Introduction
 The birth of the overthecounter swap market can be traced to a currency swap negotiated between IBM and the World Bank in 1981. The World Bank had borrowings denominated in U.S. dollars while IBM had borrowings denominated in German deutsche marks and Swiss francs. The World Bank agreed to make interest payments on IBM's borrowings while IBM in return agreed to make interest payments on the World Bank's borrowings.
 A swap is an overthecounter derivatives agreement between two companies to exchange cash flows in the future. The agreement defines the dates when the cash flows are to be paid and the way in which they are to be calculated. Usually the calculation of the cash flows involves the future value of an interest rate, an exchange rate, or other market variable. A forward contract can be viewed as a simple example of a swap. Whereas a forward contract is equivalent to the exchange of cash flows on just one future date, swaps typically lead to cashflow exchanges taking place on several future dates.
 When valuing swaps, a “discount rate” is required for cash flows. Prior to the 2008 crisis, LIBOR was used as a proxy for the riskfree discount rate. Since the 2008 credit crisis, the market has switched to using the OIS rate for discounting.
Mechanics Of Interest Rate Swaps
 By far the most common overthecounter derivative is a “plain vanilla” interest rate swap. In this a company agrees to pay cash flows equal to interest at a predetermined fixed rate on a notional principal for a number of years. In return, it receives interest at a floating rate on the same notional principal for the same period of time.
 The floating rate in most interest rate swap agreements is LIBOR. To understand how it is used, consider a fiveyear bond with a rate of interest specified as sixmonth LIBOR plus 0.5% per annum. (“Sixmonth LIBOR” means “LIBOR for a borrowing period of six months.”) The life of the bond is divided into ten periods each sixmonths in length. For each period the rate of interest is set at 0.5% per annum above the sixmonth LIBOR rate observed at the beginning of the period. Interest is paid at the end of the period.
Mechanics Of Interest Rate Swaps – Illustration
 Consider a hypothetical threeyear swap initiated on March 8, 2017, between Apple and Citigroup. Assume that Apple agrees to pay to Citigroup an interest rate of 3% per annum on a notional principal of $100 million, and in return Citigroup agrees to pay Apple the sixmonth LIBOR rate on the same notional principal. The principal itself is not exchanged. This is why it is termed the notional principal. Apple is the fixedpate payer, Citigroup is the floatingrate payer. Payments are to be exchanged every six months and that the 3% interest rate is quoted with semiannual compounding. The swap is shown in this figure.
 The first exchange of payments would take place on September 8, 2017, six months after the initiation of the agreement. Apple would pay Citigroup $1.5 million. This is the interest on the $100 million principal for six months at a rate of 3% per year. Citigroup would pay Apple interest on the $100 million principal at the sixmonth LIBOR rate prevailing six months prior to September 8, 2017that is, on March 8, 2017.
Illustration
 Suppose that the sixmonth LIBOR rate on March 8, 2017, is 2.2%.
Citigroup pays Apple
Note that there is no uncertainty about this first exchange of payments because it is determined by the LIBOR rate at the time the contract is agreed to.
 The second exchange of payments would take place on March 8, 2018, one year after the initiation of the agreement. Apple would pay $1.5 million to Citigroup. In return, Citigroup would pay interest on the $100 million principal to Apple at the sixmonth LIBOR rate prevailing six months prior to March 8, 2018that is, on September 8, 2017. Suppose that the sixmonth LIBOR rate on September 8, 2017, proves to be 2.8%.
Citigroup pays Apple

In total, there are six exchanges of payment on the swap. The fixed payments are always $1.5 million. The floatingrate payments on a payment date are calculated using the sixmonth LIBOR rate prevailing six months before the payment date. An interest rate swap is generally structured so that only net cash flows are exchanged. In our example,
 Apple would pay Citigroup _________________________________on September 8, 2017
 Apple would pay Citigroup __________________________________on March 8, 2018
 This table provides a complete example of the payments made under the swap for one particular set of LIBOR rates that could occur. Cash flows are shown from the perspective of Apple. Note that the $100 million principal is used only for the calculation of interest payments. The principal itself is not exchanged. This is why it is termed the notional principal.
Date 
LIBOR Rate (%) 
Floating Cash Flow received

