In
finance
Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of f ...
, 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
A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, ...
, benchmark products. It has associations with
forward rate agreements (FRAs), and with
zero coupon swaps (ZCSs).
In its December 2014 statistics release, the
Bank for International Settlements
The Bank for International Settlements (BIS) is an international financial institution owned by central banks that "fosters international monetary and financial cooperation and serves as a bank for central banks".
The BIS carries out its work th ...
reported that interest rate swaps were the largest component of the global
OTC derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
market, representing 60%, with the
notional amount
The notional amount (or notional principal amount or notional value) on a financial instrument is the nominal or face amount that is used to calculate payments made on that instrument. This amount generally does not change and is thus referred to a ...
outstanding in OTC interest rate swaps of $381 trillion, and the gross market value of $14 trillion.
Interest rate swaps can be traded as an index through the
FTSE MTIRS Index.
Interest rate swaps
General description
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
The London Inter-Bank Offered Rate is an interest-rate average calculated from estimates submitted by the leading banks in London. Each bank estimates what it would be charged were it to borrow from other banks. The resulting average rate is u ...
in GBP,
EURIBOR in EUR, or STIBOR in SEK.
To completely determine any IRS a number of parameters must be specified for each leg:
[Pricing and Trading Interest Rate Derivatives: A Practical Guide to Swaps]
J H M Darbyshire, 2017,
*the
notional principal amount
The notional amount (or notional principal amount or notional value) on a financial instrument is the nominal or face amount that is used to calculate payments made on that instrument. This amount generally does not change and is thus referred to a ...
(or varying notional schedule);
*the start and end dates,
value-,
trade- and
settlement dates, and date scheduling (
date rolling
Date or dates may refer to:
* Date (fruit), the fruit of the date palm (''Phoenix dactylifera'')
Social activity
* Dating, a form of courtship involving social activity, with the aim of assessing a potential partner
**Group dating
*Play date, ...
);
*the fixed rate (i.e. "
swap rate", sometimes quoted as a "
swap spread Swap spreads are the difference between the swap rate (a fixed interest rate) and a corresponding government bond yield with the same maturity ( Treasury securities in the case of the United States).
For example, if the current market rate for a f ...
" over a benchmark);
*the chosen floating interest rate index
tenor
A tenor is a type of classical male singing voice whose vocal range lies between the countertenor and baritone voice types. It is the highest male chest voice type. The tenor's vocal range extends up to C5. The low extreme for tenors is wide ...
;
*the
day count conventions for interest calculations.
Each currency has its own standard market conventions regarding the frequency of payments, the day count conventions and the end-of-month rule.
Extended description
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
The interbank market is the top-level foreign exchange market where banks exchange different currencies. The banks can either deal with one another directly, or through electronic brokering platforms. The Electronic Broking Services (EBS) and Thom ...
, however, only has a few standardised types.
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
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
In economics and finance, arbitrage (, ) is the practice of taking advantage of a difference in prices in two or more markets; striking a combination of matching deals to capitalise on the difference, the profit being the difference between t ...
opportunities they provide. Varying levels of
creditworthiness
A credit risk is risk of default on a debt that may arise from a borrower failing to make required payments. In the first resort, the risk is that of the lender and includes lost principal and interest, disruption to cash flows, and increased ...
means that there is often a positive
quality spread differential Quality spread differential (QSD) arises during an interest rate swap in which two parties of different levels of creditworthiness experience different levels of interest rates of debt obligations. A positive QSD means that a swap is in the intere ...
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
The Chicago Mercantile Exchange (CME) (often called "the Chicago Merc", or "the Merc") is a global derivatives marketplace based in Chicago and located at 20 S. Wacker Drive. The CME was founded in 1898 as the Chicago Butter and Egg Board, an ...
.
Valuation and pricing
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.
A vanilla IRS is also characterised by one leg being "fixed" and the second leg "floating" referencing an -IBOR index. The net
present value
In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has inte ...
(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
Collateral may refer to:
Business and finance
* Collateral (finance), a borrower's pledge of specific property to a lender, to secure repayment of a loan
* Marketing collateral, in marketing and sales
Arts, entertainment, and media
* ''Collate ...
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
The London Inter-Bank Offered Rate is an interest-rate average calculated from estimates submitted by the leading banks in London. Each bank estimates what it would be charged were it to borrow from other banks. The resulting average rate is u ...
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, 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
The overnight rate is generally the interest rate that large banks use to borrow and lend from one another in the overnight market. In some countries (the United States, for example), the overnight rate may be the rate targeted by the central ban ...
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
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
, 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
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost ...
: 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: usually, the most recently solved curve set.
((v) All that
need be stored are the solved
spot rates for the pillars, and the interpolation rule.)
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 "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".
LIBOR is due to be phased out by the end of 2021, with replacements including
SOFR
Secured Overnight Financing Rate (SOFR) is a secured interbank overnight interest rate. SOFR is a reference rate (that is, a rate used by parties in commercial contracts that is outside their direct control) established as an alternative to LIBOR. ...
and
TONAR.
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.
