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University Microfilms International 300 N. Zeeb Road Ann Arbor, Ml 48106 8305376
Park, Hun Young
A THEORETICAL AND EMPIRICAL INVESTIGATION OF FORWARD PRICES AND FUTURES PRICES
The Ohio State University Ph.D. 1982
University Microfilms International300 N. Zeeb Road, Ann Arbor, MI 48106
Copyright 1983 by Park, Hun Young All Rights Reserved A THEORETICAL AND EMPIRICAL INVESTIGATION OP FOBSABD
PRICES AND FUTURES PBICES
DISSERTATION
Presented in Partial Fulfillaent of the Beguireaents for
the Degree Doctor of Philosophy in the Graduate
School of The Ohio State University
BY
HUN YOUNG PARK, B-B.A., H.B.A.
The Ohio State University
1982
Beading Coaaittee: Approved ty
Andrew H. Chen
J. Huston Mcculloch
Gary C. Sanger Adviser Departaent of Finance ACKHOBLEDGBEBT
I am grateful to my dissertation committee members. Professors
Andrew H. Chen, J. Huston HcCulloch, Gary C. Sanger for their continuing encouragement and valuable comments. Ihis paper could not have been completed in a reasonable time period without their unfailing support at each stage of the research. In particular, from the bottom of my heart, 1 wish to express thanks to professor
Andrew H. Chen, who has been serving as a chairman of my dissertation committee, for suggesting the topic originally and spurring my interest in many financial area associated with the topic. 1 can never forget his spiritual guidance for my academic career in the future. Also, 1 wish to extend my special appreciation to Professor J. Huston HcCulloch for his insightful and cheerful comments at critical points during the work has made progress. I am also indebted to Professor Gary C. Sanger who has devoted time and maintained enthusiasm throughout this research.
I would like to thank participants at finance workshop at
University of Illinois, Urbana-Champaiyn, Illinois for their
helpful comments and suggestions. Special thanks are owed to
Professor C. F. Lee who provided me with a chance to present the rough draft of this paper at the University of Illinois. Finally, but assuredly not least in importance, 1 owe a special
debt of gratitude to ay lovely wife, Soo £ae, for her painstaking
support for ae to maintain a healthy franevork for Living over the
years of this work. Her good humor and patience have been shared
selflessly, although occasionally tested. I also owe a great deal
to my mother and father in Korea for their spiritual and endless supports. VITA
January 24, 1951 ------Born-Seoifl, Korea
1972-1973 ------Chaiman of the AIESEC Academic club of college students Seoul, Korea
1973-1974 ------Financial analyst, the Oriental Textile Corporation, Seoul,Korea
1974 ------Business Administration The Seoul National University Seoul, Korea
1974-1977 ------Naval officer as a Lieutenant Analyst in the Personal Administr ation Department, Korean Navy Headguarter,Seoul, Korea
1979 ------fl.B-A., The Ohio State University Columbus, Ohio
1979-1981 ------Teaching Associate, Department of Finance, The Ohio State University
1981-1982 ------Research Associate, Department of Finance, The Ohio State University
Fields of Study
Finance Economics Statistics Econometrics Quantitative analysis
- i v - TABLE OF COMTE1TS
ACKNOWLEDGMENT...... - ...... li
VITA ...... iv LIST OF T A B L E S ...... vii Chapter page
I. INTRODUCTION ...... 1
Research objectives ...... 5 Relevant literature review ...... 7
II. PRICING OF FORWARD AND FUTURES CONTRACTS IN A GENERAL EQUILIBRIUM...... 14
Pricing of forward contracts ...... 15 Pricing of futures contract ...... 24
III. RELATIONS AMONG FORWARD, FUTURES AND EXPECTED PRICES - . 34
Explicit form pricing eguations ...... 34 Relation between forward and futures prices ..... 38 The issue of Normal Backwardation or Normal Contango . 43
IV. EMPIRICAL TEST ...... 47
Data ...... 47 description of data ...... 47 Limitation of data ...... 51 Methodologies and results of the empirical test . . . 53 Tests on the relation between futures and forward prices ...... 53 Tests on Normal backwardation or Normal contango process ...... 72
V. CONCLUSION ...... 6 4
Summary ...... 84 Concluding remarks, implications, and future research areas ...... 89 Appendix page
Ar- EXCHANGES AMD UNITS OF TBADING IN FUTURES CO NIB AC IS . . . 94
B. TRADING HOURS OF FUTURES AND FORWARD CONTRACTS ..... 95
C. TESTS ON THF. RATIO OF COVARIANCE TC V A R I A N C E ...... 96
D. GROUPINGS OF COMMO D I T I E S...... - 9 9
E. DIFFERENCE BETWEEN FUTURES AND FORWARD PRICES (GROUPS) . 100
F. DIFFERENCE BETWEEN FUTURES AND FOBHARD PRICES (COMMODITIES) ...... 101
G. TESTS ON THE ANALOGY TO TERM STRUCTURES OF INTEREST RATES ...... 103
H. TESTS ON FUTURES RETURNS ...... 104
I. NORMAL BACKWARDATION OF FUTURES PBICES ...... 106
J. TESTS ON FORWARD RETURNS ...... 107
K. NORMAL BACKWARDATION OF FORWARD PBICES ...... 109
REFERENCES ...... 110
_ vi - LIST OP TABLES
Table page
1. Tests on the ratio of covariance to variance ...... 57
2. Groupings of commodities ...... 62
3. Difference between futures and forward prices (groups) . . . 64
4. Difference between futures and forward prices(commodities) 65
5. Tests on the analogy to term structure of interest rates . 71
6. Tests on futures returns ...... 74
7. Normal backwardation of futures prices ...... 77
8. Tests on forward returns ...... 79
9. Normal backwardation of forward prices ...... 82
- vii - Chapter I
IHTBODDCTION
Host of prevailing theories revolving around the underlying behaviors of forward and futures prices over tine are conflicting or at least inconclusive as to the generation of eguilibriua prices in the presence of uncertainty. Furthermore, in spite of fundamental differences between forward contracts and futures contracts, virtually no literature has distinguished one from the other.
A forward contract is an agreement between two traders, a buyer and a seller to buy or sell a specific commodity at a "specific price", called the contract or exercise price, on a specific date, called the maturity or delivery date. Ibe "specific price" in a forward contract is the forward price that is determined at the time the contract is open and this price stays fixed for the life of the contract. By this convention, the forward price on the maturity date is equal to the spot price of the commodity under consideration. Otherwise, a costless arbitrage would occur at the maturity date. In this contract, writing and settlement are not simultaneous in that the writing of this contract requires no initial changes of monny, and money transfers occur only on the maturity date. In other words, the buyer who takes a long position in a forward contract by promising to pay the fixed 2 forward price at the maturity date receives the specified comnodities at the maturity date from the seller who takes a short position now. Therefore, even though there is no initial transfer of money when the contract is open, the payoff on the maturity date is not necessarily zero but is equal to the difference between spot price on the maturity date and forward price. Of course, forward prices are fluctuating over time but in such a way that the value of the forward contract is zero at the time contract is created. Also, forward contracts are written with specific times to maturity rather than with specific maturity dates.
A futures contract is a contract between two parties where the buyer agrees to accept delivery at a "specific price" from the seller of a particular commodity, in a designated month in the future, if it is not liquidated before the contract reaches maturity. Like a forward contract, the "specific price" in a futures contract is the futures price that is determined when it is initiated, and thus the futures price on the maturity date should be equal to the spot price to prevent arbitrage opportunities.
However, the basic difference between those two contracts lies in their payment schedules because of the property of daily settling up, so called "Harking to market" in futures contracts.
In other words, at the end of each day, if the changes in futures price during a day is negative, the buyer who takes a long 3 position in the futures contract pays the full aaount of the change to the seller who takes a short position and the futures contract is rewritten at the new futures price in such a way that
■akes the value of the futures contract zero. If the change is positive, then the buyer should be paid the full aaount of the change by the seller.
These futures contracts are cleared through the clearing corporation of a commodity exchange. The clearing corpocation thereby assumes the technical responsibility to the buyer and seller respectively as a third-party guarantor of the transaction.
In effect, it becomes the buyer for every seller and the seller for every buyer.
However, it is important to notice that neither a forward nor a futures contract is an option; nothing in either is conditional.
It is an obligation. However, the seller can offset his obligation to deliver the coaaodity against the contract by liquidating the contract with an egual and opposite purchase of a futures contract. The buyer aay also offset his obligation to take delivery of the commodity through an equal and opposite sale of a futures contract.
Because futures contracts evolved from cash forward contracts, they have similar terms, but they are different in a fundamental ways. Even though the sum of the intecia payment over the life of a futures contract is equal to the futures price prevailing when it is created, the very property of daily settling up <* distinguishes futures contracts froa forward contracts in the presence of uncertainty. Also, futures contracts are traded with specific maturity dates in contrast to specific tiaes to Maturity in forward contracts.
The role of forward and futures Markets and the underlying stochastic process of their prices have attracted a lot of attention in finance and econonics. However, few articles have distinguished one fron the other because of the possible misconception of their economic properties; In fact, in the absence of solid theoretical Models, this similarity between futures contracts and forward contracts has led researchers to treat them as if they were identical. Additionally, nost of the literature xn this area tends to build upon each other as economic models were checked with empirical evidence and extentions or modifications were created only to be tested again. In other words, theories on futures or forward contracts have tended to be created to fit the observed facts, not vice versa.
In conjunction with the systematic patterns to forward or futures prices, a long-standing controversy has been whether or not these prices are systematically downward or upward biased estimates of expected future spot prices and whether this systematic bias implies market diseguilibrium or market inefficiency.
The major objective of this paper xs to analyze forward prices and futures prices and investigate sxnultaneous relations among 5 these two contract prices and expected future spot prices in equilibrium. In a sense, the lack of agreement about the systematic patterns to the forward price and the futures price is sufficient to suggest that more research based on a sore rigorous application of standard economic theory are needed.
1.1 RESEARCH OBJECTIVES
The purpose of this paper is multifold. First of all, we will attempt to clearly distinguish futures contracts from forward contracts. Secondly, we will specify theoretically under what conditions "Normal backwardation" or "Normal contango" can be accepted as an accurate description for the stochastic process of the equilibrium prices of the forward and futures contract.1
Additionally, the controversy over whether systematic bias implies market disequilibrium or market inefficiency will be examined.
The research strategy for these objectives is twofold; one is to develop theoretical models and the other is to test them empirically.
The underlying objects of the contracts in this paper include physical commodities such as Gold,Silver and foreign exchanges like Swiss Franc, German Hark and financial assets such as
1 There is a slight difference between "Normal Backwardation" and "Backwardation". "Normal Backwardation" refers to the situation where the expected future spot price is greater than the contract price while "Backwardation" refers to the situation where the current spot price is greater than the contract price. The terms "Normal Contango" and "Contango" are the reverse respectively. Ibis clarification of terminologies is due to Professor J.H. HcCulloch Treasury bill futures.
The distinction between forward contracts and futures contracts
is clearly shown by using an arbitrage argument, which says that
the value of assets must be the sane if their payoffs are the
same, or that nobody can purchase at zero cost a bundle of goods
that will strictly increase his utility in any economic
eguilibrium. Then, this paper fornulates general eguilibrium
pricing models for both forward and futures contracts in a
simplified economy in the presence of uncertainty; the basic
methodology is again the arbitrage argument saying that the
expected marginal utility from either a forward contract or a
futures contract is zero in a market eguilibrium.
In oder to deduce the clear relations and their testable
implications among the forward, futures and expected spot prices,
explicit models are developed by adding some assumptions on the
underlying process of assets (lognormality) and the utility
function (constant relative risk aversion); The general eguilibrium
models are not without cost. The cost of adding the assumptions,
however, permits us to switch from the description of a general eguilibrium model to explicit analysis of systematic patterns to
forward and futures prices, so that we can develop various academic and practical implications, which are empirically testable.
In the process, it is clearly confirmed that neither the
forward price nor the futures price is generally egual to the 7 expected spot price and this systematic bias is not inconsistent with market equilibrium or market efficiency. In other words, market equilibrium or market efficiency is neither a necessary nor a sufficient condition for forward and futures pcices to be unbiased estimates of expected spot prices.
However, in the absence of empirical supports, the interpretive value of these models in terms of explaining the systematic relations among those three prices is not complete. Thus, we will investigate those prices empirically on the basis of the theoretical models developed in this paper and examine the departures, if any, from these models. Mot only individual commodities, but several portfolios of forward contracts and futures contracts that are constructed by a grouping technique will be tested.
1.2 BELBVAHT LITEBATOBE BEVIfB
Up until the most recent working papers(Cox, lugeisoll and
Boss (1931), diehard and Sundaresan(1931), Jarrow and
Oldfield (1981) appeared, the natuie of controversies over the underlying mechanism of forward and futures prices could be divided into three hypothesis.
The first hypothesis is "normal backwardation" that is developed and supported mainly by Keynes (1930) ,Hicks (1939) and
Houthakker (1957). The supporters for this "Normal backwardation" hypothesis view forward or futures market as a semi-efficient mechanism in which risk averse speculators or hedgers insure other
risk averse investors who hold positive stocks of commodities
against changes in prices. A couple of sentences in "Value and
capital" of Hicks(1939) summarizes the major points in this
theory:
futures prices are nearly always Bade partly by speculators, who seek a profit by buying futures when the futures price is below the spot price they expect to rule on the corresponding date ------a speculator will only be willing to go on buying futures so long as the futures price remains definitely below the spot price he expects; for it is the difference between these prices which he can expect to receive as a return for his risk-bearing and it will not be worth his while to undertake the risk if the prospective return is too small
Therefore, in order to induce these speculators to hold contracts,
futures and forward prices should be downward biased estimates of
the expected spot price and are expected to increase over the
lives of the contracts since, on the maturity date, forward and
futures prices must equal the spot price.
This theory of backwardation was recently supported from a
purely theoretical perspective by Arrow (1961). According to him, the futures price at each moment tends to be below the spot price expected on the basis of currently available information and this futures price is revised as new information is available, making
the difference between the futures price and the expected spot price decrease as the maturity date is getting close; on this
basis, he argues that the futures price should be a random process with an upward drift. 9
The second is the reverse hypothesis called the "Normal contango" that is led by Hardy (1940) and lelser (1958); they view the futures market as a mechanism in which speculators are gambling rather than insuring against changes in price with risk averse investors who hold positive stocks, and thus they are willing to enter into contracts even though the futures price stays above the spot price they expect, and in turn the contract price is an upward estimate of the expected spot price.
However, neither of these conflicting theories has paid attention to the fundamental differences between forward contracts and futures contracts to explain the systematic divergence of their prices from the expected spot price.
The third hypothesis is that forward and futures prices are unbiased estimates of the future expected price. This hypothesis is suggested in efficient market theorem by Sanuelson (1965),
Mandelbrot (1963) and extended by leuthold (1972), Black(1976),
Dusak(1973) and Scholes(1981) . Especially Black(1976),
Dusak(197J) and Scholes (1981) use the CAPH of
Sharpe(1964) ,Lintner(1965) ,Mossin(1966) in order to develop pricing equations aud to compute the returns to individual speculators. They argue that forward and futures markets are not different in principle from the market for any other risky portfolio assets, in that they are all candidates for inclusion in investors* portfolios. 10
Dusak(1973) empirically examines the risk-return relationship
of three agricultural commodities, wheat, corn and soybeans from
1952 to 1967 within the context of the conventional CAPfl, and
concludes that the market risks for the commodities in futures
markets are zero and thus rejects the "Normal backwardation" of
futures prices.
However, the empirical results of Dusak are refuted by Bodie
and Hosansky{1979). In contrast to Dusak(1973), Bodie and
Kosansky(1979) shows, using the data of 23 commodities for the
longer period, that their risk-return relationship is inconsistent
with the CAPH showing the positive means and negative betas on the
average, and also partially supports the "Normal backwardation"
hypothesis by showing the positive mean excess rates of return on commodity futures.
It is questionable, however, how the CAPH assuming fixed
supplies and fixed relative prices of assets adds to our
understanding of the futures market to obtain meaningful
descriptions of stochastic prices facing the existence of price
speculators in the markets where changes in relative prices are
uncertain and infinite supply is possible inexpensively.
Grauer and Litzenberger(1979), in an attempt to incorporate changes in relative prices, developed the pricing eguatiou of
futures contracts assuming homothetic utility functions. But by assuming a two-period exchange economy, they could not distinguish
futures contracts from forward contracts or they did not pay 11 attentions to the essential difference between these two contracts.
HcCulloch (1972, 1975) showed purely mathematically that there must always be nornal backwardation and therefore positive expected profits to speculators in any forward market, in the absence of transaction costs and in the presence of uncertainty.
