Equilibrium Forward Curves for Commodities

Transcription

1 THE JOURNAL OF FINANCE VOL. LV, NO. 3 JUNE 2000 Equilibrium Forward Curves for Commodities BRYAN R. ROUTLEDGE, DUANE J. SEPPI, and CHESTER S. SPATT* ABSTRACT We develop an equilibrium model of te term structure of forward prices for storable commodities. As a consequence of a nonnegativity constraint on inventory, te spot commodity as an embedded timing option tat is absent in forward contracts. Tis option s value canges over time due to bot endogenous inventory and exogenous transitory socks to supply and demand. Our model makes predictions about volatilities of forward prices at different orizons and sows ow conditional violations of te Samuelson effect occur. We extend te model to incorporate a permanent second factor and calibrate te model to crude oil futures data. COMMODITY MARKETS IN RECENT YEARS ave experienced dramatic growt in trading volume, te variety of contracts, and te range of underlying commodities. Market participants are also increasingly sopisticated about recognizing and exercising operational contingencies embedded in delivery contracts. 1 For all of tese reasons, tere is a widespread interest in models for pricing and edging commodity-linked contingent claims. In tis paper we present an equilibrium model of commodity spot and forward prices. By explicitly incorporating te microeconomics of supply, demand, and storage, our model captures some fundamental differences between commodities and financial assets. Empirically, commodities are strikingly different from stocks, bonds and oter conventional financial assets. Among tese differences are: * Graduate Scool of Industrial Administration, Carnegie Mellon University. We tank René Stulz and an anonymous referee for teir elpful advice and David Backus, Jonatan Berk, Frank de Jong, Eric Gysels, Rick Green, Rob Heinkel, Glen Kentwell, Jon Leoczky, Bart Lipman, Kristian Miltersen, R. ~Nic! Nicolaides, Cristine Parlour, Pedro Santa Clara, Eduardo Scwartz, Steven Sreve, Sridar Tayur, and Harold Zang for comments. We also tank seminar participants at Carnegie Mellon, Nortwestern, Penn State, Toulouse, UC Berkeley, UCLA, te University of Iowa, te University of Texas at Austin, te University of Vienna, te Vienna Institute for Advanced Studies, and Yale as well as at te Sevent Annual Derivative Securities Conference at Queen s University, te 1997 NBER Summer Asset Pricing Worksop, te 1997 European Finance Association, te 1998 American Finance Association, and te 1998 Q-Group meetings. Te Decision, Risk, and Management Science Program of te National Science Foundation and te Institute for Quantitative Researc in Finance provided financial support for tis researc. Te Q-Group also awarded te paper its 1998 Roger Murray Prize. 1 For example, energy supply contracts often include so-called swing options wic give industrial consumers te flexibility to increase teir take above a baseload at a fixed price for a pre-agreed number of extra days eac mont. See Jaillet, Ronn, and Tompaidis ~1997! and Pilipovic and Wengler ~1998!. 1297

2 1298 Te Journal of Finance Commodity futures prices are often backwardated in tat tey decline wit time-to-delivery. For example, Litzenberger and Rabinowitz ~1995! document tat nine-mont futures prices are below te one-mont prices 77 percent of te time for crude oil. Spot and futures prices are mean reverting for many commodities. Commodity prices are strongly eteroskedastic ~see Duffie and Gray ~1995!! and price volatility is positively correlated wit te degree of backwardation ~see Ng and Pirrong ~1994! and Litzenberger and Rabinowitz ~1995!!. Te term structure of commodity forward price volatility typically declines wit contract orizon. Tis is known as te Samuelson ~1965! effect. However, violations of tis pattern occur wen inventory is ig ~see Fama and Frenc ~1988!!. Unlike financial assets, many commodities ave pronounced seasonalities in bot price levels and volatilities. In equilibrium, backwardation implies tat immediate ownersip of te pysical commodity entails some benefit or convenience wic deferred ownersip ~via a long forward position! does not. Tis benefit, expressed as a rate, is termed te convenience yield ~e.g., see Hull ~1997!!. A convenience yield is natural for goods, like art or land, tat offer exogenous rental or service flows over time. However, substantial convenience yields are also observed in oter commodities, suc as agricultural products, industrial metals, and energy, wic are consumed at a single point in time. Te teory of storage of Kaldor ~1939!, Working ~1948, 1949!, and Telser ~1958! explains convenience yields in terms of an embedded timing option. In particular, te older of a storable commodity ~e.g., oil, natural gas, copper! can decide wen to consume it. If it is optimal to store a commodity for future consumption, ten it is priced like an asset, but if it is optimal to consume it immediately, ten te commodity is priced as a consumption good. Tus, a commodity s spot price is te maximum of its current consumption and asset values. In contrast, forward prices derive solely from te asset value of te deferred rigt to consume after delivery. Inventory decisions are important for commodities because by influencing te relative current and future scarcity of te good tey link its current ~consumption! and expected future ~asset! values. Tis is unlike equities and bonds were outstanding quantities are fixed. Tis link is imperfect, owever, because inventory is pysically constrained to be nonnegative. Inventory can always be added to keep current spot prices from being too low relative to expected future spot prices. However, once te aggregate discretionary inventory 2 of a 2 Measured inventory is never literally driven to zero in practice since some stocks are eld as committed inputs in production ~e.g., gas or oil in transit!. By discretionary inventory we mean commodity stocks in excess of tose committed to te production process ~e.g., excange wareouse oldings or inventory eld oterwise by traders!. For example, Brennan ~1991! documents ig copper prices wen te aggregate inventory0sales ratio falls below a few weeks. Nondiscretionary inventories ave teir own convenience value by reducing production disruptions, minimizing delivery costs, etc.

