Interest rate swap

An interest rate swap is a derivative contract in which two parties exchange streams of interest payments on a notional principal for a set period. The most common form exchanges a fixed rate for a floating rate in the same currency. Variants include basis swaps, overnight index swaps (OIS), forward-start swaps and swaps with changing notionals. Since the late 2000s, collateralised swaps are typically priced and risk-managed using OIS discounting, and following the end of LIBOR new trades reference overnight risk-free rates such as the SOFR, the SONIA and the €STR. As at end-June 2024, interest rate derivatives were the largest segment of the global over-the-counter derivatives market by notional outstanding.

History

Arrangements resembling swaps emerged from back-to-back or parallel loans used in the 1970s to navigate exchange controls. A widely cited early landmark was a 1981 currency swap between IBM and the World Bank arranged by Salomon Brothers, which helped popularise the technique. The first interest rate swap is commonly dated to 1982.

Standard documentation and definitions from the ISDA in the 1990s and 2000s supported market growth and common terminology. After the 2007–2008 financial crisis, pricing for collateralised swaps shifted to OIS discounting and multi-curve approaches, reflecting the role of collateral and funding costs.

From 2021 to 2024, regulators completed the transition from LIBOR to overnight risk-free rates. Remaining synthetic sterling and United States dollar LIBOR settings ceased in 2024, which marked the end of LIBOR in mainstream use.

Mechanics

A standard interest rate swap has two legs linked to the same notional amount. The fixed leg pays a fixed rate on scheduled accrual periods. The floating leg pays a rate set at each reset date by a reference index such as the SOFR, the SONIA or the €STR, with payments exchanged on the corresponding payment dates. Day-count and business-day conventions follow market standards defined in documentation such as the ISDA Interest Rate Derivatives Definitions and, for on-venue trading, the relevant rulebooks.

Variants include forward-start swaps, amortising or accreting notionals, zero-coupon swaps, basis swaps in which both legs float, and overnight index swaps that reference a compounded overnight rate.

Types

Common structures include the following.

• Fixed-for-floating swaps exchange a fixed rate for a floating index in the same currency.
• Basis swaps exchange a floating rate for a floating rate of a different tenor of the same index.
• Overnight index swaps pay a fixed rate versus a compounded overnight risk-free rate such as SOFR, SONIA or €STR.
• Forward-start and deferred-start swaps begin on a future date.
• Amortising and accreting swaps use notionals that change over time.
• Constant-maturity swaps link one leg to a constant-maturity swap rate.

Uses

Common uses include hedging interest rate exposure, adjusting asset and liability duration, and expressing views on the level or shape of the yield curve. In United States markets, the futures and swaps ecosystem now links SOFR futures and SOFR-linked swaps after the conversion of Eurodollar futures and USD LIBOR swaps in 2023.

Valuation and pricing

A vanilla fixed-for-floating swap has a value equal to the difference between the present value of the fixed leg and the present value of the floating leg, discounted on the appropriate curve.

The present value of the fixed leg is

$P_{\text{fixed}}=N\sum _{i=1}^{n}K\,\alpha _{i}\,D(0,t_{i})$

where $N$ is notional, $K$ is the fixed rate, $\alpha _{i}$ are accrual fractions, $t_{i}$ are payment dates and $D(0,t)$ are discount factors.

Under a standard par-swap set-up, the floating leg can be written using forward rates $F_{j}$ and discount factors:

$P_{\text{float}}=N\sum _{j=1}^{m}F_{j}\,\beta _{j}\,D(0,u_{j})$

The par swap rate $S$ that sets the swap’s value to zero is

$S={\dfrac {\sum _{j=1}^{m}F_{j}\,\beta _{j}\,D(0,u_{j})}{\sum _{i=1}^{n}\alpha _{i}\,D(0,t_{i})}}$

For a quoted swap with fixed rate $R$, the mark-to-market is often written

$P_{\text{IRS}}=N\,(R-S)\,A$

with $A=\sum _{i=1}^{n}\alpha _{i}\,D(0,t_{i})$.

Following the 2008 financial crisis, collateralised swaps are commonly discounted using the overnight index swap curve that matches the collateral rate specified under the credit support annex. This leads to multi-curve frameworks that separate discounting from forward-rate projection. See next section

Multi-curve framework

Historically IRSs were valued using discount factors derived from the same curve used to forecast the -IBOR rates (i.e. the erstwhile reference rates; see below re MRRs). This has been called "self-discounted". As above, it became more apparent following the 2008 financial crisis that the approach was not appropriate, and alignment towards discount factors associated with physical collateral of the IRSs was needed. See Financial economics § Derivative pricing for further context.

