Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.1
Swaps
(互换 )
Chapter 5
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.2
Nature of Swaps
A swap is an agreement to
exchange cash flows (现金流 ) at
specified future times according
to certain specified rules
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.3
Terminology
LIBOR
the London InterBank Offer Rate
It is the rate of interest offered by
banks on
deposits from other banks in
Eurocurrency markets
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.4
An Example of a,Plain Vanilla”
Interest Rate Swap(大众型利率互换 )
An agreement by,Company B” to
RECEIVE 6-month LIBOR and
PAY a fixed rate of 5% pa
every 6 months for 3 years on a
notional principal of $100 million
Next slide illustrates cash flows,where
POSITIVE flows are revenues (inflows) and
NEGATIVE flows are expenses (outflows)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.5
---------Millions of Dollars---------
LIBOR FLOATING FIXED Net
Date Rate Cash Flow Cash Flow Cash Flow
Mar.1,1999 4.2%
Sept,1,1999 4.8% +2.10 –2.50 –0.40
Mar.1,2000 5.3% +2.40 –2.50 –0.10
Sept,1,2000 5.5% +2.65 –2.50 +0.15
Mar.1,2001 5.6% +2.75 –2.50 +0.25
Sept,1,2001 5.9% +2.80 –2.50 +0.30
Mar.1,2002 6.4% +2.95 –2.50 +0.45
Cash Flows to Company B
(See Table 5.1,page 123)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.6
More on Table 5.1
The floating-rate payments are calculated
using the six-month LIBOR rate prevailing six
month before the payment date
The principle is only used for the calculation
of interest payments,However,the principle
itself is not exchanged—Meaning for
―Notional principle‖
The swap can be regarded as the exchange
of a fixed-rate bond for a float-rate bond,
Company B (A) is long (short) a floating-rate
bond and short (long) a fixed-rate bond.
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.7
Typical Uses of an
Interest Rate Swap
Converting a liability
from a
– FIXED rate liability to a
FLOATING rate liability
– FLOATING rate liability
to a FIXED rate liability
Converting an investment
from a
– FIXED rate investment to a
FLOATING rate investment
– FLOATING rate investment
to a FIXED rate investment
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.8Transforming a Floating-rate
Loan to a Fixed-rate
Consider a 3-year swap initialized on March 1,2000 where
Company B agrees to pay Company A 5%pa on
$100 millionCompany A agrees to pay Company B 6-mth
LIBOR on $100 million
Suppose Company B has arranged to borrow $100
million LIBOR + 80bp
Company
B
Company
A
5%
LIBOR LIBOR+0.8%5.2%
Note,1 basis point (bp) = one-hundredth of 1%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.9
Transforming a Floating-rate
Loan to a Fixed-rate (continued)
After Company B has entered into the swap,they
have 3 sets of cash flows
1,Pays LIBOR plus 0.8% to outside lenders
2,Receives LIBOR from Company A in the swap
3,Pays 5% to Company A in the Swap
In essence,B has transformed its variable rate
borrowing at LIBOR + 80bp to a fixed rate of 5.8%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.10A and B Transform a Liability
(Figure 5.2,page 125)
A B
LIBOR
5%
LIBOR+0.8%
5.2%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.11Financial Institution is
Involved
(Figure 5.4,page 126)
A F.I,B
LIBOR LIBOR
LIBOR+0.8%
4.985% 5.015%
5.2%
“Plain vanilla” fixed-for-float swaps on US interest rates are
usually structured so that the financial institutions earns 3 to 4
basis points on a pair of offsetting transactions
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.12
A and B Transform an Asset
(Figure 5.3,page 125)
A B
LIBOR
5%
LIBOR-0.25%
4.7%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.13Financial Institution is
Involved
(See Figure 5.5,page 126)
A F.I,B
LIBOR LIBOR
4.7%
5.015%4.985%
LIBOR-0.25%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.14
The Comparative Advantage
Argument (Table 5.4,page 129)
Company A wants to borrow floating
Company B wants to borrow fixed
Fixed Floating
Company A 10.00% 6-month LIBOR + 0.30%
Company B 11.20% 6-month LIBOR + 1.00%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.15
The Comparative Advantage
(continued)
One possible swap is
Company A has 3 sets of cash flows
1,Pays 10%pa to outside lenders
