behavior as it relates to inventories and expected inventories, however, he does not explicitly model the convenience yield, leaving it an abstract notion. The literature has generated some empirical support for the Kaldor-Working hypothesis that the convenience yield is decreasing in aggregate inventory, but has done so without ex-plicitly measuring the observed convenience yield and available inventories. The one exception is the study by Telser (1958) which examines cotton and wheat. This research, however, has several problems. First, the inventory data used in the study is aggregate inventory, therefore government stocks, which may not be readily available to industry, have to be accounted for. Secondly, because of the time period over which the study was performed, there is some ques-tion as to the accuracy of the data, particulary in terms of matching aggregate inventory with the point when the spread in the futures prices are observed. A third problem, or at least query, is why Telser included an aggregate consumption variable on the right hand side of of the re-gression equation. No economic reason extends from the theory for doing so. Finally, perhaps due to the above problems, the results obtained by Telser are quite weak, with statistically significant results obtained only for several of the regressions relating to wheat. The more recent papers by Fama and French (1987) and Brennan (1986) come close to the 27 empirical work in this thesis. However, both leave room for additional empirical analysis. Fama and French do not collect inventory data, and therefore conclude that their results support the hypothesis that the convenience yield is decreasing in aggregate inventory based on the coefficients of seasonal dummies. Brennan does not directly measure the convenience yield but instead estimates it using his autonomous model of the convenience yield. He tests the Kaldor-Working theory with a non-linear regression of estimated convenience yields on the level of inventories normalized by a measure of sales. While the regression is somewhat 'ad hoc', Brennan makes the point that there exists no formal model of the dependence of the convenience yield on the level of inventories, so he is not constrained to estimating a specific model. Nevertheless, nomalizing inventory by aggregate sales is not an implication of the discussions of either Kaldor or Working. CHAPTER THREE A MODEL OF THE CONVENIENCE YIELD 3.0 Theoretical Model 29 3.1 Description of the Economy The economy has three dates, to,ty, and t2. In this economy there are J identical price-taking firms, each of which is endowed with inventory of some homogeneous commodity. Let ft denote the endowment of the j th firm. The aggregate supply of inventory is S / = i P — I- The commodity can be sold at to or carried over to t\ or t2. The amount of inventory carried from to to i i by each firm is ^ , and the amount of inventory carried by each firm from t\ to t2 is 4. The commodity can also be produced at t\. The amount produced by each firm is ifx and the total cost of producing q{ is C(q[). The amount a firm sells in each period is y\. The aggregate amount sold in each period is Z~2j=i Vt = Vt- A l l inventory must be sold on or before t2. Interest rates and storage costs are both zero. Random shocks to demand of hi and h2 occur at t\ and t2, respectively. These shocks have the probability functions /t(n<). The pricing function is the same in each period. Pt — Pt{ht,yt) is the price at time t and is linear and increasing in ht and decreasing in yt- Although the price in any period is decreasing in the aggregate amount sold, the individual firms treat the prices as fixed, given the shock to demand. Firms have identical beliefs about the distribution of the shocks to demand. Expectation of the shock at t2 is conditional on the shock incurred at t\. In particular, given the shock to demand at t\, the shock at t2 is h2 = pn\ + e, where, E(e) — 0 and <re = a < oo. 1 3 At to the expected shocks at ti and t2 are both zero. Firms are risk neutral. Each firm acts as a price taker and maximizes profit by selling in the period when they expect to receive the highest price. The notation is summarized on the following page: (i+P2)* ^ Appendix 1 shows the correlation coefficient between iij and h2 is — j -30 Summary of Variables V A R I A B L E ia k k Beginning Inventory £ / = i P = 1 E / = i 4 = *i Amount Produced [Cost] 0 E / = 1 = CM] 0 Amount Sold at t yo = / - «o Vi - «o + ?i - h 1/2 = ii Shocks to Demand 0 h2 = pn\ + e Spot Prices Po •Pi P 2 Futures Prices Po,2 Pl,2 -^ 2,2 3.2 Storage and Production Decisions Firms have three decision variables when maximizing profit: the amount of inventory to carry from to to t\, the amount of inventory to carry from t\ to t2, and the amount of the com-modity to produce. The problem faced by a firm at to can be solved as a dynamic programming problem. First, profit is maximized at t\ by finding the optimal inventory to carry from t\ to t2 and the optimal quantity to produce at t\, given the shock to demand, h\, and the inventory carrried from to,h\. Then, profit is maximized at to by finding the optimal inventory to carry from to to t\. Each firms profit function at t\ given n\ and «Q is (3.1) ^ i ( n i , 4 ) = ^ ( B i ; » i ) V i + ^ ( P 2 ^ « i + c , V 2 ) ) - » ; 2 - C ( ^ ( n 1 ) ) , where, y1 and y2 are taken as fixed. A firm cannot carry more inventory from t\ to t2 than carried from to to t\, plus what it produced at tj, nor can it carry less inventory than zero. Hence, the individual firm's decision problem at t\ is to maximize equation (3.1) subject to the constraints, 31 and { > 0. Aggregating equation (3.1) and the above two constraints across firms, gives the following Lagrangian at t\. L = Pi{n1,y1)-[io + qi{ni)-i1} + Ei(P2(pn1+e,y2))-i1 - C(ql(m)) - A r[z'i - qi(n{)} - A 2 -where, Ai and A 2 are the Lagrange multipliers.1 4 Maximizing the Lagrangian with respect to i\ and qi the first order conditions are: i. ) - P i ( n 1 ; y x ) + EiiPiipri! + ~e, y2)) - A 2 + A 2 = 0, ii. ) PiinuyJ - C( f t(m)) + Ai = 0, iii. ) Ai(n - to - qi(ni)) - 0, and iv. ) A 2n = 0. The optimal t\ profit for the economy is, T * ( « O , ni) = Pi(ni,y{) • [i0 + ?i(n x) - ij] + £ i ( P 2 ( / m i + V2)) • *i -C(q{(n{)) - Ax • - io - ?i(m)] - A 2 • where, *[ = t[(^, nx), q{{ni) = q\{io,ni), y{ = io + gj*(ni) - tj, and = *i-Each firm will observe P\(n\, y{) and anticipate Ei(P2(pn\+e, y2)). With unlimited produc-tion at a sufficiently low marginal cost, instantaneous recontracting on the part of the individual firms would always insure that in equilibrium, P\(n\,y\) = E\(P2(pni + e,y2)) — C'(q\(ni)). When the marginal cost of production is sufficiently high, prices may differ between periods, depending on the shocks to demand. If a firm expects the t\ price to be higher (lower) than the t2 price, it will sell (store) all its inventory at t\. Each firm observes the same equilibrium prices and when the price in one period is greater/less than the expected price in the next period, I 4 By aggregating the cost function across firms the assumption is implicitly made that there are proportional costs of production. 32 all firms will respectively sell or store their entire inventory. Assuming existence, such 'corner solutions' in the aggregate maximization problem lead to the following three cases and their respective values of y\, y 2 , A i and A 2 . 1 5 A . ) If A x > Oand A 2 = 0 then, Pi(ni,j/*) < Ex(P2(pnx+e, y*2)), y*2 = i[ = q\{ni) + io, y{ = 0, and ql(ni) is such that E\(P2(pni + ?, y 2)) = inl))-B. ) If Ai = A 2 = 0 then, P i ( m , yj) = £7i(P 2 (pn 1 + e, y*)) = C'{q*{ni)), y'2 = i\ > 0, y{ > 0 and ql(ni) > 0. C. ) If Ai = 0 and A 2 > 0, then, P i ( n 1 ; y l ) > £ i ( P 2 ( p n i + I, y2)), y'2 = = 0, y\ = io + gj(m) and q{{ni) is such that P i ( n i , y i ) = C'(ql(ni)). 3.3 Characterization of the Decision In order to characterize an optimum, it is useful to define critical values for n\ such that regions can be identified within which either case A , B , or C will obtain: i) h\ is a shock to demand such that if everything that is stored from to and everything that is produced at t\ is sold at ti, the equilibrium t\ price is exactly equal to the expected equilibrium t2 price, i.e. h\ is such that, Pi(n^, t0 + q{{ni)) = i?i(P 2 (pni + e,0)); ii) h[ is a shock to demand such that if everything that is stored from to and everything that is produced at £1 is carried over and sold at i 2 , the equilibrium ti price exactly equals the equilibrium £2 price, i.e., h[ is such that, Pi(n' 1 ;0) = £ ' i (P 2 (pni + e,i0 + ?i(«i)))-Assume that both h\ < 00 and hlx > —00 exist. The critical values, h\ and h[, imply the following equilibrium conditions over the ranges (A, B, and C) of n\: ^ Note that and are optimal quantities, but are taken as fixed amounts y± and y2 respectively in the price functions P-y{ni,yi) and E\P2{pn\ + i^!/2))-33 A . ) E1{P2{pn1 +e,io + q\(ni))) > P ( n 1 ; 0 ) <^ => m < n' l 5 B. ) P i ( n i , y J ) = £ i ( P 2 ( / m i + i , ^ ) ) = • P ^ n ^ y J ) = C ' ^ K ) ) = £^ (P 2 (pn i +• e, y2)) h[ < ni < hi, and C. ) P 1 ( n 1 , i 0 + gj(m)) > ^i(P 2 ( /9n 1 + e,0)) m > h\. The following summarizes optimal economy-wide expected profits given n\\ P\{ni,io + ?J(ni)) • [«o + gj( ni)] ~ c ' ( ? i ( r a i ) ) w h e n rai ^ "1 E\{P2{pni + e, "i(ni))) • »i(ni)+ when n'x < rex < raj ^ J5i(P 2(pni + e,io+ ql{ni))) • [to + ql{ni)} - C(gi(ni)) when nx < h[ In the interval h[ < ni < h\ , E\{P2{pni + e,i[{n\)) = Pi(rai,«o + ~ J i ( r e i ) ) - The optimal expected ti profit can be expressed as E{nl(nltio)) = [%Q + ql{ni)}^J * Ei{P2{pni + e,y2)f(ni)dni + J P i ( « i , J / i ) ) / ( n i ) r f n i } -C(? 1*(m)). Let P(t'o) = | / " c o £ ' 1 (P 2 (pn 1 + ?,3/ 2)/(ni)dni + P I ( T I 1 ; y"i))/(ni)dni j. A t to the expected profit function is To(*b) = - P o • «o + [*o + • P(*o) - C{q\(rii)), for 7 > io > 0. There are three possible situations, each implying a different optimal inventory decision: 1. ) P0{I~~io) > P(*o) = » Jo = 0 and P 0 ( / ) > P(0). 2. ) P 0(7 - io) < P{io) => io = I and P o(0) < P ( J + <7iK))-3. ) P 0(7 - = P(*o) = • 0 < io < J and P 0 ( / - %) = 34 It is assumed that Po(I - ^) = P(ZQ) and it is this equality that defines ^ . P(io) is equivalent to an expected equilibrium average price that will be received when inventory is held. Po(I — «o) is equivalent to the cost of the good stored in inventory and sold at t\ or t2. At to the expected shocks to demand in the subsequent periods are both zero. It is reasonable to assume that profit maximizing behavior on the part of the individual firms will ensure this equilibrium condition will hold in aggregate. The equilibrium commodity price at to is > Ei{P2{pni+~e,y;))f{n1)dn1+ / P^m, • -oo J a * To illustrate the dependancy of price on demand shocks three cases are considered: p = 0,p = — 1, and p — I. Figure 1 illustrates the case where p = 0 and aggregate inventory is held fixed. Notice Pi(ni,yl) is increasing in n\. Given demand shocks are uncorrelated, then demand at t\ does not imply anything about the expected shock to demand at t2. If the shock at t\ is sufficiently large (i.e. n\ > nj; region C), the price at t\ is greater than the expected price at t2 even when everything that is produced at t\ and stored from to, is sold at t\. If the shock to demand at ti is sufficiently low (i.e. ni < h[; region A ) , the price at ti is below the expected price at t2 even though nothing is sold at t\. At shocks to demand between the critical values, and h\y (region B), equilibrium inventory stored from t\ to t2 is such that the price at ti will exactly equal the expected price at t2. In this situation the commodity will be sold both at t\ and t2. Figure 2 illustrates the prices in the case where the shocks are negatively related, or p — — 1. The expected t2 price is no longer constant as in Figure 1, but is a decreasing function of n\. Figure 2 has regions A , B and C similar to those in Figure 1. In region C, the price at i\ is higher than the expected price in t2 by the amount equal to the shadow price of inventory. 35 That is P i ( » i , Vi) = E1{P2{pn1 + e, y"2)) + \ 2 , where, X2 is the lagrange multiplier of the constraint i\ > 0. Lastly, Figure 3 illustrates the case when p = 1. The expected shock at t2 is always the observed shock at t\. Profit maximizing by the firms will ensure that Pi(ni,yl) = E(P2(pn\ + £,y2)) a n < l there is no value associated with holding inventory. 3.4 Analysis of Convenience Yield When shocks to demand at t2 are conditional on the shock that occurred at t\ (i.e. p ^ 0), the convenience yield is analogous to the value of a call option with maturity t\ and stochastic exercise price E\(P2(pn\ + e,y2))- Specifically, the convenience yield is the difference between the spot price of a commodity and the futures price, less interest and storage expenses. In this model, interest and storage expenses are assumed to be zero so the convenience yield will equal C Y = P0(I-$)- ^,,2(ib). where -Pb,2(^o) denotes the futures price at to for a contract that delivers at t2. The expected price at t2 is /oo ia(P2(/»*i + e,02))/(niHni--co Given risk neutrality and no interest or storage costs, the futures price at to will equal the expected t2 spot price, i.e. f0(2(tS) = ^)(P2 ( p n i + ? , » ; ) ) • The convenience yield can then be expressed as C Y = P0(I - £) - E0(P2(pni + I, yl)). Substituting equation (3.2) for Po(I — 2Q) and equation (3.3) for Eo(P2(pn\ +e, y2)), the conve-nience yield can be restated as r oo (3.4) C Y = / P i ( m , yl) - EiiPtipni + e, ylMn^dm. 36 This is the expected difference between Pi(n\, j/J) and E(P2(n2, y^)) over the interval where this difference has a positive value. At to the expected t2 price at ti is not known since it depends on the i i shock to demand. The exercise price is, therefore, stochastic. At t\ the value of owning the commodity over owning the futures contract is Max(0, P\(ni, yl) - Ei(P2(pni+e, j/ 2))). Two securities with the same return must have the same price, so the convenience yield is equivalent to the value of a call written on the i i spot price with maturity ti and stochastic exercise price £ i ( P 2 ( / m i + e,2/2)). 1 6 Following similar logic, when p = 0, the convenience yield is like a call option on the com-modity spot price with maturity i i and constant exercise price Ei(P2(ri2, y^))• The analysis is exactly like in the case of a stochastic exercise price except that £ i (P 2 (n 2 ,2 /2) ) is not dependent on h]_. Therefore, Eo(P2(n2, y^)) = E\(P2(n2, y^)) and the exercise price is constant. Given equation (3.4) as the expression for the convenience yield, the following three propo-sitions are derived. Proposition 1. The value of the convenience yield is decreasing in aggregate inventory when p ? 1, Proof: See Appendix 2. While a formal proof of Proposition 1 has been provided, it is worthwhile to discuss the intuition behind it. As illustrated in Figure 1, the value of the convenience yield is the expected value of the difference between P\(n\,i*Q + q[) and Ei(P2(pn\ + ?,0)). The value of the shock to demand that will equate the i i price when all firms sell their entire inventory at i i , with the i 2 price when nothing is sold, is increasing in the amount sold at i i . As the total amount sold becomes larger, h\ also increases and the region over which the convenience yield has a positive value becomes smaller. Succinctly, the larger the aggregate inventory, the lower the likelihood of incurring shortages in the commodity which are necessary for a positive value associated In his unpublished dissertation, "An Analysis of the Displacement of Private Stocks by Public Stocks and of the Accumulation of Public Stocks by a Loan Program", Zulauf (1983) recognizes the option value of inventories of commodities. While no economic model is derived in the thesis, preliminary work by Cootner (which was incomplete at the time of his death) is acknowledged by Zulauf for the 'option' insight. The model in this paper was developed independently of Zulauf (1983). 37 with holding the commodity in storage. Proposition 2. The value of the convenience yield is zero when p = 1. Proof: See Appendix 3. Proposition 3. When the marginal cost of production is constant, the value of the convenience yield is decreasing in p. Proof: See Appendix 4. Propositions 2 and 3 are illustrated in Fgures 1-3. Figure 1 shows the convenience yield when p = 0 as the difference between Pi(n\,i^ + ql) and Ei(P2(pni +e,0)) when ni is greater than h\. Figure 2 illustrates the same relationship when p = — 1. Because of the negative correlation in shocks, a large positive shock at t\ implies an expected large negative shock at t2. This affects the convenience yield in two ways. First, the breakeven shock needed to equate P\{ni,io + ql) and E\(P2(pni + e,0)) is lower when p = —1. Secondly, the difference between the two prices is greater since E\(P2) is decreasing in n i . Both effects combine to increase the value of the integral in equation (3.4). It has been shown that when p = 1, h\ —> oo. In this case P i will always equal £'i(P2) a n d the convenience yield will equal zero. This result is shown graphically in figure 3. If p = 1, then P i = ^i(P2)- A t to, P0 = E0{Pi) = Eo{P2). Prices for which this is true are said to follow a random walk. Following from Proposition 3, commodities which prices follow a random walk will not have convenience yields. 