Arbitrage When A Market Forward Differs

Why a Market Forward Quote Must Match the Spot Curve

A forward rate is a rate agreed today for borrowing or investing during a future interval. Spot rates also determine the accumulation available over that interval: invest to the first date at its spot rate, then invest again at the next date, or invest directly to the final date. In a frictionless market, these alternatives must give the same final accumulation. If a quoted forward rate does not, an investor can combine spot borrowing or lending with a forward transaction to lock in a gain.

The comparison must use the same dates, compounding convention, and principal. A quoted forward rate is not tested by comparing it directly with the spot rate at the final maturity. Instead, compare the accumulation from investing through the quoted forward with the accumulation from investing directly to the final date.

Definition: Assume annual effective compounding and integer-year maturities. Let $s_m$ and $s_n$ be the spot rates for maturities $m$ and $n$, where $0<m<n$. The spot-implied annual effective forward rate for the interval from year $m$ to year $n$, denoted $f^*_{m,n}$, satisfies $$ (1+s_m)^m(1+f^*_{m,n})^{n-m}=(1+s_n)^n. $$ A market quote $f^M_{m,n}$ is consistent with no arbitrage only if $f^M_{m,n}=f^*_{m,n}$, subject to the market assumptions stated below.

The comparison is about equivalent investment strategies, not a forecast of what a future short-term rate will be. Here, a spot rate $s_j$ is an annual effective rate for investing from time 0 to year $j$. A forward investment of principal $C$, starting at year $m$ and ending at year $n$, requires investing $C$ at $m$ and returns $C(1+f^M_{m,n})^{n-m}$ at $n$.

Deriving the No-Arbitrage Comparison

Write $A_m=(1+s_m)^m$ and $A_n=(1+s_n)^n$ for the accumulated value at each maturity of one unit invested at time 0. Investing one unit to year $m$ produces $A_m$. If those proceeds are invested at the quoted forward rate, the value at year $n$ is $A_m(1+f^M_{m,n})^{n-m}$. Investing one unit directly to year $n$ instead produces $A_n$.

Both strategies cost one unit at time 0. If their final values differ, buy the strategy with the higher payoff and finance its initial cost by selling the strategy with the lower payoff. The initial cash flows cancel; there are no intermediate net cash flows if the investment at year $m$ is funded by the matching forward transaction. The final difference is then a positive, predetermined amount. This explains why the no-arbitrage condition is equality of accumulated values:

(1+s_m)^m(1+f^M_{m,n})^{n-m}=(1+s_n)^n.

Solving the equality for the implied rate gives

f^*_{m,n}= \left(\frac{(1+s_n)^n}{(1+s_m)^m}\right)^{1/(n-m)}-1.

This formula assumes that the spot investments and forward transaction can be entered at the stated rates, that borrowing and lending are available at the same rates, and that there are no transaction costs, taxes, credit losses, or restrictions on short selling. It also assumes the forward quote is for the same interval and effective annual compounding. If any of these conditions fails, a calculated mismatch alone does not establish an executable arbitrage.

Which Positions Exploit the Mismatch?

The direction of the trade follows directly from which strategy has the larger final accumulation. A higher quoted forward rate makes the strategy that rolls the investment through year $m$ more valuable. A lower quoted forward rate makes direct investment to year $n$ more valuable.

Compare final accumulations.
Calculate $A_n=(1+s_n)^n$ and $A_m(1+f^M_{m,n})^{n-m}$. Keep the initial principal the same for both.

If the market forward is too high, roll through the forward.
Invest to year $m$, then lend or invest the proceeds at the quoted forward rate. Short the direct investment to year $n$. The final surplus is the rolled accumulation minus the direct accumulation.

If the market forward is too low, invest directly and borrow forward.
Buy the direct investment to year $n$, short the investment to year $m$, and borrow at the quoted forward rate. The short investment’s year-$m$ liability is matched by the forward borrowing; the final surplus is the direct accumulation minus the rolled accumulation.

In these strategies, “short the direct investment” means receive its price at time 0 and commit to pay its accumulated value at maturity. “Borrow forward” means receive the agreed principal at the forward start date and repay that principal with interest at the forward end date. These descriptions make the timing of the cash flows explicit.