3 Equilibrium Forward Curves for Commodities 1299 commodity is driven to zero, its spot price is tied solely to te good s ~ig! immediate use consumption value. Tus, stockouts break te link between te current consumption and expected future asset values of a good. Te result is backwardation and positive convenience yields. In tis paper we follow Deaton and Laroque ~1992, 1996!, Williams and Wrigt ~1991!, and Cambers and Bailey ~1996! and use a competitive rational expectations model of storage to study te impact of te embedded timing option on commodity spot and futures0forward pricing. 3 We assume tat te immediate-use consumption value is driven by a mean-reverting Markov process and solve for te equilibrium inventory of competitive, riskneutral agents. Te sock process and inventory rule ten jointly determine te spot and forward price processes. Our main results are te following: Te equilibrium term structure of spot and forward prices is decreasing in inventory and under a natural sufficient condition increasing in te current Markov sock. Endogenous binomial price trees are constructed for pricing and edging commodity-linked futures and ~by extension! options and oter derivatives. Conditional violations of te Samuelson effect occur wen inventory is sufficiently ig in te model. In particular, forward price volatilities can initially increase wit contract orizon. Hedge ratios for long-dated forward positions using sort-dated forwards are not constant, but are conditional on te current demand sock and te endogenous inventory level. A one-factor version of te model cannot matc bot te ig unconditional volatility of long-orizon Nymex crude oil futures prices and te conditional volatilities given contango and backwardation. However, a tractable two-factor augmentation of te basic model is more successful. An alternative to modeling forward and spot prices explicitly from economic primitives is to treat te convenience yield as an exogenous dividend process. For example, Brennan ~1991!, Gibson and Scwartz ~1990!, Amin, Ng, and Pirrong ~1995!, and Scwartz ~1997! all model spot prices and convenience yields as separate stocastic processes wit a constant correlation. 4 Altoug tese models are powerful tools for derivative pricing and edging, 3 Routledge, Seppi, and Spatt ~1999! extend tis analysis to multiple commodities and, specifically, to electricity. Wile electricity itself is not directly storable, potential marginal fuels ~e.g., natural gas, coal! are storable. Heinkel, Howe, and Huges ~1990! analyze a tree-period economy in wic optionality induces convenience yields and Bresnaan and Spiller ~1986! discuss te relationsip between inventory and te slope of te forward curve. Litzenberger and Rabinowitz ~1995! model te impact of timing options on te optimal extraction pat for a depletable resource and equilibrium prices. Pirrong ~1998! studies commodity option pricing wit storage. 4 Scwartz and Smit ~2000! present a model wit two factors, a long-run price component and a transitory disturbance, wic tey sow is equivalent to te Gibson and Scwartz ~1990! convenience yield model. Miltersen and Scwartz ~1998! model a stocastic term structure of convenience yields based on Heat, Jarrow, and Morton ~1992!.

4 1300 Te Journal of Finance our approac also as several attractions. First, explicitly modeling te joint evolution of inventory and spot prices ensures te consistency of te spot price and convenience yield ~i.e., forward price! dynamics. 5 Second, our model predicts tat te correlation between spot prices and convenience yields ~or teir innovations! is unlikely to be constant due to its dependence on inventory. Tird, inventory acts as a second state variable summarizing past socks, wic allows our model to capture a stocastic convenience yield wit only one exogenous factor. Te paper is organized as follows. Section I describes te basic one-factor model, demonstrates existence of equilibrium, and derives properties of te equilibrium inventory and spot price processes. Section II investigates properties of te forward curve and endogenous implied convenience yields. We also study te forward price volatility term structure and edge ratios. Section III carries out a numerical calibration exercise and presents a tractable two-factor version of te basic model. Section IV concludes. Proofs are collected in te Appendix. I. Equilibrium Model of Commodity Prices Te goal of tis paper is to caracterize spot and forward commodity prices in an equilibrium model of inventory wit nonnegative storage. Our analysis builds on and extends Deaton and Laroque ~1992, 1996!, Cambers and Bailey ~1996!, Williams and Wrigt ~1991!, and Wrigt and Williams ~1989!. We start wit general functional forms and develop tractable numerical implementations for derivative security valuation. We also investigate properties of te term structure of forward price volatility. A. Model Structure Consider a discrete-time, infinite orizon model in wic a single omogeneous commodity is traded in a competitive market at dates t 1,2,... Current production and consumption demand for immediate use are modeled as stocastic, reduced-form functions, g t and c t, of te spot price, P t. Te commodity can be stored by a group of competitive risk-neutral inventory traders wo ave access to a costly storage tecnology. Storage is costly due to a constant proportional depreciation or wastage factor, d ~0,1#. Storage of q units of te commodity at t 1 yields ~1 d!q at t. One can interpret d as spoilage ~for agricultural commodities! or as a volumetric cost ~for metals and energy!. For tecnical reasons d is strictly positive, but we abstract, for simplicity, from any oter fixed or marginal costs. 5 Analogously, in some term structure settings assuming exogenous dynamics for interest rates of bonds of different maturities can be incompatible wit equilibrium ~i.e., admit arbitrage opportunities!. Examples along tese lines are noted in Cox, Ingersoll, and Ross ~1985!. Heat et al. ~1992! describe te restrictions on forward price processes tat are implied by te absence of arbitrage. Arbitrary joint dynamics for stock prices and dividends may also be inconsistent wit equilibrium.

