The effects of decision making on futures price volatility
David A. Hennessy
The volatility of commodity futures contracts is of economic interest to those in the agriculture industry for many reasons. Perhaps the most immediate reason is the fact that options on futures contracts, being nonlinear in the underlying futures price, have value that is dependent on second and higher moments of the futures price. Moschini and Lapan, among others, have identified nonlinearities in production whereby producers may optimally choose options as part of their hedge. The variability of futures price matters to these hedgers. It also has been found (e.g., Lence and Hayes) that basis risk or bias may induce a hedge ratio that differs from one. This ratio depends implicitly on futures price volatility, so even producers who use only futures to hedge may be interested in futures price volatility. Further, it has been demonstrated (Grossman and Stiglitz) that high volatility is required to induce the speculative interest necessary to sustain futures markets. If prices for long-term futures contracts do not vary much, then insufficient trading may occur to cover the fixed costs of setting up a market.
The existing literature on the determination of futures price volatility does not accommodate the possibility of production or consumption responses. In this paper we will show that if a futures contract is of sufficiently long duration, allowing producers to receive and act upon evolving information, a pattern in futures price volatilities emerges as maturity approaches. This pattern is consistent with observations on the seasonality of volatilities and with the maturity hypothesis. However, unlike the prevailing explanation of the seasonality phenomenon (i.e., the state variable hypothesis), seasonality is shown to arise not from the resolution of uncertainty, but rather from increasingly constrained supply and demand functions as settlement date approaches. Contrary to the state variable hypothesis, we show that the resolution of supply and demand uncertainty may increase rather than decrease volatility. We demonstrate our theory by using a rational expectations model that incorporates production flexibility. Data from a total of twenty-four contracts for five different crops are considered over the period 1985-94.
Since 1956, when Telser suggested that futures price volatility may increase as the settlement date approaches, there has been a vigorous debate on the issue. Samuelson (1965) used an autoregressive price relationship to demonstrate the plausibility of Telser’s conjecture. Also, commencing with his 1965 paper, Samuelson (1971, 1973, 1976) wrote a series of articles with the goal of reconciling the randomness of the price of claims on assets inferred by rational expectations considerations with the economic implausibility of the seemingly associated unbounded variation in prices. He proposed that, at the limit, as time to maturity increases, the settlement price becomes ergodic in distribution, i.e., independent of the initial cash price. For the futures price martingale property (Samuelson 1965) to be consistent with the ergodicity property, it is necessary that, as time to maturity increases, the volatility of futures price falls, eventually, to zero. In its weakest form, Samuelson’s hypothesis (henceforth known as the SH) holds that volatility must only eventually fall as time to maturity increases (Samuelson 1976), so empirical investigations of the hypothesis can never be absolutely conclusive.
Alternative hypotheses exist. Perhaps the most intuitive is that volatility should be negatively correlated with the availability or prospective availability of the commodity. In empirical studies, Streeter and Tomek investigated the effects of inventories on hand; additionally, both they and Kenyon et al. considered expected production. The elements for a theoretical foundation for this relationship have been laid by Samuelson (1971), who showed how stocks can affect price distributions through time, and by Deaton and Laroque, who showed that the impossibility of negative inventories can give rise to rapid, large movements in prices. A more widely studied alternative hypothesis, the state variable hypothesis, is that the flow of information matters (Richard and Sundaresan). Anderson and Danthine and also Leistikow (1989, 1990, 1993) derived theoretically that volatility should be high at times when much uncertainty about fundamentals is resolved, and low when little uncertainty is resolved.
The SH and the state variable hypothesis are not necessarily incompatible in that the ergodicity assumption can be interpreted as the restriction that as time to maturity increases, knowledge about fundamentals becomes impossible, and so no uncertainty can be resolved. Samuelson motivates his hypothesis with the following illustration:
Consider [in December 1975] the price of cocoa in July 1999. Let one month pass, as December 1975 turns into January 1976. How different will be your evaluation and mine of that ultimate 1999 price? But, by contrast, consider the price of cocoa in March 1976. Surely we will learn much about it as our three month horizon turns into a two month horizon (1976, p. 123).
As noted by Anderson and Danthine (p. 262), this reasoning seems to come very close to stating that the acquisition of information drives the time-to-maturity conjecture. Their interpretation of the SH is that the amount of information revealed about market conditions at contract expiration increases as the date of expiration draws near.
