Cost Of Carry For Storable Commodities

Why Storage Changes a Forward Price

A commodity held for future delivery may require physical storage before the delivery date. A party that buys the commodity today to meet a future forward commitment must fund both its purchase and its carrying expenses. Those costs affect the fair forward delivery price. Conversely, a known cash benefit received while holding the commodity reduces the net cost of carrying it.

The central rule is to compare the forward with a strategy that buys and carries the commodity until delivery. The strategy’s initial purchase, storage expenses, and any known cash benefits must be valued consistently with the payment made under the forward. In particular, a storage payment made before delivery is not the same as the same amount paid at delivery: it must be financed from its payment date to the delivery date.

Definition: For a storable commodity, the cost of carry is the net cost of purchasing and holding the commodity until a specified delivery date. It includes financing and storage expenses, less any known cash benefits received from holding the commodity. The forward price is the delivery price that makes entering a forward contract at inception have value zero, under the stated no-arbitrage assumptions.

Model and Forward-Price Formula

Assume the commodity can be bought and stored in the required quality and location, and that it can be delivered under the forward contract. Assume competitive markets, no taxes or transaction costs, no binding limits on borrowing or short selling, and no risk that a stated cost or benefit will differ from its specified amount. Also assume there is no additional non-cash benefit from physically holding the commodity. These assumptions allow the forward to be compared with a buy-and-carry strategy. If storage is unavailable, costs are uncertain, or market frictions bind, the simple equality below need not apply.

Let $S_0$ be the spot price, $T$ the delivery time in years, and $K$ the forward delivery price. Let $P(0,t)$ denote the time-0 price of a unit payment at time $t$, for $0\leq t\leq T$. Suppose storage expenses $C_j$ are paid at dates $t_j$, and known cash benefits $B_k$ are received at dates $s_k$, all before or on delivery. The value at time 0 of the initial purchase and the net carrying cash flows is

S_0+\sum_j C_jP(0,t_j)-\sum_k B_kP(0,s_k).

A forward with delivery price $K$ requires a payment of $K$ at time $T$, whose time-0 value is $K P(0,T)$. Equating the two values gives the fair forward price:

K=F_{0,T} =\frac{S_0+\sum_j C_jP(0,t_j)-\sum_k B_kP(0,s_k)} {P(0,T)}.

This formula accommodates a term structure of interest rates: use the appropriate discount factor for each cash-flow date. For a single effective annual rate $i$, $P(0,t)=(1+i)^{-t}$. Equivalently, a cash flow at time $t$ can be accumulated to delivery by multiplying it by $(1+i)^{T-t}$. The effective annual rate is being applied over fractional years when $t$ or $T$ is not an integer.

Why the Formula Holds

The relation follows by replicating the commodity delivery and matching cash flows. At time 0, buy one unit for $S_0$. Pay each specified storage expense when due and receive each specified benefit when due. At time $T$, the strategy delivers the commodity. A forward buyer instead pays $F_{0,T}$ at time $T$ and receives the commodity. If the initial cost of the buy-and-carry strategy differed from the present value of that forward payment, an investor could take the cheaper position and finance the difference, contradicting the no-arbitrage assumption.

F_{0,T}P(0,T) = S_0+\sum_j C_jP(0,t_j)-\sum_k B_kP(0,s_k).

The signs matter: a required storage payment raises the cost of carrying, while a cash benefit received by the holder lowers it. When all payments occur before delivery, one can also see the financing explicitly by accumulating every item to $T$:

F_{0,T} =S_0(1+i)^T +\sum_j C_j(1+i)^{T-t_j} -\sum_k B_k(1+i)^{T-s_k}.

The two versions are equivalent for a flat effective annual rate. The first discounts all cash flows to time 0; the second accumulates them to delivery. Do not discount a cash flow to time 0 and then also treat it as though it were already a delivery-date amount.

Worked Examples

Worked Example: Continuous Storage Expense at a Flat Rate

A storable metal has spot price $S_0=\$100$. A forward delivers one unit in two years. The effective annual interest rate is $i=5\%$, and storage is paid continuously at $\$2$ per year throughout the two-year period. Find the fair forward price. Assume the storage service is available continuously, the stated rate is fixed, and there are no cash benefits from holding the metal.

Method. Accumulate the purchase price and each infinitesimal storage payment to the delivery date. The force corresponding to the effective annual rate is $\delta=\ln(1.05)$, because $e^\delta=1.05$.

The spot purchase accumulates for two years to

100(1.05)^2=\$110.25.

A storage payment of $2\,dt$ at time $t$ accumulates for $2-t$ years. Thus the accumulated storage expense is

\int_0^2 2(1.05)^{2-t}\,dt = 2\frac{(1.05)^2-1}{\ln(1.05)} = \frac{0.205}{0.04879016417} \approx \$4.2016665.

Therefore,

F_{0,2}=110.25+4.2016665 \approx \$114.4516665.

The present value of the storage stream provides a cross-check. Discounting the accumulated storage amount by $(1.05)^2=1.1025$ gives

\frac{4.2016665}{1.1025}\approx \$3.8110354.

Then $(100+3.8110354)(1.1025)\approx \$114.4516665$, consistent with the accumulation calculation. Rounded to the nearest cent, the fair forward price is \$114.45. A common mistake is to add the undiscounted total storage payment of $\$4$ directly to the delivery-date value of the spot purchase; the storage is paid at different times and must be accumulated payment by payment.

Worked Example: Dated Storage Payments and a Known Benefit

A commodity has spot price $\$80$, and a forward delivers one unit in three years. The effective annual interest rate is $4\%$. Storage costs $\$1.50$ at the end of year 1 and $\$2.50$ at the end of year 2. A known cash benefit of $\$3$ is received at the end of year 2. Find the fair delivery price. Assume all stated cash flows are certain and the commodity is deliverable at year 3.

