FIXED-INCOME SECURITIES Chapter 10 Swaps
Outline Terminology Convention Quotation Uses of Swaps Pricing of Swaps Non Plain Vanilla Swaps
Terminology Definition Agreement between two parties They exchange interest payments Computed on a notional principal Principal is not exchanged Classic swap One side pays a fixed rate Counterpart pays floating rate Floating rate is usually Libor Rate is reset at every payment date
Terminology Different Kinds of Swaps Swap contract come under many different shapes and forms Standard, or plain vanilla, swaps Asset swap Accrediting swaps Amortizing swaps Roller coaster swaps «Basis» swaps Zero-coupon swaps Yield-curve swap Forward start swaps Constant maturity swap (CMS) We first focus on plain vanilla contracts
Terminology Standard Swaps Plain vanilla contracts Exchanging a fixed leg whose payments depend on a fixed rate for a floating leg whose payments depends on a floating rate in which the notional principal remains constant throughout the life of the contract and where the maturity of the variable-rate index is identical to the payment frequency of the floating-leg flows.
Terminology Example Example Current date: 01/01/01 6-months Euribor swap with 2 years maturity, fixed rate F and notional amount 100,000 euros Schedule of payments 07/01/01 01/01/02 07/01/02 01/01/03 100,000*F 100,000*F -100,000*E(01/01/01) -100,000*E(07/01/01) -100,000*E(01/01/02) -100,000*E(07/01/02) where E(t) is the 6 months Euribor rate at month t payed at month t+6
Convention Buyer and Seller Every 6 months, and prorated for the period, the borrower Receives the 6-month Euribor observed 6 months earlier multiplied by the notional Pays a fixed rate F every year multiplied by the notional In this example, the swap is structured so that the buyer receives the floating leg and pay the fixed leg Of course, the opposite swap may be structured in which the buyer receives the fixed leg and pays the floating leg
Convention Principal Amount Principal amount A swap is composed of two legs a fixed leg whose payments depends on a fixed rate and a floating leg whose payments depends on a floating rate The notional, or principal, amount allows one to calculate the exact amount of the different payments on the two legs of the swap Example Consider a 3-year swap exchanging the 1-year Libor for the fixed 5% The notional principal is $10 million Payments on the two legs are annual Then each year, the amount paid on the fixed leg is equal to 5%x10 million = $500,000 The amount received on the floating leg is the 1-year Libor multiplied by $10 million
Convention Rates Maturity date: date of termination of the swap contract Frequency Payments on the fixed-leg take place either annually or semi-annually (e.g., in the US) Payments on the floating leg match the maturity of the reference rate (e.g., 4 times a year if the reference rate is a 3 months rate) Rate and payments The floating rate for each period is fixed at the start of the period The first interest payment of the swap is known in advance by both parties Note that even if both parties pay and receive interest payments, at a payment date only the net difference between the two interest payments change hands
Pricing of Swaps Basic Principles Exchange of a fixed-rate (F) for a floating-rate (V) security Initially, both have the same value Otherwise it would not be a fair deal F 0 V 0 Later on, prices can differ depending on the evolution of the term structure Fixed-rate notes have longer duration => a rise (decline) in interest rates tends to lower (increase) the value of the fixed-leg more than that of the variable leg This raises (lowers) the value of the swap to the buyer and lowers (raises) the value of the swap to the seller Value of the swap (party that pays fixed): Ft V t
Pricing of Swaps Pricing the Fixed Leg We assume no risk of default and perfect knowledge of the term structure Value of F: present value of future payments (discounted at the spot rate) Problem with V: we know next payment but payments after that are unknown Trick: those payments will be at the prevailing market rate
Pricing of Swaps Pricing the Floating Leg We assume that the notional is also exchanged Then, the floating will pay notional plus market rate The present value of notional plus market rate is: => Notional The price of a floating rate note on each and every coupon date is equal to par
Pricing of Swaps Example Today is 1/1: remaining life of 9 months Receives 10% a year Semiannually paid coupons On 3/31 and 9/30 Pays 6-month LIBOR Notional principal of $1,000 Next payment based on LIBOR at 6% Term structure r 3m = 5% r 9m = 7%
Pricing of Swaps Example (continued) Fixed: 50 1050 F = + 1/4 3/4 (1+.05) (1+.07) = 1047.44 Floating: 30 1000 V = + 1/4 1/4 (1+.05) (1+.05) = 1017.51 Value: 1047.44-1017.51 = 29.93