Bessembinder et al. dispute this interpretation. While agreeing that the revelation of much information close to expiration could give rise to an inverse relationship between volatility and time to maturity, they claim that such a pattern of information flows is neither necessary nor sufficient to cause it. They also contend that the above quotation should not be interpreted as Anderson and Danthine do. Bessembinder et al. study the mean reverting property of the spot price process used by Samuelson (1965) and identify the covariance between spot price and cost of holding inventory as being critical in determining the validity of the SH. They calculate this covariance for several products on which futures are traded, and show that its sign correctly predicts the effect of time to maturity on volatility.
Another set of the literature has looked at the seasonal structure of volatility. That seasonality in volatility exists is well established (Anderson; Choi and Longstaff; Kenyon et al.). Because volatility peaks in the summer months, Anderson ascribed its existence to the seasonality of resolution of supply and demand uncertainty. However, Milonas (1991) also suggested that the depletion of inventories as harvest approaches may have an impact.
Much empirical analysis has been conducted to identify the determinants of the magnitude of volatility. Rutledge, testing only for the time-to-maturity effect, found an effect for silver and cocoa, but could not identify an effect for wheat or soybean oil. Milonas (1986) tested several agricultural, financial, and metal futures contracts for only the maturity effect. The results strongly favor the SH. Anderson, studying agricultural commodities as well as silver, found quite strong evidence in favor of seasonal effects, but weaker evidence in favor of the time-to-maturity effect. Choi and Longstaff investigated three soybean contracts over five years, and identified seasonality in two contracts. Castelino and Francis reported strong support for the maturity effect over the 1960-71 period for Chicago Board of Trade soybean complex and wheat contracts.
Leistikow (1989) examined Anderson and Danthine-type information flows, and found that some contract volatilities tended to be less sensitive to information the further the contracts were from maturity. However, for other contracts, the sensitivities of volatilities were not found to differ by time to maturity. For Winnipeg contracts, Khoury and Yourougou found convincing results in favor of the time-to-maturity effect. Kenyon et al. conducted a very comprehensive analysis of several agricultural contracts over the period 1974 to 1983, and found that volatilities tended to differ from year to year. They also detected pronounced seasonality in corn and soybean contracts, limited seasonality in wheat and hog contracts, and no seasonality in a cattle contract. For the grains and soybean contracts, the summer months displayed the largest volatility. This study also measured stock effects for the livestock contracts by using monthly seven-state cattle on feed numbers and quarterly ten-state market hog numbers. For hogs, the counterintuitive result that volatility increased with the stock of hogs was obtained. For cattle, no stock effect was found.
Perhaps the most comprehensive empirical study to date has been that of Streeter and Tomek on the Chicago Board of Trade March and November soybean contracts over the 1976 to 1986 period. Consistent with other studies, they found significant year and seasonality effects. Using time to maturity and the square of time to maturity as regressors, they identified the maturity effect to be significant and in accord with the SH. Three different time series representing the current and prospective availability of soybeans were used, but one was found to be statistically insignificant and the others gave counterintuitive signs.
Perhaps at this stage a word needs to be said about some of the variables included in the above-mentioned studies. Neither time-to-maturity nor seasonality effects are, in and of themselves, economically meaningful. They proxy fundamental economic phenomena about which little is known. Whether the SH is due to information, to technology, or to other factors has never been established, though Anderson and Danthine as well as Leistikow (1990), have shown that information effects can cause local derivatives of either sign. In a similar vein, the connection between information and seasonality has never been empirically validated. Information such as temperature, rainfall, forecasted production, and forecasted demand could be modeled empirically and tested to explain seasonality.
In this paper we propose a model that emphasizes the importance of production and demand inflexibilities arising from decision making. These factors have not been considered in the literature, though presumably Samuelson would include them in his “ultimate economic law” (1976, p. 120). Our model is compatible with both the maturity and the state variable hypotheses, and may explain more satisfactorily the observed patterns in the volatility data. More importantly, it provides insights into the fundamental economic meaning of both the SH and the seasonality effects. Further, it demonstrates that production and demand structures matter even in such apparently esoteric areas as option pricing. The main body of this paper is comprised of two sections. In the first, the theoretical model is developed. In the second section, the model’s implications are tested.