Fixed Cash Flow paid

Net Cash Flow 
Mar.8,2017 
2.20 



Sept.8,2017 
2.80 



Mar.8,2018 
3.30 



Sept.8,2018 
3.50 



Mar.8,2019 
3.60 



Sept.8,2019 
3.90 



 If the principal were exchanged at the end of the life of the swap, the nature of the deal would not be changed in any way. The principal is the same for both the fixed and floating payments. Exchanging $100 million for $100 million at the end of the life of the swap is a transaction that would have no financial value to either Apple or Citigroup. Let's draw a timeline of the cash flows with a final exchange of principal added in.
This provides an interesting way of viewing the swap. The cash flows in the timeline for Apple are the cash flows from a long position in a floatingrate bond where the interest rate is sixmonth LIBOR. The cash flows in the timeline for Citigroup are the cash flows from a short position in a fixedrate bond. This shows that the swap can be regarded as the exchange of a fixedrate bond for a floatingrate bond. Apple, is long a floatingrate bond and short a fixedrate bond. Citigroup is long a fixedrate bond and short a floatingrate bond. On a floatingrate bond, interest is generally set at the beginning of the period to which it will apply and is paid at the end of the period.
 For Apple, the swap could be used to transform a floatingrate loan into a fixedrate loan, as indicated in this figure. Suppose that Apple has arranged to borrow $100 million for three years at LIBOR plus 10 basis points (or LIBOR+0.1%) After Apple has entered the swap, it has three sets of cash flows:
 It pays LIBOR plus 0.1% to its outside lenders.
 It receives LIBOR under the terms of the swap.
 It pays 3% under the terms of the swap.
These three sets of cash flows net out to an interest rate payment of _________. Thus, for Apple the swap could have the effect of transforming borrowings at a floating rate of LIBOR plus 10 basis points into borrowings at a fixed rate of 3.1%.
 A company wishing to transform a fixedrate loan into a floatingrate loan would enter into the opposite swap. Suppose that Intel has borrowed $100 million at 3.2% for three years and wishes to switch to a floating rate linked to LIBOR. Like Apple it contacts Citigroup. We assume that it agrees to enter into the swap shown in this figure. It pays LIBOR and receives 2.97%.
 So after entering into the swap, its position would then be as indicated in this figure. It has three sets of cash flows:
 It pays 3.2% to its outside lenders.
 It pays LIBOR under the terms of the swap.
 It receives 2.97% under the terms of the swap.
 Swaps can also be used to transform the nature of an asset. Consider Apple in our example. Suppose that Apple owns $100 million in bonds that will provide interest at 2.7% per annum over the next three years. After Apple has entered into the swap discussed earlier, it is in the position shown in this figure. It has three sets of cash flows:
 It receives 2.7% on the bonds.
 It receives LIBOR under the terms of the swap.
 It pays 3% under the terms of the swap.
These three sets of cash flows net out to an interest rate inflow of ________________________. The swap has therefore transformed an asset earning 2.7% into an asset earning ____________________.
 Suppose that Intel has an investment of $100 million that yields LIBOR minus 20 basis points. After it has entered into the swap with Citigroup discussed earlier, it is in the position shown in this Figure. It has three sets of cash flows:
 It receives LIBOR minus 20 basis points on its investment.
 It pays LIBOR under the terms of the swap.
 It receives 2.97% under the terms of the swap.
These three sets of cash flows net out to an interest rate inflow of 2.77%. Thus, one possible use of the swap for Intel is to transform an asset earning LIBOR minus 20 basis points into an asset earning ____________________.
The ComparativeAdvantage Argument
 In the context of swaps, a comparative advantage is advantage that leads to company being treated more favorably in one debt market than in another debt market. Some companies, it is argued, have a comparative advantage when borrowing in fixedrate markets, whereas other companies have a comparative advantage when borrowing in floatingrate markets. To obtain a new loan, it makes sense for a company to go to the market where it has a comparative advantage. As a result, the company may borrow fixed when it wants floating, or borrow floating when it wants fixed. The swap is used to transform a fixedrate loan into a floatingrate loan, and vice versa.
rating; BBB Corp has a BBB credit rating. Assume that BBB Corp wants to borrow at a fixed rate of interest, whereas AAA Corp wants to borrow at a floating rate of interest linked to sixmonth LIBOR. Since BBB Corp has a worse credit rating than AAA Corp, it pays a higher rate of interest in both fixed and floating markets.
 A key feature of the rates offered to AAA Corp and BBB Corp is that the difference between the two fixed rates is greater than the difference between the two floating rates.
BBB Corp appears to have a comparative advantage in floatingrate markets, whereas AAA Corp appears to have a comparative advantage in fixedrate markets. It is this apparent anomaly that can lead to a swap being negotiated. AAA Corp borrows fixedrate funds at 4% per annum. BBB Corp borrows floatingrate funds at LIBOR plus 0.6% per annum. They then enter into a swap agreement to ensure that AAA Corp ends up with floatingrate funds and BBB Corp ends up with fixedrate funds. Let's assume (somewhat unreal istically) that AAA Corp and BBB Corp get in touch with each other directly. The sort of swap they might negotiate is shown in this figure. AAA Corp agrees to pay BBB Corp interest at sixmonth LIBOR on $10 million. In return, BBB Corp agrees to pay AAA Corp interest at a fixed rate of 4.35% per annum on $10 million.
 AAA Corp has three sets of interest rate cash flows:
 It pays 4% per annum to outside lenders.
 It receives 4.35% per annum from BBB Corp.
 It pays LIBOR to BBB Corp.
The net effect of the three cash flows is that AAA Corp pays _____________per annum. This is ________per annum less than it would pay if it went directly to floating rate markets.
 BBB Corp also has three sets of interest rate cash flows:
 It pays LIBOR+0.6% per annum to outside lenders.
 It receives LIBOR from AAA Corp.
 It pays 4.35% per annum to AAA Corp.
The net effect of the three cash flows is that BBB Corp pays _________________ per annum. This is ________________ per annum less than it would pay if it went directly to fixedrate markets.
Illustration