5 Equilibrium Forward Curves for Commodities 1301 Te spot price, P t, is determined by market clearing. At eac date t te two sources of te commodity, current production and incoming inventory g t ~1 d!q t 1, must equal te two types of demand, immediate consumption and outgoing inventory c t Q t. Tis can be rearranged to get c t ~P t! g t ~P t! Q t, ~1! were Q t Q t ~1 d!q t 1. Wen Q t is positive ~i.e., wen inventory is increased!, less of te good is left for immediate consumption. If te immediate use net-demand c t ~P t! g t ~P t! is monotone decreasing in te spot price P t, te spot market can be summarized wit an inverse net demand function P t f ~a t, Q t!. ~2! We initially abstract from permanent socks and model te net demand socks a t as realizations of a finite-dimensional, irreducible, m-state Markov process ~m 2!, wit transition probabilities p~a6a t! in a matrix. Te socks, a t, represent te transitory effects of weater and0or production disruptions on te immediate use net demand, c t g t. Tey are unaffected by current and0or past inventory. Were noted, we sometimes make an additional assumption tat p~a6a t!.0 for all a t, a ~denoted 0!, wic implies tat te demand socks ave a limiting distribution tat is independent of te current state. 6 Inventory ~or te cange in inventory Q! is te key endogenous variable. We assume tat te net demand function, f, as te following properties for all realizations a : ~A! f ~a, Q!.0 ~B! f~a, Q!is increasing in Q ~C! f ~a, Q! is continuous and unbounded from above in Q. Te first assumption states tat prices are positive. Te second says tat increased storage raises te good s marginal valuation since it reduces te amount available for immediate use. Te tird assumption, tat f is continuous and unbounded, is sufficient to ensure tat a market-clearing spot price exists. 7 Te marginal valuation, f, need not be linear. Nonlinearities in f can arise from eiter te demand, c t, or supply, g t. 6 An irreducible Markov process as te property tat is te unique ergodic set. Tis means tat all states are reacable wit positive probability in some finite number of transitions from any initial state. For example, tere are no absorbing states. Te additional assumption tat is strictly positive implies tat tere are no cyclically moving subsets. Alternatively, everywere we assume.. 0, we could instead assume some finite n exists suc tat n.. 0 ~see Karlin and Taylor ~1975!!. 7 Our analysis does not require tat Q is unbounded, but rater only tat f ~a, Q! is unbounded. Te special case in wic te maximum feasible current production output is bounded by some quantity g S ~e.g., as wit a arvest! as f ~a, Q! r ` as Q r g. S

6 1302 Te Journal of Finance Te one-period risk-free interest rate is assumed ere to be constant, r 0. Nonstocastic interest rates simplify te analysis and let us abstract from te difference between forward and futures prices. B. Equilibrium Inventory decisions of te risk-neutral traders are easily caracterized since te only motive for olding inventory is trading profit. 8 If spot prices were expected to rise by more tan te carrying costs ~i.e., wastage and interest!, additional inventory would be purcased. Tis would ten increase current ~and lower future! spot prices. Conversely, if prices were expected to fall ~or rise by less tan te carrying costs!, ten inventory would be sold. However, inventory or, more precisely, traders discretionary oldings in excess of stocks irreversibly committed to production can only be reduced to zero. Optimal trading, given tis pysical nonnegativity constraint, implies tat equilibrium spot prices and aggregate inventory must jointly satisfy P t ue t 1 # if Q t 0 ~3a! P t ue t 1 # if Q t 0, ~3b! were u ~1 d!0~1 r!, 1 and E denotes ~rational! expectations conditional on te information, a t and Q t 1, available at time t. Positive equilibrium inventory, Q t. 0, implies traders are indifferent to marginal canges in teir inventory. However, if inventory is stocked out, Q t 0, traders may want to sell additional units but are pysically constrained from doing so. One cannot consume goods tat do not yet exist. It is precisely tis asymmetry tat makes te embedded timing option in te spot commodity valuable and tat, in equilibrium, leads to option-like beavior in commodity prices. We use P~a, q! and J~a, q! to denote te equilibrium spot price and aggregate inventory functions given a current demand state, a, and previous aggregate inventory, q. Witout loss of generality, we assume Q 0 0. Definition of Equilibrium: $Q t, P t % is a stationary rational expectations equilibrium inventory and price process if inventory and spot-price functions Q 0 0, Q t J~a t,q t 1! and P t P~a t,q t 1! f ~a t, Q t ~1 d!q t 1! exist tat satisfy condition ~3! for all t. 8 Jagannatan ~1985! and Ricard and Sundaresan ~1981! derive equilibrium commodity prices under risk aversion. Tese papers do not consider te sort-sale constraint wile our model abstracts from risk aversion. Wit risk-averse traders te expectations in condition ~3! sould be taken using te risk-neutral martingale measure.

7 Equilibrium Forward Curves for Commodities 1303 Proposition 1 establises existence. Te corollaries tat follow identify some useful properties of te equilibrium inventory process. Tese are standard ~e.g., see Deaton and Laroque ~1992!! and are not te focus of our paper. PROPOSITION 1 ~Equilibrium!: A stationary rational expectations equilibrium exists and as te following properties: ~a! te equilibrium inventory J~a, q! is continuous in q and, for all a, q 0, and perturbations e 0, satisfies 0 J~a, q e! J~a,q! ~1 d!e ~4! ~b! a unique finite upper bound, Q max 0, exists suc tat J~a, q! Q max for all a and max # and J~a,Q max! Q max for some a, and ~c! te equilibrium spot price P~a, q! is continuous and decreasing in q and is bounded suc tat 0 P~a, q! ` for all a and max #. Costly storage and te assumptions on f ensure tat te slope of te inventory function J is less tan 1 d and, ence, tat equilibrium inventory is bounded. Anoter implication of te slope of J being less tan 1 d is tat equilibrium spot prices P t are smaller wen previous storage Q t 1 is iger. Prices are positive due to te proportional storage cost and finite since inventory ~and inventory canges! are bounded. If carrying costs, u, are large enoug, equilibrium inventory is always zero ~i.e., Q t Q max 0!. However, equilibrium storage can be positive if f ~a j,0! u E@ f~a,0!6a j # in some demand states a j. In equilibria wit positive storage, certain a can be identified as sell states in wic inventory is never accumulated. In particular, if te prior inventory is sufficiently low, ten all incoming inventory is sold for consumption and a stockout occurs. COROLLARY 1.1 ~Properties of J!: In a rational expectations equilibrium wit Q max 0, tere exists a nonempty set s of sell states suc tat for all a s s ~a! inventory is reduced so tat J~a s, q! ~1 d!q for all q and ~b! a critical inventory level q s 0 exists for eac a s s suc tat a stockout, J~a s, q! 0, occurs wenever previous inventory is sufficiently low, q q s. By construction, te equilibrium inventory and demand sock processes, ~a t, Q t!, are jointly Markovian. However, te sequence of demand socks, t $a t % t 0, affects te inventory level, Q t, because of te kink in te equilibrium condition ~3!. For example, a sell state followed by a buy state ~in wic inventory is accumulated! does not necessarily lead to te same ending inventory as vice versa. However, any time inventory stocks out ~i.e., is

