Converting Floating-Rate Debt to Fixed Cost with a Swap

Why the Swap Direction Matters

A borrower whose interest bill rises with a reference rate may want to replace that uncertainty with a known borrowing cost. But a swap does not hedge a liability merely because it involves the same reference rate: the direction of each swap payment matters. With floating-rate debt, a pay-floating, receive-fixed swap adds another floating-rate payment while providing a fixed-rate receipt. Unless other rate movements offset one another, the combined cost is still variable.

The calculations below use the borrower’s perspective: a positive net amount is a payment by the borrower, and a negative amount is a net receipt. Rates are annualized, while cash flows are calculated over the stated accrual periods.

Definition: Suppose a borrower owes interest on notional principal $N_D$ at floating rate $R_t+s$ during period $t$, where $s$ is a constant borrowing spread. The borrower also enters a swap with notional $N_S$, pays floating rate $S_t$, and receives fixed rate $K$. For accrual factors $\alpha_{D,t}$ on the debt and $\alpha_{S,t}$ on the swap, the combined net payment is $$ C_t=N_D\alpha_{D,t}(R_t+s)+N_S\alpha_{S,t}(S_t-K). $$ A negative $C_t$ means the borrower receives more from the swap than it pays on the debt for that period.

The floating rates $R_t$ and $S_t$ need not be the same benchmark. Likewise, the two notionals and accrual factors need not match. Keeping the debt and swap amounts separate until the final addition prevents a common sign or scaling error.

The Matched-Benchmark Case

If the debt and swap use the same floating rate $L_t$, and have the same notional $N$ and accrual factor $\alpha_t$, the combined payment simplifies to

\begin{aligned} C_t &=N\alpha_t(L_t+s)+N\alpha_t(L_t-K)\\ &=N\alpha_t(2L_t+s-K). \end{aligned}

The rate $2L_t+s-K$ is the combined annualized cost, before multiplying by the accrual factor and notional. The two appearances of $L_t$ have the same sign: the debt requires a floating payment, and the swap requires a floating payment. The swap’s fixed receipt offsets some of the total, but does not cancel the floating rate.

Key result: With a matched notional, accrual factor, and floating benchmark, floating-rate debt at $L_t+s$ combined with a pay-floating, receive-fixed swap at $L_t$ versus $K$ costs $2L_t+s-K$ on an annualized basis. If $L_t$ changes, the combined cost changes at twice the rate of that change. This combination does not ordinarily create a fixed borrowing cost.

Worked Example: Calculate Payments Across Three Resets

A hypothetical company has a \$3,000,000 loan that charges the quarterly reference rate plus 0.70%. It enters a \$3,000,000 swap in which it pays the same quarterly reference rate and receives 4.20% fixed. Each accrual factor is 0.25. The reference rate is 3.10% for the first quarter, 4.00% for the second, and 2.80% for the third. Find the net payment in each quarter and determine whether the combined annualized cost is fixed.

Method. For each period, calculate the loan interest and the swap net payment separately. The borrower’s swap payment is $N\alpha(L_t-K)$; a negative result is a net swap receipt.

Quarter 1. The loan rate is $3.10\%+0.70\%=3.80\%$, so the loan interest is $3{,}000{,}000(0.25)(0.038)=\$28{,}500$. The swap net payment is $3{,}000{,}000(0.25)(0.031-0.042)=-\$8{,}250$. Thus, the combined payment is $\$28{,}500-\$8{,}250=\$20{,}250$. The annualized combined rate is $2(3.10\%)+0.70\%-4.20\%=2.70\%$; multiplying by $0.25$ and \$3,000,000 also gives \$20,250.

Quarter 2. The loan rate is $4.00\%+0.70\%=4.70\%$, and loan interest is $3{,}000{,}000(0.25)(0.047)=\$35{,}250$. The swap net payment is $3{,}000{,}000(0.25)(0.040-0.042)=-\$1{,}500$. The total is $\$35{,}250-\$1{,}500=\$33{,}750$, which is also $3{,}000{,}000(0.25)[2(0.040)+0.007-0.042]$.

Quarter 3. The loan rate is $2.80\%+0.70\%=3.50\%$, and loan interest is $3{,}000{,}000(0.25)(0.035)=\$26{,}250$. The swap net payment is $3{,}000{,}000(0.25)(0.028-0.042)=-\$10{,}500$. The total is $\$26{,}250-\$10{,}500=\$15{,}750$, equal to $3{,}000{,}000(0.25)[2(0.028)+0.007-0.042]$.