It is in fact not until the several nost recent working papers
(Cox, Ingersoll and Boss(1981), Richard and Sundaresan(1981),
Jarrow and Oldfield(1981) appeared that the basic iaportant differences between forward prices and futures prices and their implications have coiae to attract serious attention in financial economics. Cox, Ingersoll and Ross (1981) shows the relation between forward prices and futures prices using a portfolio approach. Richard and Sundaresan (1981) formulate implicit equilibrium pricing models for both contracts using a portfolio strategy similar to Cox, Ingersoll and Ross(1981) in a continuous time , multigood identical consumer economy. Jarrow and
Oldfield (1981) makes explicit the distinction between forward and futures prices using an arbitrage argument and emphasizes stochastic interest rates as the most important factor that makes the futures price distinct from the forward price.2 Also,
2 While I was writing the previous draft of this paper, I received these three working papers in which the models for futures prices and forward prices are guite similar to each other. I independently developed the basic eguilibrium models whose implications turn out to be somewhat similar to them, but from different perspectives. later in December ,1981, these papers were published in the Journal of Financial Economics. 12
Hargrabe (1978) shows that forward prices and futures prices are not the saae generally and derived the futures price as a function of the forward price, assuming that price changes of assets are given by a stochastic differential eguation. Ihe models of the above four working papers, and their implications are substantially similar to each other. Nevertheless, the explicit expressions for the simultaneous relations among forward prices, futures prices and expected spot prices are attracted little attentions in those papers.
Deriving similar models and thus similar implications obtained from a different approach, this paper addresses the issue directly. The organization of this paper is as follows:
In chapter 2, general equilibrium models of forward contracts and futures contracts are formulated. These models are guite general but purely theoretical in the absence of the knowledge of the commodity generating process and the specification of utility function. Nevertheless, some important implications are derived which illustrate the conceptual economic rationale for “normal backwardation" and "normal contango". On the basis of models formulated in chapter 2, the explicit form models of both contracts are developed in chapter 3. Also, the simultaneous relations among forward and futures prices and expected spot prices and some testable implications are derived and analyzed by examining the variables involved. Chapter 4 adds empirical support providing the results of various tests. Finally, chapter 13
5 summarizes the paper and discuss the implications of theoretical models and empirical findings, and concludes the paper suggesting future research areas. Chapter II
PBICIMG OF FOBHABD AMD FOTOBES COITBACTS IB A GEIEBAL EQOILIBBIOfl
Throughout this paper, the consideration of aargin requirement3 is ruled out; it is important to note that margin requirements are not partial eguity payments against the market value of the commodity represented ly the contract, as it is when buying common stocks, but a guarantee in the event of adverse price movements.
Nevertheless, a controversy over margin requirements exists in connection with investors' optimal portfolio construction. For example, Telser (1981) argues that uaLgin requirement should be incorporated in the pricing equations because they may disrupt individual investor's optimal portfolio allocation and thus induce costs even though they are in the form of interest-earning securities like Treasury bills. however, as long as individual investors can borrow in a perfect capital market against their portfolios to buy Treasury bills for the purpose of posting them as margin requirements, it doesn't induce any cost. In other words, the opportunity cost for posting margin requirements is zero, assuming no transaction costs in a perfect capital market.
3 Statistically the margin deposit for a contract is less than 5% of face value.
- 14 - 15
2-1 PBICIMG OF FOBHABD COBTBACIS
In a complete market, it is veil known that the payoff on any asset can be described as the payoff of other assets or the payoff of some combinations of other assets. The payoff of an asset nay be linear or non-linear. If the payoff is linear like the one of forward contracts, the price of a portfolio of forward contracts is equal to a portfolio of forward prices and the same thing can be applied for futures contracts. If the payoff is non-linear like the one of an option, a call or a put, an option on a portfolio is not the same as a portfolio of options, as noted in
Cox, Ingersoll and Boss (1981). But even though the payoff of a call optiou alone or a put option alone is not linear, the linear payoff can be constructed by combining call options and put options, so that we can use it to identify the payoff of the forward contracts at the maturity date.
Consider the following two strategies;
A: (1) At time t, buy a European call option on 1 unit of
commodity xi whose maturity date is 1 at the exercise
price fi(t,T), which stands simultaneously for the price
of a forward contract on commodity Xi which is made at
the present time t and is going to be closed out at the
maturity date X.
(2) At the same tine, sell a European put option on 1 unit of
Xi at the same exercise price and the same maturity date
as the call option. 16
B: At tine t, enter a forward contract by promising to buy 1
unit of Xi for fi(t, 1) at the maturity date.
At the naturity date T, the payoffs of the two investment strategies A and B are the same,4 which is Pi (1)-fi (t,1) ; Pi(T ) stands for the spot price of commodity i at tiue 1 , which is random. Therefore, the value of a forward contract (strategy B) should be same as that of a portfolio of buying a call option and selling a put option (strategy A) to rule out the arbitrage opportunities; Kane (1980) used the similar strategy ("options strategy" in his paper) of selling a call option and buying a put option on the bond, both exercisable at an identical striking price, in order to explain the time-series behavior of the divergencies between the forward and futures interest rates on
U.S. Treasury securities. However, he did not realize that this strategy itself is equivalent to selling a forward contract.
On the other hand, it is a well known fact that the value of a portfolio of a long position in a European call and a short position in a European put with the same exercise price as the call is equal to the value of a portfolio of a long position in
♦ The values of each strategy of time t and their payoffs at time T can be described as following;
strategy the value at time t The payoff at time T P K T K t K t . T l PiTlII >fi It.T) A Pc (t,T)-Pp (t,T) * Pi (T)-fi (t,I) Pi (T)-fi (t,T) B Vt fi(t.T) *♦ Pi(T)-fi(t,T) Pi (T)-fi (t,T) *Pc(t,T):the call price at tine t that is exercisable only at T *Pp(t,T):the put price at time t that is exercisable only at T **Vt fi(t,l) : the value of a forward contract at time t- 17
the underlying asset and a short posxtion in a riskless bond with
the face value equal to the exercise price of options from the
arbitrage condition, assuming that borrowing rates and lending
rates are equal.5 Therefore, the value of a forward contract is
equivalent to that of a long position in the underlying commodity
and a short position in a riskless bond with the face value,
fi(t,T) in the above strategy.
Let B(t,T) be the value as of the current time t of $1 payable
at the time T, and Vt(.) be the value as of time t of (-). Note
that the value of a forward contract is zero at time t when it is
initiated by the logic stated in the introduction.
Then, Vt[px(T)j - B (t,T) fi (t,T) = 0
Thus, fi (t,T) = B(t,Tr‘ Vt[pi Equation (1) implies that the forward contract price is the value of a claim that pays B ^ l f 1 units of the commodity under consideration at its maturity date. By entering a forward contract on one unit of commodity i with a forward price fi (t,T), a person who is in the long position in the contract xs paid B(t,Tr' units of commodity i at time T. Also equation (1) has the same implication as Black (1976) and Jarrow and Oldfield (1981) argument that a forward contract is equxvalent to selling short a riskfree discount bond maturing at time T and ,by using the 5 See Smith (1976) for an excellent proof of this proposition. 18 proceeds, baying and storing the conmodity in guestion over the sane period. Re can take a look at eguation (1) from another perspective using the same arbitrage argument. In order to coapare the forward contract with the investoent reguiring fi(t,2) at tine t, consider the portfolio that invests fi (t,2) dollars at tiae t in riskless bonds naturing at tine 2 and B(t,Tfl forward contract on commodity i. The payoff of this portfolio is B (t,Tf* fi (t,T) f B (t,Tr' [pl(T)-fi(t,T)J = B (t ^jT1 Pi (2) Since there is no transfer of noney in the forward contract at time t, the portfolio requires an initial expenditure of fi(t,T). In other words, spending fi(t,2) at time t is entitled to the claim on B(t,Tfl units of coamodity i at tiae 2, which is the sane implication as eguation (1). Note also in eguation (1) that B(t,Tr* is known at the present tine t and thus the forward contract is a speculation only on the unknown future spot price at tiae 2 . Once the payoff is known in terns of the nunber of units that can be traded at the maturity date 2 , a general forward contract pricing formula can be developed through naxiaizatron of the expected utility function that is assuned the sane for all individuals. 19 Consider the n+1 goods economy composed of Xo, XI, 12 ...... Xn, where Xo serves as a numeraire good- Investors are assumed to be rational in the sense that, under uncertainty, they are capable of finding every alternatives and choosing the best ones so as to maximize a lifetime expected utility function that is time additive. where «t is the utility discount factor that is known at time t and U (.) is a Von Neuman-morgenstern utility function that is strictly increasing, concave and twice differentible with respect to Xi z u{ Xo (t ) - X1 (*) , --...Xn('r)) is denoted by 0(.) hereafter. Suppose that two people, a buyer and a seller of forward contracts commit to fi forward contracts, each contract for 1 unit of Xi at the price fi(t,T) to be paid at time 1 and the buyer is supposed to pay in terms of Xo- Then, using eguation (1), the utility function at time T can be written as Xo (T) -(t fi(t,T)B (t,l)' ,X1 (T) , Xi (t ,1) Xn(T)} ------(3) At the present time t, the market clearing condition and the first order condition for the expected utility maximization in a general equilibrium requires that the differentiation of the expected utility with respect to fi conditional fi =0 is zero. 20 Thus, where Et(.) denotes the expected value of (.) at the present time t; note that equation (4) is consistent with the no-arbitrage condition in that it reflects the essential point of arbitrage argument that the expected marginal utility from a forward contract should be zero taking no-initial-transfer of money into consideration. Also, a market clearing condition simply ceguires that demand equals supply for each commodity, in which the market is cleared of all outstanding units of assets or contracts. Thus, even though consumers face different consumption investment opportunity sets and thus have heterogeneous perceptions on the probability distributions of risk-return on each asset or claim, it poses no problem for the determination of a market equilibrium. From equation (4) -fi (t, 1) B {t,TVEt[exp (- (l-t)ot) Uo (Tj) «■ B (t,T)*Et[exp (- (T-t) ,where Ui (T) denotes the marginal utility of commodity i at time T. fi (t,T)=B fi(t,T) = Et Ui(T) / Et ub(T) 21 vhich is the general forward price of coanodity i from the market clearing condition.6 On the other hand, the relative spot price of coanodity Xi in terns of the numeraire good Xo at each tine is the ratio of narginal utility of Xi to aarginal utility of XO froa the first order condition for utility naxinization. Pi (T) = Ui(T)/U0 (r) (6) Thus, Etf Pi (T» = Et( Ui (T)/U0 Note from equations (5) and (7) that the question whether the forward price follows the "Normal backwardation" process (Keynes-Hicks- flouthakker) or "Nornal contango"(Telser) or whether the forward price is equal to the expected spot price depends on whether Etui (1)/EtUo (T) is less or greater than, or equal to Et {Di(T)/U0 (T)J . Also, following this procedure, it is very sinple to derive the relation between the forward price and the expected future spot price that is sonewhat similar to the results that Bichard and Sunderasan (1981) and Cox, Ingersoll and Boss (1981) obtained through a fairly conplicated procedure. 6 J.H. RcCulloch (1978) derived the sane pricing equation in connection with short-lived options pricing when the underlying distribution of price is log-synmetric stable. 22 fi(t,T) = Et Ui (T) /EtUO (1) fi(t,T) = Et IUO (T) - Pi (1)} /EtUO (1) from eguation (2) = (cOVt{UO (T) ,Fi (T)J + E t ub(T) £t Pi{l))/Et 0*0(1) = Et Fi (T) ♦COVtlUO(T),Fl(T)l /Et DO (T) (8) From equation (8), the relation between the forward price and the expected spot price of commodity i in connection with the controversy of "Normal backwardation" or "Normal contango" can be described as follows; if the correlation between the future spot price aud the marginal utility of a numeraire good at the maturity date is positive or negative, the forward price is geater or less than the expected spot price respectively- If there is no correlation between the spot price and the marginal utility of a numeraire good at time T , the forward price will be equal to the expected spot price at the maturity date as Black (1976), Dusak(1973) and Samuelson (1965) argue. These results are intuitively appealing in the following sense; for example, suppose that the covariauce between the spot price aud the marginal utility of a numeraire (it may be money in dollar terms or one-period discount bond price as will be assumed later in this paper) at the maturity date 1 is negative. Considering a long position in a forward contract, the spot price. Pi (T), has a positive correlation with the payoff , and thus with the return at time T. Then, the negative covariance between returns or profits and the marginal utility of money or one-period bond price implies 23 decreasing marginal utility and thus that the magnitude of the marginal utility of return or profit is less than that of the marginal utility of loss with the same dollar amount. This result reflects the notion of risk aversion of investors. Thus, in order to induce risk averse investors to commit to a forward contract, there must he compensation for hearing this risk in the form of a positive return. Ihis logic leads in turn to the conclusion that the forward price should be less that the expected spot price, which conforms with the Keyues-Hicks-Houthakker*s argument, the "Normal backwardation". It is very important to note that the expected spot price is expressed in terms of relative prices and that they are explicitly incorporated into pricing of the forward contract. This is plausible when we take into account the fact that consumption bundles of each investor depend mainly on relative prices, but not on absolute prices. Note also that in this paper no restrictions are put on the aggregate supply that is assumed fixed in the literature based on the CAPH framework; allowing random aggregate supply is plausible in the forward and futures markets since market makers in those markets can easily supply the contracts very inexpensively. Nevertheless, the general equilibrium model expressed in terms of the ratio of marginal utilities is not without cost in that in the absence of the knowledge of the specific utility function and/or the underlying process of commodities, it is uot possible to consider directly testable hypotheses. 24 An explicit aodel assuming lognormality and constant relative risk aversion will be developed in the following chapter from which some interesting testable hypothesis can be deduced in academic and practical senses. 