# Introduction to swaps

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1 Introduction to swaps Steven C. Mann M.J. Neeley School of Business Texas Christian University incorporating ideas from Teaching interest rate and currency swaps" by Keith C. Brown (Texas-Austin) and Donald J. Smith (Boston U.)

2 Swaps: contracts specifying exchange of cash flows: fixed price traded for floating price fixed rate traded for floating rate Swap structures include: interest rate swaps currency swaps commodity swaps vast assortment of variations quantro, basis, index differential, etc.

3 Interest rate swap exchange of fixed rate for a floating reference rate, or exchange of one variable rate for another. periodic net settlements made by comparing rates, adjusting for day-count conventions, then multiplying rate difference by notional principal: Settlement amount = notional principal times: (Fixed rate x #days ) - (Floating rate x #days ) year year where #days and year are determined by day-count convention

4 Day-count conventions: The number of days assumed to be in a month and in a year when calculating the interest accrued on a debt instrument. Treasury notes, bonds: actual over actual Corporate bonds: 30 over 360 money-market (Libor): actual over 360 (usual) actual days is actual number of days in period actual for years is either 365 or 366.

5 Interest rate swap "Buyer" and "Seller" payer of fixed rate is "buyer". Fixed rate payer "buys" floating rate (LIBOR), the fixed rate is the "price" Floating Rate (LIBOR) Fixed-Rate Payer Swap "Buyer" 6.75% Fixed Rate Fixed-Rate Receiver Swap "Seller"

6 Currency swaps exchange fixed (or floating) cash flows denominated in one currency for fixed (or floating) cash flows denominated in another currency. Examples: fixed \$ for fixed DM fixed \$ for floating DM floating \$ for fixed DM floating \$ for floating DM Usually principal is exchanged at origination and at maturity.

7 Fixed/floating \$/DM swap US \$ payer DM 7.25% Fixed (times DM 30 million) DM payer U.S. \$ LIBOR (times \$ 20 million) settlement dates: April 11, October 11 of each year rate set 2 days prior. payment: \$ is actual/360; DM is 30/360 Oct. 9, 1998, \$ LIBOR is 5.5%. Assume payment is in arrears on April 11, 1999: US\$ payer owes \$556, (=.055 x 182/360x \$20 million) DM payer owes DM 1,087,500 (=.0725 x 180/360 x DM 30 mm) Settlement on either gross basis or net based on spot \$/DM exchange rate on 4/11/99.

8 Swap Applications New Issue "Arbitrage" (lower borrowing costs) Access new markets lower borrowing costs (borrower) increase returns (investor) Financial restructuring (transform risks) Hedge exposures create synthetic portfolios

9 New issue "arbitrage" Straight Debt Swap-Driven Debt Structure Corporation Corporation LIBOR 6.75% Swap Counterparty 7.00% Fixed-Rate Note LIBOR % Floating-Rate Note Fixed- Income Investor Variable- Income Investor Cost of Funds with swap structure: 6.85%

10 Comments on "New issue arbitrage" Is the swap structure comparable to straight debt? not if straight debt is callable - straight debt rate is higher as payment for call option. Swap structures may have embedded options (credit triggers, e.g. Texaco & Banker's Trust) Swap structure should provide lower cost of funds, as swap contains counterparty credit risk: joint probability of default and replacement swap with higher fixed rate. Tax and accounting conventions may matter also.

11 Swaps to access new markets Straight Debt Swap-Driven Debt Structure Corporation Corporation DM 30,000, % Swap Counterparty 7.00% Fixed-rate Note in US \$ Fixed-rate note in DM 7.25% 6.85% on US\$20,000,000 US \$ Investor DM Investor Cost of Funds with swap structure: 6.85%

12 Swaps to restructure debt Straight Debt Swap-Driven Debt Structure Institutional Investor Institutional Investor LIBOR 6.85% Swap Counterparty 8.00% Fixed-Rate Note 8.00% Fixed-rate Note Debt Issuer Debt Issuer Asset return with swap S.Mann, structure: 1999 LIBOR %

13 Interpretations of swap contracts 1. Series of Forward Contracts useful to understand initial pricing interest rate swap is series of FRAs currency swap is series of FX forwards 2. Combination of Bonds useful for mark-to-market insights from duration and convexity 3. Combination of Options multiperiod put-call parity risk management insights

