# Interest rate swap

In finance, an **interest rate swap** (**IRS**) is an interest rate derivative (IRD). It involves exchange of interest rates between two parties. In particular it is a linear IRD and one of the most liquid, benchmark products. It has associations with forward rate agreements (FRAs), and with zero coupon swaps (ZCSs).

## Contents

## General description[edit]

An interest rate swap's (IRS's) effective description is a derivative contract, agreed between two counterparties, which specifies the nature of an exchange of payments benchmarked against an interest rate index. The most common IRS is a fixed for floating swap, whereby one party will make payments to the other based on an initially agreed fixed rate of interest, to receive back payments based on a floating interest rate index. Each of these series of payments is termed a 'leg', so a typical IRS has both a fixed and a floating leg. The floating index is commonly an interbank offered rate (IBOR) of specific tenor in the appropriate currency of the IRS, for example LIBOR in USD, GBP, EURIBOR in EUR or STIBOR in SEK.

To completely determine any IRS a number of parameters must be specified for each leg; the notional principal amount (or varying notional schedule), the start and end dates and date scheduling, the fixed rate, the chosen floating interest rate index tenor, and day count conventions for interest calculations.^{[1]}

## Extended description[edit]

As OTC instruments, interest rate swaps (IRSs) can be customised in a number of ways and can be structured to meet the specific needs of the counterparties. For example; payment dates could be irregular, the notional of the swap could be amortized over time, reset dates (or fixing dates) of the floating rate could be irregular, mandatory break clauses may be inserted into the contract, etc. A common form of customisation is often present in **new issue swaps** where the fixed leg cashflows are designed to replicate those cashflows received as the coupons on a purchased bond. The interbank market, however, only has a few standardised types. Each currency has its own standard market conventions regarding the frequency of payments, the day count conventions and the end-of-month rule.^{[2]}

There is no consensus on the scope of naming convention for different types of IRS. Even a wide description of IRS contracts only includes those whose legs are denominated in the same currency. It is generally accepted that swaps of similar nature whose legs are denominated in different currencies are called cross currency basis swaps. Swaps which are determined on a floating rate index in one currency but whose payments are denominated in another currency are called quantos.

In traditional interest rate derivative terminology an IRS is a **fixed leg versus floating leg** derivative contract referencing an **IBOR** as the floating leg. If the floating leg is redefined to be an overnight index, such as EONIA, SONIA, FFOIS, etc. then this type of swap is generally referred to as an **overnight indexed swap (OIS)**. Some financial literature may classify OISs as a subset of IRSs and other literature may recognise a distinct separation.

**Fixed leg versus fixed leg** swaps are rare, and generally constitute a form of specialised loan agreement.

**Float leg versus float leg** swaps are much more common. These are typically termed (single currency) basis swaps (SBSs). The legs on SBSs will necessarily be different interest indexes, such as 1M, LIBOR, 3M LIBOR, 6M LIBOR, SONIA, etc. The pricing of these swaps requires a **spread** often quoted in basis points to be added to one of the floating legs in order to satisfy value equivalence.

## Uses[edit]

Interest rate swaps are used to hedge against or speculate on changes in interest rates.

Interest rate swaps are also used speculatively by hedge funds or other investors who expect a change in interest rates or the relationships between them. Traditionally, fixed income investors who expected rates to fall would purchase cash bonds, whose value increased as rates fell. Today, investors with a similar view could enter a floating-for-fixed interest rate swap; as rates fall, investors would pay a lower floating rate in exchange for the same fixed rate.

Interest rate swaps are also popular for the arbitrage opportunities they provide. Varying levels of creditworthiness means that there is often a positive quality spread differential that allows both parties to benefit from an interest rate swap.

The interest rate swap market in USD is closely linked to the Eurodollar futures market which trades among others at the Chicago Mercantile Exchange.

## Valuation and pricing[edit]

IRSs are bespoke financial products whose customisation can include changes to payment dates, notional changes (such as those in amortised IRSs), accrual period adjustment and calculation convention changes (such as a day count convention of 30/360E to ACT/360 or ACT/365).