8 1304 Te Journal of Finance driven to zero! te inventory process regenerates or renews. 9 Tis leads to te useful property tat te distribution of long-run inventory ~and, ence, of long-run prices! does not depend on current inventory. COROLLARY 1.2 ~Regeneration!: In te rational expectations equilibrium, ~a! stockouts occur wit positive probability; in oter words, te probability of perpetually positive inventory, Prob~Q t 0, Q t 1 0, Q t 2 0,...6a t,q t 1!, is zero for any a t and Q t 1, ~b! te long-run probability distributions of future inventory and prices, Q T and P T, are independent of current inventory, Q t,astr` ~c! if te transition probabilities are all strictly positive, 0, ten ~i! limiting inventory and price distributions exist and do not depend on a t or Q t 1 in tat lim Tr` Prob~Q T q6a t,q t 1! f Q ~q! and lim Tr` Prob~P T p6a t,q t 1! f P ~ p!, and ~ii! te limiting probability of a stockout is strictly positive, f Q ~0! ~0,1#. Inventory cannot always be positive wit positive carrying costs. Hence, stockouts occur and current inventory Q t only as a temporary effect on future ~T t! inventory, Q T, and prices, P T. If, moreover, te Markov sock process as an invariant limiting distribution, ten te current net demand state a t also as only a temporary influence. C. Numerical Example To implement our model we must specify a Markov process for te netdemand socks a t, te inverse demand function f, te storage cost rate d, and interest rate r. Our baseline parameterization is a linear two-state special case: a H 1 a L 0 p~a H6a H! p~a L 6a H!! p~a H 6a L! p~a L 6a ~5! L P t f ~a, Q! a Q, wit a storage cost d 0.1 and interest rate r 0. Several features of tis example deserve comment. First, seasonality is easily added by introducing cyclically moving subsets into te Markov structure. 10 Second, future ig ~low! demand states are more likely if te current state is ig ~low!. Some 9 As is common in te study of inventory systems, te inventory process can be used to define a renewal process over te stockout event. See Capter 5 of Karlin and Taylor ~1975!. 10 For example, we could partition into four seasons k were k $A, W, S, M% wit positive transition probabilities p~a6a t!.0 only if a k~t 1! were k~t 1! denotes te season of date t 1.

9 Equilibrium Forward Curves for Commodities 1305 Figure 1. Numerical example of equilibrium inventory. Te equilibrium outgoing inventory, Q t, is plotted as a function of te previous inventory, Q t 1, in te ig and low net demand states given te baseline parameters of te numerical example: a H 1, a L 0, p~a H 6a H! p~a L 6a L! 0.75, r 0, and d 0.1. of our teoretical results on te volatility of forward prices assume tat te Markov process as tis second property. Despite its simplicity, tis twostate, linear specification is able to generate endogenous binomial trees of spot ~and forward! prices wit many interesting features. Figure 1 sows te equilibrium inventory function Q t J~a,Q t 1! for tis numerical example. 11 Given te previous inventory Q t 1, outgoing inventory Q t is greater in te low state a L tan in te ig state a H. In particular, J~a H,Q t 1! ~1 d!q t 1 J~a L,Q t 1! so tat, consistent wit Corollary 1.2 ~wit just m 2 states!, inventory is accumulated in te low state and consumed in te ig state. 11 We calculate J~a, q! using a piecewise linear approximation for eac a on an equally spaced grid, G, wit 1000 points. From an initial conjecture J 0, successive approximations J n are calculated in a fixed-point contraction mapping algoritm. For eac a and q G, J n 1 ~a, q! is te maximum of te solution to te equilibrium condition wit equality ~equation ~3a! and zero. Tis is very similar to te construction in te proof of Proposition 1. Te convergence rate is rougly 1 d, requiring typically 20 to 50 iterations to be witin tolerance of te fixed point, J~a, q!. See Deaton and Laroque ~1992! for additional details. In Section III we use a tird-order polynomial approximation of J ~in place of te discrete grid! to increase te speed of te algoritm.

