Swap And Futures Hedges With Residual Cash Flows

Combining Cash Flows Across a Hedge

A swap and a futures position can adjust a floating-rate borrowing cost or investment return, but the two instruments do not automatically eliminate every rate exposure. The swap may cover only part of the underlying notional, and the futures contract may reference a different rate or move by a different amount. A reliable calculation therefore keeps the underlying cash flow and each hedge settlement separate until the final combination.

The method in this tutorial uses a single accrual period and simple interest. A positive amount will mean an outflow for a borrower or a receipt for an investor, as appropriate; the direction of each individual settlement will be stated explicitly. We assume no fees, taxes, collateral interest, discounting, or interim variation-margin cash flows. The futures settlement formula below is the specified linear rate-change model for the problems; an actual contract may have contract-specific settlement conventions.

Model and conventions: Let $\tau$ be the year fraction for the accrual period. A floating-rate loan of principal $L$ at annual simple rate $R_L$ has interest $L\tau R_L$. A pay-fixed, receive-floating swap of notional $N$, fixed rate $K$, and floating rate $R_S$ pays the net amount $N\tau(K-R_S)$ when this is positive; a negative amount is a net receipt. For a rate futures position with total notional $P$, the settlement gain is $P\tau(F_0-F_1)$ for a long position and $P\tau(F_1-F_0)$ for a short position, where $F_0$ and $F_1$ are the initial and ending futures rates, expressed as decimals.

Build the Net Cash Flow Before Adding Futures

For a borrower with a floating-rate loan, loan interest is an outflow. A pay-fixed, receive-floating swap generates a net receipt when the floating rate exceeds the fixed rate, which reduces that outflow. If the swap floating rate is exactly the same as the loan rate, the combined amount before futures can be written compactly.

\begin{aligned} \text{Borrowing cost before futures} &=L\tau R+N\tau(K-R)\\ &=\tau\bigl[(L-N)R+NK\bigr]. \end{aligned}

The expression shows why a swap notional equal to the loan principal can remove the loan's direct rate sensitivity in this simplified matching case. When $N<L$, however, the unmatched amount $L-N$ remains exposed to the floating rate. The fixed-rate term $NK$ is still part of the cash cost.

For an investor earning a floating rate $R$ on principal $L$, a receive-fixed, pay-floating swap has net cash flow $N\tau(K-R)$. The total receipt before futures is therefore $L\tau R+N\tau(K-R)$. This expression has the same algebraic form, but its interpretation is investment income rather than a borrowing outflow.

Futures direction: Under the stated model, a long rate-futures position gains when the futures rate falls, while a short position gains when the futures rate rises. A borrower exposed to rising rates may use a short position as a hedge; an investor exposed to falling rates may use a long position. Whether the hedge is effective depends on how closely the futures rate change tracks the underlying rate change.

Why the Residual Rate Exposure Has This Form

Suppose the borrower has a loan rate $R_L$, a swap floating rate $R_S$, and a pay-fixed, receive-floating swap. The amount of interest paid on the loan plus the swap's net payment is

C=\tau\left(LR_L+NK-NR_S\right).

This follows by adding the loan interest $L\tau R_L$ to the swap net payment $N\tau(K-R_S)$. If both instruments use the same rate $R$, then $R_L=R_S=R$, so

C=\tau\left[(L-N)R+NK\right].

Thus a change $\Delta R$ in the common rate changes the pre-futures cost by $\tau(L-N)\Delta R$. If the loan and swap rates differ, their exposures must instead be tracked separately: the change is $\tau(L\Delta R_L-N\Delta R_S)$. This distinction is important when the underlying loan rate and the swap reference rate do not match.

