Consistency Of Forward Prices Across Delivery Dates

Why Forward Prices Must Agree Across Maturities

A forward curve lists prices today for delivery of the same asset at different future dates. Those prices cannot be chosen independently: if the asset can be bought at one delivery date and carried to another, the financing and carrying costs link the two prices. A violation can create a calendar arbitrage, a trade combining forward contracts with different delivery dates and the asset’s carry over the intervening period.

The comparison depends on the asset and the market assumptions. We begin with an asset that has no income and no storage cost, and assume deterministic interest rates and the ability to borrow and lend at those rates. Later, the same argument is extended to deterministic proportional income and carrying costs.

Definition: Let $F(0,T)$ be the no-arbitrage forward price set at time 0 for delivery of one unit of an asset at time $T$. Let $P(0,T)$ be the time-0 price of a zero-coupon payment of \$1 at time $T$. For an asset with no income or carrying cost, $$ F(0,T)=\frac{S_0}{P(0,T)}, $$ where $S_0$ is the spot price. For $0<T_1<T_2$, define the intervening discount factor by $P(T_1,T_2)=P(0,T_2)/P(0,T_1)$. Then $$ F(0,T_2)=\frac{F(0,T_1)}{P(T_1,T_2)}. $$

The second relation is the cross-maturity consistency condition. The forward price for the later date must equal the earlier forward price accumulated from $T_1$ to $T_2$ at the applicable financing rate. Since $P(T_1,T_2)$ is a discount factor, the accumulation factor is its reciprocal. If the market quotes a different later forward price, the difference can be compared with the cost of carrying the asset between the dates.

Why the Maturity Relation Holds

Suppose an investor enters a long forward for delivery at $T_1$, pays $F(0,T_1)$ at that date, and receives the asset. The investor can hold it until $T_2$ and deliver it into a short forward for $T_2$. The asset delivered at $T_2$ is then matched exactly by the asset received at $T_1$.

The payment at $T_1$ has an equivalent value at $T_2$ of $F(0,T_1)/P(T_1,T_2)$. Therefore the later forward price must equal that amount: otherwise the matched purchase, holding period, and delivery produce a locked-in surplus or deficit. In particular, under no-arbitrage,

F(0,T_2)-\frac{F(0,T_1)}{P(T_1,T_2)}=0.

This argument assumes that the asset is available to hold between the delivery dates, that its value is not changed by an unaccounted-for income or cost, and that the financing rate is known and tradable. If the asset pays income, incurs storage costs, or cannot be borrowed or traded as assumed, the simple relation must be adjusted or may not be directly exploitable.

Worked Example: Testing Two Forward Quotes

An asset has spot price \$120. The discount factors are $P(0,1)=0.96$ and $P(0,3)=0.90$. A dealer quotes a one-year forward price of \$125 and a three-year forward price of \$136. Assume the asset has no income or carrying costs and borrowing and lending at the given discount curve are possible. Are the quotes consistent?

Method. Derive the financing factor from year 1 to year 3, then grow the one-year forward price to the three-year date.

The intervening discount factor is $$ P(1,3)=\frac{P(0,3)}{P(0,1)}=\frac{0.90}{0.96}=0.9375. $$

Thus the intervening accumulation factor is $$ \frac{1}{P(1,3)}=\frac{1}{0.9375}=1.0666667. $$

The three-year price implied by the one-year quote is $$ \frac{125}{0.9375}=133.3333. $$

The dealer’s three-year quote of \$136 exceeds the implied price by $$ 136-133.3333=2.6667. $$ The quotes are not consistent with the stated assumptions: the later quote is too high relative to the earlier one. The cross-maturity check is $F(0,3)=F(0,1)/P(1,3)$, and the quoted prices do not satisfy it.

Check. The spot-based fair prices are $120/0.96=125$ at one year and $120/0.90=133.3333$ at three years. These agree with the one-year quote and the implied three-year price. The method avoids treating the quoted \$136 as if it were already adjusted for the financing between years 1 and 3.

