# Interest rate swap

An interest rate swap (IRS) is a popular and highly liquid financial derivative instrument in which two parties agree to exchange interest rate cash flows, based on a specified notional amount from a fixed rate to a floating rate (or vice versa) or from one floating rate to another.[1] Interest rate swaps are commonly used for both hedging and speculating.

## Structure

Party A is currently paying floating rate, but wants to pay fixed rate. Party B is currently paying fixed rate, but wants to pay floating rate. By entering into an interest rate swap, the net result is that each party can swap their existing obligation for their desired obligation.

In an interest rate swap, each counterparty agrees to pay either a fixed or floating rate denominated in a particular currency to the other counterparty. The fixed or floating rate is multiplied by a notional principal amount (say, USD 1 million) and an accrual factor given by the appropriate day count convention. When both legs are in the same currency, this notional amount is typically not exchanged between counterparties, but is used only for calculating the size of cashflows to be exchanged. When the legs are in different currencies, the respective notional amounts are typically exchanged at the start and the end of the swap.

The most common interest rate swap is one where one counterparty A pays a fixed rate (the swap rate) to counterparty B, while receiving a floating rate indexed to a reference rate (such as LIBOR or EURIBOR). By market convention, the counterparty paying the fixed rate is called the "payer" (while receiving the floating rate), and the counterparty receiving the fixed rate is called the "receiver" (while paying the floating rate).

A pays fixed rate to B (A receives floating rate)

B pays floating rate to A (B receives fixed rate)

Currently, A borrows from Market @ LIBOR +1.5%. B borrows from Market @ 8.5%.

Consider the following swap in which Party A agrees to pay Party B periodic fixed interest rate payments of 8.65%, in exchange for periodic variable interest rate payments of LIBOR + 70 bps (0.70%) in the same currency. Note that there is no exchange of the principal amounts and that the interest rates are on a "notional" (i.e. imaginary) principal amount. Also note that the interest payments are settled in net (e.g. Party A pays (LIBOR + 1.50%)+8.65% - (LIBOR+0.70%) = 9.45% net). The fixed rate (8.65% in this example) is referred to as the swap rate.[2]

At the point of initiation of the swap, the swap is priced so that it has a net present value of zero. If one party wants to pay 50 bps above the par swap rate, the other party has to pay approximately 50bps over LIBOR to compensate for this.

## Types

Normally the parties do not swap payments directly, but rather each sets up a separate swap with a financial intermediary such as a bank. In return for matching the two parties together, the bank takes a spread from the swap payments (in this case 0.30% compared to the above example)

Being OTC instruments, interest rate swaps can come in a huge number of varieties and can be structured to meet the specific needs of the counterparties. For example, the legs of the swap can be in the same currency or in different currencies. The notional of the swap could be amortized over time. The reset dates of the floating rate could be non-regular, etc. However, in the interbank market, just a few, standardized types are traded. They are listed below. Each currency has its own standard market conventions regarding the frequency of payments, the day count conventions and the end-of-month rule.[3]

### Fixed-for-floating rate swap, different currencies

Party P pays/receives fixed interest in currency A to receive/pay floating rate in currency B indexed to X on a notional N at an initial exchange rate of FX for a tenure of T years. For example, you pay fixed 5.32% on the USD notional 10 million quarterly to receive JPY 3M (TIBOR) monthly on a JPY notional 1.2 billion (at an initial exchange rate of USD/JPY 120) for 3 years. For nondeliverable swaps, the USD equivalent of JPY interest will be paid/received (according to the FX rate on the FX fixing date for the interest payment day). No initial exchange of the notional amount occurs unless the Fx fixing date and the swap start date fall in the future.

Fixed-for-floating swaps in different currencies are used to convert a fixed rate asset/liability in one currency to a floating rate asset/liability in a different currency, or vice versa. For example, if a company has a fixed rate USD 10 million loan at 5.3% paid monthly and a floating rate investment of JPY 1.2 billion that returns JPY 1M Libor +50bps monthly, and wants to lock in the profit in USD as they expect the JPY 1M Libor to go down or USDJPY to go up (JPY depreciate against USD), then they may enter into a Fixed-Floating swap in different currency where the company pays floating JPY 1M Libor+50bps and receives 5.6% fixed rate, locking in 30bps profit against the interest rate and the fx exposure.

### Floating-for-floating rate swap, same currency

Party P pays/receives floating interest in currency A Indexed to X to receive/pay floating rate in currency A indexed to Y on a notional N for a tenure of T years. For example, you pay JPY 1M LIBOR monthly to receive JPY 1M TIBOR monthly on a notional JPY 1 billion for 3 years or you pay EUR 3M EURIBOR quarterly to receive EUR 6M EURIBOR semi-annually. The second type of example, where the indexes are of the same type but with different tenors, are the most liquid and most traded same currency floating-for-floating swaps.