The combined payments are therefore \$20,250, \$33,750, and \$15,750. The annualized rates are 2.70%, 4.50%, and 2.10%, respectively, so the cost is not fixed. Common mistake avoided: subtracting the floating rate from the loan’s floating rate as if the swap received floating. This swap pays floating; its net payment is $L_t-K$, not $K-L_t$. The calculations above reconcile both the separate cash flows and the combined-rate formula.

Unequal Notionals and Benchmarks

When the swap notional differs from the debt notional, the floating-rate exposure must be weighted by the notional ratio. If the debt and swap still use the same rate $L_t$, and both accrual factors are $\alpha_t$, let $q=N_S/N_D$. Dividing the total payment by $N_D\alpha_t$ gives the combined annualized rate:

\frac{C_t}{N_D\alpha_t}=L_t+s+q(L_t-K)=(1+q)L_t+s-qK.

For positive notionals, $q>0$, so the floating-rate coefficient $1+q$ is positive. A smaller swap notional can reduce or increase the size of the added exposure, but it does not make the matched-rate combination fixed. If the accrual factors differ, use the original cash-flow formula instead of this simplified rate expression.

Worked Example: Allow for a Smaller Swap Notional

A hypothetical borrower owes \$2,000,000 at a floating rate plus 0.40%, with a half-year accrual factor of 0.50. It enters a pay-floating, receive-fixed swap with notional \$1,200,000, the same floating benchmark, the same accrual factor, and a fixed rate of 4.00%. Calculate the combined payment when the benchmark is first 3.50% and then 4.50%.

Method. Calculate the debt payment using \$2,000,000 and the swap payment using \$1,200,000. Here $q=1{,}200{,}000/2{,}000{,}000=0.60$.

At 3.50%, the debt rate is $3.50\%+0.40\%=3.90\%$, so debt interest is $2{,}000{,}000(0.50)(0.039)=\$39{,}000$. The swap net payment is $1{,}200{,}000(0.50)(0.035-0.040)=-\$3{,}000$. The net payment is $\$39{,}000-\$3{,}000=\$36{,}000$. As a check, the combined rate on the debt notional is $(1+0.60)(3.50\%)+0.40\%-0.60(4.00\%)=3.60\%$; \$2,000,000 times 0.50 times 3.60% is \$36,000.

At 4.50%, debt interest is $2{,}000{,}000(0.50)(0.049)=\$49{,}000$. The swap net payment is $1{,}200{,}000(0.50)(0.045-0.040)=\$3{,}000$. The total is \$52,000. The formula gives $1.60(4.50\%)+0.40\%-0.60(4.00\%)=5.20\%$, and $2{,}000{,}000(0.50)(0.052)=\$52{,}000$. The effective cost therefore changes from 3.60% to 5.20%. Common mistake avoided: applying the debt notional to the swap leg; the swap cash flow is based on its own \$1,200,000 notional.

With different floating benchmarks, denote the debt benchmark by $R_t$ and the swap benchmark by $S_t$. For equal accrual factors, the combined rate on the debt notional is

$$ R_t+s+q(S_t-K). $$

A fixed combined rate requires the variable part $R_t+qS_t$ to remain constant across the relevant periods. Merely having two different benchmarks does not ensure this. If their movements do not offset in the required proportion, the combined cost remains exposed to rate changes.

Worked Example: Check a Condition for a Fixed Combined Rate

A hypothetical borrower owes \$4,000,000 at benchmark $R_t$ plus 0.60%. It enters a \$2,000,000 pay-floating, receive-fixed swap that pays benchmark $S_t$ and receives 4.80%. Both accrual factors are 0.25. In the first period, $R_t=4.00\%$ and $S_t=4.00\%$. In the second, $R_t=5.00\%$ and $S_t=2.00\%$. Calculate both combined payments and assess whether these two periods have the same cost.

Method. Here $q=2{,}000{,}000/4{,}000{,}000=0.50$. The combined rate on the debt notional is $R_t+0.60\%+0.50(S_t-4.80\%)$.

In the first period, the debt rate is $4.00\%+0.60\%=4.60\%$. Debt interest is $4{,}000{,}000(0.25)(0.046)=\$46{,}000$. The swap net payment is $2{,}000{,}000(0.25)(0.040-0.048)=-\$4{,}000$. The total is \$42,000. The combined annualized rate is $4.00\%+0.60\%+0.50(4.00\%-4.80\%)=4.20\%$, which also gives $4{,}000{,}000(0.25)(0.042)=\$42{,}000$.