The contingent claims methodology that is used in this paper has not been widely employed in the agricultural economics literature. Notable applications are the valuation of the target price program (Marcus and Modest) and insurance programs (Turvey). It is also the technique that underlies the Black valuation formula for options on futures. The methodology has been extensively applied in the finance literature, and is being increasingly used in general economics to model uncertainty. To familiarize readers with the approach, we provide an introduction to the basic tools of analysis.
The purpose of this subsection is to briefly introduce the foundations of the contingent claims methodology. The approach is based on a specification of the nature of uncertainty and a law governing how that uncertainty affects the value of a function. The primary building block for the specification of uncertainty is the Wiener stochastic process. This is the continuous analog to the random walk (i.e., unit root first-order autoregressive process), and we will denote it by z(t), where t is a time index. The following three properties define the Wiener process (Dixit and Pindyck):
(a) It is Markov; i.e., the probability distribution for all future values of the process depends only on the present value.
(b) It has independent increments, i.e., cov[z(t) – z(t – j), z(t – k) – z(t – k – j)] = 0 for all nonoverlapping time intervals.
(c) The change in the process over any finite time interval is normal, with mean equal to zero and standard deviation proportional to the square root of the time interval, i.e., var[z(t) – z(t – [Delta]t)] = k[Delta]t, where k is a constant.
Continuous-time Brownian motion with drift generalizes the applicability of the Wiener process. Specifically, x(t) follows Brownian motion with drift if dx(t) = [Alpha]dt + [Sigma]dz(t), and z(t) is Wiener. The drift parameter, [Alpha], denotes the average rate of movement up or down. The [Sigma] parameter scales the variance of dx(t) from the unit variance associated with dz(t) to [[Sigma].sup.2]. A variant of this specification is geometric Brownian motion, where x(t) is replaced by ln[x(t)] to give dx(t) = [Alpha]x(t)dt + [Sigma]x(t)dz(t). This variant has the advantage of constraining the value of x(t) to being positive, a property desirable for stochastic price processes.
Ito’s lemma is the law that determines how uncertainty affects the value of a function. It involves an extension of conventional calculus. Conventionally, the total derivative of a function with two arguments, V(x, y), is given by dV(x, y) = ([Delta]V/[Delta]x)dx + ([Delta]V/[Delta]y)dy. It is an approximation, true only for small changes. It can be viewed as a Taylor’s series approximation that could be more properly written as
(1) dV(x, y) = [Delta]V / [Delta]x dx + [Delta]V / [Delta]y dy
+ 0.5 [[Delta].sup.2]V / [Delta][x.sup.2] [(dx).sup.2] + [[Delta].sup.2]V [Delta][y.sup.2] [(dy).sup.2] + [[Delta].sup.2]V / [Delta]x[Delta]y dxdy + …
Usually for small changes, the higher-order terms may be ignored. However, if either x or y are stochastic, then higher-order terms cannot be ignored. Specifically, if x follows geometric Brownian motion through time, then [(dx).sup.2] gives rise to the expression [[[Sigma]x(t)].sup.2] which is of order one and cannot be dismissed. Thus, in continuous-time Brownian motion models, one should expect a quadratic term such as [[Delta].sup.2]F/[Delta][x.sup.2] to survive in a first-order total differential. Curvature matters due to Jensen’s inequality.
Denote the harvest date by T. Within this continuous-time framework, we partition time into two intervals: the open-on-the-left interval ([Tau], T] when no production response is possible, and the interval [0, [Tau]] where one factor can be altered to change production. We propose the stochastic Cobb-Douglas production function
(2) [Mathematical Expression Omitted]
where [Q.sub.s, T] is output quantity, [J.sub.[Tau]] denotes level of input chosen at time [Tau], [k.sub.s] is the production constant of proportionality, and f is flexibility, which is equal to the elasticity of expected output with respect to input [J.sub.[Tau]]. Here, [[Psi].sub.T] is the evaluation at time T (harvest time) of the stochastic process [[Psi].sub.t], which evolves according to the geometric Brownian motion
(3) d[Psi] / [Psi] = [[Sigma].sub.[Psi]][dz.sub.[Psi]]
where [[Sigma].sub.[Psi]] is the diffusion parameter, and [z.sub.[Psi]] denotes the standard normal Wiener process. Drift is not included in the analysis because it would complicate the algebra without either changing the results or providing additional insight.