1 7 3.5 Options with stochastic exercise prices . The above results show that the convenience yield is like an option on the spot price of the commodity. Under different assumptions regarding the shocks to demand the convenience yield will be valued as an option with a constant exercise price, an option with a stochastic exercise price, or an option where the spot price will always equal the exercise price and, therefore, will always have a zero value. I 7 Note that if p > 1, then E(P2 | hy) > Pi^n^. This implies that there is no value to holding the commodity over the futures contract, and no convenience yield. 38 To show the generality of these results it is useful to compare them to a model of option pricing with a stochastic exercise price developed by Fischer (1978) and Margrabe (1978). 1 8 Using the spot price of the commodity to represent the underlying asset on which the option is written, and the expected future price at t2 to represent the exercise price, the convenience yield is expressed as: (3.5) CY{P1,E{P2))^ PiNidi)-E{P2)N{d2), where, rfi = v . , d2 = di - <J\Ai ~ k>, N(-) is the cumulative standard normal density function, and, <T2 = ap^ — 2O~P1<JE[P2)PPI,E{P2) ~^AE(p2)> * s ^ n e instantaneous proportional variance of the change in the ratio E(P2) ' In the Margrabe and Fisher models when PpLTE{P2) — 1 a n c ^ A P \ = AE(P2)> which is analagous to the case in this paper when prices follow a random walk and the option value equals zero. If AE(P2) = 0> the Margrabe Fischer formula reduces to the Black Scholes option pricing formula for European call option. The degree of correlation that maximizes the value of the option is PP1:E{P2) = — 1- m general, the lower the correlation between the spot price and the ex-pected spot price, the higher the convenience yield associated with the commodity. The results obtained in the current paper are consistent with the Margrabe Fischer option model. 3.6 Summary of Results 1.) The convenience yield is like an option. It is an option to sell the commodity at t\ if Pi > Ei(P2). The exercise price is the futures price for delivery at t2. In other words, it is what is foregone to sell the commodity at ix- The exercise price is the t2 expected spot price of the commodity. Under the assumptions of the model presented in this paper, Ex(P2) = Fi:2. ^ In their papers it is assumed that the instantaneous rate of returns on Pi and E(P2) are ~p~ = [&p^dt+ &p^dzp^\ . 3E(P2) a n d w *E(P2)dt + °E(P2)d*E(P2) , where dPi and dE(P2) are Wiener processes. The correlation between the Wiener processes dzp^ and dzE(P2) is Pp^ E(P2)' " ' s a ' 3 0 a s s u m e c ' ' h a ' aPi' aE(P2)'°Pi,anc' aE(P2) a r e c o n s ' a n t s -39 2. ) When £ ( P 2 ) is conditional on hi, the exercise price of the above option is stochastic at to. 3. ) The convenience yield is always non-negative. This follows from the fact that it is an option. The holder of the commodity has the option of selling the good at t\, but is not committed to do so. In the absence of interest charges and storage costs the spot commodity is at least as valuable as the futures contract. 4. ) The amount of inventory carried from t\ of t2 will equal zero only when Pi > Ei{P2). This amounts to a positive value of the convenience yield depending on the possibility of a corner solution in the maximization of aggregate profit. 5. ) If the marginal costs to production are low and there are no contraints on the amount produced, then market forces will always drive Po = ^b(Pi) = EQ{P2) = Eo[C{q\)\. A com-modity for which production is completely flexible would not have a convenience yield. 6. ) With constant marginal costs to production the convenience yield is decreasing in p. 40 CHAPTER FOUR EMPIRICAL ANALYSIS 41 4. E M P I R I C A L A N A L Y S I S 4.1 Data 4.1.1 Description The commodities included in the sample are Chicago wheat, Kansas wheat, Minneapolis wheat, soybeans, soybean oil, corn, copper, silver and gold. 1 9 The completeness of the data varies across the commodities. Recall that for each period the commodity's convenience yield is calculated as CYi=Pi(l + Rt+1D~Fit+1 where, C Yt* = the convenience yield of commodity i at time t, Fjt+i = the current futures price of a contract of commodity i for delivery one quarter hence (i.e. time t+ 1), Rt = the risk free rate over the period from t to t = 1, 7j = the storage cost for i over the period from t to t + 1 expressed as a percentage of the spot price, and PI = the spot price of commodity i at time t. U.S. commodity spot and futures prices are gathered from the Wall Street Journal over the post wage and price control period commencing March, 1974 and extending to March, 1986. 2 0 Prices collected are the closing prices on the first trading day of each quarter. If no trade occured in a specific commodity on the first day of the quarter, then the closing price of the previous trading day is used. The futures prices are for contracts maturing one quarter hence. For example, prices collected on December 1 are for contracts that mature in March; etc.. Two problems may arise in attempting to empirically assess the relationship between com-modity futures prices and spot prices: First, most commodity future contracts do not have a 1 9 Wheat traded on different exchanges are treated as distinct commodities due to differences in transportation costs, grade and quality specifications. Due to the potential influence of such controls on futures price movements the time period from August 15, 1971 to April, 30, 1974 is omitted. 42 specific maturity date. The delivery period is usually three to four weeks after the beginning of the maturity month. As commodities are received during the delivery period they are allocated to the long trader whose contracts have been outstanding the greatest period of time. To deal with this problem, it is assumed that contracts mature on the first trading day of the delivery month. For example, the December 1, 1973 futures price for the March, 1973 contract is used as a 3-month futures price. Second, for non-financial commodities futures prices may be affected by such things as terms of delivery and grades of the specific commodity. Comparing a futures price for immediate delivery to the current spot price may indicate an arbitrage opportunity when, in fact, the difference is due to the specific terms of the futures contract. To deal with this problem, the futures price of maturing contracts are substituted for spot prices. 2 1 Fol-lowing from the example above, the price used to represent the December 1, 1973 spot price of a commodity is the closing price on December 1, 1973 of a futures contract for delivery in December. The model of the convenience yield derived in Chapter 3 looks at the difference between the spot price and a forward price. In a world of zero interest rates, a forward contract is equivalent to a futures contract since there is no return associated with the daily 'marking to market' of the futures contract. When testing the model, futures prices are used instead of forward prices due to the availabiltiy of data. Since the futures price is affected by the return from marking to market, the results of the analysis may not be as strong as they might be if forward prices were used. A test of the reliability of using futures prices for immediate delivery to represent current spot prices is performed by calculating the correlation coefficient between spot prices and futures prices for immediate delivery. The spot price data used for this calculation are obtained from the Commodity Research Bureau, which in turn collects prices from the Wall Street Journal. The correlation coefficients are reported in Table 1. As can be seen, the correlation between the prices is very high (over .9870) for every commodity with the exception of copper. This 2 1 This practice is also employed by Fama and French (1987) for the same reason. 43 evidence lends support for the use of the proposed proxies for current spot prices. 2 2 Delivery months for the commodities in the sample with the exception of gold are March, July, September and December. Therefore, the period extending from March to July is used to represent the first quarter and the period from July to September is used to represent the second. This results in two three-month intervals, one two-month interval and one four-month interval. In the case of gold, there is a June contract but no July delivery date and the quarters are each three months long. Soybeans have no December delivery date so only two quarterly convenience yields can be calculated per year. Estimates of storage costs are obtained from Fama and French (1987). These costs include both handling fees (i.e. costs of loading and unloading) and monthly warehousing cost per dollar of the June, 1984 spot price. Inventory data are gathered from two sources: For wheat, corn, soybeans, soybean oil, and copper, aggregate U.S. commercial stock data are taken from the Commodity Yearbook. Because of the uneven quarters described above, monthly numbers are needed to exactly match the inventory observations with convenience yield observations. For corn, only 15 monthly inventory observations are available over the period from 1974 to 1978 with several quarters missing. Hence, while quarterly data are available for three-month intervals over the entire period, the corn inventory and convenience yield observations are misaligned by one month. For silver and gold, aggregate U.S. industrial stocks are obtained from the U.S. Bureau of Mines Industrial Surveys. The data are quarterly with the quarters starting March, June, September and December. The convenience yield observations for silver do not align with the inventory data since the second quarter starts in July instead of June. Hence, only three quarters per year are used for silver. United States treasury bill data are collected from the C R S P Government Bond File. Prices obtained are for bills that have the latest maturity date in each quarter, where the quarters 2 2 See Hughes and Holthausen (1978) and McNicol (1975) for theories and evidence pertaining to the two-price system for copper. 44 correspond with those used to calculate the convenience yields. 2 3 4.1.2 Time Series Plots Time series plots of the inventory data for each of the commodities are shown in Fiqures 4 to 10. A l l quarterly observations are included in these plots, notwithstanding the alignment problems mentioned earlier. The purpose of these plots is to display the seasonality or lack of seasonality which occurs in the inventory data. The commodities for which seasonality is obvious are wheat, corn and soybeans. Seasonality is much less obvious in soybean oil and appears non-existent in the metals. Plots of the weekly spot prices for each commodity are shown Figures 11 to 17. Several features of these plots are worth noting. The first is the distinct change in behavior of copper spot prices in 1978 when copper pricing changed to being determined by market forces. The period prior to 1978 is not relevant for analysis since these were not free market prices. The second is the sharp increase in the prices of all three of the metals in December, 1979. The increase in silver prices is associated with the large purchases of silver by the Hunt brothers in the last quarter of 1979. The unstable political situation in the Middle East at that time is a possible explanation for the increase in the other metals prices as well as silver. 4.1.3 Summary Statistics Tables 2 through 7 summarize the data which are used in the empirical test of the model proposed in section three. For the seasonal commodities (namely, wheat, corn, and soybeans) the quarters are arranged according to the crop year with the first quarter being the harvest quarter. Table 2 presents a summary of the inventory data. The numbers shown are beginning of period inventories, and hence the effect of the harvest shows up in the post harvest quarter. In the case of wheat, that is the quarter from September to December. The statistics for corn are based on very few observations but still reflect the impact of the harvest, which occurs from September to December. The seasonality is more difficult to observe in Soybeans given only two quarters are available. Since the harvest for Soybeans takes place from September to 2 3 See Appendix 9 for a list of data sources. 45 December, the quarters available are the two pre-harvest quarters. Summary statistics of the commodity spot prices are presented in Table 3. Seasonality is evident in the agricultural commodities with the mean spot price rising over the crop year. The largest standard deviation for wheat occurs in the pre-harvest quarter, while the largest standard deviation for corn occurs in the harvest quarter. The large standard deviation in the harvest quarter for corn could be caused by the fact that the harvest for corn does not fall entirely in the September to December quarter, which is the quarter observed. The U.S. Department of Agriculture reports the harvest as starting at the beginning of October and lasting until the beginning of January. Since the spot prices are observed at the beginning of September, it is really a pre-harvest period. In the case of metals, differences across quarters in the means of the spot prices are likely a result of outliers in the data caused by such events as the transactions by the Hunt brothers mentioned previously. The summary statistics of the quarterly futures prices are reported in Table 4. The sea-sonality in the futures prices differs from that in the spot prices as a result of the one quarter lag in delivery (i.e. The March futures price is for delivery in July). The standard deviations of the quarterly futures prices are similar to those of the spot prices. Table 5 reports the summary statistics for the quarterly convenience yields. In light of Gay and Manaster (1986), it is possible that for the agricultural commodities the size of the convenience yield is overstated due to a quality option embodied in the futures price. 2 4 While the impact of this option may inflate the size of the convenience yield, it will not have an impact on tests of seasonality in the convenience yield since the quality option itself is not seasonal. Because there is little variation in the quality of the delivered commodity for metals, the quality option has little or no value. For the wheats and corn, both the largest mean convenience yield and the largest variance occurs in the pre-harvest quarter. Table 6 shows the ratios of the convenience yield to the spot prices. The largest convenience 2 4 Recall from Chapter Two the 'quality option' arises from the discretion over the quality of the commodity delivered to fulfill the obligation of a futures contract. 46 yield in percentage terms is for Chicago wheat and is 14 percent in the pre-harvest quarter. The convenience yield as a percentage of spot price is just over 6 percent in the pre-harvest quarter for soybeans and ranges from approximately 3.9 percent to 6.2 percent for soybean oil. The relative size of the mean convenience yield for corn falls between those of the wheat contracts and soybeans. The average pre-harvest ratio of the convenience yield to spot price for corn is about 11.6 percent. For the seasonal commodities, the ratio of convenience yield to spot price is largest in the pre-harvest quarter; the quarter when inventories are at their lowest levels. Inventory levels for metals do not exhibit seasonality and, therefore, there should be no seasonality in the convenience yield ratio. The mean convenience yield for copper changes from approximately 1.2 percent to 2.3 percent and is always positive. The mean convenience yields for gold and silver are small and negative in three of the four quarters and small and positive in the remaining quarter. The negative values for the convenience yields for silver and gold are most likely a result of mismeasurement of the storage costs. The largest negative number reported is the September to December quarter for silver and is -.655 percent with a standard deviation of .01869. Casual examination of the ratios for metals reveals no obvious seasonality. The ratios are also much smaller for the metals than for either the seasonal commodities or soyoil. Lastly, Table 7 shows the futures returns for the different commodities. The future returns are similar to the ratio of the convenience yield to the spot price, however, since they are not adjusted for carrying costs, they take on negative values. 4.2 Econometric Design 4.2.1 General Linear Model While Proposition 1 from Chapter Three provides a prediction on the direction of the effects of a change in inventory levels on the convenience yield, it does not provide a functional form of the relationship between those variables. Accordingly, the specifications of an appropriate regression model for capturing that relationship empirically is to some extent dependent on 47 what the data will support. A useful starting point is to characterize the relationship between inventory levels and convenience yields as a grouped equations ordinary least squares regression: Y — j3X - t - u where, y' = [j/i, 2/2, •••,</;], yj = a T x 1 vector of convenience yields for the j th commodity, / X i 0 . . . C M 0 X2 . . . 0 V 0 0 . . . Xj) Xj = a K X T matrix of inventory levels for the j t h commodity, u' = \ui,u2, ...,Uj], u}: = a T X 1 vector of independent N (0 ; <7?) distributed disturbances for the j th commodity (and independent across commodities), and /? = [a1,f31,a2,/32,... ,aj,j3j], The assumptions which underlie this model are considered next. 4.2.2 Heteroscedasticity The use of data from unequal quarters is likely to introduce heteroscedasticity within commodities even when other sources of heteroscedasticity are not present.2 5 Assuming that the variance of the convenience yield is linearly related to the length of the period over which it is calculated, the data should be transformed by weighting the observations of the March to July quarter by the square root of | and weighting the observations of the July to September quarter by the square root of | . 