Converting Fixed-Rate Debt to Floating Cost with a Swap

Introduction

A borrower with fixed-rate debt may want its cost to move with a floating benchmark. A swap can change the borrower’s net interest payments, but only if the swap has the appropriate direction. Keeping the debt payment and each swap leg separate before combining them is the safest way to check the result.

There is an important sign issue in this topic: a borrower that pays fixed on its debt generally needs to receive fixed and pay floating on the swap to obtain a floating net cost. A pay-fixed, receive-floating swap has the opposite effect on the debt cash flows. The calculations below make that distinction explicit. The treatment is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: A borrower’s net interest payment for an accrual period is the debt interest payment plus the swap payments the borrower makes, minus the swap payments the borrower receives. A positive result is a net outflow; a negative result is a net receipt. Unless stated otherwise, assume no principal is exchanged, the debt and swap use the same payment dates and accrual factors, and the floating rate for a period is set at the start of that period and paid at its end.

Set the Cash-Flow Signs Before Calculating

Let $N_D$ be the debt principal, $c$ the annual fixed debt rate, and $\alpha_j$ the year fraction for period $j$. The debt interest outflow in that period is $N_Dc\alpha_j$. For the swap, let $N_S$ be its notional, $K$ its annual fixed rate, and $L_j$ the annual floating rate applicable to period $j$. The swap’s interest amounts are based on $N_S\alpha_j$.

If the borrower pays fixed and receives floating on the swap, the borrower’s net outflow is the debt payment plus the swap’s fixed payment less the floating receipt. With matching debt and swap notionals, accrual factors, and dates, this is:

N_Dc\alpha_j+N_DK\alpha_j-N_DL_j\alpha_j =N_D\alpha_j(c+K-L_j).

That expression is generally not the floating rate $L_j$ times the notional and accrual factor. In contrast, if the borrower receives fixed and pays floating, the net outflow is:

N_Dc\alpha_j+N_D L_j\alpha_j-N_DK\alpha_j =N_D\alpha_j(c+L_j-K).

When the debt and swap notionals match and $K=c$, the fixed amounts cancel, leaving $N_D L_j\alpha_j$, the floating interest payment for the period.

Why the Fixed Amounts Cancel

The cancellation follows directly from adding the signed cash flows. In the receive-fixed, pay-floating position, the borrower owes the debt coupon $N_Dc\alpha_j$, owes the swap floating amount $N_D L_j\alpha_j$, and receives the swap fixed amount $N_DK\alpha_j$. A receipt reduces the net outflow, so the total is $N_D\alpha_j(c+L_j-K)$. If $K=c$, then $c-K=0$, and only the floating amount remains.

This cancellation requires the debt and swap fixed amounts to match in each period. Equal annual rates are not enough if the notionals or accrual factors differ. More generally, with potentially different notionals and accrual factors, the receive-fixed, pay-floating net outflow is:

N_Dc\alpha_{D,j}+N_S L_j\alpha_{S,j}-N_SK\alpha_{S,j}.

This formula also shows why a partial-notional swap leaves some fixed-rate exposure. The debt continues to generate its full fixed payment; the swap offsets only its own notional times its fixed rate.

Worked Example: Check the Swap Direction

A borrower has a hypothetical \$10,000,000 loan with a 5.2% annual fixed rate and quarterly payments. The borrower enters a \$10,000,000 quarterly swap with a 5.2% fixed rate. The floating rates set for the next three quarters are 4.6%, 5.0%, and 5.4%. Each quarter has accrual factor 0.25. Find the net outflows if the borrower pays fixed and receives floating on the swap. Then identify the position that would produce floating-rate borrowing cost.

Method and sign convention. Treat payments as positive outflows and receipts as amounts subtracted from outflows. The loan payment each quarter is:

10{,}000{,}000(0.052)(0.25)=\$130{,}000.

The swap fixed payment is also \$130,000 per quarter. The floating receipts are calculated period by period:

\begin{aligned} \text{Quarter 1: }&10{,}000{,}000(0.046)(0.25)=\$115{,}000,\\ \text{Quarter 2: }&10{,}000{,}000(0.050)(0.25)=\$125{,}000,\\ \text{Quarter 3: }&10{,}000{,}000(0.054)(0.25)=\$135{,}000. \end{aligned}

With the stated pay-fixed, receive-floating swap, the net outflows are debt plus swap fixed payment less swap floating receipt:

\begin{aligned} \text{Quarter 1: }&130{,}000+130{,}000-115{,}000=\$145{,}000,\\ \text{Quarter 2: }&130{,}000+130{,}000-125{,}000=\$135{,}000,\\ \text{Quarter 3: }&130{,}000+130{,}000-135{,}000=\$125{,}000. \end{aligned}

Those are not the floating loan payments: the first net outflow is \$145,000, not \$115,000. To obtain the floating cost, the borrower should instead receive fixed and pay floating. Then the swap cash flows offset the loan’s \$130,000 payment by a \$130,000 fixed receipt and add the floating payment. The resulting outflows are \$115,000, \$125,000, and \$135,000. Check: each corrected outflow equals the corresponding floating rate multiplied by \$10,000,000 and 0.25. The common mistake avoided is treating “pay fixed, receive floating” as the direction that converts fixed debt to floating.

