Interest rate swap

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.[2] 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:

${\displaystyle P_{\text{fixed}}=NR\sum _{i=1}^{n_{1}}d_{i}v_{i}}$

where ${\displaystyle N}$ is the notional, ${\displaystyle R}$ is the fixed rate, ${\displaystyle n_{1}}$ is the number of payments, ${\displaystyle d_{i}}$ is the decimalised day count fraction of the accrual in the i'th period, and ${\displaystyle v_{i}}$ 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:

${\displaystyle P_{\text{float}}=N\sum _{j=1}^{n_{2}}r_{j}d_{j}v_{j}}$

where ${\displaystyle n_{2}}$ is the number of payments of the floating leg and ${\displaystyle r_{j}}$ 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:

${\displaystyle P_{\text{IRS}}=P_{\text{fixed}}-P_{\text{float}}}$

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 approach is the multi-curve 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.[4]

Regarding the curve build, see: [5] [6] [2] 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, one for each IBOR-tenor "forecast curve", and the build is then based on quotes for IRSs and OISs, with FRAs included as before. 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 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—very often a modification of Newton's method. The forecast-curves for other tenors can be solved in a "second stage", bootstrap-style, with discounting on the now-solved OIS curve.

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, often using Hermitic splines. (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. [7] [8] [9] (iv) The initial estimate: usually, the most recently solved curve set. ((v) All that need be stored are the solved spot rates for the pillar points, and the interpolation rule.)

A CSA could allow for collateral, and hence interest payments on that collateral, in any currency.[10] 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 "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".[11]

LIBOR is being phased out, with replacements including SOFR and TONAR ("market reference rates", MRRs, as 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.[12][13][14]

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:

${\displaystyle r_{j}={\frac {1}{d_{j}}}\left({\frac {x_{j-1}}{x_{j}}}-1\right)}$ where ${\displaystyle x_{i-1}}$ and ${\displaystyle x_{i}}$ 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, ${\displaystyle S}$ of an IRS (defined by the value of fixed rate ${\displaystyle R}$ that gives a net PV of zero), the above formula is re-arranged to:

${\displaystyle S={\frac {\sum _{j=1}^{n_{2}}r_{j}d_{j}v_{j}}{\sum _{i=1}^{n_{1}}d_{i}v_{i}}}}$

In the event old methodologies are applied the discount factors ${\displaystyle v_{k}}$ can be replaced with the self discounted values ${\displaystyle x_{k}}$ and the above reduces to:

${\displaystyle S={\frac {x_{0}-x_{n_{2}}}{\sum _{i=1}^{n_{1}}d_{i}x_{i}}}}$

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

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

where ${\displaystyle A}$ is the so-called Annuity factor ${\displaystyle A=\sum _{i=1}^{n_{1}}d_{i}v_{i}}$ (or ${\displaystyle A=\sum _{i=1}^{n_{1}}d_{i}x_{i}}$ 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 P&L Attribution, and hedging, 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.

Interest rate swaps expose users to many different types of financial risk.[2] 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. Regardless, due to regulations set out in the Basel III Regulatory Frameworks, trading interest rate derivatives commands a capital usage. The consequence of this is that, 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.

Given these concerns, banks will typically calculate a credit valuation adjustment, as well as other x-valuation adjustments, which then incorporate these risks into the instrument value.

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; although, see the discussion above re hedging in a multi-curve environment. The other, aforementioned risks must be hedged using other systematic processes.

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.[16]

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.[16] 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.[17] 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).[16]

The controversy surrounding interest rate swaps reached a peak in the UK during the financial crisis where banks sold unsuitable interest rate hedging products on a large scale to SMEs. The practice has been widely criticised[18] by the media and Parliament.

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

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.

• 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
• G4 LIBOR Swap Calculator