A vanilla IRS is the term used for standardised IRSs. Typically these will have none of the above customisations, and instead exhibit constant notional throughout, implied payment and accrual dates and benchmark calculation conventions by currency.^{[1]} A vanilla IRS is also characterised by one leg being 'fixed' and the second leg 'floating' referencing an -IBOR index. The net present value (PV) of a vanilla IRS can be computed by determining the PV of each fixed leg and floating leg separately and summing. For pricing a mid-market IRS the underlying principle is that the two legs must have the same value initially; see further under Rational pricing.

Calculating the fixed leg requires discounting all of the known cashflows by an appropriate discount factor:

where is the notional, is the fixed rate, is the number of payments, is the decimalised day count fraction of the accrual in the i'th period, and is the discount factor associated with the payment date of the i'th period.

Calculating the floating leg is a similar process replacing the fixed rate with forecast index rates:

where is the number of payments of the floating leg and are the forecast -IBOR index rates of the appropriate currency.

The PV of the IRS from the perspective of receiving the fixed leg is then:

Historically IRSs were valued using discount factors derived from the same curve used to forecast the -IBOR rates. This has been called 'self-discounted'. Some early literature described some incoherence introduced by that approach and multiple banks were using different techniques to reduce them. It became more apparent with the 2007–2012 global financial crisis that the approach was not appropriate, and alignment towards discount factors associated with physical collateral of the IRSs was needed.

Post crisis, to accommodate credit risk, the now-standard pricing framework is the *multi-curves framework* where forecast -IBOR rates and discount factors exhibit disparity.
Note that the economic pricing principle is unchanged: leg values are still identical at initiation. See Financial economics § Derivative pricing for further context.
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.^{[3]}

Regarding the curve build, see
^{[4]}
^{[5]}
^{[1]}.
Under the old framework a single self discounted curve was "bootstrapped", i.e. solved such that it exactly returned the observed prices of selected instruments, 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, one for each forecast curve.
Here, since the observed average overnight rate is swapped for the -IBOR rate over the same period (the most liquid tenor in that market), and the -IBOR swaps 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 — very often a modification of Newton's method. Other tenor's curves can be solved in a "second stage", bootstrap-style.
(
Under both frameworks, the following apply.
(i) Maturities for which rates are solved directly are referred to as "pillar points", these correspond to the input instrument maturities; other rates are interpolated.
(ii) The objective function: prices must be "exactly" returned, as described.
(iii) The penalty function will weigh: that forward rates are positive (to be arbitrage free) and curve "smoothness"; both, in turn, a function of the interpolation method.
(iv) The initial estimate: often, the most recently solved curve set.
)

A CSA could allow for collateral, and hence interest payments on that collateral, in any currency.^{[6]}
Typically then, banks include in their curve-set a USD discount-curve, sometimes called the "basis-curve".
This curve 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;
a pre-solved (external) USD LIBOR curve is therefore an input into the curve build.
Each currency's curve-set will then include a local-currency discount-curve and its USD basis-curve.
As required, a third-currency discount curve — i.e. for local trades collateralized in a currency other than local or USD — can then be constructed from the third-currency and local-currency basis-curves as combined via an arbitrage relationship known as "FX Forward Invariance".^{[7]}

The complexities of modern curvesets mean that there may not be discount factors available for a specific -IBOR index curve. These curves are known as 'forecast only' curves and only contain the information of a forecast -IBOR index rate for any future date. Some designs constructed with a discount based methodology mean forecast -IBOR index rates are implied by the discount factors inherent to that curve:

- where and are the start and end
*discount factors*associated with the relevant forward curve of a particular -IBOR index in a given currency.

To price the mid-market or par rate, of an IRS (defined by the value of fixed rate that gives a net PV of zero), the above formula is re-arranged to:

In the event old methodologies are applied the discount factors can be replaced with the self discounted values and the above reduces to:

In both cases, the PV of a general swap can be expressed exactly with the following intuitive formula:

where is the so-called Annuity factor (or for self-discounting). This shows that the PV of an IRS is roughly linear in the swap par rate (though small non-linearities arise from the co-dependency of the swap rate with the discount factors in the Annuity sum).