2,Receives 9.95%pa from B Pays LIBOR + 0.05%
3,Pays LIBOR to B a 25bp gain
Company B has 3 sets of cash flows
1,Pays LIBOR + 1.00%pa to outside lenders
2,Receives LIBOR from A Pays 10.95%pa
3,Pays 9.95% to A a 25bp gain
Company
B
Company
A
9.95%
LIBOR10% LIBOR + 1%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.16
The Swap (Figure 5.6,page 130)
A B
LIBOR
LIBOR+1%
9.95%
10%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.17
The Swap when a Financial
Institution is Involved
(Figure 5.7,page 130)
A F.I,B
10%
LIBOR LIBOR
LIBOR+1%
9.93% 9.97%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.18
Total Gain from an
Interest Rate Swap
The total gain from an interest rate swap is always
|a-b| where
a is the difference between the interest rates in the
fixed-rate market for the two parties,and
b is the difference between the interest rates in the
floating-rate market for the two parties
In this example a=1.20% and b=0.70%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.19
Criticism of the Comparative
Advantage Argument
The 10.0% and 11.2% rates available to
A and B in fixed rate markets are 5-year
rates
The LIBOR+0.3% and LIBOR+1% rates
available in the floating rate market are
six-month rates
B’s fixed rate depends on the spread
above LIBOR it borrows at in the future
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.20
Valuation of an Interest Rate
Swap
Interest rate swaps can be valued as
the difference between
--the value of a fixed-rate bond &
--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,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.21
Valuation of an Interest Rate Swap
as a Package of Bonds
The fixed rate bond is valued in the
usual way (page 132)
The floating rate bond is valued by
noting that it is worth par immediately
after the next payment date (page 132)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.22
Valuation of an Interest Rate Swap
as a Package of Bonds (continued)
Define
Vswap,value of the swap to the financial institution
Bfix,value of the fixed-rate bond underlying the swap
Bfl,value of the floating-rate bond underlying the swap
L,notional principal in a swap agreement
ti,time when the ith payments are exchanged
ri,LIBOR zero rate for a maturity ti
Then,Vswap = Bfix - Bfl and if k is the fixed-rate coupon and k* is the floating
111111 )e(ee
ee
**
fl
1
f i x
trtrtr
tr
n
i
tr
kLkLB
LkB nnii
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.23Example of an Interest Rate Swap
Valued as a Package of Bonds
Suppose,that you agreed to pay 6-month LIBOR and receive 8%
pa (with semiannual compounding) on a notional amount of
$100 million,The swap has a remaining life of 15 months and
the next payment is due in 3 months,The relevant rates for
continuous compounding over 3,9,and 15 months are 10.0%,
10.5%,and 11.0%,respectively,The six-month LIBOR rate at
the last payment was 10.2% (with semi-annual compounding).
In this case k = $4 million and k* = $5.1 million,so that
Bfix= 4e-0.25x0.10 + 4e-0.75x0.105 + 104e-1.25x0.11
= $ 98.24 million
Bfl = 5.1e-0.25x0.10 + 100e-0.25x0.10
= $102.51 million
Hence,Vswap = 98.24 - 102.51 = -$4.27 million
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.24
Valuation in Terms of FRAs
Each 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 (See section 4.6,page 97)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.25
Valuation of an Interest Rate Swap
as a Package of FRAs
A simple three step process
1,Calculate each of the forward rates for each of the
LIBOR rates that will determine swap cash flows
2,Calculate swap cash flows on the assumption
that the LIBOR rates will equal the forward rates
3,Set the swap rates equal to the present value of
these cash flows
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.26
Example of an Interest Rate Swap
Valued as a Package of FRAs
Same problem as before.
The cash flows for the payment in 3 months have already been set,A
rate of 8% will be exchanged for a rate of 10.2%,The NPV of this
transaction is
0.5 * 100 * (0.08 - 0.102)e-0.1*0.25 = -1.07
To figure out the NPV of the remaining two payments,we first need to
calculate the forward rates corresponding to 9 and 15 months
or 10.75% with continuous compounding which corresponds to
11.044% with semi-annual compounding.
