UNIVERSITY OF 
 
 ILLINOIS LIBRARY 
 
 AT URBANA-CHAMPAIGN 
 
 BOOKSTACKS 
 
o ^^ ^ ^ o 
 
V ^: 
 
 
 FACULTY WORKING 
 PAPER NO, 927 
 
 Ca' 
 
 usa! and Systematic Reiaticns Among 
 Forvvard, Futures and Expected Spot Prices 
 
 Hun Y. Park 
 
 3Hi JO AMVMan 3H1 
 
 CcMegs cf Comrnerrie 'ind Business AdiT!ir!isT''ation 
 ^' -ail of Economic and Sus'ness Research 
 .^rsih/ oi iiiinoiS, IJrtana-Cfiampaign 
 

 ;.^.;t'v:;;;^jf1^1i 
 
BEBR 
 
 FACULTY WORKING PAPER NO. 927 
 College of Commerce and Business Administration 
 University of Illinois at Urbana-Champaign 
 February 1983 
 
 Causal and Systematic Relations Among 
 Forward, Futures and Expected Spot Prices 
 
 Hun Y. Park, Assistant Professor 
 Department of Finance 
 
 Acknowledgment: This paper is a part of my dissertation. 
 
 I am grateful to my dissertation committee members. Professors 
 
 Andrew H. Chen, J. Huston McCulloch, Gary C. Sanger for their 
 
 continuing encouragement and valuable comments. Special 
 
 thanks are owed to Professor Edward J. Ray for his insightful 
 
 comments. 
 
Digitized by the Internet Archive 
 
 in 2011 with funding from 
 
 University of Illinois Urbana-Champaign 
 
 http://www.archive.org/details/causalsystematic927park 
 
Abstract 
 
 In the absence of transaction costs and in the presence of uncer- 
 tainty, this paper derives implicit and explicit pricing equations of 
 forward and futures contracts, from which causal and systematic rela- 
 tions can be derived among forward, futures and expected prices. The 
 implicit pricing models of both contracts alE developed by using basic 
 raicroeconomic theories, i.e., the market clearing condition and the 
 first order condition for the expected utility maximization. These 
 models conject economic rationale for "normal backwardation" and 
 "normal contango" processes. Adding two assumptions, lognorraality and 
 constant relative risk aversions, permits us to switch from the implicit 
 description of a general equilibrium model to the explicit analysis of 
 systematic patterns to the contract prices, from which empirically test- 
 able hypothesis can be derived in terras of causal relations among 
 futures, forward and expected spot prices. Especially, it will be exa- 
 mined under what conditions forward and futures prices are systemati- 
 cally different and, at the same time, "normal backwardation" and 
 "normal contango" are accepted as accurate descriptions of the contract 
 prices in market equilibrium. 
 
I. INTRODUCTION 
 
 Most recently, several papers have attributed the fundamental 
 difference between forward and futures prices to the different payment 
 schedule due to the property of marking-to-market in futures contracts 
 (see Margrabe [11], Black [2], Cox, Ingersoll and Ross [4], Richard and 
 Sundaresan [15], Jarrow and Oldfield [9], Chen and Park [3], and Park 
 [14]). Employing different approaches, they have consistently shown 
 that each price is the value of an asset which will pay a specific 
 number of units of the underlying good on the maturity date. Spe- 
 cifically, the number in the forward price is the total return from 
 "going long" strategy in a default-free discount bond maturing at the 
 same time as the forward contract. The number in the futures price is 
 die total return from "rolling over" strategy in one-period bonds up 
 to the contract maturity date. Based upon this result, the purpose 
 of this paper is to derive implicit and explicit pricing equations of 
 forward and futures contracts, from which causal and systematic relations 
 fan be derived among forward, futures and expected prices. Especially, 
 it will be examined under what conditions forward and futures prices are 
 systematically different, and, at the same time, under wbat conditions 
 
