### Bank secrecy (Ofer Abarbanel online library)

Banking secrecy,[1][2] alternately known as financial privacybanking discretion, or bank safety,[3][4] is a conditional agreement between a bank and its clients that all foregoing activities remain secure, confidential, and private.[5][6] Continue reading “Bank secrecy (Ofer Abarbanel online library)”

### 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).

In 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%, with the notional amount outstanding in OTC interest rate swaps of $381 trillion, and the gross market value of$14 trillion.[1]

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

## Interest rate swaps

### General description

Graphical depiction of IRS cashflows between two counterparties based on a notional amount of EUR100mm for a single (i’th) period exchange, where the floating index {\displaystyle r_{i}} will typically be an -IBOR index.

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 GBP, EURIBOR in EUR, or STIBOR in SEK.

To completely determine any IRS a number of parameters must be specified for each leg: [2]

• the notional principal amount (or varying notional schedule);
• the start and end dates, value-, trade- and settlement dates, and date scheduling (date rolling);
• the fixed rate (i.e. “swap rate”, sometimes quoted as a “swap spread” over a benchmark);
• the chosen floating interest rate index tenor;
• 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.[3]

### Extended description

 There are several types of IRS, typically: “Vanilla” fixed for floating Basis swap Cross currency basis swaps Amortising swap Zero coupon swap Constant maturity swap Overnight indexed swap

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.

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 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

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”, 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. 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. 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, 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 pillars, and the interpolation rule.)

A CSA could allow for collateral, and hence interest payments on that collateral, in any currency.[10] To address this banks include in their curve-set, a USD discount-curve — sometimes called the “basis-curve” — to be used for discounting local-IBOR trades with USD collateral. 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 (the basis-curve may be solved in the “third stage”). Each currency’s curve-set will then 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 as “FX Forward Invariance”.[11]

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.

## Risks

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.

## Quotation and Market-Making

### ICE Swap Rate

ICE Swap rate[12] 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.[2]

## Controversy

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.[13] 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.[14] 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)

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[15] by the media and Parliament.

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

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

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.
• Henrard M. (2007). The Irony in the Derivatives Discounting, Wilmott Magazine, pp. 92–98, July 2007. 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:

• 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. (2010). The Irony in the Derivatives Discounting Part II: The Crisis, Wilmott Journal, Vol. 2, pp. 301–316, 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.
• Kijima M., Tanaka K., and Wong T. (2009). A Multi-Quality Model of Interest Rates, Quantitative Finance, pages 133-145, 2009.

## References

1. ^ “OTC derivatives statistics at end-December 2014” (PDF). Bank for International Settlements.
2. Jump up to:a b c d e Pricing and Trading Interest Rate Derivatives: A Practical Guide to Swaps, J H M Darbyshire, 2017, ISBN 978-0995455528
3. ^ “Interest Rate Instruments and Market Conventions Guide” Quantitative Research, OpenGamma, 2012.
4. ^ Multi-Curve Valuation Approaches and their Application to Hedge Accounting according to IAS 39, Dr. Dirk Schubert, KPMG
5. ^ M. Henrard (2014). Interest Rate Modelling in the Multi-Curve Framework: Foundations, Evolution and Implementation. Palgrave Macmillan ISBN 978-1137374653
6. ^ See section 3 of Marco Bianchetti and Mattia Carlicchi (2012). Interest Rates after The Credit Crunch: Multiple-Curve Vanilla Derivatives and SABR
7. ^ P. Hagan and G. West (2006). Interpolation methods for curve construction. Applied Mathematical Finance, 13 (2):89—129, 2006.
8. ^ P. Hagan and G. West (2008). Methods for Constructing a Yield Curve. Wilmott Magazine, May, 70-81.
9. ^ P du Preez and E Maré (2013). Interpolating Yield Curve Data in a Manner That Ensures Positive and Continuous Forward Curves.SAJEMS 16 (2013) No 4:395-406
10. ^ 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.
11. ^ Burgess, Nicholas (2017). FX Forward Invariance & Discounting with CSA Collateral
12. ^ ICE Swap Rate. [1]
13. ^ 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.
14. ^ [1996] UKHL 12, [1996] AC 669
15. ^ “HM Parliament Condemns RBS GRG’s Parasitic Treatment of SMEs Post date”.