In this example, the swap has been structured so that the net gain to both sides is the same, 0.25%. This need not be the case. However, the total apparent gain from this type of interest rate swap arrangement is always
, where
is the difference between the interest rates facing the two companies in fixedrate markets, and
is the difference between the interest rates facing the two companies in floatingrate markets. In this case,
= 1.2% and
= 0.7%, so that the total gain is 0.5%.
 If the transaction between AAA Corp and BBB Corp were brokered by a financial institution, an arrangement such as that shown in this figure might result. In this case, AAA Corp ends up borrowing at __________________, BBB Corp ends up borrowing at _____________, and the financial institution earns a spread of ______________ per year. The gain to AAA Corp is _______________________; the gain to BBB Corp is __________________; and the gain to the financial institution is ___________. The total gain to all three parties is 0.5% as before.
Criticism Of The Comparative Advantage Argument

The comparativeadvantage argument in the context of swaps is open the following criticism –
 Why should the spreads between the rates offered to AAA Corp and BBB Corp be different in fixed and floating markets? Now that the interest rate swap market has been in existence for a long time, it can be reasonably expected that the differentials that allow for the comparative advantage in the first place, should be arbitraged away.
The reason that spread differentials appear to exist is due to the nature of the contracts available to companies in fixed and floating markets. The fixed rates available in fixed rate markets are fiveyear rates. The floating rates available in floatingrate markets are shorter term rates. In the floatingrate market, the lender usually has the opportunity to review the spread above LIBOR every time rates are reset. (In our example, rates are reset semiannually.) If the creditworthiness of AAA Corp or BBB Corp has declined, the lender has the option of increasing the spread over LIBOR that is charged. The providers of fixedrate financing do not have the option to change the terms of the loan in this way.
 The comparative advantage argument assumes floating rates will not adjust and converge, an assumption, which in practice does not hold up.
 Credit risk taken by AAA is ignored.
 The spreads between the rates offered to AAA Corp and BBB Corp are a reflection of the extent to which BBB Corp is more likely to default than AAA Corp. During the next six months, there is very little chance that either AAA Corp or BBB Corp will default. Default statistics show that on average the probability of a default by a company with a BBB credit rating increases faster than the probability of a default by a company with a AAA credit rating. This is why the spread between the fiveyear rates is greater than the spread between the sixmonth rates.
Valuation Of Interest Rate Swaps