Floating-for-floating rate swaps are used to hedge against or speculate on the spread between the two indexes widening or narrowing. For example, if a company has a floating rate loan at JPY 1M LIBOR and the company has an investment that returns JPY 1M TIBOR + 30bps and currently the JPY 1M TIBOR = JPY 1M LIBOR + 10bps. At the moment, this company has a net profit of 40bps. If the company thinks JPY 1M TIBOR is going to come down (relative to the LIBOR) or JPY 1M LIBOR is going to increase in the future (relative to the TIBOR) and wants to insulate from this risk, they can enter into a float-float swap in same currency where they pay, say, JPY TIBOR + 30bps and receive JPY LIBOR + 35bps. With this, they have effectively locked in a 35bit/s profit instead of running with a current 40bps gain and index risk. The 5bps difference (w.r.t. the current rate difference) comes from the swap cost which includes the market expectations of the future rate difference between these two indices and the bid/offer spread which is the swap commission for the swap dealer.

Floating-for-floating rate swaps are also seen where both sides reference the same index, but on different payment dates, or use different business day conventions. This can be vital for asset-liability management. An example would be swapping 3M LIBOR being paid with prior non-business day convention, quarterly on JAJO (i.e. Jan, Apr, Jul, Oct) 30, into FMAN (i.e. Feb, May, Aug, Nov) 28 modified following・

### Floating-for-floating rate swap, different currencies

Party P pays/receives floating interest in currency A indexed to X to receive/pay floating rate in currency B indexed to Y on a notional N at an initial exchange rate of FX for a tenure of T years. The notional is usually exchanged at the start and at the end of the swap. This is the most liquid type of swap with different currencies. For example, you pay floating USD 3M LIBOR on the USD notional 10 million quarterly to receive JPY 3M TIBOR quarterly on a JPY notional 1.2 billion (at an initial exchange rate of USDJPY 120) for 4 years; at the start you receive the notional in USD and pay the notional in JPY and at the end you pay back the same USD notional (10 million) and receive back the same JPY notional (1.2 billion).

To explain the use of this type of swap, consider a US company operating in Japan. To fund their Japanese growth, they need JPY 10 billion. The easiest option for the company is to issue debt in Japan. As the company might be new in the Japanese market without a well known reputation among the Japanese investors, this can be an expensive option. Added on top of this, the company might not have appropriate debt issuance program in Japan and they might lack sophisticated treasury operation in Japan. To overcome the above problems, it can issue USD debt and convert to JPY in the FX market. Although this option solves the first problem, it introduces two new risks to the company:

• FX risk. If this USDJPY spot goes up at the maturity of the debt, then when the company converts the JPY to USD to pay back its matured debt, it receives less USD and suffers a loss.
• USD and JPY interest rate risk. If the JPY rates come down, the return on the investment in Japan might go down and this introduces an interest rate risk component.

The first exposure in the above can be hedged using long dated FX forward contracts but this introduces a new risk where the implied rate from the FX spot and the FX forward is a fixed rate but the JPY investment returns a floating rate. Although there are several alternatives to hedge both the exposures effectively without introducing new risks, the easiest and the most cost effective alternative would be to use a floating-for-floating swap in different currencies. In this,

### Fixed-for-fixed rate swap, different currencies

Party P pays/receives fixed interest in currency A to receive/pay fixed rate in currency B for a term of T years. For example, you pay JPY 1.6% on a JPY notional of 1.2 billion and receive USD 5.36% on the USD equivalent notional of 10 million at an initial exchange rate of USDJPY 120.

### Other variations

A number of other variations are possible, although far less common. Mostly tweaks are made to ensure that a bond is hedged "perfectly", so that all the interest payments received are exactly offset by the swap. This can lead to swaps where principal is paid on one or more legs, rather than just interest (for example to hedge a coupon strip), or where the balance of the swap is automatically adjusted to match that of a prepaying bond (such as RMBS Residential mortgage-backed security)

## Uses

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

### Speculation

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 very popular due to the arbitrage opportunities they provide. Due to varying levels of creditworthiness in companies, there is often a positive quality spread differential which 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.

### British local authorities

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 controller of the commission, Howard Davies realised that the council had put all of its positions on interest rates going down; he sent a team in to investigate.

By January 1989 the commission obtained legal opinions from two Queen's Counsel. Although they did not agree, the commission preferred the opinion which made it ultra vires for councils to engage in interest rate swaps. Moreover interest rates had gone up from 8% to 15%. The auditor and the commission then went to court and had the contracts declared illegal (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, although Hammersmith was unusual in betting all one way.[4] This left a train of cases, where generally the banks lost their claims for compound interest on debts to councils, finalised in Westdeutsche Landesbank Girozentrale v Islington London Borough Council.[5]

## Valuation and pricing

Further information: Rational_pricing § Swaps

The valuation of vanilla swaps was often done using the so-called textbook formulas using a unique curve in each currency. Some early literature described some incoherence introduced by that approach and multiple banks were using different techniques to reduce them. It became even more apparent with the 2007–2012 global financial crisis that the approach was not appropriate. The now standard pricing framework is the multi-curves framework.