The value of the 9 month FRA is
0.5 * 100 * (0.08 - 0.11044)e-0.105*0.75 = -1.41
1 0 7 5.05.0 10.0*25.01 0 5.0*75.093f
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.27
Example Swap as Valued as FRAs
(continued)
The 15 month forward rate is
or 11.75% with continuous compounding which corresponds
to 12.102% with semi-annual compounding.
The value of the 15 month FRA is
0.5 * 100 * (0.08 - 0.12102)e-0.11*1.25 = -1.79
Hence,the total value of the swap is -1.07 - 1.41 - 1.79 = -4.27
1 1 7 5.05.0 75.0*1 0 5.025.1*11.0159f
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.28
An Example of a Currency
Swap
An agreement to
-- pay 11% on a sterling principal of
￡10,000,000 &
-- receive 8% on a US$ principal of
$15,000,000
-- every year for 5 years
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.29
Exchange of Principal
In an interest rate swap
the principal is not exchanged
In a currency swap
the principal is exchanged at
-- the beginning &
-- the end of the swap
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.30
The Cash Flows (Table 5.5,page 137)
Years
Dollars Pounds
$
------millions------
0 –15.00 +10.00
1 +1.20 –1.10
2 +1.20 –1.10
3 +1.20 –1.10
4 +1.20 –1.10
5 +16.20 -11.10
￡
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.31
Typical Uses of a
Currency Swap
Conversion
from a liability in
one currency
to a liability in
another currency
Conversion
from an investment in
one currency
to an investment in
another currency
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.32
Comparative Advantage
Arguments for Currency Swaps
(Table 5.6,pages 137-139)
Company A wants to borrow AUD
Company B wants to borrow USD
USD AUD
Company A 5.0% 12.6%
Company B 7.0% 13.0%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.33
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.34
Example for a Currency Swap
Suppose that the term structure of interest
rates is flat in both US and Japan,Further
suppose the interest rate is 9% pa in the US
and 4% pa in Japan,Your company has
entered into a three-year swap where it
receives 5% pa in yen on 1,200 million yen
and pays 8% pa on $10 million,The current
exchange rate is 110 yen = $1,Evaluate the
swap under the assumption that payments
are made just once per year.
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.35
Example Valued as Bonds
Here we have a domestic and a foreign bond
BD = 0.8e-0.09x1 + 0.8e-0.09x2 + 10.8e-0.09x3
= $ 9.644 million
BF = 60e-0.04x1 + 60e-0.04x2 + 1260e-0.04x3
= ￥1,230.55 million
Thus,the value of the swap is Vswap= S0BF -BD
m il li o n55.1$644.9110 55.230,1
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.36
Example Valued as FRAs
The current spot rate is 110 yen per dollar or 0.009091
dollars per yen,Because the interest rate differential
is 5% the one,two,and three year exchange rates
are (from eq (3.13))
0.009091e0.05x1 = 0.0096
0.009091e0.05x2 = 0.0100
0.009091e0.05x3 = 0.0106
The value of the forward contracts corresponding to the
exchange of interest are therefore
((60 * 0.0096) - 0.8)e-0.09x1 = -0.21
((60 * 0.0100) - 0.8)e-0.09x2 = -0.16
((60 * 0.0106) - 0.8)e-0.09x3 = -0.13
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.37
Example Valued as FRAs
(continued)
The final exchange of principal involves
receiving 1,200 million yen for $10 million,
The value of the forward contract
corresponding to this transaction is
((1,200 * 0.0106) - 10)e-0.09x3 = 2.04
Hence,the total value of the swap is
2.04 -0.13 - 0.16 - 0.21 = 1.54 million
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.38
Swaps & Forwards
A swap can be regarded as a convenient
way of packaging forward contracts
The ―plain vanilla‖ interest rate swap in our
example consisted of 6 FRAs (page 133)
The ―fixed for fixed‖ currency swap in our
example consisted of a cash transaction
& 5 forward contracts (ex.5.4)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.39
Valued as Forward Contracts
The value of the swap is the sum of the values
of the forward contracts underlying the swap