Tracing a Calendar Cash-and-Carry Trade

If the later forward price is above the amount required by the cross-maturity relation, the long-$T_1$, short-$T_2$ calendar position can lock in a surplus. At time 0, enter a long forward for $T_1$ and a short forward for $T_2$. Arrange financing for the $T_1$ payment. At $T_1$, pay the first forward price and receive the asset; carry the asset to $T_2$, deliver it into the short forward, and receive the second forward price. Compare that receipt with the financing repayment due at $T_2$.

Enter the calendar position.
At time 0, take a long forward for delivery at $T_1$ and a short forward for delivery at $T_2$. A newly entered forward has no initial payment under the usual fair-contract convention.

Fund the earlier purchase.
Set aside or borrow $F(0,T_1)P(0,T_1)$ at time 0. Accumulated to $T_1$, this amount is $F(0,T_1)$, the payment due on the long forward.

Carry the asset.
At $T_1$, use the accumulated funds to pay for and receive the asset. Hold it until $T_2$, then deliver it into the short forward.

Compare the locked cash flows.
At $T_2$, receive $F(0,T_2)$ and repay $F(0,T_1)/P(T_1,T_2)$. The difference is the terminal surplus.

The surplus under these assumptions is

\text{surplus at }T_2 = F(0,T_2)-\frac{F(0,T_1)}{P(T_1,T_2)}.

A positive amount identifies a high later forward price relative to the earlier one. The reverse calendar exposure is relevant when the later quote is below the implied price; it requires that the underlying can be borrowed or shorted and that the reversed carry and financing transactions are available. The direction and feasibility of a trade therefore depend on market access as well as the quoted prices.

Worked Example: Calculating the Locked-In Surplus

Use the discount factors from the preceding example. The one-year forward price is \$125, and the three-year forward price is quoted at \$136. Assume no income or storage cost, and assume the investor can borrow and lend at the stated curve. Calculate the surplus at year 3 from the long-one-year, short-three-year calendar position.

Method. Finance the year-1 payment and compare the year-3 forward receipt with the financing repayment.

The year-1 payment requires a time-0 amount of $$ 125P(0,1)=125(0.96)=120. $$ Borrow \$120 at time 0 and invest \$120 at the one-year rate. The investment accumulates to $$ \frac{120}{0.96}=125 $$ at year 1, exactly the amount required to pay the long forward. The investor receives the asset at year 1 and holds it until year 3.

The \$120 borrowing is repaid at year 3 in the amount $$ \frac{120}{0.90}=133.3333. $$ This is also the year-3 value of the year-1 forward payment, since $$ \frac{125}{P(1,3)}=\frac{125}{0.9375}=133.3333. $$

At year 3, deliver the asset into the short forward and receive \$136. After repaying the borrowing, the surplus is $$ 136-133.3333=2.6667. $$ Rounded to the nearest cent, the locked-in surplus is \$2.67 per unit.

Check. The asset received at year 1 is the same asset delivered at year 3, so there is no remaining exposure to its market price. The cash receipt and repayment are both at year 3; their difference is positive. This method avoids subtracting the two forward prices directly, since the \$125 payment occurs two years earlier and must be accumulated before comparison.

Including Deterministic Income and Carrying Costs

For an asset with deterministic proportional income yield $q(t)$ and deterministic proportional carrying cost $u(t)$, the no-arbitrage forward price under continuous compounding is

F(0,T)=S_0\exp\left(\int_0^T [r(t)+u(t)-q(t)]\,dt\right),

where $r(t)$ is the deterministic force of interest. The yield reduces the forward price because holding the asset produces income; carrying costs increase it because holding the asset requires additional outlay. Taking the ratio of the prices for two maturities cancels the spot price and leaves only the net carry over the interval:

\frac{F(0,T_2)}{F(0,T_1)} = \exp\left(\int_{T_1}^{T_2}[r(t)+u(t)-q(t)]\,dt\right).

This version assumes that the rates and proportional carry are known and that the income and costs can be incorporated in the stated continuous-compounding model. It does not directly cover uncertain income, discrete cash distributions, or assets with constraints on storage or short selling. Those features require their own cash-flow treatment.