Worked Arbitrage Examples

Worked Example: A Forward Quote Above the Implied Rate

The one-year spot rate is 4% and the two-year spot rate is 5%, both annual effective. A dealer offers a one-year forward investment from year 1 to year 2 at 7%. For a principal of $100,000, identify the arbitrage and calculate its year-2 profit. Assume the investments and short sale are available on the stated terms without transaction costs or default risk.

Method. Compare the direct two-year accumulation with the accumulation from investing for one year and then using the quoted forward. The relevant growth factors are

A_1=(1.04)^1=1.04,\qquad A_2=(1.05)^2=1.1025.

The spot-implied one-year forward rate is

f^*_{1,2}=\frac{1.1025}{1.04}-1 =1.0600961538\ldots-1 =0.0600961538\ldots.

Thus the implied rate is approximately 6.0096%, below the quoted 7%. The quoted-forward strategy has the larger accumulation.

Positions and cash flows. At time 0, invest $100,000 in the one-year spot investment, costing $100,000, and short the two-year spot investment, receiving $100,000. The initial net cash flow is $-\$100{,}000+\$100{,}000=\$0$. At year 1, the spot investment pays $ \$100{,}000(1.04)=\$104{,}000$. Invest that amount at the forward rate; this uses the full $104,000, so the year-1 net cash flow is zero. At year 2, the forward investment pays

\$104{,}000(1.07)=\$111{,}280.

The short two-year position requires paying

\$100{,}000(1.05)^2=\$110{,}250.

The net year-2 profit is $ \$111{,}280-\$110{,}250=\$1{,}030$. The check is that the initial net cost and the year-1 net cash flow are both zero, while the final cash flow is positive. The common mistake this avoids is comparing 7% with the two-year spot rate of 5%; the correct comparison is with the one-year forward rate implied by both spot rates.

Worked Example: A Forward Quote Below the Implied Rate

Use the same annual effective spot rates: 4% for one year and 5% for two years. The available one-year forward borrowing rate from year 1 to year 2 is 5%. For a $100,000 principal, determine the arbitrage and its year-2 profit. Assume the same frictionless trading conditions.

Method. The spot-implied rate is still $1.1025/1.04-1=0.0600961538\ldots$, or approximately 6.0096%. Since the quoted 5% is lower, the direct two-year investment accumulates more than the investment rolled through the quoted forward.

Positions and cash flows. At time 0, buy the two-year spot investment for $100,000 and short the one-year spot investment, receiving $100,000. The initial net cash flow is zero. At year 1, the short position requires repayment of $ \$100{,}000(1.04)=\$104{,}000$. Borrow $104,000 through the forward transaction at 5%; the borrowing proceeds exactly meet that repayment, leaving no net year-1 cash flow. At year 2, the direct investment pays

\$100{,}000(1.05)^2=\$110{,}250.

The forward borrowing must be repaid at

\$104{,}000(1.05)=\$109{,}200.

The year-2 profit is $ \$110{,}250-\$109{,}200=\$1{,}050$. The initial and year-1 cash flows cancel, and the year-2 cash flow is positive, confirming the arbitrage. The common mistake is to lend forward when the quoted forward rate is low; the profitable direction here is to borrow forward and invest directly to the later maturity.

Worked Example: A Two-Year Forward Interval

The one-year and three-year spot rates are both 5%, annual effective. A market offers a two-year forward investment beginning at year 1 at 6%. For a $20,000 principal, find the arbitrage profit at year 3. Assume all positions can be entered without cost or default risk.

Method. First compare the investment paths over the same three years. Direct investment to year 3 accumulates by

$$ (1.05)^3=1.157625. $$

Investing to year 1 and then at the two-year quoted forward accumulates by

$$ (1.05)(1.06)^2 =1.05(1.1236) =1.17978. $$

The forward route is larger. The implied annual effective forward rate is 5%, because the one-year and three-year spot rates are equal; equivalently, $(1.05)^3/(1.05)=1.05^2$. Therefore the quoted 6% is above the implied rate.

Positions and cash flows. At time 0, invest $20,000 at the one-year spot rate and short the three-year spot investment with the same $20,000 time-0 value. The initial net cash flow is zero. At year 1, the investment pays $ \$20{,}000(1.05)=\$21{,}000$, which is invested through the forward. At year 3, the forward investment pays

\$21{,}000(1.06)^2 =\$21{,}000(1.1236) =\$23{,}595.60.