2.2 PBICIHG OF FtJTUHES C0NXBAC1 In this section, not only the pricing eguation of futures contracts will be derived but also its implication in connection with forward contracts will be discussed. Especially the issue of how to construct the futures contract that is eguivalent to the forward contract in economic sense will be discussed. In the previous section, the portfolio of a long position in a call option and a short position in a put option that are exercisable only at the maturity date were used to develop the forward contract pricing eguation. This was possible since the exchange of commodities in question for the forward contract price occurs only on the specified maturity date 1. However, in futures contracts, because of the characteristics of daily settling up so that the value of the futures contract is zero at the end of each day, we can not derive the futures contract price simply by combining a call and a put option exercisable only at the maturity date with the same exercise price. Let Fi(t,T) be the futures contract price on commodity i that is initiated at time t and will be closed out at time T. Instead of the simple call and put that was used in deriving the forward 25 contract pricing equation, consider the call and put options that are exercisable at the price Fi(t,2) at tiae T, but are settled up daily during the contract period by paying or receiving the difference between today's futures price and the previous day's futures price, Fi (t+1,T)-Fi (t,T), and rewritten at the new exercise price Fi(t+1,T), until the original aaturity date T- Except for this daily settling up, other properties are the sane as call and put options used in the forward contract. From this point, the logic developing the futures contract pricing model is the same as the one employed in the forward contract case. Thus, by identifying a futures coutract promising to buy 1 unit of Xi for Fi(t,T) at tiae T (strategy B) and a portfolio of a long and a short position in a call and a put respectively on 1 unit of Xi at the exercise price Fi(t,T) but renewed at the exercise price Fi(t+1,l) at time t+1 aud Fi(t*2,T) at time t+2 in this way up to the aaturity date 1 (strategy A), the futures pricing model can be easily developed from the arbitrage argument. Note once again that the portfolio of buying a call and at the same time selling a put is equivalent to the portfolio of a long position in the underlying asset and a short position in the bond with the face value equal to the exercise price of the option from the no-arbitrage condition. The only difference from the case of the forward contract is that one-period(it may be one-day) bond which is reinvested until the maturity date should be employed because of the daily marking-to- market property. From the same argument employed to arrive at eguation (1), it follows using the same notations that VtfPi(T)}- VtjFi (t,l)/exp£ r (T)} = 0 r*t where r(r) is the continuously compounded interest rate from tine 7 to time t ♦1. Thus, Fi (t,T) = Vt{exp jr| r(T)Pi(T)} ------(9) Tst This implies that the futures price is the value of a claim that pays exp r (t ) units of the commodity under consideration at the maturity date. Also, like the case of forward contracts, in order to compare the futures contract with the investment reguiring the dollar amount Fi(t,T) at time t, consider the portfolio as of time t that invests Fi(t,T) in one-period bond and continuously reinvest the gross return in one-period bond every day until the aaturity date, and at the same time, buying the futures contracts of exp r(r) tit at each time £ until the maturity date. The payoff of this portfolio at the maturity date can be easily T»l ^ seen as expi 2£ r(r)J- Pi(T). Since there is no initial change of tst money from the buyer to the seller of the futures contract, this portfolio reguires the dollar amount of only Fi(t,T). In other words, the buyer of a future's contract on coanodity i whose maturity date is T can obtain exp jE? r(r) units of commodity i at r*t tiae T by giving up Fi (t,l) at the present tine t. 27 Note that exp 5; r(T) units of the coanodity is unknown at tine Tst t, and thus engaging in futures contract is a speculation not only on the future spot price but. on the number of units of coaaodities than can be paid for the futures contract at the aaturity date. Fron the same logic that was employed in the forward contract pricing model, if a person engages in $ units of futures contracts, each unit of the contract for 1 unit of Xi at the price Fi(t,T) by promising to pay for it with Xo, then his utility function at time T is U{Xo(T)-p- Fi (t,T) exp z^r (T) ,X1(T),X2(T) ,,,,,,,,,, ,,,,Xi(T) ♦ A-expl'r (T) ,,,,,,, Xn(l)}------(10) T«t From equation (4) , -Fi (t, T) Etls: expUT).exp2rr (t) Uo (T))*Et{5: exp(-*T)-exp £r (r) Ui (T)}=0 T*t Tst Tst ?*t Thus, Fi(t,T> = Et expi^rw-aj.ui (T)} /Et exp{Z(m-)-*)-U0 During the process deriving equations (1) and (9) by using a simple call or put option, or continuously rewritten options, it is obvious that the difference between futures and forward 28 contracts is analogous to the difference between "rolling over" and "going long" strategies in bond investment respectively. 7 Ihis is another intuitive reason why forward contracts can be described as a speculation only on the future spot price while futures contracts are described as a speculation not only on the future spot price, but also on the number of units that will be paid for the contract at the maturity date. Ihis difference, in turn, results in the forward contract being easier hedging instrument than the futures contract, or this is why futures contracts need more information than forward contracts and thus causing the former costly instrument tor hedging. However, it is easy to coustruct a futures contract that is equivalent to a forward contract.0 If a person who is engaged in a futures contract is paid at the end of each day s the present value that the difference between the futures prices, F(s,l) F(s-1,T), would have if it were paid at the aaturity date, this type of futures contract is equivalent to forward contracts for the following reasons. It was already confirmed that the portfolio of a long position in a call and a short position in a put is equivalent to a forward contract from the arbitrage argument. let P* and C* be the put 7 See Cox, Ingersoil and boss (1981). 0 This is called "quasi-futures contract" in Cox, Ingersoil and Boss (1981). However, 1 have come up with the same conclusion from a quite different intuition. Also, the interpretative value in economic sense of this type of futures contracts was provided by Professor J.H. McCulloch. 29 and call price respectively with the same aaturity 7 and the saae exercise price X*. Then it can be easily shown (for example, fro* the Black-Scholes option pricing formula) that d P*/d X* = dC*/ dX* ♦ exp(-rl) thus, dC*/ dX* - dP*/ dx* = d(C*-P*)/ dX* = -exp (-rl) d (value of a forward contract) /d X* = -exp(-rT) (12) Equation (12) implxes that the change in the value of the portfolio of a long and a short position in a call and put respectively when the exercise price changes by $1 is the ainus present value of $1 payable at the aaturity date T. however, the exercise price in the call and put options used in the futures contract was equal to the futures price. Therefore, when a person in a futures contract pays or is paid the present value that the changes in the futures prices would have if it were paid at time T rather than the full amount, it is obvious that this futures contract is equivalent to the forward contract. This kind of futures contracts, in fact, aakes more economic sense taking it into consideration that the magnitude of the marginal utility of the profit (i.e., favorable price changes) is usually less than that of the marginal utility of the loss(i.e, unfavorable price changes) assuming risk averse investors. In other words, in ordinary futures contracts, if futures prices 30 increase, it tends to be overpaid by the seller and if they decrease, it tends to be overcharged by the buyer in the daily settling up. Besides, the relation between the expected spot price and the futures price can be easily derived from eguation (11) as9 Fi (t, 1) = Et Pi (T) + C0v{Pi(T) ,U0(l)exp2Tr ( t ) } /EtiU0(l)exp^r (t)> r*t Trt (13) It is obvious now that the relation between the futures price and the expected spot price depends on the covariance between the spot price and the marginal utility of a numeraire good multiplied T-l by exp 2^ r (t) at the maturity date. It depends no longer on the x~t simple covariance between the spot price and the marginal utility of a numeraire good due to the possibility of stochastic one-period interest rates. 9 Fi (t,T) = Et{ Ui(T)-exp^r(T)) /Et^UO(T) ■ exp2^r(7)l since oc is deterministic = Et I Pi (T) • U0 (T) expr'r (T)}/Et{ U0 (T) exp£'r(r)} from eguation (2) Xmt =£cov(pi (T) ,U0 (T) exp^rn)! ♦ Et Pi(T). E t U0 (T) expz'r = Et Pi (T) ♦COv{pi (T) ,00 (1) expj’r (r)l/Etl U0 (T)exp (T)) Tst T»t which is the eguation (13). This eguation (13) is guite similar to the models of Bichard and Sundaresan(1981) obtained from a guite different approach. 31 It is important to note that the oarket efficiency concept does not say anything about equations (8) and (13). In ether words, the guestion whether the futures and forward prices are greater than, or equal to, or less than the expected spot price is irrelevent to the question whether or not those markets, forward and futures, are efficient in processing information. The results expressed in equations (8) and (13) are quite general and depart from the existing literature in that they were developed by the knowledge of the number of units of commodities, obtained through an arbitrage argument, that will be paid for a forward or futures contract at the maturity date. The number of units of commodities in the forward contracts is the gross return obtained through an investment in a discount bond maturing at tiae T, while the number in the futures contract is the gross return obtained through continuous reinvestments in one-period bond up to the maturity date. In a sense, the fact that we can express the payoff of the forward or the futures contracts in terms of the number of units of commodities is a fairly important result, which gives us a good deal to go on and from which we can deduce some interesting and positive implications. The desirability of the procedure employed in this paper is attributed also to the following; Pirst, the implications of the models developed so far are strong theoretically since they are based on dominance or arbitrage arguments. 32 Second, since changes in relative prices can be implicitly incorporated into the pricing equations, changes in investor's consumption-investaent opportunity sets due to the changes in relative prices are no longer relevant. Thus, the coaplicated procedure used by Grauer and Litzenberger (1979) to derive the futures contract pricing equation incorporating the changes in relative prices under different inflation rates can be avoided easily; note, however, that they draw no distinction between forward contracts and futures contracts and that the futures contract pricing equation in their paper is in fact the forward contract pricing equation. In this sense, this paper shows also good contrasts to the literatures based on the conventional CAPA. One of the critical assumptions underlying the CAPA is that the relative prices are constant over time. Mote however that this simple assumption can be easily violated in case of price speculation even in a single period. From this perspective, Scboles (1981) hinted that short or long positions in forward or futures contracts might be profitable because of changes in relative prices even though his basic argument was based on the CAPH framework. On the other hand, because of the assumption of fixed relative prices in the CAPA, if the absolute price of an asset does not fluctuate over tiae, it is regarded as a riskless asset and thus fails to take into account the possibility that utility itself is fluctuating while the absolute price remains fixed but relative 33 prices changes. In other words, the CAPH is not appropriate, in general, for explaining the behavior of speculators in forward and futures markets. Third, in a purely theoretical sense, futures or forward prices can be stated as the sum of the current spot price plus the marginal opportunity costs and the marginal carrying costs from the no- arbitrage condition in eguilibrium (see Scholes (1981)). However, we are not concerned with changes in storage and opportunity costs either in absolute or relative prices, and the storage induced changes in the mean and variance of spot prices and futures prices and forward prices, since all of those costs are incorporated implicitly into the number of units of commodities that can be paid for a contract at the maturity date. Fourth, the procedure employed in this paper is plausible also in an empirical sense since any commodity rates of interest can be easily transformed into money rates of interest as can be seen in eguation (1) and (9). Chapter III RELATIONS AHONG FOBBABD, FOTOfiES AND EIPECTED PRICES 3-1 EXPLICIT FOBH PRICING j?J2fllUQJS In order to derive empirically testable hypotheses and their implications, we assume that the joint distribution of n + 1 commodities is multivariate lognormal, which implies that the joint distribution of any two commodities is Livariate lognormal; in fact, we need means and variances only fcr Xi and Xo and their covariances assuming that a buyer of forward or futures contract on Xi pays for it in terms of Xo. If Xi is lognormally distributed, then log Xi is normally distributed, and the density of the lognormal distribution is given by * (Xi.-yU.ff*) = (1/xi VIT5-)exp{-1/26l(logXi -11 f) where E(Xi) = exp (U +1/2<5A) ,and Var (Xi) = exp(2/U *26) - exp E(logXi) = XX and Var(logXi) = 6 * (1 In addition to this lognormality assumption of commodity distribution, we assume that the consumption-investment decision of each individual on n+1 commodities is based on a power utility - 34 - 35 function such as (k/1~tf)x'~9 , where the parameter showing the degree of risk aversion is positive and characterized by decreasing absolute risk aversion and constant relative risk aversion.10 Note that the power utility function has desirable properties in terms of providing a reasonacle description of indxvxdual investor behavior. For example, the implication that the commodities in question are normal goods follows directly from the properties of decreasing absolute risk aversion and constant relative risk aversion. Thxs implication of the power utility function contrasts well to the one based on other utility functions such as a quadratic utility function which xmplies that every commodities are inferior goods due to its undesirable properties of the increasing absolute risk aversion and the increasing relative risk aversion (for more desirable properties of the power utility function, see Butinstein (1974), Grauer(1978) and Kraus and litzenberger (1975). »o If U (X) = (k/1-*)x'“9 , U» - X'* , U« =-k/ff X*4“ Thus, the absolute risk aversion, «A (X) = -D'VU' = -k^ /X'* = k/£ X"' Ba'(X) = -ky* X* < 0 ,which implies the decreasing absolute risk aversion. Also, the relative risk aversion, Rx (X) = RA (X) X = k/s X* X = kz BR'(X) = 0 ,which implies the constant relative risk aversion. 3b Notation used in tbis section is as follows. X 6 i (t ) : variance of log Xi(T) at time T - So (t ) : variance of log Xo (T) at timer- 6r (r); variance of f: r(T). r=t Pio : correlation between log Xi<7) and log lo(T) T-l Pir : correlation between log Ui(r) and r(T)- & for : correlation between log Uo(T) and r (T) - tst 0(T) = log Ui (7) 1(7) = log UO (T) ^ T-l T-l 0'(T) = log Ui (T) expjl r (t )= log Ui (7) ♦ r (X) Tst £ 1(7) = log Uo (*r) exp^ r (7 ) = log Uo (T) ♦ r (7) Tst 7«tit Z(7) = 0 (T) - $ (t) = 0 (T) “ i'(t) • From the assumption of power utility function, Utlxi(T) ,Xo(T)) = (1/ (1-a) ) ?i(T)-A ♦ (1/(1-b)) jTo {T)--— (15) where positive a and b show the degree of risk aversion on commodity Xi and XO repectively.11 Ui (T) - Xi(lfa , Uo (T) = Xo(T fb 0 (T) = log Ui (T) - -a logXi (T) (T) = log Uo(T) * -b logXo(T) i-tf »» The general power utility function is given as (Ki/1-y ) Xi In this paper, I assume Ki egual to 1 for simplicity with no loss of generality for comparison between forward and futures prices and expected spot prices. 