We consider a futures contract that matures at harvest, and assume that futures price and physicals price are identical at maturity,
(4) [F.sub.T] = [P.sub.T]
where [F.sub.T] is futures price at settlement time T and [P.sub.T] is physicals price at settlement time T.
The demand equation is considered to possess the following iso-elastic form:
(5) [Mathematical Expression Omitted]
where [Q.sub.d,T] represents output demand at harvest time, [[Phi].sub.T] denotes the stochastic demand shock evaluated at harvest, [k.sub.d] designates the demand constant of proportionality, and [Epsilon] is absolute elasticity of demand. Here, [[Phi].sub.T] has evolved over time according to the geometric Brownian motion,
(6) d[Phi] / [Phi] = [[Sigma].sub.[Phi]][dz.sub.[Phi]]
where [[Sigma].sub.[Phi]] represents the diffusion parameter, and [z.sub.[Phi]] denotes the standard normal Wiener process.
Let the two processes be correlated as follows: cor(d[Phi]/[Phi], d[Psi]/[Psi]) = [Rho], where [Rho] is a constant. Let acres-planted be the production decision under consideration. Now, given that the acreage decision has been made (i.e., over the semi-open interval ([Tau], T]), the sole determinants of the futures price stochastic process are the evaluations of the stochastic processes ([[Phi].sub.t], [[Psi].sub.t]). Information on their harvest time values (i.e., values at maturity of the futures contract under consideration) are obtained by observing ([[Phi].sub.t], [[Psi].sub.t]) evolve. We will first examine how ([[Phi].sub.t], [[Psi].sub.t]) affect the evolution of the futures price stochastic process over ([Tau], T], that is, after the input [J.sub.[Tau]] has been chosen. By market clearance and assuming no storage, harvest date supply must equal harvest date demand. Equating equation (2) with equation (5) at time T, we get
(7) [Mathematical Expression Omitted].
By multiplying equations (7) and (2), we now have a value of harvest at time T contingent on the input choice [J.sub.[Tau]]. The value of harvest is a function of the time T realizations of ([[Phi].sub.t], [[Psi].sub.t]), and we will write it as V([[Phi].sub.T], [[Psi].sub.T], T; [J.sub.[Tau]]) where
(8) [Mathematical Expression Omitted].
Here, [K.sub.1] is the appropriate constant of proportionality. This harvest time valuation is stochastic because both [[Phi].sub.T] and [[Psi].sub.T] are the realizations of stochastic processes.
We are interested in the expected value of the crop before harvest because input decisions are made before harvest. Thus, we seek to value V([[Phi].sub.t], [[Psi].sub.t]; [J.sub.[Tau]]) at time points earlier than T. Specifically, we seek V([[Phi].sub.[Tau]], [[Psi].sub.[Tau]], [Tau]; [J.sub.[Tau]]), the value at the time that [J.sub.[Tau]] is chosen. To do this, we use Ito’s lemma to expand V([[Phi].sub.t], [[Psi].sub.t], t; [J.sub.[Tau]].
(9) dV([[Phi].sub.t], [[Psi].sub.t], t; [J.sub.[Tau]])
= [Delta]V / [Delta]t dt + [Delta]V / [Delta][Phi] d[Phi] + [Delta]V / [Delta][Psi] d[Psi]
+ 0.5 [[Delta].sup.2]V / [Delta][[Phi].sup.2] [(d[Phi]).sup.2] + 0.5 [[Delta].sup.2]V / [Delta][[Psi].sup.2] [(d[Psi]).sup.2]
+ [[Delta].sup.2]V / [Delta][Phi][Delta][Psi] d[Phi]d[Psi] + … .
Here, t is nonstochastic, so partial own and cross derivatives in higher than the first partial can be ignored. Substitute for d[Phi] and d[Psi] from equations (3) and (6) above, take the expectation of the right- and left-hand sides, and divide through by dt to get
(10) [Mathematical Expression Omitted].
The expected change in the value of the crop per unit time should equal the opportunity cost,
(11) 1 / dt E[dV([[Phi].sub.t], [[Psi].sub.t], t: [J.sub.[Tau]])] = rV([[Phi].sub.t], [[Psi].sub.t], t; [J.sub.[Tau]]).
