Interest rate swaps · RiskAtoms Academy

Know these first

L1.7.02 Forwards
L1.4.03 Pricing a bond
L1.6.05 Benchmark rates: from LIBOR to SOFR, €STR and SONIA

Math Atoms

∑ Geometric series

By the end you will be able to

1Describe the fixed and floating legs of an interest rate swap and who pays what.
2Value a simple swap as the difference between the present values of its two legs.
3Find the par swap rate that makes a new swap worth zero.
4Explain how a company uses a swap to turn a floating-rate loan into a fixed-rate one.

Why it matters

A company borrows at a floating rate because that is what its bank offers. Its board wants to know what interest it will pay for the next five years. A single contract, the interest rate swap, turns one into the other. Interest rate swaps are the largest part of the derivatives market.

Core explanation

Swapping fixed for floating

In an interest rate swap, two parties agree to pay each other interest on the same notional amount for a number of years. One side pays a rate fixed at the start: the fixed leg. The other pays a rate that resets each period to a benchmark such as SOFR in US dollars, €STR in euros or SONIA in sterling (L1.6.05): the floating leg. The notional itself never changes hands; it only sets the size of the payments.

The side that pays fixed is the payer; the side that receives fixed is the receiver. On each payment date only the difference is paid, which is called netting.

How a company fixes its loan

A company owes SOFR + 1.50% on a USD 10m loan. It enters a swap to pay 4.00% fixed and receive SOFR.

To the lender: SOFR + 1.50%
To the bank in the swap: 4.00%
From the bank in the swap: SOFR
Net cost: SOFR + 1.50% + 4.00% − SOFR = 5.50% fixed, whatever SOFR does.

Here is what that looks like if SOFR falls over three years:

Year	SOFR	Loan interest	Swap net to company	Total cost
1	4.30%	580,000	+30,000	550,000 (5.50%)
2	3.80%	530,000	−20,000	550,000 (5.50%)
3	3.20%	470,000	−80,000	550,000 (5.50%)

When SOFR falls, the company pays less on its loan but makes up the difference in the swap. The total never moves. It has given up any saving from lower rates in return for certainty.

Pay fixed or receive fixed?

4 cards

A property company with a floating-rate loan wants a known interest bill.
A bank has lent fixed-rate mortgages but pays floating rates on its deposits.
A life insurer has promised fixed payments 25 years ahead and wants more long-dated fixed income.
A company has issued a fixed-rate bond but expects rates to fall and wants to pay floating.

Valuing a swap. Each leg is a stream of cash flows, so it can be valued like a bond (L1.4.03): discount each payment and add them up. The swap is worth the difference between the two legs. The fixed leg is easy because its payments are known. For the floating leg there is a neat shortcut: if you add the notional at the end, the floating leg becomes a floating-rate loan, which always pays the going rate and so is worth its face value today. When a swap is first agreed, the fixed rate is set so that both legs are worth the same. That rate is the par swap rate, and the swap starts out worth zero. As rates move, it gains value for one side and loses it for the other.

The formal version

We use $N$ for the notional, $K$ for the fixed rate, $T$ for the number of annual payments and $D_t$ for the discount factor for year $t$ . With a single rate $y$ , $D_t = (1+y)^{-t}$ .

On a flat curve. If every year uses the same rate $y$ , the sum of discount factors is a geometric series with the closed form you met as the annuity factor:

\sum_{t=1}^{T} (1+y)^{-t} = \frac{1-(1+y)^{-T}}{y} = A

So the floating leg is $N(1-D_T) = N \times y \times A$ , the fixed leg is $N \times K \times A$ , and the value collapses to one line:

V_{\text{pay fixed}} = N \times (y - K) \times A

The par rate on a flat curve is $y$ itself. A payer who locked in $K$ gains if new swaps now pay more than $K$ , and loses if they pay less.

Conventions here

Annual payments on both legs, each period counted as exactly one year, annual compounding. Real SOFR swaps usually pay annually on both legs, count days on an Actual/360 basis and compound SOFR daily over each period, which changes the numbers a little but not the method.

Try it

A USD 10m swap. Set the fixed rate equal to the market rate and the value is zero. Then move the market rate: the projected floating payments rise or fall with it, while the fixed payments stay put. Hover a bar to see its value today.

Try itSwap valuer

Open the interest rate swap model →

Fixed rate in the swap, K4.00%

Market rate today (flat curve), y3.50%

Years remaining, T5

You

fixed payment projected floating paymentUSD thousands, notional USD 10m

Annuity factor4.5151

PV fixed leg1,806,021

PV floating leg1,580,268

Value to you (USD)

−225,753

Par swap rate today: 3.50%. You pay 4.00%, so you are behind.