37 Z*(T) = 0(1) -j£ (T) = log t Gi(T)/UQ(l)} = logPi (1) EtZ(T) = Et0(T) - Etil(T) Var 0{T) = Varjlog Ui (I)} = a' <5^ Var |§ (T) = Var[log Uo (T)) = J>* ■ tf0*(T) Var 0'(I) = Var Jlog Ui(l) ♦ lr(7)| = ax- (T) ♦ 6*c (T) ♦ 2 fir a 6i(T)6r(T) Var ^ (T) = Var{logUo (T) ♦ ’£fr(T)J ret = bx• From eguations (14) and (16), fi(t,T) = EtOi (T)/EtU0 (T) =exp J Et 0 (T) ♦ 1/2a4. =exp{ EtZ (X) ♦ 1/2(a*^i(T) - b‘ (fb(T)))------(17) Fi (t,T) = EtJ Ui (T) expEr(T)) /Et{ ITo (1) expSrft)) T:t =exp{E 0'(l)*V2(a‘ 6i(T) + fir (1) *2 P,r a 6r(T) 5r (1))) / e x p j E i ’(T) ♦V2(b1 =exp{ EtZ(T) ♦ 1/2 ((a*6^T) ♦2<$‘r(ftra (Si (T)-p.rb 6 o (T)) -b*6o (T))) (18) Also, EtPi (T) = Et( Ui (T)/00 (1)| =Et Jexp 0(T)/exp^(T))=Et(exp (0(T)-5(TJ > Var J logPi (T )} = Var|0(T) - B (T)} = ax Thus, from (14), (16) and (19) EtPi (T) = exp[EtZ(T) ♦sr{a‘-<5 r (T) ♦^ - 6 0 (1) - 2 (io at (20) Equations (17), (18) and (20) show the explicit fora pricing models for forward contract, futures contract and the expected future spot price under the given assumptions. The most important implication from eguation (17), (18) and (20)is that we can not only specify the causal relations among those three prices, hut also derive testable hypotheses. In other words, by examining the variables in the eguations, we can investigate under what conditions a "Normal backwardation" or a "Normal contango" process of forward or futures prices is possible aud under what conditions the forward contract price is eguivalent to the futures contract price. 3.2 RELATION BETWEEN FORWARD AID F0I0BES PRICES From eguation (17) and (18), log Fi (t,T) - logfi(t,T) =6rlfir a6i(T) -for brfo(l)} (21) Then, the following implications follow directly from eguation (21). 1) if In other words, if the one-period interest rate is nonstochastic, a futures contract is equivalent to a forward contract regardless of the degree of risk aversion and the variance of each coanodity. Ihis is intuitively plausible in that a nonstochastic interest rate implies that the hedge ratio for both futures and forward contract is given at time t when the contracts are open, and thus the contracts should te equivalent under rational expectations. However, note that zero variance of the one period interest rate is sufficient but not necessary for the equivalence of forward and futures contracts. 2) If the variance of the interest rate is non-zero, the question of whether Fi(t,T) is greater than or equal to or less than fi(t,T) depends on the magnitude of fir a6i (T) and ?or btfo(T). Hereafter, in this paper, the numeraire good as assumed to be a one- period riskfree discount bond. T-i t-1 Then, 50(1) =Sr(T); note that Var XT r CO = 21 Var i£ T=t t*t interest rate at period "t is independent of that at period t ♦!. Also, noting that the bond price is negatively correlated with the interest rate, the following causal relations can be derived; If ab f io 6"i (1) d o (I) = b* , then Fi (t,T) | fi (t,T) 40 Additionally, we assume, for simplicity, the logarithmic utility function, which is a limiting case of power utility functions, so that the parameters of the degree of risk aversion on each commodity are equal to 1. Then, the above relation can be simplified as follows; If fio6i(T) 6 0 (T) = 60 (T) , then Fi(t,T) | fi(t,T)------(22) This implies that if the covariance between the price of the commodity i and the price of one-period riskfree discount bond is less than the variance of the price of the bond, the futures price is greater than the forward price; Cox, Ingersoil and Boss(1981) showed the similar relation between the forward price and the futures price in terms of covariances with bond prices rather than in terms of the covariance between commodity spot prices and the bond prices. Generally, if interest rates go up, it becomes more costly for speculators to buy commodities and for firms to build up inventories, thus increasing commodity prices. Considering the negative correlation between interest rates and the bond prices, the covariance between bond prices and storable commodity prices tends to be negative, thus the futures price tends to be greater than the forward price. On the other hand, financial futures such as Treasury bills, are expected to have a high correlation with the one-period bond prices, so that futures prices for treasury bills tend to be lower than their forward prices. 41 It is important to note that if the commodity i serves as a hedging instrument against changes in one-period bond prices, futures price is expected to be greater than forward price. Ihis result is independent of whether the markets, forward and futures, are efficient or not; the difference between forward and futures contracts depends only on the difference in risk level that is summarized in relation (22). The choice of the one-period riskfree discount bond as a numeraire good in this paper is not coincidental for the following reasons. First, as J. B. [licks noted, as long as the terms ou which commodities can be converted into money are given, we can draw up a determinate indifference curve between any two commodities, for example, commodity Xi and purchasing power of money or any commodity like a discount bond. In micro economic theory, consumer chooses between spending his money on any commodity or keeping it available for expenditure on other commodities. However, the commodities to be bought and sold need not be physical goods. In other words, if once they are objects of desire that can be arranged in an order of preference, any commodity like money or a one-period riskfree discount bond can be a candidate for the numeraire good. Secondly, by choosing the one-period discount bond as numeraire good, we can easily investigate how differently changes in marginal opportunity costs and marginal carrying costs are 42 reflected in forward and futures prices and thus their risks; it is interesting to note that the difference between the futures or forward price and the spot price is equivalent to the sua of marginal opportunity cost and Marginal carrying cost. Ihus, forwrd contracts can be distinguished from futures contracts by the correlation between the sum of the unexpected changes in the Marginal opportunity cost and the unexpected changes in the marginal carrying cost ,and the changes in interest rate. Some recent articles provides casual empirical support for the sensitxvity of contracts price to changes in interest rate; for example futures prices recently have tended to move in response to any signal of the direction of the economy(High interest rate discourage commodity speculation and inventory buildiug, attracting capital to the money markets instead) (See the Hall Street Journal, 9/29/81) Third, the choree of a discount bond is consistent with an arbitrage argument in a complete market. Suppose that there is a futures contract in every state of the world. In the other words, a futures contract is an asset with a payoff in the next period that is equal to the state price that is uncertain, assuming the existence of a futures contract on each state that expires at every instaut and another created that matures in the next instant. Then it is well known that a set of futures contract plus a risk free one-period discount bond can achieve the complete market in Arrow and Oebrew sense. In the light of the above reasoning, the choice of the one-period discount bond is believed to be a reasonable choice as a numeraire good. 43 3.3 THE ISSUE OF MOBHAL BACKMABDAT101 OB JS1SA& £SiliISQ By comparing equation (17) and (18) vitb eguation (20), we can specify explicitly under what conditions, the forward or futures price is an unbiased estimate of the expected future spot price and when they are supposed to follow the "Normal Backwardation" or "Normal Contango" process. Substracting eguation (20) from (17) and (18) respectively after taking logarithm log f i (t,T) - log EtPi(T) = b6o (T)t fio a tfi(T)- b log Fi(t,T) - log EtPi(I) = -^*6(1) ♦ 6 r {fir a£i(T) -Por b6o (T)}+fio ab£i (T) (To (T) = 6o|a fio 6*i(l) - b tf'o(T)} (b - 1) ------(24) assuming the one-period discount bond as a numeraire good;note that if we substract equation (23) from equation (24), it turns out to be equal to equation (21). From eguatious (23) and (24), following implications can be derived immediately. / 1) First, if the interest rate is nonstochastic and thus £o(I)=0, then fi(t,T) = Fi(t,T) = EtPi(T); both forward and futures contract prices are unbiased estimates of future expected spot prices. This has intuitive explanation because of the reasons 44 stated in section 3.1. Note however that this is not a necessary condition for the equivalence of forward and futures contract, nor are they unbiased estimates of the expected spot price. 2) Second, assuming that b > 1 ,which is the case of the generalized negative power utility function, a fairly striking result follows from the equations (21), (23) and (24); the comparison of futures prices or forward prices with expected spot prices is equivalent to that of forward prices with futures prices. In other words, the simultaneous relations among those three prices hold as follows: If ab fio 6i(1) 6o(?) | b*-6o(T) , then EP (T) | F (t,T) | f(t,T) ------(25) In words, it the covariance between the price of commodity i and the discount bond price is less than the variance of the bond price, both prices, futures aud forward prices, will follow a "Normal Backwardation" process and if the former is greater than the latter, both prices will follow a "Normal Contango" process. Concerning the possibility of hedging instrument of the contracts: If commodity i and the one-period discount bond are negatively correlated, so that the commodity is a good candidate for a hedging instrument against changes in the price of the one-period discount bond, both futures and forward price are downward biased U5 ■easure of the expected spot price. "Moraal Backwardation" is a natural deduction for the commodity. It is very important to notice that "Ncrxal Backwardation" is not inconsistent with market efficiency or with general aarket clearing coudition. It depends only on the risk factors that are measured by variance and covariance with a numeraire good, one-period discount bond in this paper. In other words, the fact that futures and forward contracts are sold at prices lover than the expected spot price has nothing to do with aarket disequilibrium or market inefficiency. In addition to this, it is interesting to note that if the utility function is logarithmic and thus a=b=1, then the futures price is always an unbiased measure of the future expected spot price, regardless of the correlation between the commodity and the numeraire, one period discount bond. So far we have examined forward contract prices, futures contract prices and expected spot prices and their relations with each other in a general equilitriua framework. First, we formulated the implicit pricing eguations of the forward and futures contracts through the optimization of the expected utility function and the market clearing condition. In turn, on the basis of these implicit pricing formula, the explicit pricing equation for those two contracts have been developed by specifying the underlying process of coaaodities and the utility function. In 46 the process, the relations among those prices have teen investigated theoretically in conjunction with the issue of "Normal Backwardation" or "Noraal Contango" of forward or futures prices. In particular, it was clearly confirmed that "iioraal Backwardation" or "Normal contango" of forward and futures contract price is not inconsistent at all with aarket efficiency or market equilibrium. In the absence of empirical investigation, the practicality of the models developed in this paper is indeternrnate or at least heuristic in terms of explaining the behaviors of forward and futures prices. The following chapter wrll add empirical support and investigate departures, if any, from the models developed. Chapter IV EHPIBICAL TEST 4-1 M l h 4.1-1 description of data Forward Prices Data Forward price data is very limited since forward contracts have been traded only on several underlying objects only on one exchange, the American Board of Trade in Mew York. The forward contracts offered and traded on the Aaerican Board of Trade include six basic commodities, two financial securities, five foreign currencies and two foreign currency "Backet" contracts: Gold Bullion, Silver Bullion, Silver Coins, Platinua, Copper, Plywood, CDs, 3-month Treasury bills, Swiss francs, Geraan Harks, Japanese yen, British pounds, Canadian dollars, FI-5 Basket, and FX-5-5 Basket programs.12 12 "FX-5Basket" program is a foreign currency forward contract, in which five aajor currencies are combined iuto a homogenized standard unit comprising 10,000 Swiss francs, 10,000 German marks, 1,000 British pounds, 2,500 Canadian dollars and 1,000,000 Japanese yen. Trading action is in the form of establishing five long commitments that are tied to five specific short positions, all initiated and liquidated simultaneously. The "FI-5.5 Basket" program is essentially identical in function and scope to the "FI-5 Basket" program. - 47 - However, this paper will investigate six tasic commodities (Gold,Silver,Silver Coin, Platinum, Copper and Plywood), four foreign currencies (Swiss Franc, Gernan Hark, British Pound and Japanese Yen), and one financial security(Treasury bills), since none of the data on Canadian Dallars and little on CDs are publicly available. The tines to maturity of the forward contracts on the six commodities and the four foreign currencies are strictly standardized with one, two, three, six and twelve months, while the forward contracts on Treasury-bills are standardized with one, two, three, and six months. However, even though the American Board of Trade was founded in 1969, it began to trade the forward contracts only from July, 1977. Furthermore, until November, 1979, the forward price data are restricted to three commodities, Gold, Silver and Silver coins with maturities of three, six and twelve months. These data are available in the Journal of Commerce only in terms of the closing prices. Since November, 1979, the Hall Street Journal has provided daily closing prices for forward contracts on the six commodities (Gold, Silver, Silver coins. Copper, Platinum and Plywood) and the four foreign currencies (Swiss Francs, German Harks, British Pounds and Japanese Yen) with maturities of one, two, three, six and twelve months. Additionally, the forward closing prices for T-bills with maturities of one, two, three and but with trades conducted in one-half size of the NFX-5 Basket". 99 six months ace listed each day in the Hall Street Journal.13 Another available data source is the National Spot flarket HEEKLK PRICE BULLETIN, a publication of the Journal of Investment Finance, Inc: This is a weekly summary of forward closing prices that is published each Friday. Futures Price Data In contrast to forward prices, futures price data are available over a longer period. However, the theoretical models in the previous chapters basically assume the simultaneous existence of both forward and futures markets. Thus, it would be reasonable to observe prices of futures and forward contracts with the same commodities and the same date of maturities for the same observation period. As we discussed in the introduction, futures contracts are traded with specific maturity dates (typically in terms of maturity month) while forward contracts are traded with specific times to maturity. Especially, the seller who is in the short position of the futures contracts is typically permitted to choose any delivery date within the specific maturity month. Therefore, it is not easy to make the specific tine to maturity in the forward contract coincide with the maturity date in the futures contract. 13 These data on the Journal of Commerce and the Hall street Journal are available in the microfilm of the Ohio State University library. 