Here, r can be the risk-free rate or a risk-augmented rate. Equating the two expressions, we obtain
(12) [Mathematical Expression Omitted].
This partial differential equation must satisfy condition (8) at time T. Differential equations of this form are called Euler equations, and often have polynomial solutions. This is true in our case, so we solve by substituting in candidate polynomial solutions with arbitrary indices.
Solving equation (12) subject to equation (8), and subject to the boundary conditions that if either d[Phi] or d[Psi] ever equals zero, then V([[Phi].sub.t], [[Psi].sub.t]. t; [J.sub.[Tau]]) must equal zero, we get
(13) [Mathematical Expression Omitted]
where [K.sub.2] is independent of [[Phi].sub.t], [[Psi].sub.t], and [J.sub.[Tau]], and g is a constant. This valuation pertains to all time points in the semi-open interval ([Tau], T]. However, at time [Tau] the level of [J.sub.[Tau]] is chosen to maximize the farmer’s welfare in a competitive market scenario. Cost is w[J.sub.[Tau]], where w is the unit price of [J.sub.[Tau]]. It is not appropriate to maximize the difference between V([[Phi].sub.[Tau]], [[Psi].sub.[Tau]], [Tau]; [J.sub.[Tau]]) and w[J.sub.[Tau]] because this would give a monopoly solution. Instead, one should partition V([[Phi].sub.[Tau]], [[Psi].sub.[Tau]], [Tau]; [J.sub.[Tau]]) into two components, one associated with price and the other with quality. Viewing equations (13) and (2), it is appropriate to write
(14) [Mathematical Expression Omitted].
Here, [K.sub.3] and [K.sub.4] are terms independent of [[Phi].sub.[Tau]], [[Psi].sub.[Tau]], and [J.sub.[Tau]], while the first term in parentheses is an expression representing the time-discounted unit present value of output. This first term is exogenous to the producer. Call it H, and pose the producer’s problem as
(15) [Mathematical Expression Omitted].
This solves to give [J.sub.[Tau]] = [(H[K.sub.4][[Psi].sub.[Tau]]w).sup.1/(1 – f)].
The second-order condition is satisfied if 0 [less than] f [less than] 1, a condition that is required if decreasing returns to scale is to be satisfied. Substitute for H to obtain
(16) [Mathematical Expression Omitted]
where [K.sub.5] and [K.sub.6] are terms independent of [[Phi].sub.[Tau]], [[Psi].sub.[Tau]], and [J.sub.[Tau]]. Substituting this expression into equation (7) gives the following expression for futures price evaluated at harvest time:
(17) [Mathematical Expression Omitted]
where [K.sub.7] is independent of [[Phi].sub.[Tau]] and [[Psi].sub.[Tau]].
There are two time-points of interest in the expression: the harvest or maturity date, and the input decision date. The stochastic process governing the motion of futures price after planting differs from that governing motion before planting. To understand how they differ, denote the futures price at time [Tau] as F([[Phi].sub.t], [[Psi].sub.t], t). Apply Ito’s lemma to get
(18) [Mathematical Expression Omitted].
Substitute in for d[Phi] and d[Psi] from equations (3) and (6) above, take the expectation of the right- and left-hand sides, and divide through by dt to obtain
(19) [Mathematical Expression Omitted].
As futures contracts are settled each day, there is no net capital outlay. If one assumes that the risk premium is zero, then (1/dt)E[dF] = 0, and we have the partial differential equation
(20) [Mathematical Expression Omitted].
We solve this equation subject to equation (17), and subject to the boundary conditions that if either [[Phi].sub.t] or [[Psi].sub.t] ever equals zero, then F([[Phi].sub.t], [[Psi].sub.t], t) must equal zero. There are two parts to the solution: one for the time interval ([Tau], T] when [[Phi].sub.[Tau]] and [[Psi].sub.[Tau]] are fixed, and one for the time interval before [Tau] when [[Phi].sub.[Tau]] and [[Psi].sub.[Tau]] are stochastic. The solution over the time interval ([Tau], T] is
(21) [Mathematical Expression Omitted]
where [K.sub.8] does not depend on either [[Phi].sub.t] or [[Psi].sub.t]. Logging both sides of this equation and then taking the variance gives
[TABULAR DATA FOR TABLE 1 OMITTED]
(22) [Mathematical Expression Omitted]
where [Mathematical Expression Omitted] is the volatility of futures price. Prior to time [Tau], the solution is
(23) [Mathematical Expression Omitted]
where [K.sub.9] does not depend on either [[Phi].sub.t], or [[Psi].sub.t]. Log variance is
(24) [Mathematical Expression Omitted].