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
IAS 39: Financial Instruments: Recognition and Measurement was an international accounting standard which outlined the requirements for the recognition and measurement of financial assets, financial liabilities, and some contracts to buy or sell ...
for jurisdictions following
IFRS, and
FAS 133
Launched prior to the millennium, (and subsequently amended) FAS 133 ''Accounting for Derivative Instruments and Hedging Activities'' provided an "integrated accounting framework for derivative instruments and hedging activities."
FAS 133 Overvi ...
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.
Risks
Interest rate swaps expose users to many different types of financial risk.
Predominantly they expose the user to
market risk
Market risk is the risk of losses in positions arising from movements in market variables like prices and volatility.
There is no unique classification as each classification may refer to different aspects of market risk. Nevertheless, the most ...
s and specifically
interest rate risk
In finance and economics, interest is payment from a borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, the amount borrowed), at a particular rate. It is distinc ...
.
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)
In mathematical finance, the Greeks are the quantities representing the sensitivity of the price of derivatives such as options to a change in underlying parameters on which the value of an instrument or portfolio of financial instruments is de ...
,
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 Credit valuation adjustments (CVAs) are accounting adjustments made to reserve a portion of profits on uncollateralized financial derivatives. They are charged by a bank to a risky (capable of default) counterparty to compensate the bank for taking ...
, 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.
Quotation and market-making
ICE Swap rate
ICE Swap rate replaced the rate formerly known as ISDAFIX in 2015. Swap Rate benchmark rates are calculated using eligible prices and volumes for specified interest rate derivative products. The prices are provided by trading venues in accordance with a “Waterfall” Methodology. The first level of the Waterfall (“Level 1”) uses eligible, executable prices and volumes provided by regulated, electronic, trading venues. Multiple, randomised snapshots of market data are taken during a short window before calculation. This enhances the benchmark's robustness and reliability by protecting against attempted manipulation and temporary aberrations in the underlying market.
Market-making
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.
Controversy
In June 1988 the
Audit Commission was tipped off by someone working on the swaps desk of
Goldman Sachs
Goldman Sachs () is an American multinational investment bank and financial services company. Founded in 1869, Goldman Sachs is headquartered at 200 West Street in Lower Manhattan, with regional headquarters in London, Warsaw, Bangalore, Ho ...
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.
[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''.]
By January 1989 the Commission obtained legal opinions from two
Queen's Counsel
In the United Kingdom and in some Commonwealth countries, a King's Counsel (post-nominal initials KC) during the reign of a king, or Queen's Counsel (post-nominal initials QC) during the reign of a queen, is a lawyer (usually a barrister o ...
. 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
The House of Lords, also known as the House of Peers, is the upper house of the Parliament of the United Kingdom. Membership is by appointment, heredity or official function. Like the House of Commons, it meets in the Palace of Westminst ...
failed in ''
Hazell v Hammersmith and Fulham LBC O
''Hazell v Hammersmith and Fulham LBC'' 9922 AC 1 is an English administrative law case, which declared that local authorities had no power to engage in interest rate swap agreements because they were beyond the council's borrowing powers, an ...
''); the five banks involved lost millions of pounds. Many other local authorities had been engaging in interest rate swaps in the 1980s.
[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''.] This resulted in several cases in which the banks generally lost their claims for
compound interest on debts to councils, finalised in ''
''. 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).
[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''.]
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
by the media and Parliament.
See also
*
Constant maturity swap A constant maturity swap, also known as a CMS, is a swap that allows the purchaser to fix the duration of received flows on a swap.
The floating leg of an interest rate swap typically resets against a published index. The floating leg of a consta ...
*
Equity swap
*
Eurodollar
*
FTSE MTIRS Index
*
Inflation derivative
*
Interest rate cap and floor
*
Swap rate
*
Total return swap
References
Further reading
General:
*
*
* Richard Flavell (2010)
''Swaps and other derivatives''(2nd ed.) Wiley.
* Miron P. & Swannell P. (1991). ''Pricing and Hedging Swaps'', Euromoney books.
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 & Managemen
SSRN preprint.* Henrard M. (2007). ''The Irony in the Derivatives Discounting'', Wilmott Magazine, pp. 92–98, July 2007
SSRN preprint.* Henrard M. (2010). ''The Irony in the Derivatives Discounting Part II: The Crisis'', Wilmott Journal, Vol. 2, pp. 301–316, 2010
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
Lehman Brothers Holdings Inc. ( ) was an American global financial services firm founded in 1847. Before filing for bankruptcy in 2008, Lehman was the fourth-largest investment bank in the United States (behind Goldman Sachs, Morgan Stanley, ...
Multi-curves framework:
* 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. .
* Kijima M., Tanaka K., and Wong T. (2009). ''A Multi-Quality Model of Interest Rates'', Quantitative Finance, pages 133-145, 2009.
External links
Pricing and Trading Interest Rate Derivativesby J H M Darbyshire
Understanding Derivatives: Markets and InfrastructureFederal Reserve Bank of Chicago, Financial Markets Group
- 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.
Historical LIBOR Swaps data ''WorldwideInterestRates.com''
Interest Rate Swap Calculators and Portfolio Management ToolG4 LIBOR Swap Calculator
{{Derivatives market
Derivatives (finance)
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Swaps (finance)