During the life of the swap the same valuation technique is used, but since, over time, both the discounting factors and the forward rates change, the PV of the swap will deviate from its initial value. Therefore, the swap will be an asset to one party and a liability to the other. The way these changes in value are reported is the subject of IAS 39 for jurisdictions following IFRS, and FAS 133 for U.S. GAAP. Swaps are marked to market by debt security traders to visualize their inventory at a certain time. As regards hedging, and P&L Attribution, the new framework adds complexity in that the trader's position is now potentially affected by numerous instruments not obviously related to the trade in question.

## Risks[edit]

Interest rate swaps expose users to many different types of financial risk.^{[1]} Predominantly they expose the user to market risks and specifically interest rate risk. The value of an interest rate swap will change as market interest rates rise and fall. In market terminology this is often referred to as delta risk. Interest rate swaps also exhibit gamma risk whereby their delta risk increases or decreases as market interest rates fluctuate. See Greeks (finance), Value at risk #Computation methods, Value at risk #VaR risk management. Other specific types of market risk that interest rate swaps have exposure to are basis risks (where various IBOR tenor indexes can deviate from one another) and reset risks (where the publication of specific tenor IBOR indexes are subject to daily fluctuation).

Uncollateralised interest rate swaps - those executed bilaterally without a CSA in place - expose the trading counterparties to funding risks and credit risks. Funding risks because the value of the swap might deviate to become so negative that it is unaffordable and cannot be funded. Credit risks because the respective counterparty, for whom the value of the swap is positive, will be concerned about the opposing counterparty defaulting on its obligations. Collateralised interest rate swaps, on the other hand, expose the users to collateral risks: here, depending upon the terms of the CSA, the type of posted collateral that is permitted might become more or less expensive due to other extraneous market movements. Credit and funding risks still exist for collateralised trades but to a much lesser extent. Due to regulations set out in the Basel III Regulatory Frameworks, trading interest rate derivatives commands a capital usage. Dependent upon their specific nature interest rate swaps might command more capital usage and this can deviate with market movements. Thus capital risks are another concern for users. See Credit valuation adjustment, XVA.

Reputation risks also exist. The mis-selling of swaps, over-exposure of municipalities to derivative contracts, and IBOR manipulation are examples of high-profile cases where trading interest rate swaps has led to a loss of reputation and fines by regulators.

Hedging interest rate swaps can be complicated and relies on numerical processes of well designed risk models to suggest reliable benchmark trades that mitigate all market risks. The other, aforementioned risks must be hedged using other systematic processes.

## Market-Making[edit]

The market-making of IRSs is an involved process involving multiple tasks; curve construction with reference to interbank markets, individual derivative contract pricing, risk management of credit, cash and capital. The cross disciplines required include quantitative analysis and mathematical expertise, disciplined and organized approach towards profits and losses, and coherent psychological and subjective assessment of financial market information and price-taker analysis. The time sensitive nature of markets also creates a pressurized environment. Many tools and techniques have been designed to improve efficiency of market-making in a drive to efficiency and consistency.^{[1]}

## Trivia[edit]

On its December 2014 statistics release, the Bank for International Settlements reported that interest rate swaps were the largest component of the global OTC derivative market representing 60% of it, with the notional amount outstanding in OTC interest rate swaps of $381 trillion, and the gross market value of $14 trillion.^{[8]}

Interest rate swaps can be traded as an index through the FTSE MTIRS Index.

## Controversy[edit]

In June 1988 the Audit Commission was tipped off by someone working on the swaps desk of Goldman Sachs that the London Borough of Hammersmith and Fulham had a massive exposure to interest rate swaps. When the commission contacted the council, the chief executive told them not to worry as "everybody knows that interest rates are going to fall"; the treasurer thought the interest rate swaps were a "nice little earner". The Commission's Controller, Howard Davies, realised that the council had put all of its positions on interest rates going down and ordered an investigation.