 '^Jormal Backwardation" and "Normal Contango" are accepted as accurate 
 
 2 
 descriptions of underlying contract prices in market equilibrium. Note 
 
 that "Normal Backwardation" and "Normal Contango" are referred to the 
 
 processes in which contract prices (forward and futures) are systematically 
 
 downward and upward biased estimates of expected spot prices over time 
 
 respectively. To the author's knowledge, the explicit expressions for 
 
-2- 
 
 che simultaneous relations among forward, futures and expected spot 
 prices have been attracted little attentions so far. In section II, in 
 the absence of transaction costs and in the presence of uncertainty, the 
 implicit pricing models of forward and futures contracts are formulated 
 through the market clearing condition and the first order condition for 
 the expected utility maximization. Some important implications are 
 developed from these models in terms of the economic rationale for 
 "Normal Backwardation" and "Normal Contango." On the basis of models 
 formulated in section II, the explicit form pricing equations of both 
 contracts are developed in section III, adding two assumptions, log- 
 normality and constant relative risk aversion. The simultaneous and 
 causal relations among forward, futures and expected spot prices are 
 derived and analyzed. Finally, section IV summarizes and concludes 
 the paper. 
 
 II. IMPLICIT PRICING MODELS OF FORWARD 
 AND FUTURES CONTRACTS.^ 
 
 Notations used in this section are as follows: 
 
 Xi: commodity X (subscript i represents the specific 
 commodity i) 
 
 T: maturity date of forward and futures contracts 
 
 fi(t,T): forward price at current time t on commodity i (T > t) 
 
 Fi(t, T): futures price at time t on commodity i 
 
 ?i(T): spot price of commodity i at time t 
 
 B(t, t): price as of time t of a riskless bond paying one 
 dollar at time t(t > t) 
 
 rC":): continuously compounded interest rate from time t 
 to time T + 1 
 
-3- 
 
 COV: covariance 
 
 VAR = z^: variance 
 
 £t(«): expected value of the argument (•) at time t 
 
 (•): randomness of the argument (•) 
 
 As noted in the section I, the forward price is the value of a 
 claim that pays B(t,T)~ units of the commodity under consideration 
 at its maturity date T: i.e. 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 is paid B(t,T) units of commodity i at time T. 
 Note however that B(t,T) is known at time t. 
 
 Once the payoff is known in terns of the number of units that 
 can be paid at time T, a general forward contract pricing model can 
 be developed through the market clearing condition and maximization 
 of the expected utility function that is assumed the same for all 
 individuals. 
 
 Consider the n+1 goods economy composed of Xo, XI, X2.... Xn, 
 where Xo ser</es 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. 
 
 E exp(-aT)U{Xo(T), XI (t) Xnd)} 
 
 ■=t 
 
 (1) 
 
 where a is the utility discount factor that is known at time t and 
 U(*) is a Von Neuman-mor gens tern utility function that is strictly 
 increasing, concave and twice dif ferentiable with respect to Xi: 
 U{Xo(-)«Xl(-),... Xn(T)} is denoted by U(«) hereafter. 
 
-4- 
 
 Suppose that two people, a buyer and a seller of forward contracts 
 conmit to 2 forward contracts, each contract for 1 unit of Xi at the 
 price fi(t,T) to be paid at time T and the buyer is supposed to pay in 
 terms of Xo. Then, the utility function at time T can be written as 
 
 U{Xo(T) - 6fi(t,T)B(t,T)"-'-, X1(T) 
 
 Xi(T) + 6-B(t,T)"-'-, Xn(T)} (2) 
 
 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 6 conditional S = is zero. Thus, 
 
 d Et 
 
 T-1 
 i: exp(-aT)U(T) 
 
 /de 
 
 = (3) 
 
 8=C 
 
 where Et(«) denotes the expected value of (•) at the present time t; 
 note that equation (3) is consistent with the no arbitrage condition 
 in that it reflects the essential point of arbitrage argximent 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 requires 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. 
 
-5- 
 
 From equation (2) and (3) 
 -fi(t,T)B(t,T)"^Et[exp(-(T-C)a)Uo(T)] + BCt.D'-'-Et [exp(-(T-t )a)Ui(T) ] = 0, 
 where Ui(T) denotes the marginal utility of commodity i at time T. 
 
 fi(t,T) = B(t,T)-^Et[(exp(-(T-t)a)Ui(T)]/B(t,T)-^Et[(exp(-(T-t)ct)Uo(T)] 
 fi(t,T) - EtUi(T)/EtUo(T) (4) 
 
 viiich is the generl forward price of commodity i from the market clearing 
 condition. Cn the other hand, the relative spot price of commodity Xi 
 in terms of the numeraire good Xo at each time is the ratio of marginal 
 utility of Xi to marginal utility of Xo from the first order condition 
 for utility maximization. 
 