### Bankers’ bonuses (Ofer Abarbanel online library)

Bankers’ bonuses are traditionally paid or awarded to some workers in the finance industry at the end of the bank’s financial year. They are intended to reward employee behavior during that year that has increased the profits of the bank or some relevant part of its business, as shown by the annual accounts. Continue reading “Bankers’ bonuses (Ofer Abarbanel online library)”

### Arranger (banking) (Ofer Abarbanel online library)

In investment banking, an arranger is a provider of funds in the syndication of a debt. They are entitled to syndicate the loan or bond issue, and may be referred to as the “lead underwriter”. This is because this entity bears the risk of being able to sell the underlying securities/debt or the cost of holding it on its books until such time in the future that they may be sold. They do not necessarily acquire all the debt – this may be split into various parts and sold to a variety of Arrangers. Continue reading “Arranger (banking) (Ofer Abarbanel online library)”

### Christmas club (Ofer Abarbanel online library)

The Christmas club is a savings program that was first offered by various banks and credit unions in the United States beginning in the first half of the 20th century, and including the Great Depression. The concept is that bank customers deposit a set amount of money each week into a special savings account, and receive the money back at the end of the year for Christmas shopping. Continue reading “Christmas club (Ofer Abarbanel online library)”

### 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).

In 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%, with the notional amount outstanding in OTC interest rate swaps of $381 trillion, and the gross market value of$14 trillion.[1]

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

## Interest rate swaps

### General description

Graphical depiction of IRS cashflows between two counterparties based on a notional amount of EUR100mm for a single (i’th) period exchange, where the floating index {\displaystyle r_{i}} will typically be an -IBOR index.

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 GBP, EURIBOR in EUR, or STIBOR in SEK.

To completely determine any IRS a number of parameters must be specified for each leg: [2]

• the notional principal amount (or varying notional schedule);
• the start and end dates, value-, trade- and settlement dates, and date scheduling (date rolling);
• the fixed rate (i.e. “swap rate”, sometimes quoted as a “swap spread” over a benchmark);
• the chosen floating interest rate index tenor;
• 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.[3]

### Extended description

 There are several types of IRS, typically: “Vanilla” fixed for floating Basis swap Cross currency basis swaps Amortising swap Zero coupon swap Constant maturity swap Overnight indexed swap

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.

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 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

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”, 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. 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. 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, 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 pillars, and the interpolation rule.)

A CSA could allow for collateral, and hence interest payments on that collateral, in any currency.[10] To address this banks include in their curve-set, a USD discount-curve — sometimes called the “basis-curve” — to be used for discounting local-IBOR trades with USD collateral. 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 (the basis-curve may be solved in the “third stage”). Each currency’s curve-set will then 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 as “FX Forward Invariance”.[11]

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.

## Risks

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.

## Quotation and Market-Making

### ICE Swap Rate

ICE Swap rate[12] 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.[2]

## Controversy

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.[13] 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.[14] 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)

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[15] by the media and Parliament.

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

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

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.
• Henrard M. (2007). The Irony in the Derivatives Discounting, Wilmott Magazine, pp. 92–98, July 2007. 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:

• 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. (2010). The Irony in the Derivatives Discounting Part II: The Crisis, Wilmott Journal, Vol. 2, pp. 301–316, 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.
• Kijima M., Tanaka K., and Wong T. (2009). A Multi-Quality Model of Interest Rates, Quantitative Finance, pages 133-145, 2009.