2 6 The statistic that is used to test for heteroscedasticity, beyond that captured by the above transformation, depends on the form that the heteroscedasticity is believed to take. For ex-or Tables 8 and 9 give the values of the two tests for heteroscedasticity both before and after the data are transformed for the uneven quarter lengths. While the values of the statistics before the transformation are not discussed, as expected they are in general larger than those for the transformed data. 2 ^ See Appendix 5 for an explanation of this transformation. 48 ample: the standard deviation of the disturbances may be a linear function of the inventory levels, the variance may be constant within but not between quarters, or there may be other forms of non-stationarity. The most likely source of heteroscedasticity is seasonality in the data. Accordingly, two statistics are reported to test for heteroscedasticity of this type: the Bartlett test and a Univariate F-statistic. The Bartlett test is a test of homogeneity of variance across sub-sets of observations. In the present application the null hypothesis is that the variance of the convenience yield is equal across quarters. If the null hypothesis is true, then s2 = —j-*-, where s2 is the variance of the ith quarter with fi degrees of freedom, / = ^2t=i /<> a n < ^ s 2 provides an estimate of the common population variance a2. Barlett has shown that the null hypothesis can be tested by the ratio g , which is approximately x2 with k-1 df, where The univariate F-statistic is similar in spirit to that proposed by Goldfeld and Quandt (1965). A derivation of this statistic is provided in Appendix 6. The advantage of using the univariate F-statitic is that is isolates the quarters where the variance is predicted to differ. For example, the pre-harvest and harvest quarters are tested for homogeneity of variance. The values of the test statistics for heteroscedasticity in the convenience yields are shown in Table 8. The post-transformation Bartlett statistic is significant at the five-percent level or better for the wheats, copper, soybeans and gold, implying the presence of heteroscedasticity. However, homoscedasticity cannot be rejected for corn, soyoil and silver. The results of the F-tests are similar to that of the Bartlett test with two differences: the F-test fails to reject the null hypothesis for Minneapolis wheat and rejects the null hypothesis for silver. 2 7 2 7 The Bartlett test and the F-test are both performed for the commodities futures returns and are reported in Table 9. Not surprisingly, since the futures return is closely related to the convenience yield, the results of these tests are almost identical to those for the convenience yields. A=f-ln(s2)-JLfrln(s2) and 49 The tests for heteroscedasticity indicate that, while the different lengths of the quarters are responsible for some of the heteroscedasticity, other factors also contribute to the problem. For the agricultural commodities the source of the heteroscedasticity may be due to the seasonality in the convenience yields. The heteroscedasticity in the metals data is likely a result of outliers. Overall, the analysis of the data indicates that a generalized least squares procedure should be used to examine the relationship between the convenience yield and inventory. In addition to tests of heteroscedasticity across seasons, tests of heteroscedasticity within pre-harvest and post-harvest are performed on the convenience yields of the wheats and soy-beans. The tests are performed on these commodities since wheat and soybeans are the only two seasonal commodities with enough within season observations. Goldfeld-Quandt statistics are calculated for each period. 2 8 These statistics are reported in Table 10. For the wheat data the null hypothesis of homoscedastic errors cannot be rejected in either period, whereas for soybeans the null hypothesis is rejected in the pre-harvest quarter. The large value of the statistic is due to two outliers which account for 95% of the residual variance. A n OLS procedure assumes £(e,-e|.) = a2 IT, (ie. <j2t = a2 for £ = 1, 2,... , T). Using an OLS procedure to estimate the parameters when the errors are heteroscedastic yields consistent but inefficient estimates. However, estimates of the covariance matrix are inconsistent. To address this issue the models are also estimated using a GLS procedure which incorporates the White (1980) Heteroskedastic Consistent Covariance Matrix into the estimation procedure. This procedure does not affect the parameter estimates, but does provide consistent estimates of their errors in computing test statistics. 4.2.3 Serial Correlation The normal statistic of a runs test is used to examine the data for first order serial cor-relation. 2 9 The runs test is a non-parametric test. The motivation for using this test is that it makes no assumption of normality of the distribution of residuals; an assumption that is 2 8 See Appendix 6 for an explanation of this statistic. 2 9 See Gujarati (1978) p.246 for a description of this statistic. 50 unlikely to hold with this data. 3 0 This test looks at first order serial correlation. The statistic is calculated for each commodity using the residuals from three different OLS regressions. • The three equations estimated are Yj = ct\j + XjPij + B\j, Yj = d2j+ (y;)P2] +£2j, Yj = a3j + ln(Xj)p3j + e3j, T X 1 vector of convenience yields for commodity j, T X 1 vector of aggregate inventory levels, the intercept parameter of equation i for commodity j, the slope parameter of equation i for commodity j, and T X 1 vector of disturbances with E(eije'~) = S,y. Table 11 reports statistics for the different equations, (4.1) to (4.3). The null hypothesis of no serial correlation is rejected for each equation in the case of Minneapolis wheat, copper and silver. For corn and gold the null hypothesis is rejected for equations (4.2) and (4.3) but not for equation (4.1). The null hypothesis is rejected when the residuals from equation (4.1) are tested for Kansas wheat and is rejected for Soyoil when the residuals from equation (4.2) are tested. The only commodities for which there is no evidence of serial correlation in each of the regressions are Chicago wheat and Soybeans. A GLS procedure can also be implemented to correct the for the problem of inefficient parameter estimates and inconsistent esitmates of the covariance matrix resulting from using 3 0 See subsection 4.2.4 below. (4.1) (4.2) and (4.3) where 51 an OLS when the disturbances are autocorrelated. In particular, the GLS procedure employed estimates the covariance matrix by replacing I with where / 1 P ••• P T - ' \ 1 P2 \ \pt-' PT~2 . . . i ; and E[etEt-t] Ps = 2 is the correlation between two disturbances s periods apart. 4.2.4 Normality The residuals from the estimation of the above equations are examined for normality using a chi-square goodness of fit test. The coefficient of skewness and the coefficient of kurtosis are also calculated. These statistics are" displayed in Table 11. Based on the chi-square statistic the hypothesis that the residuals are normal can be rejected at the 5% level for Soybeans, Soyoil and the metals data. The residuals from the estimation of equation (4.1) for Chicago wheat and Kansas wheat also have distributions which are not normal based on the chi-square statistic. The expected value of the coefficient of Kurtosis is zero for a normal distribution. The large values indicate that the distributions are leptokurtic (i.e. fat-tailed and highly-peaked). Brown and Warner (1985) examine the properties of test statistics when the data are not normally distributed. They find that for large samples, conventional test statistics are well specified and approximately unit normal under the null hypothesis. Moreover, even in small samples, the specification of the test statistic is not dramatically altered, although Kurtosis of the t-distribution does increase. 4.2.5 Outliers A number of regression diagnostics are available for the identification of outliers. 3 1 These 3 1 See Belsey, D.A., Kuh, E . , and Welsch, B.E. (1980) for a discussion of these techniques. 52 techniques usually involve deletion or perturbation of rows in the X matrix in a sequential manner. The criteria for identifying influential observations, however, is ultimately answered by judgement and intuition. For this reason, the method followed in this study is to examine the plots for data points which are obvious outliers, check the data source for recording errors, and check for an economic explanation underlying each unusual observation. Looking at Graphs 15 to 23, each commodity has at least one such observation. Starting with Figure 18, which is Chicago wheat, there are three outliers labeled A , B , and C . 3 2 Point A is a combination of a very high convenience yield and a very low inventory level. The date corresponding to this point is March, 1974. During 1974 there was a world drought and the inventories of many agricultural products fell to historical lows. In March it was anticipated that the size of that year's wheat harvest would be very low. Point B on Figure 18 corresponds to the date July, 1974, the post-harvest period. As expected, the harvest was very poor and the inventory levels remained at extremely low levels. Point C on Figure 18 is the March, 1975 data point. This point is unusual in that not only is the inventory level very low, the convenience yield is also quite low. In March, 1975, inventory levels were still very low due to the previous year's drought, however, ideal weather conditions lead to the expectation of a 'bumper' crop. The anticipation of a large increase in supply may have influenced the size of the convenience yield. The outliers (each denoted as point A) for Kansas wheat ( Figure 19), Minneapolis wheat ( Figure 20), Corn ( Figure 21) and Soyoil ( Figure 23) are all the March, 1974 observation, which was influenced by the drought. The March, 1975 observation also appears as an outlier for Kansas wheat (point B). Figure 22 is a plot of the soybeans data and two outliers are identified. Points A and B correspond to the dates July, 1977 and July, 1978, respectively. The points are characterized by very high convenience yields and average or above average inventory levels. An examination of The information regarding the outliers for the agricultural commodities was obtained from the U.S. Department of Agriculture's monthly publication "Agricultural Outlook". 53 the events in the soybean market that might have influenced the value of the convenience yield revealed that foriegn producer's supply was severely curtailed. During 1977-1978, the Brazilian harvest (the U.S.'s major competitor in the production of soybeans) was virtually eliminated by a drought. Therefore, while the Unitied States's inventory levels were not unusually low, world levels were. Turning to the metals, only one outlier is identified for copper. 3 3 This is depicted by point C on Figure 24 and is the March, 1974 observation. In 1974, refined copper demand was extremely strong. In the midst of this demand the copper industry was faced with the expectation of a large labour strike in mid to late 1974. While there were some stoppages, all were settled within a month or so, and all producers had resumed operations by Labour Day. There are two obvious outliers identified for Silver (shown on Figure 25). Point A is the December, 1979 observation. This is the time when the Hunt brothers initiated massive purchases of silver. The point is characterized by a very high convenience yield and relatively low inventory level. Point B is the other outlier and is the March, 1983 observation. This point also has low inventory, but has a low convenience yield as well. While no event pertaining directly to silver could be identified, the economic environment was being affected by sharp cuts in oil prices by the O P E C countries, which in turn fueled expectations of low inflation and put downward pressure on precious metal prices. The March, 1983 observation is also identified as an outlier for gold (point E on Figure 26). Four other outliers (points A to D) are the consecutive observations from March, 1980 to December, 1980. This was a period of 'sky-rocketing' interest rates, high inflation, and a very tumultuous situation in the Middle East. Together, these events created an unusual environment which, apparently, was reflected in the market for gold. 4.2.6 Tests of Cross-Dependencies It is reasonable to expect that there may be cross-dependencies in the residuals from a 3 3 The information regarding the outliers for the metals was gathered from the U.S. Bureau of Mines Industrial Surveys, The Wall Street Journal, and Metai Statistics. 54 regression of the convenience yield projected on aggregate inventory for certain related com-modities. This is especially true for the three wheat contracts given the similarity of the underlying commodities. Soybeans and soyoil, as well as gold and silver, may also have some cross-dependency. However, due to the mismatching of quarters and different sample periods, it is only useful to examine the cross-correlation for the wheats and soybeans and soyoil. In the case of the wheats, the cross-correlations of residuals are .534 for Chicago and Kansas wheat, .300 for Chicago and Minneapolis wheat and .486 for Kansas and Minneapolis wheat. For soybeans and soyoil the cross-correlation of residuals is -.0046. These correlation coefficients imply that although wheat contracts are subject to institu-tional differences, there is still contemporaneous correlation between the residuals. In the case of soybeans and soyoil, such correlation is not evident. For test of hypotheses involving restrictions across commodities each equation is aggregated into a pooled cross-sectional model of the form, (4.4) Y = X~i + e where Y = (lh o . . . 0 \ 0 I i i . . . 0 a, \0iJ V 0 0 . . . 17,7 Within this structure E(es') = V, although several simplifications regarding the form of V are imposed during the estimation. In the absence of autocorrelation it is assumed that V is of the form Cl <£> IT where Q is the contemporaneous i X i covariance matrix of disturbances across commodities. 4.2.7 Non-Constant Parameters One of the assumptions made in deriving the model is that all inventory is sold by t2- For seasonal commodities the inventory changes dramatically over the harvest cycle. It is possible that the parameters of the model also change, since the assumption of marking clearing will be 55 more realistic in some quarters than in others. A comparison of the fit of the model between the pre-harvest and non pre-harvest periods will indicate whether the parameters are constant across seasons. 4.3 Results 4.3.1 OLS and GLS Regression The parameter estimates and relevant statistics from estimating equations (4.1) to (4.3), both by using an OLS procedure and by taking into account heteroscedasticity through the implementation of a GLS procedure, are reported in Tables 13 to 15. The slope coefficients on the aggregate inventory variable are negative across equations for every commodity and are significant at the 5% level for every commodity with the exception of soybeans and silver. The largest R2 (.6049) is from the estimation of equation (4.2) for Chicago wheat. Estimation of equation (4.2) also provides the best fit of the data for Kansas and Minneapolis wheat, soyoil, and copper (with R2,s of .3041, .4239, .3927 and .4414 respectively). The t-ratios from the GLS regressions are generally slightly lower than the t-ratios from the OLS regressions for these commodities, but are still significant. The estimation of equation (4.1) provides the best fit of the models for corn (i? 2 = .5602) and gold (i? 2 = .1764). The t-ratios for these commodities are significant when either the OLS and GLS procedure is used. The two commodities for which the regressions have very little explanatory power are soybeans and silver. The lack of fit for silver is not surprising. First, since silver generally has very high inventory levels, the convenience yield is predicted to be small or near zero. In addition, the equations are estimated with data that includes the period when the Hunt brothers attempted to control the market and, thus, may not be indicative of general market conditions. 3 4 The results for soybeans are much harder to explain. The t-ratios are negative, however, they are not significant and the R2's are all below .02. While two of the four quarterly convenience yields are missing each year, the two quarters that are observed are the pre-harvest quarters and should have the largest convenience yields. The explanation for the poor results 3 4 An analysis of the data omitting extreme outliers is contained in sub-section 4.3.2. 56 is likely due to a complex relationship between soybeans, soybean products (soybean oil and soybean meal) and foriegn producers of soybeans. For example, when the convenience yield of soybeans is projected on the aggregate inventory of soybean oil in estimating equation (4.1) the R2 increases to over .11. Finally, two extreme outliers in the soybean data, which appear related to shocks to foreign producers' inventory, add to the problem of identifying a relationship between the convenience yield and aggregate U.S. inventory. 