Residual Fixed Spreads

In practice, the swap fixed rate may not exactly equal the debt rate. With matching notionals and accrual factors, a receive-fixed, pay-floating swap gives:

N\alpha_j(c+L_j-K) =N\alpha_j\bigl(L_j+(c-K)\bigr).

Thus, the net rate is the floating rate plus the fixed difference $c-K$. If $c>K$, the borrower pays the floating rate plus a positive residual spread. If $c<K$, the expression is below the floating rate. This is a cash-flow calculation under the stated assumptions, not a guarantee that the borrower’s full financing cost will match a particular market index. For example, a debt margin over a benchmark, different reset terms, or different payment dates can prevent an exact conversion.

Worked Example: A Coupon Mismatch Leaves a Spread

A borrower has \$6,000,000 of hypothetical fixed-rate debt at 5.65% per year, paid semiannually. It enters a \$6,000,000 swap in which it receives 5.40% fixed and pays the floating rate set for each period. The next three floating rates are 4.80%, 5.60%, and 6.20%. Each period has accrual factor 0.5. Find the net payment in each period and the corresponding annualized net rate.

Method. The borrower receives fixed and pays floating, so net outflow equals debt interest plus swap floating interest less swap fixed interest. The debt payment each period is:

6{,}000{,}000(0.0565)(0.5)=\$169{,}500.

The fixed swap receipt is:

6{,}000{,}000(0.0540)(0.5)=\$162{,}000.

The floating swap payments are:

\begin{aligned} \text{At }4.80\%:\;&6{,}000{,}000(0.0480)(0.5)=\$144{,}000,\\ \text{At }5.60\%:\;&6{,}000{,}000(0.0560)(0.5)=\$168{,}000,\\ \text{At }6.20\%:\;&6{,}000{,}000(0.0620)(0.5)=\$186{,}000. \end{aligned}

Combining the flows gives:

\begin{aligned} \text{First period: }&169{,}500+144{,}000-162{,}000=\$151{,}500,\\ \text{Second period: }&169{,}500+168{,}000-162{,}000=\$175{,}500,\\ \text{Third period: }&169{,}500+186{,}000-162{,}000=\$193{,}500. \end{aligned}

To express each payment as an annualized rate on the debt notional, divide by $6{,}000{,}000(0.5)=\$3{,}000{,}000$. The rates are $151{,}500/3{,}000{,}000=5.05\%$, $175{,}500/3{,}000{,}000=5.85\%$, and $193{,}500/3{,}000{,}000=6.45\%$. As a check, each is the floating rate plus $5.65\%-5.40\%=0.25\%$. The common mistake avoided is assuming that a swap with a fixed rate different from the debt coupon cancels all of the fixed-rate component.

Partial-Notional Swaps

A swap need not cover the full debt principal. In that case, apply each rate only to the notional and accrual factor of its own contract. A convenient expression, when the debt and swap share the same period accrual factor $\alpha_j$, is:

\text{Net outflow}_j =\alpha_j\left[N_Dc+N_S(L_j-K)\right].

Dividing by $N_D\alpha_j$ gives the net rate relative to the full debt principal:

c+\frac{N_S}{N_D}(L_j-K).

The factor $N_S/N_D$ matters: applying the swap’s floating rate to the entire debt notional would overstate the hedged amount when $N_S<N_D$.

Worked Example: Converting Only Part of the Debt

A borrower has \$8,000,000 of hypothetical debt at a 4.80% annual fixed rate, paid quarterly. It enters a \$5,000,000 swap with a 5.10% fixed rate, receiving fixed and paying floating. The next three quarterly floating rates are 4.20%, 5.00%, and 5.80%. Each accrual factor is 0.25. Calculate each net payment and the annualized net rate measured against the full debt principal.

Method. First calculate the full debt interest for each quarter:

8{,}000{,}000(0.048)(0.25)=\$96{,}000.

The swap’s fixed receipt in each quarter is:

5{,}000{,}000(0.051)(0.25)=\$63{,}750.