10 1306 Te Journal of Finance II. Properties of Forward and Spot Prices Define F t, t n ~a t,q t 1! as te forward price agreed to at date t for one unit of te commodity to be paid for and delivered at a future date t n given te information, a t and Q t 1, available at t. Since traders are risk neutral and te interest rate is nonstocastic, market clearing requires F t, t n ~a t,q t 1! t n 6a t,q t 1 #. ~6! Since forward contracts only involve payment at delivery t n, te carrying costs, d and r, enter forward prices only indirectly troug teir effect on te inventory and spot price processes. For concreteness, Figure 2 illustrates te range of forward-price term structures possible in our numerical example. 12 Eac forward curve is for a different combination of net demand sock, a L or a H, and previous inventory, Q t 1. Te forward curves are upward sloping in te low demand state a L and slope downward in te ig demand state a H wit a low ~or zero! prior inventory. Wen demand is ig and te incoming inventory is at a moderate level, ten te forward curve can be ump saped. In particular, forward prices initially rise ~i.e., P t u E t 1 # E t 1 # F t, t 1 since outgoing inventory, Q t, is positive!, but eventually decline to a state-independent ~constant!, long-term forward price, F` ~see Proposition 3 below!. One artifact of tis two-state example is tat te forward curves are identical in a stockout since outgoing inventory is 0 ~by definition! and te state is always a H. Te stockout spot prices differ, owever, because of differences in te level of incoming inventory. Te result is te fan at F t, t 1 in te stockout states. A. Inventory and Forward Prices Inventory plays a crucial role as an endogenous state variable summarizing te cumulative impact of past socks, a t 1, a t 2,... in our model. To understand its impact on prices, consider two equilibrium inventory and price sequences tat differ only by an ~exogenous! perturbation to te initial 12 Calculating exact forward prices requires a tree of all possible future spot prices. Since te inventory process is not recombining, eac iteration to determine forward prices to orizon n potentially requires, given m demand states, calculations on te order of m n. Wit a discrete grid of 1000 levels of inventory, tis produces a large ~but sparse! Markov transition matrix tat can be used to calculate forward prices of any orizon as well as te limiting ~unconditional! distribution. Te impact of approximation errors on prices from rounding te inventory level are limited by Proposition 2. One can improve te accuracy ~or reduce te number of discrete inventory levels! by coosing te inventory grid based on te equilibrium inventory function. Te renewal property implies tat one only needs to track pats until inventory reaces a stockout. Since stockouts appen wit non-zero probability, most of te time inventory is witin only a few steps ~i.e., demand realizations! from a stockout. For example, Figure 2 was produced wit just 100 points ~carefully cosen! on te inventory grid. Te 60 most likely forward curves are sown.

11 Equilibrium Forward Curves for Commodities 1307 Figure 2. Forward curves. Te 60 most common forward curves are depicted in te ~A! ig, a H, and ~B! low, a L, demand states in te numerical example wit baseline parameters: a H 1, a L 0, p~a H 6a H! p~a L 6a L! 0.75, r 0, and d 0.1. inventory. In particular, te pat of demand realizations, $a t %, is identical for te two sequences. Te following proposition sows tat spot and forward prices are decreasing in initial inventory, but tat te effect is only temporary.

12 1308 Te Journal of Finance PROPOSITION 2 ~Increase Inventory!: Consider two inventory processes $Q t % and a perturbed process $Q x t % were Q t J~a t,q t 1! and Q x x t J~a t,q t 1!. If Q 0 0and Q x ` 0 x 0, ten for every realization of $a t % t 1 ~a! te perturbed inventory is (weakly) larger, but te difference srinks wit time, 0 Q x t Q t ~1 d! t x, ~b! te perturbed spot and forward prices are weakly lower, P x t P t and x F t, t n F t, t n for all n 0, ~c! lim tr`~q x t Q t! 0, lim tr`~p x x t P t! 0, lim tr`~f t, t n F t, t n! 0, and ~d! for all e ~0,1!, tere exists a date t suc tat Prob~Q x t Q t ) e for all t t. Since te slope of J is less tan 1 d, any positive perturbation x reduces te current and future net inventory canges Q t. Tus spot and forward prices are lower. However, te perturbation is strictly temporary and dies out at a rate ~1 d! t. Tis is wy statements ~a! to ~c! old pat-by-pat for every sequence of demand socks. Furtermore, after te first stockout in te Q process, bot inventory processes regenerate and are identical tereafter. Since inventory does not affect te probabilities of future demand socks, canges in inventory simply sift te probability distribution of future spot prices. Altoug our focus is primarily on forward prices, Proposition 2 as an obvious but important implication for option prices, as stated in Corollary 2.1. COROLLARY 2.1 ~Derivative Prices!: Te prices of European calls (and all oter derivatives wose payoffs are increasing in future spot or forward prices) are decreasing in inventory in te stationary rational expectations equilibrium. Since current inventory as no long-term impact, te equilibrium prices preserve many of te features of te Markov structure of te underlying demand socks. For example, if all demand socks ave strictly positive probability ~ 0!, ten forward prices, F t, t n, are bounded in a range tat tigtens wit te contract orizon, n, and a limiting forward price, F`, exists tat is invariant to te current demand state and inventory level. 13 PROPOSITION 3 ~Forward Prices!: In te stationary rational expectations equilibrium of an economy wit 0, two monotonic sequences, $F n min % ` n 0 (wic n ` is increasing in n) and $F max % n 0 (wic is decreasing in n), bound forward prices F t, t n n n n min, F max # for all t and n wit lim nr` F min lim nr` F max F`. Te limiting forward price, F`, is te unconditional mean spot price. In general, it is ard to determine te effect of storage on te mean spot price. Since storage is costly, less of te good is available for consumption, wic increases its price. However, storage also smoots socks tat may, de- 13 Wit seasonal cyclical subsets in, tere is not a global limiting price distribution, but rater one for eac season.

13 Equilibrium Forward Curves for Commodities 1309 pending on te curvature of f, reduce te mean price. However, even if te net-demand function f is linear, te effect of moving from efficient storage ~d 0! to no-storage ~d 1! on F` is not monotonic. Increasing d, olding inventory constant, raises spot prices since more of te good is lost to te storage cost. However, equilibrium storage also canges wit d. At low values of d~close to 0!, an increase in storage costs raises te mean spot price since te mean equilibrium inventory levels are large. However, for ig values of d, and, ence, lower equilibrium inventory levels, te mean spot price decreases since less is lost to storage costs. In te limit as d r 1, noting is stored and noting is lost to storage cost, te limiting forward price, F`, ise@f~a t,0!#. B. Net Demand Socks and Forward Prices Te a t realizations influence spot and forward prices directly, troug current net demand, and indirectly, via te probability distribution over future socks. Due to tese two influences, two furter assumptions ~in addition to ~A!, ~B!, and ~C! in Section I.A! are used to give an unambiguous order to te m demand states of : ~D! For all Q, te inverse net demands are ordered wit f ~a 1, Q! {{{ f ~a m, Q!. ~E! First-Order Stocastic Dominance: for all a, a were f ~a, Q! f ~a, Q!, k ( j 1 k p~a j 6a! ( p~a j 6a! ~7! for all k 1,...,m ~wit strict inequality for at least one k!. Te first assumption, ~D!, says tat te spot prices corresponding to te net demand states ave te same order regardless of te cange in inventory. Te second, ~E!, states tat te transition probabilities from ig states, a, first-order stocastically dominate tose from lower states, a.inour two-state example, tis implies tat socks are persistent in tat p~a H 6a H!, p~a L 6a L!.0.5. Under tese two ~sufficient! assumptions forward prices ave a particularly simple structure. PROPOSITION 4 ~Increase Demand!: In te stationary rational expectations equilibrium of an economy satisfying (D) and (E), ~a! spot prices are ordered: P~a 1, q! {{{ P~a m, q!, ~b! forward prices are ordered: F t, t n ~a 1, q! {{{ F t, t n ~a m, q! for all orizons t n t. Part ~a! establises tat if ~D! and ~E! old, ten inventory just smoots spot prices in tat te demand-state order of socks in ~D! is preserved in te equilibrium spot prices. However, ~D! alone is not sufficient for tis j 1