3 5 4.3.2 Trimmed Samples As noted earlier, the plots of the convenience yield against aggregate inventory indicate that outliers are most serious for the metals data. In light of this, equation (4.1) is estimated for copper, silver, and gold with the data trimmed for outliers. The results are reported in Table 16. Once again the slope coefficient is negative for each commodity and is significant for copper and silver. The R2 improves for silver, increasing to .1605, but drops to .1107 for gold. The results of these regressions are more in line with what the model would predict. The fit is poorest for gold; the commodity which has the highest inventory levels. Copper has the largest t-ratio on the slope parameter, as it did when the outliers were included. 3 6 4.3.3 Seasonal Parameters To examine whether there is a difference in the fit of the models between seasons, the wheat and soybean data are divided into pre-harvest and post-harvest periods. 3 7 The parameter estimates and statistics from the estimation of equation (4.1) using an OLS framework are summerized in Tables 17 and 18. Separating the data into seasons does not improve the explanatory power of the model for soybeans. The slope coefficients are still insignificant and the R2,s are very small. For the wheat 3 ^ An examination of the plot of soybeans's convenience yield against aggregate inventory in Figure 22 highlights thse outliers. 3 6 The results of estimating equation (4.1) for copper with the speculative period omitted gives a t-ratio on the slope parameter of-2.615 and an of .1529. 3 7 The post-harvest season in actually the non pre-harvest season and includes the harvest period. 57 contracts the R2,s are all higher in the pre-harvest season. The slope coefficient on inventory, however, is no longer significant for Kansas wheat. The Durbin-Watson statistics from the pre-harvest regressions indicate that with annual observations, autocorrelation is a problem. The existence of autocorrelation in the wheat data is difficult to explain, particularly in light of the fact that it is negative for Chicago wheat and positive for Kansas and Minneapolis wheat. Recognizing that autocorrelation exists in the annual wheat data, the regressions are also estimated with a GLS framework which calculates the correlation between the error terms and modifies the covariance matrix appropriately. Table 19 to 21 summerize the results from this estimation procedure. The R2's improve when using a GLS procedure to adjust for autocorrelation. The slope coefficient is negative and significant for Chigaco and Minneapolis wheat. It is negative but insignificant for Kansas wheat. 4.3.4 Seemingly Unrelated Regressions (SUR) As mentioned earlier, cross-dependencies are present for the three wheat contracts. Ac-cordingly, the Zellner SUR technique is used to estimate the parameters for equations (4.1) to (4.3) for the wheat contracts. The advantage of this procedure is that allows for hypothesis testing across commodities. The estimates and F-statistics from the S U R estimation are contained in Tables 22 to 27. Once again equation (4.2) has the highest R2. The t-statistic on each slope parameters is negative and significant. For each equation, the largest t-ratio is for Chicago wheat. Hypothesis tests of the equality of the slope parameters across the different wheat con-tracts indicate that the slope coefficient is different for Chicago and Minneapolis wheat in the estimation of equation (4.1). F-tests performed on the estimated parameters of equation (4.2) reject that the slope parameters for all wheats are equal, reject that the slope parameters for Chicago and Kansas wheats are equal, and reject that the slope parameters for Chicago and Minneapolis wheats are equal. The F-tests performed on the estimates of the coefficients from 58 equation (4.3) reject that the slope parameter is equal for Chicago and Minneapolis wheat. According to these tests, only the Kansas and Minneapolis wheat contracts are similar, thus there is justification for treating the three wheat contracts as distinct commodities. 4.3.5 Dummy Variable Regressions The commodities convenience yields are tested for seasonality using a dummy variable regression. The advantage of using dummy variables to test for seasonality in commodities convenience yields emanates from the difficulty in measuring inventories. For a commodity such as gold, determining what constitutes available aggregate inventory is very difficult. Even in the case of agricultural products, there is a great deal of discretion in defining inventories. Using dummy variables helps to circumvent the problem of determining how to accurately measure inventory. In effect, the dummy variables approach assumes inventories are constant across similar quarters from different years. Alternatively, one could employ an instumental variables approach in which the seasonal dummies serve as the instuments. The qualitative distinction is that average (across years) quarterly inventories would replace the ones in each quarterly dummy. The model that is estimated is (4.5) CY^Dipi + ei, where CYi = T x 1 vector of convenience yields for commodity i, D = Tx matrix of dummy variables identifying the quarter of the year, M I , where I4 is and identity matrix of order 4, h i ?/>,•= 4 X 1 vector of intercept paramters for commodity i and s,• = T X 1 vector of distur-bances with jE(e,-£|-) = S,-. To operationalize the tests of seasonality in the convenience yields the following null hy-potheses are tested: 59 HI : The coefficients for each quarterly dummy variable are equal. H2: The coefficients for the non pre-harvest dummies are equal. H3: The dummy parameters for the two pre-harvest quarters are equal. H4: The dummy paprameters for the two non pre-harvest quarters are equal. The restrictions imposed by hypotheses H1-H4 are tested with the standard F-statistic described in Theil(1971, p.313). The model is estimated separately for each commodity, with and without constraints, and the F-statistic is computed as, where, T — the number of observations, K = the number of parameters, Q = the number of constraints , SSEC = the sum of squared errors constrained, and SSEU = the sum of squared errors uncontstrained. The parameter estimates and F-statistics from equation (4.5) are contained in Tables 28 and 29 respectively. Significant seasonals in the convenience yields are evident for the wheat con-tracts but not for corn or soybeans. There is also no evidence of seasonality in the convenience yields for soyoil and the metals. 4.3.6 Replication of Fama and French Seasonality in futures returns and implicitly in convenience yields is tested for with a methodology similar to that of Fama and French (1987). Again, seasonal dummy variables proxy inventory levels. The seasonality hypothesized to exist in the commodity's futures returns will depend on the production charactersitics of the specific commodity. Agricultural commodities, where the inventories vary throughout the crop year, will have parameter estimates on the dummy variables which also vary throughout the year. This hypothesis can be tested with a standard F-statistic described in section 4.3.5. The parameter estimates for the dummy F = T - K Q SSEU + (T - K)\' S S Ec — S S Eu 60 variables of the metals are hypothesized to display no seasonality since inventory does not vary systematically. The unconstrained model assumed to generate commodity futures returns is of the form, (4.6) Y^Ehi+R^ + ei where Y,• = T x 1 vector of futures returns for commodity i, FRU FRTi. D — T x 4 matrix of dummy variables identifying the quarter of the year, h h R = T x 1 matrix of T-bill rates, n 7,- = 4 X 1 vector of intercept parameters for commodity i, <j>i = slope parameter for commodity i, and = T x 1 vector of disturbances with E(et-£,i) = £,-. The model of section 3 hypothesizes that the difference between the current spot price of a commodity and its futures price will equal the commodity's convenience yield less carrying costs over the term of the futures contract. Assuming that storage costs are a constant percentage of the current spot price, seasonality in the quarterly parameter of the dummy variable will capture the seasonality in the commodity's convenience yield. That is, under this assumption the differences in estimates will be attibutable to differences in convenience yields. The parameter estimates of equation (4.6) for the commodities are reported in Table 30 and the F-statistics from the hypothesis tests are reported in Table 31. The commodities that display significant seasonality in their futures returns are Chicago wheat, corn and soybeans. In the case of Kansas and Minneapolis wheat the hypothesis that the coefficients for each quarterly dummy are equal cannot be rejected, although the F-statistic is very close to the critical value. 61 The values of the F-statistics examining the seasonality in the futures returns are very low for soyoil, copper, silver and gold. The results are consistent with those of Fama and French (1987). In their paper seasonals in the difference between a commodity's futures price and spot price are found for corn, soybeans and wheat but not for metals and soyoil. 4.3.7 Autoregression in Demand Shocks Proposition 2 states that the convenience yield is zero when p = 1 and proposition 3 states that the convenience yield is decreasing in p when the marginal cost of production is constant. Ceterus paribus, the closer p is to 1, the smaller the convenience yield. It is difficult (if not impossible) to observe p directly, however, Appendix 7 proves that the covariance between the t\ and t2 spot prices is increasing in p. Interpreting a quarter of a year as one complete cycle of the model, the correlation coefficient between the spot price at the end of the quarter and the spot price observed on the 15th of the previous month, can serve as a proxy for p. These correlation coefficients are calculated and reported in Table 12. Cross-sectional differences, such as the level of aggregate inventory, cannot be controlled for so an interpretation of the association between p and the relative size of the convenience yield must be conditional on this fact. It is felt that an examination of the correlation coefficients of the commodities may still reveal information that is useful in interpreting further results. The commodity that has the largest correlation coefficient between its spot prices is gold. The value of this coefficient is .98444. Silver has the second highest correlation coefficient at .97440. The ratios of convenience yield to spot price for these commodities are each under one percent; the lowest values for all commodities. Copper prices have a correlation coefficient of .91423 and convenience yields of between 1 to 2 percent of its spot price. The correlation coefficients for soybeans and soybean oil are .73615 and .80592 respectively. These are the two lowest correlation coefficients across commodities. For these two goods the convenience yields are roughly between 4 and 6 percent of the spot price. Corn and Kansas wheat are the two commodities that, when compared to the sample, appear to have both high correlation 62 coefficients (.86390 and .92306 respectively) and high convenience yields (ranging from approx-imately 7.5% to 14%). A number of factors could explain these results. For example, these two grain products may have a higher probability of shortages than the other goods. It is also possible that the quality option modeled by Gay and Manaster (1986) may be larger for grains than for metals, soybeans and soybean oil. Finally, the correlation coefficient in spot prices may not be an accurate proxy for p. Even in light of the corn and Kansas wheat results, there is a general relationship between the correlation in spot prices and the size of the convenience yield, with metals having the high-est correlations and the smallest convenience yields and agricultural products having relatively smaller correlation coefficients and higher convenience yields. 4.4 Summary The empirical results obtained lend support to the model of the convenience yield derived in Chapter Three. The strongest result is the negative association between a commodity's convenience yield and the level of aggregate inventory. This result is robust with respect to the type of estimation procedure used. Indirect support for proposition 3, that the convenience yield is decreasing in p, is also provided. 63 CHAPTER FIVE CONCLUSION 64 5.0 Conclusion This study had two primary objectives. The first was to provide a model of the convenience yield. The second was to empirically test the implications emanating from the model. A model is derived in Chapter Three that draws an analogy between the convenience yield and a call option with a stochastic exercise price. The 'option' only has value when there is a non-zero probability of the spot price exceeding the futures price (which is like the exercise price). This occurs when a 'corner solution' is obtained (i.e. all the inventory is sold in one period and no inventory is sold in the subsequent period). The above result implies that there must be a non-zero probability of shortages in the commodity in order for the convenience yield to have value. The model has three major implications: 1. ) The convenience yield is decreasing in aggregate inventory. 2. ) The convenience yield is zero when p = 1. 3. ) When the marginal cost of production is constant, the convenience yield is decreasing in p. The empirical analysis focuses mainly on the first implication, that the convenience yield is decreasing in aggregate inventory. Futures prices and aggregate inventory data are collected for eleven commodities; three metals and eight agricultural products. Using estimates of storage costs and T-Bill rates, a time series of convenience yields is calculated for for the different commodities. The data are examined for irregularities, and adjustments are made in the estimation procedure for departures from the assumptions of the general linear model. Using OLS, GLS, and SUR estimation techniques, support is generated for the hypothesis that the convenience yield is decreasing in aggregate inventory. There is some evidence that the relationship is non-linear, and the model appears to have the best fit when inventories are low. This last observation may imply that when inventories are above some 'critical level' the convenience yield approaches zero. This notion is supported by a simple examination of the summary statistics for convenience 65 yields. The mean values of the convenience yields are found to vary across seasons for seasonal commodities, and also across commodities, being the smallest in the pre-harvest month. In the case of metals, the convenience yields are close to zero. The last two implications of the model are difficult to test due to the fact that it is impossible to observe the correlation between shocks to demand. Using a proxy for this correlation, the empirical evidence does lend some support for the hypothesis that the convenience yield is decreasing in the correlation between such shocks, although a statistical test is not provided to test this relationship. 