Futures Prices Under Cost Of Carry

Why Financing and Carrying Costs Affect a Futures Quote

A futures contract sets a price today for a transaction at a future date. A buyer who takes delivery later avoids paying the spot price now, but the asset must still be available at delivery. If an investor can buy and store the asset today, the funds used for that purchase have a financing cost. Storage may add further costs, while income from holding the asset offsets some of those costs.

The cost-of-carry relation combines these effects into a no-arbitrage futures quote. It is a pricing relation under stated market assumptions, not a prediction of what the spot price will be at the contract date. The relation is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: The cost of carry is the net cost of holding an asset from today to a specified future date. It includes financing and carrying costs, less income received from holding the asset. A fair futures price is the quote consistent with the cost of acquiring and carrying the underlying asset, subject to the model assumptions.

The Cost-of-Carry Model

Let $S_0$ be the asset's spot price today and let $T$ be the time to the futures contract's delivery date, measured in years. For the first model, assume a constant, known effective annual financing rate $i$, so an amount invested or borrowed for $t$ years accumulates by the factor $(1+i)^t$. Assume no arbitrage, no taxes or transaction costs, and that the asset can be bought, held, and—if needed—short-sold. Assume also that all income and carrying costs during the contract term are known and can be replicated.

If there are no interim income payments or carrying costs, buying the asset now and financing the purchase costs $S_0(1+i)^T$ at the delivery date. The fair futures price is therefore:

$$ F_0(T)=S_0(1+i)^T $$

This formula uses an annual-effective rate and a term measured in years. For example, a term of nine months is $T=0.75$, not $T=9$. If the financing rate varies but is known in advance, the single factor $(1+i)^T$ must be replaced by the appropriate known accumulation factor from today to $T$.

Worked Example: Financing an Asset for Nine Months

A hypothetical metal is available for $S_0=\$120$ per unit. A futures contract requires delivery in nine months. The effective annual financing rate is $i=4\%$, with no income or carrying costs before delivery. Find the fair futures price per unit.

Method. Accumulate the spot price for the contract term. The term is $T=9/12=0.75$ years.

F_0(0.75)=120(1.04)^{0.75} =120(1.0298524\ldots) =\$123.5823\ldots

Rounded to the nearest cent, the fair futures price is \$123.58 per unit. The check is that the quote exceeds the current spot price: with positive financing cost and no offsetting income, holding the metal until delivery costs more than buying it today.

Known Income and Discrete Carrying Costs

Suppose the asset pays known income amounts before delivery and incurs known carrying costs, such as storage charges, on specified dates. Each amount must be valued at the delivery date, because the futures quote is a delivery-date amount. With a constant effective annual financing rate $i$, let $I_j$ be an income payment at time $t_j$, and let $C_k$ be a carrying cost at time $s_k$. Then:

F_0(T) = S_0(1+i)^T +\sum_k C_k(1+i)^{T-s_k} -\sum_j I_j(1+i)^{T-t_j}

An income payment reduces the cost of carrying the asset, so its accumulated value is subtracted. A carrying cost increases the cost, so its accumulated value is added. This expression assumes every payment date is no later than $T$ and that each amount is known. If a cash flow is paid at delivery, its accumulation period is zero.

Why the formula holds: Buying the asset now requires financing $S_0$ to delivery. A cost paid before delivery must also be financed from its payment date to $T$, while income received before delivery can be used to reduce the net amount financed. Thus the delivery-date cost of holding the asset is the accumulated spot price plus accumulated costs less accumulated income.

This accumulated-cost argument also gives the no-arbitrage condition. A cash-and-carry position buys and holds the asset, pays its carrying costs, receives its income, and takes the opposite futures position. At delivery, the asset can be delivered against the short futures position. If the futures quote exceeds the accumulated cost of this position, the difference is an arbitrage profit under the assumptions above. If the quote is below that cost, the reverse position—requiring feasible short sales and replicable cash flows—provides the corresponding comparison. A quote inconsistent with the cost of the replicating position cannot persist in the frictionless model.

Set the timeline.
Write the delivery time $T$ in years and place every income payment and carrying cost at its stated payment time.

Accumulate to delivery.
Accumulate the spot price from time 0 to $T$, and each interim cash flow from its payment date to $T$.

Apply the signs.
Add carrying costs and subtract income. Then state the per-unit quote and any requested rounding.

Worked Example: A Storage Charge and a Rebate

A hypothetical storable asset has a spot price of $\$800$. A one-year futures contract is quoted per unit. The effective annual financing rate is $5\%$. At month 6, holding one unit incurs a known storage cost of $\$15$ and provides a known rebate of $\$10$. Find the fair futures price.