Both swaps and forwards are normally
“at-the-money” initially
– This means that it costs NOTHING to enter into
a forward or swap
– It does NOT mean that each forward contract
underlying a swap is,at-the-money” initially
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.40
Credit Risk (page 143)
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.41
Examples of Other Types of
Swaps
Amortizing & step-up swaps
(本金分期减少方式互换与本金逐步增加的互换 )
Extendible & puttable swaps (可延长与可赎回互换 )
Index amortizing rate swaps (指数递减比率互换 )
Swaption,Options on swaps (互换权 )
Equity swaps (股权的互换 )
Commodity swaps (商品的互换 )
Differential swaps (差异互换 )
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.42
Assignments
5.1,5.2,5.3,5.4,5.8,5.9,5.10,5.11,5.12,
5.15
Assignment Questions
Tang Yincai,Shanghai Normal University
5.1
Swaps
(互换 )
Chapter 5
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.2
Nature of Swaps
A swap is an agreement to
exchange cash flows (现金流 ) at
specified future times according
to certain specified rules
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.3
Terminology
LIBOR
the London InterBank Offer Rate
It is the rate of interest offered by
banks on
deposits from other banks in
Eurocurrency markets
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.4
An Example of a,Plain Vanilla”
Interest Rate Swap(大众型利率互换 )
An agreement by,Company B” to
RECEIVE 6-month LIBOR and
PAY a fixed rate of 5% pa
every 6 months for 3 years on a
notional principal of $100 million
Next slide illustrates cash flows,where
POSITIVE flows are revenues (inflows) and
NEGATIVE flows are expenses (outflows)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.5
---------Millions of Dollars---------
LIBOR FLOATING FIXED Net
Date Rate Cash Flow Cash Flow Cash Flow
Mar.1,1999 4.2%
Sept,1,1999 4.8% +2.10 –2.50 –0.40
Mar.1,2000 5.3% +2.40 –2.50 –0.10
Sept,1,2000 5.5% +2.65 –2.50 +0.15
Mar.1,2001 5.6% +2.75 –2.50 +0.25
Sept,1,2001 5.9% +2.80 –2.50 +0.30
Mar.1,2002 6.4% +2.95 –2.50 +0.45
Cash Flows to Company B
(See Table 5.1,page 123)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.6
More on Table 5.1
The floating-rate payments are calculated
using the six-month LIBOR rate prevailing six
month before the payment date
The principle is only used for the calculation
of interest payments,However,the principle
itself is not exchanged—Meaning for
―Notional principle‖
The swap can be regarded as the exchange
of a fixed-rate bond for a float-rate bond,
Company B (A) is long (short) a floating-rate
bond and short (long) a fixed-rate bond.
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.7
Typical Uses of an
Interest Rate Swap
Converting a liability
from a
– FIXED rate liability to a
FLOATING rate liability
– FLOATING rate liability
to a FIXED rate liability
Converting an investment
from a
– FIXED rate investment to a
FLOATING rate investment
– FLOATING rate investment
to a FIXED rate investment
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.8Transforming a Floating-rate
Loan to a Fixed-rate
Consider a 3-year swap initialized on March 1,2000 where
Company B agrees to pay Company A 5%pa on
$100 millionCompany A agrees to pay Company B 6-mth
LIBOR on $100 million
Suppose Company B has arranged to borrow $100
million LIBOR + 80bp
Company
B
Company
A
5%
LIBOR LIBOR+0.8%5.2%
Note,1 basis point (bp) = one-hundredth of 1%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.9
Transforming a Floating-rate
Loan to a Fixed-rate (continued)
After Company B has entered into the swap,they
have 3 sets of cash flows
1,Pays LIBOR plus 0.8% to outside lenders
2,Receives LIBOR from Company A in the swap
3,Pays 5% to Company A in the Swap
In essence,B has transformed its variable rate
borrowing at LIBOR + 80bp to a fixed rate of 5.8%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.10A and B Transform a Liability
(Figure 5.2,page 125)
A B
LIBOR
5%
LIBOR+0.8%
5.2%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.11Financial Institution is
Involved
(Figure 5.4,page 126)