In the second period, the debt rate is 5.60%, giving $4{,}000{,}000(0.25)(0.056)=\$56{,}000$. The swap net payment is $2{,}000{,}000(0.25)(0.020-0.048)=-\$14{,}000$. The total is again $\$56{,}000-\$14{,}000=\$42{,}000$. The combined annualized rate is $5.00\%+0.60\%+0.50(2.00\%-4.80\%)=4.20\%$. Thus the rate relationship is satisfied in these periods: $R_t+0.50S_t=6.00\%$ in each. Common mistake avoided: assuming the two benchmarks cancel just because they move; their weighted sum must actually stay constant. This result establishes a fixed cost only for the stated rates and periods, not a guarantee that the relationship will continue.

Derivation and Practical Checks

The payment formula follows directly from adding the cash obligations for one accrual period. The debt interest is its principal multiplied by the annualized rate and the fraction of a year represented by the period. The swap’s net payment is the floating amount paid less the fixed amount received, each calculated on the swap notional and accrual factor. Adding those two net outflows yields the combined payment:

\begin{aligned} C_t &=\underbrace{N_D\alpha_{D,t}(R_t+s)}_{\text{debt payment}} +\underbrace{N_S\alpha_{S,t}S_t-N_S\alpha_{S,t}K}_{\text{swap payment less receipt}}\\ &=N_D\alpha_{D,t}(R_t+s)+N_S\alpha_{S,t}(S_t-K). \end{aligned}

This reasoning assumes the stated interest rates apply for the whole accrual period, the payment dates align for the combined calculation, the notionals are known, and there are no fees, taxes, default losses, or other cash flows. If payment dates or accrual periods differ, calculate each actual cash flow on its own date and period rather than treating the rates as if they shared one accrual factor. If the floating rates are not known in advance, a future combined payment may not be known, even if its formula is.

For exam calculations, use this sequence:

Write each leg from the borrower’s perspective.
Debt interest is an outflow. For this swap direction, floating is an outflow and fixed is an inflow.

Use each leg’s own terms.
Apply the correct notional, benchmark, spread, and accrual factor to the debt and swap separately.

Add signed cash flows.
A net swap receipt is negative as a borrower payment; it reduces the debt cash outflow.

Test whether the result is fixed.
Collect the variable-rate terms. A fixed cost requires their combined value to be constant, not merely a swap that has a fixed leg.

Common Exam Trap: Assuming the Floating Rates Cancel

For a debt rate $L_t+s$ and a same-notional swap that pays $L_t$ and receives $K$, an incorrect calculation is $L_t+s-L_t-K=s-K$. That expression treats the swap as though it received floating and paid fixed. For the stated pay-floating, receive-fixed position, its signed payment is $L_t-K$, so the correct combined rate is

$$ (L_t+s)+(L_t-K)=2L_t+s-K. $$

A negative swap net payment should not be mistaken for a negative debt payment: it is a receipt that offsets the loan interest for that period. Always check the result by comparing the combined-rate calculation with the sum of the separately calculated cash flows. To obtain a fixed borrowing cost from ordinary matched floating-rate debt, the swap must offset—not add to—the debt’s floating-rate exposure. The direction of the floating leg is therefore essential.

Check Your Understanding

For each problem, use the borrower’s perspective and state the net payment for the specified period.

A \$1,500,000 loan charges the same reference rate plus 0.50% as a pay-floating, receive-fixed swap. The swap notional is \$1,500,000, its fixed rate is 3.00%, the reference rate is 2.40%, and the accrual factor is 0.25. Find the net payment. (Answer: \$8,625.)
A \$800,000 loan charges a floating rate plus 0.40%. A same-notional swap pays that same rate and receives 4.10%. The rate is 3.60% and the accrual factor is 0.50. Find the combined payment and annualized cost. (Answer: \$14,000; 3.50%.)
A \$2,000,000 loan charges reference rate $L$ plus 0.30%. A pay-floating, receive-fixed swap has notional \$1,500,000, uses the same reference rate, and receives 5.00%. If $L=4.20\%$ and the accrual factor is 0.25, find the combined payment and annualized cost on the loan notional. (Answer: \$19,500; 3.90%.)
A \$1,000,000 loan charges benchmark $R$ plus 0.80%. A \$1,000,000 swap pays benchmark $S$ and receives 4.00%. For one half-year period, $R=3.90\%$, $S=4.40\%$, and both accrual factors are 0.50. Find the combined payment. (Answer: \$25,500.)
A \$5,000,000 loan charges benchmark $R$ plus 0.20%. A \$2,500,000 swap pays benchmark $S$ and receives 4.20%. Both accrual factors are 0.50. In two periods, the pairs $(R,S)$ are $(3.50\%,5.00\%)$ and $(4.50\%,3.00\%)$. Do the periods have the same combined annualized cost, and what is each period’s payment? (Answer: Yes; 4.10% and \$102,500 in each period.)

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