14 1310 Te Journal of Finance spot-price ordering wen ~te number of states! m. 2. Witout an assumption on te conditional transition probabilities suc as ~E!, a low demand state ~according to ~D!! could be associated wit a large probability of ig future demand. In suc a case te resulting large equilibrium inventory demand migt induce a ig spot price despite te low current demand. Te stocastic dominance assumption in ~E!, owever, is sufficient to ensure tat low states lead to low equilibrium spot prices. Part ~b! of te proposition establises tat, given ~D! and ~E!, te forward prices also preserve te demand-state order. In oter words, given any fixed level of incoming storage, te forward curve in any state, a, lies everywere below ~i.e., does not cross! te forward curve in eac iger-demand state, a 1,...,a m. A key implication of Proposition 4~b! is tat forward prices all move in te same direction and, tus, tat edge ratios ~using one maturity forward contract to edge anoter! are all positive. A perturbation of te demand state from a low state a to a iger state a as two effects on te distribution of future spot prices. First, te firstorder stocastic dominance assumption of ~E! improves te distribution of future demand states. Second, te perturbation affects current equilibrium inventory. In te case were inventory is reduced, J~a,Q t 1! J~a,Q t 1!, te effect on forward prices is reinforced. However, if inventory is increased, J~a,Q t 1! J~a,Q t 1!, tis partially offsets te improved demand state distribution. 14 To see wat can appen witout an assumption like ~E!, consider our simple two-state example. Wit just two states, $a L, a H %, spot prices at eac date t are ordered, P~a H,Q t 1! P~a L,Q t 1!, even witout ~E!. Te Markov process violates ~E! if p~a H 6a H! and p~a L 6a L! are less tan 0.5. In tis case a realization of a L at time t implies a iger conditional probability of a H at t 1 tan does a H at t. Tus, perturbing te current demand state from a L to a H can lead to lower forward prices at odd orizons and iger prices at even orizons. Tis will occur if te difference in Q t is small ~e.g., if d is large!. Terefore witout assumption ~E!, forward prices can move in opposite directions. C. Numerical Example (Continued) Our numerical example as only two net-demand states, a L and a H, but te dynamics of inventory induce an endogenous binomial tree of forward curves over time. Figure 3 illustrates part of te tree. Four features deserve comment. First, since te starting node ere at t is a stockout wit zero incoming inventory, tis node and its two successors are repeated at date t 1 if te next sock is anoter ig realization a H. Tis is te renewal feature of inventory and prices ~see Corollary 1.2!. Second, te degree of 14 In economies wit more tan two demand states ~m 3!, it is possible tat iger demand states may lead to iger current inventory levels. For example, if te lowest state as a very ig conditional probability of recurring, ten equilibrium storage in tat state can be zero wile storage in an intermediate state is strictly positive. Despite tis, spot prices and forward prices increase wit te demand state.

15 Equilibrium Forward Curves for Commodities 1311 Figure 3. Binomial tree of forward curves. Part of te binomial tree of forward curves is sown, starting in te ig state, a~0! a H, following a stockout, Q~0! 0, in te previous period. Te baseline parameters are used ere: a H 1, a L 0, p~a H 6a H! p~a L 6a L! 0.75, r 0, and d 0.1.

16 1312 Te Journal of Finance backwardation or contango of te forward curve depends on ow muc inventory as been accumulated. Tus, an a L at t 1 ~making inventory ceap! can flip te curve from backwardation to contango. Tird, te model can also produce ump-saped curves. Moreover, te ump moves furter back along te forward curve if a run of low socks leads to a buildup of inventory and tus puses te possibility of a stockout furter into te future. Fourt, te tree is nonrecombining. Te forward curve given a sock sequence $a H, a L % is very different from te forward curve given te reverse sequence $a L, a H % due to te equilibrium implication of te nonnegative inventory constraint. D. Spanning Altoug spot prices are binomial ere, te inability to sort te commodity in a stockout limits te payoffs tat can be spanned wit te spot good. 15 However, our example is still dynamically complete using a bond and a oneperiod forward contract. Terefore, our framework can determine a market price of convenience yield risk in a more complex model wit risk-averse agents. Tis property can be extended to economies wit binomial ~rater tan just two-state! sock processes. 16 PROPOSITION 5 ~Spanning!: Te market is dynamically complete using a oneperiod forward contract and a riskless bond if at eac date t te Markov process is binomial and satisfies (D) and (E). Unfortunately, extending tis result to general Markov processes is not straigtforward. Te pat dependence of equilibrium inventory prevents te use of standard arguments of generic completeness. Te pat dependence implies tat tere are uncountably many pats. Terefore, in order to ceck te dynamic completeness, one needs sufficient conditions for te payoff matrix using forward prices and a riskless bond to be invertible for all incoming inventory and demand state pairs. Fortunately, te two-state and binomial settings are quite flexible. E. Forward Curve Slopes Te relationsip between contemporaneous spot and forward prices is usually described eiter in terms of te slope of te forward curve or wit implied convenience yields. We consider slopes first and ten restate tese results for convenience yields in Section II.F. 15 In addition, replication wit a long spot position is a dominated strategy in a stockout ~from equilibrium condition ~3!!. Ross ~1976! and Breeden and Litzenberger ~1978! discuss using derivative securities to complete te market and Raab and Scwager ~1993! consider replication wit sort-sale restrictions on some assets. 16 By binomial we mean tat, given any sock a t at any time t, only two socks a j ave positive probability, p~a j 6a t!.0. In contrast to our ongoing example, tese need not be te same two socks for eac a t.

