Cash-And-Carry Arbitrage For Overpriced Forwards

Recognizing an Overpriced Forward

A forward contract fixes a price today for an exchange of an asset at a future date. If the market delivery price is higher than the amount needed to finance ownership of that asset until delivery, an investor can buy and carry the asset while agreeing to sell it forward. The delivery-date proceeds then exceed the cost of acquiring and carrying the asset. This strategy is called cash-and-carry arbitrage.

The comparison must use the same asset and delivery date as the forward contract. It must also account for financing and any known income or carrying costs over the period. A high-looking quoted price is not enough by itself: the relevant question is whether it exceeds the no-arbitrage delivery price under the stated assumptions.

Definition: A cash-and-carry arbitrage for an overpriced forward consists of buying and holding the underlying asset, financing its purchase and any carrying costs, and entering a short forward contract to deliver the asset at the quoted delivery price. If that price exceeds the no-arbitrage delivery price, the strategy has zero net initial investment and a positive, known profit at delivery, assuming the model conditions hold.

No-Arbitrage Price and the Strategy

First consider an asset that pays no income and has no carrying costs. Let $S_0$ be its spot price, $i$ the effective annual risk-free interest rate, $T$ the time to delivery in years, and $K$ the forward delivery price. Assume the rate is fixed and the same for borrowing and investing, the asset can be bought and sold at the stated prices, and there are no transaction costs, taxes, short-sale constraints, or default risk. The forward is newly entered and requires no initial payment.

The amount needed at time $T$ to repay a loan of $S_0$ is $S_0(1+i)^T$. Thus the no-arbitrage forward delivery price is $S_0(1+i)^T$. If the quoted delivery price $K$ is greater, the cash-and-carry trades are: borrow $S_0$, buy one unit of the asset, and take a short position in one forward contract with delivery price $K$. The initial borrowing pays for the asset, so the net initial cash flow is zero.

$$ K^*=S_0(1+i)^T $$

At delivery, deliver the asset under the short forward and receive $K$. Use part of that receipt to repay the loan. The remaining amount is $K-S_0(1+i)^T$, which is positive when $K>K^*$. The short forward position is essential: it fixes the sale price in advance rather than leaving the investor exposed to the asset’s market price at delivery.

Compare prices.
Calculate the no-arbitrage delivery price for the asset and delivery date. Check whether the market quote is higher.

Buy and finance the asset.
Borrow the funds needed for the purchase and acquire the asset at the spot price.

Sell forward.
Enter a short forward contract for the same number of units and delivery date, using the quoted delivery price.

Settle at delivery.
Deliver the asset, collect the forward price, and repay the accumulated financing. The difference is the arbitrage profit.

Why the No-Arbitrage Comparison Works

The pricing argument follows by comparing two ways to obtain one unit of the asset at time $T$. One way is to buy it then, paying the forward delivery price $K$. The other is to buy it now and finance the purchase until $T$, which costs $S_0(1+i)^T$. These alternatives provide the same asset at the same date. If their costs differ in a frictionless market, the cheaper route can be used while a forward position locks in the more expensive route’s price.

For an overpriced forward, buying the asset now and selling it forward produces the cash-and-carry strategy. With no income or costs, its initial cash flow is $-S_0+S_0=0$: the first term is the asset purchase and the second is the loan proceeds. At delivery, the asset is delivered for $K$, while the loan repayment is $S_0(1+i)^T$. Hence the net delivery-date amount is $K-S_0(1+i)^T$. It is positive precisely when the market forward price exceeds the financed spot price.

Worked Example: An Overpriced Forward on a No-Income Asset

An asset is available for $S_0=\$100$. It pays no income and has no carrying costs. A two-year forward contract has a quoted delivery price of $K=\$116.50$. The effective annual borrowing and investment rate is $i=5\%$, fixed for both years. Determine whether cash-and-carry arbitrage is available and calculate its profit.

Method. Compare the quoted price with the financed spot price, then list the trades and the delivery-date cash flows.

The no-arbitrage delivery price is $100(1.05)^2=100(1.1025)=\$110.25$. Since $\$116.50>\$110.25$, the forward is overpriced under the stated assumptions.

At time zero, borrow $\$100$, buy one unit of the asset for $\$100$, and enter a short forward to deliver that unit in two years for $\$116.50$. The loan proceeds and purchase price offset, so the net initial cash flow is $-\$100+\$100=\$0$.

At time two, deliver the asset and receive $\$116.50$. The loan repayment is $\$100(1.05)^2=\$110.25$. The profit is $\$116.50-\$110.25=\$6.25$.

Check. The profit is also the quoted forward price less its no-arbitrage value: $\$116.50-\$110.25=\$6.25$. The common mistake to avoid is comparing the forward price with the spot price of $\$100$ rather than with the spot price accumulated to the delivery date.

Known Income and Carrying Costs

The same comparison applies when the asset produces known cash distributions or requires known carrying payments before delivery. For clarity, suppose all such amounts are deterministic, occur at stated dates $t_j$ between time zero and $T$, and can be invested or financed at the same effective annual rate $i$. Let $D_j$ be an income payment received at time $t_j$, and let $C_j$ be a carrying cost paid then. Accumulating each cash flow to delivery gives the no-arbitrage price:

K^*=S_0(1+i)^T+\sum_j C_j(1+i)^{T-t_j}-\sum_j D_j(1+i)^{T-t_j}

The spot purchase and its financing contribute $S_0(1+i)^T$. Carrying costs add to the amount that must be funded, so each cost is accumulated forward and added. Income received while holding the asset helps fund the position, so each distribution is accumulated to delivery and subtracted. Equivalently, the initial cost can be adjusted by the present values of those known cash flows and then accumulated to $T$.