5.1 Discussion of Extensions and Limitations The model Chapter 3 assumes a three date, two period world implying that all inventory is sold by t2, the last date. When testing the model the assumption that all the inventory must be sold by t2 is violated. In extending the model to a continuous time or repeated period world, the implication of relaxing this assumption must be addressed. If firms are allowed to costlessly carry inventory over from the maturity of the futures contract to the following period, then the value of the commodity delivered to the holder of the futures contract will encompass a convenience yield associated with the following period. For example, if the cycle of tQ,t\, and t2 were repeated, and if firms could continue to carry over inventory, then there would be a convenience yield in the spot commodity delivered at t2. The convenience yield can be thought of as a flow of goods (like a dividend) over a period. Nevertheless, holding the commodity in inventory would still yield the benefit of the convenience yield which accrues during the maturity of the futures contract. There are two factors which would reduce the value of the future stream of benefits implicit in the spot commodity delivered at the maturity of the futures contract. The first is the interest rate. With high interest rates the value of the future flow of benefits is discounted, widening the spread between the futures price and the spot price. The second factor is related to the perishability of the commodity. If the good which is delivered deteriorates with time, the value 66 of the convenience yield contained in the commodity which has been stored for a period of time will be -less as the commodity deteriorates. In fact, the assumption that the good must be sold at t2 could be replaced with an assumption that the commodity is perishable, and significantly diminishes in value before the next production period. While these assumptions are more reasonable for some commodities than others, a revised model of the convenience yield could have the rate of deterioration as one of its components. Intuitively, it would seem that the convenience yield of a commodity held from one production period until prior to the next production period would have a value which is increasing in perishability. Although the model in the thesis is developed in a two period world, the concepts can be generalized to a repeated period world. An analogy can be drawn between the C A P M and the proposed model of the convenience yield. Even though two periods were used in the derivation of the model of the convenience yield, the intuition extends beyond the constraints of the model. The idea that the holder of the commodity has the option to take advantage of transitory increases in the spot price can be applied to a multi-period framework. Similarly, the basic intuition of the C A P M , which is that investors are only compensated for systematic risk, also extends to a multi-period world. Like the C A P M , however, there are important underlying assumptions. In the derivation of the model of the convenience yield the assumption of no instantaneous production is made. The importance of this assumption is discussed in Chapter 3 but it is worth emphasizing. With instantaneous production at a low marginal cost, profit maximizing behavior of the firms would always force Po = -Eb(Pi) = ^(Pi) = Eo[C'(qi)]. Since the convenience yield is expressed as Po - EQ[P2\, with instantaneous production at a low marginal cost it would always be zero. Instantaneous production at increasing costs is consistent with the existence of a convenience yield. For exam-67 pie, increasing the rate at which copper is produced would likely result in additional expenses such as: paying a higher marginal wage to attract more labour, employing less productive machinery, and opening mines with a lower ore concentration. In the case of agricultural prod-ucts, increasing production in a non-growing season could be achieved, but at the expense of growing the commodity in greenhouses or in a country where it is currently the growing sea-son and transporting it back to meet domestic demand. Obviously, different commodities will have different costs associated with changing the rate of production. Likewise, the size of the convenience yield will vary across commodities. Assumptions have been made in the derivation of the model to simplify the expression of the convenience yield. These assumptions are: zero interest rates, no storage costs and risk neutral firms. Relaxing these assumptions would not affect the results in the paper. The outcome of relaxing these assumptions would be to add components to the representation of the futures price and to complicate the model of the convenience yield. There would still be an option value to holding inventories of a commodity, and the value of this option would be decreasing in aggregate inventory. 5.2 Direction of Future Research The obvious direction in which to extend this research is to move from a two period world to a continuous time model. The advantage of this extension is that continuous time more closely approximates reality. While continuous observations of price movements is not available, daily data is. The maturity of the futures contract would form a boundry condition which would be used to derive an expression for the convenience yield. Such an expression could be more easily related to the standard Black-Scholes option pricing model. The extension would aid in the interpretation of the model but will not have a significant impact on the major results of the paper for the reasons which have been previously discussed. The intuition of an option value associated with an inventory of a commodity is robust with respect to relaxing the assumption of a three date world. 68 APPENDICES 69 Appendix 1 Proof: The correlation between hi and h2 is — - — j - , Assuming that h\ ~ E(hi) = 0; with ox = a, and recalling that h2 = pni + e, where, E{e) = 0, a2 = o{\ + /?2)2"and er then, h\h2 — hi(phi + e) = /?ra2 + rai?. Taking the expectation, E(hih2) = pa1. Therefore, the covariance of h\ and h2 is Cov(ni,ra2) = po1 - E{hx)E{h2), and the correlation coefficient is Corr (n i ; h2) = P<T <7 2(l+p 2)? (l + p2)? Q.E.D. 70 Appendix 2 Proof: The Convenience Yield is decreasing in aggregate inventory. Let q[(ni) = ql and ^ ( « i ) = i{. The convenience yield is expressed as: r oo C Y = / [Pi(»1;«5 + q{) - E{P2{pnl + e,0))}f{n1)dn1. Jh\ Then the convenience yield will be affected by the aggregate inventory through the change in the optimal amount of the commodity carried over to t\; i.e., a c Y _ d C Y a*o dl ~ 3% ' dl' In signing the comparative static ^jj- the first step is to determine how a change in ^  impacts P i and Ei(P2) and then to find how changing I affects ijj. There are three different equilibrium conditions over the range of it\ and the comparative statics T J T and -yi- must be found for each region. d,Q d,Q 5 A. ni < hlx: The equilibrium condition in this region is (3.5) C'{ql) = j P2{pni +e,zl+ ql)g{e)de. Let C" = C"(ql) = a positive constant, <j> = dEg^2^ = 'j^g(e)de. Prices are decreasing in yt, therefore <j> < 0. Taking the total derivative of equation (3.5) gives the expression (C - <t>)dq{ = <l>d% dq* and the derivative ^J- is greater than —1 arid less than zero; B . h[ < ni < raj: 71 In this range there are two equilibrium conditions, i.) C'(ql) = Pi(n 1 ; io + q\ - t[), «•) ' <?(?!) = / P2{pnl + Z,ii)g(e)ck. and By h), or, Therefore, From i , Rearranging terms, C dq^ = (fidi'i, C" dh= (~^)d^i-dz* C" «?i 9 d'dql = wXK + d q l ~ { T ) d q ^ ) ' dyi dyi V 9 «?l _ dyi In the region n x < ni < h\, + f g  (C"-<£) _ 1 < 7 7 * ~ < °-C. ni > h\: 72 In this range the equilibrium condition is C'{q\) = Pi[nlt$ + ql). Taking the total derivative, the expression is n» , * dPi <, dP\ „ c d 9 l = — d l Q + — i 9 l . The sign of the comparative static is aPi dq{ _ dSl C" -and thus, 1 < —*V <o. The equilibrium commodity price at to is, -oo + / 1 ^ i ( P 2 0 ' « l - r : ? , * l ) ) / ( n i ) r f n i TOO The total derivative of Po is, <9P0 d i n + / ^ ( ^ o + ^ ^ ) / ( n i ) r f « i -Jfh dyi = d*o n ^ ( i + M ) / M ( / n i •/-oo <"o 73 +/,1"+IH-dp Let [•] = Z. Because 4> and are less than zero, Z < 0. Grouping the terms gives ^2/0 \ 3 y 0 A n Thus, the comparative static is > 0. Taking the total derivative of the convenience yield with respect to the amount of inventory carried from to to t\, the expression is, dCY din d ~ Pi(hh1,%+ ql) - E1(P2(phh1 + ?,0)) }(h\)dni + / —Lf(ni)dni. Jo L J Jn\ dyi By definition of h*, Pl{nhv il + ql) - Ei{P2[ph\ + 6,0)) = 0, therefore, a c Y Jh1} dyi The comparative static of the convenience yield with respect to inventories is then S C Y d C Y din 31 di*0 dl < 0. QE.D 74 Appendix 3 Proof: The Convenience Yield is zero when p = 1. To examine the relationship between the convenience yield and p the following assumptions are made: gp. i . ) = a, a constant, so Pt = ant + K{yt), i i . ) IC < 0 and #(0) = M , and i i i . ) n2 = + ? ; where i£(e) = 0, and a e = cr = <7n . The t\ price when all available inventory is sold is, The expected t2 price where nothing is sold is Ex{P2{n2,)) = a p n ! + M . The shock to demand that equates these two prices is raj. The breakeven value of raj is that value where a{l-p)h\ = M-K($i+ql). or (3.6) a{\-p)h\ = M - K?0-Kql{h\). The left hand side (LHS) of equation (3.6) is increasing in n\ and is zero when n\ is zero. The right hand side (RHS) of equation (3.6) is positive when n\ is zero. From the first order condition, ^ = 2 e / /_^ > 0, therefore, the RHS is increasing in n\. As p —• 1, the value of the LHS approaches zero with a slope equal to zero. As p —> 1, the intercept of the RHS drops but is always positive. 3 8 The slope of the RHS remains unchanged as p —* 1, therefore, raj —• oo. Appendix 4 proves that < 0. 75 The convenience yield is expressed as roo CY= ^1(^,11 + q{) - E{P2(pni +~e,0))]f{nl)dn1. Therefore, h\ oo => C Y 0. Q.E.D 76 Appendix 4 Proof: When the marginal cost of production is constant the convenience yield is decreasing in p. Let a 2 = J[ j£g{e)de > 0. The convenience yield is expressed as poo C Y = / [P^in + ql)-E{P2{pnl+~e,0))\f{n1)dnl. The derivative of the convenience yield with respect to p is dp Jhh Ldyi d% dp It has already been illustrated that ( l + -Jj-) = -prz^ when ni > nj. When the marginal costs to production are constant, c" = 0. Since a 2 > 0, a-jJ- < 0 Q.E.D. 77 Appendix 5 Adjustment for Heteroscedasticity When the elements of the random error vector e are uncorrelated and have identical vari-ances the covariance matrix of errors is, E[ee']= a21. In the case when the e's have different vari-ances, (ie. the data is heteroscedastic) the covariance matrix of the error vector is E[ee']=cr2,I' where \& is a real positive definite symmetric matrix. The estimate of the coefficients becomes, J3 = {X'^~lX)~1X'^~1y. The appropriate transformation of the data is, Py = PX/3 + Pe, such that P'P = According to the theory, ^ is a diagonal matrix with | the first element of the first row and | the second element of the second row. The third and fourth rows have ones in the third and fourth positions. 78 Appendix 6 Derivation of the Univariate F-Statistic Consider the problem of a single commodity where there are four (sub) samples generated from the model Yj = otj + Ijj3j + £j where, i = 1,4 Assume that within each subsample s « iV(0,<r2) (a2.1) and that there are T observations in each subsample. Further, assume that the subsamples are from independent time periods. The hypothesis Ho: cr,2 = cr2 is tested against the alternative H A : a, > Cj for i = 1 and j — 2. A n F-statistic is suggested as a possible test procedure. 2'e-Under assumption (a2.l), the quadratic form is distributed x 2(7 1 — 2) where ai Si=yi-XiPi and pi={X'lXiy1X,iyi. Proof: (A2.1) et = yi-Xi(X'iXi)-1X'iyi = Xifli + Si - X^X^X^X,^ + e,-) where M = [/ - Xi{Xi{X\Xt)-lX'i]. Thus, {A2.2) 79 Each of the et- vectors in (A2.2) can be used to test the hypothesis that a2 equals any set value. For this data the hypothesis of interest is Ho: cr2 = <r2 for the subsamples i = 1 and j = 2. For subsample j the analogous test statistic is (A2.3) - 2 _ | ^ x 2 ( T - 2 ) . Under the null hypothesis af = a2. The ratio of these two quadratic forms has an F(T-2, T-2) distribution. Proof: The ratio of two independent x 2 variates times the ratio of the inverse of their respective degrees of freedom has an F distribution, or The ratio of equations (A2.2) and (A2.3) has this form because the statistics are independent and have the same degrees of freedom: _ i a2 (A2.4) T~2 « F{T - 2, T - 2). Under the null hypothesis a2 = a2 and (A2.4) reduces to e'Me • (42.5) ' ' ' w F{T - 2, T - 2). Ej MjSj Q E . D . Sufficiently large values of the F-statistic in excess of a critical value, represents a rejection of the null hypothesis. The alternative is one tailed. Discussion: The statistic proposed is similar in spirit to the Goldfeld-Quandt test (1965). The Goldfeld-Quandt (G-Q) test depends on the ability to rank observations according to increasing variance. To implement the test r central observations are omitted and two separate regressions are 80 run on the first and last (T-r)/2 observations. The G-Q statistic is calculated as R = ^ where si and s2 are the residual sums of squares from the first and second regressions. The decision to accept or reject the hypothesis is based on R which has an F distribution with [(T - r - 2K)/2, (T - r - 2K)/2\ degrees of freedom. The Goldfeld-Quandt test is based on exact finite sample properties. Because two separate regressions are run, lack of independence of the least squares residuals is not a problem. The decision regarding what r to use, however, is quite arbitrary. The optimum value of r is not obvious. Large values are likely to increase the power of the test through an increase in the value of the F-statistic but decrease the power through a reduction in degrees of freedom. The statistic proposed in this paper is similar to the Goldfeld-Quandt statistic in that it compares the variance of two regressions, but the observations which are omitted are done so according to theory. Assuming that the variance of the convenience yield is linearly related to the length of the period over which it is calculated, the first and the second quarters will be affected by herteroscedasticity. The third and fourth quarters are each three months long and there is no theory that states there should be any difference between their variances. In performing the test, s\ is the variance of the first quarter and s2, is the variance of the second quarter. Omitting the third and fourth quarters observations is analogous to omitting central observations. 81 A P P E N D I X 7 Proof: The Covariance Between P i and P 2 is Increasing in p. Assume: P\ = ani + K(io — i\) and P 2 = c*n2 + A"(ti), where n 2 = pn\ + e . In the region n'i < n\ < h\ (-41.1) ccni -+- K(io + *i) = cxpn\ + K[t[yit[ is the definition of %[. This implies dii _ a ( l ~ P) 0 dm 2 / f (y 2 ) Proof: Taking the derivative of equation (1) with respect to n\ yields, Rearranging terms gives the result, dii = <*(! ~ l ) < Q rfni 2K'(-) di* Having shown that ^ < 0, the relation between Cov(Pi, P 2 ) and p will be derived. The C o v ( P i , P 2 ) can be expressed as, (A1.2) Cov{Pl,P2) = Cov(ani + Kfo - %),apni + ae + K(%)), where by definition of %[, an\ + K{IQ — i^) = apn\ + K{t£). Using the defintion of i[, equation (A 1.2) can be rewritten as Co t ; (P i ; P 2 ) = Cov(apni, apn{) + Cov{apn\, K(i\)) + Cov[apni, K(iD) + Cov{K{i\), K{i\)). Rearranging terms, (41.3) Cov(P1;P2) = {aP)2a2ni + 2<xpCov{n1,K{il)) + Var{K{i[)). 82 Assuming that then K{y2) = M - By2 = A - Bi[, Jf(-) = -B. (A1.4) The derivative of with respect to n\ is di[ a(l — p) 2B < 0. Integrating equation (4) over ni gives a(l-p) » i ( « i ) = * 2B~ni' where K is a constant of integration. Using the boundry condition that «x(™i) = 0> K *S found to be equal to q ( l - p) A «; = —— n and 25 ft*- '5*> 1; <*{I-P)' The constant of integration equals \ io. The optimal amount of inventory carried from t\ to t2 is (A1.5) „, 1 a ( i - /?) *i(»i) = 2 5 — W l " The variance of /f(tj(ni)) is equal to the variance of equation (A1.5), or V ar(K(il(ni)) = Q ( A a 2 (25) The Co«(m,if(fI)) = Covinu^p^ni) = ^ ^ j ^ , and the Var{K{i\)) With these results, equation (A1.3) can be written as, Cov(Pl, P2) = (apfcr^ + a2P(l - p ) < + ^ ~ a(i-p) 2 "1" = ^ l 1 + «>[\p* + p(i-p) + \(i-P)*K = ^ < + y [ p 2 + 2p ( l -^ ) + ( l - p ) 2 ] < where [•] = 1. Therefore, Cov(Pl!p2)= 2" ' nK The derivative of the covariance between Pi and P2 is then equal to dCovjP^P,) _ 2 2 TP 01 pa»i > a Q.E.D. 84 Appendix 8 Proof: $ < 0. The equilibrium commodity price at tn is • a' Po(/" ib) = / 1 MP2{pni + ~e,i0' + ql))f{n1)dn1 J —oo + / P i ( « i , £ + ?i - *[))/(ni)rfni /•oo + / L Pi(«i;«b + q\)f{n\)dni. The total derivative of PQ is V —oo "Vi + r v 1 ( i + ^ - ^ - ^ K 0 / ( n l ) ^ y ^ «*o d ? o Jhh diQ Rearranging terns gives the expression a 2n 1/(n 1)rfrai dp = rfzo|^ + y ^2(1 + - ^ - ) / ( « i ) r f » i + / , ^ 1 ( i + [ l - & . M ) / ( B l ) d l l l a?i a«o Jfj} DLQ 85 Let [•] = Li and {•} = L2. Then the above equation can be written as L\ dp = L2diQ. Since, Li > 0, and L2 the derivative, f = h < 0. dp L2 Q.E.D. 86 Appendix 9 Data Sources Treasury Bi l l Rates - Treasury Bi l l Rates - 1986 C R S P Government Bond Tape Inventory Data Silver - U.S. Bureau of Mines Mineral Industrial Surveys (Quarterly), end of period Industry Stocks. Gold - U.S. Bureau of Mines Mineral Industrial Surveys (Quarterly) Commercial Stocks, end of period stocks. Copper - Commodity Yearbook, 1986 (p. 55). Refined copper stocks in the U.S.A. Wheat - Commodity Yearbook, 1982 (p. 369), 1986 (p. 287). Commercial Stocks of Domestic Wheat in the U.S.. Corn - Commodity Yearbook, 1978 (p. 133). U.S. Commercial Stocks of Corn. Soybeans - Commodity Yearbook, 1986 (p. 242). U.S. Commercial Stocks of Soybeans. Soyoil - Commodity Yearbook, 1977 (p. 314), 1986 (p. 238). U.S. Stocks of Soyoil at Factories and Warehouses. Price Data - A l l futures prices and spot prices are gathered from the Wall Street Journal. Prices collected are the closing prices of the first of trading day of each quarter. If no trade occured in a specific commodity on the first day of the quarter, then the closing price of the previous trading day is used. A P P E N D I X 10 T A B L E 1 Correlation Coefficients Between Commodity Spot Prices and Futures Prices of Contracts for Immediate Delivery Commodity Wheat (Kansas) Corn (Chicago) Soybeans (Chicago) Soyoil (Chicago) Copper (CMX) Silver (CMX) Gold (IMM) Correlation Coefficient .98837 .98704 .99817 .98984 .83345 .99883 .99853 89 T A B L E 2 Summary Statistics U.S. Commercial Stocks of Commodities* Name Obs. Mean St. Dev. Variance Mmimum Maximum Wheat millions of bu. July 12 264.56 63.224 3997.3 162.90 383.90 Sept. 12 421.84 88.546 7840.4 217.40 526.50 Dec. 12 382.27 81.485 6639.8 246.90 473.90 March 12 260.39 80.470 6475.4 128.40 370.00 Corn millions of bu. Sept. 3 33.3 9.3 86.3 25.6 46.4 Dec. 4 140.7 15.35 235.8 119.5 155.1 March 5 132.8 6.5 42.0 122.7 152.9 July 3 52.1 12.7 162.5 34.8 65.1 Soybeans millions of bu. March 12 52.667 13.104 171.71 35.3 73.200 July 12 29.967 11.233 126.19 12.700 53.200 Soyoil millions of lbs. July 12 1165.8 488.36 .23850E+06 530.60 2139.0 Sept. 12 1051.1 367.34 .13494E+06 567.10 1783.0 Dec. 12 1097.0 464.25 .21553E+06 580.10 1884.0 March 12 1244.5 535.76 .28704E+06 633.70 2141.0 Copper thousands of short tons July 12 340.09 135.94 18481 39.800 519.40 Sept. 12 337.52 133.58 17843 54.840 522.70 Dec. 12 346.21 124.27 15442 134.70 505.20 March 12 347.97 140.47 19733 43.800 501.60 Silver thousands of troy oz. Sept. 12 25337 7518.4 .56527E+08 15587 39551 Dec. 12 26910 10046.0 .10093E+09 15937 49329 March 12 28245 9284.2 .86196E+08 17240 49062 Gold thousands of troy oz. July 8 3167.1 1389.6 .19310E+07 1082.0 4766.0 Sept. 8 3140.4 1399.2 .19577E+07 800.0 5096.0 Dec. 8 3495.0 1363.5 .18590E+07 1119.0 5927.0 March 8 3077.6 1295.8 .16790E+07 1050.0 4290.0 * The second quarter inventory measurement for silver does not correspond to the futures and spot price data and was omitted. Likewise, difficulty in corresponding the inventory data with the price data is responsible for so few observations for corn. Since there are no Soybean futures contracts for delivery in December, the September and December inventories are not reported. 