For a short futures position, let $\Delta F=F_1-F_0$. Its gain under the model is $P\tau\Delta F$, reducing the borrower's cost. The change in cost after futures is therefore

\Delta C_{\text{after futures}} =\tau\left[(L-N)\Delta R-P\Delta F\right]

when the loan and swap share a rate. The residual is zero only if the futures position's gain offsets the unmatched loan-and-swap exposure for the rate moves in question. Matching just the notionals is not sufficient if the rates move differently.

A Reliable Calculation Sequence

Write the underlying cash flow.
For a loan, calculate its interest outflow. For an investment, calculate its interest receipt. Apply the stated rate and accrual fraction.

Calculate the swap settlement.
For pay fixed and receive floating, compute $N\tau(K-R_S)$ as a payment; a negative result is a net receipt. Reverse the direction for receive fixed and pay floating.

Calculate the futures settlement separately.
Use the position direction and the change in futures rate. Do not infer the sign from the change in futures price without checking the stated quotation convention.

Combine and interpret.
Add receipts and payments with their signs, then compare the result with a baseline or calculate the remaining sensitivity if requested.

Worked Examples

Worked Example: Partially Swapped Borrowing With a Short Futures Hedge

A borrower has a $ \$2{,}000{,}000$ floating-rate loan. For one quarter, its loan rate is $5.0\%$, and the loan interest is based on that same rate as the swap's floating leg. The borrower pays fixed at $4.5\%$ and receives floating on a $ \$1{,}500{,}000$ swap. It also shorts three rate-futures contracts, each with notional $ \$100{,}000$; the futures rate rises from $4.2\%$ to $4.8\%$. Assume simple interest, $\tau=0.25$, and the settlement model stated above. Find the outflow after both hedges and compare it with the cost at the initial $4.0\%$ loan rate before futures.

Method. Calculate the loan interest, the swap's net cash flow, and the futures gain separately. The total futures notional is $P=3(100{,}000)=\$300{,}000$.

Loan interest. The rate is $0.050$, so

2{,}000{,}000(0.25)(0.050)=\$25{,}000.

Swap. Its net payment, with positive meaning a payment by the borrower, is

1{,}500{,}000(0.25)(0.045-0.050) =-\$1{,}875.

The negative sign means a net receipt of $ \$1{,}875$. The borrowing outflow after the swap but before futures is $25{,}000-1{,}875=\$23{,}125$.

Futures. The short position gains because the futures rate rises:

300{,}000(0.25)(0.048-0.042)=\$450.

This gain reduces the borrowing outflow, giving $23{,}125-450=\boxed{\$22{,}675}$.

Baseline check. At the initial loan rate of $4.0\%$, loan interest is $2{,}000{,}000(0.25)(0.040)=\$20{,}000$. The swap net payment is $1{,}500{,}000(0.25)(0.045-0.040)=\$1{,}875$, so the pre-futures baseline is $20{,}000+1{,}875=\$21{,}875$. The final cost is $22{,}675-21{,}875=\$800$ above that baseline. This also follows from the unmatched-rate exposure: $500{,}000(0.25)(0.010)=\$1{,}250$ additional pre-futures cost, reduced by the $ \$450$ futures gain, leaving $ \$800$. The check confirms both the sign and the amount.

Common mistake avoided: Subtracting the futures gain twice, or treating the swap as if it covered the full $ \$2{,}000{,}000$ loan. Only $ \$1{,}500{,}000$ is swapped, leaving $ \$500{,}000$ directly exposed to the loan rate.

Worked Example: Floating Investment, Receive-Fixed Swap, and Long Futures

An investor earns a floating rate on $ \$1{,}200{,}000$ for one quarter. The rate is $3.6\%$ at the start and $4.4\%$ for the investment period. The investor receives fixed at $4.1\%$ and pays floating on a $ \$900{,}000$ swap. It is long two rate-futures contracts of $ \$100{,}000$ each; the futures rate rises from $4.0\%$ to $4.5\%$. Assume simple interest and $\tau=0.25$. Find the combined receipt and its change from the initial-rate receipt before futures.

