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)
7.6
Nature of Swaps
A swap is an agreement to exchange cash flows at specified future times according to certain specified rules
7.7
An Example of a “Plain Vanilla” Interest Rate Swap
An agreement by Microsoft to receive 6-month LIBOR & 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
7.8
---------Millions of Dollars---------LIBOR FLOATING FIXED Net
Date Rate Cash Flow Cash Flow Cash FlowMar.5, 2012 4.2%
Note: these CFs are not quite precise because of day count conventions
3
7.9
Cash Flow Diagram of Swap: MS Pay Fixed at 5%
Receive Floating at LIBOR Payment at ti, fixed at ti-1
Net Cash exchanged
+
-
t1 t2 t3 t4 t5 t6t0
+
-
7.10
Swap Characteristics 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, BFIX
Long a Floating-Rate Bond, BFLT
For a New Swap, by Definition, BFIX = BFLT
Be Aware of Day Count Conventions, Fixed vs. Floating LIBOR: 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”
7.11
Typical Uses of anInterest Rate Swap
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
7.12
Intel and Microsoft (MS) Transform a Liability
MS
LIBOR
5%
LIBOR+0.1%
5.2%Intel
- 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 LIBOR+20bps
- Micro has borrowed $100mm for 3-years at LIBOR+10bps-- Net of the Swap, Micro has transformed this to a fixed, 3-year
borrowing at 5.1%
4
7.13
Financial Institution is Involved- The Swap Dealer (AAA)
F.I.
LIBOR LIBORLIBOR+0.1%
4.985% 5.015%
5.2%Intel MS
Net of the Dealer:- Intel has LIBOR+21.5bps financing- MS has 5.115% fixed financing
7.14
Intel and Microsoft (MS) Transform an Asset
Intel MS
LIBOR
5%
LIBOR-0.2%
4.7%
Intel has an investment that pays LIBOR-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 LIBOR-30bps
7.15
Financial Institution is Involved
Intel F.I. MS
LIBOR LIBOR
4.7%
5.015%4.985%
LIBOR-0.2%
Net of the Dealer:- Intel has a 4.78% fixed rate asset- MS has a LIBOR-31.5bps floating rate asset
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
7.23
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
AAACorp 4.0% 6-month LIBOR – 0.10%
BBBCorp 5.20% 6-month LIBOR + 0.60%
A=120bps B=70bpsOne can always structure swap so the pick-up is:
A - B = 50bps 7.24
The Comparative AdvantageDirectly between AAA & BBB
AAA
Corp
BBB
Corp
LIBOR
LIBOR+60bps
4.35%
4%
- AAACorp Creates a Synthetic LIBOR-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
7
7.25
The Comparative Advantage With a Swap Dealer
AAACorp
F.I. BBBCorp
4%
LIBOR LIBOR
LIBOR+60bps
4.33% 4.37%
- AAACorp Creates LIBOR-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 BBBCorp7.26
The Comparative Advantage
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 LIBOR-0.1% and LIBOR+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 LIBOR at which it can borrow in the future If short term borrowing cost goes up to LIBOR+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
7.27
The Nature of Swap Rates – They are Lower than Term Rates
Six-month LIBOR 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 LIBOR to AA borrower and collect LIBOR on
principal After 6-months, relend at 6-month LIBOR – maybe to a different AA
borrower; repeat as often as needed Enter into 5-year Swap and collect swap rate on principal
7.28
The Nature of Swap Rates – They are Lower than Term Rates
Six-month LIBOR 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 LIBOR; 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 LIBOR 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
7.29
Using Swap Rates to Bootstrap the LIBOR/Swap Zero Curve Consider a new Swap where Fixed side is the Swap Rate, SN
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
This shows that Swap rates define “Par Yield” (Swap Curve) Bonds: to therefore bootstrap the LIBOR/Swap Zero Curve …
1 1
1
* *0
1 1002
, where: 360
and , 0,1,..., are the LIBOR/Swap zero (swap) rates for each
i i i i N N
NR t R t R t
FIX i N FLOATi
R tFLOAT
i i
