Reverse Cash-And-Carry Arbitrage For Underpriced Forwards

Why an Underpriced Forward Creates an Opportunity

A forward contract sets a price today for an asset to be exchanged at a future date. If the delivery price is below the asset’s no-arbitrage forward price, an investor can arrange to buy the asset through the forward and sell an equivalent asset now. The sale proceeds are invested, and the asset received at maturity is used to return the borrowed asset. This is called a reverse cash-and-carry arbitrage.

The key idea is to compare two ways of obtaining one unit of the asset at time $T$: buying it in the market at time $T$, or locking in a purchase through a forward contract today. A forward price that is too low makes the second route cheaper. The reverse trade turns that price difference into a nonnegative, predetermined terminal cash flow under the assumptions below.

Definition: Consider an asset with spot price $S_0$, no income or carrying costs, and a forward contract maturing at time $T$. Let $i$ be the effective annual risk-free rate, assumed deterministic, and let $K$ be the contract’s delivery price. The no-arbitrage forward price is $F_{0,T}=S_0(1+i)^T$. If $K<F_{0,T}$, the forward is underpriced in this model.

The model assumes that the asset can be borrowed and sold short, that the short-sale proceeds can be invested at the stated rate, and that the forward is entered without an initial payment. It also assumes no taxes, transaction costs, default, or restrictions on borrowing and short selling. These assumptions matter: without them, the theoretical price difference may not be an achievable arbitrage profit.

The Reverse Cash-and-Carry Positions

For each unit of the asset, the investor borrows one unit, sells it at the spot price, invests the proceeds, and enters a long forward with delivery price $K$. The long forward requires a payment of $K$ at time $T$, in exchange for one unit of the asset. That unit is returned to the lender to close the short position.

Borrow and sell the asset.
Borrow one unit and sell it for $S_0$. The short sale creates a repayment obligation: one unit of the asset must eventually be returned.

Invest the proceeds.
Invest $S_0$ at the risk-free rate. At time $T$, the investment is worth $S_0(1+i)^T$.

Enter a long forward.
Agree to pay $K$ at time $T$ and receive one unit of the asset.

Settle and return the asset.
Use $K$ from the investment account to pay for the forward-delivered unit, then return that unit to the lender. The remaining cash is the arbitrage profit.

\text{Profit at }T=S_0(1+i)^T-K=F_{0,T}-K

The short-sale proceeds are not the profit. They are invested to fund the forward purchase, and the relevant comparison is between the accumulated proceeds and the delivery payment. If the delivery price is below the no-arbitrage price, the difference is positive.

Why the Formula Holds

The formula follows by tracking the cash account and the asset obligation separately. The short sale supplies $S_0$ at time 0. With deterministic interest and annual effective rate $i$, that amount accumulates for $T$ years to $S_0(1+i)^T$. At maturity, the forward requires a payment of $K$, and supplies exactly the one asset unit needed to repay the asset loan. Therefore the asset positions cancel, leaving only the cash difference:

\begin{aligned} \text{Terminal cash} &= \text{accumulated short-sale proceeds} - \text{forward delivery payment}\\ &= S_0(1+i)^T-K. \end{aligned}

There is no uncertainty about the terminal cash amount in this idealized setup: neither the short-sale proceeds’ growth nor the forward delivery payment depends on the asset’s future spot price. The condition for a positive profit is therefore $S_0(1+i)^T-K>0$, or equivalently $K<F_{0,T}$.

Worked Example: An Underpriced Forward on an Asset With No Income

A hypothetical asset sells for $120$ dollars today. The effective annual risk-free rate is $5\%$, and a two-year forward has a delivery price of $126$ dollars per unit. Assume the asset pays no income and can be borrowed and sold short. Construct the reverse cash-and-carry trade and calculate its profit per unit.

Method. Compare the market delivery price with the no-arbitrage forward price, then track the short-sale account through settlement.

