Equilibrium Forward Curves for Commodities

This paper presents an equilibrium model of the term structure of forward prices for storable commodities. Our approach differs from Brennan (1991) and Schwartz (1997) in that we do not explicitly assume an exogenous "convenience yield." Rather, our spot commodity has an embedded timing option that is absent in forward contracts, which arises from a non-negativity constraint on inventory. The value of this option changes over time as a function of both the endogenous inventory and exogenous transitory shocks to supply and demand. Our model makes predictions about the volatilities of forward prices at different horizons and shows how conditional violations of the "Samuelson effect" can occur. We address the related issue of dynamically trading near-dated forward contracts to hedge a long-dated position. We also present a tractable extension of the model with a permanent second factor and a calibration of the model to crude oil futures price data. Equilibrium Forward Curves for Commodities Commodity markets in recent years have experienced dramatic growth in trading volume, the variety of contracts, and the range of underlying commodities. Market participants are also increasingly sophisticated about recognizing and exercising operational contingencies embedded in delivery contracts. For all of these reasons there is a widespread interest in models for pricing and hedging commodity-linked contingent claims. In this paper we present an equilibrium model of commodity spot and forward prices. By explicitly incorporating the microeconomics of supply, demand, and storage, our model captures some fundamental differences between commodities and financial assets. Empirically, commodities are strikingly different than stocks, bonds and other conventional financial assets. Among these differences are: • Commodity futures prices are often "backwardated" in that they decline with time-to-delivery. For example, Litzenberger and Rabinowitz (1995) document that nine-month futures prices are below the one-month prices 77 percent of the time for crude oil. • Spot and futures prices are mean reverting for many commodities. • Commodity prices are strongly heteroscedastic (see Duffie and Gray (1995)) and price volatility is positively correlated with the degree of backwardation (see Ng and Pirrong (1994) and Litzenberger and Rabinowitz (1995)). • The term structure of commodity forward price volatility typically declines with contract horizon. This is known as the "Samuelson (1965) effect." However, violations of this pattern occur when inventory is high (see Fama and French (1988)). • Unlike financial assets, many commodities have pronounced seasonalities in both price levels and volatilities. In equilibrium, backwardation implies that immediate ownership of the physical commodity entails some benefit or convenience which deferred ownership (via a long forward position) does not. This benefit, expressed as a rate, is termed the "convenience yield" (e.g., see Hull (1997)). A convenience yield is natural for goods, like art or land, that offer exogenous rental or service flows over time. However, substantial convenience yields are also observed in other commodities, like agricultural products, industrial metals, and energy, that are consumed at a single point in time. 1 For example, energy supply contracts often include so-called "swing" options which give industrial consumers the flexibility to increase their "take" above a baseload at a fixed price for a pre-agreed number of extra days each month. See Jaillet, Ronn and Tompaidis (1997) and Pilipovic and Wengler (1998). 1 Equilibrium Forward Curves for Commodities April 1999 The "theory of storage" of Kaldor (1939), Working (1948, 1949) and Telser (1958) explains convenience yields in terms of an embedded timing option. In particular, the holder of a storable commodity (e.g., oil, natural gas, copper) can decide when to consume it. If it is optimal to store a commodity for future consumption, then it is priced like an asset, but if it is optimal to consume it immediately, then the commodity is priced as a consumption good. Thus, a commodity's spot price is the maximum of its current consumption and asset values. In contrast, forward prices derive solely from the asset value of the deferred right to consume after delivery. Inventory decisions are important for commodities because – by influencing the relative current and future scarcity of the good – they link its current (consumption) and expected future (asset) values. This is unlike equities and bonds where outstanding quantities are fixed. This link is imperfect, however, because inventory is physically constrained to be non-negative. Inventory can always be added to keep current spot prices from being too low relative to expected future spot prices. However, once the aggregate discretionary inventory of a commodity is driven to zero, its spot price is tied solely to the good’s (high) "immediate use" consumption value. Thus, "stockouts" break the link between the current consumption and expected future asset values of a good. The result is backwardation and positive convenience yields. In this paper we follow Deaton and Laroque (1992, 1996), Williams and Wright (1991), and Chambers and Bailey (1996) and use a competitive rational expectations model of storage to study the impact of the embedded timing option on commodity spot and futures/forward pricing. We assume the "immediate-use" consumption value is driven by a mean-reverting Markov process and solve for the equilibrium inventory of competitive, risk-neutral agents. The shock process and inventory rule then jointly determine the spot and forward price processes. Our main results are: • The equilibrium term structure of spot and forward prices is decreasing in inventory and – under a natural sufficient condition – increasing in the current Markov shock. • Endogenous binomial price trees are constructed for pricing and hedging commodity-linked futures and (by extension) options and other derivatives. 