Thus, the now-standard pricing approach is the multi-curve framework. Note that the economic pricing principle is unchanged: leg values are still identical at initiation. Here, overnight index swap (OIS) rates are typically used to derive discount factors, since that index is the standard inclusion on Credit Support Annexes (CSAs) to determine the rate of interest payable on collateral for IRS contracts. As regards the rates forecast, since the basis spread between LIBOR rates of different maturities widened during the crisis, forecast curves are generally constructed for each LIBOR tenor used in floating rate derivative legs.

Regarding the curve build, see: Under the old framework a single self-discounted curve was "bootstrapped" for each tenor; i.e.: solved such that it exactly returned the observed prices of selected instruments—IRSs, with FRAs in the short end—with the build proceeding sequentially, date-wise, through these instruments. Under the new framework, the various curves are best fitted to observed market prices as a "curve set": one curve for discounting, and one for each IBOR-tenor "forecast curve"; the build is then based on quotes for IRSs and OISs, with FRAs included as before. Here, since the observed average overnight rate plus a spread is swapped for the -IBOR rate over the same period (the most liquid tenor in that market), and the -IBOR IRSs are in turn discounted on the OIS curve, the problem entails a nonlinear system, where all curve points are solved at once, and specialized iterative methods are usually employed (see further following). The forecast-curves for other tenors can be solved in a "second stage", bootstrap-style, with discounting on the now-solved OIS curve.

A CSA could allow for collateral, and hence interest payments on that collateral, in any currency. To accommodate this, banks include in their curve-set a USD discount-curve to be used for discounting local-IBOR trades which have USD collateral; this curve is sometimes called the (Dollar) "basis-curve". It is built by solving for observed (mark-to-market) cross-currency swap rates, where the local -IBOR is swapped for USD LIBOR with USD collateral as underpin. The latest, pre-solved USD-LIBOR-curve is therefore an (external) element of the curve-set, and the basis-curve is then solved in the "third stage". Each currency's curve-set will thus include a local-currency discount-curve and its USD discounting basis-curve. As required, a third-currency discount curve — i.e. for local trades collateralized in a currency other than local or USD (or any other combination) — can then be constructed from the local-currency basis-curve and third-currency basis-curve, combined via an arbitrage relationship known here as "FX Forward Invariance".

Various approaches to solving curves are possible. Modern methods tend to employ global optimizers with complete flexibility in the parameters that are solved relative to the calibrating instruments used to tune them. These optimizers will seek to minimize some objective function - here matching the observed instrument values - and this assumes that some interpolation mode has been configured for the curves; the approach ultimately employed may be a modification of Newton's method. Maturities corresponding to input instruments are referred to as "pillar points"; often, these are solved directly, while other spot rates are interpolated. (Then, once solved, all that need be stored are the pillar point rates and the interpolation rule.)

Starting in 2021, LIBOR is being phased out, with replacements including other "market reference rates" (MRRs) such as SOFR and TONAR. (These MRRs are based on secured overnight funding transactions). With the coexistence of "old" and "new" rates in the market, multi-curve and OIS curve "management" is necessary, with changes required to incorporate new discounting and compounding conventions, while the underlying logic is unaffected; see.

Market structure and regulation

A large share of plain-vanilla swaps is centrally cleared, with clearing mandates and reporting rules in major jurisdictions. In the United States, the Commodity Futures Trading Commission updated the clearing requirement in 2022 to reflect the transition to risk-free rates and added SOFR overnight index swaps across standard maturities. In the European Union, reforms under the EMIR 3.0 framework introduce an active account requirement intended to ensure EU market participants maintain and use accounts at EU central counterparties for specified interest rate derivatives.

Market convention summaries for on-venue trading are published by swap execution facilities and multilateral trading facilities.

Conventions by currency

Typical fixed-leg conventions for vanilla swaps vary by currency. Actual terms depend on documentation and venue rules.