Step 1: Calculate the no-arbitrage forward price. Here $S_0=120$, $i=0.05$, and $T=2$:

F_{0,2}=120(1.05)^2=120(1.1025)=132.30.

Since $126<132.30$, the forward is underpriced by $6.30$ dollars per unit.

Step 2: Establish the positions at time 0. Borrow one unit and sell it for $120$ dollars. Invest the $120$ dollars, and enter a long forward requiring payment of $126$ dollars in two years.

Step 3: Settle at time 2. The invested proceeds grow to $120(1.05)^2=132.30$ dollars. Pay $126$ dollars to receive the asset under the forward, then return the asset to the lender. The remaining account balance is $132.30-126=6.30$ dollars.

Check. The result equals $F_{0,2}-K=132.30-126=6.30$. The asset received and the asset owed cancel, leaving a positive cash amount. The common mistake avoided here is calling the initial $120$-dollar short-sale receipt the profit; it must first be grown to maturity and used to pay the forward price.

Known Cash Distributions Before Delivery

If the asset pays a known cash distribution before time $T$, the investor who has borrowed and sold the asset must compensate the lender for that distribution. The short seller does not own the asset when the distribution is paid, but the lender would have received it if the asset had not been lent. The distribution is therefore a cash outflow in the reverse trade.

Suppose known distributions $D_j$ are paid at times $t_j$, where $0<t_j<T$. Under the same deterministic effective annual rate $i$, each distribution payment reduces the invested balance at its payment date. Its effect at time $T$ is $D_j(1+i)^{T-t_j}$. Thus the no-arbitrage forward price and the reverse-arbitrage terminal profit per unit are:

Result: For a known-income asset with cash distributions $D_j$ at times $t_j$, no other income or carrying costs, and a deterministic effective annual rate $i$, $$ F_{0,T}=S_0(1+i)^T-\sum_j D_j(1+i)^{T-t_j}. $$ If a market forward delivery price $K$ is below this value, the reverse trade’s terminal profit per unit is $F_{0,T}-K$, provided the short seller pays the distributions owed to the lender.

This result can also be read from the cash account. The initial short-sale proceeds grow to $S_0(1+i)^T$ if left invested. A payment $D_j$ made at $t_j$ would otherwise have accumulated to $D_j(1+i)^{T-t_j}$ by maturity. Subtracting the accumulated value of each required payment gives the terminal value available for the forward purchase. The investor can fund a payment from the account or borrow to make it; either way, its terminal cost is the same under the stated rate assumption.

Worked Example: Accounting for a Distribution Owed to the Lender

A hypothetical share has a spot price of $200$ dollars. It will pay a known cash distribution of $8$ dollars exactly one year from now. A two-year forward has a delivery price of $204$ dollars. The effective annual risk-free rate is $4\%$, and the share can be borrowed and sold short. Calculate the reverse-arbitrage profit per share.

Method. Accumulate the short-sale proceeds to year 2, subtract the year-1 distribution’s year-2 value, and then pay the forward delivery price.

Step 1: Calculate the accumulated short-sale proceeds.

$$ 200(1.04)^2=200(1.0816)=216.32. $$

Step 2: Account for the distribution paid at year 1. The short seller pays $8$ dollars to the lender at year 1. Its value at year 2 is $8(1.04)=8.32$ dollars.

Step 3: Find the cash available at year 2 before the forward purchase.

$$ 216.32-8.32=208.00. $$

Step 4: Settle the forward and return the share. Pay $204$ dollars under the long forward, receive one share, and return it to the lender. The remaining cash is $208-204=4$ dollars per share.

Check. The no-arbitrage forward price is $200(1.04)^2-8(1.04)=208$, so the profit is $208-204=4$. The common mistake avoided is ignoring the distribution because the short seller does not own the share on the payment date; the lender is still owed that cash.