50 To circumvent this problem, the data observed on two aaturity dates, the first trading date and the fifteenth calandar date of each nonth of the forward contracts will he used for comparing forward contracts with the corresponding futures contracts. For example, on the first trading date and the fifteenth calendar date of January, one-month forward contract prices are coapared with February futures contract prices, and two-month forward prices with (larch future's prices, and three-month forward prices with April futures prices and so on. Treating the price of a different maturity of a commodity as the price of a different coaacdity, each forward contract has a total of five price pairs with a futures contract since each forward contract is standardized with one, two, three, six and twelve month maturities. The data used in the tests include closing prices of futures and forward contracts between July, 1977 and December, 1981. All of these observations are obtained from the Hall Street Journal and the Journal of Commerce. The details of futures contracts in terms of the unit and the exchanges on which they are traded are given in the appendix A. v One-dav interest rate This paper will use overnight federal fund rates obtained from "Federal Deserve Statistical fielease" provided by the Federal Deserve Bank of New fork. 51 Monthly interest rates One, two, three, six and twelve month 1-hill rates Mill le used for corresponding monthly interest rates. All of these data were obtained from "An Analytical fiecord of fields and Iield Spreads", published by "Solomon Brothers", which is a member of the New lock Stock Exchange, Inc.1* 4.1.2 Limitation of data It was already mentioned in the previous section that there is a problem of ambiguous maturity dates of futures. There are several other factors that might affect observing true aarket equilibrium prices even in trivial ways. First, if the trading is active, the closing prices would reflect the market prices at the last few moments of trading. However, daring a day of lrttle or no trading in an extreme case, the Clearing Corporation of the exchange should estimate the approximate market price. Furthermore, no trading volume or open interest is available in forward contracts. Also, even though the trading is active, if there is a range of closing price, a settlement price should be determined by the Clearing Corporation, usually the midpoint of the range. This might cause a problem in observing true market equilibrium prices. Two month T-bill rates that have been obtained from the Hall Street Journal were generously provided by Professor J.H. McCulloch. 52 Secondly, in order to compare futures prices with forward prices, they should be the market prices that are observed at the same point of time. However, as we can see in Appendix fi, the trading time periods during a day of futures contracts are different from those of forward contracts. Then, it is unlikely that the reported closing prices of Loth contracts reflect the same information because of the time discrepancy. Third, the delivery process varying from one exchange to another, or from one commodity to another night have a effect ou the price at which the contract trades. The market trades with the knowledge that delivery can occur at contract maturity even though the traders may not actually make or take delivery of the contracts. Thus, depending upon the delivery specifications for a specific contract, the price may act differently from another contract. All of these problems imply that the relationship between futures and forward prices is measured with error, even though they would not cause a systematic bias ru the relationship. However, we can get around this measurement error problem by assuming that the errors are random, not moving systematically and satisfying the full ideal conditions for usual disturbances; it is well known that errors in the dependeut variable do not cause a serious problem except increasing the standard errors of estimations of the coefficients in the regression. 53 4.2 METHODOLOGIES ABD BESPITS OF IHE £S££11£11 I£ST 4.2.1 Tests or the relation betvgen futures and forward jogiges Methodology 2 The model developed in the previous chapter connects that the difference between futures prices and forward prices depends on the comparative magnitude of the covariance of commodity prices with one-period bond prices relative to the variance of the one-period bond prices themselves. In other words, if the covariance between the price of the commodity and the price of one-period riskfree discount boud is less thau the variance of the price of the bond so that the commodity provides a hedging investment against changes in the one-period bond price, the futures price of the commodity is greater thau its forward price. However, it is important to note that only ex-ante unanticipated portion of changes in prices would shift the consumption-investment opportunities of investors in an efficient market. Thus, the covariance and the variance should be interpreted as the covariance of the unexpected changes in commodity prices with the unexpected changes in one-period bond prices, and the variance of the unexpected changes in one-period bond prices respectively. The first testing method used in this section involves the following three steps; 5 < * 1. Tests on whether the covariance is greater or less than or equal to the variance for each commodity (5 kinds of maturities for 11 commodities each) 2. Classification of commodities into three groups based on the criterion provided by the step one. 3. Tests on whether differences between futures and forward prices are positive or negative, both on group basis and individual commodity basis. This last step involves testing the null hypothesis that the difference between futures and forward prices depends on the ratio of the covariance to the variance. In this paper, one, two, three, six and one year 1-bill prices are used as the corresponding period bond prices. After taking logarithms of the spot prices and their expected prices in order to be consistent with the assumption of the lognormal distribution of prices, we can compare the magtitude of the covariance and the variance, by computing the ratio COV{logP(T) - log £t(P(T)) , log (T-BILL (T)- log(Et (T-BILL(1))| VAB {log (T-BILL (T) ) - log Et (T-BILL (T)) J ,where P (T) and T-bill (T) are spot prices of commodities and T-bills respectively at time T, when their contracts, futures and forward, are matured, and Et(P(T)) and Et (T-bill (T)) are their expected prices as of time t today. Mote, however, that the above ratio is just the beta coefficient B1 of the independent variable of the following regression equation. 55 logP (T) - logE (P (T) ) = B0+ B 1 {log (1-till (T))-logE^T-till (1) )}♦ t (26) ,where the disturbance tern £ satisfies the full ideal conditions, i.e. independently, identically and noraally distributed with the ■ean zero and the constant variance 6*. The estimation of B1 from an QLS (Ordinary Least Sguares) regression is B1 = Cov (W,Z)/Var (Z) (27) where ll= logP(T) - logE (P (1)) Z= log (T-till(T))- logE(T-bill(T)) (Hereafter, H and Z will be used as defined here) Therefore, under the null hypotheses, HO, that B1=1, we can test whether the true coefficient E1 is significantly greater than or egual to, or less than one, using *T' statistics; if B1 is greater or less than one, the covariance of the unexpected changes in commodxty prices with the unexpected changes in T-till prices is greater or less than the variance of the unexpected changes in I-bills respectively. In the regression eguation (26), forward prices with one, two, three and six mouths maturities were used as expected prices for corresponding time periods, and the futures prices with twelve months maturities were used as the expected prices after one year since the T-bill forward contract with one year tines to maturity 56 was not available. The reason that forward prices rather than fatures prices with one, two, three and six months maturities were used was not only due to the effort of obtaining the observations of commodity contract prices and T-brll contract prices for the same period as much as possible but also due to no trading of one of commodity futures contract (Silver-coin futures contracts) since the end of 1980. As stated in the previous section, because of the ambiguity of the actual maturity dates of futures contract, two different maturity dates were considered for each futures contracts, the first trading dates and the fifteenth calendar dates or the closest dates to the fifteenth dates if there were no trading on those dates. Table 1 -Table 9 present the results of various tests involving futures and forward contracts which nature on the first trading dates. The results of the same tests involving the same contracts which mature on the fifteenth calendar dates are presented in Appendix C -Appendix K corresponding to Table 1- Table 9 respectively. However, the results based on the fifteenth calendar dates do not deviate significantly from those based on the first trading dates. Thus, examination of results of empirical tests will be restricted to the case of the first trading dates. Table 1 summarizes the results of the test whether B1 defined in eguation (27) is greater or less than, or egual to one as the first step of tests. TABLE 1 Tests on the ratio of covariance to variance logX-logEX= Bo + B 1 {log (T-bill) -logE (T-fcill)} Ho : B 1= 1 Gold Hat"(T) obs Bo To B 1 11 d B( 1 19 0.1616 2.06 -115218 -470118 < (0.0703) (0.6286) c 2 18 0.2368 1.95 -2.6525 -3.4107 < (0. ny 1) (1.0709) 3 17 0.2145 1.81 -3.2562 -3.4357 < (0. 1186) (1.2388) 6 14 -0.0410 -0.49 -4. 1391 -3. 1839 < (0.0830) (1.6141) 12 30 0-2723 1.77 -3.4984 -2.3879 < (0- 154 1) (1.8838) Silver 1 17 0. 1798 1.40 -1.6558 -2.4807 < (0.1285) (1.0706) 2 16 0- 3904 2.25 -4.0998 -3.2259 < (0. 1736) (1.5809) 3 15 0-5148 2.79 -6-5602 -3.9372 < (0. 1847) (1-9202) 6 12 0. 111 5 0.66 -8.9489 -3.1197 < (0- 1691) (3. 1891) 12 30 0.2239 0.90 -8.8865 -3.0334 < (0.2484) (3.2592) Silver Coin 1 19 0. 1791 1.92 -1.6961 -3.5919 < (0.0935) (0.7506) 2 18 0. 3121 2.53 -3.4031 -3.9711 < (0. 1233) (1.1088) 3 17 0.2879 2.06 -4.1312 -3.5157 < (0. 1397) (1.4595) » 6 14 -0.0106 © t © -5.0868 -2-0929 < (0. 1163) (2.2603) 12 30 0.2564 1.15 3.1550 0.3830 = (0.2233) (5.6260) Platinum 1 17 0. 1159 1.28 -1.0769 -2.7480 < (0.0907) (0-7558) 2 16 0. 1873 1.43 -2.1473 -2.6339 < (0- 1312) (1.1949) 3 15 0-2193 1.59 -3.3861 -3.0625 < (0. 1378) (1.4322) 6 12 -0.0677 -0.61 -3.8216 -2.3070 < (0. 1108) (2.0901) Table 1 (continued) 12 10 -0.6171 -3.66 -4. 1614 -2.1860 (0. 1684) (2.3590) Copper 1 19 0.0722 1.46 -0.7565 -4.4267 (0.0494) (0.3968) 2 18 0.0951 1.98 -1.2664 -5.2427 (0.0481) (0.4323) 3 17 0.1338 1.74 -2.0027 -3.7282 (0.0771) (0.8054) i 0 ui U3 6 14 -0.0206 1 -2. 1933 -4.69d 1 (0.0350) (0.6797) 12 10 -0.3277 -3.97 -2.0470 -4.7402 (0.0824) (0.6428) Plywood 1 19 0. 1331 2.92 -1.1924 -5.9853 (0.0456) (0.3663) 2 18 0. 1858 3.24 -1.9692 -5.7532 (0.0456) (0.3663) 3 17 0.2131 3.33 -2.7392 -5.5951 (0.0640) (0.6683) 6 14 0. 1190 2.65 -4.2647 -6.0271 (0.0449) (0.8735) 12 10 -0.1487 -1.59 -1.2634 -1.2104 (0.0938) (1.8699) S.F. (Swiss Frauc) ~1 " 19 -oTo 30 8 -0.68 0.3376 -1.8565 (0.0445) (0.3568) 2 18 0.0186 0.24 0-0337 -1.4057 (0.0764) (0.6874) 3 17 0.0533 0.60 -0.1312 -1.2118 (0.0894) (0.9335) 6 31 -0.0517 -0.91 1.5281 0.4395 (0.0568) (1.2016) 12 10 0- 1244 5.01 -0.2825 -0.8579 (0.0248) (1.4590) G. H. (German Hark) 1 19" -070253 -0.70 0.3326 -2.29C3 (0.0360) (0.2914) 2 18 0.0042 0-07 0.2576 -1.3236 (0.0624) (0.5609) 3 17 0.0428 0.59 0.1147 -1.1672 (0.0726) (0.7585) 0 CO 1 6 31 -0.0351 ( 1.5575 0.6275 (0.0417) (0.8884) 12 10 0- 1720 5.45 1.8076 0.9731 (0.0316) (0.8299) 59 lable 1 (continued) B.P. (British Pound) 1 19 -0.0877 -2.28 0.7749 -0.7301 S (0.0384) (0.3083) 2 16 -0. 1069 -1.90 1. 1730 0.3424 .S (0.0562) (0.5053) 3 17 -0. 1250 -2.04 1.7772 1.2147 (0.0613) (0.6398) 6 30 -0.0524 -1.12 0.6208 -0.3881 = (0.0467) (0.9770) 12 10 0.1014 1.17 -0.1976 -0.6910 - (0.0869) (1.7331) J.Y. (Japanese Yen) 1 19 -0.0426 -0.94 0.3640 -1.7530 =: (0.0452) (0.3628) 2 18 -0.0555 -0.97 0.5846 -0.8107 s s (0.0569) (0.5124) 3 17 -0.1030 -1.99 1.2802 0.5179 = (0.0518) (0.5410) 6 27 0.0055 0. 10 0. 1477 -0.7182 = (0.0564) (1.186e) 12 10 -0.0260 -0.40 -0.6219 -1.2369 = (0.0658) (1.3113) ^ X represents commodity prices. b The symbol •<• represents that B1 is significantly less than 1. The symbol *=’ represents that B1 is not significantly different from 1. The symbol ■>• represents that B1 is significantly greater than 1. c The figures in brackets show standard errors of estimations, d *t* statistics of BO and B1 are against 0 and 1 respectively. The beta coefficients of all physical commodities. Gold, Silver, 60 Silver Coin, Platinum, Copper and Plywood in every maturities, one, two, three, six and twelve months, turned out to he significantly less than one at 5% level as expected except only twelve months maturity Silver Coin and Plywood.15 In the light of the arguments of previous chapters, this implies that those commodities might provide good hedging instruments against the unexpected changes in l-hill prices, since the covariances (W,Z) are less than the Variances (Z), where h and Z are as defined in (27); it is also noticeable that all of the beta coefficients of those commodities regardless of their maturities are negative implying the negative correlations of the unexpected changes in those commodities with the unexpected changes in T-bills. On the other hand, all of the beta coefficients of non-physical commodities, the foreign exchange rates such as Swiss Franc, German Mark, British Pound and Japanese Yen turned out to be insignificantly different from one at 5* level except one-month German Bark. It is also interesting to note that none of commodities with all maturities has beta coefficient which is significantly greater than one, so that the covariance (H,Z) is significantly greater than the variance (Z). However, taking it into account that T-bill futures contract is itself a discount bond maturing at some time later than T, the contract maturity date, its covariance is expected to be greater than the variance; 15 The coefficient B1 of twelve months Platinum is less than and equal to one at 10% and 5% significance levels respectively. 61 the beta coefficient of this T-bill couldn't be conputed since the independent variable of the regression equation involved the T-bill itself. Based on the results of Table 1, Table 2 shows the classification of commodities of each maturities into two groups as the second step of tests, in terms of the magnitude of the difference of the covariance(H#Z) and variance (Z). In each maturity, the first group consists of commodities whose cov(W,Z) are less than Var(Z), while the second group with commodities whose cov(H,Z) are equal to Var(Z). for example, in one-month maturity case, the commodities belonging to the first group are Gold, Silver, Silver Coin, Platinum, Copper, Plywood and German Hark and the second group is composed of Swiss franc, British Pound and Japanese Yen. Concerning with the theory developed in the previous chapter, the difference of futures and forward prices of the first group should be positive and the one of the second group should be close to zero, if the model describes the futures and forward contract prices correctly. So, the next step is to test the null hypothesis that the true difference between the futures and forward price is zero. This test was done by examining the *t' statistics of the estimation of beta coefficient of the following eguation; Futures (T)- Forward(T) = B (1) (28) 62 TABLE 2 Groupings of coamodities I. One-month maturity Group 1; Cov < Var Gold, Silver, Silver Coins, Flatinum, Copper, Plywood, G.n. Group 2; Cov = Var S-F-, B. P. , J. Y. II. Two-month maturity Group 1; Cov< Var Gold, Silver, Silver Coins, Platinum, Copper, Plywood Group 2; Cov = Var B.P., S.F., G.M., J.Y. III. Three-month maturity Group 1; Cov < Var Gold, Silver, Silver Coins, Platinum, Copper, Plywood Group 2; Cov = Var I). Pi I S.F., G.Ha, J. Y. IV. Six-month maturity Group 1; Cov < Var Gold, Silver, Silver Coins, Platinum, Copper, Plywood Group 2; Cov = Var B.P., S.F., G.H., J.Y. V. Twelve-month maturity Group 1; Cov < Var Gold, Silver, Platinum, Copper Group 2; Cov = Var Silver Coin, Plywood B.P., J.Y., S.F., G-fl. 63 where futures (1) and forward (1) are futures and forward contract prices with tines to naturity 1, and E is the coefficient of the independent variable, which is one. The results of the tests on the differences tetween futures prices and forward prices of the two groups in each naturity are presented in table 3. As expected, the magnitudes of the differences of the second group are not significantly different from zero throughout all maturities. Also, the differences between the futures and forward prices of the first group are significantly greater than zero at 5% level except the two-month maturity commodities. The difference between the futures and forward prices of commodities with two-month maturities in the first group turned out to be 1.1292 which is not significantly different from zero. Overall, these significant results provide strong evidence that the futures aud forward prices can be described by the theoretical models developed in the previous chapters. In addition to these results, involving two groups based on the ratio of Cov(W,Z) to Var(Z), we can get another results supporting the model by examining the beta coefficients of the regression eguation (28) in individual commodities. The results of these tests on individual commodities are reported on table 4. In Gold, Silver, Platinum, Copper, the differences between the futures and forward prices are entirely consistent with the hypothesis at 5X significant level that they are positive or 64 TABLE 3 Difference between futures and forward prices(groups) Futures (T) - Forward (T) = B(1) ;roups 1 2 3 6 J 2 & Group 1 est 3.9310 0.1292 13.1414 10.1129 4.0508 Cov = > > > t obs 98 65 182 180 134 Group2 est 0.000937 -0.00354 0.000306 -0.00209 -0.4373 Cov-Var S.E 0.00434 0.00 466 0.00523 0.00359 2.6696 t 0.22 0.76 0.06 -0.58 -0. 