A F.I,B
LIBOR LIBOR
LIBOR+0.8%
4.985% 5.015%
5.2%
“Plain vanilla” fixed-for-float swaps on US interest rates are
usually structured so that the financial institutions earns 3 to 4
basis points on a pair of offsetting transactions
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.12
A and B Transform an Asset
(Figure 5.3,page 125)
A B
LIBOR
5%
LIBOR-0.25%
4.7%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.13Financial Institution is
Involved
(See Figure 5.5,page 126)
A F.I,B
LIBOR LIBOR
4.7%
5.015%4.985%
LIBOR-0.25%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.14
The Comparative Advantage
Argument (Table 5.4,page 129)
Company A wants to borrow floating
Company B wants to borrow fixed
Fixed Floating
Company A 10.00% 6-month LIBOR + 0.30%
Company B 11.20% 6-month LIBOR + 1.00%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.15
The Comparative Advantage
(continued)
One possible swap is
Company A has 3 sets of cash flows
1,Pays 10%pa to outside lenders
2,Receives 9.95%pa from B Pays LIBOR + 0.05%
3,Pays LIBOR to B a 25bp gain
Company B has 3 sets of cash flows
1,Pays LIBOR + 1.00%pa to outside lenders
2,Receives LIBOR from A Pays 10.95%pa
3,Pays 9.95% to A a 25bp gain
Company
B
Company
A
9.95%
LIBOR10% LIBOR + 1%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.16
The Swap (Figure 5.6,page 130)
A B
LIBOR
LIBOR+1%
9.95%
10%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.17
The Swap when a Financial
Institution is Involved
(Figure 5.7,page 130)
A F.I,B
10%
LIBOR LIBOR
LIBOR+1%
9.93% 9.97%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.18
Total Gain from an
Interest Rate Swap
The total gain from an interest rate swap is always
|a-b| where
a is the difference between the interest rates in the
fixed-rate market for the two parties,and
b is the difference between the interest rates in the
floating-rate market for the two parties
In this example a=1.20% and b=0.70%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.19
Criticism of the Comparative
Advantage Argument
The 10.0% and 11.2% rates available to
A and B in fixed rate markets are 5-year
rates
The LIBOR+0.3% and LIBOR+1% rates
available in the floating rate market are
six-month rates
B’s fixed rate depends on the spread
above LIBOR it borrows at in the future
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.20
Valuation of an Interest Rate
Swap
Interest rate swaps can be valued as
the difference between
--the value of a fixed-rate bond &
--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,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.21
Valuation of an Interest Rate Swap
as a Package of Bonds
The fixed rate bond is valued in the
usual way (page 132)
The floating rate bond is valued by
noting that it is worth par immediately
after the next payment date (page 132)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.22
Valuation of an Interest Rate Swap
as a Package of Bonds (continued)
Define
Vswap,value of the swap to the financial institution
Bfix,value of the fixed-rate bond underlying the swap
Bfl,value of the floating-rate bond underlying the swap
L,notional principal in a swap agreement
ti,time when the ith payments are exchanged
ri,LIBOR zero rate for a maturity ti
Then,Vswap = Bfix - Bfl and if k is the fixed-rate coupon and k* is the floating
111111 )e(ee
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**
fl
1
f i x
trtrtr
tr
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i
tr
kLkLB
LkB nnii
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.23Example of an Interest Rate Swap
Valued as a Package of Bonds
Suppose,that you agreed to pay 6-month LIBOR and receive 8%
pa (with semiannual compounding) on a notional amount of
$100 million,The swap has a remaining life of 15 months and
the next payment is due in 3 months,The relevant rates for
continuous compounding over 3,9,and 15 months are 10.0%,
10.5%,and 11.0%,respectively,The six-month LIBOR rate at
the last payment was 10.2% (with semi-annual compounding).