14 Forward Rate Agreements (FRAs) FRA is one-date interest rate swap usually LIBOR for fixed. cash-settled, OTC forward contract (no margin account, mark-to market) Settlement and maturity dates part of FRA name: 3x6 FRA is 3-month LIBOR, 3 months forward 6x9 FRA is 3-month LIBOR, 6 months forward 6x12 FRA is 6-month LIBOR, 6 months forward

15 FRAs Dates x 6 6 x 9 9 x 12 FRAs at Date 0 3 x 9 6 x 12 0 x 3 Cash Market at Date 0 0 x 6 0 x 9 0 x 12

16 FRA payoffs Payoff to long FRA holder (pays fixed) is: (Notional Principal) times: (LIBOR - FRA rate) x ( days/360) Example: Notional principal = \$10,000,000 3 x 6 FRA : 3 month forward, 90 day FRA FRA is 7.00 today. Assume that 3 months later, 3 month LIBOR is 7.10% Then Payoff to long is: \$10 million x ( )(90/360) = \$2500 Payoff per million on any forward 90 day FRA ( X x 3) is \$25 per basis point change in forward LIBOR

17 Pricing FRAs 0 x 3 rate is 5.00 % ; 0 x 6 rate is 5.25 % What is 3 x 6 FRA rate? Unless (1+ r (0x6) ).5 = (1+ r (0x3) ).25 (1+ r (3x6) ).25 there is arbitrage. So (1+ r (0x6) ).5 4 r (3 x 6) = { } - 1 (1+ r (0x3) ).25 Thus r (3x6) = % \$ FRAs usually priced off Eurodollar futures

18 Swaps versus series of FRAs Series of FRAs: each FRA will likely have a different rate (fixed payer pays different rates) according to the shape of the spot yield curve. Swap: usually has the same fixed rate for all settlement periods. At Origination: Each FRA has zero economic value Swap has zero value, but some settlement dates have positive value and some negative

19 Series of FRAs (pay-fixed) Swap vs. FRA series 6.00% June LIBOR Sept. LIBOR 6.00% 6.87% 7.32% Dec. LIBOR 7.68% Pay-Fixed Interest rate Swap 6.96% 6.96% 6.96% 6.96%

20 Pricing Interest rate swaps swap fixed rate found by treating the swap as a series of "off market" FRAs, and setting the swap price such that the present value of the "off market" FRAs is equal to zero. The June (3 x 6) FRA rate is 6.87%. Let SFR be the Swap Fixed Rate, Present value of the off market June FRA is: ( SFR) (1+ r (0x3) ).25 (1+ r (3x6) ).25 x (Notional principal) x (90/360)

21 "Off market" FRA values June swap leg value ("off market" FRA value): ( SFR) x (Notional principal) x (90/360) (1+ r (0x3) ).25 (1+ r (3x6) ).25 or ( SFR) (1.0600).25 (1.0687).25 x (Notional principal) x (90/360) Note that: ( SFR) (1.0600).25 (1.0687).25 = ( SFR) = ( ).50 ( SFR) (1 + r (0x6) ).50

22 Set PV of Swap legs equal to zero (.06 - SFR) 0 = + (1+r (0x3) ).25 ( SFR) (1+r (0x6) ).50 + ( SFR) (1+r (0x9) ).75 + ( SFR) (1+r (0x12) ) 1.00 solve for SFR to find SFR = 6.96% Note: eliminated notional principal term (same multiple for each term) each term is simply the current value of a forward contract: value = PV (contract price - current forward price)

23 Value of swap legs FRAs 6.00% June LIBOR Sept. LIBOR Dec. LIBOR Forward rates 6.00% 6.87% 7.32% 7.68% swap rate too high swap rate too low Swap Swap Fixed rate 6.96% 6.96% 6.96% 6.96%

24 Swaps on a balance sheet If swap pieces were placed on balance sheets: Swap fixed payer Swap fixed receiver Assets Liabilities Assets Liabilities 6 x 9 0 x 3 0 x 3 6 x 9 9 x 12 3 x 6 3 x 6 9 x 12 credit risk is front-loaded for fixed-receiver, back-loaded for fixed-payer (if forward curve is upward S.Mann, sloping) 1999

