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Chapter 7Swaps
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 1
Nature of Swaps
A swap is an agreement to exchange cash flows at specified future times according to certain specified rules
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 2
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 that could occur (Day count conventions are not considered)
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 3
One Possible Outcome for Cash Flows to Microsoft (Table 7.1, page 150)
Date LIBOR Floating Cash Flow
Fixed Cash Flow
Net Cash Flow
Mar 5, 2012
4.20%
Sep 5, 2012 4.80% +2.10 −2.50 −0.40
Mar 5, 2013
5.30% +2.40 −2.50 −0.10
Sep 5, 2013 5.50% +2.65 −2.50 + 0.15
Mar 5, 2014
5.60% +2.75 −2.50 +0.25
Sep 5, 2014 5.90% +2.80 −2.50 +0.30
Mar 5, 2015
+2.95 −2.50 +0.45Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 4
Typical Uses of an Interest Rate Swap
Converting a liability fromfixed rate to floating rate floating rate to fixed rate
Converting an investment from fixed rate to floating ratefloating rate to fixed rate
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 5
Intel and Microsoft (MS) Transform a Liability (Figure 7.2, page 151)
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 6
Intel MS
LIBOR
5%
LIBOR+0.1%
5.2%
Financial Institution is Involved(Figure 7.4, page 152)
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 7
F.I.
LIBOR LIBORLIBOR+0.1
%
4.985% 5.015%
5.2%Intel MS
Financial Institution has two offsetting swaps
Intel and Microsoft (MS) Transform an Asset (Figure 7.3, page 152)
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 8
Intel MS
LIBOR
5%
LIBOR-0.2%
4.7%
Financial Institution is Involved(See Figure 7.5, page 153)
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 9
Intel F.I. MS
LIBOR LIBOR
4.7%
5.015%4.985%
LIBOR-0.2%
Quotes By a Swap Market Maker (Table 7.3, page 154)
Maturity Bid (%) Offer (%) Swap Rate (%)
2 years 6.03 6.06 6.045
3 years 6.21 6.24 6.225
4 years 6.35 6.39 6.370
5 years 6.47 6.51 6.490
7 years 6.65 6.68 6.665
10 years 6.83 6.87 6.850
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 10
Day Count
A day count convention is specified for for fixed and floating payment
For example, LIBOR is likely to be actual/360 in the US because LIBOR is a money market rate
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 11
Confirmations
Confirmations specify the terms of a transaction
The International Swaps and Derivatives has developed Master Agreements that can be used to cover all agreements between two counterparties
Governments now require central clearing to be used for most standardized derivatives
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 12
The Comparative Advantage Argument (Table 7.4, page 156)
• AAACorp wants to borrow floating• BBBCorp wants to borrow fixed
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 13
Fixed Floating
AAACorp 4.0% 6 month LIBOR − 0.1%
BBBCorp 5.2% 6 month LIBOR + 0.6%
The Swap (Figure 7.6, page 157)
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 14
AAACorp BBBCorp
LIBOR
LIBOR+0.6%
4.35%
4%
The Swap when a Financial Institution is Involved (Figure 7.7, page 157)
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 15
AAACorp F.I.
BBBCorp4%
LIBOR LIBOR
LIBOR+0.6%
4.33% 4.37%
Criticism of the Comparative Advantage Argument
The 4.0% and 5.2% rates available to AAACorp and BBBCorp in fixed rate markets are 5-year rates
The LIBOR−0.1% and LIBOR+0.6% rates available in the floating rate market are six-month rates
BBBCorp’s fixed rate depends on the spread above LIBOR it borrows at in the future
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 16
The Nature of Swap RatesSix-month LIBOR is a short-term AA borrowing rate
The 5-year swap rate has a risk corresponding to the situation where 10 six-month loans are made to AA borrowers at LIBOR
This is because the lender can enter into a swap where income from the LIBOR loans is exchanged for the 5-year swap rate
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 17
Using Swap Rates to Bootstrap the LIBOR/Swap Zero Curve
Consider a new swap where the fixed rate is the swap rateWhen 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 bondThe floating-rate rate bond is worth par. The swap is worth zero. The fixed-rate bond must therefore also be worth par This shows that swap rates define par yield bonds that can be used to bootstrap the LIBOR (or LIBOR/swap) zero curve
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 18
Example of Bootstrapping the LIBOR/Swap Curve (Example 7.1, page 160)
6-month, 12-month, and 18-month LIBOR/swap rates are 4%, 4.5%, and 4.8% with continuous compounding.
Two-year swap rate is 5% (semiannual)
The 2-year LIBOR/swap rate, R, is 4.953%Options, Futures, and Other Derivatives, 8th Edition,
Copyright © John C. Hull 2012 19
1005102525252
2
51048001045050040
Reeee
.... ......
