Introduction

• The birth of the over-the-counter 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 over-the-counter 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 cash-flow 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 risk-free 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 over-the-counter 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 five-year bond with a rate of interest specified as six-month LIBOR plus 0.5% per annum. (“Six-month LIBOR” means “LIBOR for a borrowing period of six months.”) The life of the bond is divided into ten periods each six-months in length. For each period the rate of interest is set at 0.5% per annum above the six-month 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 three-year 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 six-month 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 fixed-pate payer, Citigroup is the floating-rate 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 six-month LIBOR rate prevailing six months prior to September 8, 2017-that is, on March 8, 2017.

Illustration

• Suppose that the six-month 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 six-month LIBOR rate prevailing six months prior to March 8, 2018-that is, on September 8, 2017. Suppose that the six-month 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 floating-rate payments on a payment date are calculated using the six-month 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 floating-rate bond where the interest rate is six-month LIBOR. The cash flows in the timeline for Citigroup are the cash flows from a short position in a fixed-rate bond. This shows that the swap can be regarded as the exchange of a fixed-rate bond for a floating-rate bond. Apple, is long a floating-rate bond and short a fixed-rate bond. Citigroup is long a fixed-rate bond and short a floating-rate bond. On a floating-rate 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.

Using The Swap To Transform A Liability

• For Apple, the swap could be used to transform a floating-rate loan into a fixed-rate 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:

1. It pays LIBOR plus 0.1% to its outside lenders.
2. It receives LIBOR under the terms of the swap.
3. 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 fixed-rate loan into a floating-rate 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:

1. It pays 3.2% to its outside lenders.
2. It pays LIBOR under the terms of the swap.
3. It receives 2.97% under the terms of the swap.

Using The Swap To Transform Asset

• 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:

1. It receives 2.7% on the bonds.
2. It receives LIBOR under the terms of the swap.
3. 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:

1. It receives LIBOR minus 20 basis points on its investment.
2. It pays LIBOR under the terms of the swap.
3. 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 Comparative-Advantage 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 fixed-rate markets, whereas other companies have a comparative advantage when borrowing in floating-rate 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 fixed-rate loan into a floating-rate 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 six-month 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 floating-rate markets, whereas AAA Corp appears to have a comparative advantage in fixed-rate markets. It is this apparent anomaly that can lead to a swap being negotiated. AAA Corp borrows fixed-rate funds at 4% per annum. BBB Corp borrows floating-rate funds at LIBOR plus 0.6% per annum. They then enter into a swap agreement to ensure that AAA Corp ends up with floating-rate funds and BBB Corp ends up with fixed-rate 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 six-month 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:

1. It pays 4% per annum to outside lenders.
2. It receives 4.35% per annum from BBB Corp.
3. 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:

1. It pays LIBOR+0.6% per annum to outside lenders.
2. It receives LIBOR from AAA Corp.
3. 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 fixed-rate 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 fixed-rate markets, and
  
  is the difference between the interest rates facing the two companies in floating-rate 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 comparative-advantage 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 five-year rates. The floating rates available in floating-rate markets are shorter term rates. In the floating-rate 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 fixed-rate 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 five-year rates is greater than the spread between the six-month 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 floating-rate bond and a short position in a fixed-rate 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 floating-rate bond and a short position in a fixed-rate 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:
    1. Calculate forward rates for each of the LIBOR rates that will determine swap cash flows.
    2. Calculate the swap cash flows on the assumption that LIBOR rates will equal forward rates.
    3. Discount the swap cash flows at the risk-free 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 6-month 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 risk-free rates with continuous compounding for maturities of 2 months, 8 months, and 14 months are given in this table. The 6-month 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 6-month, 12-month, and 18-month rates (with continuous compounding) are 3.8%, 4.3%, and 4.6%, respectively. Suppose the two-year swap rate is 5%.

Fixed-For-Fixed Currency Swaps

• A fixed-for-fixed 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.

Fixed-For-Fixed Currency Swaps – Example

• Consider a hypothetical five-year 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 fixed-for-fixed 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

Use Of A Currency Swap To Transform Liabilities And Assets

• 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 UK-denominated investment. The swap has the effect of transforming the U.S. investment into a
  
  investment in the U.K. yielding 4%.

Valuation Of Fixed-For-Fixed Currency Swaps

• Like plain vanilla interest rate swaps, we can use the following two methods to value currency swaps.

1. Bond Method
2. FRA Method

Currency Swaps Valuation In Terms Of Bond Prices

• A fixed-for-fixed 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 fixed-for-fixed 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 fixed-for-fixed 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:

1. Fixed-for-floating where a floating interest rate in one currency is exchanged for a fixed interest rate in another currency
2. Floating-for-floating 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 fixed-for-floating 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 6-month 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 floating-for floating interest rate swaps (known as basis swaps) are negotiated. For example, the three-month CP rate plus 10 basis points might be exchanged for three-month 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 LIBOR-in-arrears 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 step-up 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 six-month LIBOR applied to a certain principal for the 10-year 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 10-year 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 three-month LIBOR observed in the United States is exchanged for three-month 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 six-month 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 pre-agreed 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.