16 HO obs 38 22 33 63 51 a The first row shows five different maturities. "E The symbol '<* represents that B is significantly less than 0. The symbol *=* represents that B xs not srgnificantly different from 0. The symbol *>' represents that B is significantly greater than 0. c Computed against 0. TABLE 4 Differencei between futures and forward1 prices(coaaodii Futures - Forward = B(1) : (individual commodities) Mat. J 2 3 6 J2 Gold est 1.4548 1.3000 3.6550 3.8243 3.0632 S. E 0.6716 0.6281 0.b897 0.7615 0.9301 t 2-17 2.07 5-30 5.02 3.29 Ho= 0 b >0 >0 a. >0 >0 >0 Cov/Var < < < < < c obs 21 17 50 53 53 Silver est 3.7550 6.5125 10.5380 7.6280 6.8082 S-E 1.0306 2. 1313 1.7844 1.5003 1.9257 t 2.05 3.06 5.91 5.08 3.54 Ho >0 a. >0 >0 >0 >0 Cov/Var < < < < < obs 20 16 50 50 49 Silver est 0.8433 -0.6200 0.6542 0.5155 -0.0125 Coir. S.E 0.1822 0.0800 0.2095 0.2593 0.0705 t 4.63 -7.75 3.12 1.99 -0. 18 Ho >0 <0 >0 >0 a = 0 Cov/Var < < < < = obs 3 2 24 22 20 Platinum est 2.4125 1.4000 4.5133 5.2179 4/6570 S-E 1.2630 0.6083 1.2230 1.3066 2.0639 t 1.91 2. 30 3.69 3.99 2.27 Ho >0 a >0 a >0 >0 >0 Cov/Var < < < < < obs 8 5 15 14 10 Copper est 0.1933 0.2485 0-6295 0.6333 0.4705 S.E 0.0766 0.1074 0. 1488 0.1366 0. 1363 t 2.53 2.31 4.23 4.64 3.45 Ho >0 >0 >0 >0 >0 Cov/Var < < < < < obs 21 20 21 21 22 Plywood est 0.6591 -0.1900 2.0114 1.4525 0.2300 S.E 0.4586 0.2472 0.4385 0.2780 0.3714 t 1.44 -0.77 4.59 5.22 0.62 Ho -=0 = 0 >0 >0 = 0 Cov/Var < < < < •= obs 11 5 22 20 10 S. F. est -0.0086 0.2257 0.0966 0.0624 -0.0769 S.E 0.0694 0.0255 0.0664 0.0351 0. 1497 t -0. 12 8.84 1.45 1.78 -0-51 G.H. G.H. .. s -.24 0.0614 -0.0264 est J.Y. 0-1333 0.0470 est B.P. Tesmo in The thissymbol thesamerowisas one in seventh colume c Itrlcu infcn a 10X level atSignificant h ycl ■Thesymcol >• represents that B issignificantly greater fTbe 1.ofTable •='The representssymbolthat Bnot is siguificantly than 0. fromdifferent 0. h ybl '<•The represents that Bsymbol is than significantly 0. less o/a = = = Cov/Var = < Cov/Var o/a = = = Cov/Var o/a = = = Cov/Var b 1 3 10 obs o 0 >0 =0 Ho b 1 5 14 obs 0.0303 0.0298 S.E b 1 7 14 obs b 1 7 14 obs . 001 0.0479 0.0411 S.E 0.0167 0.1130 S.E >0 =0 Ho o 0 =0 =0 Ho o 0 =0 =0 Ho s -.07 0.0500 -0.0057 est -.4 1.28 -0.64 t 8.00 0.42 t -.9 1.65 -0.19 t al 4 (continued) 4 Table .31 .31 0.0321 0.0341 0.0351 .9 0.94 4.59 0.0291 0.0190 0.0273 0.0871 -0.90 1.01 0.26 .98 .45 -0.1400 0.0425 0.0978 .50 .43 -0.9500 0.1413 0.0500 .87 .40 1.05 0.1400 0.1887 .8 .5 -1.70 1.25 2.78 ■=0 = >0 15 7 1 2 15 6 8 16 9 >0 1 13 17 9 0 =0 =0 =0 =0 =0 =0 -0 66 negative depending upon whether Cov(W,Z) is less or greater than Var(Z) respectively.** In silver Coin, Swiss franc, Geraan hart, British Pound and Japanese Yen, they are consistent with the hypothesis with one exception in each commodity- for exaaple, in one-month German Hark, there is no significant difference between the two contract prices even though the Cov(H,Z) is significantly less than Var(Z).17 In Plywood case, the results are not conclusive relative to other commodities; two out of five different cases (five kinds of maturities) are not consistent with the hypothesis. Specifically, the differences of the contract prices, of one-month and two-month cases are not significantly different from zero although the Cov(H,Z) are significantly less than Var(Z) in both cases. Overall, thrrty-six and forty-one commodities out of forty-nine are consistent with the hypothesis developed earlier at 5% and 10* significance levels repectively, that the difference between the two different contract prices depends on the Cov(W,Z) and Var(Z).1® Moreover, interestingly enough, the number of *6 The differences of two-aonth Gold, one-nonth Silver, six-aonth Silver Coin, one- month, two-aonth Platinuas are significantly greater than zero at 10* level. *7 The differences of the three-month Geraan Hark are consistent with the hypothesis at 2% significance level. is Ten commodities have five kinds of maturities. Counting the different maturity of a commodity as a different coaaodity, the total number of commodities is 50. Actually, fifty prxce pairs are examined, five price pairs for ten commodities each. However, no observation is available in twelve-month Japanese Yen in terms of the difference between its futures and forward prices. 68 observations of those commodities which are not consistent with the hypothesis is less than ten in general. Substracting those commodities whoSe number of observations is less than ten, twenty-nine and thirty-two commodities out of thirty-four are consistent with the hypothesis at 5% and 10J4 significance levels repectively. Again these results strongly support the models describing futures and forward contract prices. Methodology 2 Another way to test the models developed in previous chapters is to use the analogy of the relation between futures and forward prices to term structure arguments. Eguation (5) and (11) can be rewritten as follows; fi (t,T) = E[exp(-(T-t)«i) .Ui (1)] /E exp{-(T-t)* . UO (1)} = B",£(exp{-(T-t)«) .Ui (T)}]-/B‘‘E£exp{-(T-t)*} .UO (T)] = BH E[exp{-(T-t)«} .Ui (1)]/E[exp(-(T-t)<4 -UO (T) .B~’] = B-'E exp(T-t)aj -Ui(t) /U0(t) = | Pi (t)/Ui (t)> B’'exp{- (T-t)w) .E Ui (1)------(29) ,using UO (t) = Ui(t)/Pi(t) On the other hand. Fi (t,T) = E Ui(T)exp zi|r (7)-o<) /E UO (T) expj^l r (T) - <*} = E Ui(*r)expgtr(T)-*l /UO (t) = I Pi (t) /Ui (t)) . E Ui (T) exp ZT|r (r)-oC) ---- — (30) T *t 69 By dividing equation (30) with equation (29), the ratio of futures prices to forward prices can he rewritten as; Fi(t,T) (Pi(t)/Ui(t)} E[ui(T). expjr{c(T)-«l] fi (t,T) {Pi(t)/Ui(T)} B"'(eKpJ-(l-t)oi})l Ui (T) exp{-(T-t)«<} • E{ Ui (T) expz'r(T)} . ______B~'exp{-(T-t)*}-E Ui (T) Cov{ Ui(T)» expr'r (?)} ♦ E Ui(l)- E expzVtt) = ______T s t ______T-.t B~*E Ui (T) E expzr (t ) Covl Ui (I) , expl'r (t )} ______T:t + T it B'1 B*'E Ul(T) In words, if the covariance between the Marginal utility of the commodity i and interest rates is zero, the ratio of futures prices to forward prices narrows down to the ratio of the expected return from "rolling over" investment in one-day bonds to the gross return from "going long" investment in T period bonds. But unfortunately, neither the ex-ante expected return from the rolling over strategy in bond investments nor the marginal utility of commodity can be observed. To get around this problem, the actual return that was observed at the maturity date T was used for the proxy for the expected return. Mote, also, the theoretical deduction that the differences between futures prices and forward prices depends on the covariance between the 70 unexpected changes in commodity prices and the unexpected changes in bond prices. Thus, utilizing the dependency, the following regression equation with a dummy variable can be constructed as a substitute tor equation (31). Fi (t,ij/fi(t,T) = B 1 ♦ 021E expz'r(T) / B‘‘}+ B3 D (32) k T--t where P = 1 if the covariance is significantly less than the variance as defined above. D = 0 if otherwise. If the models of previous chapters correctly describe futures and forward prices, B2 should not be significantly different from one and B3 should be different from zero in equation (32). Thus, in running the regression, we can set up the joint null hypothesis as follows; HO : 02 =1 and B3 =0 TaLle 5 presents the results on this test. As expected, the null hypothesis that 02 is close to one is accepted in all maturities except the one-month maturity at 5% significance level. Additionally, the hypothesis that 03-0 is rejected iu three, six and twelve maturity cases. These results provide additional evidence that futures and forward prices can be described by the model developed earlier. Note however that the test on B3 is a indirect test on the implication that the TABLE 5 Tests oa the analogy to tecs structure of interest rates ( T - l F/f = B1 ♦ B2{exp(£ rttyexp {T—t) U ) ♦ B3.D r*t HO: B2=1, B3=0 Haturity obs B1 B2 B3 1 150 est 10.9899 -9 .9629 ■0.00057 S.E 3.2863 3.2763 0.0093 t 3-39 -3.3961 •0.06 * Ho *1 = 0 96 est 0.9912 0.0532 0.0117 S.E 2.2809 2.2799 0.0137 t 0.91 -0 .9159 0.86 Ho = 1 = 0 211 est 0.1131 0.8898 0.0267 S.E 1.2738 1.2696 0.0119 t 0.09 -0.0911 2.25 HO =1 *0 220 est -0.2692 1.2969 0.0911 S.E 0.5290 0;5 152 0.0105 t -0.51 0.9792 3.89 Ho =1 40 12 193 est 0.9317 0.5998 0.0991 S.E 0.3211 0.3099 0.0189 t 1.39 -1 .9712 2.66 ho = 1 *0 * *t' statistics of B1 and B3 are against 0 and *t' of B2 is against 1. 72 difference of the contract prices depends on the covariance between commodity prices and interest rates, since the tine series of the covariance tern is not available. %.2.2 Tests on Normal backwardation or Normal contango process In deriving causal relations among the futures, forward and expected prices, it was connected in the previous chapter (see equation (25)) that the futures and forward contract prices follow the normal-backwardation process if Cov(H,Z) is less than Var(Z). If Cov (W,Z) is not significantly different from Var(Z), neither Normal Backwardation nor Normal Contango process is appropriate for describing the futures or forward prices and thus these contract prices are unbiased estimates of the expected spot prices. One strategy to test this issue, which is employed in this paper, is to use the returns as follows; if the Normal Backwardation is true underlying process of futures and forward prices, EP (T) -F (t , T) EP(T) -F (t-1,T) EP (T)-F (I-1,T) > > > ------F(t,T) F (t- 1, T) F(T-1,T) For example, assuming December 1st, 1982 as the maturity date, using one, two, three, six and twelve months times to maturity. 73 |EP (Dec/1982) —P (Dec/198 1, Dec/ 1982)|/F (Dec/1981,Dec/1982) > lEP(Dec/1982)—F(June/1981,Dec/1982)} /F (June/1982,Dec/1982) > {EP(Dec/1982)-F(Sep/1982,Dec/1982)) /F > jEP (Dec/1982)-F (Oct/1982,Dec/1982)} /F (Oct/1982,Dec/1982) > {EP (Dec/1982)-F (Nov/1982,Dec/1982)} /F(Nov/1982,Dec/1982) In other words, if the Normal Backwardation is the true process for futures prices, the return defined as |lP (I)-F (t,I)j /F(t,I) should be a function of times to aaturity. In this paper, the ez-post observed spot prices were eaployed as the expected spot price at each tine. Table 6 provides the results of tests on whether the returns of futures contracts defined as above are positive or not in conjunction with the criterion based on the ratio of Cov(H,Z) to Var(Z) . Except one commodity. Plywood, the return distributions of futures contracts strongly tend to be consistent with the basic hypothesis that they depend on their covariances with one-period bond prices(T-bill in thxs paper). Thirty-one coaaodities out of forty-nine are consistent with the hypothesis; if plywood is excluded, thirty-one out of firty-four coaaodities are consistent. These results add supports for the model. TABLE 6 Tests on futures returns Futures return; ( Spot-F)/F x 100 ,vhere F stands for futures prices nat. 1 U Gold Mean 1.7183 2-6673 7.3091 11.2895 34.0844 S. D 9.9766 15. 1810 19.9583 32.7156 54.1600 obs 48 36 47 47 41 T 1. 19 1.05 2.51 2.37 4.03 a_ Ho > > > t Silver Mean 1.5021 4.6289 8.6289 26.3712 52.0149 S.D 16.0581 30. 1168 39.2153 79.8181 121.1796 obs 48 43 47 44 39 T 0.65 1-01 1.51 2. 19 2.68 HO > > Silver Mean 10.1353 26. 1386 14.2014 40.8408 88.6825 Coin S.E 35. 7442 35.7098 25.7421 75.6812 115.8682 obs 13 12 24 22 20 T 1.02 2.54 2.70 2.53 3.42 Ho > > > > Pla - Mean -2.7427 2-7497 8.7860 14.5346 33.4428 tinum S. D 10.4907 21.3144 20.3147 32.5748 43.3807 obs 19 14 31 28 24 T -1. 14 0.48 2-41 2.36 3.78 HO > > > Copper Mean 2.3144 0. 8937 2.2677 6.8264 10.0351 S. D 9.7423 13.8908 15.1547 20.1526 28.4230 obs 46 3 7 43 42 37 T 1.61 0. 39 0.98 2.20 2. 15 Ho > > Ply- Mean -2.3121 -2.8359 -4.9918 -5.5802 -8.2999 wood S.D 5.8155 8-4108 8.9507 9.7088 9.6971 obs 11 6 2 4 20 21 T -1.32 -0.83 -2.73 -2.57 -3.92 Ho = < < < S.F. Mean -0.0071 -2. 3225 0.2193 -0.5714 12.2776 S.D 4.8733 7.4841 9. 1654 13.236? 12.2776 obs 23 17 24 31 20 T - 0.01 -1.28 0. 12 -0.24 -0.30 Ho G.M. Mean -0.4414 -1.5567 -1.0509 -2.6595 -2.1869 S.D 3.3313 3.9651 6.1288 9.3656 11. 1761 75 Table 6 (continued) obs 23 15 25 31 17 T -0.64 -1.52 -0.86 -1.58 -0.81 Ho B.P. Bean 0.2614 -0.3442 1.2592 2.9467 6.5500 S.D 2.8244 3.5062 6.2819 9.2760 12.5847 obs 19 13 23 30 15 T 0-40 -0.35 0.96 1.74 2.02 Ho = J.Y. Hean -0.3479 -1.5395 -0.3767 -0.4777 S.D 4.4665 6.2192 8.7806 12.0836 obs 23 17 23 27 T -0.37 -1.02 -0.21 -0.21 Ho = a Computed against 0. T The symbol '<• represents that the mean is significantly less than 0. The symbol * = * represents that the mean is not significantly different from 0. The symbol ■>• represents that the mean is significantly greater than 0. 76 More interesting and convincing results concerning the issue of Normal Backwardation were obtained by running the following regression, taking it into account that the return distribution of futures or forward contracts is a positive function of times to maturity if the Normal Backwardation is the true underlying process. Return = BO + B1 (Time) ------(33) where Time =1 if the return is for one month Time -2 if the return is for two months Time =3 if the return is for three months Time =6 if the return is for six months Tine -12 if the return is for twelve months If the Normal Backwardation is the true process and thus the return is a positive function of times to maturity, El should be positive significantly. So, by examining the coefficient of the regression equation (33), we can test the null hypothesis; HO ; B1 = 0 The results of this test involving futures contracts are presented on table 7. As expected in the light of the results of table 1 through table Copper, are significantly greater thao zero, and thus show that the returns of futures contracts are positive functions of times TABLE 7 Normal backwardation of futures prices Futures Return = Bo+ B1 Time. BO to B1 tl b Ho c Gold -3.2525 -1.01 2. 9646 5.65 > 0.1282 (219)a (3.2139) (0.5249) Silver — 4.5894 -0.67 4.6275 4.00 > 0.0681 (221) (6.8893) (1. 1565) Silver -0.2366 -0.02 7. 2348 3.91 > 0.1464 Coin (9 1) (12. 1992) (1.8519) Platinum -6.1696 -1.41 3.2580 4.80 > 0.1680 (116) (4.3771) (0.6792) Copper 0.6923 0.35 0.6852 2.09 > 0.0155 (205) (1.9699) (0.3274) Plywood -2.7879 -1.65 -0.4701 -1.95 = 0.0452 (82) (1.6864) (0.24 16) S.F. 1.0262 0.66 -0.4524 1.79 — 0.0407 (115) (1.5443) (0.2525) G.M. -0.8486 -0.74 -0.1645 -0.84 0.0065 (111) (1.1523) (0. 1951) B- P- -0.7025 -0.54 0.4041 1.86 = 0.0731 (100) (1-2890) (0.2173) J. Y. 1.8498 1.09 -0.8012 -1.94 = 0-0507 (92) (1.6916) (0.4111) £ The figure in the bracket of this colume is the number of observations, b Computed against 0. £ The symbol •<• represents that B1 is significantly less than The symbol ' = * represents that £1 is not significantly different from 0. The symbol *>* represents that El is significantly greater than 0. 78 to Maturity and in turn, supporting the "Noraal Backwardation " process. Only exception is Plywood again. On the other hand, the beta coefficients of foreign exchange rates, Swiss Franc, Geraan nark, British Pound and Japanese ten are not significantly different fron zero, iaplying that the futures contract prices of these foreign exchanges are unbiased estimates of the future expected spot prices regardless of tiaes to maturity. These results are once again consistent with the basic hypothesis and provide additional supports for the aodel of futures contracts developed earlier. The same implications and conclusions can be obtained using the returns on forward contracts. Table 8 presents the results of tests on whether the forward contract returns are positive or negative depending upon the criterion of Cov (li,Z)/Var (Z). In connection with table 8, table 9 shows the results of tests on the issue "Normal Backwardation " of forward contracts • In table 8, thirty-two out of fifty coaaodities are consistent with the basic hypothesis at 5* significance level. The only one comaodity which is totally inconsistent with the hypothesis is Plywood again. Excluding Plywood, thirty-two coaaodities out of forty-five are entirely consistent with the h y p o t h e s i s . I t is interesting also to note that ,if Cov(ii,Z) is less than Var (Z), the coaaodities whose nean return is not ** Using 10X significance level rather than 5X, one-month Silver Coin and two-month Copper are consistent with the hypothesis. 79 TABLE 8 Tests on forward returns Forward return; (Spot-FF)/FF x 100 ,where FP stands £or forward prices Hat. 1 J I Gold Bean 1. 