This formula assumes that the asset and the cash flows can be obtained as specified and that the investor can fund interim payments. For an overpriced forward, carry out the same trades as before, but incorporate the actual income and costs into the financing plan. The delivery-date profit is $K-K^*$. If income or costs are uncertain, or if borrowing and investment rates differ, this simple deterministic calculation may not establish a risk-free arbitrage.

Worked Example: A Known Distribution Before Delivery

A share costs $\$200$ and will pay a known cash distribution of $\$6$ exactly one year from now. A two-year forward contract is quoted at $K=\$213$. The effective annual borrowing and investment rate is $i=4\%$, with no transaction costs or other income or costs. Calculate the cash-and-carry profit.

Method. Accumulate the spot purchase cost to year two and subtract the year-one distribution accumulated from its payment date to year two.

The spot cost accumulated for two years is $\$200(1.04)^2=\$200(1.0816)=\$216.32$. The distribution grows for one year after receipt, so its delivery-date value is $\$6(1.04)=\$6.24$. Therefore, the no-arbitrage delivery price is $\$216.32-\$6.24=\$210.08$.

Because $\$213>\$210.08$, borrow $\$200$, buy one share, and short the two-year forward at $\$213$. The initial loan funds the share purchase, giving zero net initial cash flow. At year one, receive the $\$6$ distribution and use it to reduce the outstanding loan balance. The balance after the first year is $\$200(1.04)=\$208$; after paying $\$6$, it is $\$202$. It grows to $\$202(1.04)=\$210.08$ by year two.

At year two, deliver the share under the forward and receive $\$213$. Repay $\$210.08$, leaving a profit of $\$213-\$210.08=\$2.92$.

Check. Accumulating the year-one distribution to year two gave $\$6.24$, exactly the reduction from the $\$216.32$ financed spot cost to the $\$210.08$ no-arbitrage price. Avoid subtracting the original $\$6$ at delivery without accounting for the interest earned or debt avoided after the distribution date.

Worked Example: Financing Deterministic Carrying Costs

A storable asset costs $\$50$. Holding one unit requires a known storage payment of $\$1$ at the end of each of years one and two. The asset pays no income. A two-year forward is quoted at $K=\$59$. The effective annual borrowing and investment rate is $i=4\%$. Find the arbitrage profit and show how the storage payments are financed.

Method. Accumulate the spot purchase and each storage payment to the delivery date, then compare that total carrying cost with the forward quote.

The spot purchase accumulates to $\$50(1.04)^2=\$50(1.0816)=\$54.08$. The year-one storage payment accumulates for one year to $\$1(1.04)=\$1.04$. The year-two payment contributes $\$1$. Thus the no-arbitrage delivery price is $\$54.08+\$1.04+\$1=\$56.12$.

The market quote exceeds this amount by $\$59-\$56.12=\$2.88$. At time zero, borrow $\$50$, buy one unit of the asset, and short the two-year forward at $\$59$. The initial net cash flow is zero. Pay the first $\$1$ storage charge at year one, financing it with an additional borrowing if necessary. Its year-two repayment is $\$1.04$. Pay the second $\$1$ storage charge at year two. The total year-two debt is $\$54.08+\$1.04+\$1=\$56.12$.

At delivery, deliver the asset and receive $\$59$. After repaying $\$56.12$, the remaining profit is $\$2.88$.

Check. Both storage payments are included, but they are accumulated for different lengths of time. Treating both as year-zero costs, or accumulating both for two years, would give an incorrect financing requirement.

Common Exam Traps

The calculation is short once the cash flows are organized, but several errors can reverse or distort the result. Keep the contract position and the funding cash flows distinct.

Comparing with spot instead of the delivery-date value. For a no-income asset, compare $K$ with $S_0(1+i)^T$, not just $S_0$.
Taking the wrong forward position. In cash-and-carry, buy the asset and take the short forward position. At delivery, the short position requires delivery of the asset in exchange for the contracted price.
Subtracting income at the wrong date. A distribution received before delivery reduces the financing need, but its effect at delivery includes accumulation from its payment date.
Omitting carrying costs. If the strategy requires paying storage or another deterministic cost, those payments must be included in the amount funded.
Claiming an arbitrage without checking assumptions. The positive difference is guaranteed only under the model’s assumptions, including known cash flows and the ability to borrow and trade as required.

Check Your Understanding

For each problem, assume borrowing and investment are available at the stated fixed effective annual rate, all stated cash flows are certain, and there are no transaction costs or default risk. Determine the cash-and-carry profit when the quoted forward is overpriced.

An asset with no income or carrying costs has spot price $\$80$. A two-year forward is quoted at $\$90$, and $i=5\%$. Find the no-arbitrage price and profit. (Answer: no-arbitrage price $\$88.20$; profit $\$1.80$ per unit)
An asset costs $\$150$, pays a known distribution of $\$5$ at year one, and has no other income or costs. A two-year forward is quoted at $\$160$, and $i=4\%$. Find the delivery-date profit. (Answer: $\$2.96$ per unit)
An asset costs $\$60$. It produces known income of $\$1$ at year one and has a known carrying cost of $\$2$ at year one. There are no other cash flows. A two-year forward is quoted at $\$69$, and $i=5\%$. Find the profit. (Answer: $\$1.80$ per unit)
An asset costs $\$125$, pays known distributions of $\$4$ at the ends of years one and two, and has no carrying costs. A three-year forward is quoted at $\$142$, and $i=6\%$. Find the profit to the nearest cent. (Answer: $\$1.86$ per unit)
An asset costs $\$90$ and pays no income. It has known carrying costs of $\$1$ at year one and $\$2$ at year two. A three-year forward is quoted at $\$105.50$, and $i=4\%$. Find the profit to the nearest cent. (Answer: $\$1.10$ per unit)

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