Worked Example: Comparing Quotes With Net Carry

A hypothetical storable asset has a one-year forward price of \$206.09. From year 1 to year 2.5, assume a constant force of interest of 4% per year, a proportional storage cost rate of 1% per year, and a proportional income yield of 2% per year. A market quote for the 2.5-year forward is \$220.00. Test the quote and find the calendar surplus per unit if the underlying can be carried and the stated rates are deterministic.

Method. Calculate net carry over the 1.5-year interval, use it to obtain the implied later price, and compare that price with the quote.

The net carry rate is $$ r+u-q=0.04+0.01-0.02=0.03. $$ The time from year 1 to year 2.5 is $2.5-1=1.5$ years, so the intervening accumulation factor is $$ \exp(0.03(1.5))=\exp(0.045)=1.0460279. $$

Using the stated one-year price, the implied 2.5-year forward price is $$ 206.09(1.0460279)=215.5759. $$ The quoted later price is higher by $$ 220.00-215.5759=4.4241. $$ Rounded to the nearest cent, the quote exceeds the implied price by \$4.42 per unit. Under the assumed carry model, the long-earlier, short-later calendar position produces that terminal surplus per unit after accounting for net carry.

Check. Positive net carry means the later price should be greater than the earlier price, and $215.5759$ is indeed above $206.09$. The quote of \$220.00 is higher still. This calculation avoids applying the income yield and storage cost with the wrong signs.

Common Exam Traps

A frequent error is to use the intervening discount factor in the wrong direction. In the first example,

P(1,3)=\frac{0.90}{0.96}=0.9375, \qquad \frac{1}{P(1,3)}=\frac{0.96}{0.90}=1.0666667.

The incorrect calculation $125(0.9375)=117.1875$ discounts the year-1 payment again rather than accumulating it to year 3. The correct comparison is $125/0.9375=133.3333$. A useful check is that, with positive interest and no income or cost, a later forward price should be higher than an earlier one.

Comparing cash flows at different dates without accumulating them. Convert the earlier payment to the later date before subtracting.
Using the total maturity discount factor instead of the interval factor. For two maturities, $P(T_1,T_2)=P(0,T_2)/P(0,T_1)$.
Ignoring income or costs between deliveries. Apply the net carry for the interval, not automatically for the full period from time 0.
Assuming a price discrepancy is automatically tradable. The trade relies on the ability to finance, hold, or borrow the asset and to realize the assumed carry.

Key takeaway: For an asset with no income or carrying costs, consistency requires $F(0,T_2)=F(0,T_1)/P(T_1,T_2)$. With deterministic proportional carry, multiply the earlier forward price by the net-carry accumulation factor between the delivery dates. A quote discrepancy is measured only after cash flows are placed at a common date.

Check Your Understanding

For each question, use the stated assumptions and compare cash flows at a common date.

An asset with no income or carrying costs has $P(0,1)=0.95$ and $P(0,2)=0.88$. Find the two fair forward prices if the spot price is \$80. (Answer: $F(0,1)=\$84.21$, $F(0,2)=\$90.91$, to the nearest cent.)
For an asset with no income or carrying costs, the one-year forward price is \$150 and $P(1,2)=0.92$. What two-year price is consistent with the one-year quote? (Answer: $\$163.04$, to the nearest cent.)
Two forward quotes for the same no-income asset are $F(0,1)=\$51$ and $F(0,2)=\$54$. Given $P(0,1)=0.98$ and $P(0,2)=0.94$, find the two-year price implied by the one-year quote and state whether the quotes are consistent. (Answer: implied $F(0,2)=\$53.17$; the \$54 quote is too high.)
A deterministic net carry rate $r+u-q$ is 2% per year with continuous compounding from year 0.5 to year 2. A quoted forward price for year 0.5 is \$104. Find the consistent year-2 forward price. (Answer: $\$107.17$, to the nearest cent.)
For an asset with no income or carrying costs, $F(0,1)=\$90$, $P(1,2)=0.90$, and the quoted two-year forward price is \$103. Assuming borrowing, lending, and asset holding are feasible, find the year-2 surplus per unit from the long-one-year, short-two-year calendar position. (Answer: $\$3.00$ per unit.)

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