Forward Prices With Income And Carrying Costs

Why Income and Carrying Costs Change a Forward Price

A forward contract sets a price today for an asset to be exchanged at a future date. The spot price alone does not determine that forward price when holding the asset before delivery produces income or requires expenses. An owner may receive income before delivery, while also paying costs to store, insure, or maintain the asset. A fair forward price must reflect both effects.

The central method is to compare the forward contract with a financed position in the underlying asset. Buying and carrying the asset produces the same delivery asset as a long forward, but also creates interim cash flows. Discounting or accumulating those cash flows consistently gives the no-arbitrage price.

Definition: Let $S_0$ be the spot price at time 0 and $F_{0,T}$ the delivery price of a forward contract maturing at time $T$, measured in years. A known income is a specified cash amount received by the asset holder before or at delivery. A carrying cost is a specified cash amount paid to maintain the asset during that period. The forward price is the delivery price that makes the forward and the financed carry strategy equivalent under the stated market assumptions.

Forward Price with Known Cash Flows

First use a deterministic interest model: the effective annual rate $i$ is constant, positive or zero, and the same rate is available for borrowing and investing. Assume there are no taxes, transaction costs, or default risk; the asset and its income can be traded or replicated; and the specified income and carrying-cost cash flows are certain. For an exact equality, assume the carry cash flows and the positions needed to compare strategies are feasible on consistent terms.

Suppose income $C_j$ is received and carrying cost $K_j$ is paid at time $t_j$, for $0<t_j\leq T$. A cash flow received at the delivery time is included only if the holder of the asset receives it before handing over the asset. The discount factor for a cash flow at time $t$ is $v^t=(1+i)^{-t}$. Define the present values at time 0:

PV_0(C)=\sum_j C_j(1+i)^{-t_j}, \qquad PV_0(K)=\sum_j K_j(1+i)^{-t_j}.

The no-arbitrage forward price is the future value at time $T$ of spot cost, less the value of income, plus the value of carrying costs:

F_{0,T}=\bigl(S_0-PV_0(C)+PV_0(K)\bigr)(1+i)^T.

The same result can be calculated directly at time $T$: accumulate the spot purchase to $T$, accumulate each income receipt from its payment date to $T$, and accumulate each cost in the same way. This version makes the signs and timing especially visible.

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

These two expressions agree because accumulating a time-0 present value to time $T$ gives the same amount as accumulating each dated cash flow directly to $T$. The formula does not apply unchanged when cash flows are uncertain, financing rates differ materially from investment rates, or relevant trading and carrying positions are unavailable. In those settings, further assumptions or no-arbitrage bounds may be needed.

Worked Example: Known Income and a Storage Payment

A hypothetical asset has a spot price of $1,000. A two-year forward is being priced using an effective annual interest rate of 10%. The owner receives $50 at the end of year 1 and $40 at the end of year 2, immediately before delivery. A storage payment of $10 is due at the end of year 1. Find the no-arbitrage forward price.

Method. Accumulate every cash flow to year 2. This keeps the income reduction and cost increase on the same valuation date.

The spot purchase accumulated for two years is:

$$ 1000(1.10)^2=1000(1.21)=1210. $$

The $50 income received at year 1 accumulates for one year, and the $40 received at year 2 is already at the focal date:

50(1.10)^{2-1}=55, \qquad 40(1.10)^{2-2}=40.

The $10 storage payment at year 1 accumulates for one year:

10(1.10)^{2-1}=11.

Therefore the delivery price is:

F_{0,2}=1210-55-40+11=\$1{,}126.

Check. At year 2, the financed spot purchase costs $1,210. The accumulated income reduces that cost by $95, and the accumulated storage payment increases it by $11, giving $1,210-$95+$11=$1,126. The method avoids treating a year-1 cash flow as though it were paid at delivery.

Why the Replication Formula Holds

The formula follows from comparing what it costs to obtain the asset at time $T$ in two ways. A long forward requires payment of $F_{0,T}$ at $T$ and delivers the asset. Alternatively, buy the asset now using financing, hold it until $T$, collect its income, and pay its carrying costs. At $T$, this second strategy also has the asset available for delivery.

The carry strategy’s net cost at $T$ is the accumulated spot purchase, minus accumulated income, plus accumulated carrying costs. If the forward price differs from that net cost under the stated replication assumptions, one strategy costs less than the other for the same delivery asset. Buying the cheaper strategy and selling the more expensive one creates an arbitrage. Thus the two costs must agree:

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

This argument also explains the signs. Income lowers the cost of carrying the asset, so it lowers the forward price. Carrying costs raise the cost of obtaining the asset for delivery, so they raise the forward price.

Choose the focal date.
Use time 0 for present values or time $T$ for accumulated values. Do not mix dates in one sum.

List the asset cash flows.
Record each income receipt and carrying payment with its amount, time, and sign from the asset holder’s perspective.

Accumulate or discount consistently.
Use the same interest model for the spot investment and every interim cash flow.

Check the direction.
Income should reduce the result; carrying costs should increase it. Confirm that a cash flow at $T$ has no further accumulation.

Proportional Income and Carrying Rates

Some models express income and carrying costs as continuous proportions of the asset value rather than as known fixed-dollar payments. Suppose the asset pays a continuous proportional income yield $q$, incurs a continuous proportional carrying-cost rate $c$, and the financing force of interest is constant at $\delta$. Under deterministic rates and continuous compounding, the net cost of carry is $\delta+c-q$. The corresponding formula is:

F_{0,T}=S_0e^{(\delta+c-q)T}.