## References

1. ^ “OTC derivatives statistics at end-December 2014” (PDF). Bank for International Settlements.
2. Jump up to:a b c d e Pricing and Trading Interest Rate Derivatives: A Practical Guide to Swaps, J H M Darbyshire, 2017, ISBN 978-0995455528
3. ^ “Interest Rate Instruments and Market Conventions Guide” Quantitative Research, OpenGamma, 2012.
4. ^ Multi-Curve Valuation Approaches and their Application to Hedge Accounting according to IAS 39, Dr. Dirk Schubert, KPMG
5. ^ M. Henrard (2014). Interest Rate Modelling in the Multi-Curve Framework: Foundations, Evolution and Implementation. Palgrave Macmillan ISBN 978-1137374653
6. ^ See section 3 of Marco Bianchetti and Mattia Carlicchi (2012). Interest Rates after The Credit Crunch: Multiple-Curve Vanilla Derivatives and SABR
7. ^ P. Hagan and G. West (2006). Interpolation methods for curve construction. Applied Mathematical Finance, 13 (2):89—129, 2006.
8. ^ P. Hagan and G. West (2008). Methods for Constructing a Yield Curve. Wilmott Magazine, May, 70-81.
9. ^ P du Preez and E Maré (2013). Interpolating Yield Curve Data in a Manner That Ensures Positive and Continuous Forward Curves.SAJEMS 16 (2013) No 4:395-406
10. ^ 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.
11. ^ Burgess, Nicholas (2017). FX Forward Invariance & Discounting with CSA Collateral
12. ^ ICE Swap Rate. [1]
13. ^ 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.
14. ^ [1996] UKHL 12, [1996] AC 669
15. ^ “HM Parliament Condemns RBS GRG’s Parasitic Treatment of SMEs Post date”.

### Forward rate

The forward rate is the future yield on a bond. It is calculated using the yield curve. For example, the yield on a three-month Treasury bill six months from now is a forward rate.[1]

## Forward rate calculation

To extract the forward rate, we need the zero-coupon yield curve.

We are trying to find the future interest rate {\displaystyle r_{1,2}} for time period {\displaystyle (t_{1},t_{2})}{\displaystyle t_{1}} and {\displaystyle t_{2}} expressed in years, given the rate {\displaystyle r_{1}} for time period {\displaystyle (0,t_{1})} and rate {\displaystyle r_{2}} for time period {\displaystyle (0,t_{2})}. To do this, we use the property that the proceeds from investing at rate {\displaystyle r_{1}} for time period {\displaystyle (0,t_{1})} and then reinvesting those proceeds at rate {\displaystyle r_{1,2}} for time period {\displaystyle (t_{1},t_{2})} is equal to the proceeds from investing at rate {\displaystyle r_{2}} for time period {\displaystyle (0,t_{2})}.

{\displaystyle r_{1,2}} depends on the rate calculation mode (simpleyearly compounded or continuously compounded), which yields three different results.

### Simple rate

{\displaystyle (1+r_{1}t_{1})(1+r_{1,2}(t_{2}-t_{1}))=1+r_{2}t_{2}}

Solving for {\displaystyle r_{1,2}} yields:

Thus {\displaystyle r_{1,2}={\frac {1}{t_{2}-t_{1}}}\left({\frac {1+r_{2}t_{2}}{1+r_{1}t_{1}}}-1\right)}

The discount factor formula for period (0, t) {\displaystyle \Delta _{t}} expressed in years, and rate {\displaystyle r_{t}} for this period being {\displaystyle DF(0,t)={\frac {1}{(1+r_{t}\,\Delta _{t})}}}, the forward rate can be expressed in terms of discount factors: {\displaystyle r_{1,2}={\frac {1}{t_{2}-t_{1}}}\left({\frac {DF(0,t_{1})}{DF(0,t_{2})}}-1\right)}

### Yearly compounded rate

{\displaystyle (1+r_{1})^{t_{1}}(1+r_{1,2})^{t_{2}-t_{1}}=(1+r_{2})^{t_{2}}}

Solving for {\displaystyle r_{1,2}} yields :

{\displaystyle r_{1,2}=\left({\frac {(1+r_{2})^{t_{2}}}{(1+r_{1})^{t_{1}}}}\right)^{1/(t_{2}-t_{1})}-1}

The discount factor formula for period (0,t{\displaystyle \Delta _{t}} expressed in years, and rate {\displaystyle r_{t}} for this period being {\displaystyle DF(0,t)={\frac {1}{(1+r_{t})^{\Delta _{t}}}}}, the forward rate can be expressed in terms of discount factors:

{\displaystyle r_{1,2}=\left({\frac {DF(0,t_{1})}{DF(0,t_{2})}}\right)^{1/(t_{2}-t_{1})}-1}

### Continuously compounded rate

EQUATION→ {\displaystyle e^{{(r}_{2}\ast t_{2})}=e^{{(r}_{1}\ast t_{1})}\ast \ e^{\left(r_{1,2}\ast \left(t_{2}-t_{1}\right)\right)}}