 Pi(T) = Ui(T)/U0(T) (5) 
 
 Thus, 
 
 Et{Pi(T)} = Et{Ui(T)/Uo(T)} (6) 
 
 Note from equations (A) and (6) that the question whether the forward 
 price follows the "Normal backwardation" process (Keynes [10] , Hicks 
 [7], Houthakker [3]) or "Normal Contango" (Telser [17]) or whether the 
 forward price is equal to the expected spot price depends on whether 
 EtUi(T) /EtUo(T) is less or greater than, or equal to Et{Ui(T) /Uo(T)}. 
 Also, following this procedure, it is very simple to derive the 
 relation between the forward price and the expected future spot price 
 that is somewhat similar to the results that Richard and Sundaresan 
 [15] obtained through a fairly complicated procedure. 
 
-6- 
 
 fi(t,T) = EtUi(T)/EtUo(T) 
 
 = Et:{Uo(T)«Pi(T)}/EtUo(T) From equacion (1) 
 = [COVt{Uo(T),Pi(T)} + EtUo(T)«Etii(T)]/EtUo(T) 
 = EtPi(T) + COVt{Uo(T),Pi(T)}/EtUo(T) (7) 
 
 From equation (7), 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 and the 
 marginal utility of a numeraire good at the maturity date is positive 
 or negative, the forward price is greater 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 [2], Dusak [5], and Samuelson [16] argue. 
 
 These results are intuitively appealing in the following sense: 
 for example, suppose that the covariance between the spot price and 
 die 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 T 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 decreasing marginal utility and thus 
 that the magnitude of the marginal utility of return or profit Is less 
 t±ian that of the marginal utility of loss with the same dollar amount. 
 
-7- 
 
 This result reflects the notion of risk aversion of investors. Thus, 
 in order to induce risk averse investors to commit to a forward con- 
 tract, there must be compensation for bearing this risk in the form of 
 a positive return. This logic leads in turn to the conclusion that 
 the forward price should be less than the expected spot price, which 
 conforms with the Keynes-Hicks-Houthakker's argument, the "Normal 
 Backwardation. " 
 
 It is very important to note that the expected spot price is 
 expressed in terns 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. 
 
 On the same line of logic, futures contract pricing models can 
 
 be formulated. As discussed in the introduction, the futures price is 
 
 T-1 
 the value of a claim that pays exp I r(T) units of the commodity under 
 
 T=t 
 
 consideration at time T. 
 
 T-1 
 Note that exp Z r(i) units of the commodity is unknown ac time t, 
 
 T=t 
 
 and thus engaging in futures contract is a speculation not only on the 
 future spot price but on the number of units of commodities than can 
 be paid for the futures contract at the maturity date. From the same 
 logic that was employed in the forward contract pricing model, if a 
 person engages in 3 units of futures contracts, each unit of the con- 
 tract 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 
 
-8- 
 
 T-1 
 U{Xo(T)-2-Fi(t,T)exp Z r (t) .Xl (T) ,X2 (T) , 
 
 T = t 
 
 T-1 
 Xi(T)+6«exp Z rd) Xn(T)} (8) 
 
 T=t 
 
 From equation (3), 
 
 T-1 T-1 . T-1 T-1 
 
 -Fi(t,T)Et{ E exp(-aT)«exp Z r (t)Uo(T) }+Et{ I exp(-aT)«exp Z r(T)Ui(T)} » 
 
 T=t T=t T=t T=t 
 
 Thus, 
 
 T-1 _ T-1 
 
 Fi(t,T) = Et exp{ Z (r (T)-a) 'UKT) }/Et exp{ Z (r (T)-a) -UoCT) } 
 
 T=t T=t 
 
 (9) 
 
 which is the general futures contract pricing equation that is deduced 
 
 from the market clearing condition. 
 
 Besides, the relation between the expected spot price and the 
 
 futures price can be easily derived from equation (9) as follows (See 
 
 footnote 6 for proof) 
 
 T-1 . T-1 
 Fi(t,T) = Et Pi(T) + COV{Pi(T),Uo(T)exp Z r(T)}/Et Uo(T)exp I rCt) 
 
 T=l T-1 
 
 (10) 
 
 It is obvious now that the relation between the futures price and 
 
 the expected spot price depends on the covariance between the spot price 
 
 T-1 
 and the marginal utility of a numeraire good multiplied by exp E rd) 
 
 T=t 
 
 at the maturity date. It depends no longer on the simple covariance 
 
 between the spot price and the marginal utility of a numeraire good 
 due to the oossibility of stochastic one-period interest rates. 
 