Scaling the Trade and Avoiding Sign Errors

If a contract covers $N$ units and the delivery price is quoted per unit, multiply the per-unit profit by $N$. If a contract specifies a different unit multiplier, apply that multiplier to the per-unit cash flows. The trade’s initial cash flows and its maturity cash flows should not be mixed: the short-sale proceeds arrive at time 0, while the forward payment and the calculated arbitrage profit are measured at time $T$.

Worked Example: Calculating Profit for Multiple Units

A hypothetical commodity has a spot price of $48$ dollars per unit and pays no income. The effective annual risk-free rate is $5\%$. An investor can borrow 25 units, sell them, and enter a two-year long forward for all 25 units at a delivery price of $51.80$ dollars per unit. Calculate the total arbitrage profit at maturity.

Method. First calculate the fair forward price per unit, then find the per-unit and aggregate differences.

Step 1: Calculate the no-arbitrage forward price per unit.

F_{0,2}=48(1.05)^2=48(1.1025)=52.92.

Step 2: Find the profit per unit. The market delivery price is $51.80$, which is $52.92-51.80=1.12$ dollars below the fair forward price.

Step 3: Scale to 25 units.

$$ 25(1.12)=28.00. $$

Settlement check. The 25 short-sale proceeds accumulate to $25(52.92)=1{,}323$ dollars. The 25-unit forward payment is $25(51.80)=1{,}295$ dollars. Returning the 25 units received under the forward closes the borrowing, and the account retains $1{,}323-1{,}295=28$ dollars. The common mistake avoided is comparing a total contract delivery payment with a per-unit forward value.

Common Exam Traps

Using the wrong inequality. Reverse cash-and-carry applies when $K<F_{0,T}$. If $K>F_{0,T}$, the reverse trade has a negative terminal cash flow under this model; it is not the underpriced-forward arbitrage.
Forgetting to borrow the asset. Selling an asset the investor does not own requires a short sale supported by an asset borrow. The long forward supplies the unit used to return the borrowed asset at maturity.
Treating the short-sale proceeds as immediate profit. The proceeds are invested and later used to pay for the forward-delivered asset. Only the remaining cash after settlement is profit.
Omitting distributions. When a short seller owes distributions to the lender, subtract their accumulated values from the maturity account. Do not treat the borrowed asset as if it earned distributions for the short seller.
Mixing time units. Use $T$ in years when $i$ is an effective annual rate. For a payment at $t_j$, accumulate it for the remaining $T-t_j$ years, not for the full contract term.

Key takeaway: Under the idealized no-arbitrage assumptions, borrow and sell the asset, invest the proceeds, and go long the underpriced forward. At maturity, use the forward-delivered asset to repay the borrow; the cash left is the accumulated short-sale proceeds minus any accumulated distribution obligations and the forward delivery payment. This method is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, assume the stated effective annual rate is deterministic, the required asset borrow and short sale are available, and there are no transaction costs unless distributions are specified.

An asset with no income has spot price $90$. The effective annual rate is $4\%$, the term is two years, and the forward delivery price is $92$ per unit. Find the reverse-arbitrage profit per unit at maturity. (Answer: $5.34$ dollars per unit.)
An asset has spot price $150$, pays a known distribution of $9$ at time $0.5$, and has a one-year forward delivery price of $148$. The effective annual rate is $6\%$. Calculate the reverse-arbitrage profit per unit, rounding to the nearest cent. (Answer: $1.73$ dollars per unit.)
An asset pays no income, has spot price $64$, and has a three-year forward delivery price of $67.50$. The effective annual rate is $3\%$. Calculate the profit per unit at maturity, rounded to the nearest cent. (Answer: $2.43$ dollars per unit.)
An asset has spot price $240$, pays a known distribution of $12$ at time 1, and has a two-year forward delivery price of $249$. The effective annual rate is $5\%$. If the investor can short 8 units, calculate the total reverse-arbitrage profit at maturity. (Answer: $24.00$ dollars.)

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