Note that the before- and after-planting values of [Mathematical Expression Omitted] are equal when f = 0. Note also that [(1 – f).sup.2]/[(f + [Epsilon] – f[Epsilon]).sup.2] is decreasing infwhen 0 [less than] f [less than] 1 and 0 [less than] [Epsilon] [less than] 1. These conditions represent decreasing returns to scale and inelastic demand, respectively, and are realistic assumptions. Under these assumptions, 1/[(f + [Epsilon] – f[Epsilon]).sup.2] is also decreasing in f. Therefore, [Mathematical Expression Omitted] under these conditions. Similarly, -2(1 – f)[Rho][[Sigma].sub.[Phi]][[Phi].sub.[Psi]]/[(f + [Epsilon] – f[Epsilon]).sup.2] [less than or equal to] -2[Rho][[Sigma].sub.[Phi]][[Sigma].sub.[Psi]]/[[Epsilon].sup.2] holds under these conditions if [Rho] [less than or equal to] 0. It is possible that the before-planting value of [Mathematical Expression Omitted] exceeds the after-planting value when [Rho] [greater than] 0, but it is unlikely to occur for small positive [Rho] values. In a diversified economy such as that of the United States, the most plausible value of [Rho] is zero because the demand for food is driven by forces unrelated to the stochastic determinants of supply. Thus, it is expected that futures price volatility before planting will be lower than after planting. This result is always true if there is no demand uncertainty, or if there is no supply uncertainty, or if there is zero correlation between the two sources of uncertainty.
Introduction of inventories makes the problem dynamic in the interyear sense. Samuelson (1971) showed the problem to be one requiring a dynamic programming solution. The ability to substitute intertemporally will reduce volatilities. Thus, if the commodity is nonperishable, then equations (22) and (24) represent upper bounds on the magnitudes of volatilities.
Testing the Flexibility Hypothesis
The contracts considered are presented in table 1. In all cases, all consistently traded contracts for each commodity are considered. These commodities were chosen because they are among the most widely traded, they trade far from maturity, there are several contracts in each year, and not all planting dates occur in the spring. The time period 1985-94 was chosen for all Chicago Board of Trade contracts because all contracts were traded for at least fifteen months during these years. The Kansas City soft red winter wheat contract did not consistently trade for longer than twelve months prior to 1986, while the Minneapolis hard red spring wheat contract did not consistently trade for longer than eleven months prior to 1987. These years were chosen as cutoff points because of the need to have consistently long data series. The contracts for each commodity are estimated in their own system. Thus, for example, five corn contract equations are estimated in a system. The large number of equations and unequal observations render it computationally impossible to estimate all equations in one system. In addition, due to the twelve-month length of the Minneapolis contract, a combined system would have involved dropping an inordinate number of (scarce) observations that are far from the maturity date.
To test for a planting effect, we specify a comprehensive model that allows for economic state variables as well as the seasonal 1-0, annual 1-0, and time-to-maturity structural proxy variables. The model is specified as
(25) V(i, t, T) = [[Beta].sub.0] + [[Beta].sub.1] TTM + [[Beta].sub.2]Plant
+ [[Beta].sub.3]Stock + [[Beta].sub.4]Rain + [[Beta].sub.5]Temp
+ [[Beta].sub.6]RainTemp + [[Beta].sub.7]June + [[Beta].sub.8]July
+ [[Sigma].sub.Years] [[Beta].sub.Year] Annual Dummies.
[TABULAR DATA FOR TABLE 2 OMITTED]
The variables are described in table 2. Futures data were obtained from the Futures Industry Institute Data Center. The formula for V(i, t, T) is given by
(26) V(i, j, k) = 365var[log([F.sub.i,j,k.t]/[F.sub.i,j,k,t-1])]
where [F.sub.i,j,k,t] is the futures price for commodity i on trading day t of month j of the contract with expiration at time k. Expression V(i, j, k) is the annualized daily volatility calculated from observations in month j. The TTM (time to maturity) variable is included to test whether a maturity effect survives after taking account of the other variables. Since acres planted is not the only source of flexibility, TTM may continue to be significant in regressions even if flexibility is the fundamental determinant of the time-to-maturity effect.