Quotation Swap Rate For a given maturity, the convention for Quotes in the market is for the swap market maker to set the floating leg at Libor and then quote the fixed rate, called the swap rate, that makes the value of the swap equal to zero The swap rate is then the value of the fixed rate that makes the swap's fixed leg equal to its floating leg because the value of the swap is very simply the difference between the sum of the discounted cashflows of one leg and the sum of the discounted cashflows of the other leg (see above)
Quotation Example Example Consider a seven year 3-month Libor swap quoted by a market maker Floating-rate payer: pays 3-month Libor and receives fixed rate of 6% Fixed-rate payer: pays fixed rate of 6.05% and receive 3-month Libor The bid price quoted by the market maker is 6% to pay the fixedrate and the ask price to receive the fixed rate is 6.05%
Quotation Swap Spread A swap is also quoted as a swap spread The swap spread of a swap with a given maturity is equal to the difference between the fixed rate of the swap and the benchmark treasury bond yield of the same maturity It is expressed as a number of basis points Example: a seven year 3-month Libor swap A market maker quotes 45-50 Means that he is willing to enter a swap paying fixed 45 points above the seven-year benchmark bond yield and receiving the Libor And receiving fixed 50 basis points above the seven-year bond yield and paying the Libor
Quotation (US)
Uses of Swaps Motivation Swaps may be used to (1) Optimize the financial conditions of a debt (2) Convert the financial conditions of a debt (3) Create new synthetic assets (4) Hedge a bond or another fixed-income security against any change of the yield curve (1) and (3) are discussed below (4) is related to dynamic hedging (2) To finance their needs Most of the firms issue long-term maturity fixed bonds because of the large liquidity of these bonds A treasurer may anticipate a decrease of rates and wish to transform its debt at a fixed rate into a floating-rate debt to take profit of the future decrease of rates Enter a swap in which the firm will pay the floating and receives the fixed
Uses of Swaps (1): Comparative Advantage Fixed Floating A 8% 6-month LIBOR +.5% B 10% 6-month LIBOR + 1% A would prefer floating and B fixed A has comparative advantage at fixed 200 basis points better than B for fixed 50 basis points better than B for floating A borrows at fixed and B at floating They enter a swap
Uses of Swaps Comparative Advantage (Cont ) Terms of the swap A will pay 6-month Libor B will pay 8% After the swap A pays: Libor + 8% - 8% = Libor B pays: 8% + Libor + 1% - Libor = 9% Advantage A saves 50 basis points (pays Libor instead of Libor +.5%) B saves 100 basis points (pays 9% instead of 10%) Other agreements are possible
Uses of Swaps (3): Create New Assets Swaps may be used to create new assets that do not exist in the market The transaction is called an asset swap Example A firm with a rating BBB has issued bonds with a 10% fixed coupon and maturity 4 years An investor likes the coupon paid by this firm but in the same time anticipates a rise of short-term rates May create a synthetic bond of this firm that delivers a 1-year Libor coupon plus a margin For that, buy the 10% fixed coupon bond with maturity 4 years and enter a swap where the investor receives 1-year Libor and pays the fixed Assume the swap rate for such a swap quoted by the market is 6% (default-free rate) The synthetic bond delivers 1-year Libor + 4%
Uses of Swaps Institutional Aspects Value: over 17 trillion Very efficient market with low spreads Mostly commercial banks Mostly unregulated No secondary market Need the counterpart to close Risk of default is asymmetric
INTEREST RATE SWAPS Currency(1) 1996 1997 % Change 1997 Amounts Full Year Full Year as US $ Equiv US $ Equiv % of Total US Dollar 4,213,547 4,386,963 4% 25.7% Australian Dollar 142,771 198,524 39% 1.2% Belgian Franc 109,459 158,040 44% 0.9% British Sterling 817,646 1,034,098 26% 6.1% Canadian Dollar 230,944 359,111 55% 2.1% Danish Krone 55,910 105,079 88% 0.6% Deutschemark 1,821,076 2,593,347 42% 15.1% Dutch Guilder 105,533 139,901 33% 0.8% Euro Currency Unit 149,907 166,062 11% 1.0% French Franc 1,669,580 2,518,697 51% 14.8% Hong Kong Dollar 64,215 76,959 20% 0.5% Italian Lira 564,353 952,960 69% 5.6% Japanese Yen 2,832,659 2,899,832 2% 17.0% New Zealand 19,081 29,421 54% 0.2% Spanish Peseta 224,679 326,922 46% 1.9% Swedish Krona 124,561 137,446 10% 0.8% Swiss Franc 288,012 327,581 14% 1.9% Other Currencies (2) 244,240 656,163 169% 3.8% TOTALS(3) $13,678,173 $17,067,106 25% 100.00% 1.All Amounts in US $ whole millions 2.Other Currencies include all swaps not included in the specific currency categories listed. 3.Numbers may not foot exactly due to rounding. Source: ISDA.