17 Equilibrium Forward Curves for Commodities 1313 Since forward prices are positive, we can define te one-period ~scaled! slope wit respect to delivery orizon, n, as ' F t,t n F t,t n 1 F t,t n F t,t n. ~8! A forward curve is in contango if it is upward sloping ~F '. 0! and in backwardation if it is downward sloping ~F ', 0!. 17 Storage and interest carrying costs make forward curves slope upward; negative slopes are possible due to current or future potential stockouts. PROPOSITION 6 ~Forward Slopes!: Te scaled slopes of forward prices in te ' stationary rational expectations equilibrium are bounded by F t, t n ~r d!0~1 d! for all t and orizons n 1. Additionally, if 0, ten ' ' ~a! if F t, t n ~r d!0~1 d! for a delivery orizon n, ten F t, t ~r d!0~1 d! for all later delivery orizons n, ~b! for every date t tere is a future date t * t suc tat te probability ' of future backwardation, Prob~F t, t 1 06a t,q t 1!, is strictly positive for all dates t t *, and ' ~c! te forward curve at long orizons is flat wit lim nr` F t, t n 0. Part ~a! follows from te equilibrium condition ~3! and iterated expectations. If outgoing inventory is positive, ten condition ~3a! olds and te forward slope is positive and at its maximum, ~r d!0~1 d!, between P t and F t, t 1. If, instead, current inventory is zero and condition ~3b! olds strictly, ten te slope is less tan te maximum. Using iterated expectations, te argument extends to longer orizons. Parts ~a! troug ~c! all use te assumption tat p~a6a t!.0. If te probability of a stockout at a future date t * t is positive, ten it is also positive at all longer orizons since sell states ave a positive probability of recurring. Strictly positive transition probabilities also imply tat te combination of te igest possible demand state and zero incoming inventory as a positive probability. Since tis produces te igest possible spot price ~P max!, te forward curve must ave a negative slope in tis situation. Finally, a limiting forward slope of zero is an immediate consequence of Proposition Some autors ~e.g., Litzenberger and Rabinowitz ~1995!! distinguis between strong and weak backwardation. Strong backwardation occurs wen te forward price is lower tan te spot price ~i.e., F t, t n P t!. Weak backwardation is wen te forward price is lower tan te spot price adjusted for time value of money ~i.e., F t, t n P t ~1 r! n!. Oter autors ~e.g., Keynes ~1930!! use te term backwardation to refer to te ~average! difference between forward prices and te future spot price. Given risk neutrality ere, we restrict attention to te contemporaneous spot-forward relationsip.

18 1314 Te Journal of Finance F. Convenience Yields Anoter way to describe forward curves is in terms of convenience yields, wic express te interim benefits accruing to te pysical owners of a commodity as a rate. In our model, tis benefit is te timing option to consume te good in ig demand states ~or to sell it for consumption by someone else! and ten buy it back at a lower expected price in te future. Given any forward curve in our model, we can calculate te corresponding implied endogenous convenience yields using te cost-of-carry relation. Tus, convenience yields are an output of our model rater tan an input as in Brennan ~1991!, Scwartz ~1997!, and Amin et al. ~1995!. Te one-period convenience yield, y t, t ~a t,q t 1!, over te interval t to t 1 given te spot and forward prices at date t is defined implicitly by F t, T ~a t,q t 1! P~a t,q t 1! 1 r T 1 1 d T t t t ) ~1 y t, t ~a t,q t 1!!. ~9! Since interest rates and storage costs are constant ere, tis can be written as 18 ~1 d! y t,t 1 ~1 r! F t,t 1 F t,t. ~10! Using tis definition, te results in Proposition 6 can be rewritten to sow tat te implied convenience yield in our model is always well defined and nonnegative, and tat y t, t, 1. PROPOSITION 7 ~Convenience Yield!: In te stationary rational expectations equilibrium, ~a! implied convenience yields are nonnegative, y t, t 0, ~b! te convenience yield is positive, y t, t 0, if and only if te stockout probability is positive, Prob~Q t 0! 0, and ~c! in an economy wit 0, ~i! if y t, t. 0, ten y t, t' 0 for all later delivery dates, t' t, and ~ii! te limiting convenience yield is constant wit lim tr` y t, t ~d r!0~1 r!. Convenience yields inerit teir option-like quality from te embedded timing ~i.e., storage! option in spot prices. Convenience yields are strictly positive only if tere is a positive probability of a stockout ~wic is like an option exercise!. As in Proposition 6, if sell states repeat wit positive prob- 18 Tis is te discrete-time analogue to te instantaneous convenience yield in continuoustime models suc as Amin et al. ~1995! or Scwartz ~1997!. Convenience yields are typically expressed net of storage. However, since storage costs are constant ere, tis distinction is not important.