-9- 
 
 It is important to note chat the market efficiency concept does not 
 say anything about equations (7) and (10). In other words, the question 
 whether the futures and forward prices are greater than, or equal to, or 
 less than the expected spot price is irrelevant to the question whether 
 or not those markets, forward and futures, are efficient in processing 
 information. 
 
 III. RELATIONS AMONG FORWARD, FUTURES AND EXPECTED PRICES 
 3.1 Explicit Form Pricing Equations 
 
 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 bivariate lognormal; in fact, we need means 
 and variances only for 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 lognonaally distributed, then log Xi is normally distrib- 
 uted, and the density of the lognormal distribution is given by 
 
 F(Xi;u,a2) = (l/Xi/2TO")exp{-l/2a2(iogXi-M)2} 
 
 viiere E(Xi) = exp(u+l/2'a^) , and Var(Xi) = exp(2u+2a^) - exp(2M+<j2) 
 
 E(logXi) = u and Var(logXi) = a^ (11) 
 
 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 function such as 
 (k/l-y)^ '^ , where the parameter showing the degree of risk aversion 
 
-10- 
 
 is positive and characterized by decreasing absolute risk aversion and 
 constant relative risk aversion. 
 
 Notation used in this section is as follows: 
 
 a^id): variance of log XiCr) at time t. 
 
 ct'^o(t): variance of log Xo(t) at time t. 
 
 T-1 
 a^rCr): variance of Z r(T). 
 
 T=t 
 
 Cio: correlation between log Xi(T) and log Xo(t). 
 
 T-1 
 pir: correlation between log UiCr) and Z rCr). 
 
 T=t 
 
 T-1 
 por: correlation between log Uo(t) and Z rCt). 
 
 T=t 
 
 ^(t) = log Ui(T) 
 
 Ht) = log Uo(t) 
 
 T-1 - T-1 
 
 ^^(t) = log Ui(T)exp Z r(T) = log Ui(T) + Z rd) 
 
 T=t T=t 
 
 -1 - T-1 . T-1 
 
 $ (t) = log Uo(T)exp Z r(T) = log Uo(t) + Z t(t) 
 
 T=t T=t 
 
 Z(t) = 0(t) - <&(t) = dUt) - $^(t). 
 
 From the assumption of power utility function, 
 
 Ut{Xi(T),Xo(T)} = (l/(l-a))Xi{T)^'^ + (l/(l-b))Xo(T)^'^ (12) 
 
 where positive a and b show the degree of risk aversion on commodity Xi 
 
 8 
 and Xo respectively. 
 
 Ui(T) = Xi(T)"^, Uo(T) = Xo(T)~'' 
 ?(T) = log Ui(T) = -a log Xi(T) 
 *(T) = log Uo(T) =■ -b log Xo(T) 
 
-11- 
 
 Z(T) = ^(T) - ^(T) = log{Ui(T)/Uo(T)} = log Pi(T) 
 
 EtZ(T) = EtD(T) - Et$(T) 
 
 Var ;(T) = VarClog Ui(T)} = a-«a-i(T) (13) 
 
 Var $(T) = VarClog Uo(T)} = b--a-o(T) 
 
 T-1 
 
 Var 4(^(1) = Vardog Ui(T) + E rd)} 
 
 T=t 
 
 = a^«a^i(T) + a^rCT) + 2pir a ai(T)ar(T) 
 
 -1 .T-1 
 
 Var 4 (T) = VarClog Uo(T) + Z r(T)} 
 
 T = t 
 
 = b^'G^od) + a2r(T) + 2oor b0o(T)ar(T) 
 
 From equations (11) and (13), 
 
 fi(t,T) = EtUi(T)/EtUo(T) 
 
 = expCEt^(T) + l/2a2.a2i(T)}/expCEtHT) + l/2b2.a2o(T)} 
 
 = expCEtZd) + l/2(a2.a-l(T) - b^-a^od))} (14) 
 