Method. Accumulate the spot price for one year. Since the storage cost and rebate occur at the same date, first find their net effect: the carrying cost is $\$15-\$10=\$5$ at $t=0.5$. Accumulate that net cost for the remaining half-year.

\begin{aligned} \text{Accumulated spot price} &=800(1.05)=840,\\ \text{Accumulated net month-6 cost} &=5(1.05)^{0.5} =5(1.0246951\ldots) =5.123475\ldots,\\ F_0(1) &=840+5.123475\ldots\\ &=\$845.123475\ldots \end{aligned}

Rounded to the nearest cent, the fair futures price is \$845.12 per unit. The check is that the net interim cost is positive, so the fair quote is slightly higher than the $\$840$ accumulated spot price. The month-6 cash flow is accumulated for only half a year, not for the full year.

Proportional Income and Continuous Carrying Rates

Some models describe income and carrying costs as rates proportional to the asset's value rather than as fixed dollar payments. Assume the financing rate is a constant force of interest $\delta$, the asset provides a continuous proportional income yield $y$, and it incurs a continuous proportional carrying-cost rate $c$. All rates are expressed per year, and the rates and asset value follow this model continuously over $T$ years. Under these assumptions, the cost-of-carry formula is:

F_0(T)=S_0e^{(\delta+c-y)T}

Financing and carrying costs increase the delivery-date cost, while proportional income offsets it. This continuous-rate formula is not a replacement for the discrete-cash-flow formula when the asset pays known fixed-dollar amounts on specific dates. The model must match how the cash flows are specified.

Worked Example: A Continuous Income Yield and Storage Rate

A hypothetical commodity has spot price $\$250$ per unit and a two-year futures maturity. The constant force of financing interest is $\delta=0.06$, the continuous proportional income yield is $y=0.02$, and the continuous proportional carrying-cost rate is $c=0.01$. Find the model price.

Method. Use the continuous-rate model. The net annual rate in the exponent is financing plus carrying cost less income:

\delta+c-y=0.06+0.01-0.02=0.05

Therefore:

F_0(2)=250e^{(0.05)(2)} =250e^{0.10} =250(1.1051702\ldots) =\$276.2926\ldots

Rounded to the nearest cent, the model price is \$276.29 per unit. Since the net carrying rate is positive, the price is above spot. The common mistake avoided here is to subtract the income yield and carrying-cost rate together; the carrying cost is an addition, while only income is subtracted.

Pricing Traps and Model Limits

The relation is useful because it converts a quoted futures price into a cost comparison that can be checked systematically. Before calculating, confirm what the problem means by “income,” “cost,” and “rate.” The word “yield” usually describes proportional income; a fixed-dollar payment is a discrete cash flow and should be accumulated from its payment date.

Using the wrong time unit. With an effective annual rate, convert months to a fraction of a year before using $(1+i)^T$. Six months is $T=0.5$.
Giving a cash flow the wrong sign. Income reduces the net cost of holding the asset; storage and other carrying costs increase it.
Accumulating an interim cash flow for the full term. A payment at time $t$ is accumulated for $T-t$ years, not $T$ years.
Mixing rate conventions. Use $(1+i)^T$ for a constant effective annual rate and $e^{\delta T}$ for a constant force of interest. Do not insert an effective annual rate into a continuous-rate exponent.
Treating the fair price as a forecast. The cost-of-carry price follows from a replicating strategy and no-arbitrage assumptions. It does not state what the future spot price will be.

The simple relation may not apply unchanged when cash flows are uncertain, the asset cannot be stored or short-sold, or transaction costs and market restrictions prevent replication. With futures, daily marking to market also matters: the standard relation used here assumes known financing conditions so that the timing of settlement gains and losses does not create an uncertain financing effect. If rates are uncertain, the futures quote need not equal the corresponding forward quote.

Key takeaway: Under the stated no-arbitrage assumptions, a fair futures price is the delivery-date cost of the spot asset after accounting for financing, known carrying costs, and income. Match the formula to the cash-flow pattern and rate convention before calculating.

Check Your Understanding

For each problem, use the model and assumptions stated in that problem. Prices are per unit.

A hypothetical asset has spot price $\$1{,}000$, no income or carrying costs, and a six-month maturity. The effective annual financing rate is $6\%$. Find the fair futures price to the nearest cent. (Answer: $\$1{,}029.56$)
A hypothetical asset has spot price $\$400$ and a one-year maturity. The effective annual financing rate is $5\%$. It pays known income of $\$12$ at month 6 and incurs a known carrying cost of $\$4$ at month 3. Find the fair futures price to the nearest cent. (Answer: $\$411.85$)
A hypothetical commodity has spot price $\$300$, a two-year maturity, constant force of interest $\delta=0.05$, continuous income yield $y=0.015$, and continuous carrying-cost rate $c=0.01$. Find the continuous-model price to the nearest cent. (Answer: $\$328.25$)
A hypothetical asset has spot price $\$75$, a 1.5-year maturity, and an effective annual financing rate of $4\%$. It pays known income of $\$1.50$ at month 6 and incurs a known carrying cost of $\$0.80$ at year 1. Find the fair futures price to the nearest cent. (Answer: $\$78.80$)

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