Introduction to Financial Derivatives

Transcription

1 Introduction to Financial Derivatives Week of October 22, 2012 Swaps Where we are Previously: Interest Rate Futures (Chapter 6, OFOD) and FRAs Last Week: Mid Term (& Review) This Week: Swaps (Chapter 7, OFOD) Next Week: Option Markets and Stock Options (Chapter 8-9, OFOD) 7.1 Final Exam Thursday, Dec 20th; 9:00am Noon Gilman Assignment Assignment For This Week (October 22 nd ) Read: Hull Chapter 7 Problems (Due Oct 29 th ) Chapter 7: 1, 3, 5,6, 9, 12, 18; 22, 23 Chapter 7 (7e): 1, 3, 5, 6, 9, 12, 18; 20, For Next Week (October 29 th ) Read: Hull Chapter 9-10 Problems (Due Oct 29 th ) Chapter 7: 1, 3, 5,6, 9, 12, 18; 22, 23 Chapter 7 (7e): 1, 3, 5, 6, 9, 12, 18; 20, 21 Problems (Due November 5 th ) Chapter 10: 7, 14, 15, 18, 19; 23 Chapter 9(7e): 7, 14, 15, 18, 19; 23 Look at DerivaGem problems & (7e) 9.21 &

2 Plan for This Week Swaps Interest Rate Swap Structure and Uses The Swap Curve Valuation of IR Swap Bond Viewpoint & as PF of FRA Currency Swap Valuation of Currency Swap As bond & PF of Forward FX Contracts Other Swaps and Compounding Swaps What s Ahead: Options (9-10); Binomial Trees (12); Wiener Process & Ito Lemma (13); Black-Scholes-Merton Model (14); BSM for Options on Indexes, Currencies & Futures (16-17); The Greeks (18) Nature of Swaps A swap is an agreement to exchange cash flows at specified future times according to certain specified rules An Example of a Plain Vanilla Interest Rate Swap An agreement by Microsoft to receive 6-month & pay a fixed rate of 5% per annum every 6 months for 3 years on a notional principal of $100 million Next slide illustrates cash flows Cash Flows to Microsoft (MSFT pays Fixed) Millions of Dollars FLOATING FIXED Net Date Rate Cash Flow Cash Flow Cash Flow Mar.5, % Sept. 5, % Mar.5, % Sept. 5, % Mar.5, % Sept. 5, % Mar.5, % Note: these CFs are not quite precise because of day count conventions 7.8 2

3 Cash Flow Diagram of Swap: MS Swap Characteristics t 0 t 1 t 2 t 3 t 4 t 5 t 6 Pay Fixed at 5% Receive Floating at Payment at t i, fixed at t i-1 Net Cash exchanged Cash Flows (Fixed and Floating) could be described (augmented) by showing notional amount exchanged at maturity Allows for the description of the Swap as Short a Fixed-Rate Bond, B FIX Long a Floating-Rate Bond, B FLT For a New Swap, by Definition, B FIX = B FLT Be Aware of Day Count Conventions, Fixed vs. Floating : Act/360 Fixed: 30/360 or Act/360 or etc. For the most part, these issues are suppressed during this introduction for ease of exposition Typical Uses of an Interest Rate Swap Intel and Microsoft (MS) Transform a Liability Converting a liability from fixed rate to floating rate floating rate to fixed rate Converting an investment from fixed rate to floating rate floating rate to fixed rate % Intel 5% MS +0.1% - Intel has issued a 3-year, $100mm note at a rate of 5.2% -- Net of the Swap, Intel has transformed this to a borrowing of +20bps - Micro has borrowed $100mm for 3-years at +10bps -- Net of the Swap, Micro has transformed this to a fixed, 3-year borrowing at 5.1%

4 Financial Institution is Involved - The Swap Dealer (AAA) Intel and Microsoft (MS) Transform an Asset 4.985% 5.015% 5.2% Intel F.I. MS +0.1% -0.2% Intel 5% MS 4.7% Net of the Dealer: - Intel has +21.5bps financing - MS has 5.115% fixed financing 7.13 Intel has an investment that pays -20bps over the next 3-years - Net of the Swap, Intel has transformed this into an investment paying a fixed 4.80% Micro has an investment that pays a fixed rate of 4.7% over 3-years - Net of the Swap, MS has converted this into a floating rate deposit at -30bps 7.14 Financial Institution is Involved Swap Confirmation 4.985% 5.015% Intel F.I. MS -0.2% Net of the Dealer: - Intel has a 4.78% fixed rate asset - MS has a -31.5bps floating rate asset 4.7%