 T-1 , T-1 
 
 Fi(t,T) = EtCUi(T)-exp Z r (t) }/EtCUo(T) exp Z r(T)} 
 
 T=t T=t 
 
 = expCE^l(T) + l/2(a-«a-i(T)+o-r(T)+2oir aai(T)or (T)) }/ 
 
 exp{E*^T) + l/2(b2a^o(T)-Kr2r(T)+2oor bao(T)ar(T)) } 
 » expCEtZ(T) + l/2((a^a2i(T)+2ar(pir aai(T)-oor bao(T)) 
 -b^ffSoCT))} (15) 
 
 Also, 
 
 EtPi(T) = EtCUi(T)/Uo(T)} = EtCexp-J) (T) /exp* (T) } = EtCexp(6 (T)-<f>(T) } 
 VarClog Pi(T)} = VarCl)(T) - * (T) } 
 
 = a-«a2i(T) + b2.a^o(T) - 2Dio abai(T)(jo(T) 
 
 (16) 
 
-12- 
 
 Thiis, from (11), (13), and (16) 
 
 EcPi(T) = expTEtZd) + l/ZCa^.a^KT) + b^'O-o(T) - 
 
 2pio abai(T)ao(T)}] (17) 
 
 Equations (14), (15), and (17) show the explicit form pricing models for 
 forward contract, futures contract and the expected future spot price 
 under the given assumptions. 
 
 The most important implication from equation (14), (15), and (17) 
 is that we can not only specify the causal relations among those three 
 prices, but also derive testable hypotheses. In other words, by exam- 
 ining the variables in the equations, we can investigate under what 
 conditions a "Normal Backwardation" or a "Normal Contango" process of 
 forward or futures prices is possible and under what conditions the 
 forward contract price is equivalent to the futures contract price. 
 
 3.2 Relation Between Forward and Futures Prices 
 From equation (14) and (15) , 
 
 log Fi(t,T) - log fi(t,T) = ar{pir aai(T) - por bao(T)} 
 
 (18) 
 Then, the following implications follow directly from equation (18) , 
 
 1) if ar = 0, then Fi(t,T) = fi(t,T). 
 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 commodity. This is intuitively 
 plausible in chat a nonstochastic interest rate implies that the hedge 
 ratio for both futures and forward contract is given at time t when the 
 
-13- 
 
 coatracts are open, and chus the contracts should be 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 pir aai(T) and por bao(T) . Assuming a 
 cne-period riskfree discount bond as the numeraire good, following 
 implications can be deduced. Hereafter, in this paper, the numeraire 
 
 Q 
 
 good is assumed to be a one-period riskfree discount bond.^ 
 
 T-1 T-1 
 
 Then, c;o(T) = cr(T) ; note that Var Z r(T) = Z Var r(T) if 
 
 T=C T=t 
 
 interest rate at period t is independent of that at period x + 1. 
 
 Also, noting that the bond price is negatively correlated with 
 the interest rate, the following causal relations can be derived: 
 If ab pio ci(T) ao(T) = b^«a^o(T), 
 then Fi(t,T) = fi(t,T) 
 
 This implies that if the covariance between the price of the com- 
 aodity 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. This result is intuitively plausible in Chat futures 
 prices depend on the correlation of spot prices and interest rates, while 
 forward prices do not. 
 
 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 
 
-14- 
 
 prices and storable commodicy 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. 
 
 3.3 The Issue of Normal Backwardation or Normal Contango 
 
 By comparing equation (14) and (15) with equation (17), 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 . 
 
 Subtracting equation (17) from (14) and (15) respectively after 
 
 taking logarithm 
 
 leg fi(t,T) - log EtPi(T) = bao(T){pio aai(T) - bao(T)} 
 
 (19) 
 
 log Fi(t,T) - log EtPi(T) = -b^'a^o(T) + ar{pir aai(T) 
 
 -por b0o(T)} + pio abai(T)ao(T) 
 = ao{a pio 0i(T) - bao(T)}(b-l) 
 
 (20) 
 
 assuming the one-period discount bond as a numeraire good; note that if 
 ^,-e subtract equation (19) from equation (20), it turns out to be equal 
 to equation (18) . 
 
 From equations (19) and (20), following implications can be 
 
 derived immediately. 
 
-15- 
 
 1) First, if the interest rate is nonstochastic and thus ao(T) = D, 
 then fi(t,T) = Fi(t,T) = EtPi(T); bcth forward and futures contract 
 prices are unbiased estimates of future expected spot prices. This has 
 intuitive explanation because of the reasons stated in section 3.1. 
 