To justify the exponent, over a small time interval $dt$, financing increases the carried value at rate $\delta$, the proportional carrying charge adds rate $c$, and the income yield offsets rate $q$. The net proportional growth rate is therefore $\delta+c-q$. Solving the constant-rate growth equation $dS/S=(\delta+c-q)\,dt$ over the interval from 0 to $T$ gives $S_T=S_0e^{(\delta+c-q)T}$, which is the delivery price for the replicated carry position. This model is not the fixed-cash-flow formula: the yield and cost rates must apply proportionally and continuously as specified.

Worked Example: Continuous Income Yield and Carrying Rate

A hypothetical asset has spot price $80. Its continuous income yield is 1% per year, its proportional carrying-cost rate is 2% per year, and the constant financing force of interest is 4% per year. Find the one-and-a-half-year forward price.

Method. Use the continuous proportional-rate model. The net cost-of-carry rate is:

\delta+c-q=0.04+0.02-0.01=0.05.

Over $T=1.5$ years, the exponent is $0.05(1.5)=0.075$. Thus:

F_{0,1.5}=80e^{0.075}=80(1.07788415)=\$86.2307.

Rounded to the nearest cent, the forward price is $86.23. The result exceeds spot because the financing and carrying rates together exceed the income yield. The method avoids using an effective annual rate formula for rates explicitly stated as continuous forces.

Checking a Quoted Forward Price

A calculated fair price can also check a market quote. If a quoted forward price is above the cost of buying and carrying the asset, and the model’s financing and cash-flow assumptions hold, a trader can buy and carry the asset and sell the forward. The asset is delivered at maturity, and the higher forward receipt covers the carry cost. A price difference is not automatically an arbitrage in an actual market: transaction costs, short-sale restrictions, uncertain cash flows, or unequal financing rates may prevent the replication.

Worked Example: Testing a Forward Quote

A hypothetical asset has spot price $600. The effective annual financing rate is 5%, and a two-year forward is quoted at $660. The asset pays known income of $12 at the end of year 1. Storage costs $3 at the end of each year. Under the assumptions for exact replication, determine whether the quote permits an arbitrage and calculate the terminal profit.

Method. First compute the fair forward price using terminal values. The financed spot purchase costs:

$$ 600(1.05)^2=600(1.1025)=661.50. $$

The income accumulates one year to $12(1.05)=12.60$. The first storage payment accumulates to $3(1.05)=3.15$, while the year-2 storage payment is already at maturity and equals $3. The fair delivery price is:

F_{0,2}=661.50-12.60+3.15+3.00=\$655.05.

The market quote of $660 exceeds the fair price by $4.95. The carry strategy is to borrow $600, buy and hold the asset, pay its storage costs, and sell it forward for $660. At maturity, the forward provides $660; repaying the financing costs $661.50; accumulated income provides $12.60; and accumulated storage costs total $6.15. The net result is:

660-661.50+12.60-6.15=\$4.95.

Check. The arbitrage profit equals the quote minus the calculated fair price: $660-655.05=4.95$. This check confirms the sign and amount of the discrepancy. The method avoids calling a quote mispriced before accounting for both interim income and storage payments.

Common Forward-Pricing Traps

A frequent error is to add income and subtract costs, reversing their effects. Another is to subtract or add the cash amounts without first bringing them to the same date as the spot investment. For example, in the first worked example, the incorrect calculation $1210+55+40-11=1294$ adds income to the cost and subtracts storage. The correct calculation is $1210-55-40+11=1126$: income offsets the cost of ownership, while storage adds to it.

Timing at maturity also matters. An income payment at $T$ is subtracted without further accumulation only when the asset holder receives it before delivery. If delivery occurs first, the forward buyer does not receive that payment, and it must not be counted as the holder’s income. Similarly, a cost paid at $T$ is added without accumulation only if it is required to make the asset available for delivery.

Key takeaway: A forward price reflects the future value of financed spot ownership, adjusted downward for income received and upward for carrying costs paid before delivery. State the timing, rate convention, and replication assumptions before applying the formula.

Check Your Understanding

For each problem, use the stated model and keep all cash flows on the same focal date.

An asset costs $200 spot. With no income or carrying costs, find its two-year forward price at an effective annual interest rate of 5%. (Answer: $220.50)
An asset costs $800 spot. It pays known income of $24 at time $t=0.5$, has no carrying costs, and matures under a one-year forward. The effective annual interest rate is 4%. Find the forward price, rounded to the nearest cent. (Answer: $807.52)
An asset costs $300 spot. A two-year forward uses an effective annual rate of 10%. The asset pays $15 at year 1; carrying costs are $8 at year 1 and $5 at year 2. Find the fair forward price. (Answer: $360.30)
An asset costs $150 spot. Its continuous income yield is 2% per year, its proportional carrying-cost rate is 1% per year, and the financing force is 3% per year. Find the two-year forward price, rounded to the nearest cent. (Answer: $156.12)
An asset costs $1,000 spot. At an effective annual rate of 5%, a two-year forward includes known income of $20 at year 1 and storage costs of $5 at year 1 and $4 at year 2. Find the fair forward price. (Answer: $1,090.75)

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