The variable Plant is equal to zero from commencement of a contract through the last month of planting for that crop, and is equal to one thereafter. The month of planting is considered to be April for spring crops and October for the winter wheat contracts. The coefficient is expected to be positive. One would expect that the planting effect would be stronger for corn and soybeans than for the wheat contracts for several reasons. First, wheat plantings occur in both the winter and the spring, so it is unclear where to place the planting dummy. Second, in an average year, 15%-20% of winter wheat plantings succumb to winter weather, and these “winterkill” acres are planted again in the spring. Third, wheat tends to be less geographically centralized than the corn and soybean crops. This geographical dispersion gives rise to a wide range of planting dates even within the winter and spring crops. Fourth, while the United States dominates world production of corn and soybeans, international competition dampens the influence of the United States on wheat prices. In all, one would expect greater planting effects for corn and soybeans than for wheat, and for spring wheat than for winter wheat.
Stocks demonstrate a sawtooth pattern over time, being high after harvest and low prior to harvest. If, as in some prior studies, we had looked at only one contract per commodity, then it would have been difficult to distinguish between a stock effect and a time-to-maturity effect. This was a primary reason for using multiple contracts per commodity. Stocks data were obtained from quarterly Economic Research Service (ERS) crop outlook reports (Feed Outlook, Oil Crops Outlook, Wheat Outlook). The data was linearly interpolated to give a monthly estimate of current stocks. Thus the stock variable is stocks in inventory and not past or prospective stock levels. The stock coefficient is expected to be negative because low supplies will reduce the ability of the marketplace to cope with a disappointing harvest.
Monthly precipitation and temperature variables were introduced as absolute deviations from monthly means over the 1950 to 1994 period to capture weather stress placed on plants. Crops can be stressed by unseasonably high or low levels of precipitation or temperature. Because corn and soybeans tend to be grown in the Corn Belt, temperature and precipitation indices for that region were used for these crops. As the wheats tend to be grown in the Northern Great Plains, temperature and precipitation indices for that region were used for these crops. Historical weather data was obtained from the ERS (USDA 1992). An update of this data file was obtained from the ERS (Teigen) to cover more recent years. It is expected that volatility will increase when plants are stressed, so that the coefficients of these variables should be positive. To capture interaction effects, the product of the rain and temperature indices was also included. To capture seasonality, seasonal dummies were introduced for the two months where seasonality is most pronounced, June and July (see Anderson, Milonas 1991). The coefficients are expected to be positive. Annual dummies are also introduced to screen out purely annual phenomena.
The system for each commodity was estimated using the seemingly unrelated regressions method. Because of low Durbin-Watson statistics, these systems were corrected for first-order autocorrelation. Since we have studied twenty-four contracts with up to twenty regression coefficients per contract, space does not permit presentation of all results and statistics. Of primary interest to this study are the coefficients for the planting, time-to-maturity, seasonality, stock, and weather variables. These coefficients and t-statistics are presented in table 3 as well as equation [R.sup.2] values within the system. The means, by time to expiration, of actual and predicted values for the harvest contracts are presented in figures 1-3. The correlation coefficients between the actual and predicted values presented in the graphs vary from 0.717 for Chicago wheat to 0.96 for corn. As expected, the fits are better for the spring crops than for the winter-sown crops.
From table 3, it can be seen that in all cases the Plant coefficients are positive, and seven are statistically significant at the 5% level. Statistical significance of the plant coefficients tends to be strongest for the spring-planted crops (in accord with expectations). Corn and soybean results were found to be robust with respect to the choice of a planting dummy that shifts in April or May. However, the significance levels for the spring wheat results decrease for the earlier shift in the planting dummy. The coefficients for time-to-maturity are mixed and only one at the 5% level and one at the 10% level are statistically significant. They are not uniformly positive for any of the wheats. Thus, the time-to-maturity effect does not appear to be significant. Stock coefficients are negative for all contracts, but only three are statistically significant at the 5% level. All rain coefficients are positive and six are significant [TABULAR DATA FOR TABLE 3 OMITTED] at the 5% level. Three temperature coefficients are negative. The coefficients on the interaction terms between rain and temperature are always negative, and generally significant, especially for spring-planted crops. The negative sign is somewhat intuitive in that plentiful rain and high temperatures suggest high yields, and so volatility is low. Low temperatures together with high rainfall (floods) or high temperatures and low rainfall (drought) increase volatility. It is not clear what impact low temperatures together with low rainfall should have on volatility.