Method. Treat investment interest and the receive-fixed, pay-floating swap as receipts with signs. The futures position is long, so a rise in the futures rate produces a loss under the specified model.

Investment income.

1{,}200{,}000(0.25)(0.044)=\$13{,}200.

Swap settlement. For this receive-fixed, pay-floating swap, net cash flow to the investor is $N\tau(K-R)$:

900{,}000(0.25)(0.041-0.044)=-\$675.

The negative result is a net payment of $ \$675$. The combined investment and swap receipt is $13{,}200-675=\$12{,}525$.

Futures settlement. Total notional is $P=2(100{,}000)=\$200{,}000$. The long position loses as the rate rises:

200{,}000(0.25)(0.040-0.045)=-\$250.

The final receipt is $12{,}525-250=\boxed{\$12{,}275}$.

Baseline check. At the initial rate of $3.6\%$, investment income is $1{,}200{,}000(0.25)(0.036)=\$10{,}800$. The swap pays the investor $900{,}000(0.25)(0.041-0.036)=\$1{,}125$, making the pre-futures baseline $ \$11{,}925$. The final receipt is $12{,}275-11{,}925=\$350$ higher. Before futures, the rate rise adds $300{,}000(0.25)(0.008)=\$600$ because only $ \$300{,}000$ remains unmatched; the futures loss reduces that improvement by $ \$250$, leaving $ \$350$. The calculation checks.

Common mistake avoided: A long rate-futures position does not gain when the rate rises under this settlement convention. Its loss of $ \$250$ must be subtracted from the investment-and-swap receipt.

Worked Example: Different Loan and Swap Rates Leave Basis Exposure

A borrower has a $ \$1{,}000{,}000$ loan for half a year. Its reference rate rises from $4.1\%$ to $5.1\%$. The borrower pays fixed at $4.2\%$ and receives floating on a $ \$700{,}000$ swap, but the swap rate rises only from $3.9\%$ to $4.8\%$. It shorts two futures contracts of $ \$100{,}000$ each; the futures rate rises from $4.0\%$ to $4.6\%$. Assume $\tau=0.5$, simple interest, and the stated futures settlement model. Find the final outflow and its change from the initial-rate pre-futures cost.

Method. Since the loan and swap use different rates, calculate each cash flow using its own rate. Do not substitute the loan rate into the swap formula. The futures total notional is $P=\$200{,}000$.

Loan interest at the ending rate:

1{,}000{,}000(0.5)(0.051)=\$25{,}500.

Swap settlement at the ending swap rate:

700{,}000(0.5)(0.042-0.048)=-\$2{,}100.

This is a net receipt of $ \$2{,}100$, so the outflow before futures is $25{,}500-2{,}100=\$23{,}400$.

Futures settlement:

200{,}000(0.5)(0.046-0.040)=\$600.

The short position gains $ \$600$, reducing the final outflow to $23{,}400-600=\boxed{\$22{,}800}$.

Baseline and residual check. Initially, loan interest is $1{,}000{,}000(0.5)(0.041)=\$20{,}500$. The swap payment is $700{,}000(0.5)(0.042-0.039)=\$1{,}050$, so the initial pre-futures cost is $20{,}500+1{,}050=\$21{,}550$. The final outflow is $22{,}800-21{,}550=\$1{,}250$ higher. The underlying loan cost rose by $1{,}000{,}000(0.5)(0.010)=\$5{,}000$; the swap's net cash flow improved by $700{,}000(0.5)(0.009)=\$3{,}150$. The pre-futures cost therefore rose by $5{,}000-3{,}150=\$1{,}850$. After the futures gain of $ \$600$, the residual increase is $1{,}850-600=\$1{,}250$, matching the direct comparison.

Common mistake avoided: Assuming the floating leg of the swap exactly offsets the loan. Here the loan rate rises by $1.0$ percentage point while the swap rate rises by $0.9$ percentage point, so both notional mismatch and rate-basis mismatch remain relevant.