The short three-year position requires paying $ \$20{,}000(1.05)^3=\$23{,}152.50$. The net profit is $ \$23{,}595.60-\$23{,}152.50=\$443.10$. The check is that the initial cost is zero, the intermediate cash is fully invested, and the final net payment is positive. The common mistake is to apply the quoted 6% for three years; it applies only to the two-year forward interval.

Common Exam Traps and Practical Checks

A mismatch should be treated as a cash-flow problem, not merely as a difference between two percentages. Before naming an arbitrage, verify that its initial cost is zero, that all intermediate obligations are funded, and that its final payoff is positive. Scaling every position by a positive amount scales the final profit by the same amount, but does not change which direction is profitable.

Comparing the quote with the wrong spot rate. The forward quote is not generally equal to the spot rate at the forward end date. Derive the implied rate from both relevant spot maturities.
Ignoring interval length. For a forward interval lasting $n-m$ years, the forward accumulation factor is $(1+f)^{n-m}$, not $1+f$ unless the interval is one year.
Comparing rates without matching the investment amount. Use the same time-0 investment value, or explicitly adjust the principal. Otherwise the final amounts are not comparable.
Getting the trade direction backward. If the quoted-forward route pays more, invest through the forward and short the direct route. If the direct route pays more, buy it and borrow through the forward.
Claiming arbitrage without stating assumptions. Transaction costs, different borrowing and lending rates, counterparty risk, or inability to short may remove the apparent profit.

Key takeaway: For the same initial investment, compare $(1+s_n)^n$ with $(1+s_m)^m(1+f^M_{m,n})^{n-m}$. If the second is larger, roll through the quoted forward and short the direct investment. If the first is larger, invest directly and borrow forward. A valid arbitrage has zero initial net cost, funded intermediate cash flows, and a positive final surplus under the stated market assumptions.

Check Your Understanding

For each problem, assume annual effective compounding, no transaction costs or default risk, and that the stated borrowing, lending, and short positions are available. Compare accumulations over matching dates before deciding the trade direction.

The one-year spot rate is 2% and the two-year spot rate is 3%. A one-year forward investment from year 1 to year 2 is quoted at 4%. For a $10,000 principal, identify the profitable strategy and its final profit. (Answer: The implied forward rate is approximately 4.0196%, so the quote is too low; invest directly for two years and borrow forward. Profit: $1.00 at year 2.)
The one-year spot rate is 4% and the three-year spot rate is 6%. A two-year forward investment beginning at year 1 is quoted at 7%. For a $100,000 principal, state the trade direction and final profit. (Answer: The forward route accumulates to $1.04(1.07)^2=1.190696$, below $1.06^3=1.191016$; invest directly and borrow forward. Profit: $32.00 at year 3.)
The two-year spot rate is 3% and the four-year spot rate is 4%. The two-year forward rate from year 2 to year 4 is 5%. For a $250,000 principal, find the arbitrage direction and profit. (Answer: $1.04^4=1.16985856$ exceeds $1.03^2(1.05)^2=1.16964225$; invest directly and borrow forward. Profit: $54.08 at year 4.)
The one-year spot rate is 5% and the three-year spot rate is 4%. A two-year forward borrowing rate beginning at year 1 is quoted at 3%. For a $10,000 principal, determine whether to use the forward borrowing or lending position and calculate the final profit. (Answer: Direct accumulation is $1.04^3=1.124864$, compared with $1.05(1.03)^2=1.113945$; buy the three-year investment, short the one-year investment, and borrow forward. Profit: $109.19 at year 3.)
The two-year spot rate is 4% and the five-year spot rate is 5%. A three-year forward investment beginning at year 2 is quoted at 5.5%. Is the quote above or below the spot-implied rate, and which pair of strategies gives the arbitrage direction? (Answer: The implied rate is $\left((1.05)^5/(1.04)^2\right)^{1/3}-1$, approximately 5.67%, so the quote is below it; invest directly to year 5 and borrow forward, with the year-2 spot investment shorted to fund the forward borrowing.)

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