5 Quotes By a Swap Market Maker March 01, 2007 Swap Curve March 01, 2007 Maturity Bid (%) Offer (%) Swap Rate (%) 2 years years years years years years US Treasury March 01, 2007 Swap Spreads March 01,

6 SWAP ED Analysis March 01,07 The Comparative Advantage - Why Swaps are so Popular One Curious Property Underscoring the Utility of Swaps is the Ability of Counterparties to take advantage of their Comparative Advantage in the Debt Markets Different Credit Quality Borrowers may have a Comparative Advantage in Fixed or Floating Markets For New Financing, Borrowers go to the Cash Market where they have a Comparative Advantage As a Result a Company may Borrow Fixed when it wants Floating Borrow Floating when it wants Fixed BUT, It can Adjust the result through the Swap Market The Comparative Advantage AAACorp wants to borrow $10 million for 5-years Has a Comparative Advantage in Fixed Market 120bps better than BBBCorp BBBCorp wants to borrow $10 million for 5-years Has a Comparative Advantage in Floating Market Only 70bps worse than AAACorp Fixed Floating The Comparative Advantage Directly between AAA & BBB 4% AAA Corp 4.35% BBB Corp +60bps AAACorp 4.0% 6-month 0.10% BBBCorp 5.20% 6-month % A=120bps B=70bps One can always structure swap so the pick-up is: - AAACorp Creates a Synthetic -35(bps) Floater through the Swap Market -- 25bp improvement over the Cash Market - BBBCorp Creates a Synthetic 4.95% Fixed Rate through the Swap Market -- 25bp improvement over the Cash Market A - B = 50bps

7 The Comparative Advantage With a Swap Dealer The Comparative Advantage 4% AAA Corp 4.33% 4.37% F.I. BBB Corp +60bps - AAACorp Creates -33(bps) with Dealer Charges included - BBBCorp Creates 4.97% with Dealer Charges included - The 4bps the Dealer takes out reduces AAACorp credit exposure and makes the facility available to BBBCorp 7.25 The 4.0% and 5.2% rates available to AAACorp and BBBCorp in fixed rate markets are 5-year rates Spread reflects credit issues for the long haul The -0.1% and +0.60% rates available to a borrower in the floating rate market are six-month rates Credit issue is diminished as any credit weakening is mitigated by the lenders (typical) prerogative to charge a higher spread or not to roll BBBCorp s apparent fixed rate depends on the spread over 6-month at which it can borrow in the future If short term borrowing cost goes up to +160bps Fixed rate borrowing goes up to 5.97% Well above the 4.97% that appeared locked-in (via Comparative Advantage) if Cash Market spreads had not widened 100bps 7.26 The Nature of Swap Rates They are Lower than Term Rates Six-month is a short-term AA borrowing rate A Dealer can earn the 5-year swap rate on a given principal by doing the following (at very low risk) Loan at 6-month to AA borrower and collect on principal After 6-months, relend at 6-month maybe to a different AA borrower; repeat as often as needed Enter into 5-year Swap and collect swap rate on principal 7.27 The Nature of Swap Rates They are Lower than Term Rates Six-month is a short-term AA borrowing rate The 5-year swap rate has a risk of 10 six-month loans made to AA borrowers at ; and the risk is higher lending in the 5-year Term (Cash) Market This is because the lender can enter into a swap where income from the loans is (paid) exchanged for the 5-year swap rate (rec d) This is more favorable than (lending in) the 5-year Cash Market where the borrower may be AA only at the beginning of the 5-years Also, the Swap Dealer is only exposed to the net of the exchange on the Swap, not the notional amount No Arbitrage on Swap-to-Cash since there is no risk on the Notional Amount (to Principal Amount in the Cash Mkt.) 5-year swap rates are therefore lower than 5-year AA rates

