Hedging a Future Loan Rate with a Forward-Starting Swap

Why Hedge a Loan That Has Not Started?

A borrower may know that it plans to take out a floating-rate loan on a future date but not yet know the benchmark rates that will apply during the loan. A forward-starting interest rate swap can reduce this uncertainty: the borrower agrees now to pay a fixed rate and receive a floating rate over future periods. Once the loan begins, the floating receipt from the swap offsets the loan’s floating benchmark charge, provided the loan and swap cash-flow terms match.

The hedge depends on more than choosing a swap with the right general maturity. Its start date, reset dates, payment dates, accrual factors, benchmark, and notional schedule must correspond to the loan. This tutorial assumes a hypothetical borrower, no upfront swap payment, no default or transaction costs, and a single consistent benchmark and discounting framework unless stated otherwise.

Definition: A forward-starting swap is a swap agreed at an initial valuation date but with its first accrual period beginning at a specified future date. In a standard interest rate swap, the parties exchange interest amounts calculated on a reference notional; they do not exchange that notional. A borrower hedging a floating-rate loan typically pays fixed and receives floating.

Set the Dates Before Calculating

Let the loan and swap begin on a future date $T_0$, with payment dates $T_1,\ldots,T_n$. The first floating rate is normally set at $T_0$ for the period from $T_0$ to $T_1$, and its interest is paid at $T_1$. Subsequent rates are set at the beginning of their respective periods and paid at the end. The time between $T_{j-1}$ and $T_j$, measured using the contract’s day-count convention, is the accrual factor $\alpha_j$.

For a loan with a constant principal balance $N$, benchmark rate $L_j$, and contractual spread $s$, the borrower’s interest payment for period $j$ is $N\alpha_j(L_j+s)$. If the swap fixed rate is $K$, the borrower pays $N\alpha_jK$ on its fixed leg and receives $N\alpha_jL_j$ on its floating leg. Thus, on matching dates and terms, the combined interest payment is:

N\alpha_j(L_j+s)+N\alpha_jK-N\alpha_jL_j =N\alpha_j(K+s).

The benchmark component cancels period by period. The loan spread remains: the swap exchanges the benchmark rate, not the lender’s credit or other contractual spread. The borrower’s principal repayments also remain loan cash flows; a swap’s notional is a reference amount, not a principal exchange.

Worked Example: Mapping a Future Loan to Swap Cash Flows

A hypothetical borrower will take a one-year loan of $1,200,000 at time 1, with interest paid annually at times 2 and 3. The loan rate is the annual benchmark rate plus 0.40%. The borrower enters a forward-starting swap that starts at time 1, pays fixed at 4.25%, and receives the same benchmark annually. The benchmark rates actually set for the two loan periods are 3.80% and 5.10%. Calculate each loan payment, swap net payment, and combined interest cost.

Method. Align the periods first, then compute loan interest and the borrower’s swap cash flow separately. A swap net payment is fixed paid minus floating received; a negative amount means a net receipt.

Period from time 1 to time 2. The loan rate is $3.80\%+0.40\%=4.20\%$. Loan interest is $1{,}200{,}000(0.042)=\$50{,}400$. The swap fixed payment is $1{,}200{,}000(0.0425)=\$51{,}000$, and the floating receipt is $1{,}200{,}000(0.038)=\$45{,}600$. The swap’s net payment is $\$51{,}000-\$45{,}600=\$5{,}400$. Combined interest cost is $\$50{,}400+\$5{,}400=\$55{,}800$.

Period from time 2 to time 3. The loan rate is $5.10\%+0.40\%=5.50\%$, so loan interest is $1{,}200{,}000(0.055)=\$66{,}000$. The swap fixed payment remains $\$51{,}000$, while the floating receipt is $1{,}200{,}000(0.051)=\$61{,}200$. The swap’s net payment is $\$51{,}000-\$61{,}200=-\$10{,}200$, a net receipt. Combined interest cost is $\$66{,}000-\$10{,}200=\$55{,}800$.

Check. The fixed-rate result is $1{,}200{,}000(0.0425+0.0040)=\$55{,}800$ for each annual period, matching both cash-flow calculations. The common mistake avoided here is treating the floating receipt as an additional payment rather than as an offset to loan interest.

Choosing the Fixed Rate at the Start Date

If the swap is entered at fair value with no initial payment, its fixed rate is chosen so that the present value of the fixed leg equals the present value of the floating leg. For a standard swap with a constant notional of 1, the same accrual factors on both legs, and a single curve used to derive forwards and discount payments, write $D_j=P(0,T_j)$ for the discount factor to $T_j$. The forward benchmark rate for period $j$ is $F_j$. The par fixed rate for a swap beginning at $T_0$ is:

K=\frac{\displaystyle\sum_{j=1}^{n}D_j\alpha_jF_j} {\displaystyle\sum_{j=1}^{n}D_j\alpha_j}.