B C e S e Le B
nB L k e k Lr
R i N t
7.30
Using Swap Rates to Bootstrap the LIBOR/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 R4 is the 2-year zero rate, then
And if we have the Libor/Swap zero curve, then we can determine the successive Swap Rates (vice versa)
4 4
4
4
1
2.00.04 0.5 0.045 1.0 0.048 1.5
4
1 5 100 1002
2.5 2.5 2.5 102.5 1004.953%
i i i iR t R t R tFIX i
i
R
B C e e e
e e e eR
7.31
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 VSwap=BFIX - BFLOAT Alternatively, they can be valued as a portfolio of forward rate
agreements (FRAs) where Fi = Forward rate, i>0 F0 = Current LIBOR C = Swap Rate (fixed side)
EXAMPLE: Consider the SWAP Pay 6M LIBOR; Receive 8% PA on $100mm 15-months to go
Each exchange of payments in an interest rate swap is an FRA (receive the contract rate pay LIBOR)
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.25xeF(C3)x.5 = e.105x.75 => F(C3) = 10.75 9x15 (CC) rate: e.105x.75xeF(C9)x.5 = e.11x1.25 => F(C9) = 11.75
Converting continuous compounding back to SA:
So 3x9 (SA) rate: 10.75 => 11.0442 And 9x15 (SA) rate: 11.75 => 12.102
12 2/2 cFeF
7.34
1st Row: CF in 3-mos, already determined 2nd & 3rd Row: CF in 9- & 15-mos assumes forward rates 2nd Row: FRA is (8-11.044)(.5)xe-0.105x0.75 = -1.407 3rd Row: FRA is (8-12.102)(.5)xe-0.11x1.25 = -1.787
As Noted above: FRAs are receive contract rate (swap rate) and pay LIBOR
VSwap= -4.267 (pay LIBOR receive fixed) Same as in Terms of Bonds
TIME FXD CF FLT CF NET CF DISCOUNT PV NET CF0.25 4.00 -5.10 -1.100 0.9753 -1.0730.75 4.00 -5.522 -1.522 0.9243 -1.4071.25 4.00 -6.051 -2.051 0.8715 -1.787
TOTAL -4.267
Valuation in Terms of FRAs
7.35
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 …
7.36
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 $18 million
10
7.37
An Example of a Currency Swap An agreement to pay 5% on a sterling principal of £10,000,000 &
receive 6% on a US$ principal of $18,000,000 every year for 5 years The cash flows to IBM:
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
DATE $ CF (mm) £ CF (mm)
Feb 1, 2011 -18.00 +10.00
Feb 1, 2012 +1.08 -.50
Feb 1, 2013 +1.08 -.50
Feb 1, 2014 +1.08 -.50
Feb 1, 2015 +1.08 -.50
Feb 1, 2016 +19.08 -10.50
7.38
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
7.39
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 AUDGeneral Electric 5.0% 7.6%
Quantas 7.0% 8.0%A = 200 B = 40
Can always structure swap pick-up as A-B = 160 bps 7.40
The Comparative Advantage with US Swap Dealer – Xform a Liability
GeneralElectric
+1.3%-------------------
US SwapDealer---------------------
-1.1%
Quantas
USD5%
AUD 6.9% AUD 8.0%
AUD 8.0%
USD 5.0% USD 6.3%
- 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 foreach year to lock-in the $30,000 spread: 195,000-220,000x0.75=30,000
<<AUD 20mm<<
>>USD 15mm>>
11
7.41
Valuation of Currency Swaps
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
Similarly, for a portfolio of forward FX contracts where USD is received
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: LIBOR/Swap Curve: flat in US (at 9%PA) & Japan (at 4%PA) FX spot market has Y110 = $1
FDSWAP BSBV 0
0( ) ( )
fi i i i i
R R T R TSWAP i iV USD T FC T S e e
7.42
Valuation in Terms of Bonds
USD Fixed-Rate Bond (short) is valued in the usual way Using the USD LIBOR/Swap Zero Curve
JPY Fixed-Rate Bond (long) is valued the same way Using the JPY LIBOR/Swap Zero Curve
Current Spot FX is Y110=$1 or S0 = $0.009091 Forward FX: F(Ti)=S0 x exp[.05 x Ti] ( .05=0.09-0.04 R$ - RY ) PV the Net USD CF:
PV(Ti)=[ – USD(Ti) + JPY(Ti) x F(Ti)] exp[-Ri x Ti] ( Ri = .09 )
TIME USD CF JPY CF FWD FX USD CL3 NET USD PV NET CF1 -.80 60 0.009557 0.5734 -0.2266 -0.20712 -.80 60 0.010047 0.6028 -0.1972 -0.16473 -.80 60 0.010562 0.6337 -0.1663 -0.12693 -10.0 1,200 0.010562 12.6746 12.6746 2.0417
TOTAL Paid Received 1.5430
7.44
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
7.45
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