The seasonality coefficients are strongly positive for the corn, soybean, and some Minneapolis spring wheat contracts. The t-statistics for the winter wheats are low, and negative coefficients occur for the May Kansas City contract. Overall, these results suggest that there is substantially more noise in the winter contracts than in the spring contracts. The [R.sup.2] values in table 3 confirm this. The equation [R.sup.2] values for the spring crop contracts are around 0.45-0.5, while for the winter wheat contracts they are around 0.25-0.35.
The Wald statistics for collective significance of different sets of variables within a system are presented in table 4. The plant variable is strongly significant for all the spring contracts, but, as expected, is not significant for the winter contracts. The seasonal dummy variables are strongly significant for the spring contracts, but are not significant for the winter wheat contracts. As is to be expected from table 3, time-to-maturity is never collectively significant. Despite the almost uniformly negative signs in table 3, the stock variable is statistically significant at the 10% level only for Chicago wheat. The annual dummies and weather variables are collectively significant in all systems.
Each system was also tested for equality of Plant coefficients across equations, and the hypothesis was rejected. Similarly, the hypothesis that TTM coefficients are equal across all equations in each system was rejected. In fact, of all the sets of coefficients delineated horizontally in table 3, equality of the coefficients within a system was rejected for all systems. This implies that the term structures of volatilities for contracts on the same commodity but with different expiration months differ fundamentally from each other.
In this paper we suggest that the structures of technology and preferences are among the primary determinants of futures price volatility. A contingent claims methodology is applied to provide theoretical support for this hypothesis. An analysis of futures price volatilities for multiple contracts of five major crops grown in the United States provides empirical support for the hypothesis. Planting date was used to indicate decreased flexibility and was found to be collectively significant at the 5% level for all of the spring crops considered. Plant was not significant for the winter wheat contracts, as expected. No time-to-maturity effect was found, which indicates that the time-to-maturity effect may be a proxy for what are truly consequences of decreasing flexibility. However, volatilities continued to display seasonality. Also, weather variables were found to have a causal effect on volatilities. A stock effect was not found, but the preponderance of negative stock coefficients suggests a negative effect on volatilities.
An interesting aspect of the theoretical model is that volatility bears a functional relationship with supply and demand parameters. Noting that the parameter in equation (2) can be easily transformed into an elasticity of expected supply with respect to own price, it should be possible to extract supply and demand elasticities from observations of futures price volatilities.
As the model does not completely explain seasonality, the flexibility effect is not likely to replace the information hypotheses that are well established in the literature. The effects are related in that flexibility is the aspect of technology required to transform information into increased expected profit (or expected utility). More research is warranted on the relationships between the information flow and flexibility effects and their relative importance. Also, it is not clear whether other fundamental economic forces may drive the Samuelson Hypothesis. Because statistical data may be more readily available for agricultural commodities than for metals, stocks, or financial instruments, perhaps commodities such as corn and wheat are most suitable for further study of the interactions between financial market phenomena and the structure of technology and preferences.
[TABULAR DATA FOR TABLE 4 OMITTED]
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David A. Hennessy is assistant professor in the Department of Economics at Iowa State University. Thomas I. Wahl is associate professor in the Department of Agricultural Economics at Washington State University.
COPYRIGHT 1996 American Agricultural Economics AssociationExisting literature on commodity futures price volatility emphasizes time to expiration and the resolution of uncertainty. In this paper we stress the supply and demand inflexibilities arising from decision making. A decision made on the supply (demand) side makes future supply (demand) responses less elastic. Therefore, a shock arising after a decision is made is more effective in changing the futures price than a shock before the decision is made. The results support the time-to-maturity hypothesis, but do not conflict with the state variable hypotheses of futures price volatility. Evidence supporting the impacts of inflexibilities are presented for corn, soybean, and wheat contracts.
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