Common Traps and Practical Checks

When cash flows from several instruments appear in one problem, the arithmetic is usually manageable; the main risk is assigning a sign or rate to the wrong item. Before finalizing a result, check the following:

Keep the notional specific to each instrument. Loan interest uses the loan principal; swap cash flows use the swap notional; futures settlements use the total futures notional.
Apply the accrual fraction to every interest-based amount. A quarterly period uses the stated $\tau$, not a full year. Use the problem's convention rather than silently assuming a day-count fraction.
Respect the swap direction. Pay fixed and receive floating has net payment $N\tau(K-R_S)$. A negative result is a receipt, not an additional payment.
Respect the futures direction and quote convention. In this model, short futures gain when the futures rate rises; long futures gain when it falls.
Do not treat a futures rate change as the loan-rate change unless specified. A different reference rate or a different rate movement creates basis risk and a residual cash flow.

For example, a common incorrect approach in Worked Example 1 would be to set the swap receipt equal to the entire loan's interest-rate increase, treating the full loan as hedged. The swap covers only $ \$1{,}500{,}000$, so the unmatched principal is $ \$500{,}000$. Its rate-driven cost increase is $500{,}000(0.25)(0.010)=\$1{,}250$, not zero. Subtracting the actual futures gain of $ \$450$ leaves an $ \$800$ increase, in agreement with the full cash-flow calculation.

Key takeaway: Calculate the underlying cash flow, swap settlement, and futures settlement independently before combining them. Then compare with a clearly defined baseline or calculate the remaining exposure from the unmatched notionals and rate movements. A hedge is exact only when its cash flows offset the relevant exposure under the stated assumptions.

Check Your Understanding

For each problem, use simple interest, the stated period fraction, and the futures settlement model defined above. Assume no fees or other cash flows.

A borrower has a $ \$800{,}000$ loan for one quarter at $5.2\%$. It pays fixed at $4.5\%$ and receives the same floating rate on a $ \$500{,}000$ swap. It shorts one $ \$100{,}000$ futures position as the futures rate rises from $4.7\%$ to $5.1\%$. Find the final borrowing outflow. (Answer: $ \$9{,}425$)
An investor earns $3.2\%$ on $ \$1{,}500{,}000$ for half a year. It receives fixed at $3.8\%$ and pays the same floating rate on a $ \$1{,}000{,}000$ swap. It is long futures with total notional $ \$250{,}000$, and the futures rate falls from $4.0\%$ to $3.6\%$. Find the combined receipt. (Answer: $ \$27{,}500$)
A borrower has a $ \$1{,}800{,}000$ loan for one quarter at $4.9\%$. It pays fixed at $4.3\%$ and receives the same floating rate on a $ \$1{,}200{,}000$ swap. It shorts four $ \$50{,}000$ futures contracts as the futures rate rises from $4.1\%$ to $4.6\%$. Find the final outflow. (Answer: $ \$20{,}000$)
A borrower has a $ \$2{,}000{,}000$ loan for half a year. The loan rate rises from $4.2\%$ to $5.0\%$. A pay-fixed, receive-floating swap at $4.5\%$ covers $ \$1{,}500{,}000$ and uses the loan rate. A short futures position of total notional $ \$400{,}000$ gains as the futures rate rises from $4.3\%$ to $4.9\%$. Find the final outflow and its difference from the initial-rate cost before futures. (Answer: $ \$45{,}050$; $ \$800$ above the $ \$44{,}250$ baseline)
A borrower has $ \$600{,}000$ of loan principal left unmatched after its swap. Over one quarter the loan rate rises by $0.40$ percentage point. A short futures position of total notional $ \$250{,}000$ gains when its futures rate rises by $0.35$ percentage point. Under the matched-rate model, find the remaining increase in borrowing cost after the futures gain. (Answer: $ \$381.25$)

What You'll Learn