Solving for {\displaystyle r_{1,2}} yields:

STEP 1→ {\displaystyle e^{{(r}_{2}\ast t_{2})}=e^{{(r}_{1}\ast t_{1})+\left(r_{1,2}\ast \left(t_{2}-t_{1}\right)\right)}}
STEP 2→ {\displaystyle \ln {\left(e^{{(r}_{2}\ast t_{2})}\right)}=\ln {\left(e^{{(r}_{1}\ast t_{1})+\left(r_{1,2}\ast \left(t_{2}-t_{1}\right)\right)}\right)}}
STEP 3→ {\displaystyle {(r}_{2}\ast \ t_{2})={(r}_{1}\ast \ t_{1})+\left(r_{1,2}\ast \left(t_{2}-t_{1}\right)\right)}
STEP 4→ {\displaystyle r_{1,2}\ast \left(t_{2}-t_{1}\right)={(r}_{2}\ast \ t_{2})-{(r}_{1}\ast \ t_{1})}
STEP 5→ {\displaystyle r_{1,2}={\frac {{(r}_{2}\ast t_{2})-{(r}_{1}\ast t_{1})}{t_{2}-t_{1}}}}

The discount factor formula for period (0,t{\displaystyle \Delta _{t}} expressed in years, and rate {\displaystyle r_{t}} for this period being {\displaystyle DF(0,t)=e^{-r_{t}\,\Delta _{t}}}, the forward rate can be expressed in terms of discount factors:

{\displaystyle r_{1,2}={\frac {1}{t_{2}-t_{1}}}(\ln DF(0,t_{1})-\ln DF(0,t_{2}))}

{\displaystyle r_{1,2}} is the forward rate between time {\displaystyle t_{1}} and time {\displaystyle t_{2}},

{\displaystyle r_{k}} is the zero-coupon yield for the time period {\displaystyle (0,t_{k})}, (k = 1,2).

## Related instruments

• Forward rate agreement
• Floating rate note

• Forward price
• Spot rate

## References

1. ^ Fabozzi, Vamsi.K (2012), The Handbook of Fixed Income Securities (Seventh ed.), New York: kvrv, p. 148, ISBN 0-07-144099-2.

### Covered interest arbitrage

Covered interest arbitrage is an arbitrage trading strategy whereby an investor capitalizes on the interest rate differential between two countries by using a forward contract to cover (eliminate exposure to) exchange rate risk.[1] Using forward contracts enables arbitrageurs such as individual investors or banks to make use of the forward premium (or discount) to earn a riskless profit from discrepancies between two countries’ interest rates.[2] The opportunity to earn riskless profits arises from the reality that the interest rate parity condition does not constantly hold. When spot and forward exchange rate markets are not in a state of equilibrium, investors will no longer be indifferent among the available interest rates in two countries and will invest in whichever currency offers a higher rate of return.[3] Economists have discovered various factors which affect the occurrence of deviations from covered interest rate parity and the fleeting nature of covered interest arbitrage opportunities, such as differing characteristics of assets, varying frequencies of time series data, and the transaction costs associated with arbitrage trading strategies.

## Mechanics of covered interest arbitrage

A visual representation of a simplified covered interest arbitrage scenario, ignoring compounding interest. In this numerical example the arbitrageur is guaranteed to do better than would be achieved by investing domestically.

An arbitrageur executes a covered interest arbitrage strategy by exchanging domestic currency for foreign currency at the current spot exchange rate, then investing the foreign currency at the foreign interest rate. Simultaneously, the arbitrageur negotiates a forward contract to sell the amount of the future value of the foreign investment at a delivery date consistent with the foreign investment’s maturity date, to receive domestic currency in exchange for the foreign-currency funds.[4]