The present value of a plain vanilla (i.e. fixed rate for floating rate) swap can be computed by determining the present value (PV) of the fixed leg and the floating leg.

The value of the fixed leg is given by the present value of the fixed coupon payments known at the start of the swap, i.e.

$PV_\text{fixed} = N \times C \times \sum_{i=1}^n \left( \tilde \delta_i \times P^D(\tilde t_i) \right)$

where C is the swap rate, n is the number of fixed payments, N is the notional amount, $\tilde \delta_i$ is the accrual factor according to the day count convention for the fixed rate period and $P^D(\tilde t_i)$ is the discount factor for the payment time $\tilde t_i$.

The value of the floating leg is given by the present value of the floating coupon payments determined at the agreed dates of each payment. However, at the start of the swap, only the actual payment rates of the fixed leg are known in the future, whereas the forward rates are unknown. The forward rate for each floating payment date is calculated using the forward curves. The forward rate for the period $[\tilde t_{j-1}, t_j]$ with accrual factor $\delta_i$ is given by

$F_j = \frac{1}{\delta_j}\left( \frac{P^I(t_{j-1})}{P^I(t_j)} - 1 \right)$

where I is the market index, such as USD LIBOR, and $P^I(t_j)$ is the discount factor associated to the relevant forward curve. The value of the floating leg is given by the following:

$PV_\text{float} = N \times \sum_{j=1}^m ( F_j \times \delta_j \times P^D(t_j) )$

where m is the number of floating payments, $\delta_i$ is the accrual factor according to the floating leg day count convention.

The fixed rate offered in the swap is the rate which values the fixed rates payments at the same PV as the variable rate payments using today's forward rates, i.e.:

$C = \frac{PV_\text{float}}{\sum_{i=1}^n ( N \times \tilde\delta_i \times P^D(\tilde t_i))}$[6]

Therefore, at the time the contract is entered into, there is no advantage to either party, i.e.,

$PV_\text{fixed} = PV_\text{float} \,$

Thus, the swap requires no upfront payment from either party.

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.

## Risks

Interest rate swaps expose users to interest rate risk and credit risk.

• Market Risk: A typical swap consists of two legs, one fixed, the other floating. The risks of these two component will naturally differ. Newcomers to market finance may think that the risky component is the floating leg, since the underlying interest rate floats, and hence, is unknown. This first impression is wrong. The risky component is in fact the fixed leg and it is very easy to see why this is so.[7]

(Comment: the above comment is not entirely accurate. Normally people will assume a hypothetical notional exchange at the end. After this hypothetical assumption, the swap can be understood as a floating rate bond vs a fixed rate bond. The risks on the floating bond side are small compared to the risks from the fix rate bond side. However, without this hypothetical notional exchange at the end, purely the floating cash flow from the coupon payments will have higher risks than the cash flow from the fixed coupon payments. Both understanding have their merit and fully understand these two views are important to see the risks, particularly for floating floating swaps)

The discussion of pricing interest rate swaps illustrated an important point. Regardless of what happens to future Libor rates, the value of a rolling deposit or FRN always equals the notional amount N at the reset dates. Between the reset dates this value may be different from N, but the discrepancy cannot be very large since the δ will be 3 or 6 months. Interest rate fluctuations have minimal effect on the values of fixed instruments with such maturities. In other words, the value of the floating leg changes very little during the life of a swap.

On the other hand the fixed leg of a swap is equivalent to a coupon bond and fluctuations of the swap rate may have major effects on the value of the future fixed payments.

• Credit risk on the swap comes into play if the swap is in the money or not. If one of the parties is in the money, then that party faces credit risk of possible default by another party. However, when the swap is negotiated through an intermediary financial institution, usually the intermediary assumes the default risk in exchange for a fixed percentage of the transaction (the bid-ask spread). In an intermediated swap, the two parties are not typically even aware of the identity of the second party to the transaction, making a quantification of the other party's credit risk not only irrelevant, but impossible.

## Market size

The Bank for International Settlements reports that interest rate swaps are the largest component of the global OTC derivative market. The notional amount outstanding as of June 2009 in OTC interest rate swaps was $342 trillion, up from$310 trillion in Dec 2007. The gross market value was $13.9 trillion in June 2009, up from$6.2 trillion in Dec 2007.

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

## References

1. ^ "Interest Rate Swap". Glossary. ISDA.
2. ^ "Interest Rate Swap" by Fiona Maclachlan, The Wolfram Demonstrations Project.
3. ^ "Interest Rate Instruments and Market Conventions Guide" Quantitative Research, OpenGamma, 2012.
4. ^ 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.
5. ^ [1996] UKHL 12, [1996] AC 669
6. ^ "Understanding interest rate swap math & pricing". California Debt and Investment Advisory Commission. January 2007. Retrieved 2007-09-27.
7. ^ http://chicagofed.org/webpages/publications/understanding_derivatives/index.cfm
• Pricing and Hedging Swaps, Miron P. & Swannell P., Euromoney books 1991

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

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

Multi-curves framework:

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