By January 1989 the Commission obtained legal opinions from two Queen's Counsel. Although they did not agree, the commission preferred the opinion that it was *ultra vires* for councils to engage in interest rate swaps (ie. that they had no lawful power to do so). Moreover, interest rates had increased from 8% to 15%. The auditor and the commission then went to court and had the contracts declared void (appeals all the way up to the House of Lords failed in *Hazell v Hammersmith and Fulham LBC*); the five banks involved lost millions of pounds. Many other local authorities had been engaging in interest rate swaps in the 1980s.^{[9]} This resulted in several cases in which the banks generally lost their claims for compound interest on debts to councils, finalised in *Westdeutsche Landesbank Girozentrale v Islington London Borough Council*.^{[10]} Banks did, however, recover some funds where the derivatives were "in the money" for the Councils (ie, an asset showing a profit for the council, which it now had to return to the bank, not a debt)

## See also[edit]

- Swap rate
- Interest rate cap and floor
- Equity swap
- Total return swap
- Inflation derivative
- Eurodollar
- Constant maturity swap
- FTSE MTIRS Index

## Further reading[edit]

- J H M Darbyshire (2017).
*Pricing and Trading Interest Rate Derivatives*(2nd ed. 2017 ed.). Aitch and Dee Ltd. ISBN 978-0995455528. - 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.

## References[edit]

- ^
^{a}^{b}^{c}^{d}^{e}Pricing and Trading Interest Rate Derivatives: A Practical Guide to Swaps, J H M Darbyshire, 2017, ISBN 978-0995455528 **^**"Interest Rate Instruments and Market Conventions Guide Archived 2012-06-17 at the Wayback Machine" Quantitative Research, OpenGamma, 2012.**^**Multi-Curve Valuation Approaches and their Application to Hedge Accounting according to IAS 39, Dr. Dirk Schubert, KPMG**^**M. Henrard (2014).*Interest Rate Modelling in the Multi-Curve Framework: Foundations, Evolution and Implementation.*Palgrave Macmillan ISBN 978-1137374653**^**See section 3 of Marco Bianchetti and Mattia Carlicchi (2012).*Interest Rates after The Credit Crunch: Multiple-Curve Vanilla Derivatives and SABR***^**Fujii, Masaaki Fujii; Yasufumi Shimada; Akihiko Takahashi (26 January 2010). "A Note on Construction of Multiple Swap Curves with and without Collateral".*CARF Working Paper Series No. CARF-F-154*. SSRN 1440633.**^**Burgess, Nicholas (2017).*FX Forward Invariance & Discounting with CSA Collateral***^**"OTC derivatives statistics at end-December 2014" (PDF). Bank for International Settlements.**^**Duncan Campbell-Smith, "Follow the Money: The Audit Commission, Public Money, and the Management of Public Services 1983-2008", Allen Lane, 2008, chapter 6*passim*.**^**[1996] UKHL 12, [1996] AC 669

- Pricing and Hedging Swaps, Miron P. & Swannell P., Euromoney books 1991

Most recent, industry standard literature on the evolution of the swaps market to incorporate credit and collateral risks:

- Pricing and Trading Interest Rate Derivatives: A Practical Guide to Swaps, J H M Darbyshire, 2017

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

- Interest rate parity, money market basis swaps and cross-currency basis swaps, Tuckman B. and Porfirio P., Fixed income liquid markets research, Lehman Brothers, 2003.
- Cross currency swap valuation, Boenkost W. and Schmidt W., Working Paper 2, HfB - Business School of Finance & Management, 2004. SSRN preprint.
- The Irony in the Derivatives Discounting, Henrard M., Wilmott Magazine, pp. 92–98, July 2007. SSRN preprint.

Multi-curves framework:

- A multi-quality model of interest rates, Kijima M., Tanaka K., and Wong T., Quantitative Finance, pages 133-145, 2009.
- Two Curves, One Price: Pricing & Hedging Interest Rate Derivatives Decoupling Forwarding and Discounting Yield Curves, Bianchetti M., Risk Magazine, August 2010. SSRN preprint.
- The Irony in the Derivatives Discounting Part II: The Crisis, Henrard M., Wilmott Journal, Vol. 2, pp. 301–316, 2010. SSRN preprint.

## External links[edit]

- 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
- Glossary - 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