An interest rate swap is worth close to zero when it is first initiated. But after that, its value may be positive or negative. Two methods can be used for valuation –

Bond Method Valuation – As already discussed, if two companies enter into an interest rate swap arrangement, then the position of one of the companies can be considered equivalent to a long position in floatingrate bond and a short position in a fixedrate bond. So the value of the swap for that company becomes
. For the other company (i.e. the counterparty), it's position can be considered equivalent to a long position in floatingrate bond and a short position in a fixedrate bond. So the value of the swap for that company becomes

FRA Method Valuation – Because an interest rate swap is nothing more than a portfolio of FRAs, it can also be valued by assuming that forward rates are realized. The procedure is:
 Calculate forward rates for each of the LIBOR rates that will determine swap cash flows.
 Calculate the swap cash flows on the assumption that LIBOR rates will equal forward rates.
 Discount the swap cash flows at the riskfree rate.
Valuation Of Interest Rate Swaps – Bond Method
 Suppose that some time ago a financial institution entered into a swap where it agreed to make semiannual payments at a rate of 5% per annum and receive 6month LIBOR on a notional principal of $100 million. The swap now has a remaining life of 1 year and 2 months. Payments will therefore be made at 2 months, 8 months and 14 months from today. The riskfree rates with continuous compounding for maturities of 2 months, 8 months, and 14 months are given in this table. The 6month LIBOR at the last coupon payment date was 4.5%. Find value of swap today.
t 
LIBOR

2m 
4.7% 
8m 
4.9% 
14m 
5.8% 
Valuation Of Interest Rate Swaps – Fra Method
 The floating rate bond resets itself at every coupon payment date.
A 
B 
C 
D 


T 
Fixed CF 
Floating CF 
PV factor 
PV of fixed CF 
PV of floating CF 
2m 





8m 





14m 





Calculating Discount Rates In A Plain Vanilla Swap
 The bootstrap method for calculating zero rates has already been covered in the chapter – Interest Rates. A similar method is used here. Suppose that the 6month, 12month, and 18month rates (with continuous compounding) are 3.8%, 4.3%, and 4.6%, respectively. Suppose the twoyear swap rate is 5%.
FixedForFixed Currency Swaps
 A fixedforfixed currency swap involves exchanging principal and interest payments at a fixed rate in one currency for principal and interest payments at a fixed rate in another currency.
 A currency swap agreement requires the principal to be specified in each of the two currencies. The principal amounts in each currency are usually exchanged at the beginning and at the end of the life of the swap. Usually the principal amounts are chosen to be approximately equivalent using the exchange rate at the swap's initiation. But when they are exchanged at the end of the life of the swap, their values may be quite different.
FixedForFixed Currency Swaps – Example

Consider a hypothetical fiveyear currency swap agreement between British Petroleum and Barclays entered into on February 1, 2017. We suppose that British Petroleum pays a fixed rate of interest of 3% in dollars to Barclays and receives a fixed rate of interest of 4% in British pounds (sterling) from Barclays. Interest rate payments are made once a year and the principal amounts are $15 million and
. This is termed a fixedforfixed currency swap because the interest rate in both currencies is fixed. The swap is shown in this figure. Initially, the principal amounts flow in the opposite direction to the arrows in this figure. The interest payments during the life of the swap and the final principal payment flow in the same direction as the arrows.
 At the outset of the swap, British Petroleum pays ___________ and receives __________. Each year during the life of the swap contract, British Petroleum receives _________________________and pays _______________________. At the end of the life of the swap, it pays __________________ and receives_________________. These cash flows are shown in this table. The cash flows to Barclays are the opposite to those shown here.
DATE 
Dollar Cash Flow (millions)