Suppose the share price rises to $115. The net value is now only$5 (the previous net value of $20 minus the share’s$15 rise in price), so, to maintain the broker’s minimum margin, Jane needs to increase this net value to $10 or more, either by buying the share back or depositing additional cash. Short selling refers to the selling of securities that the trader does not own, borrowing them from a broker, and using the cash as collateral. This has the effect of reversing any profit or loss made on the securities. The initial cash deposited by the trader, together with the amount obtained from the sale, serve as collateral for the loan. The net value—the difference between the cash amount and the value of loan security—is initially equal to the amount of one’s own cash used. This difference has to stay above a minimum margin requirement, the purpose of which is to protect the broker against a rise in the value of the borrowed securities to the point that the investor can no longer cover the loan. ## Types of margin requirements • The current liquidating margin is the value of a security’s position if the position were liquidated now. In other words, if the holder has a short position, this is the money needed to buy back; if they are long, it is the money they can raise by selling it. • The variation margin or mark to market is not collateral, but a daily payment of profits and losses. Futures are marked-to-market every day, so the current price is compared to the previous day’s price. The profit or loss on the day of a position is then paid to or debited from the holder by the futures exchange. This is possible, because the exchange is the central counterparty to all contracts, and the number of long contracts equals the number of short contracts. Certain other exchange traded derivatives, such as options on futures contracts, are marked-to-market in the same way. • The seller of an option has the obligation to deliver the underlying security associated with the option when it is exercised. To ensure they can fulfill this obligation, they have to deposit collateral. This premium margin is equal to the premium that they would need to pay to buy back the option and close out their position. • Additional margin is intended to cover a potential fall in the value of the position on the following trading day. This is calculated as the potential loss in a worst-case scenario. • SMA and portfolio margins offer alternative rules for U.S. and NYSE regulatory margin requirements.[clarification needed] ## Margin strategies Enhanced leverage is a strategy offered by some brokers that provides 4:1 or 6+:1 leverage. This requires maintaining two sets of accounts, long and short. Example 1 An investor sells a put option, where the buyer has the right to require the seller to buy his 100 shares in Universal Widgets S.A. at 90¢. He receives an option premium of 14¢. The value of the option is 14¢, so this is the premium margin. The exchange has calculated, using historical prices, that the option value will not exceed 17¢ the next day, with 99% certainty. Therefore, the additional margin requirement is set at 3¢, and the investor has to post at least 14¢ (obtained from the sale) + 3¢ = 17¢ in his margin account as collateral. Example 2 Futures contracts on sweet crude oil closed the day at$65. The exchange sets the additional margin requirement at $2, which the holder of a long position pays as collateral in his margin account. A day later, the futures close at$66. The exchange now pays the profit of $1 in the mark-to-market to the holder. The margin account still holds only the$2.
Example 3
An investor is long 50 shares in Universal Widgets Ltd, trading at 120 pence (£1.20) each. The broker sets an additional margin requirement of 20 pence per share, so £10 for the total position. The current liquidating margin is currently £60 “in favour of the investor”. The minimum margin requirement is now -£60 + £10 = -£50. In other words, the investor can run a deficit of £50 in his margin account and still fulfil his margin obligations. This is the same as saying he can borrow up to £50 from the broker.

## Initial and maintenance margin requirements

The initial margin requirement is the amount of collateral required to open a position. Thereafter, the collateral required until the position is closed is the maintenance requirement. The maintenance requirement is the minimum amount of collateral required to keep the position open and is generally lower than the initial requirement. This allows the price to move against the margin without forcing a margin call immediately after the initial transaction. When the total value of the collateral dips below the maintenance margin requirement, the position holder must pledge additional collateral to bring their total balance back up to or above the initial margin requirement. On instruments determined to be especially risky, however, either regulators, the exchange, or the broker may set the maintenance requirement higher than normal or equal to the initial requirement to reduce their exposure to the risk accepted by the trader. For speculative futures and derivatives clearing accounts, futures commission merchants may charge a premium or margin multiplier to exchange requirements. This is typically an additional 10%–25%.

## Margin call

The broker may at any time revise the value of the collateral securities (margin) after the estimation of the risk, based, for example, on market factors. If this results in the market value of the collateral securities for a margin account falling below the revised margin, the broker or exchange immediately issues a “margin call”, requiring the investor to bring the margin account back into line. To do so, the investor must either pay funds (the call) into the margin account, provide additional collateral, or dispose some of the securities. If the investor fails to bring the account back into line, the broker can sell the investor’s collateral securities to bring the account back into line.