8148 2.0038 7.7404 11.9592 35.6322 S.D 12.0954 18.6572 19.8983 32.8167 54.3 128 obs 24 23 51 48 42 t 0.74 0.52 2.78 2.52 4.25 Ho = * > > > Silver Mean 1.5084 5.0735 1 1.2334 27.2874 52.7620 S. D 26.686 2 30.9138 38-5807 78.4222 119.2293 OHS 22 21 49 46 41 t 0.27 0.75 2.04 2.36 2-83 Ho — s > > > Sxlver Hean 10.5929 25.7304 18.2041 41.3252 87.8519 Coin S.D 26.0341 24.0100 32.9860 67.3536 106.3507 obs 24 23 51 48 41 t 1. 99 5.14 3.94 4.25 5-35 Ho => _a > > > > Platinum Hean 0.2765 3.2425 10.3300 16.7233 37.9940 S.D 15.6445 21-1623 24.5196 18.5592 18.4153 obs 22 21 20. 17 12 t 0.08 0.76 1.88 3.72 7. 15 Ho = > > > Copper Hean 2-3909 4.3550 6.6823 12.7537 22.9675 S. D 8.7313 11.7176 12.3614 8.8755 8.9958 obs 24 23 22 19 13 t 1. 34 1.7 a 2.54 6.26 9.21 Ho = =>* > > > Plywood Hean -1.1315 -2.5422 -3.8314 -5. 1761 -7.3305 S.D 5.7191 7.9001 9. 1702 11.0063 12.8907 obs 24 23 22 19 13 t -0. 97 -1.54 -1.96 -2.05 -2.05 Ho = =< ji =< _a =< S. F. Hean -1.1517 2.2968 3.7346 2.6252 -1.3423 S.D 5.1115 7.7772 9.5695 9.8462 5.8927 obs 24 23 22 19 13 t -1. 10 1.42 0.83 1. 16 0.82 Ho = = — G. H. Hean -1.4344 -1.9223 -1.5904 -2.1633 S.D 4.1855 6.1476 7.4173 7.7358 5.7334 80 Table 8 (continued) obs 24 23 22 19 13 t - 1.68 -1.50 - 1.01 - 1.22 -2.56 Ho < B.P. Hean -0.2881 -1.0290 -2-1654 4.7343 8.5510 S.D 4.1432 5.9459 7.3381 11.6217 12.6732 obs 24 23 22 19 13 t -0.34 -0.83 -1.38 1.78 2-43 Ho > J.Y Hean 0.0206 -0.1874 -0.3885 0.0021 1.4402 S.D 4.4352 6. 1398 6-9325 10.6726 10.5439 obs 24 23 22 19 13 t 0.02 -0.15 -0. 26 0.0 0.50 Ho a '<• or •>• for 10% significance level T All statistics are on the same basis as table 6. 81 consistent with the hypothesis show positive return distributions still even though they are not significant at 5X level. If Cov(W,Z) is not significantly different fros Var(Z) like in Swiss Franc, German Hark, British Pound, and Japanese Ten, there is no consistent sign of their returns as expected. Table 9 presents the results of tests on the issue "Normal Backwardation" of forward contract prices in the same manner as used in futures contract case, i.e. by running the regressxon eguation (33). Gold, Silver, Silver Coin, Platinum, and copper show significantly positive beta coefficient at 5% level, supporting the hypothesis that the forward returns of those commodities are positive functions of times to maturity, and in turn the hypothesis that those contract prices follow the "Normal Backwardation" process. On the other hand, the Swiss Franc, British Pound and Japanese Yen, have beta coefficients whxch are not different from zero, implying again that there are no significant trends of return distributions in terms of tines to maturity, and thus that the forward prices are unbiased estimates of the expected future prices. All of these results are totally consistent with the hypothesis tests presented on table 1 - table 4. The only exceptions on table 9 are Plywood and German Bark; the beta coefficients of Plywood and German Bark are significantly negative even though their Cov(W,Z)/Var (Z) are significantly less than and egual to one respectively- TABLE 9 Normal backwardation of forward prices Forward returns = BO ♦ B1 Time * Bo To B_1 11 Ho I** Gold -5. 1803 -1.26 3. 1038 5.04 > 0.1199 (188) (4. 1082) (0.6164) Silver -6.1428 -0.65 4.6801 3.35 > 0-0595 (179) (9-4090) (1.3981) Silver -7.9952 -1.00 4.6432 3.89 > 0.0752 Coin (186) (7.9573) (1.1939) Platinum 4.2468 1.36 3.5096 6.02 > 0.2873 (92) (3.1130) (0-5827) Copper -0.6403 -0.41 1.9054 6.54 > 0.3019 (101) (1.5522) (0.2912) Plywood -1.4267 -1.04 -0.5405 -2. 10 < 0-0427 (101) (1.3709) (0-2572) S.F. 1. 1657 0.99 -0.3455 -1.56 •= 0.3844 (101) (1. 1834) (0.2220) G.E. 0.8054 0.85 -1.8819 -10.59 < 0.5312 (101) (0.9471) (0.1777) B.P. 0.3124 0.25 -0.3657 -1.57 = 0.0982 (101) (1.2435) (0.2333) J. Y. -0.4738 -0.41 0-1336 0.62 0.0039 (101) (1.1457) (0.2149) * All statistics are on the same basis as table 7. 83 In general, even with these two exceptions, the results support the nodels describing futures and forward prices and their derived implications. In other words, Noraal Backwardation is an accurate description of forward prices in eguilibriua for those coaaodities which can provide hedging instruments against unexpected changes in one-period bond prices. Also, for those coaaodities, where Cov(V,Z) is not significantly different froa Var(Z), forward prices are unbiased estimates of the expected spot prices in eguilibriua. Chapter V C01CLD5I0V 5-1 SOHHABY Since futures contracts evolved originally from cash forward contracts, they have siailar terms. Ihis similarity between the two contracts has led researchers to treat then as if they were identical even though the role of forward and futures aarkets and the underlying stochastic process of their prices have attracted a lot of attention in financial economics. Another lony-standiuy controversy concerning with systematic patterns to forward and futures prices has been whether or not these contract prices are systematically downward or upward biased estimates of expected spot prices. In connection with the first and the second issues, it has keen also controversial whether or not the differences between forward and futures prices, and their systematic biases against expected future spot prices imply market disequilibrium or market inefficiency. However, the theories in these area have tended to be created to fit the observed facts, not vice versa, so that most of the literature tends to build upon each other as economic models were - 84 - 85 checked with empirical evidence and modifications or extensions were created only to he tested again, without solid theoretical models. In the absence of a formal theoretical model within which the results can be interpreted, one can not say anything convincingly about causality. The research strategy adopted in this paper is twofold; one is to address these three controversial issues theoretically by developing eguilibriua models, and the other is to support the models empirically. Using an arbitrage argument saying that no investor can purchase at zero cost a bundle of goods that will strictly increase his utility in eguilibriua, this paper has developed general eguilibriua pricing models for both forward and futures contracts in the presence of uncertainty and in the absence of transaction costs. In the process • of developing the theoretical models, it was clearly demonstrated that futures contracts could be distinguished from forward contracts by the number of commodities that could be paid for a contract on the maturity date, and this difference between those two contracts was due to the different payment schedule because of the property of marking-to-market in futures contracts. Also, it was shown convincingly that this difference between the two contracts is not inconsistent with market efficiency or market equilibrium. Additionally, using a simple mathematics, the implicit relations between the contract prices, forward and futures, and 86 the expected spot prices were derived connecting the following; if the correlation between the commodity price and the marginal utility of a numeraire good at the maturity date is positive (negative)# the forward price is greater (less) than the expected spot price# so that the normal Contango and the Normal Backwardation correspondingly can be accepted as accurate descriptions of underlying contract prices. Concerning futures prices# their relations with expected spot prices# and thus the issue of Normal Backwardation or Normal Contango does not depend on the simple covariance of the commodity spot price with the marginal utility of a numeraire good# but on the covariance between the spot price and the marginal utility of a numeraire T-l good multiplied by exp 2Z * (?) at the maturity date# because of the possibility of stochastic interest rates. It was confirmed that neither the forward price nor the futures price is generally equal to the expected spot price# and that market eguilibrium or market efficiency is neither necessary nor sufficient condition for the contract prices to be unbiased estimates of the expected spot prices. In the process# it was shown also how to construct a futures contract that is equivalent to the forward contract; if a person who is engaged in a futures contract is paid at the end of each day S the present value that the difference between the futures prices, F (3,T)-F (S-1,1), would have if it were paid at the maturity date# this type of futures contract is equivalent to forward contract. 87 Besides* the clear analogy was shown between the ratio of futures prices to forward prices* and the tern structure of interest rates. For example, assuming* for simplicity, that the marginal utility of commodities is not correlated with interest rates* the ratio of futures prices to forward prices boils down to the ratio of the gross return from "rolling over" strategy in bond investment to the gross return from "going long" strategy. Adding two assumptions on the underlying process of assets (lognormality) and the utility function(constant relative risk aversions) permitted us to switch from the implicit description of a general equilibrium model to the explicit analysis of systematic patterns to the two contract praxes* from which empirically testable hypothesis could be derived in terms of causal relations among futures* forward and expected spot prices. The following implications were iunediate; if the interest rate is nonstochastic* futures contracts are equivalent to forward contracts regardless of the degree of risk aversion and the variance of commodities* and at the same time* both contract prices are unbiased estimates of future expected spot prices. If the covariance between the changes in commodity prices and the changes in discount oond prices is less than the variance of the changes in bond prices* futures prices are greater than forward prices* and simultaneously both of these contract prices are downward biased estimates of the expected spot prices* so that the Normal Backwardation is a natural deduction for describing the 86 contract prices. If the covariance is equal to or greater than the variance, futures prices are equal to or less than forward prices respectively, and at the sane tiae, toth contract prices are unbiased, or upward biased estimates of expected spot prices correspondingly. This iaplies that if a coaaodity provides a hedging instrument against changes in bond prices, normal Backwardation process can be said to be an accurate description of both futures and forward prices. Based on these theoretical models and their implications, various hypotheses about the causal relations among futures, forward and expected spot prices were tested on six basic commodities (Gold, Silver, Silver Coin, Platinum, Copper and Plywood), four foreign exchange rates (Swiss franc, German Bark, British Pound, Japanese Yen) . Eecause of ambiguity of actual maturity dates of futures contracts, the data observed on two different dates were used, one on the first trading dates and the other on the fifteenth calendar dates of months. The first method for testing the relations between futures and forward prices involved three steps, testing on the ratio of the covariance to the variance, grouping commodities based on the results of the first step test, and testing on the difference between futures and forward prices both on group and individual basis. The results on both basis leave little doubt about the hypothesis that futures and forward prices can be described by the 89 models developed in this paper. Noticeablely, 13.3% or 83-7% of commodities(thirty-six or forty-one price pairs out of forty-nine) are consistent with the hypothesis at 5% or 10X significance level, that the difference between futures and forward depends on the Cov (H,Z) and Var(Z). The second method was to utilize the analogy of the relation of futures and forward prices to the term structure of interest rates. Using actual returns as proxies for the ex-ante expected returns, additional evidence was provided that the analogy holds between the relation of futures and forward prices, and the term structure arguments, and thus that those two contract prices can be explained by the models in this paper. Tests on the hypothesis about the issue, "Normal Backwardation" or "Normal Contango" are also satisfactory except Plywood. Running the regression of commodity returns on times to maturity, indicates that beta coefficients of the independent variable depend on the covariance between the changes in commodity prices and the changes in riskfree discount bond prices. 5.2 CONCLUDING REMARKS, IMPLICATIONS. AND PUTONE RESEARCH AREAS The theoretical models and the various empirical findings are consistent with each other, in terms of the magnitude of the difference between forward and futures prices, the analogy of their relations to the term structure of interest rates, and the issue of Normal Backwardation or Normal Contango. 90 The most important implication is that ve can reject the conventional hypothesis that the systematic difference between futures and forward prices or the systematic bias of those prices against expected future prices may be due to market disequilibrium or market inefficiency. This is obvious from the fact that the models have been developed from the simple uo-arnitrage conditions in the absence of any help of market-efficiency arguments. The model with empirical tests supports the conjectures that futures prices are greater than forward prices, if the covariance of the changes in commodity prices with the changes in riskfree discount bond prices is less than the variance of tbe changes in the bond itself and thus the commodity serves as a hedging instrument against fluctuations in bond prices as we can see in commodities, Gold, Silver, and other physical commodities. Then, one might ask how to interpret these differences of prices in conventional risk-return framework. However, one possible and reasonable answer was yielded during the process of developing models, i.e. liquidity or risk premium in the lignt of the analogy of the relations between futures and forward prices to the relation between rolling-over strategy and going-long strategy in bond investments. Nevertheless, the implication of the difference between futures and forward prices should be interpreted with caution. For example, the fact that futures prices of a commodity are greater than forward prices does not imply that we can get a lower price 91 if a futures market is replaced by a forward aarket, since such a replacement could result in changes in eguilibriua prices of all other assets. lie aay not exclude the possibility of a higher price to the contrary. On the other hand, the empirical evidence that both futures and forward foreign exchange rates are unbiased estimates of expected spot rates has reasonable and plausible implications in the following sense: Consider two foreign exchange markets, U.S.A. and U.K., and suppose that British Pounds in American markets and U.S. dollar in British markets follow normal backwardation processes to the contrary. Then, British speculators can aake pound profits by takiug short positions in the pound and American speculators can make dollar profits by taking short positions in the dollar. Note however, the exchange rate in U.K. is the other side of the same coin in U.S.A. Then, the guestion following naturally might be why speculators go to the futures aarket rather than the forward aarket which has longer history in foreign exchanges. This guestion is beyond the scope of this study and is left unanswered. However, one possible explanation which is institutional rather than theoretical, appears to be that the futures foreign currency aarket nay be catering to saall or aedium speculators who are excluded from the interbank aarket. Since the models are formulated in a simplified econoay, the analysis in this paper can never be perfect in explaining the sources of deviations from the aodels. Also the problem 92 associated with empirical results is compounded by the fact that they are largely determined by the availability of data. The purpose of this paper, however, is not so much to introduce all the many factors that could theoretically influence futures and forward prices simultaneously into one equation as it is to find the best explanation for the causal relations anong those prices and expected spot prices in the simple economy. In order to construct a more comprehensive model, the following related areas might be suggested; First: how do institutional factors such as differential tax treatment tetveen normal income and capital gain, and regulation on daily