Sterling Cash Flow (millions)

Feb 1, 2017 


Feb 1, 2018 


Feb 1, 2019 


Feb 1, 2020 


Feb 1, 2021 


Feb 1, 2022 


Feb 1, 2023 


Cash Flows to British Petroleum in Currency Swap

A swap such as the one just considered can be used to transform borrowings in one currency to borrowings in another currency. Suppose that British Petroleum can borrow
at 4% interest. The swap has the effect of transforming this loan into one where it has borrowed $15 million at 3% interest. The initial exchange of principal converts the amount borrowed from sterling to dollars. The subsequent exchanges in the swap have the effect of swapping the interest and principal payments from sterling to dollars.

The swap can also be used to transform the nature of assets. Suppose that British Petroleum can invest $15 million to earn 3% in U.S. dollars for the next five years, but feels that sterling will strengthen (or at least not depreciate) against the dollar and prefers a UKdenominated investment. The swap has the effect of transforming the U.S. investment into a
investment in the U.K. yielding 4%.
Valuation Of FixedForFixed Currency Swaps
 Like plain vanilla interest rate swaps, we can use the following two methods to value currency swaps.
 Bond Method
 FRA Method
Currency Swaps Valuation In Terms Of Bond Prices

A fixedforfixed currency swap can also be valued in a straightforward way as the difference between two bonds. If we define
as the value in U.S. dollars of an outstanding swap where dollars are received and a foreign currency is paid, that is,
where
is the value, measured in the foreign currency, of the bond defined by the foreign cash flows on the swap and is the value of the bond defined by the domestic cash flows on the swap, and
is the spot exchange rate (expressed as number of dollars per unit of foreign currency). The value of a swap can therefore be determined from LIBOR rates in the two currencies, the term structure of interest rates in the domestic currency, and the spot exchange rate.
 Similarly, the value of a swap where the foreign currency is received and dollars are paid is
Currency Swaps Valuation In Terms Of Bond Prices – Example

Suppose that two companies, A and B, enter into a fixedforfixed currency swap with periodic payments annually. Initially, Company A pays a principal amount to B of $150 million, and B pays
100 million to A at the outset of the swap. Company A pays 5% in Euros () to Company B and receives 7% in US Dollars ($) from Company B. The initial exchange rate was $1.5 = 1 when the swap was exchanged. Find the value of the swap using Bond Method when the remaining life of the swap is 2 years and the exchange rate is now $1.3 = 1
Currency Swaps Valuation In Terms Of Fra – Example

Suppose that two companies, A and B, enter into a fixedforfixed currency swap with periodic payments annually. Initially, Company A pays a principal amount to B of $150 million, and B pays
100 million to A at the outset of the swap. Company A pays 5% in Euros () to Company B and receives 7% in US Dollars ($) from Company B. The initial exchange rate was $1.5 = 1 when the swap was exchanged. Find the value of the swap using FRA method when the remaining life of the swap is 2 years and the exchange rate is now $1.3 = 1
Other Currency Swaps
 Two other popular currency swaps are:
 Fixedforfloating where a floating interest rate in one currency is exchanged for a fixed interest rate in another currency
 Floatingforfloating where a floating interest rate in one currency is exchanged for a floating interest rate in another currency.
Credit Risk In A Swap
 When swaps and other derivatives are cleared through a central counterparty there is very little credit risk. But standard swap transactions between a nonfinancial corporation and a derivatives dealer can be cleared bilaterally, which means that both sides are then potentially subject to credit risk. Consider the bilaterally cleared transaction between Intel and Citigroup discussed earlier. This would be netted with all other bilaterally cleared derivatives between Intel and Citigroup. If Intel defaults when the net value of the outstanding transactions to Citigroup is greater than the collateral (if any) posted by Intel, Citigroup will incur a loss. Similarly, if Citigroup defaults when the net value of the outstanding transactions to Intel is greater than the collateral (if any) posted by Citigroup, Intel will incur a loss.
 It is important to distinguish between the credit risk and market risk to a financial institution in any contract. The credit risk arises from the possibility of a default by the counterparty when the value of the contract to the financial institution is positive. The market risk arises from the possibility that market variables such as interest rates and exchange rates will move in such a way that the value of a contract to the financial institution becomes negative. Market risks can be hedged by entering into offsetting contracts; credit risks are less easy to hedge.
Variations On The Standard Interest Rate Swap