90 T A B L E 3 Summary Statistics Quarterly Spot Prices Name Obs. Mean St. Dev. Variance Minimum Maximum Wheat (Chicago) cents per bu. July-Sept. 12 355.60 59.924 3590.0 240.24 443.25 Sept.-Dec. 12 358.92 68.414 4680.5 220.38 467.50 Dec.-March 12 368.80 74.779 5591.9 255.88 495.25 March-July 12 375.35 89.573 8023.3 259.13 593.50 Wheat (Kansas) cents per bu. July-Sept. 12 359.40 59.516 3542.2 236.50 433.00 Sept.-Dec. 12 363.74 61.801 3819.3 231.63 450.00 Dec .-March 12 375.19 73.012 5330.7 256.88 481.00 March-July 12 377.64 80.760 6522.1 266.75 571.00 Wheat (Minneapolis) cents per bu. July-Sept. 12 388.76 61.228 3748.9 251.00 466.00 Sept.-Dec. 12 381.40 63.739 4062.7 248.50 455.25 Dec .-March 12 383.19 73.352 5380.6 275.25 535.00 March-July 12 386.46 79.725 6356.0 269.50 575.00 Corn (Chicago) cents per bu. Sept.-Dec. 12 278.72 56.494 3191.6 183.88 362.75 Dec .-March 12 277.45 55.937 3129.0 219.88 389.00 March-July 12 276.19 38.770 1503.1 225.75 351.00 July-Sept. 12 288.72 37.771 1426.7 216.13 356.25 Soybeans (Chicago) cents per bu. March-July 12 633.70 98.950 9791.1 478.63 783.75 July-Sept. 12 650.33 76.941 5919.9 500.50 747.00 Soyoil (Chicago) cents per lb. Sept.-Dec. 12 25.184 5.1073 26.084 17.410 34.875 Dec .-March 12 24.030 6.0462 36.557 16.830 38.025 March-July 12 23.986 5.0817 25.823 16.560 33.850 July-Sept. 12 24.865 4.9140 24.147 18.670 33.425 Copper ( C M X ) cents per lb. July-Sept. 12 69.508 12.705 161.43 53.400 90.600 Sept.-Dec. 12 68.546 11.586 134.23 54.600 90.000 Dec .-March 12 67.883 16.224 263.21 52.300 108.00 March-July 12 75.625 20.980 440.16 55.400 116.00 Silver ( C M X ) dollar per troy oz. July-Sept. 12 7.4410 3.7050 13.727 4.3950 16.810 Sept.-Dec. 12 7.6945 3.8647 14.936 3.7600 16.335 Dec .-March 12 8.5980 5.4107 29.275 4.0630 20.050 March-July 12 9.3295 8.5131 72.473 4.1620 35.200 Gold ( IMM) dollars per troy oz. July-Sept. 8 311.65 160.14 25644 128.30 566.00 Sept.-Dec. 8 333.88 175.65 30853 105.20 639.00 Dec .-March 8 348.41 170.77 29161 131.00 624.50 March-July 8 321.74 176.36 31103 131.90 630.50 * The spot prices reported are the prices of a future contracts for immediate delivery. 91 T A B L E 4 Summary Statistics Quarterly Futures Prices* Name Obs. Mean St. Dev. Variance Minimum Maximum Wheat (Chicago) cents per bu. July-Sept. 12 364.33 60.845 3702.1 247.38 448.75 Sept.-Dec. 12 374.22 73.102 5343.9 230.13 492.50 Dec .-March 12 381.48 80.889 6543.0 267.00 526.00 March-July 12 369.51 77.526 6010.2 267.38 541.00 Wheat (Kansas) cents per bu. July-Sept. 12 364.83 60.964 3716.7 242.50 433.00 Sept.-Dec. 12 375.24 67.524 4559.5 236.50 473.00 Dec .-March 12 381.01 77.642 6028.3 266.25 507.00 March-July 12 374.95 78.836 6215.1 263.50 543.50 Wheat (Minneapolis) cents per bu. July-Sept. 12 386.19 58.889 3467.9 257.75 454.00 Sept.-Dec. 12 386.81 67.572 4566.0 248.75 474.75 Dec .-March 12 391.07 78.212 6117.1 281.88 543.00 March-July 12 387.75 75.026 5628.9 272.25 554.00 Corn (Chicago) cents per bu. Sept.-Dec. 12 279.48 54.859 3009.5 192.63 360.00 Dec .-March 12 287.23 57.018 3251.1 227.75 404.50 March-July 12 288.59 37.744 1424.6 230.13 365.75 July-Sept. 12 283.29 36.029 1298.1 221.25 342.75 Soybeans (Chicago) cents per bu. March-July 12 659.58 101.02 10204 496.25 791.50 July-Sept. 12 645.98 77.810 6054.4 490.25 750.25 Soyoil (Chicago) cents per lb. Sept.-Dec. 12 24.720 4.7367 22.436 17.950 32.930 Dec .-March 12 23.903 5.5792 31.127 17.200 36.520 March-July 12 23.525 3.7016 13.702 17.125 28.100 July-Sept. 12 24.059 3.8102 14.518 19.010 29.550 Copper (C MX) cents per lb. July-Sept. 12 70.521 12.622 159.31 56.100 92.000 Sept.-Dec. 12 70.508 12.122 146.95 55.700 92.300 Dec .-March 12 69.025 15.181 230.47 53.800 103.100 March-July 12 77.358 19.997 399.88 57.400 120.800 Silver ( C M X ) dollars per troy oz. July-Sept. 12 7.5645 3.7742 14.244 4.4280 17.100 Sept.-Dec. 12 7.9192 3.9791 15.833 4.0950 16.870 Dec .-March 12 8.7922 5.5325 30.608 4.1380 20.070 March-July 12 9.7287 8.9108 79.403 4.2520 36.690 Gold ( IMM) dollars per troy oz. July-Sept. 8 320.14 166.23 27632 130.15 582.20 Sept.-Dec. 8 343.32 182.87 33443 105.20 660.70 Dec .-March 8 358.95 179.03 32050 132.65 665.50 March-July 8 333.01 186.40 34746 132.15 662.50 * Futures prices are for delivery one quarter in the future. 92 T A B L E 5 Summary Statistics Quarterly Convenience Yields* Name Obs. Mean St. Dev. Variance Minimum Maximum Wheat (Chicago) cents per bu. July-Sept. 12 22.625 7.6256 58.150 12.030 36.001 Sept.-Dec. 12 26.336 9.3058 86.597 13.862 44.179 Dec.-March 12 29.962 10.462 109.44 15.516 51.897 March-July 12 55.488 31.354 483.09 19.582 122.33 Wheat (Kansas) cents per bu. July-Sept. 12 27.060 7.544 56.915 13.084 39.865 Sept.-Dec. 12 30.697 8.250 68.211 19.942 51.631 Dec-March 12 37.531 10.875 118.260 17.371 57.589 March-July 12 53.091 21.425 459.040 20.689 97.761 Wheat (Minneapolis) cents per bu. July-Sept. 12 39.586 13.775 189.750 13.418 58.562 Sept.-Dec. 12 38.777 7.295 53.211 26.106 46.408 Dec-March 12 36.264 8.606 74.062 23.156 51.824 March-July 12 50.743 17.877 319.600 24.712 92.650 Corn (Chicago) cents per bu. Sept.-Dec. 12 31.871 11.566 133.770 11.172 49.111 Dec-March 12 22.539 8.535 72.843 14.031 38.512 March-July 12 26.682 9.008 81.150 15.561 44.326 July-Sept. 12 34.047 17.236 297.080 12.607 65.707 Soybeans (Chicago) cents per bu. March-July 12 25.703 9.4609 89.509 11.825 .42.969 July-Sept. 12 39.369 25.742 662.67 13.245 91.248 Soyoil (Chicago) cents per lb. Sept.-Dec. 12 1.4620 1.3262 1.7588 -.11442 3.7018 Dec .-March 12 1.0675 1.3116 1.7202 -.04018 4.0640 March-July 12 1.5003 2.3565 5.5532 .02656 8.3885 July-Sept. 12 1.7630 2.0781 4.3187 -.67965 5.8743 Copper ( C M X ) cents per lb. July-Sept. 12 1.0202 .8755 .7666 .3900 3.0293 Sept.-Dec. 12 .8633 .3974 .1579 .2097 1.6253 Dec-March 12 1.6537 2.7088 7.3378 -.0581 10.0610 March-July 12 2.1744 4.2254 17.8540 .2330 15.4090 Saver ( C M X ) dollars per troy oz. July-Sept. 12 -.00883 .03092 .00090 -.05851 .04987 Sept.-Dec. 12 -.03024 .07565 .00572 -.24297 .05219 Dec .-March 12 .03237 .18305 .03351 -.14899 .59533 March-July 12 -.03289 .16337 .02669 -.43386 .26381 Gold ( IMM) dollars per troy oz. July-Sept. 8 -.44722 1.6783 2.8187 -4.420 .6951 Sept.-Dec. 8 -.58646 1.8675 3.4875 -4.7039 1.483 Dec-March 8 -.90950 2.7108 7.3482 -6.7659 2.8537 March-July 8 -1.8722 4.9072 24.0800 -10.188 3.9865 * The convenience yield is calculated as CY — P<(1 + Rt + 7) - Ft. The convenience yields reported in this table are adjusted for the uneven lengths of the quarters. 93 T A B L E 6 Summary Statistics Quarterly Ratios of Convenience Yield to Spot Price* Name Obs. Mean St. Dev. Variance Mmirnum Maximum Wheat (Chicago) July-Sept. 12 .06297 .01552 .00024 .04358 .08366 Sept-Dec 12 .07324 .02235 .00050 .04896 .13004 Dec-March 12 .08121 .02536 .00064 .05986 .13848 March-July 12 .14092 .05741 .00329 .06292 .26016 Wheat (Kansas) July-Sept. 12 .07491 .01582 .00025 .05532 .09434 Sept-Dec 12 .08518 .02155 .00046 .06555 .13796 Dec .-March 12 .09998 .02335 .00055 .06762 .13163 March-July 12 .13862 .04102 .00168 .07666 .18740 Wheat (Minneapolis) July-Sept. 12 .09980 .02835 .00080 .05246 .14595 Sept-Dec 12 .10234 .01504 .00023 .07556 .12865 Dec .-March 12 .09533 .01786 .00032 .06684 .12058 March-July 12 .12878 .02439 .00059 .08507 .16755 Corn (Chicago) Sept-Dec 12 .11188 .02984 .00089 .06075 .16564 Dec .-March 12 .07987 .01922 .00037 .05265 .11338 March-July 12 .09498 .02112 .00045 .06067 .13381 July-Sept. 12 .11574 .05185 .00269 .05731 .19842 Soybeans (Chicago) March-July 12 .03971 .01063 .00011 .02346 .05483 July-Sept. 12 .06019 .03623 .00131 .01965 .12505 Soyoil (Chicago) Sept-Dec 12 .05454 .04528 .00205 -.00553 .12957 Dec .-March 12 .03928 .04360 .00190 -.00199 .14447 March-July 12 .05191 .07042 .00496 .00144 .24781 July-Sept. 12 .06236 .06720 .00452 -.02766 .17574 Copper (CMX) July-Sept. 12 .01380 .00959 .00009 .00659 .03381 Sept-Dec 12 .01245 .00525 .00003 .00379 .02437 Dec .-March 12 .01999 .02443 .00060 -.00101 .09316 March-July 12 .02345 .03524 .00124 .00207 .13283 Silver (CMX) July-Sept. 12 -.00123 .00502 .00003 -.01289 .00841 Sept-Dec 12 -.00655 .01859 .00035 -.06462 .00474 Dec .-March 12 .00062 .01013 .00010 -.01016 .02969 March-July 12 -.00502 .01346 .00018 -.04212 .00749 Gold (EVLM) July-Sept. 8 -.00050 .00400 .00002 -.00781 .00475 Sept-Dec 8 .00037 .00627 .00004 -.00736 .01410 Dec .-March 8 -.00219 .00521 .00003 -.01083 .00701 March-July 8 -.00364 .01174 .00014 -.02470 .01188 * The ratio of the convenience yield to the spot price is calculated as ^ = 94 T A B L E 7 Summary Statistics Quarterly Futures Returns* Name Obs. Mean St. Dev. Variance Mmiirrum Maximum Wheat (Chicago) July-Sept. 12 -.03039 .01793 .00032 -.06297 -.00673 Sept.-Dec 12 -.04209 .02382 .00057 -.07487 .017660 Dec-March 12 -.03336 .02607 .00068 -.06209 .02268 March-July 12 .00721 .05837 -.00340 -.06806 .12444 Wheat (Kansas) July-Sept. 12 -.01845 .01600 .00026 -.04822 .00400 Sept.-Dec 12 -.03015 .02228 .00050 -.60832 .01737 Dec-March 12 -.01458 .02255 .00051 -.05405 .01417 March-July 12 .00491 .04148 .00172 -.05657 .05642 Wheat (Minneapolis) July-Sept. 12 .00643 .02743 .00075 -.03294 .05189 Sept.-Dec 12 -.01299 .01581 .00025 -.04283 .01725 Dec-March 12 -.01923 .02138 .00046 -.06376 .00509 March-July 12 -.00493 .02498 .00062 -.04378 .03250 Corn (Chicago) Sept.-Dec 12 -.00466 .02996 .00090 -.05060 .04385 Dec-March 12 -.03590 .02098 .00044 -.06848 .00102 March-July 12 -.04012 .02502 .00063 -.08694 -.00261 July-Sept. 12 .02139 .05316 .00283 -.05022 .10915 Soybeans (Chicago) March-July 12 -.03581 .01599 .00026 -.06351 -.00856 July-Sept. 12 .00799 .03902 .00152 -.03539 .07958 Soyoil (Chicago) Sept.-Dec 12 .01510 .04778 .00228 -.06335 .08715 Dec-March 12 .00061 .04459 .00198 -.54260 .10594 March-July 12 .00584 .07424 .00551 -.05992 .20595 July-Sept. 12 .03097 .06852 .00469 -.05208 .14565 Copper ( C M X ) July-Sept. 12 -.01857 .011571 .00013 -.03781 .00273 Sept.-Dec 12 -.02819 .00888 .00008 -.03016 -.01233 Dec-March 12 -.01988 .02234 .00050 -.03813 .04537 March-July 12 -.02400 .03801 .00145 -.06183 .08959 Silver ( C M X ) July-Sept. 12 -.01964 .00804 .00006 -.03338 -.00702 Sept.-Dec 12 -.03008 .02025 .00041 -.08910 -.01336 Dec-March 12 -.02215 .01073 .00012 -.04683 -.00010 March-July 12 -.03273 .01664 .00028 -.06710 -.01693 Gold (TMM) July-Sept. 8 -.02397 .00952 .00009 -.03983 -.00901 Sept.-Dec 8 -.02364 .01292 .00017 -.04465 .00000 Dec-March 8 -.02670 .01128 .00013 -.04964 -.01260 March-July 8 -.02859 .01553 .00024 -.05075 -.00189 * The futures return is calculated as FRt = Pt-Ft Pt • 95 TABLE 8 Tests For Heteroscedasticity In Convenience Yields* Bartlett Test F-Test Commoditiy Obs. Before Transformation After Transformation Before After Transformation Transformation Wheat (Chicago) 12 40.643* 29.963* 16.401* 8.200* Wheat (Kansas) 12 24.622* 15.970* 22.965* 11.483* Wheat (Minneapolis) 12 14.504* 10.467* 3.219 1.609 Corn (Chicago) 12 2.734 7.053 N / A N / A Soybeans (Chicago) 12 4.217 9.144* 3.740 7.479* Soyoil (Chicago) 12 4.811 1.390 4.545* 2.273 Copper (CMX) 12 59.862* 50.250* 44.298* 33.224* Silver (CMX) 12 8.682* 7.708 10.215* 7.662* Gold (IMM) 8 10.018* N / A 11.045* N / A * The Bartlett test statistic is distributed approximately Chi-squared. The critical value at the 5 percent level with 3 degrees of freedom is 7.81473. The F-statistic is distributed F with (T-2,T-2) degrees of freedom. The critical value at the 5 percent level for the F-statistic with (10,10) degrees of freedom is 2.95. The critical value of the F-statistic at the 5 percent level with (6,6) degress of freedom is 4.28. The F-statistic is not calculated for corn since the convenience yield must be regressed on inventory to do so and no suitable data is available. 96 TABLE 9 Tests For Heteroscedasticity In Futures Returns* Bartlett Test F-Test Commoditiy Obs. Before Transformation After Transformation Before Transformation After Transformation Wheat (Chicago) 12 27.609* 18.394* 29.167* 14.584* Wheat (Kansas) 12 18.496* 10.820* 13.690* 6.847* Wheat (Minneapolis) 12 3.719 3.351 1.548 0.774 Corn (Chicago) 12 5.6390 11.436* 0.331 0.166 Soybeans (Chicago) 12 2.987 7.452 4.995* 9.988* Soyoil (Chicago) 12 6.059 4.020 2.052 1.026 Copper (CMX) 12 34.590* 25.519* 26.620* 13.311* Silver ( C M X ) 12 14.736* 10.163* 14.489* 7.245* Gold (IMM) 8 1.6947 N / A 8.673* N / A * The Bartlett test statistic is distributed approximately Chi-squared. The critical value at the 5 percent level with 3 degrees of freedom is 7.81473. The critical value at the 5 percent level for the F-statistic with (10,10) degrees of freedom is 2.95. The critical value of the F-statistic at the 5 percent level with (6,6) degress of freedom is 4.28. The statistics are not calculated for the transformed gold data since the quarters for gold are each three months, therefore, the length of the quarter should not induce the type of heteroscedasticity that the transformation is attempting to correct. 97 T A B L E 10 Goldfeld-Quandt Test Statistics of Within Season Heteroscedasticity Commodity Statistic (dl) Pre-Harvest Post-Harvest Wheat (Chicago) 8.66 1.5 (3,3) (13,13) Wheat (Kansas) 1.423 1.088 (3,3) (13,13) Wheat (Minneapolis) 6.29 1.42 (3,3) (13,13) Soybeans (Chicago) 285.19* 1.42 (3,3) (3,3) The Goldfeld-Quandt statistic has an F distribution. The critical value at the 95% confidence level is 9.28 with (3,3) degrees of freedom and 2.58 with (13,13) degrees of freedom.. 98 T A B L E 11 Tests of Serial Correlation and Normality of the Data* Commodity Normal Statistic Chi-Square Coefficient Coefficient Variables of Runs Test Goodness of fit of Skewness of Kurtosis Wheat (Chicago) C Y inv -.5724 6.0246* 1.8200 3.2917 C Y =± tnv -.5388 4.3432 -.2624 1.0359 C Y log(inv) -.5836 1.6519 .4283 1.3427 Wheat (Kansas) C Y inv -2.1246* 9.1632* .9600 1.1956 C Y Tmii) -1.6017 4.8726 .6543 .6104 C Y log{inv) -2.2610* 4.4001 .7672 .6461 Wheat (Minneapolis) C Y inv -2.8898* .8411 .4417 1.0011 C Y T ^ v [tnv) -3.4956* 4.5249 -.0692 -.3706 C Y log(inv) -3.4776* 1.3294 .1475 -.0239 Corn (Chicago) C Y inv -.7898 2.5961 .5778 .4739 -2.4052* 2.2920 .8617 .3615 C Y log(inv) -2.9436* 2.2920 .6261 .1709 Soybeans (Chicago) C Y inv -1.7317 14.4437* 1.9335 3.7058 C Y r^K (inv) -1.0049 12.2462* 1.9241 3.6583 C Y log[inv) -1.0049 12.2462* 1.9313 3.6823 Soyoil (Chicago) C Y inv -1.8193 12.5228* 1.7772 4.4775 -2.2237* 16.5786* 1.5725 3.6445 C Y log(inv) -1.2086 13.4044* 1.6633 4.1200 Copper (CMX) C Y inv -2.9612* 17.1602* 3.4172 15.4561 C Y T=± (inv) -2.7353* 18.3679* 1.7205 9.6799 C Y log(inv) -2.5576* 12.2750* 2.2324 9.7751 Silver ( CM X) C Y inv -3.0438* 6.6063* 1.0658 9.1595 C Y r ^ y (tnv) -3.0438* 6.6063* 1.0958 9.1141 C Y log{inv) -2.3561* 6.9319* 1.0631 9.1030 Gold ( IMM) C Y tnv -1.9211 12.3954* -1.3268 4.1413 -2.5733* 12.8623* -1.3992 3.1327 C Y log(inv) -2.7738* 10.9305* -1.3307 3.3830 * An asterisk indicates significance at the 95 level. 99 T A B L E 12 Indirect Tests of Autoregression in Demand Shocks From Spot Prices Correlation Between Mid-Month and Month End Spot Prices Commodity Wheat (Kansas) Corn Soybeans Soyoil Copper Silver Gold Correlation Coefficient .92306 .86390 .73615 .80592 .91423 .97440 .98444 100 T A B L E 13 Regressions of the Convenience Yield Projected on Inventory*  Coefficients (t-ratio OLS) [t-ratio GLS] Residual Commodity Obs. Constant Inventory R2 D.W. Variance Wheat (Chicago) 48 77.814 (9.8396) [5.9736] -0.13065 (-5.8590) [-3.9135] .4273 1.6317 268.85 Wheat (Kansas) 48 63.252 (9.2389) [6.8004] -0.07730 (-4.0042) [-3.2502] .2585 1.3434 201.49 Wheat (Minneapolis) 48 65.260 (12.0630) [9.1616] -0.07067 (-4.6333) [-3.7282] .3182 .8385 125.82 Corn (Chicago) 15 46.317 (10.493) [7.9889] -0.17226 (-4.0691) [-3.6237] .5602 1.7498 48.179 Soybeans (Chicago) 24 40.399 (2.8807) [5.5922] -.19105 (-.58744) [-1.0783] .0154 1.7079 420.43 Soyoil (Chicago) 48 3.8791 (6.9945) [5.8073] -0.00209 (-4.7488) [-4.5325] .3290 .8824 2.1856 Copper (CMX) 48 4.3368 (4.8565) [2.3579] -0.00830 (-3.5019) [-1.8789] .2105 1.4779 5.1558 Silver ( C M X ) 36 .049239 (.61644) [.48134] -.000002 (-.78303) [-.71035] .0177 1.3948 .02189 Gold ( IMM) 32 2.1265 (1.6254) [2.9504] -0.00096 (-2.5350) [-3.3382] .1764 1.7977 7.5227 * The GLS procedure implements the White (1980) Heteroskedastic Consistent Covariance Matrix. 101 TABLE 14 Regressions of the Convenience Yield Projected on the Negative of the Reciprocal of Inventory* Coefficients (t-ratio OLS) [t-ratio GLS] Residual Commodity Obs. Constant - l Inventory R2 D.W. Variance Wheat (Chicago) 48 -4.6261 (.93234) [-.64042] -11353 (-8.3914) [-4.5946] .6049 1.3936 185.51 Wheat (Kansas) 48 16.474 (3.2886) [2.5968] -6123.9 (-4.4832) [-2.9878] .3041 1.2811 189.10 Wheat (Minneapolis) 48 21.279 (5.6648) [4.8367] -5958.6 (-5.8175) [-4.4728] .4239 .8725 106.32 Corn (Chicago) 15 22.671 (5.6309) [7.3633] -494.72 (-2.1029) [-2.1343] .2700 1.4148 79.963 Soybeans (Chicago) 24 28.540 (2.5426) (3.3855) -145.66 (-.38392) (-.78239) .0067 1.6931 424.19 Soyoil (Chicago) 48 -1.3826 (-2.4808) (-2.6588) -2797.0 (-5.4540) (-4.5401) .3927 .8988 1.9780 Copper (CMX) 48 -.05128 (-.13683) [-.10105] -950.31 (-6.0287) [-2.2125] .0075 1.3853 .02212 Silver (CMX) 36 -.05128 (-.60559) (-.53417) .950.31 (-.50673) (-.36524) .0075 1.3853 .02212 Gold (EV1M) 32 -2.3126 (-2.5493) [-2.5483] -3304.9 (-1.8069) [-3.0298] .0981 1.7693 8.2377 * The GLS procedure implements the White (1980) Heteroskedastic Consistent Covariance matrix. 