8 Using Swap Rates to Bootstrap the /Swap Zero Curve Consider a new Swap where Fixed side is the Swap Rate, S N When principals are added to both sides on the final payment date the Swap is the exchange of a Fixed-Rate Bond for a Floating-Rate Bond The Floating-Rate bond is worth Par (at fixing) The Swap is worth zero Therefore, the Fixed-Rate Bond must also be worth par N Rt 1 ii Rt ii RNtN BFIX Ce i SNe Le 100 BFLOAT i 1 2 * Rt 11 * n BFLOAT L k e, where: k Lr0 360 and R, i 0,1,..., N are the /Swap zero (swap) rates for each t i This shows that Swap rates define Par Yield (Swap Curve) Bonds: to therefore bootstrap the /Swap Zero Curve 7.29 i Using Swap Rates to Bootstrap the /Swap Zero Curve Suppose the 6-, 12-, and 18-month Libor/Swap zero rates have been determined as 4%, 4.5%, and 4.8% (continuous compounding) And the 2-year swap rate (for semiannual payments) is 5% The 5% swap rate means A bond with $100 principal and a semiannual coupon of 5% per annum sells for Par If R 4 is the 2-year zero rate, then 4 Rt 1 ii 5 Rt ii 100 Rt 44 BFIX Ce i e e 100 i R e 2.5e 2.5e 102.5e 100 R % And if we have the Libor/Swap zero curve, then we can determine the successive Swap Rates (vice versa) 7.30 Valuation of an Interest Rate Swap that is NOT New Interest Rate Swap can be valued as the difference between the value of a fixed-rate bond and a floating-rate bond V Swap =B FIX -B FLOAT Alternatively, they can be valued as a portfolio of forward rate agreements (FRAs) where F i = Forward rate, i>0 n ti ti 1 Riti F 0 = Current VSwap L( C Fi 1 ) e C = Swap Rate (fixed side) i EXAMPLE: Consider the SWAP Pay 6M ; Receive 8% PA on $100mm 15-months to go Market (cc): 3M=10%; 9M=10.5%; 15M=11% 6M 3M ago: 10.2% First Bonds then FRAs 7.31 Valuation in Terms of Bonds The Fixed-Rate Bond is valued in the usual way Using the /Swap Zero Curve The Floating-Rate Bond is valued by noting that it is worth par when the coupon is re-set in 3-months (.25-yr) V Swap =B FIX -B FLOAT TIME FXD CF FLT CF DISCOUNT PV FXD CF PV FLT CF TOTAL V Swap = (pay receive fixed)

9 Valuation in Terms of FRAs Valuation in Terms of FRAs Each exchange of payments in an interest rate swap is an FRA (receive the contract rate pay ) The FRAs can be valued on the assumption that today s 6-mo forward rates (in 3- & 9-mos) are realized 3x9 (CC) rate: e.1x.25 xe F(C3)x.5 = e.105x.75 => F(C3) = x15 (CC) rate: e.105x.75 xe F(C9)x.5 = e.11x1.25 => F(C9) = Converting continuous compounding back to SA: Fc / 2 F2 2 e 1 So 3x9 (SA) rate: => And 9x15 (SA) rate: => st Row: CF in 3-mos, already determined 2 nd & 3 rd Row: CF in 9- & 15-mos assumes forward rates 2 nd Row: FRA is ( )(.5)xe x0.75 = rd Row: FRA is ( )(.5)xe -0.11x1.25 = As Noted above: FRAs are receive contract rate (swap rate) and pay TIME FXD CF FLT CF NET CF DISCOUNT PV NET CF TOTAL V Swap = (pay receive fixed) Same as in Terms of Bonds 7.34 The Currency Swap Exchange of Cash Flows across 2 Currencies Principal and Interest in one Currency for Principal and Interest in another Currency Principal Amounts are (usually) Exchanged at the Beginning and at the End of the life of the Swap At Initiation, the market exchange rate is used to set the principal amounts At the End, the principal amount in each currency is the same, but the value can be quite different due to the then-prevailing exchange rates Consider the Interest in a Fixed-for-Fixed Currency Swap payments are made once a year 7.35 An Example of a Currency Swap Suppose IBM and British Petroleum enter into a 5-year currency swap agreement on Feb 1, 2011 IBM agrees to pay a fixed rate of 5% in Sterling and receives a fixed rate of 6% in Dollars from BP Payments are made once a year and the principal amounts are $18 million and 10 million This is termed a fixed for fixed currency swap Initially, the principal amounts are exchanged in the opposite direction to the annual payments, shown above At the end of the term, IBM pays back the 10 million and receives back its $18 million