The swap floating payments are \$52,500, \$62,500, and \$72,500, calculated respectively as $5{,}000{,}000(0.042)(0.25)$, $5{,}000{,}000(0.050)(0.25)$, and $5{,}000{,}000(0.058)(0.25)$. Combining the debt payment with each swap payment and receipt:

\begin{aligned} \text{First period: }&96{,}000+52{,}500-63{,}750=\$84{,}750,\\ \text{Second period: }&96{,}000+62{,}500-63{,}750=\$94{,}750,\\ \text{Third period: }&96{,}000+72{,}500-63{,}750=\$104{,}750. \end{aligned}

The annualized rates relative to the full debt are these payments divided by $8{,}000{,}000(0.25)=\$2{,}000{,}000$: $4.2375\%$, $4.7375\%$, and $5.2375\%$. As a check, the rate formula gives $4.80\%+(5/8)(L_j-5.10\%)$. For the first period, this is $4.80\%+(5/8)(-0.90\%)=4.2375\%$; the other periods give $4.7375\%$ and $5.2375\%$. The common mistake avoided is treating the \$5,000,000 swap as if it covered all \$8,000,000 of debt.

A Reliable Cash-Flow Check

For each period, write down the debt payment, the swap payment, and the swap receipt as separate signed amounts before netting them. This prevents a change in swap direction from being hidden by an otherwise correct rate calculation.

Write the debt outflow.
For fixed-rate debt, calculate $N_Dc\alpha_{D,j}$. This remains an outflow regardless of the swap position.

Identify each swap leg from the borrower’s perspective.
A leg the borrower pays is an outflow; a leg the borrower receives reduces the net outflow. Do not infer the direction from the word “convert.”

Use the correct notional and accrual factor for each leg.
Calculate each amount from its own contract terms and the rate applicable to that accrual period.

Add signed amounts and interpret the result.
A positive total is a net payment. Compare it with the intended floating amount only after the debt and swap cash flows have been combined.

Common Mistakes and When the Formula Does Not Apply

Using the wrong swap direction. Fixed debt plus pay-fixed, receive-floating swap produces debt fixed interest plus swap fixed interest less floating receipts. It does not generally produce a floating payment. For the usual fixed-to-floating conversion, use receive-fixed, pay-floating.
Subtracting a payment or adding a receipt. Under the outflow-positive convention, swap payments are added and swap receipts are subtracted.
Ignoring a rate mismatch. If $c\ne K$, the matching-notional net rate is $L_j+(c-K)$, not just $L_j$.
Applying the swap rate to the wrong principal. Swap amounts use $N_S$, while debt amounts use $N_D$. These are equal only when the contract notionals match.
Combining cash flows with different dates or accrual bases as though they matched. The simple cancellation requires aligned payment dates, compatible accrual factors, and appropriate matching notionals. If these conditions fail, calculate the actual dated cash flows separately; a residual exposure may remain.

The basic conversion formula describes contractual interest cash flows under its assumptions. It does not include fees, principal repayments, counterparty default, or any debt margin not included in $c$. If a floating debt benchmark differs from the swap’s floating index, the two rates do not cancel or combine into a single benchmark without accounting for that difference.

Check Your Understanding

For each problem, use outflow-positive signs and combine the debt and swap cash flows period by period.

A \$2,000,000 loan charges 4.8% fixed interest quarterly. A matching swap has the borrower receive 4.8% fixed and pay floating at 5.3%. The accrual factor is 0.25. What is the net quarterly payment? (Answer: \$26,500)
A borrower has \$3,000,000 of fixed debt at 6.2%, paid semiannually. A matching swap has the borrower receive 6.0% fixed and pay floating at 5.4%. The accrual factor is 0.5. Find the net payment for the period. (Answer: \$84,000)
A \$1,200,000 loan charges 5.0% fixed interest semiannually. The borrower pays fixed at 5.0% and receives floating at 4.4% on a matching swap. The accrual factor is 0.5. Find the net payment and state whether it equals the floating debt amount. (Answer: \$33,600; no, the floating debt amount would be \$26,400)
A borrower has \$10,000,000 of debt at 4.5% fixed, paid quarterly. A \$4,000,000 swap has the borrower receive 4.7% fixed and pay floating at 5.2%. The accrual factor is 0.25. Find the combined net payment and annualized rate relative to the full debt. (Answer: \$117,500; 4.70%)
A borrower has \$5,000,000 of fixed debt at 5.1%, paid semiannually. On a matching notional and accrual factor of 0.5, the borrower receives 5.3% fixed and pays floating. The floating rates for two periods are 4.9% and 6.1%. Find the net payment in each period. (Answer: \$117,500 and \$147,500)

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