 Note however that this is not a necessary condition for the equivalence 
 of forward and futures contract, nor for them to be unbiased estimates 
 of the expected spot price. 
 
 2) Second, assuming that b > 1, which is the case of the gener- 
 alized negative power utility function, a fairly striking result follows 
 from the equations (18), (19), and (20); 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 oio cri(T) cto(T) = b^-a^od), 
 
 then EP(T) = F(t,T) = f(t,T) (21) 
 
 In words, if 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 and 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 corre- 
 lated, so that the commodity is a good candidate for a hedging instru- 
 ment against changes in the price of the one-period discount bond, both 
 
-16- 
 
 futures and forward price are doT>mward biased estimates of the expected 
 spot price. "Normal Backwardation" is a natural deduction for the 
 
 ccirmodity . 
 
 IV. SUMMARY AND CONCLUSION 
 
 The fact that we can express the payoff of forward and 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 implications. 
 
 Based on the number of units of commodities that can be paid for a 
 futures and a forward contracts, this paper developed implicit pricing 
 models of both contracts through the market clearing condition and the 
 first order condition for the expected utility maximization. These 
 models conject economic rationale for "Normal Backwardation" and "Normal 
 Contango." 
 
 Adding two assiimptions on the underlying process of assets (log- 
 normality) 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 prices, from which empirically testable hypothesis 
 could be derived in terms of causal relations among futures, forward 
 and expected spot prices. 
 
 The following implications were immediate; 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 
 
-17- 
 
 future expected spot prices. If the covariance between the changes in 
 coniLodity prices and the changes in discount bond 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 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 same time, both contract prices are unbiased, or upward 
 biased estimates of expected spot prices correspondingly. This implies 
 that if a commodity 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. 
 
 In the process, it was clearly confirmed that the systematic dif- 
 ference between futures and forward prices, and "Normal Backwardation" 
 or "Normal Congango" are not inconsistent with market equilibrium, or 
 narket efficiency. Nevertheless, since the models are formulated in a 
 simplified economy, the analysis in this paper can never be perfect in 
 explaining the sources of deviations from the models. The purpose of 
 this paper, however, is not so much to introduce all the many factors 
 that can theoretically influence futures and forward prices simulta- 
 neously into one equation as it is to find the best explanation for 
 the causal relations among futures, forward and expected spot prices 
 in the simple economy. 
 
-18- 
 
 Footnotes 
 
 For descriptions of fundamental differences between futures 
 and forward contracts, see Black [2], Cox, Ingersoll and Ross [4], 
 Margrobe [11], Jarrow and Oldfield [9], Richard and Sundaresan [15], 
 Chen and Park [3], and Park [l^]. 
 
 2 
 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. This clarification of 
 
 terminologies is due to Professor J. H. McCulloch. See Keynes [10], 
 
 Hicks [7], Houthakker [8], Arrow [1], McCulloch [12] for normal 
 
 backwardation, and Hardy [6] and Telser [17] for normal contango. 
 
 Throughout this paper, the consideration of margin requirement 
 is rilled out; it is important to note that margin requirements are 
 v£)t partial equity payments against the market value of the commodity 
 represented by 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 [18] argues that margin 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. 
 
 J. H. McCulloch [13] derived the same pricing equation in connection 
 with short-lived options pricing when the underlying distribution of price 
 is log-symmetric stable. 
 
 This was pointed out also by Cox, Ingersoll and Ross [4], and 
 Richard and Sundaresan [15]. 
 
 T-1 _ T-1 
 °Fi(t,T) = Et{Ui(T)«exp Z r(T) }/Et{Uo(T) -exp E r(T) 
 
 T"t T=t 
 
 since a is deterministic 
 
 T-1 , T-1 
 
 = Et{Pi(T)«Uo(T)-exp Z rCx) }/Et{Uo(T) exp Z rd)} 
 
 T=t T»t 
 
 from equation (1) 
 
-19- 
 
 r - - T-i _ _ T-i " 
 
 = COV{Pi(T), Uo(T) exp L r(T)} + Et Pi(T)'EC Uo(T) exp L r(T) 
 
 |_ T=t T=t 
 
 T-1 
 
 /Et{Uo(T) exp Z r(T)} 
 
 T=C 
 
 T-1 . T-1 
 
 = Et Pi(T) + COV{Pi(T),Uo(T) exp E r(T) }/Et{Uo(T) exp I r(T)} 
 
 T=t T=t 
 
 which is Che equation (10). This equation (10) is quite similar to the 
 ncdels of Richard and Sundaresan [15] obtained from a quite different 
 approach. 
 