10 An Example of a Currency Swap An agreement to pay 5% on a sterling principal of 10,000,000 & receive 6% on a US$ principal of $18,000,000 every year for 5 years The cash flows to IBM: DATE $ CF (mm) CF (mm) Feb 1, Feb 1, Feb 1, Feb 1, Feb 1, Feb 1, In an interest rate swap the principal is not exchanged In a currency swap the principal is usually exchanged at the beginning and the end of the swap s life 7.37 Typical Uses of a Currency Swap Conversion of a liability in one currency to a liability in another currency Conversion of an investment in one currency to an investment in another currency 7.38 Comparative Advantage Arguments for Currency Swaps GE wants to borrow 20 million in AUD for 5-years Has a Comparative Advantage in USD 200bps better than Quantas Qantas wants to borrow 15 million in USD for 5-years Has a Comparative Advantage in AUD Only 40bps worse than GE Implied FX is 1.33 AUD per USD or.75 USD per AUD Borrowing Rates for each 5-year rates USD AUD General Electric 5.0% 7.6% The Comparative Advantage with US Swap Dealer Xform a Liability General Electric >>USD 15mm>> USD 5.0% USD 6.3% +1.3% US Swap Dealer Quantas 7.0% 8.0% A = 200 B = 40 Can always structure swap pick-up as A-B = 160 bps 7.39 each year to lock-in the $30,000 spread: 195, ,000x0.75=30, USD 5% % Quantas AUD 8.0% AUD 6.9% AUD 8.0% <<AUD 20mm<< - GE Creates AUD 6.9%, Pick-up of 70bps over AUD borrowing - Quantas Creates USD 6.3%, Pick-up of 70bps over USD costs - The Dealer gains 1.3% (195,000) on USD leg and looses 1.1% (220,000) on AUD leg -Dealer takes out 20bps (So all 160bps is accounted for) - FX risk can be avoided by buying AUD 220,000 PA in the Forward FX market for 10

11 Valuation of Currency Swaps Valuation in Terms of Bonds Like interest rate swaps, currency swaps can be valued either as the difference between 2 bonds or as a portfolio of forward contracts Let the Swap be where USD is received and foreign currency is paid VSWAP BD S0BF Similarly, for a portfolio of forward FX contracts where USD is received f Ri Ri Ti RiTi VSWAP USD( Ti ) FC( Ti ) S0e e EXAMPLE SWAP (legacy): 3-year tenor, annual pay Pay: USD at 8%PA on $10 million Receive: YEN at 5%PA on Y1,200 million Market: /Swap Curve: flat in US (at 9%PA) & Japan (at 4%PA) FX spot market has Y110 = $ USD Fixed-Rate Bond (short) is valued in the usual way Using the USD /Swap Zero Curve JPY Fixed-Rate Bond (long) is valued the same way Using the JPY /Swap Zero Curve Since the USD is Paid, V Swap = S 0 B F B D TIME USD CF PV $ JPY CF PV JPY ,200 1, TOTAL Short Long 1, V Swap = 1,230.55/ = $1.543 million 7.42 Valuation as a Portfolio of Forward FX Contracts Current Spot FX is Y110=$1 or S 0 = $ Forward FX: F(T i )=S 0 x exp[.05 x T i ] (.05= R$ - RY ) PV the Net USD CF: PV(T i )=[ USD(T i ) + JPY(T i ) x F(T i )] exp[-r i x T i ] ( R i =.09 ) TIME USD CF JPY CF FWD FX USD CL3 NET USD PV NET CF , TOTAL Paid Received Swaps & Forwards A swap can be regarded as a convenient way of packaging forward contracts The plain vanilla interest rate swap in our (MS-Intel) example consisted of 6 FRAs The fixed for fixed currency swap in our (GE- Quantas) example consisted of a cash transaction & 5 forward contracts The value of the swap is the sum of the values of the forward contracts underlying the swap Swaps are normally at the money initially This means that it costs nothing to enter into a swap It does not mean that each forward contract underlying a swap is at the money initially

12 Credit Risk A swap is worth zero to a company initially At a future time its value is liable to be either positive or negative The company has credit risk exposure only when its value is positive The credit risk exposure is limited to the value of the contract, not the notional amount. Other Types of Swaps Floating-for-floating interest rate swaps Amortizing swaps Step up swaps Forward swaps Constant maturity swaps Compounding swaps -in-arrears swaps Accrual swaps Differential swaps Cross currency interest rate swaps Equity swaps Extendable swaps & Puttable swaps Swaptions Commodity swaps Volatility swaps