102 TABLE 15 Regressions of the Convenience Yield Projected on the Natural Log of Inventory* Coefficients (t-ratio OLS) [t-ratio GLS] Residual Commodity Obs. Constant log(inventory) R2 D.W. Variance Wheat (Chicago) 48 280.43 (8.1674) [4.6554] -42.810 (-7.2028) [-4.1972] .5300 1.5313 220.64 Wheat (Kansas) 48 175.24 (5.4462) [3.7979] -23.960 (-4.3018) [-3.0607] .2869 1.2947 193.77 Wheat (Minneapolis) 48 172.17 (6.9567) [5.0812] -22.691 (-5.2966) [-3.9169] .3788 .8249 114.63 Corn (Chicago) 15 78.628 (5.0446) [4.0534] -11.032 (-3.1530) [-2.6753] .4333 1.5386 62.074 Soybeans (Chicago) 24 52.769 (1.1889) [2.4318] -5.5255 (44.385) [-.86107] .0094 1.6931 420.43 Soyoil (Chicago) 48 20.026 (5.6697) [5.0666] -2.6632 (-5.2687) [-4.8651] .3763 .8871 2.0312 Copper ( C M X ) 48 15.495 (5.3808) [2.0704] -2.4544 (-4.9115) [-1.9451] .3440 1.6802 4.2838 Silver (CMX) 36 .50844 (.64263) [.50844] -.05136 (-.65590) [-.52287] .0125 1.3875 .02201 Gold ( IMM) 32 14.933 (1.9953) [3.0773] -1.9953 (-2.1275) [-3.0470] .13311 1.7884 7.9367 * The GLS procedure implements the White (1980) Heteroskedastic Consistent Covariance matrix. 103 T A B L E 16 OLS Regressions of the Convenience Yield Projected on Inventory Omitting Outliers* Coefficients (t-ratio OLS) Commodity Obs. Constant Inventory R2 D.W. Variance Copper (CMX) 46 1.5945 -.00182 .1345 1.1633 .3792 (5.9594) (-2.6151) Silver ( C M X ) 30 .048616 -.000002 .1605 1.9909 .0021 (1.7889) (-2.3138) Gold ( IMM) 25 .82486 -.00026 .1107 2.1183 .8405 (1.7000) (-1.6919) * For Copper the March, 1974 and December, 1977 observations are omitted. For both Gold and Silver, the March, 1983 observation is omitted, as well as the period from December, 1979 to March, 1981. 104 TABLE 17 Pre-Harvest Quarter Regressions of the Convenience Yield Projected on Inventory Coefficients (t-ratio OLS) Commodity Obs. Constant Inventory 722 Residual D.W. Variance Wheat (Chicago) 12 138.80 (7.2680) -.36943 (-4.5494) .6742 2.8334 352.28 Wheat (Kansas) 12 75.285 (3.4757) -.09814 (-1.0686) .1025 1.0907 453.20 Wheat (Minneapolis) 12 87.176 (5.8828) -.16156 (-2.5639) .3966 1.2400 212.12 Soybeans (Chicago) 12 37.720 (1.6354) .04494 (.07597) .0006 1.5128 728.51 TABLE 18 Post-Harvest Quarter Regressions of the Convenience Yield Projected on Inventory Coefficients (t-ratio OLS) Residual Commodity Obs. Constant Inventory R2 D.W. Variance Wheat (Chicago) 36 41.402 (6.4457) -.040139 (-2.4144) .1464 1.4443 78.311 Wheat (Kansas) 36 41.903 (5.9985) -.02697 (-1.4915) .0614 1.7511 92.630 Wheat (Minneapolis) 36 54.936 (8.0595) -.04448 (-2.5214) .1575 .8426 88.194 Soybeans (Chicago) 12 29.643 (2.4111) -.08639 (-.32949) .0107 2.2921 97.402 105 T A B L E 19 Pre-Harvest Quarter Regressions of the Convenience Yield Projected on Inventory Using a GLS Model * Coefficients [t-ratio GLS1 Commodity Obs. Constant Inventory R Residual Variance Wheat (Chicago) 12 Wheat (Kansas) 12 Wheat (Minneapolis) 12 127.63 [10.445] 80.399 [3.6770] 86.281 [5.6746] -.32478 [-6.2738] -.11336 [-1.2386] -.15528 [-2.4185] .7920 -.55326 .2303 .35771 .4444 .25536 224.89 388.67 195.33 T A B L E 20 Pre-Harvest Quarter Regressions of the Convenience Yield Projected on The Negative of the Reciprocal of Inventories Using a G L S Model * Coefficients [t-ratio GLS] Commodity Obs. Constant -1 Inventory R2 Residual Variance Wheat (Chicago) 12 Wheat (Kansas) 12 Wheat (Minneapolis) 12 -2.0898 [-.17671] 25.615 [1.2430] 18.864 [1.4190] -11477 [-4.9223] -5992.7 [-1.6968] -6558.6 [-2.7359] .7065 -.48095 .3156 .47159 .5079 .31856 317.36 345.58 173.01 T A B L E 21 Pre-Harvest Quarter Regressions of the Convenience Yield Projected on the Natural Log of Inventory Using a GLS Model * Coefficients [t-ratio GLS] Commodity Obs. Constant log(Inventory) R2 P Variance Wheat (Chicago) 12 398.82 -64.145 .7590 -.52610 260.57 [6.5425] [-5.6774] Wheat (Kansas) 12 200.56 -27.112 .2677 .40696 369.78 [1.9832] [-1.4357] Wheat (Minneapolis) 12 230.71 -33.426 .4743 .28448 184.80 [3.2977] [-2.5626] * The estimation procedure estimates p with a Cochrane-Orcutt technique and adjusts the covariance matrix appropriately. 106 T A B L E 22 Regression of Convenience Yield Projected On Inventory Using a S U R Framework Coefficients Cross-Section Obs. Constant Inventory Wheat (Chicago) 48 78.005 -.13108 (9.3184) (-5.560) Wheat (Kansas) 48 65.797 -.08395 (8.5973) (-3.9536) Wheat (Minneapolis) 48 66.112 -.0708 (11.296) (-4.5889) Buse #2=.3669 p=. 18944 T A B L E 23 F-Statistic For Tests of Cross-Sectional Differences in the Slope Parameter for Inventory Hypothesis F-Statistic (di-) HI 2.3108 (2,138) H2 2.2072 (1,138) H3 4.5828* (1,138) H4 .25197 (1,138) HI : The slope parameters for all wheats are equal. H2: The slope parameters for Chicago and Kansas wheat are equal. H3: The slope parameters for Chicago and Minneapolis wheat are equal. H4: The slope parameter for Kansas and Minneapolis wheat are equal. Critical value of the F-statistic with (1,138) degrees of freedom is 3.84. Critical value of the F-statistic with (2,138) degrees of freedom is 3.00. 107 T A B L E 24 Regression of Convenience Yield Projected On the Negative of the Reciprocal of Inventory Using a S U R Framework Coefficients Cross-Section Obs. Constant Inventory Wheat (Chicago) 48 -7.0131 -12146 (-1.2481) (-8.3262) Wheat (Kansas) 48 13.744 -7008.4 (2.4068) (-4.8227) Wheat (Minneapolis) 48 22.063 -5821.9 (4.2256) (-4.3091) Buse #2=.4791 ^?=.31298 T A B L E 25 F-Statistic For Tests of Cross-Sectional Differences in the Slope Parameter for the Negative of the Reciprocal of Inventory Hypothesis F-Statistic d.f. HI 5.5442* (2,138) H2 6.2261* (1,138) H3 10.1172* (1,138) H4 .3575 (1,138) HI : The slope parameters for all wheats are equal. H2: The slope parameters for Chicago and Kansas wheat are equal. H3: The slope parameters for Chicago and Minneapolis wheat are equal. H4: The slope parameter for Kansas and Minneapolis wheat are equal. Critical value of the F-statistic with (1,138) degrees of freedom is 3.84. Critical value of the F-statistic with (2,138) degrees of freedom is 3.00. 108 T A B L E 26 Regression of Convenience Yield Projected On the Natural Log of Inventory Using a SUR Framework Coefficients Cross-Section Obs. Constant Inventory Wheat (Chicago) 48 288.29 -44.139 (7.7304) (-6.8378) Wheat (Kansas) 48 194.01 -27.161 (5.5331) (-4.4761) Wheat (Minneapolis) 48 154.28 -19.551 (5.1045) (-3.7287) Buse fl2=.4056 ^=.23896 T A B L E 27 F-Statistic For Tests of Cross-Sectional Differences in the Slope Parameter for the Natural Log of Inventory Hypothesis F-Statistic d.f. H I 4.4147* (2,138) H2 3.6726 (1,138) H3 8.7416* (1,138) H4 .90049 (1,138) HI : The slope parameters for all wheats are equal. H2: The slope parameters for Chicago and Kansas wheat are equal. H3: The slope parameters for Chicago and Minneapolis wheat are equal. H4: The slope parameter for Kansas and Minneapolis wheat are equal. The critical value of the F-statistic with (1,138) degrees of freedom is 3.84. The critical value of the F-statitic with (2,138) degress of freedom is 3.00. 109 TABLE 28 GLS Regressions of the Convenience Yield on Quarterly Seasonal Dummies* C O E F F I C I E N T S Quarterly Parameters (t-ratio) March- July- Sept.- Dec- R E S I D U A L Commodity July Sept. Dec. March R2 V A R I A N C E Wheat 55.488 22.625 26.336 29.962 .3698 309.32 (Chicago) (10.929) (4.4564) (5.1872) (5.9015) Wheat 53.091 27.060 30.697 37.531 .3818 175.61 (Kansas) (13.879) (7.0738) (8.0244) (9.8109) Wheat 50.743 39.586 38.777 36.264 .1750 159.16 (Minneapolis) (13.933) (10.870) (10.648) (9.957) Corn 26.682 34.047 31.871 22.539 .1308 146.21 (Chicago) (7.6440) (9.7540) (9.1306) (6.4570) Soybeans 25.703 39.369 N / A N / A .1193 376.09 (Chicago) (4.5912) (7.0324) Soyoil 1.5003 1.7630 1.4620 1.0675 .0198 3.3377 (Chicago) (2.8447) (3.3428) (2.7721) (2.0241) Copper 2.1744 1.0202 .86329 1.6537 .0437 6.5290 (CMX) (2.9479) (1.3831) (1.1704) (2.2419) Silver -.03289 -.00883 -.03024 .03238 .0426 .01672 (CMX) (-.88124) (-.23668) (-.81005) (.86737) Gold -1.8722 -.44715 -.58646 -.90950 .0361 9.4332 (IMM) (-1.7241) (-.41178) (-.54008) (-.83757) * With the exception of gold each regression has 48 observations. The regression for gold has 32. R2 is the coefficient of determination. In the case of gold July is replaced by June. The regressions are run using the White(1980) Heteroskedastic Consistent covariance matrix. 110 T A B L E 29 F-Statistics for Tests of Seasonality in the Quarterly Convenience Yield* F-Statistics (di.) Commodity H I H2 H3 H4 Wheat (Chicago) 8.6063 .52211 N / A N / A (3,44) (2,44) Wheat (Kansas) 9.0590 1.9311 N / A N / A (3,44) (2,44) Wheat (Minneapolis) 3.1122 .2263 N / A N / A (3,44) (2,44) Corn (Chicago) 2.2065 1.7945 .1943 .7045 (3,44) (2,44) (1,44) (1,44) Soybeans (Chicago) 2.9800 N / A N / A N / A (1,22) Soyoil (Chicago) .2959 N / A N / A N / A (3,44) Copper ( C M X ) .6698 N / A N / A N / A (3,44) Silver ( C M X ) .6533 N / A N / A N / A (3,44) Gold ( IMM) .3497 N / A N / A N / A (3,28) * Significant at the 95confidence level. HI Hypothesis One - The coefficients for each quarterly dummy are equal. H2 Hypothesis Two - The coefficients for the non-preharvest quarterly dummies are equal. H3 Hypothesis Three- The dummy parameters for the two preharvest quarters are equal. H4 Hypothesis Four - The dummy parameters for the two non-preharvest quarters are equal. Critical Values of the F-Distribution at the 95% Confidence Level Degrees of Freedom Critical Value (1,22) 4.30 (1,44) 4.08 (2,44) 3.23 (3,28) 2.95 (3,44) 2.83 I l l TABLE 30 GLS Regressions of the Futures Return on the Quarterly Nominal Yield and Quarterly Seasonal Dummies* C O E F F I C I E N T S ~ Quarterly Parameters Nominal Yield (t-ratio) (t-ratio) Commodity March-July July-Sept. Sept.-Dec. Dec-March Rt R2 R E S I D U A L Variance Wheat (Chicago) .04170 (1.5585) -.00919 (-1.1710) -.01756 (-1.3448) -.00968 (-.7370) -1.0980 (-2.1915) .2841 .00120 Wheat (Kansas) .02324 (1.0190) -.00684 (-.7365) -.01724 (-1.2156) -.00212 (-.1487) -.57795 (-.9431) .2116 .00075 Wheat (Minneapolis) .01809 (1.1407) .01639 (1.7524) .00447 (.4438) -.00236 (-.2586) -.78240 (-1.9760) .2228 .00038 Corn (Chicago) -.00682 (-.4282) .03597 (2.5373) .02437 (1.6563) -.00786 (-.6471) -1.2999 (-2.4822) .4184 .00112 Soybeans (Chicago) -.00752 (-.89901) .018955 (1.4873) N / A N / A -1.0141 (-4.3467) .3222 .00100 Soyoil (Chicago) .08651 (2.2483) .06263 (3.2033) .07372 (3.5929) .05722 (2.7519) -2.6243 (-3.4427) .1518 .00326 Copper (CMX) -.00035 (-.0191) -.00235 (-.4421) -.00807 (-1.0250) -.00045 (-.0640) -.90054 (-2.6137) .1185 .00050 Silver (CMX) -.00957 (-1.4911) .-.00282 (-1.0422) -.00935 (-1.5439) -.00213 (-.64882) -.92833 (-5.555) .3431 .000167 Gold (IMM) -.00236 (-.6827) 0.0000 (.6462) .00155 (.4558) -.00096 (-.2781) -1.1043 (-11.607) .6709 .00006 * The regression run corresponds to equation(4.4) and is = Dipi + Rj3i + e,-. With the exception of gold each regression has 48 observations. The regression for gold has 32. R2 is the coefficient of determination.In the caseof gold July is replaced by June.The regressions are run using the White(1980) Heteroskedastic Consistent covariance matrix. 112 T A B L E 31 F-Statistics for Tests of Seasonality in the Quarterly Dummy Variables* F-Statistics (di-) Commodity H I H2 H3 H4 Wheat (Chicago) 3.072* .5752 N / A N / A (3,43) (2,43) Wheat (Kansas) 2.8161 1.6306 N / A N / A (3,43) (2,43) Wheat (Minneapolis) 2.7886 2.8804 N / A N / A (3,43) (2,43) Corn (Chicago) 6.8304* 5.9440* .6078 .0130 (3,43) (2,43) (1,43) (1,43) Soybeans (Chicago) 4.3423* N / A N / A N / A (1,21) Soyoil (Chicago) .5152 N / A N / A N / A (3,43) Copper ( C M X ) 1.2800 N / A N / A N / A (3,43) Silver ( C M X ) 1.0366 N / A N / A N / A (3,43) Gold ( IMM) .3941 N / A N / A . N / A (3,27) * Significant at the 95 percent confidence level. HI Hypothesis One - The coefficients for each quarterly dummy are equal. H2 Hypothesis Two - The coefficients for the non-preharvest quarterly dummies are equal. H3 Hypothesis Three- The dummy parameters for the two preharvest quarters are equal. H4 Hypothesis Four - The dummy parameters for the two non-preharvest quarters are equal. Critical Values of the F-Distribution at the 95% Confidence Level Degrees of Freedom Critical Value (1,21) 4.32 (1,43) 4.07 (2,43) 3.22 (3,27) 2.96 (3,43) 2.83 103 T A B L E 16 OLS Regressions of the Convenience Yield Projected on Inventory Omitting Outliers* Coefficients (t-ratio OLS) Commodity Obs. Constant Inventory R2 D.W. Variance Copper ( C M X ) 46 1.5945 -.00182 .1345 1.1633 .3792 (5.9594) (-2.6151) Silver ( C M X ) 30 .048616 -.000002 .1605 1.9909 .0021 (1.7889) (-2.3138) Gold ( IMM) 25 .82486 -.00026 .1107 2.1183 .8405 (1.7000) (-1.6919) * For Copper the March, 1974 and December, 1977 observations are omitted. For both Gold and Silver, the March, 1983 observation is omitted, as well as the period from December, 1979 to March, 1981. 104 TABLE 17 Pre-Harvest Quarter Regressions of the Convenience Yield Projected on Inventory Coefficients (t-ratio OLS) Commodity Obs. Constant Inventory Residual R2 D.W. Variance Wheat (Chicago) 12 138.80 (7.2680) -.36943 (-4.5494) .6742 2.8334 352.28 Wheat (Kansas) 12 75.285 (3.4757) -.09814 (-1.0686) .1025 1.0907 453.20 Wheat (Minneapolis) 12 87.176 (5.8828) -.16156 (-2.5639) .3966 1.2400 212.12 Soybeans (Chicago) 12 37.720 (1.6354) .04494 (.07597) .0006 1.5128 728.51 TABLE 18 Post-Harvest Quarter Regressions of the Convenience Yield Projected on Inventory  Coefficients (t-ratio OLS) Residual Commodity Obs. Constant Inventory R D.W. Variance Wheat (Chicago) 36 41.402 (6.4457) -.040139 (-2.4144) .1464 1.4443 78.311 Wheat (Kansas) 36 41.903 (5.9985) -.02697 (-1.4915) .0614 1.7511 92.630 Wheat (Minneapolis) 36 54.936 (8.0595) -.04448 (-2.5214) .1575 .8426 88.194 Soybeans (Chicago) 12 29.643 (2.4111) -.08639 (-.32949) .0107 2.2921 97.402 105 TABLE 19 Pre-Harvest Quarter Regressions of the Convenience Yield Projected on Inventory Using a GLS Model * Coefficients [t-ratio GLS] Commodity Obs. Constant Inventory Ri Residual Variance Wheat (Chicago) 12 127.63 -.32478 .7920 -.55326 224.89 [10.445] [-6.2738] Wheat (Kansas) 12 80.399 -.11336 .2303 .35771 388.67 [3.6770] [-1.2386] Wheat (Minneapolis) 12 86.281 -.15528 .4444 .25536 195.33 [5.6746] [-2.4185] TABLE 20 Pre-Harvest Quarter Regressions of the Convenience Yield Projected on The Negative of the Reciprocal of Inventories Using a GLS Model * Coefficients [t-ratio GLS] Commodity Obs. Constant - l Inventory R2 Residual Variance Wheat (Chicago) 12 Wheat (Kansas) 12 Wheat (Minneapolis) 12 -2.0898 [-.17671] 25.615 [1.2430] 18.864 [1.41901 -11477 [-4.9223] -5992.7 [-1.6968] -6558.6 .7065 -.48095 .3156 .47159 .5079 .31856 317.36 345.58 173.01 TABLE 21 Pre-Harvest Quarter Regressions of the Convenience Yield Projected on the Natural Log of Inventory Using a GLS Model * Coefficients [t-ratio GLS] Residual Commodity Obs. Constant log(Inventory) R2 p Variance Wheat (Chicago) 12 398.82 -64.145 .7590 -.52610 260.57 [6.5425] [-5.6774] Wheat (Kansas) 12 200.56 -27.112 .2677 .40696 369.78 [1.9832] [-1.4357] Wheat (Minneapolis) 12 230.71 -33.426 .4743 .28448 184.80 [3.2977] [-2.5626] * The estimation procedure estimates p with a Cochrane-Orcutt technique and adjusts the covariance matrix appropriately. 106 T A B L E 22 Regression of Convenience Yield Projected On Inventory Using a SUR Framework Coefficients Cross-Section Obs. Constant Inventory Wheat (Chicago) 48 78.005 -.13108 (9.3184) (-5.560) Wheat (Kansas) 48 65.797 -.08395 (8.5973) (-3.9536) Wheat (Minneapolis) 48 66.112 -.0708 (11.296) (-4.5889) Buse #2=.3669 p=. 18944 T A B L E 23 F-Statistic For Tests of Cross-Sectional Differences in the Slope Parameter for Inventory Hypothesis F-Statistic (d.f.) HI 2.3108 (2,138) H2 2.2072 (1,138) H3 4.5828* (1,138) H4 .25197 (1,138) HI : The slope parameters for all wheats are equal. H2: The slope parameters for Chicago and Kansas wheat are equal. H3: The slope parameters for Chicago and Minneapolis wheat are equal. H4: The slope parameter for Kansas and Minneapolis wheat are equal. Critical value of the F-statistic with (1,138) degrees of freedom is 3.84. Critical value of the F-statistic with (2,138) degrees of freedom is 3.00. 107 T A B L E 24 Regression of Convenience Yield Projected On the Negative of the Reciprocal of Inventory Using a SUR Framework Coefficients Cross-Section Obs. Constant Inventory Wheat (Chicago) 48 -7.0131 -12146 (-1.2481) (-8.3262) Wheat (Kansas) 48 13.744 -7008.4 (2.4068) (-4.8227) Wheat (Minneapolis) 48 22.063 -5821.9 (4.2256) (-4.3091) Buse #2=.4791 ^=.31298 T A B L E 25 F-Statistic For Tests of Cross-Sectional Differences in the Slope Parameter for the Negative of the Reciprocal of Inventory Hypothesis F-Statistic d.f. HI 5.5442* (2,138) H2 6.2261* (1,138) H3 10.1172* (1,138) H4 .3575 (1,138) HI : The slope parameters for all wheats are equal. H2: The slope parameters for Chicago and Kansas wheat are equal. H3: The slope parameters for Chicago and Minneapolis wheat are equal. H4: The slope parameter for Kansas and Minneapolis wheat are equal. Critical value of the F-statistic with (1,138) degrees of freedom is 3.84. Critical value of the F-statistic with (2,138) degrees of freedom is 3.00. 