Non Plain Vanilla Swaps Accrediting, Amortizing, Roller Coaster Swaps In a plain vanilla swap, the notional principal remains unchanged during the life of the swap Thus it is referred as a bullet swap On the contrary An accrediting swap is one in which the notional principal increases in amount over time An amortizing swap is one in which the notional principal decreases in a predetermined way over the life of the swap A roller coaster swap is one in which the notional principal can rise or fall from period to period Example: One year ago, a firm issued a 1-year Libor amortizing bond with maturity 5 years and $10 million notional principal The notional principal of the bond is equally amortized so that each year $2 million is paid off
Non Plain Vanilla Swaps Basis Swaps and Differential Swaps A basis swap is a floating-for-floating interest rate swap That exchanges floating rate of two different markets or that exchanges the same floating rate but with different maturities or that exchanges floating rate of two different markets and with different maturities Example The 6-month Libor exchanged for the 3-month CD rate is an example of basis swap The 6-month Libor exchanged for the 1-month Libor is another example of basis swap When one of the legs is calculated in a different currency, the basis swap is called a differential swap
Non Plain Vanilla Swaps Constant Maturity Swaps A constant maturity swap is a floating-for-floating interest rate swap exchanging a Libor rate for a particular swap rate A constant maturity treasury swap is a floating-forfloating interest rate swap exchanging a Libor rate for a particular government bond rate Example The 3-month Libor exchanged for the 10-year swap rate The 6-month Libor exchanged for the 5-year government bond rate It is also possible to exchange a constant swap rate against a constant government bond rate
Non Plain Vanilla Swaps Forward-Start Swaps A forward start swap allows the counterparties to initiate it at a specified deferred date Firms typically use this kind of swap when they want to fix a hedge or cost of borrowing for a specified period starting in the future Example: We are now on 05/06/01 One year and a half ago a firm issued a 3-month Libor floating rate debt with maturity 5 years The firm now anticipates a rise in the 3-month Libor but only in six months Solution: enters today a 3-year forward start swap beginning on 05/12/01 where it will pay the fixed and receive the 3-month Libor
Non Plain Vanilla Swaps Yield-Curve Swaps A yield curve swap is a floating-for-floating interest rate swap where the counterparties exchange the difference between interest rates at two points on a given yield curve This swap enables a firm to make bets on a spread between two rates of a given yield curve Example The 6-month T-bill rate exchanged for the 5-year CMT rate is an example of yield curve swap Suppose on 05/08/01 that the yield curve is flat at 5% Firm A that anticipates for a certain time a decrease of the short-term rates and an increase of the medium-term rates enters into a 3-year swap where it pays the 6-month T-bill rate and receives the 5-year CMT on a yearly basis One year later, the 6-month T-Bill rate is 4% as the 5-year CMT is 6%. Firm A has gained 2%.
Non Plain Vanilla Swaps Zero-Coupon Swaps A zero-coupon swap is a swap which allows a counterparty to exchange a fixed or floating index that delivers regular coupon for an index that delivers only one coupon at the beginning or at the end of the swap A zero-coupon swap is used to insure a rate for a given period because it avoids the problem of reinvesting coupons at future dates Example On 05/06/01, a 4-year zero-coupon swap exchanges a 7% fixed with an annual frequency for a unique F payment on 05/06/05