The numerator is the present value of the floating payments per unit notional; the denominator is the present value of one unit of fixed rate paid over the swap periods. For a standard floating-rate leg under these assumptions, the numerator telescopes to the discount-factor difference between the start and end dates.

Write each forward rate.
For a single-curve model, $F_j=(D_{j-1}/D_j-1)/\alpha_j$. This expresses the period rate implied by the discount factors.

Discount each floating payment.
Multiplying by its accrual factor and payment-date discount factor gives $D_j\alpha_jF_j=D_{j-1}-D_j$.

Add the period values.
The intermediate discount factors cancel: $\sum_{j=1}^{n}D_j\alpha_jF_j=D_0-D_n$, where $D_0=P(0,T_0)$.

Set the swap’s initial value to zero.
Equating the floating-leg value to $K\sum_{j=1}^{n}D_j\alpha_j$ gives the par-rate formula.

This derivation explains why the forward-starting swap rate is not generally just one forward rate: it is a discounted, accrual-weighted average over all the swap periods. The result assumes standard floating payments, consistent dates and accrual conventions, and the stated single-curve framework. It is not a justification for using that formula unchanged when the contract’s floating leg or market valuation assumptions differ.

Worked Example: Finding a Fair Rate for a Forward-Starting Swap

A hypothetical annual-pay swap begins at time 2 and ends at time 4. The discount factors are $D_2=0.92$, $D_3=0.88$, and $D_4=0.84$. The notional is constant, the accrual factors are 1, and the single-curve assumptions above apply. Find the par fixed rate. Then suppose the borrower takes a $5,000,000 loan over the same dates at the benchmark plus 0.60%, and the swap rate is fixed at the calculated par rate. Calculate the combined annual interest cost in each of two periods if the realized benchmark rates are 4.20% and 5.00%.

Method. First calculate the swap rate from the discounted-leg formula. Use realized rates, not the forward rates, when calculating the eventual cash payments.

The floating-leg present value per unit notional is $D_2-D_4=0.92-0.84=0.08$. The fixed-leg annuity factor is $D_3+D_4=0.88+0.84=1.72$. Therefore:

K=\frac{0.08}{1.72}=0.0465116279\ldots=4.6512\%.

For the first loan period, loan interest is $5{,}000{,}000(0.042+0.006)=\$240{,}000$. The swap’s fixed payment is $5{,}000{,}000(0.0465116279\ldots)=\$232{,}558.14$, and its floating receipt is $5{,}000{,}000(0.042)=\$210{,}000$. The net swap payment is $\$22{,}558.14$, making the combined cost $\$262{,}558.14$.

For the second period, loan interest is $5{,}000{,}000(0.050+0.006)=\$280{,}000$. The swap’s floating receipt is now $\$250{,}000$, so the net swap payment is $\$232{,}558.14-\$250{,}000=-\$17{,}441.86$. Combined interest cost is $\$280{,}000-\$17{,}441.86=\$262{,}558.14$.

Check. In either period, the cost equals $5{,}000{,}000(0.0465116279\ldots+0.006)=\$262{,}558.14$, rounded to the nearest cent. The forward rates implied by discount factors determine the fair swap rate at the contract date; they do not guarantee the rates that will later be realized.

Matching an Amortizing Loan

A loan that repays principal over time does not have one constant outstanding balance. To hedge its benchmark interest component, the swap’s reference notional for each period should match the loan balance on which that period’s interest is calculated. If the loan’s balance falls after a principal repayment, the swap notional should fall at the same point. The principal repayment itself remains due under the loan and is not exchanged through the swap.

Worked Example: Hedging a Loan with Scheduled Principal Repayments

A hypothetical borrower takes a $900,000 loan at time 1. It repays $300,000 of principal at each of times 2, 3, and 4, so the opening balances for the three annual periods are $900,000, $600,000, and $300,000. The loan charges the benchmark plus 0.25%. A forward-starting pay-fixed, receive-floating swap starts at time 1 with a fixed rate of 5.00% and matching annual notionals of $900,000, $600,000, and $300,000. The realized benchmark rates are 4.00%, 6.00%, and 5.00%. Find the combined interest cost in each period and state the principal cash flow separately.

Method. Apply the loan and swap rates to each period’s opening balance. Keep the principal repayments separate from interest because they are not swap cash flows.