price changes affect the eguilibrium futures and forward prices.2« Second: whether or not the futures and forward prices are consistent with the generalized CAPM or Arbitrage pricing models.21 Third: what is the role of 20 However, the Economic Becovery Tax Act of 1981 seems to effectively put an end to the use of commodity straddle for favorable tax treatment. For example, before this tax act was effective, the holding period for a favorable long ters capital gain treatment was & months. Also, the investor did not realize any gain or loss until th position was closed. However, under the new law, the more than 6 months holding period for the favorable capital gain treatment is uo longer relevant due to the requirement of the meriting to market of all futures position open at the er?d- of the investor's taxable year. Additionally, any gain from futures contracts will be taxed at a maximum effective tax rate of 32* in 1982 regardless of the length of time a position has been held open aud whether it is from a short or long position: note that in the past, gains from short positions were taxed as short term capital gain regardless of how long they were held open. 21 t o my knowledge, no academicians in the finance have attempted to apply modern portfolio pricing models (generalized CAPfl and Arbitrage or Factor models) based on exchange equilibrium and portfolio choice among rational investors for pricing the forward and futures contract simultaneously. Dusak(1973) and Bodie and Bosansky(1979) examined investor's return only on the 93 futures markets and forward markets in terms of the efficiency of the financial system as a whole, or efficient allocation of risk. Since the research on this futures market is at the stage of infancy of Finance, there are ample guestions unanswered, in addition to the area suggested above. Thus, while the theoretical models and the empirical results presented in this paper are consistent with each other, more theoretical and empirical work is needed before they can be given definite economic interpretations. futures coutract by using the conventional Capital Asset Pricing flodel, showing conflicting results. However, the results based on the conventional CAFfl basically assume the fixed relative prices. Breeden (1980) attempted to measure the risk of only futures contracts by using his consumption-based intertemporal CAPM. Furthermore, even though it is well known fact that the arbitrage model(Boss(1976)) should describe relative pricing, nobody has utilized the useful properties of the model to incorporate investors desire to hedge against changes in relative prices. Thus, it may be useful to examine the forward contract and the futures contract aggregating the Breeden's intertemporal CAPA and the notions of the arbitrage model of Boss(1976). Appendix A EXCHANGES AND ONUS OF SHADING IN FOTOBBS CON1BAC1S Futures contracts Exchange UNit of trading Gold Chicago aerc Exch 100 troy oz. Silver Chicago Bd of Trade 1000 troy oz. Silver Coin N.Y. Here Exch $1000 F.A. bag Platinum N. Y. Here Exch 50 Troy oz. Copper Chicago Here Exch 25000 Its. Plywood Chicago Bd of Trade 76032 sguare feet Swiss Franc Int'l Honetary Harket 125000 Franc Gernan Hack Int'l Honetary Harket 125000 Harks British Pound Int'l Honetary Harket 25000 Pounds Japanese Yen Int'l Honetary Harket 12.5 aillio Yen T-bills Iut'l Honetary Harket $1 Billion - 94 - Appendix fi TRADING HOURS OF FOTOBES AND FOR8ARD CONTRACTS Futures Forward * Gold 8:25-1;: 30 CT 10:00-2:15 Silver 8:40-1;: 25 CT 10:00-2:15 Silver Coin 9:40-1:: 25 ET 10:00-2:15 PLatinum 9:30-2;; 30 ET 10:00-2:15 Copper 9:50-2;; 00 ET 10:00-2:00 Plywood 9:00-1:; 00 CT 10:00-2:00 Swiss Franc 8:15-1:; 16 CT 10:30-2:00 German Hark 8: 15-1:; 20 CT 10:30-2:00 British Pound 8:15-1::24 CT 10:30-2:00 Japanese Yen 8:15-1:; 26 CT 10:30-2:00 T-bills 8:35-1:: 35 CT 10:30-2:00 * The exchanges of each futures contracts are shown in Appendix A The exchange of forward contracts is the Aserican Board of Trade in New York.. - 95 - Appendix C TESTS OH THE BATIO OF COFABIAICE TO TAHIAICE logX - logEX = Ho ♦ B1 logT-bill -logE(I-bill) (On the basis o£ fifteenth calendar dates of each aenth)* Mat (T) obs Bo to B1 tj Ho fi Gold 1 19 0-1051 1.53 -1.0376 -3.7059 < 0. 1731 (0.0685) (0.5999) 2 18 0- 1797 1.90 -2.1055 -3.7520 < 0.2880 (0.0920) (0.8277) 3 17 0.1579 1.69 -2.6090 -3.5899 < 0- 3099 (0.0963) (1.0053) 6 19 -0.0899 -1.21 -2.8772 -2.6873 < 0.2989 (0.0792) (1.9928) 12 28 0.9991 3.57 -2.9672 -2.5590 < 0.2218 (0. 1398) (1.5503) Silver 1 19 0.1875 1.51 -1.7867 -2-8035 < 0.1597 (0. 1238) (0.9990) O 2 18 0.3936 1 -3.8838 -3.2719 < 0.2973 (0. 1660) (1.9929) 3 17 0.9 106 2.99 -5.7069 -3.8199 < 0.9132 (0. 1681) (1.7560) 6 19 -0.0013 -0. 10 -6.8295 -2.9827 < 0.3606 (0. 1350) (2.623 3) o CO 12 28 0.5576 2.61 -6.7852 -2.6725 < • (0.2133) (2.9131) Silver Coin 1 17 0.1517 1.55 -1.3159 -2.9739 < 0- 1598 (0.0978) (0.7789) 2 16 0.1831 1.35 -2.1919 -2.6232 < 0. 1182 (0. 1357) (1.2166) 3 16 0.1953 1.30 -3.2118 -2.7298 < 0.2369 (0.1505) (1.5929) 6 13 -0.1837 -2. 19 -1.7901 -1.7579 - 0.1036 (0.0890) (1.5872) 12 28 0.5630 2.93 -2.9851 -1.9367 0. 1773 (0. 1923) (2.9292) - 96 - Appendix C (continued) PLATINUM 1 19 0-0974 1-31 -1.0234 -3.3791 < 0. 1466 (0.0746) (0.5988) 2 18 0.1535 1-60 -2.0298 -3.5157 < 0.2575 (0.0958) (0.8618) 3 17 0. 1182 1. 16 -2.3991 -3.1857 < 0.2521 (0. 1022) (1.0670) 6 14 -0.1457 -1.76 -2.3216 -2.0652 < 0-1479 (0.0827) (1.6084) 'N 0 CO 1 12 9 -0.4593 • -3.0205 -2.4286 < 0.0010 (0. 1643) (1.6555) COPPER 1 19 G.0368 0.67 -0.4723 -3-3431 < 0.0634 (0-0549) (0.4404) 2 18 0.0690 1.08 -1.0090 -3.5006 < 0.1619 (0.0638) (0.5739) 3 17 0.0346 0.56 -1.0849 -3.2026 < 0.1562 (0.0623) (0.6510) 6 14 -0.0961 -2.70 -0.8244 -2.6345 < 0.1056 (0-0356) (0.6925) 12 9 -0.2688 -4.50 -1.4410 -2.5306 < 0.0290 (0.0597) (0.9646) PLYWOOD 1 19 0.0353 0.63 -0.4167 -3.1679 < 0.0486 (0.0557) (0.4472) 2 18 0.0 906. 1-50 -1. 1070 -3.8075 < 0.2068 (0.0603) (0.5420) 3 17 0-1413 2.29 -2.0176 -4.6835 < 0.3953 (0.0617) (0.6443) 6 14 0.0283 0. 53 -2.5423 -3.4100 < 0.3330 (0-0534) (1.0388) 12 9 -0.0994 -0.83 -2.0864 -1.5976 = 0.0029 (0.1195) (1.9319) SWISS FRANC 1 19 -0.0103 -0.35 0.0908 -3.3488 < 0.0655 (0.0296) (0.2715) 2 15 -0.0110 -0.17 0.2234 -0.9645 0.0589 (0.0666) (0.8052) 3 24 -0.1044 -1.60 1.5074 0.5712 0.1149 (0.0654) (0.8918) 6 32 -0.0186 -0.35 0.6944 -0.2721 0.0126 (0.0532) (1. 1231) 12 23 -0.0643 -1. 35 1.6364 0.5222 s 0.3348 (0.0477) (1-2107) GERMAN MARK 1 2U -0-0078 -0-33 0.0558 -4.3612 < 0-0360 Appendix C (continued) (0.0233) (0.2105) 2 14 0.0042 0.09 0.0733 -1.5466 — 0.0125 (0.0476) (0.5992) 3 22 -0.0821 -2.07 1.2010 0.3680 = 0.1947 (0.0397) (0.5462) 6 30 -0.0160 -0.39 0.9543 -0.0522 — 0.0406 (0.0409) (0.8763) 12 16 -0.0859 -1.93 1.4830 0.4446 = 0.4234 (0.0446) (1.0863) BRITISH POUND 1 15 -1.3436 -2.08 9.6370 1.6952 = 0.2158 (0.6465) (5.0951) 2 10 -1.3948 -1.60 11.3538 1.2735 — 0.1960 (0.8692) (8. 1301) 3 13 -0.0195 -0.32 0.4274 -0.8815 = 0.0379 (0.0615) (0.6496) 6 16 -0.5659 -1.26 12.3791 1.4073 — 0. 1434 (0.4505) (8.0856) 12 9 0.2792 0-42 7.0790 0.5626 — 0.0578 (0.6686) (10.8054) JAPANESE YEN 1 15 -0.0504 -1.03 0.3033 -1.8087 = 0.0455 (0.0489) (0.3852) 2 10 0.0992 1-24 -1.0598 -2.7497 < 0.2001 (0.0801) (0.7491) 3 17 0.0175 0.29 -0. 1972 -1.8922 — 0.0875 (0.0599) (0.6327) 6 16 -0.0123 -0.21 0. 1117 -0.8425 = 0.0080 (0.0587) (1.0544) 12 9 -0.0033 -0.04 -0.2546 -0.8771 = 0.0450 (0.0685) (1.4304) * Appendix C - Appendix K are corresponding to Table 1 - Table on the basis of observations of fifteenth calendar dates of each nonth. Appendix D GROUPINGS OF COflflODITISS I. ONE MONTH MATURITY GROUP 1; COV < VAR Gold, Silver, Silver Coin, Platinum, Copper,Plywood Swiss tranc, German nark GROUP 2; COV = VAR British Pound, Japanese Yen II. TWO MONTH MATURITY GROUP 1; COV < VAR Gold, Silver, Silver Coin, Platinun, Copper, Plywood, Japanese Yen GROUP 2; COV = VAR Swiss Franc, German Mark, British Pound III. THREE MONTH MATURITY GROUP 1; COV < VAR Gold, Silver, Silver Coin, Platinum, Copper, Plywood GROUP 2; COV = VAR British Pound, Swiss Franc, German Mark, Japanese Yen IV. SIX MONTH MATURITY GROUP 1; COR < VAR Gold, Silver, Platinum, Copper, Plywood GROUP 2; COR = VAR Silver Coin, Swiss Franc, German Mark, British Pound, Japanese Yen V. TWELVE MONTH MATURITY GROUP 1; COR < VAR Gold, Silver, Platinum, Copper, Plywood GROUP 2; COE = VAR Silver Coin, Swiss Franc, German Mark, British Pound, Japanese Yen - 99 - Appendix £ DIFFEBENCE BETWEEN FUTDBBS ABD F0B8ABD PBICES (GBODPS) FUTUBES - FOBWABD(Groups) Group 1 ""MAT.” \ 2 3 6 _12 ~E5T. -2.2055 3.2469 1312814 3-2266 1.5791 S.E 2.4550 2.1595 4-5917 0-5804 0-7445 t -0.90 1-50 2.89 5.56 2.12 HO =0 =0 >0 >0 >0 DBS 130 108 188 169 142 Group 2 EST. -0.0017 0-0867 -0.0012 110-5811 -1.3653 S.E 0.006 0.0880 0-0004 85-2119 2.5519 t -2.61 0.99 -3-11 1-30 -0.54 HO <0 =0 <0 =0 -0 OBS 26 24 40 98 63 - 100 - Appendix F DIFFERENCE BETBEEH F010RES ABD FOBBABD PRICES(COHHODIIIES) FUTURES - FORBABD = B (1) : (individual connodities) Hat. \ 2 3 6 12 EST -1.1360 0. 1320 3. 1250 2-1692 2.2925 S.E 0-5028 0.6891 0-6163 0.4741 0.8127 GOLD t -1-95 0.19 5-07 9.58 2-76 HO =0 = 0 >0 >0 >0 COV/VAB < < < < < OBS 25 25 50 52 53 EST 2.9800 1.0792 9.7520 6.1538 0-6373 S.E 3.3387 2.8874 4.2674 1.6654 1.7889 SILVER t -0.79 0.37 2.29 3.70 0.36 HO =0 =0 >0 >0 = 0 COV/VAB < < < < < OBS 25 24 50 52 51 EST 0.2457 0.8225 0.7028 1-0300 -0-0440 S.E 0.4632 0.4357 0-3160 0.5386 0.0813 SILVER t -0.53 1.89 2.22 1.91 -0.54 COIN HO = 0 =0 >0 >=0 = 0 COV/VAR < < < ■= = OES 7 4 25 25 20 EST 2.2731 -1-2909 3.4500 3-8125 4.8533 S.E 1-1179 1.4879 1-6145 1.6416 1.9890 PLATINUH t -2.03 -0.87 2- 14 2.32 2-44 HO = 0 =0 >0 >0 >0 COV/VAB < < < < < OBS 13 11 16 16 15 EST -0-3020 -0-0771 0.4886 0.4146 0.3870 S.E 0.0501 0.0418 0.1174 0.1076 0. 1043 COPPER t -6.02 -1-85 4.16 3.85 3.71 HO <0 =0 >0 >0 >0 COV/VAR < < < < < OBS 25 24 22 24 23 EST 0.4450 0.5146 1.6300 2.0560 0-1367 S.E 0.4180 0-2510 0.4226 0-5356 0.3483 PLYWOOD t 1.06 2.05 3.86 3.84 0.39 - 101 - 102 Appendix t (continued) HO = 0 =0 >0 >0 =0 COV/VAB < < < < OBS 12 13 25 25 15 EST -0-2391 0.2000 0.1311 -0.1765 0.0313 S.E 0.3996 0.6455 0.5885 0.4007 0.3659 SWISS t -0.60 0.31 0.22 -0.44 0.09 FBANC HO — 0 =0 — 0 = 0 = 0 COV/VAB < — — — OBS 11 6 9 17 16 EST -0.4036 -0.3380 -0.0163 -0.2994 0.2300 S.E 0.3147 0.2471 0.4710 0.2775 0.3699 GERMAN t -1.28 -1.37 -0.03 -1.09 0.62 MARK HO = 0 =0 = 0 — 0 =0 CCV/VAR < = — — OBS 11 5 8 17 7 EST -0.2067 -26.1625 -2.0500 -12.6611 3-1167 S.E 1.7878 26.3499 1. 1522 11.4546 2.1218 BRITISH t -0. 12 -0.99 -1.78 -1. 11 1.47 POUND HO = 0 = 0 = 0 = 0 =0 COV/VAB — — — - OBS 12 8 10 18 3 EST 0.4218 -0.1514 0.6030 0.2450 0.2400 S.E 0.3752 0. 1585 0.3005 0-2156 0.7700 JAPANESE t 1. 12 -0.96 2.01 1. 14 0.31 YEN HO = 0 = 0 = 0 = 0 =0 COV/VAB = < = = OBS 11 7 10 18 2 EST -0.1085 -0.0457 -0.0300 -0.0594 S.E 0.0546 0.0623 0.0268 0.0228 T-BILL t -1.98 -0.73 -1.12 -2.60 HO = 0 = 0 = 0 <0 COV/VAR > > > > OBS 13 7 12 16 Appendix 6 TESTS OH THE ANALOGY TO IEBH SIB0C10BES OF IHIBBESX BATES F/t- BUB2{expZlrvyexp (T-t) B| ♦ B3-D Maturity OBS B1 B2 B 3 1 179 EST -0.4104 1.4105 0.0072 S.E 2.9443 2.9368 0.0065 T -0-14 0-14 1 - 1 HO =1 =0 2 143 EST 9.3301 -8.2265 -0.0338 S.E 21.0208 20.9223 0.0341 T 0.44 -0.44 -0.99 HO =1 = 0 3 235 EST 1.3S76 -0.3857 0.0119 S.E 0-7293 0.7250 0.0055 T 1.92 -1.91 2. 18 HO =1 *0 6 2-51 EST -2.8185 3.8080 -0.0701 S.E 3.4627 3-4143 0.0295 T -0.81 0.82 -2-38 HO =1 *0 12 162 EST 1.0875 -0.0828 0.0067 S.E 0.1091 0-1065 0.0120 T 9.96 -10.17 0.56 HO *1 = 0 - 103 - Appendix fl TESTS OH FOTOBES RETURNS Futures return ; (Spot-F)/F x 100 Hat. 1 12 Gold Hean 1.9801 3.7584 4.0599 11.8887 33.5578 S.D 10.6239 17. 1536 19.8819 32-2659 54.7373 obs 52 49 47 46 41 t 1.34 1.53 1-40 2-50 3-93 Ho = = > > Silver Hean 2.7944 4.27 07 6.8721 17-4605 46-5772 S-D 16.9495 28.2462 35-9155 70-7491 109.8512 obs 52 49 47 46 40 t 1. 19 1.06 1-31 1.67 2.68 Ho > Silver Hean 4.2719 14.0107 14.3887 26.1585 77-2855 Coin S.D 20.5688 37.9070 28.7557 63.5486 107-6141 obs 17 15 25 25 20 t 0.86 1.43 2-50 2-07 3-21 Ho > > > PLATINUM HEAN -7.0426 -2.9341 0.4421 7. 1740 30.7779 S.D 17.4310 20.3573 18.4687 29.3871 44.6167 OBS 24 21 32 31 27 T -1.98 -0.66 0. 14 1.36 3.58 HO = < = — — > COPPEB HEAN 3.2585 1.7943 3.1243 3.4789 7.2181 S.D 8.9274 11.6121 14.8462 18.6410 26. 1850 OBS 50 45 43 44 38 T 2.50 1.04 1.38 1.24 1.70 HO > s = = => PLYWOOD HEAN 0.2980 -2.3270 -4.5907 -6.4887 -6.7478 S.D b.4389 6.3416 8.1319 10.0742 10.1919 OES 13 13 26 26 25 T 0.17 -1.32 -2-88 -3.28 -3.31 HO - = < < < SWISS HEAN -0.4444 -0.7964 -0.1994 -0.0734 -5.6452 FRANC S.D 4.1732 6.4153 8.7869 12. 1894 15.3612 - 104 - 105 Appendix H (continued) OBS 20 16 25 33 24 T ■0.05 -0.50 - 0 . 11 -0.03 -1.80 HO GERMAN MEAN 0.0058 •1.1749 •0.5048 1.9518 -2.5363 HARK S.D 3.3936 4.1423 5.5741 8.7375 11.0317 OBS 21 15 23 31 17 T 0.01 1.10 •0.43 ■1.24 -0.95 HO BRITISH HEAN •0.3616 50.9569 0.8131 26.6262 3.9517 POUND S.D 2.5883 217.1404 6.0715 162.9225 25.9325 OBS 22 18 26 33 17 T ■0.66 1.00 0.68 0.94 0.63 HO JAPANESE HEAN 0.5969 ■0.8524 -0.4599 -0.4594 YEN S. D 3.2506 4.1319 8.2570 12.4686 OBS 21 17 25 32 T 0-84 ■0.85 -0.26 - 0.21 HO Appendix I NORMAL BACKWARDATION Of FUTURES PBICES RETURN = BO ♦ B1.TIME :futures return BO 10 E1 T 1 HO GOLD -2.8354 -0.95 2.8961 5.73 > 0.1237 (235) (2.9890) (0.5050) SILVER -3.7891 -0.64 4.0406 3.99 > 0.0642 (234) (5.9520) (1.0125) SILVER -4.5972 -0.47 6.4669 4. 16 > 0.1472 COIN (9.8460) (1.5564) (102) PLATINUM -10.4923 -2.69 3.3532 5.44 > 0.1821 (1350) (3.8971) (0.6162) COPPER 1.8164 1.06 0.4081 1.40 = 0.0089 (220) (1.7179) (0.2909) PLYWOOD -2.1121 -1.42 -0.4635 -2.13 < 0.0429 (103) (1.4925) (0.2178) SWISS 1.2248 0.75 -0.3911 -1.55 = 0.0315 FRANC (1.6387) (0.2529) (110) GERMAN -0.1749 -0. 15 -0.2236 -1.17 0.0130 MARK (1.1414) (0.1904) (107) *9 BRITISH 18.4919 0.99 -0.4996 -0.16 0.0002 POUND (18.6399) (3. 1886) (116) JAPANESE 1.9955 1-21 -0.1635 -0.45 0.0434 YEN (1.6526) (0.3639) (99) T-BILL 12.0337 31.20 -1.2965 -21.62 < 0.7758 (137) (0.3857) (0.0599) - 106 - Appendix J TESTS Of FOB!ABO BEZOBIS FORWARD RETURNS; (SPOT - FI)/FP* 100 Hat, 1 3 12 GOLD HEAN 072358 -0.2035 5.6177 11.8234 32.0466 S.D 13.6693 22.4534 19.7352 32.5288 54.8894 OBS 25 24 51 48 42 T 0.09 -0.04 2.03 2.52 3.78 HO > > > SILVER HEAN -1.3026 2-4758 7.1492 17.4995 42.5358 S.D 22.1842 37.6892 35.1072 69.5412 107.6454 OBS 25 24 51 48 42 T -0.29 0-32 1.45 1.74 2.56 HO -> > SILVER HEAN 0.1198 1.3079 7.1911 17.9332 44. 1617 COIN S.D 19.8509 33.3783 30.8679 60.0655 96.7915 OBS 23 22 50 47 41 T 0.03 0. 18 1.65 2.05 2.92 HO > > PLATI- HEAN 0.7938 2.0675 5.4711 16.7570 33.2371 NUH S.D 13. 1571 22.8536 24.4060 15.9848 20.6776 OBS 25 24 23 20 14 T 0. 30 0.44 1.08 4.69 6.01 HO > > COPPER HEAN 1.7073 3.7863 6. 1933 13.4907 22.4018 S.D 9.3565 12.1833 13.5461 7.4171 9.0091 OBS 25 24 23 20 14 T 0.91 1.52 2.19 8. 13 9.30 HO > > > PLYWOOD HEAN ■1. 1063 -2.2948 ■3.6926 -5.1616 -7.2890 S.D 6.7511 7.6732 8.4743 10.8286 12.7784 OBS 25 24 23 20 14 T ■0. 82 -1.47 -2.09 -2.13 -2.13 HO < < < SWISS HEAN -0.6947 - 1.0222 ■0.3614 -0.9726 -5.4251 FRANC S.D 5. 1221 6.8085 8.4365 9.4609 5.8282 OBS 25 24 23 20 14 T -0. 66 -0.74 -0. 18 -0.46 -3.48 HO < - 107 - 108 Appendix J (co tinued) GERMAN MEAN 0.3491 -2.8223 -4.3465 -9.7278 -21.5445 HARK S.D 4.0976 5.6122 6.7255 7.5468 5-4463 OBS 25 24 23 20 14 T 0.43 -2.46 -3.10 -5.76 -14.80 HO < < < < BRITISH MEAN 3.2009 3.3196 3.4089 3.7099 4.4516 POUND S.D 18.2785 18.9625 19.6313 21.1508 19.5116 OBS 25 24 23 20 14 T 0.88 0.86 0.83 0.78 0-85 HO JAPAN HEAN 0.0743 -0.0095 -0.2346 0.3170 1.5175 YEN S.D 3.7747 5.7281 6-5150 10.0160 9.9489 OBS 25 24 23 20 14 T 0.10 -0.01 -0.17 0-14 0.57 HO Appendix K ■OfiflAL BACKHABDAlIOli OF FOBHABO FBICES (FORWARD RETURNS); BETURN = BO ♦ B1.TIHE BO IQ. Ml TJ HO R* GOLD -4.3318 -1.05 2.9706 4.80 > 0.1091 (235) (4.1064) (0.6192) SILVER -6.6912 -0.83 4.0967 3.36 > 0.0568 (234) (8.0702) (1.2178) SILVER — 6.4566 -0.87 4. 1590 3.75 > 0.0722 COIN (7.3972) (1.1080) (102) PLATINUM 3.0613 1.06 3.0704 5.70 > 0.2382 (135) (2.8945) (0.5384) COPPEB -0.2762 -0.18 1.9306 6.60 > 0.2954 (220) (1.5718) (0.2924) PLYWOOD -1.2937 -0.97 -0.5484 -2-21 < 0.0450 (103) (1.3314) (0.2476) SWISS 1.4510 1.36 -0.6940 -3.49 < 0.4110 FBANC (1.0690) (0. 1988) (118) GERMAN 0.8762 1. 10 -1.8383 -11.35 < 0.5532 MARK (0.8709) (0.1620) (107) BRITISH 3.7933 1.09 1. 1211 0.21 0.0004 POUND (2.8371) (5.2769) (106) JAPANESE -0.3335 -0.32 0-1376 0.70 0.0048 YEN (1.0494) (0.1952) (106) T-BILL 14.6657 22.33 -1.6545 -8.36 N < 0.5141 (68) (0.6567) (0. 1980) - 109 - REFEBEICES Anderson, Ronald W. and Danthine Jean-Pierre(1981): "The time pattern of hedging and the volatility of futures PRICES" Working paper of CSPH, Columbia Business School Arrow, Kenneth J. 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