2 Measured inventory is never literally driven to zero in practice since some stocks are held as committed inputs in production (e.g., gas or oil in transit). By discretionary inventory we mean commodity stocks in excess of those committed to the production process (e.g., exchange warehouse holdings or inventory held otherwise by traders). For example, Brennan (1991) documents high copper prices when the aggregate inventory/sales ratio falls to a within a few weeks. Non-discretionary inventories have their own convenience value by reducing production disruptions, minimizing delivery costs, etc. 3 Routledge, Seppi and Spatt (1998) extends this analysis to multiple commodities and, specifically, to electricity. While electricity itself is not directly storable, potential marginal fuels (e.g., natural gas, coal) are storable. Heinkel, Howe and Hughes (1990) analyzes a three-period economy in which optionality induces convenience yields and Bresnahan and Spiller (1986) discuss the relationship between inventory and the slope of the forward curve. Litzenberger and Rabinowitz (1995) model the impact of timing options on the optimal extraction path for a depletable resource and equilibrium prices. Pirrong (1998) studies commodity option pricing with storage. 2 Equilibrium Forward Curves for Commodities April 1999 • Conditional violations of the "Samuelson effect" occur when inventory is sufficiently high in the model. In particular, forward price volatilities can initially increase with contract horizon. • Hedge ratios for long-dated forward positions using short-dated forwards are not constant, but are conditional on the current demand shock and the endogenous inventory level. • A one-factor version of the model cannot match both the high unconditional volatility of longhorizon Nymex crude oil futures prices and the conditional volatilities given contango and backwardation. However, a tractable two-factor augmentation of the basic model is more successful. An alternative to modelling forward and spot prices explicitly from economic primitives is to treat the convenience yield as an exogenous "dividend" process. For example, Brennan (1991), Gibson and Schwartz (1990), Amin, Ng and Pirrong (1995), and Schwartz (1997) all model spot prices and convenience yields as separate stochastic processes with a constant correlation. While these models are powerful tools for derivative pricing and hedging, our approach also has several attractions. First, explicitly modelling the joint evolution of inventory and spot prices ensures the consistency of the spot price and convenience yield (i.e., forward price) dynamics. Second, our model predicts that the correlation between spot prices and convenience yields (or their innovations) is unlikely to be constant due to its dependence on inventory. Third, inventory acts as a second state variable summarizing past shocks, which allows our model to capture a stochastic convenience yield with only one exogenous factor. The paper is organized as follows. Section 1 describes the basic one-factor model, demonstrates existence of equilibrium, and derives properties of the equilibrium inventory and spot price processes. Section 2 investigates properties of the forward curve and endogenous implied convenience yields. We also study the forward price volatility term structure and hedge ratios. Section 3 carries out a numerical calibration exercise and presents a tractable two-factor version of the basic model. Section 4 concludes. Proofs are collected in the Appendix. 1. Equilibrium Model of Commodity Prices The goal of this paper is to characterize spot and forward commodity prices in an equilibrium model 4 Schwartz and Smith (1997) presents a model with two factors, a "long-run" price component and a transitory disturbance, which they show is equivalent to the Gibson and Schwartz (1990) convenience yield model. Miltersen and Schwartz (1998) model a stochastic term structure of convenience yields based on Heath, Jarrow and Morton (1992). 5 Analogously, in some term structure settings assuming exogenous dynamics for interest rates of bonds of different maturities can be incompatible with equilibrium (i.e., admit arbitrage opportunities). Examples along these lines are noted in Cox, Ingersoll and Ross (1985). Heath, Jarrow and Morton (1992) describe the restrictions on forward price processes that are implied by the absence of arbitrage. Arbitrary joint dynamics for stock prices and dividends may also be inconsistent with equilibrium. 3 Equilibrium Forward Curves for Commodities April 1999 of inventory with non-negative storage. Our analysis builds on and extends Deaton and Laroque (1992, 1996), Chambers and Bailey (1996), Williams and Wright (1991) and Wright and Williams (1989). We start with general functional forms and develop tractable numerical implementations for derivative security valuation. We also investigate properties of the term structure of forward price volatility. 1.1 Model Structure Consider a discrete-time, infinite horizon model in which a single homogeneous commodity is traded in a competitive market at dates t=1,2,.... Current production and consumption demand for "immediate use" are modelled as stochastic, reduced-form functions, gt and ct, of the spot price, Pt. In addition, the commodity can be stored by a group of competitive risk-neutral inventory traders who have access to a costly storage technology. Storage is costly due to a constant proportional depreciation or wastage factor, δ∈(0,1]. Storage of q units of the commodity at t-1 yields (1-δ)q at t. One can interpret δ as spoilage (for agricultural commodities) or as a volumetric cost (for metals and energy). For technical reasons δ is strictly positive, but we abstract, for simplicity, from any other fixed or marginal costs. The spot price, Pt, is determined by market clearing. At each date t the two sources of the commodity, current production and incoming inventory gt + (1-δ)Qt-1, must equal the two types of demand, immediate consumption and outgoing inventory ct + Qt. This can be rearranged to get (1) ct (Pt ) gt (Pt ) ∆Qt where ∆Qt = Qt (1-δ)Qt-1. When ∆Qt is positive (i.e., inventory is increased), less of the good is left for immediate consumption. If the "immediate use" net-demand ct(Pt) gt(Pt) is monotone decreasing in the spot price Pt, the spot market can be summarized with an inverse net demand function (2) Pt f (at,∆Qt) We initially abstract from permanent shocks and model the net demand shocks at∈Ω as realizations of a finite-dimensional, irreducible, m-state Markov process (m ≥ 2), with transition probabilities π(a at) in a matrix Π. The shocks, at, represent the transitory effects of weather and/or production disruptions on the "immediate use" net demand, ct gt. They are unaffected by current and/or past inventory. Where noted, we sometimes make an additional assumption that π(a at)>0 for all at, a∈Ω (i.e., Π>>0) which implies that the demand shocks have a limiting distribution that is independent of