 'if u(x) = (k/i-Y)x^"^, u-^ = x"^, ir^ = -k/Y x""^"-^ 
 
 Thus, the absolute risk aversion, 
 
 R, (X) = -U^/V'^ = -k/Y X"^"^/X"^ = k/Y X"-"- 
 
 r/ (X) = -k/Y X^ < 0, 
 which implies the decreasing absolute risk aversion. 
 Also, the relative risk aversion, 
 
 R (X) = R (X)X = k/Y X"-"- X = k/Y 
 
 r a 
 
 R^^ (X) = 0. 
 
 Miich implies the constant relative risk aversion. 
 
 The general power utility function is given by (Ki/l-Y)Xi ^^. 
 In this paper, Ki is assumed to equal one for simplicity with no loss 
 of generality for comparison between forward and futures prices and 
 expected spot prices. 
 
 9 
 The choice 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 
 
 en each state that expires at every instant 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 Debrew 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. 
 
 This was indicated by Cox, Ingersoll and Ross [4], 
 
-20- 
 
 References 
 
 Arrow, Kenneth J., "Futures Markets: Some Theoretical Perspectives," 
 Journal of Futures Markets , Vol. 1, Number 2, 1981. 
 
 I. Black, F., "The Pricing of Commodity Contracts," Journal of Financial 
 Economics , January, 1976. 
 
 3. Chen, Andrew and Park, Hun, "Divergencies Between Futures and Forward 
 Prices: A Test on Mar king- To- Market Effects of Futures Contracts," 
 unpublished working paper, September, 19S2. 
 
 +. Cox, Ingersoll and Ross, "The Relation Between Forward Prices and 
 Futures Prices," Journal of Financial Economics , December, 1981. 
 
 >. Dusak, "Futures Trading and Investor Return: An Investigation of 
 Commodity Markets Risk Premiums," Journal of Political Economy , 
 October, 1973. 
 
 3. Hardy, Charles, "Risk and Risk Bearing," University of Chicago Press, 
 
 1940. 
 
 7. Hicks, "Value and Capital: An Inquiry into Some Fundamental Principles 
 of Economic Theory," Second Edition, Oxford Press, 1939. 
 
 3. Houthakker, H. S., "Can Speculators Forecast Prices," Review of Economics 
 and Statistics , May, 1957. 
 
 9. Jar row and Oldfield, "Forward Contracts and Futures Contracts," Journal 
 of Financial Economics , December, 1981. 
 
 3. Keynes, J. M. , "A Treatise on Money," Harcourt, Brace and Company, New 
 
 York, 1930. 
 
 1. Margrabe, W., "A Theory of Forward and Futures Prices," unpublished 
 working paper, The Wharton School, University of Pennsylvania, 1978. 
 
 2. McCulloch, J. H., "Comment on 'Risk, Interest and Forward Exchange'," 
 Quarterly Journal of Economics , 1975. 
 
 , "The Pricing of Short-lived Options When Price Uncertainty 
 
 is Log-synmetric stable," working paper of NBER and Boston College, 1978. 
 
 A. Park, H. Y., "A Theoretical and Empirical Investigation of Forward 
 and Futures Prices," unpublished dissertation thesis, the Ohio State 
 
 University, August, 1982. 
 
 i. Richard, S. F. and Sundaresan, M., "A Continuous Time Equilibrium 
 Model of Forward Prices and Futures Prices in a Multigood Economy," 
 Journal of Financial Economics, December, 1981. 
 
-21- 
 
 16. Samuelscn, D. A., "Proof chat Properly Anticipated Prices Fluctuate 
 Randomly," Industrial Management Review , Spring, 1965. 
 
 17. Telser, Lester G., "Futures Trading and the Storage of Cotton and 
 Wheat," Journal of Political Economy , June, 1958. 
 
 18. , "Margins and Futures Contracts," Journal of Futures 
 
 Markets, Vol. 1, No. 2, Summer, 1981. 
 
 M/E/314 
 
;';-,t:l,. 
 
 ::l,. 
 
..\t''' 
 
 !•■.'. 
 
 
HECKMAN 
 
 BINDERY INC. 
 
 JUN95 
 
 : _ . KL MANCHESTER. I