Worked example

The company's swap, one year on

The company above pays 4.00% fixed on USD 10,000,000 for five years and receives SOFR. Its loan costs SOFR + 1.50%. Annual payments, annual compounding.

Cash flows in year 1, then the value

Step 1 of 5

1
Year 1: SOFR works out at 4.30%
Loan interest: 5.80% × 10m = USD 580,000. In the swap the company pays USD 400,000 and receives USD 430,000, so it receives a net USD 30,000.

Regulatory lens

After 2008, standard interest rate swaps between financial firms were made subject to mandatory central clearing: in the US under the Dodd-Frank Act, with the CFTC's first clearing requirements in 2013, and in the EU under EMIR, with the first clearing obligation in 2016. Swaps that are not cleared carry margin requirements (L1.7.11). Banks also use swaps to manage the interest rate risk in their banking book (IRRBB, L3.5), and companies can use hedge accounting under IFRS 9 to report a swap alongside the loan it hedges.

Sparks

SparkGoodbye, LIBOR

For decades, the floating leg of most swaps was linked to LIBOR, a rate based on what panel banks said they could borrow at. After scandals over banks submitting false rates, regulators moved markets to rates based on real transactions. The last US dollar LIBOR settings used by the market stopped being published on a representative basis after 30 June 2023. Most new dollar swaps now reference SOFR.

SparkHuge notional, small payments

Interest rate contracts make up the large majority of the OTC derivatives market by notional, according to BIS statistics. Yet a swap on USD 10m with rates 0.30% apart moves only USD 30,000 a year. The notional measures size, not money changing hands.

Key takeaways

An interest rate swap exchanges fixed payments for floating payments on a notional that is never exchanged.
The payer pays fixed and gains when rates rise; the receiver gains when rates fall.
A company with a floating loan pays fixed in a swap to lock in its total interest cost.
Value a swap as PV of the floating leg minus PV of the fixed leg: N(1 − D_T) − K × N × ΣD_t.
The par swap rate makes a new swap worth zero. On a flat curve, value = N × (y − K) × annuity factor.

Reaction

Check what stuck

Question 1 · objective 1

A company has a floating-rate loan and wants to fix its interest cost. What swap should it enter?

APay floating, receive fixedBPay fixed, receive floatingCA currency swapDNo swap; it should sell its loan

Question 2 · objective 1

A swap has notional USD 20m. You pay fixed at 3.0% and receive floating, which for this year works out at 3.6%. With annual payments, what net amount do you receive, in dollars?

USD

Question 3 · objective 2

You receive fixed at 4% on USD 1m for 3 more years, with annual payments. The market rate is now a flat 3% (annual compounding). What is the swap worth to you, in dollars, to the nearest dollar?

USD

Question 4 · objective 3

The 1-year and 2-year discount factors are 0.97 and 0.94. What is the 2-year par swap rate with annual payments, in %, to two decimal places?

Question 5 · objective 3

Why is a newly agreed swap usually worth zero?

ABecause no notional is exchangedBBecause the fixed rate is set at the par rate, where both legs have the same present valueCBecause floating rates are always equal to fixed ratesDBecause the bank pays the company an upfront fee

Glossary

Interest rate swap: A contract to exchange fixed interest payments for floating interest payments on the same notional amount.
Fixed leg: The stream of payments at the fixed rate agreed at the start.
Floating leg: The stream of payments at a benchmark rate, such as SOFR, reset for each period.
Payer swap: The side that pays fixed and receives floating. It gains when rates rise.
Receiver swap: The side that receives fixed and pays floating. It gains when rates fall.
Netting: Paying only the difference between what each side owes, instead of both amounts in full.
Discount factor: What 1 paid at a future date is worth today: 1 ÷ (1 + rate) raised to the number of years.
Par swap rate: The fixed rate that makes a new swap worth zero to both sides.

Deep Dives & Origins

Deep Dives

John C. Hull, Options, Futures, and Other Derivatives, 11th edition (Pearson, 2021), the chapter “Swaps”. Intermediate.
Bruce Tuckman and Angel Serrat, Fixed Income Securities: Tools for Today's Markets, 3rd edition (Wiley, 2012), the chapter on swaps. Intermediate.

Origins

John C. Hull, Options, Futures, and Other Derivatives, 11th edition, Pearson, 2021.
Tuckman and Serrat, Fixed Income Securities, 3rd edition, Wiley, 2012.
Financial Conduct Authority, announcements on the end of LIBOR (2021 to 2023).
Bank for International Settlements, OTC derivatives statistics.

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Futures and margining

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Currency and cross-currency swaps