25 Swaps after first settlement Swap fixed payer Swap fixed receiver Assets Liabilities Assets Liabilities 6 x 9 3 x 6 3 x 6 6 x 9 9 x 12 9 x 12 swap becomes an asset (unless rates drop substantially) swap becomes a liability (unless rates drop substantially)

26 Swaps as Combination of Bonds value swaps after origination (marking to market) calculate swap duration and convexity Example: 5-year swap: receive fixed 10%, pay LIBOR. \$10 million Notional Principal with Semi-annual settlement Net cash flows will be same as if corp had Buys 5-year, \$10 million, 10% coupon bond Issues \$10 million floating rate note (FRN) at LIBOR flat If bonds have same value, then swap is "at market" otherwise swap is "off market" with initial payment

27 Swap as bond combination \$10,500,000 Buy \$10 million, 10% fixed coupon, 5-year bond \$500,000 coupons Years market value Issue \$10 million, 5-year floating note at LIBOR flat Years LIBOR x (# of days/360) x \$10,000,000

28 Swap as bond combination Gross Settlement Flows on 10% versus LIBOR receive-fixed swap Years Net Settlement Cash Flows Years

29 Swap fixed payer Swaps as bonds: Mark to market At origination: Swap fixed receiver Assets Liabilities Assets Liabilities Floater Fixed Note Fixed Note Floater mark to market is zero (if "on market") If swap fixed rate rises: Swap fixed payer Swap fixed receiver Assets Liabilities Assets Liabilities Fixed Fixed Floater Floater Note Note swap value positive for fixed-payer, negative for fixed-receiver

30 Swaps as Option Combinations Cap (interest rate cap agreement) series of European, cash settled put options underlying asset is hypothetical debt security Eurodollar time deposit for caps on LIBOR gains value as rates rise (underlying asset price drops with rate increase) Floor (interest rate floor agreement) series of European, cash settled call options underlying asset is hypothetical debt security Eurodollar time deposit for LIBOR floors gains value as rates drop (underlying asset price rises as rates fall)

31 Cap example firm pays 250 basis points times Notional Principal for a 5-year, 8% cap on six-month LIBOR Cap buyer receives settlement payments whenever LIBOR exceeds cap strike rate The settlement payment: if LIBOR <= 8.00% if LIBOR > 8.00% 0 (LIBOR -.08) x (180/360)x Notional Principal example : LIBOR is 8.50% at settlement, payment received is (.005)(1/2)(NP) = \$2500 per \$1 million cap owner receives \$50 per million for each basis point above the strike

32 Floor example firm pays 150 basis points times Notional Principal for a 5-year, 4.50% floor on six-month LIBOR Floor buyer receives settlement payments whenever LIBOR is less than floor strike rate The settlement payment: if LIBOR < 4.50% if LIBOR > = 4.50% (.045-LIIBOR) x (180/360) x ( Notional Principal) example : LIBOR is 4.25% at settlement, payment received is (.0025)(1/2)(NP) = \$1250 per \$1 million floor owner receives \$50 per million for each basis point below the strike 0

33 Caplet and Floorlet payoffs For any particular settlement period, the payoff for individual caplets or floorlets: 0 8% LIBOR 0 4.5% LIBOR Buy 8% cap Buy 4.5% floor Write 8% cap Write 4.5% cap 0 8% LIBOR 0 4.5% LIBOR

34 Interest rate collar gains Buy a cap floor premium { cap premium { LIBOR cap strike losses write a floor floor strike If cap premium = floor premium, it is zero-cost collar

35 Swap is Zero cost collar with same strike gains pay-fixed swap floor premium { = cap premium { LIBOR cap strike = floor strike losses Pay fixed swap is buying cap, writing floor to create zero-cost collar with same strikes

36 Swaps as options on the balance sheet assume swap fixed rate (SFR) is 7.00 % Swap fixed payer Swap fixed receiver Assets Liabilities Assets Liabilities cap with 7% strike floor with 7% strike floor with 7% strike cap with 7% strike swap fixed payer buys cap and writes floor swap fixed receiver writes cap and buys floor

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