From time 1 to time 2, loan interest is $900{,}000(0.040+0.0025)=\$38{,}250$. The swap pays fixed interest of $900{,}000(0.050)=\$45{,}000$ and receives floating interest of $900{,}000(0.040)=\$36{,}000$, for a net swap payment of $\$9{,}000$. Combined interest cost is $\$38{,}250+\$9{,}000=\$47{,}250$. The borrower also repays $300,000 of principal at time 2.

From time 2 to time 3, loan interest is $600{,}000(0.060+0.0025)=\$37{,}500$. The swap’s fixed payment is $\$30{,}000$, and its floating receipt is $\$36{,}000$, so it produces a net receipt of $\$6{,}000$. Combined interest cost is $\$37{,}500-\$6{,}000=\$31{,}500$. The borrower repays another $300,000 of principal at time 3.

From time 3 to time 4, loan interest is $300{,}000(0.050+0.0025)=\$15{,}750$. The fixed and floating swap amounts are both $300{,}000(0.050)=\$15{,}000$, so the swap’s net interest payment is zero. Combined interest cost is $\$15{,}750$, and the final $300,000 principal repayment is due at time 4.

Check. The period costs also equal the fixed swap rate plus loan spread applied to each opening balance: $900{,}000(0.0525)=\$47{,}250$, $600{,}000(0.0525)=\$31{,}500$, and $300{,}000(0.0525)=\$15{,}750$. The common mistake avoided is applying the first-period notional to every period despite the loan’s scheduled amortization.

Common Timing and Hedge-Direction Traps

The direction of the swap follows the loan exposure. A borrower whose payments rise when the benchmark rises pays fixed and receives floating: higher floating receipts offset higher loan interest. A lender receiving floating income and seeking to fix that income would generally take the opposite side. Always write the cash-flow signs from the named party’s perspective before combining amounts.

A frequent timing error is to place the first swap payment on the future start date. In a standard swap, the start date begins the first accrual period; the first interest payment is made at that period’s end. Another error is to include a swap principal exchange. The notional determines interest amounts but is not itself paid between the swap parties.

A matched swap fixes the benchmark component only when the loan and swap use the same benchmark, reset timing, payment dates, accrual factors, and relevant notionals. If the loan charges a spread, that spread remains payable. If the borrower has not yet secured the loan, the swap can still create obligations even if the borrowing does not occur; the hedge therefore depends on the planned borrowing actually taking place. These are reasons to check the transaction terms and the future cash-flow schedule before relying on the hedge.

Key takeaway: To hedge a planned floating-rate loan, align the swap’s future start date and each period’s benchmark, reset date, payment date, accrual factor, and notional with the loan. Paying fixed and receiving floating converts matching benchmark interest into a fixed rate, while the loan spread and principal repayments remain outside that offset.

Check Your Understanding

For each problem, identify the dates and cash flows before calculating. Assume the stated swap and loan terms match unless the problem says otherwise.

A borrower takes a $2,000,000 loan at time 2 for two annual periods, paying the benchmark plus 0.30%. A forward-starting swap begins at time 2 and pays fixed at 4.60%, receiving the same benchmark. The realized benchmark rates are 4.00% and 5.20%. What is the borrower’s combined interest cost in each period? (Answer: $98,000 in each annual period.)
A $1,500,000 floating-rate loan begins at time 1 and has two semiannual periods. It charges the benchmark plus 0.40%. The matching forward-starting swap pays fixed at 5.20% and receives the benchmark. What is the combined interest cost per period and over both periods? (Answer: $42,000 per half-year; $84,000 total.)
An annual-pay, constant-notional swap starts at time 1 and ends at time 3. Its discount factors are $D_1=0.97$, $D_2=0.94$, and $D_3=0.90$; accrual factors are 1. Under the single-curve assumptions in this tutorial, what is its par fixed rate? (Answer: $(0.97-0.90)/(0.94+0.90)=0.0380435$, or approximately 3.8043%.)
A loan begins at time 1 and has annual opening balances of $1,200,000, $800,000, and $400,000. The loan charges the benchmark plus 0.20%; a matching swap pays fixed at 4.50% and receives the benchmark, with the same period notionals. What are the combined interest costs for the three periods? (Answer: $56,400, $37,600, and $18,800, respectively.)
A borrower plans an annual floating-rate loan from time 2 through time 5. It pays interest at times 3, 4, and 5, with each benchmark rate set at the beginning of its accrual period. State the matching swap’s start date, payment dates, and benchmark reset dates, and say whether the loan principal is exchanged in the swap. (Answer: Swap starts at time 2; payments are at times 3, 4, and 5; resets are at times 2, 3, and 4; loan principal is not exchanged through the swap.)

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