Currency	Typical fixed-leg frequency	Typical fixed-leg day count	Common floating index
USD	Semi-annual	30/360	SOFR (compounded)
EUR	Annual	30/360	€STR (compounded) or Euribor (legacy)
GBP	Semi-annual	ACT/365F	SONIA (compounded)
JPY	Annual	ACT/365F	TONA (compounded)

Risks

Further information: Financial risk management § Investment banking

See also: Derivative (finance) § Risks, and Corporate bond § Risk analysis

Interest rate swaps expose users to several categories of financial risk. The main market risk is interest rate risk, since changes in discount factors and forward rates alter present value and can turn a position from an asset into a liability. Basis risk can arise when cash flows reference different floating rates or tenors, including in the post-LIBOR environment where differences between risk-free rates can be material.

Swaps also create counterparty credit risk. Banks measure and manage the possibility that a counterparty may default, as well as changes in the value of expected exposures. Under the Basel framework, a credit valuation adjustment capital charge applies to capture the risk of CVA changing with credit spreads.

Collateral and margining mitigate bilateral credit exposure but introduce funding and liquidity risks. Requirements at central counterparties and in bilateral agreements can amplify margin calls during stress. Central clearing reduces bilateral counterparty and liquidity risk through multilateral netting, but concentrates exposures between clearing members and the CCP.

Financial reporting for swaps reflects these risks. Under IFRS 9, hedge accounting requirements replaced IAS 39 and align reporting more closely with risk management. Under US GAAP, ASC 815 governs derivatives and hedging, including targeted improvements issued since 2017.

Benchmarks and market data

ICE Swap Rate, formerly ISDAFIX, is a benchmark for swap rates in major currencies and is used in the valuation of some interest rate swaps and swaptions and for certain close-out calculations.

See also

• Overnight index swap
• Secured Overnight Financing Rate
• Sterling Overnight Index Average
• Euro short-term rate
• International Swaps and Derivatives Association
• Swap execution facility

Further reading

General:

• Leif B.G. Andersen, Vladimir V. Piterbarg (2010). Interest Rate Modeling in Three Volumes (1st ed. 2010 ed.). Atlantic Financial Press. ISBN 978-0-9844221-0-4. Archived from the original on 2011-02-08.
• J H M Darbyshire (2017). Pricing and Trading Interest Rate Derivatives (2nd ed. 2017 ed.). Aitch and Dee Ltd. ISBN 978-0995455528.
• Richard Flavell (2010). Swaps and other derivatives (2nd ed.) Wiley. ISBN 047072191X
• Miron P. & Swannell P. (1991). Pricing and Hedging Swaps, Euromoney books. ISBN 185564052X

Early literature on the incoherence of the one curve pricing approach:

• Boenkost W. and Schmidt W. (2004). Cross Currency Swap Valuation, Working Paper 2, HfB - Business School of Finance & Management SSRN preprint.
• Tuckman B. and Porfirio P. (2003). Interest Rate Parity, Money Market Basis Swaps and Cross-Currency Basis Swaps, Fixed income liquid markets research, Lehman Brothers

Multi-curves framework:

• Henrard M. (2007). The Irony in the Derivatives Discounting, Wilmott Magazine, pp. 92–98, July 2007. SSRN preprint.
• Kijima M., Tanaka K., and Wong T. (2009). A Multi-Quality Model of Interest Rates, Quantitative Finance, pages 133-145, 2009.
• Henrard M. (2010). The Irony in the Derivatives Discounting Part II: The Crisis, Wilmott Journal, Vol. 2, pp. 301–316, 2010. SSRN preprint.
• Bianchetti M. (2010). Two Curves, One Price: Pricing & Hedging Interest Rate Derivatives Decoupling Forwarding and Discounting Yield Curves, Risk Magazine, August 2010. SSRN preprint.
• Henrard M. (2014) Interest Rate Modelling in the Multi-curve Framework: Foundations, Evolution, and Implementation. Palgrave Macmillan. Applied Quantitative Finance series. June 2014. ISBN 978-1-137-37465-3.

External links

• Pricing and Trading Interest Rate Derivatives by J H M Darbyshire
• Understanding Derivatives: Markets and Infrastructure Federal Reserve Bank of Chicago, Financial Markets Group
• Bank for International Settlements - Semiannual OTC derivatives statistics
• Interest rate swap glossary
• Investopedia - Spreadlock - An interest rate swap future (not an option)
• Basic Fixed Income Derivative Hedging - Article on Financial-edu.com.
• Hussman Funds - Freight Trains and Steep Curves
• Historical LIBOR Swaps data
• "All about money rates in the world: Real estate interest rates", WorldwideInterestRates.com
• Interest Rate Swap Calculators and Portfolio Management Tool
• G4 LIBOR Swap Calculator

and are regarded as an