Method. Accumulate each item from its own date to year 3. The year-1 expense accumulates for two years. Both year-2 cash flows accumulate for one year, with the benefit entered as a negative carrying cost.

\begin{aligned} F_{0,3} &=80(1.04)^3+1.50(1.04)^2+2.50(1.04)-3(1.04)\\ &=80(1.124864)+1.50(1.0816)+2.50(1.04)-3(1.04)\\ &=89.98912+1.6224+2.60-3.12\\ &=\$91.09152. \end{aligned}

The fair delivery price is \$91.09 to the nearest cent. As a check, the net year-2 cash flow is a benefit of $\$0.50$, since $\$3-\$2.50=\$0.50$; accumulating that net benefit one year gives $\$0.52$, so the result is also $89.98912+1.6224-0.52=91.09152$. The method avoids treating the year-1 storage payment as if it were paid at year 3, or adding the known benefit instead of subtracting it.

Worked Example: Pricing with Term-Structure Discount Factors

A specialty resin has spot price $\$120$. A forward delivers one unit in three years. The relevant discount factors are $P(0,1)=0.96$, $P(0,2)=0.90$, and $P(0,3)=0.82$. Storage costs $\$4$ at the end of year 1 and $\$5$ at the end of year 2. A known benefit of $\$2$ is received at the end of year 2. Find the fair forward price. Assume the discount factors price the specified certain cash flows and no other carrying costs or benefits apply.

Method. Discount each dated cash flow with its own discount factor, then divide the present value of the buy-and-carry strategy by the discount factor for the delivery payment.

\begin{aligned} \text{Present value of carrying strategy} &=120+4(0.96)+5(0.90)-2(0.90)\\ &=120+3.84+4.50-1.80\\ &=\$126.54. \end{aligned}

The delivery payment has present value $F_{0,3}(0.82)$, so

F_{0,3}=\frac{126.54}{0.82} \approx \$154.3170732.

The fair forward price is \$154.32 to the nearest cent. The calculation is consistent with the time-0 value equation because $154.3170732(0.82)\approx126.54$. Using a single annual rate inferred from one discount factor for every cash flow would generally be incorrect when the term structure is not flat.

Interpreting Storage Costs Carefully

A storage expense can be specified in different ways, and the form of the data determines the calculation. A fixed cash amount due on a stated date belongs in the dated-cash-flow formula. A continuous cash expense at a constant rate $c$ dollars per year is integrated over time. For a flat force of interest $\delta$, its accumulated value to delivery is

\int_0^T c e^{\delta(T-t)}\,dt = c\frac{e^{\delta T}-1}{\delta}, \qquad \delta\ne 0.

If $\delta=0$, the expression is $cT$, as follows directly from the integral. This is a cash storage cost, not a percentage charge on the commodity’s value. If a problem instead specifies a proportional carrying charge, its base and compounding convention must be identified before using a rate-based formula; it should not be silently treated as a fixed dollar payment.

Known cash benefits are handled by the same timing rule as costs, but with the opposite sign. A benefit paid at a known date reduces the present value of carrying the commodity. If the payment date is delivery, it should be included only if the holder actually receives it under the contract and ownership terms. Uncertain or non-cash benefits are not the known cash flows in the formula above.

Common Exam Traps

Ignoring payment timing. A cost paid before delivery must be financed from its payment date to delivery, or discounted using its own date and the delivery discount factor.
Using the wrong sign. Storage expenses are added; known cash benefits are subtracted.
Mixing present and future values. Either discount all flows to time 0 and divide by $P(0,T)$, or accumulate all flows to delivery. Do not mix the two methods.
Using one rate for a non-flat curve. With term-structure discount factors, use $P(0,t)$ at each cash-flow date and $P(0,T)$ for the delivery payment.
Forgetting the model assumptions. The equality describes the no-arbitrage price under the stated storage, trading, and cash-flow assumptions; it is not guaranteed when those assumptions fail.

The cost-of-carry framework is useful because it converts a physical holding problem into a set of dated cash flows. Before calculating, write down the spot purchase, every storage payment, every known cash benefit, and the delivery date. Then choose one valuation date and apply it consistently. This is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, assume the commodity is storable and deliverable as specified, and that the stated interest rates and cash flows are certain.

A commodity with spot price $\$50$ has a two-year forward. The effective annual interest rate is $4\%$. Storage costs $\$1$ at the end of year 1 and $\$2$ at the end of year 2. Find the fair forward price. (Answer: $\$57.12$)
A commodity has spot price $\$75$, a two-year delivery date, and an effective annual interest rate of $5\%$. Storage costs $\$2$ at the end of year 1, when a known cash benefit of $\$1$ is also received. Find the fair forward price. (Answer: $\$83.74$)
A commodity has spot price $\$100$, a two-year delivery date, and a flat force of interest $\delta=0.05$. Storage is paid continuously at $\$2$ per year. Find the fair forward price to the nearest cent. (Answer: $\$114.72$)
A commodity has spot price $\$200$ and a three-year delivery date. The discount factors are $P(0,1)=0.97$, $P(0,2)=0.93$, and $P(0,3)=0.88$. Storage costs $\$6$ at year 1 and $\$5$ at year 2; a known benefit of $\$2$ is received at year 2. Find the fair forward price to the nearest cent. (Answer: $\$237.06$)
A commodity has spot price $\$140$ and a three-year delivery date. The effective annual interest rate is $3\%$. Storage costs $\$2$ at the end of year 1 and $\$3$ at the end of year 2; a known benefit of $\$4$ is received at the end of year 2. Find the fair forward price to the nearest cent. (Answer: $\$154.07$)

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