Valuation of an Interest Rate SwapInitially interest rate swaps are worth close to zero
At later times they can be valued as the difference between the value of a fixed-rate bond and the value of a floating-rate bond
Alternatively, they can be valued as a portfolio of forward rate agreements (FRAs)
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 20
Valuation in Terms of Bonds
The fixed rate bond is valued in the usual way
The floating rate bond is valued by noting that it is worth par immediately after the next payment date
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 21
Valution of Floating-Rate Bond
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 22
0 t*
Valuation Date
First PmtDate
Floating Pmt =k*
SecondPmt Date Maturity
Date
Value = LValue = L+k*
Value = PV of L+k* at t*
Example
Pay six-month LIBOR, receive 8% (s.a. compounding) on a principal of $100 million
Remaining life 1.25 years
LIBOR rates for 3-months, 9-months and 15-months are 10%, 10.5%, and 11% (cont comp)
6-month LIBOR on last payment date was 10.2% (s.a. compounding)
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 23
Valuation Using Bonds (page 161)
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 24
Time Bfix cash flow
Bfl cash flow
Disc factor
PV Bfix
PV Bfl
0.25 4.0 105.100 0.9753 3.901 102.505
0.75 4.0 0.9243 3.697
1.25 104.0 0.8715 90.640
Total 98.238 102.505
Swap value = 98.238 − 102.505 = −4.267
Valuation in Terms of FRAsEach exchange of payments in an interest rate swap is an FRA
The FRAs can be valued on the assumption that today’s forward rates are realized
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 25
Valuation of Example Using FRAs (page 163)
Time Fixed cash flow
Floating cash flow
Net Cash Flow
Disc factor
PV Bfl
0.25 4.0 -5.100 -1.100 0.9753 -1.073
0.75 4.0 -5.522 -1.522 0.9243 -1.407
1.25 4.0 -6.051 -2.051 0.8715 -1.787
Total -4.267
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 26
Overnight Indexed SwapsFixed rate for a period is exchanged for the geometric average of the overnight rates
Should OIS rate equal the LIBOR rate? A bank can
Borrow $100 million in the overnight market, rolling forward for 3 months
Enter into an OIS swap to convert this to the 3-month OIS rate
Lend the funds to another bank at LIBOR for 3 months
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 27
Overnight Indexed Swaps continued
...but it bears the credit risk of another bank in this arrangement
The OIS rate is now regarded as a better proxy for the short-term risk-free rate than LIBOR
The excess of LIBOR over the OIS rate is the LIBOR-OIS spread. It is usually about 10 basis points but spiked at an all time high of 364 basis points in October 2008
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 28
An Example of a Currency SwapAn agreement to pay 5% on a sterling principal of £10,000,000 & receive 6% on a US$ principal of $18,000,000 every year for 5 years
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 29
Exchange of PrincipalIn 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
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 30
The Cash Flows (Table 7.7, page 166)
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 31
Date Dollar Cash Flows
(millions)
Sterling cash flow
(millions)
Feb 1, 2011 -18.0 +10.0
Feb 1, 2012 +1.08 −0.50
Feb 1, 2012 +1.08 −0.50
Feb 1, 2014 +1.08 −0.50
Feb 1, 2015 +1.08 −0.50
Feb 1, 2016 +19.08 −10.50
Typical Uses of a Currency Swap
Convert a liability in one currency to a liability in another currencyConvert an investment in one currency to an investment in another currency
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 32
Comparative Advantage May Be Real Because of Taxes
General Electric wants to borrow AUD
Quantas wants to borrow USD
Cost after adjusting for the differential impact of taxes
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 33
USD AUD
General Electric 5.0% 7.6%
Quantas 7.0% 8.0%
Valuation of Currency SwapsLike interest rate swaps, currency swaps can be valued either as the difference between 2 bonds or as a portfolio of forward contracts
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 34
Example
All Japanese LIBOR/swap rates are 4%
All USD LIBOR/swap rates are 9%
5% is received in yen; 8% is paid in dollars. Payments are made annually
Principals are $10 million and 1,200 million yen
Swap will last for 3 more years
Current exchange rate is 110 yen per dollarOptions, Futures, and Other Derivatives, 8th Edition,
Copyright © John C. Hull 2012 35
Valuation in Terms of Bonds (Table 7.9, page 169)
Time Cash Flows ($)
PV ($) Cash flows (yen)
PV (yen)
1 0.8 0.7311
60 57.65
2 0.8 0.6682
60 55.39
3 0.8 0.6107
60 53.22
3 10.0 7.6338
1,200 1,064.30
Total 9.6439
1,230.55
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 36
Value of Swap = 1230.55/110 − 9.6439 = 1.5430
Valuation in Terms of Forwards (Table 7.10, page 170)
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 37
Time $ cash flow
Yen cash flow
Forward Exch rate
Yen cash flow in $
Net Cash Flow
Present value
1 -0.8 60 0.009557 0.5734 -0.2266
-0.2071
2 -0.8 60 0.010047 0.6028 -0.1972
-0.1647
3 -0.8 60 0.010562 0.6337 -0.1663
-0.1269
3 -10.0 1200 0.010562 12.6746 +2.6746
2.0417
Total 1.5430
Swaps & Forwards
A swap can be regarded as a convenient way of packaging forward contracts
Although the swap contract is usually worth close to zero at the outset, each of the underlying forward contracts are not worth zero
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 38
Credit RiskA 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
Some swaps are more likely to lead to credit risk exposure than others
What is the situation if early forward rates have a positive value?
What is the situation when the early forward rates have a negative value?
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 39
Other Types of Swaps
Floating-for-floating interest rate swaps, amortizing swaps, step up swaps, forward swaps, constant maturity swaps, compounding swaps, LIBOR-in-arrears swaps, accrual swaps, diff swaps, cross currency interest rate swaps, equity swaps, extendable swaps, puttable swaps, swaptions, commodity swaps, volatility swaps……..
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 40
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