108 T A B L E 26 Regression of Convenience Yield Projected On the Natural Log of Inventory Using a SUR Framework Coefficients Cross-Section Obs. Constant Inventory Wheat (Chicago) 48 288.29 -44.139 (7.7304) (-6.8378) Wheat (Kansas) 48 194.01 -27.161 (5.5331) (-4.4761) Wheat (Minneapolis) 48 154.28 -19.551 (5.1045) (-3.7287) Buse fl2 = .4056 ^=.23896 T A B L E 27 F-Statistic For Tests of Cross-Sectional Differences in the Slope Parameter for the Natural Log of Inventory Hypothesis F-Statistic d.f. H I 4.4147* (2,138) H2 3.6726 (1,138) H3 8.7416* (1,138) H4 .90049 (1,138) HI : The slope parameters for all wheats are equal. H2: The slope parameters for Chicago and Kansas wheat are equal. H3: The slope parameters for Chicago and Minneapolis wheat are equal. H4: The slope parameter for Kansas and Minneapolis wheat are equal. The critical value of the F-statistic with (1,138) degrees of freedom is 3.84. The critical value of the F-statitic with (2,138) degress of freedom is 3.00. 109 TABLE 28 GLS Regressions of the Convenience Yield on Quarterly Seasonal Dummies* C O E F F I C I E N T S Quarterly Parameters (t-ratio) March- July- Sept.- Dec- R E S I D U A L Commodity July Sept. Dec. March R2 V A R I A N C E Wheat 55.488 22.625 26.336 29.962 .3698 309.32 (Chicago) (10.929) (4.4564) (5.1872) (5.9015) Wheat 53.091 27.060 30.697 37.531 .3818 175.61 (Kansas) (13.879) (7.0738) (8.0244) (9.8109) Wheat 50.743 39.586 38.777 36.264 .1750 159.16 (Minneapolis) (13.933) (10.870) (10.648) (9.957) Corn 26.682 34.047 31.871 22.539 .1308 146.21 (Chicago) (7.6440) (9.7540) (9.1306) (6.4570) Soybeans 25.703 39.369 N / A N / A .1193 376.09 (Chicago) (4.5912) (7.0324) Soyoil 1.5003 1.7630 1.4620 1.0675 .0198 3.3377 (Chicago) (2.8447) (3.3428) (2.7721) (2.0241) Copper 2.1744 1.0202 .86329 1.6537 .0437 6.5290 (CMX) (2.9479) (1.3831) (1.1704) (2.2419) Silver -.03289 -.00883 -.03024 .03238 .0426 .01672 (CMX) (-.88124) (-.23668) (-.81005) (.86737) Gold -1.8722 -.44715 -.58646 -.90950 .0361 9.4332 (IMM) (-1.7241) (-.41178) (-.54008) (-.83757) * With the exception of gold each regression has 48 observations. The regression for gold has 32. R2 is the coefficient of determination. In the case of gold July is replaced by June. The regressions are run using the White(1980) Heteroskedastic Consistent covariance matrix. 110 T A B L E 29 F-Statistics for Tests of Seasonality in the Quarterly Convenience Yield* F-Statistics (d.f.) Commodity H I H2 H3 H4 Wheat (Chicago) 8.6063 (3,44) .52211 (2,44) N / A N / A Wheat (Kansas) 9.0590 (3,44) 1.9311 (2,44) N / A N / A Wheat (Minneapolis) 3.1122 (3,44) .2263 (2,44) N / A N / A Corn (Chicago) 2.2065 1.7945 .1943 .7045 (3,44) (2,44) (1,44) (1,44) Soybeans (Chicago) 2.9800 (1,22) N / A N / A N / A Soyoil (Chicago) .2959 (3,44) N / A N / A N / A Copper ( C M X ) .6698 (3,44) N / A N / A N / A Silver ( C M X ) .6533 (3,44) N / A N / A N / A Gold ( IMM) .3497 (3,28) N / A N / A N / A * Significant at the 95confidence level. HI Hypothesis One - The coefficients for each quarterly dummy are equal. H2 Hypothesis Two - The coefficients for the non-preharvest quarterly dummies are equal. H3 Hypothesis Three- The dummy parameters for the two preharvest quarters are equal. H4 Hypothesis Four - The dummy parameters for the two non-preharvest quarters are equal. Critical Values of the F-Distribution at the 95% Confidence Level Degrees of Freedom Critical Value (1,22) 4.30 (1,44) 4.08 (2,44) 3.23 (3,28) 2.95 (3,44) 2.83 I l l TABLE 30 GLS Regressions of the Futures Return on the Quarterly Nominal Yield and Quarterly Seasonal Dummies* C O E F F I C I E N T S Quarterly Parameters Nominal Yield (t-ratio) (t-ratio) Commodity March-July July-Sept. Sept.-Dec. Dec-March Rt R2 R E S I D U A L Variance Wheat .04170 -.00919 -.01756 -.00968 -1.0980 .2841 .00120 (Chicago) (1.5585) (-1.1710) (-1.3448) (-.7370) (-2.1915) Wheat .02324 -.00684 -.01724 -.00212 -.57795 .2116 .00075 (Kansas) (1.0190) (-.7365) (-1.2156) (-.1487) (-.9431) Wheat .01809 .01639 .00447 -.00236 -.78240 .2228 .00038 (Minneapolis) (1.1407) (1.7524) (.4438) (-.2586) (-1.9760) Corn -.00682 .03597 .02437 -.00786 -1.2999 .4184 .00112 (Chicago) (-.4282) (2.5373) (1.6563) (-.6471) (-2.4822) Soybeans -.00752 .018955 N / A N / A -1.0141 .3222 .00100 (Chicago) (-.89901) (1.4873) (-4.3467) Soyoil .08651 .06263 .07372 .05722 -2.6243 .1518 .00326 (Chicago) (2.2483) (3.2033) (3.5929) (2.7519) (-3.4427) Copper -.00035 -.00235 -.00807 -.00045 -.90054 .1185 .00050 (CMX) (-.0191) (-.4421) (-1.0250) (-.0640) (-2.6137) Silver -.00957 -.00282 -.00935 -.00213 -.92833 .3431 .000167 (CMX) (-1.4911) (-1.0422) (-1.5439) (-.64882) (-5.555) Gold -.00236 0.0000 .00155 -.00096 -1.1043 .6709 .00006 (IMM) (-.6827) (.6462) (.4558) (-.2781) (-11.607) * The regression run corresponds to equation(4.4) and is Y,- = Dvb{ + Rj3{ + e,-. With the exception of gold each regression has 48 observations. The regression for gold has 32. R2 is the coefficient of determination.In the caseof gold July is replaced by June.The regressions are run using the White(l980) Heteroskedastic Consistent covariance matrix. 112 T A B L E 31 F-Statistics for Tests of Seasonality in the Quarterly Dummy Variables* F-Statistics (d.f.) Commodity H I H2 H3 H4 Wheat (Chicago) 3.072* .5752 N / A N / A (3,43) (2,43) Wheat (Kansas) 2.8161 1.6306 N / A N / A (3,43) (2,43) Wheat (Minneapolis) 2.7886 2.8804 N / A N / A (3,43) (2,43) Corn (Chicago) 6.8304* 5.9440* .6078 .0130 (3,43) (2,43) (1,43) (1,43) Soybeans (Chicago) 4.3423* N / A N / A N / A (1,21) Soyoil (Chicago) .5152 N / A N / A N / A (3,43) Copper ( C M X ) 1.2800 N / A N / A N / A (3,43) SUver ( CM X) 1.0366 N / A N / A N / A (3,43) Gold ( IMM) .3941 N / A N / A N / A (3,27) * Significant at the 95 percent confidence level. HI Hypothesis One - The coefficients for each quarterly dummy are equal. H2 Hypothesis Two - The coefficients for the non-preharvest quarterly dummies are equal. H3 Hypothesis Three- The dummy parameters for the two preharvest quarters are equal. H4 Hypothesis Four - The dummy parameters for the two non-preharvest quarters are equal. Critical Values of the F-Distribution at the 95% Confidence Level Degrees of Freedom Critical Value (1,21) 4.32 (1,43) 4.07 (2,43) 3.22 (3,27) 2.96 (3,43) 2.83 113 A P P E N D I X I I F I G U R E S F I G U R E 1 E q u i l i b r i u m P r i c e s . W h e n n - 0 . Price The shaded region of the graph Is the t o t a l value of s e l l i n g the cc*nmodity at t, when the price at that time i s greater than the t a p r i c e . 114 FIGURE 2 Equil ibrium Prices When / ? = - ] . Price t± shocks to demand qi-<U<ni>. i!=ii<ni) The shaded a r e o f the graph r e p r e s e n t s the t o t a l v a l u e o f the c o n v e n i e n c e y i e l d . 115 FIGURE 3 E q u i l i b r i u m P r i c e s VJhen f Price 6 . shocks t o demand i h l ( n i ) , l t - l l ( " l ) ---With the correlation equal to l ^ p r o f l t maximizing behavior by a l l firma insures that P t = E ( P ; l ) and there i s no value associated with holding the commodity. 116 FIGURE 4 Wheat I n v e n t o r i e s ( m i l l i o n s of bushels) 48 OBSERVATIONS I n v e n t o r y 526.SO 516.08 505.69 495.23 484.81 474.38 463.96 453.54 443. 12 432.69 422.27 411.8S 401.42 391.00 380.58 370.15 359.73 349.31 338.88 328.46 318.04 307.62 297.19 286.77 276.35 265.92 255. 50 245.08 234.65 224.23 213.81 203.38 192.96 182.54 172. 12 161.69 151.27 140.85 130.42 120.00 1 9 7 4 . O O O 1 9 7 5 . 5 0 0 1 9 7 7 . 0 0 0 1 9 7 8 . 5 0 0 1 9 8 0 . 0 0 0 1 9 8 1 . 5 0 O 1983 .OOO 1 9 8 4 . 5 0 0 1 9 8 6 . O O O Time M - March J - J u l y S - September D - December 117 F I G U R E 5 C o r n I n v e n t o r i e s ( m i l l i o n s o f . b u s h e l s ) 46 OBSERVATION?; Inventory 8 6 1 5 . 0 8 3 9 4 . 1 8 1 7 3 . 2 7 9 5 2 . 3 7 7 3 1 . 4 7 5 1 0 . 5 7 2 8 9 . 6 7 0 6 8 . 7 6 8 4 7 . 8 6 6 2 6 . 9 6 4 0 6 . 0 6 1 8 5 . 1 5964 . 2 5 7 4 3 . 3 5 5 2 2 . 4 5301 . 5 5 0 8 0 . 6 4 8 5 9 . 7 4 6 3 8 . 8 44 1 7 . 9 4 1 9 7 . 1 397G .2 3 7 5 5 . 3 3 5 3 4 . 4 3 3 1 3 5 3 0 9 2 . 6 287 1 .7 2 6 5 0 . 8 2 4 2 9 . 9 2 2 0 9 . 0 1988 . 1 1 7 6 7 . 2 1546 . 3 1 3 2 5 . 4 1 1 0 4 . 5 8 8 3 . 5 9 6 6 2 . 6 9 44 1 .79 2 2 0 . 9 0 - 0 . 1 1 8 4 5 E - 1 0 , 9 7 4 . 0 O 0 1975.500 .977.000 .978.500 1 9 8 0 . 0 0 0 1 8 B 1 . * » T i m e 0 J A J O c t o b e r J a n u a r y A p r i l J u n e 118 FIGURE 6 Soybean Inventories ( m i l l i o n s of b u s h e l s ) 48 O B S E R V A T I O N S Inventory 8 4 . 9 0 0 8 2 . 7 2 3 8 0 . 5 4 6 7 8 . 3 S 9 7 6 . 1 9 2 7 4 . 0 1 5 7 1 . 8 3 8 G 9 . S G 2 6 7 . 4 8 5 6 5 . 3 0 8 6 3 . 131 6 0 . 9 5 4 5 8 . 7 7 7 5 6 . 6 0 0 5 4 . 4 2 3 5 2 . 2 4 6 5 0 . 0 6 9 4 7 . 8 9 2 4 5 . 7 1 5 4 3 . 5 3 8 41 . 3 9 . 362 185 3 7 . 0 0 8 3 4 . 8 3 1 3 2 . 6 5 4 3 0 . 4 7 7 2 8 . 3 0 0 2 6 . 123 2 3 . 9 4 6 2 1 . 769 1 9 . 5 9 2 1 7 . 4 1 5 1 5 . 2 3 8 1 3 . 0 6 2 1 0 . 8 8 5 8 . 7 0 7 7 6 . 5 3 0 8 4 . 3 5 3 8 2 . 1769 - 0 . 3 5 5 2 7 E - 13 1 9 7 4 . 0 0 0 1 9 7 5 . 5 0 0 1 9 7 7 . 0 0 0 1 9 7 8 . 5 0 0 M - March J - July S - September D - December 1 9 8 0 . 0 0 0 Time 198 1 . 5 0 0 119 Soyoil Inventories (millions of pounds) 48 OBSERVATIONS Inventory 2141.0 2093.8 204S .6 1999.4 1952.2 1905.0 1857.8 1810.6 1763.4 1716.2 1668.9 1621.7 1574.5 1527.3 1480. 1 1432 .9 1385.7 1338.5 1291 .3 1244. 1 1 196.9 1149.7 1102.5 1055.3 1008. 1 960.87 913.67 866.46 819.26 772.05 724.85 677.64 630.44 583.23 536.03 488.82 441.62 394.41 347.21 300.00 1974.OOO 1975.500 1977.000 1978.500 M J S D 1980.OOO Time 1981.500 1983.000 1984.500 1986.OOO March July September December 120 FIGURE 8 Copper Inventories (thousands of short tons) 48 O B S E R V A T I O N S Inventory 5 2 2 . 7 0 5 0 9 . 3 0 4 9 5 . 8 9 4 8 2 . 4 9 4 6 9 . 0 9 4 5 5 . 6 9 4 4 2 . 2 8 4 2 8 . 8 8 4 1 5 . 4 8 4 0 2 . 0 8 3 8 8 . 6 7 3 7 5 . 2 7 361 . 8 7 3 4 8 . 4 7 3 3 5 . 0 6 32 1 .66 3 0 8 . 2 6 2 9 4 . 8 6 2 8 1 . 4 5 2 6 8 . 0 5 2 5 4 . 6 5 2 4 1 . 2 5 2 2 7 . 8 4 2 1 4 . 4 4 2 0 1 . 0 4 1 8 7 . 6 4 1 7 4 . 2 3 1 6 0 . 8 3 1 4 7 . 4 3 1 3 4 . 0 3 1 2 0 . 6 2 1 0 7 . 2 2 9 3 . 8 1 8 8 0 . 4 1 5 6 7 . 0 1 3 5 3 . 6 1 0 4 0 . 2 0 8 2 6 . 8 0 5 1 3 . 4 0 3 0 . 4 7 5 1 8 E - 1 3 1 9 7 4 . 0 0 0 1 9 7 5 . 5 0 0 1 9 7 7 . O O O 1 9 7 8 . 5 0 0 1 9 8 0 . 0 0 0 198 1 . 5 0 0 Time 1 9 8 3 . 0 0 0 1 9 8 4 . 5 C O 1 9 8 6 . 0 0 0 M J S D March July September December 1 2 1 FIGURE 9 S i l v e r Inventories (thousands of Troy ounces) 48 OBSERVATION": Inventory 49329. 48321 . 47312. 46304. 4529S. 44287. 43278. 42270. 41262. 40253. 39245. 38236. 37228. 36219. 35211. 34202. 33194. 32186. 31177 . 30169. 29160. 28152. 27143. 26 135. 25127. 24118. 23110. 22101. 21093. 20084. 19076. 18067. 17059. 16051. 15042. 14034. 13025. 12017. 11008. 10000. Time M J S D March June September December 122 FIGURE 10 Gold Inventories (thousands of Troy ounces) 32 OBSERVATIONS Inventory 5927.0 5794.3 566 1 .5 5528.8 5396.O 5263.3 5130.5 4997 . 8 4865.1 4732.3 4599.6 4466.8 4334.1 4201.3 4068.6 3935.8 3803.1 3670. 3537 . 3404 . 3272. 3139. 3006. 2873.9 274 1 .2 2608 . 2475 . 2342. 2210. 2077. 1944 . 18 11. 1679. 1546 . 1413. 128 1 . 1 148. 1015. 882.74 750.00 J 5 1976.000 1977.000 1978.000 1 9 7 9 . O O O 1980.000 1 9 8 1 . O O O 1982.000 1983.000 1984.000 Time M - March J - June S - September D - December F I G U R E 11 S33ldd CORN PRICES FROM 1974 TO 1986 400-1 150H 1 , 1 1 1 1 1 . 1 9 7 4 1 9 7 6 1 9 7 8 1 9 8 0 1 9 8 2 1 9 8 4 1 9 8 6 1 9 8 8 TIME SOYBEAN PRICES FROM 1974 TO 1986 1 3 0 0 - 1 1 2 0 0 -1 1 0 0 -4-00 H 1 : 1 1 : 1 1 1 1 9 7 4 1 9 7 6 1 9 7 8 19 8 0 1 9 8 2 19 8 4 19 8 6 TIME SOYOIL PRICES FROM 1974 TO 1986 5 0 n 10 H 1 1 1 1 1 1 1974 1976 1978 19 80 1982 19 8 4 198 6 TIME F I G U R E 15 SILVER PRICES FROM 1974 TO 1986 5 0 n 40 H OH —i 1 1 1 1 " 1 1 1974 1975 1978 1980 1982 1984 1986 1988 TIME 129 FIGURE 17 ro ro ID o o o o o o o o o o o o o o o o S33lcJd 130 FIGURE 18 Wheat (Chicago) Convenience Yie l d vs Inventory 4B OBSERVATIONS Convenience Y i e l d 1 2 2 . 3 3 1 19 . 19 1 1 S . 0 5 1 1 2 . 9 2 \ 1 0 9 . 7 8 \ 1 0 6 . 6 4 1 0 3 . S I l O O . 3 7 9 7 . 2 3 3 9 4 . 0 9 6 9 0 . 9 6 0 8 7 . 8 2 3 8 4 . 6 8 7 8 1 . 5 5 0 7 8 . 4 1 4 7 5 . 2 7 7 7 2 . 141 ! 6 9 . 0 0 4 6 5 . 8 6 7 6 2 . 7 3 1 5 9 . 5 9 4 5 6 . 4 5 8 5 3 . 3 2 1 5 0 . 1 8 5 4 7 . 0 4 8 4 3 . 9 1 2 4 0 . 7 7 5 3 7 . 6 3 9 3 4 . 5 0 2 3 1 . 3 6 5 2 8 . 2 2 9 2 5 . 0 9 2 2 1 . 9 5 6 1 8 . 8 1 9 1 5 . 6 8 3 1 2 . 5 4 6 9 . 4 0 9 6 6 . 2 7 3 1 3 . 1 3 6 5 - 0 . 3 8 1 9 2 E - 1 3 M=MULTIPLE POINT • * * * * * * M 1 2 0 . 0 0 0 1 8 0 . 0 0 0 2 4 0 . 0 0 0 A - March, 1974 B - July, 1974 C - March, 1975 3 0 0 . 0 0 0 3 6 0 . 0 0 0 4 2 0 . 0 0 0 4 8 0 . 0 0 0 5 4 0 . 0 0 0 6 0 0 . 0 0 0 Inventory 131 FIGURE 19 Wheat (Kansas) Convenience Y i e l d vs Inventory 48 O B S E R V A T I O N S Convenien 9 7 . 95 . 9 3 . 91 . 8 9 . 86 . 84 . 82 . 8 0 . 7 8 . 7 5 . 7 3 . 71 . 6 9 . 67 . 6 4 . 6 2 . 6 0 . , 5 8 . 5 6 . ' 5 4 . 51 . 49 . 47 . 4 5 . 4 3 . 4 0 . 38 . 3 6 . 3 4 . 3 2 . 2 9 . 2 7 . 2 5 . 2 3 . 21 . 19 . 16 . 14 . 12 . ce 761 575 388 202 0 1 6 8 3 0 644 458 271 0 8 5 8 9 9 713 527 34 1 154 968 782 596 4 IO 223 037 85 1 6 6 5 479 293 106 9 2 0 734 548 362 176 989 803 617 431 245 0 5 9 872 686 5 0 0 Y i e l d M=MULTIPLE POINT ' A 1 2 0 . 0 0 0 1 8 0 . 0 0 0 2 4 0 . 0 0 0 3 0 0 . 0 0 0 3 6 0 . 0 0 0 4 2 0 . 0 0 0 4 8 0 . 0 0 0 5 4 O . 0 0 O 6 0 0 . 0 0 0 Inventory A - March, 1974 B - March, 1975 132 FIGURE 20 Wheat (Minneapolis) Convenience Y i e l d vs Inventory 48 O B S E R V A T I O N S Convenie 92 9 0 88 86 84 82 80 78 76 74 72 7 0 67 65 63 61 59 57 55 53 51 49 47 45 43 41 39 37 35 33 30 28 26 24 22 20 18 16 14 12 nee . 6 5 0 . 5 9 4 . 5 3 9 . 484 . 4 2 9 . 3 7 4 . 3 1 9 . 264 . 2 0 9 . 154 . 0 9 8 04 3 . 988 . 9 3 3 . 8 7 8 . 8 2 3 . 7 6 8 . 7 1 3 . 6 5 7 . 6 0 2 . 5 4 7 . 492 . 4 3 7 . 3 8 2 . 327 . 2 7 2 . 2 1 7 . 161 . 106 .051 . 9 9 6 .941 . 886 .831 . 7 7 6 . 7 2 0 . 6 6 5 . 6 1 0 . 5 5 5 . 5 0 0 Y i e l d " ' M U L T I P L E POINT * A * M 1 2 0 . 0 0 0 1 8 0 . 0 0 0 2 4 0 . 0 0 0 3 0 0 . 0 0 0 3 6 0 . O K 4 2 0 . 0 0 0 4 8 0 . 0 0 0 5 4 0 . 0 0 0 6 0 O . 0 0 0 Inventory A - March, 1974 133 FIGURE 21 Corn (Chicago) Convenience Yield vs Inventory IS OBSERVATION" : Convenience Yie l d M=MULTIPLE POINT 54.268 53:261 52.254 5 1 . 247 50.240 49.233 48.226 47 .220 4S.213 45.206 44.199 43.192 42.185 41.178 40.172 39.165 38.158 37.151 36. 144 35.137 34.130 33. 123 32. 1 17 ** 31 . 1 IO 30.103 29.096 28.089 27.082 26.075 25.069 24.062 23.055 22.048 21.041 20.034 19.027 18.021 17.014 16.007 15.OOO 20.000 40.000 60.000 80.000 100.000 120.000 140.000 Inventory 160.000 180.000 134 FIGURE 22 Soybeans (Chicago) Convenience Yie l d vs Inventory 24 O B S E R V A T I O N S Convenience 9 1 . 2 4 8 8 8 . 9 0 9 8 6 . 5 S 9 8 4 . 2 2 9 8 1 . 8 9 0 7 9 . 5 5 0 7 7 . 2 I O 7 4 . 8 7 1 7 2 . 5 3 I 7 0 . 1 9 1 6 7 . 8 5 1 6 5 . 5 1 2 6 3 . 1 7 2 6 0 . 8 3 2 5 8 . 4 9 3 5 6 . 1 5 3 5 3 . 8 1 3 5 1 . 473 4 9 . 1 3 4 4 6 . 7 9 4 4 4 . 4 5 4 4 2 . 1 1 5 3 9 . 7 7 5 3 7 . 4 3 5 3 5 . 0 9 6 3 2 . 7 5 6 3 0 . 4 1 6 2 8 . 0 7 6 2 5 . 7 3 7 2 3 . 3 9 7 21 . 0 5 7 1 8 . 7 1 8 1 6 . 3 7 8 1 4 . 0 3 8 1 1 . 6 9 9 9 . 3588 7 . 0 1 9 1 4 . 6 7 9 4 2 . 3 3 9 7 - 0 . 9 9 0 3 2 E -Y i e l d M = M U L T I P L E POINT 13 1 5 . O O O 2 2 . 5 0 0 3 0 . 0 0 0 3 7 . 5 0 0 4 5 . 0 0 0 5 2 . 5 0 0 Inventory S O . 0 0 0 6 7 . 5 0 0 7 5 . 0 0 0 A - July, 1977 B - July, 1978 135 FIGURE 23 Soyoil (Chicago) Convenience Y i e l d vs Inventories 48 O B S E R V A T I O N S Convenience Y i e l d M MULTIPLE 8 .3885 8 .14 13 7 .8942 7 .6470 7 . 3999 7 . 1528 6 .9056 6 .6585 6 .4113 6 . 1642 5 .9171 5 . 6699 5 . 4228 5 . 1756 4 . 9285 4 . 68 14 4 .4342 4 . 187 1 3 .9399 3 . 6928 3 .4457 3 . 1985 2 .9514 2 .7042 2 . 457 1 2 .2 100 1 .9628 1 .7 157 1 . 4685 1 .2214 0. 97426 0. 727 12 0. 47998 0. 23284 - 0 . 14300E - 0 . 26144 - o . 50858 - 0 . 75572 -1 .0029 -1 . 2500 POINT 300.000 GOO.000 900.000 1200.000 1500.000 1800.000 2 100.000 2400.000 2700.000 I n v e n t o r y A - March, 1974 136 FIGURE 24 Copper (CMX) Convenience Yield,vs Inventory 48 OBSERVATIONS Convenience Yie l d M-MUDPLE POINT 15.409 14.937 14.4G5 13.993 13.521 13.049 12 .577 12.105 1 1.633 1 1 . 161 10.689 10.216 9.7445 9 .2725 8 . 8004 8.3284 7 .8564 7.3844 6.9124 6.4404 5 .9683 5.4963 5.0243 4 . 5523 4 .0803 3.6082 3.1362 2 . 6642 2. 1922 1.7202 1 . 2482 0 .77614 O.30412 -0 .16789 -O.63991 - 1 . 1 1 1 9 - 1.5839 - 2 . 0 5 6 0 - 2 . 5 2 8 0 -3.OOOO • • » * • M h * * 0 . 0 7SD00 150.000 225.OOO 300.OOO 375.OOO Inventory A - March, 1974 1 3 7 FIGURE 2 5 S i l v e r (CMX) Convenience Yie l d vs Inventory 36 OBSERVATIONS Convenience 0 . 5 3 5 3 3 0 . 5 6 8 5 2 0 . 5 4 1 7 2 O . 5 1 4 9 2 0 . 4 8 8 1 1 0 . 4 6 1 3 1 0 . 4 3 4 5 1 0 . 4 0 7 7 0 0 . 3 8 0 9 0 0 . 3 5 4 10 0 . 3 2 7 2 9 0 . 3 0 0 4 9 0 . 2 7 3 6 9 0 . 2 4 6 8 9 0 . 2 2 0 0 8 0 . 19328 0 0 0 O O O O Y i e l d 16648 13967 1 1287 8 6 0 6 5 E - 0 1 5 9 2 6 2 E - 0 1 3 2 4 5 9 E - O I 5 6 5 5 6 E - 0 2 - O . 2 1 1 4 8 E - 0 1 - O . 4 7 9 5 l E - O I - 0 . 7 4 7 5 4 E - 0 1 - O . 1 0 1 5 6 - O . 1 2 8 3 6 - O . 1 5 5 1 6 - O . 1 8 1 9 7 - O . 2 0 8 7 7 - O . 2 3 5 5 7 - 0 . 2 6 2 3 8 - O . 2 8 9 1 8 - O . 3 1 5 9 8 - 0 . 3 4 2 7 9 - O . 3 6 9 5 9 - 0 . 3 9 6 3 9 - O . 4 2 3 2 0 - 0 . 4 5 0 0 0 M = MULTI PL E POINT O . 1 0 E + 0 5 O . 1 5 E * 0 5 0 . 2 0 E * 0 5 0 . 2 5 E + 0 5 0 . 3 0 E + 0 5 0 . 3 5 E + 0 5 0 . 4 0 E + 0 5 0 . 4 5 E * 0 5 0 . 5 0 E * 0 5 Inventory A - December, 1979 B - March, 1983 138 FIGURE 26 Gold (IMM) Convenience Yie l d vs Inventory 32 O B S E R V A T I O N S Convenience 3 . 9 8 S 5 3 . 5 7 6 S 3 . 1667 2 . 7 5 6 8 2 . 3 4 6 9 1 . 9 3 7 0 1 .5271 1. 1 172 0 . 7 0 7 2 5 0 . 2 9 7 3 4 - 0 . 11257 - 0 . 5 2 2 4 8 - O . 9 3 2 4 0 - 1 .3423 - I - 2 - 2 - 2 YieldM = MULT I P L E POINT . 7 5 2 2 . 162 1 . 5 7 2 0 . 9 8 2 0 - 3 . 3 9 1 9 - 3 . 8 0 1 8 - 4 . 2 1 1 7 - 4 . 6 2 1 6 - 5 . 0 3 1 5 - 5 . 4 4 14 - 5 . 8 5 1 3 - 6 . 2 6 12 - 6 . 6 7 1 2 - 7 . 0 8 1 1 - 7 . 4 9 1 0 - 7 . 9 0 0 9 - 8 . 3 1 0 8 - 8 . 7 2 0 7 - 9 . 1 3 0 6 - 9 . 5 4 0 5 - 9 . 9 5 0 4 - 1 0 . 3 6 0 - 1 0 . 7 7 0 - 11 . 180 - 1 1 . 5 9 0 - 1 2 . 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