19 Equilibrium Forward Curves for Commodities 1315 Figure 4. Convenience yields versus spot price. Te convenience yield is plotted at different orizons against te corresponding spot price given te baseline parameters: a H 1, a L 0, p~a H 6a H! p~a L 6a L! 0.75, r 0, and d 0.1. ability ~e.g., if.. 0!, ten a positive convenience yield at one orizon implies positive convenience yields at all longer orizons. Finally, te limiting convenience yield, like te limiting forward price, does not depend on te current demand state or inventory level. Comparing te implied ~equilibrium! convenience yields in our model wit te usual ~exogenous! convenience yield processes reveals some differences. First, our model does not reduce to a one-factor Brennan ~1991! model were convenience yield is exogenously specified as a function of te spot price. Here, convenience yields are functions of bot te exogenous demand state and te endogenous inventory level. Second, Scwartz ~1997! assumes tat innovations in spot prices and instantaneous spot convenience yields ave a constant correlation. Tis in turn implies tat te correlation between spot prices and convenience yields at longer orizons, y t, t is also constant. In our model, owever, te correlation between te spot price and te one-period spot convenience yield, y t, t, depends on te endogenously determined inventory level, Q t 1 ~Proposition 7~b!!. In particular, y t, t is nonzero only wen outgoing inventory is zero. At longer orizons te convenience yield y t, t depends ~from Proposition 7~b!! on bot te probability and likely severity of a stockout at date t and, terefore, on te current inventory Q t. Figure 4 illustrates tese points by plotting te implied convenience yields versus te corresponding spot price in our numerical example. Wen te current spot price is ig at t, te corresponding one-period convenience yield is also ig. Hig spot prices are te result of bot ig current de-

20 1316 Te Journal of Finance mand, a H, and low incoming storage. Tis situation leads to a stockout. P t is decreasing in incoming inventory since te inventory buffers te ig demand sock. However, F t, t 1 is unaffected in a stockout since outgoing inventory is zero. Tis is te renewal feature seen in Figures 2 and 3. Te result is tat y t, t is decreasing in incoming inventory and is maximized wen Q t 1 0. At low spot prices, outgoing inventory Q t is positive and, terefore, y t, t 0. At longer orizons, n. 1, te spot-price0convenience-yield relation is also state-dependent, but te nonlinearity is muc less dramatic. Compare, for example, te convenience yield at te seven-period orizon, y t, t 7, wit y t, t in Figure 4. Since our limiting convenience yield is constant, te spot-price0 convenience-yield correlation falls to zero ~i.e., a constant! as te orizon lengtens. Te convenience yield y t, t n depends on te probability of a future stockout at date t n and its likely severity. Te probability and severity depend on te current state and inventory. However, due to te renewal feature, te dependence on te current state at very long orizons is small. 19 G. Conditional and Unconditional Volatilities Te volatility of forward prices at different orizons is important for bot derivative security pricing and dynamic edging. Empirically, commodities typically exibit a pattern of forward price volatility wic is declining wit contract orizon known as te Samuelson ~1965! effect. Tis is usually attributed to te smooting of expectations over a mean-reverting process. However, in our model te Samuelson effect need not old conditionally in all states at all orizons. PROPOSITION 8 ~Volatility!: In te stationary rational expectations equilibrium of an economy wit 0, te volatility of forward prices satisfies te following: ~a! Tere is a orizon N suc tat for all a t and q t 1, te conditional forward price variance, var~f t 1, t 1 n 6a t, q t 1!, is decreasing in te orizon lengt n for all n N. ~b! Tere is a demand/inventory/contract-orizon combination ~a t, q *, N! suc tat te conditional forward price volatility, var~f t 1, t 1 n 6a t, q t 1!, is increasing in n for sort orizons n N wen inventory is sufficiently ig, q t 1 q *. Part ~a! says tat, for sufficiently long orizons, forward price volatilities decline wit maturity regardless of te current demand state and inventory. Tis follows immediately from Proposition 3 were te tigtening bounds on 19 In Figure 4 te fact tat te convenience yield can take two possible values at some intermediate spot price levels ~e.g., around 0.42 for y t, t 2! is an illustration tat our convenience yield cannot be represented as a function of only te spot price as in Brennan ~1991!. A low state a L and low inventory or a ig state a H and ig inventory can bot lead to te same spot price. However, since te forward prices are different in tese situations, so are te implied convenience yields. Of course in our two-state example, very ig spot prices are only consistent wit a H. Note tis feature can also be observed in y t, t 5 and y t, t 7.

21 Equilibrium Forward Curves for Commodities 1317 Figure 5. Conditional dispersions. Te conditional dispersion, F t 1, t n ~a H, q t! F t 1, t n ~a L, q t!, of date t 1 futures prices is plotted for different orizons, n, as a function of te date t inventory, q t, given te baseline parameters: a H 1, a L 0, p~a H 6a H! p~a L 6a L! 0.75, r 0, and d 0.1. forward prices and existence of a limiting forward price, F`, are consequences of te finite Markov structure ~wit.. 0! and te regeneration property of inventory. However, part ~b! states tat conditional violations of te Samuelson effect can occur at sorter orizons. In particular, wen current inventory is ig, te spot price is less volatile tan te sort orizon forward. Fama and Frenc ~1988! document empirically suc a relation between inventory and te conditional volatilities of different orizon forward prices. Te relation between volatility and inventory is easily seen in our twostate numerical example. Consider te difference F t, t n ~a H,Q t 1! F t, t n ~a L,Q t 1! as a measure of conditional dispersion. Since te probability at t 1 of te forward prices F t, t n ~a H,Q t 1! and F t, t n ~a L,Q t 1! at t are just p~a H 6a t 1! and p~a L 6a t 1! regardless of te particular forward orizon n, conditional dispersion is one-to-one increasing in volatility in tis example. In Figure 5 te conditional dispersions are ordered by contract orizon n at low inventory levels. In particular, if inventory is zero ~Q t 1 0!, te dispersion ~volatility! is determined solely by te Markov demand process and, terefore, declines wit maturity. However, at ig levels of current inventory Q t 1, forward price volatilities are potentially increasing in maturity at several orizons. Wit enoug inventory, stockouts may not be possible for n periods so tat te n-period forward price is simply te spot price scaled up by te cost-of-carry relation, F t, t n u n P t, were u n 1 ~see equation ~3a!!. In suc cases, te longer-term ~n pe-