If a margin call occurs unexpectedly, it can cause a domino effect of selling, which will lead to other margin calls and so forth, effectively crashing an asset class or group of asset classes. The “Bunker Hunt Day” crash of the silver market on Silver Thursday, March 27, 1980, is one such example. This situation most frequently happens as a result of an adverse change in the market value of the leveraged asset or contract. It could also happen when the margin requirement is raised, either due to increased volatility or due to legislation. In extreme cases, certain securities may cease to qualify for margin trading; in such a case, the brokerage will require the trader to either fully fund their position, or to liquidate it.

## Price of stock for margin calls

The minimum margin requirement, sometimes called the maintenance margin requirement, is the ratio set for:

• (Stock Equity − Leveraged Dollars) to Stock Equity
• Stock Equity being the stock price multiplied by the number of shares bought, and leveraged dollars being the amount borrowed in the margin account.
• E.g., An investor bought 1,000 shares of ABC company each priced at $50. If the initial margin requirement were 60%: • Stock Equity:$50 × 1,000 = $50,000 • Leveraged Dollars or amount borrowed: ($50 × 1,000) × (100% − 60%) = $20,000 The maintenance margin requirement uses the variables above to form a ratio that investors have to abide by in order to keep the account active. Assume the maintenance margin requirement is 25%. That means the customer has to maintain a net value equal to 25% of the total stock equity. That means they have to maintain net equity of$50,000 × 0.25 = $12,500. So at what price would the investor be getting a margin call? For stock price P the stock equity will be (in this example) 1,000P. • (Current Market Value − Amount Borrowed) / Current Market Value = 25% • (1,000P – 20,000) / 1000P = 0.25 • (1,000P – 20,000) = 250P • 750P =$20,000
• P = $20,000/750 =$26.66 / share

So if the stock price drops from $50 to$26.66, investors will be called to add additional funds to the account to make up for the loss in stock equity.

Alternatively, one can calculate P using {\displaystyle \textstyle P=P_{0}{\frac {(1-{\text{initial margin requirement}})}{(1-{\text{maintenance margin requirement}})}}} where P0 is the initial price of the stock. Using the same example to demonstrate this:

{\displaystyle P=\$50{\frac {(1-0.6)}{(1-0.25)}}=\$26.66.}

## Reduced margins

Margin requirements are reduced for positions that offset each other. For instance spread traders who have offsetting futures contracts do not have to deposit collateral both for their short position and their long position. The exchange calculates the loss in a worst-case scenario of the total position. Similarly an investor who creates a collar has reduced risk since any loss on the call is offset by a gain in the stock, and a large loss in the stock is offset by a gain on the put; in general, covered calls have less strict requirements than naked call writing.

## Margin-equity ratio

The margin-equity ratio is a term used by speculators, representing the amount of their trading capital that is being held as margin at any particular time. Traders would rarely (and unadvisedly) hold 100% of their capital as margin. The probability of losing their entire capital at some point would be high. By contrast, if the margin-equity ratio is so low as to make the trader’s capital equal to the value of the futures contract itself, then they would not profit from the inherent leverage implicit in futures trading. A conservative trader might hold a margin-equity ratio of 15%, while a more aggressive trader might hold 40%.

## Return on margin

Return on margin (ROM) is often used to judge performance because it represents the net gain or net loss compared to the exchange’s perceived risk as reflected in required margin. ROM may be calculated (realized return) / (initial margin). The annualized ROM is equal to

(ROM + 1)(1/trade duration in years) – 1

For example, if a trader earns 10% on margin in two months, that would be about 77% annualized

Annualized ROM = (ROM +1)1/(2/12) – 1

that is, Annualized ROM = 1.16 – 1 = 77%

Sometimes, return on margin will also take into account peripheral charges such as brokerage fees and interest paid on the sum borrowed. The margin interest rate is usually based on the broker’s call.

• Collateral management
• Credit default swap
• Leverage (finance)
• LIBOR
• MVA, the x-Valuation Adjustment related to Margin
• Portfolio margin
• Repurchase agreement
• Special memorandum account
• Short selling
• Badla system (Indian stock markets)

## References

1. Jump up to:a b Cundiff, Kirby R. (January 2007). “Monetary-Policy Disasters of the Twentieth Century”. The Freeman Online. Retrieved 10 February 2009.
2. ^ Rappoport, Peter; White, Eugene N. (March 1994). “Was the Crash of 1929 Expected”. The American Economic Review. United States: American Economic Association. 84 (1): 271–281. JSTOR 2117982.