Commodity forwards storage and convenience yield

Definition A commodity forward agreed at time $0$ with delivery date $T$ and delivery price $K$ obliges the long party to pay $K$ at $T$ and to take delivery at $T$ of one unit of a commodity of a stated grade at a stated location, and the short party to make that delivery. Prices are in currency units per unit of the commodity. Nothing is paid when the contract is agreed, and the long position pays

at $T$, where $S_T$ is the spot price at $T$ of the deliverable grade at the delivery location. The forward price $F_{0,T}$ is the delivery price that gives a newly agreed contract zero value. If default-free zero-coupon bonds with prices $P(t,T)$ trade and new forwards for the same delivery can be agreed at $F_{t,T}$, an existing contract with delivery price $K$ is worth

at time $t < T$, because adding a short forward at $F_{t,T}$ leaves the sure payment $F_{t,T} - K$ at $T$. Throughout this lesson the commodity is homogeneous and storable, interest rates are deterministic, with $P(0,T) = e^{-rT}$ when the continuously compounded rate $r$ is constant, and there are no transaction costs or defaults. Unlike a share, a unit of a commodity costs money to hold and may give its holder services, and units of different grades or at different locations are different assets. A nonstorable product such as electricity generally cannot be carried from one delivery date to another by holding inventory, so the carry relations below do not apply to it, and its forwards need models of production and demand that this lesson does not treat.

Definition Storage costs are the payments needed to hold a unit of inventory until $T$, such as warehouse fees, insurance and allowances for spoilage. Two models are used here. In the first, known amounts $c_k$ are due at dates $t_k \le T$, with present value

In the second, storage is charged at a constant continuous rate $u \ge 0$ on the market value of the inventory, and it can be paid by selling the fraction $u\,dt$ of the inventory over each short interval $dt$, so a holding that starts with $e^{uT}$ units ends with one unit. Assume that the commodity can be bought at $S_0$ and stored until $T$ at these costs, that the stored unit meets the delivery terms of the contract, and that money can be borrowed and lent at the bond prices. Then no arbitrage requires the cash-and-carry bounds

the first for known amounts and the second for proportional storage. If the forward price exceeded the first bound, borrowing $S_0 + U_0$, buying one unit, putting $U_0$ into zero-coupon bonds that pay each storage bill when it is due and selling the forward would cost nothing today. At $T$ the unit is delivered for $F_{0,T}$ and the loan is repaid with $(S_0 + U_0)/P(0,T)$, which leaves a positive surplus in every state. The proportional case is the same trade with $e^{uT}$ units bought at the start. The bounds hold whatever services the inventory gives its holder, because the trade does not need them.

Definition Reverse cash-and-carry is the trade opposite to cash-and-carry. A holder of inventory sells a unit at $S_0$, avoids the storage payments it would otherwise make, invests $S_0 + U_0$ at the riskless rate and buys the forward, so that a replacement unit is delivered at $T$. The same trade is open to someone who can borrow inventory on terms that pass the avoided storage costs to the borrower and require an equivalent unit of the same grade and location to be returned at $T$. Compared with keeping the inventory, the trade ends at $T$ with the same unit and

more cash. So if some participant holds inventory, values it only for what it can be sold for, and can sell and replace it on these terms, no arbitrage also requires $F_{0,T} \ge (S_0 + U_0)/P(0,T)$, and together with the cash-and-carry bound

In the proportional model the equality reads $F_{0,T} = S_0 e^{(r + u)T}$. If inventory cannot be borrowed or sold and replaced on these terms, or if its holders value the services it gives, only the upper bound remains, and a forward below it does not by itself signal an arbitrage.

Definition In the model with constant financing rate $r$ and proportional storage rate $u$, the convenience yield is the constant $y$ for which

so that the yield implied by a quoted forward is

With known storage payments the analogous definition is $F_{0,T} = (S_0 + U_0)\,e^{(r - y)T}$. The convenience yield models the benefit of holding physical inventory, such as keeping production running or meeting unexpected demand, as an effective proportional rate. It is usually not a cash dividend, and it is usually not available to every participant or traded on its own. If the benefit were paid in units of the commodity at rate $y$ and could be reinvested, the replication argument used for a continuous dividend yield would give the formula, provided the reverse trade is also available. Otherwise the formula is either a definition of $y$ from observed prices or a model that gives the benefit an equilibrium or shadow value. When cash-and-carry is feasible, the upper bound $F_{0,T} \le S_0 e^{(r + u)T}$ implies $y \ge 0$. An implied $y$ measures the gap between the forward and full carry, and it absorbs everything the carry model leaves out, such as misstated storage costs, limited storage capacity or delivery options, so it does not by itself identify the economic value of holding inventory.

Definition The net carrying rate is

the financing rate plus the proportional storage rate minus the convenience yield. Over a short interval of length $dt$, carrying inventory worth one currency unit costs about $c\,dt$, net of the modeled inventory benefit. With constant rates the model forward is

and with deterministic rates $r(s)$, $u(s)$ and $y(s)$ that vary with time it is $F_{0,T} = S_0 \exp\bigl(\int_0^T c(s)\,ds\bigr)$ with $c(s) = r(s) + u(s) - y(s)$. The net carrying rate plays the role that $r - q$ plays for a stock with continuous dividend yield $q$. Because the convenience yield is modeled, so is the net carrying rate, and its value inherits every assumption made about $y$.

Definition Fix the date $0$ and consider the forward curve $T \mapsto F_{0,T}$. On this page the curve is in contango over a range of maturities on which it increases with $T$, and in backwardation over a range on which it decreases. The terms are also used for a comparison of one forward with spot, with $F_{0,T} > S_0$ called contango and $F_{0,T} < S_0$ called backwardation, so the comparison meant has to be stated. In the constant-rate model the slope of the curve is

and since $F_{0,T} > 0$ the curve increases at every maturity when $c > 0$, decreases at every maturity when $c < 0$ and is flat when $c = 0$. In that model $F_{0,T} - S_0$ has the sign of $c$ for every $T > 0$, so the two usages agree. With deterministic carry that varies with time, $\partial_T \ln F_{0,T} = c(T)$ at every maturity where $c$ is continuous, so the curve can rise over some maturities and fall over others, as it does when storage capacity or the usefulness of nearby inventory changes with the season. Neither shape says by itself what the physical expected spot price is or what return an investor should expect.