In fixedforfloating interest rate swaps, LIBOR is by far the most common reference floating interest rate.
 Swaps where the tenor of LIBOR is one month, three months, and 12 months also trade regularly instead of the 6month LIBOR examples used throughout this chapter.
 The tenor on the floating side does not have to match the tenor on the fixed side.
 Floating rates such as commercial paper (CP) rate are occasionally used.
 Sometimes floatingfor floating interest rate swaps (known as basis swaps) are negotiated. For example, the threemonth CP rate plus 10 basis points might be exchanged for threemonth LIBOR with both being applied to the same principal. (This deal would allow a company to hedge its exposure when assets and liabilities are subject to different floating rates.)
 In a compounding swap, interest on one or both sides is compounded forward to the end of the life of the swap according to pre agreed rules and there is only one payment date at the end of the life of the swap. In a LIBORinarrears swap the LIBOR rate observed on a payment date is used to calculate the payment on that date. In an accrual swap, the interest on one side of the swap accrues only when the floating reference rate is in a certain range.
Variations On The Standard Interest Rate Swap (Self Study)

The principal in a swap agreement can be varied throughout the term of the swap to meet the needs of a counterparty.
 In an amortizing swap, the principal reduces in a predetermined way. (This might be designed to correspond to the amortization schedule on a loan.)
 In a stepup swap, the principal increases in a predetermined way. (This might be designed to correspond to drawdowns on a loan agreement.)
 Forward swaps (sometimes referred to as deferred swaps) where the parties do not begin to exchange interest payments until some future date are also sometimes arranged.
 Sometimes swaps are negotiated where the principal to which the fixed payments are applied is different from the principal to which the floating payments are applied.
 A constant maturity swap (CMS swap) is an agreement to exchange a LIBOR rate for a swap rate. An example would be an agreement to exchange sixmonth LIBOR applied to a certain principal for the 10year swap rate applied to the same principal every six months for the next five years. A constant maturity Treasury swap (CMT swap) is a similar agreement to exchange a LIBOR rate for a particular Treasury rate (e.g., the 10year Treasury rate).
Other Type Of Swaps (Self Study)
 Quantos – Sometimes a rate observed in one currency is applied to a principal amount in another currency. One such deal would be where threemonth LIBOR observed in the United States is exchanged for threemonth LIBOR in Britain with both principals being applied to a principal of 10 million British pounds. This type of swap is referred to as a diff swap or a Quanto.
 An equity swap is an agreement to exchange the total return (dividends and capital gains) realized on an equity index for either a fixed or a floating rate of interest. For example, the total return on the S&P 500 in successive sixmonth periods might be exchanged for LIBOR with both being applied to the same principal. Equity swaps can be used by portfolio managers to convert returns from a fixed or floating investment to the returns from investing in an equity index, and vice versa.
 Options – Sometimes there are options embedded in a swap agreement. For example, in an extendable swap, one party has the option to extend the life of the swap beyond the specified period. In a puttable swap, one party has the option to terminate the swap early. Options on swaps, or swaptions, are also available. These provide one party with the right at a future time to enter into a swap where a predetermined fixed rate is exchanged for floating.
 Commodity swaps are in essence a series of forward contracts on a commodity with different maturity dates and the same delivery prices.
 In a volatility swap, there are a series of time periods. At the end of each period, one side pays a preagreed volatility while the other side pays the historical volatility realized during the period. Both volatilities are multiplied by the same notional principal in calculating payments.
 Swaps are limited only by the imagination of financial engineers and the desire of corporate treasurers and fund mangers for exotic structures.