In this case k = $4 million and k* = $5.1 million,so that
Bfix= 4e-0.25x0.10 + 4e-0.75x0.105 + 104e-1.25x0.11
= $ 98.24 million
Bfl = 5.1e-0.25x0.10 + 100e-0.25x0.10
= $102.51 million
Hence,Vswap = 98.24 - 102.51 = -$4.27 million
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.24
Valuation in Terms of FRAs
Each 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 (See section 4.6,page 97)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.25
Valuation of an Interest Rate Swap
as a Package of FRAs
A simple three step process
1,Calculate each of the forward rates for each of the
LIBOR rates that will determine swap cash flows
2,Calculate swap cash flows on the assumption
that the LIBOR rates will equal the forward rates
3,Set the swap rates equal to the present value of
these cash flows
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.26
Example of an Interest Rate Swap
Valued as a Package of FRAs
Same problem as before.
The cash flows for the payment in 3 months have already been set,A
rate of 8% will be exchanged for a rate of 10.2%,The NPV of this
transaction is
0.5 * 100 * (0.08 - 0.102)e-0.1*0.25 = -1.07
To figure out the NPV of the remaining two payments,we first need to
calculate the forward rates corresponding to 9 and 15 months
or 10.75% with continuous compounding which corresponds to
11.044% with semi-annual compounding.
The value of the 9 month FRA is
0.5 * 100 * (0.08 - 0.11044)e-0.105*0.75 = -1.41
1 0 7 5.05.0 10.0*25.01 0 5.0*75.093f
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.27
Example Swap as Valued as FRAs
(continued)
The 15 month forward rate is
or 11.75% with continuous compounding which corresponds
to 12.102% with semi-annual compounding.
The value of the 15 month FRA is
0.5 * 100 * (0.08 - 0.12102)e-0.11*1.25 = -1.79
Hence,the total value of the swap is -1.07 - 1.41 - 1.79 = -4.27
1 1 7 5.05.0 75.0*1 0 5.025.1*11.0159f
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.28
An Example of a Currency
Swap
An agreement to
-- pay 11% on a sterling principal of
￡10,000,000 &
-- receive 8% on a US$ principal of
$15,000,000
-- every year for 5 years
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.29
Exchange of Principal
In an interest rate swap
the principal is not exchanged
In a currency swap
the principal is exchanged at
-- the beginning &
-- the end of the swap
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.30
The Cash Flows (Table 5.5,page 137)
Years
Dollars Pounds
$
------millions------
0 –15.00 +10.00
1 +1.20 –1.10
2 +1.20 –1.10
3 +1.20 –1.10
4 +1.20 –1.10
5 +16.20 -11.10
￡
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.31
Typical Uses of a
Currency Swap
Conversion
from a liability in
one currency
to a liability in
another currency
Conversion
from an investment in
one currency
to an investment in
another currency
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.32
Comparative Advantage
Arguments for Currency Swaps
(Table 5.6,pages 137-139)
Company A wants to borrow AUD
Company B wants to borrow USD
USD AUD
Company A 5.0% 12.6%
Company B 7.0% 13.0%
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.33
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.34
Example for a Currency Swap
Suppose that the term structure of interest
rates is flat in both US and Japan,Further
suppose the interest rate is 9% pa in the US
and 4% pa in Japan,Your company has
entered into a three-year swap where it
receives 5% pa in yen on 1,200 million yen
and pays 8% pa on $10 million,The current
exchange rate is 110 yen = $1,Evaluate the
swap under the assumption that payments
are made just once per year.
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.35
Example Valued as Bonds
Here we have a domestic and a foreign bond
BD = 0.8e-0.09x1 + 0.8e-0.09x2 + 10.8e-0.09x3
= $ 9.644 million
BF = 60e-0.04x1 + 60e-0.04x2 + 1260e-0.04x3
= ￥1,230.55 million
Thus,the value of the swap is Vswap= S0BF -BD
m il li o n55.1$644.9110 55.230,1
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.36
Example Valued as FRAs
The current spot rate is 110 yen per dollar or 0.009091
dollars per yen,Because the interest rate differential
is 5% the one,two,and three year exchange rates
are (from eq (3.13))
0.009091e0.05x1 = 0.0096
0.009091e0.05x2 = 0.0100
0.009091e0.05x3 = 0.0106
The value of the forward contracts corresponding to the
exchange of interest are therefore
((60 * 0.0096) - 0.8)e-0.09x1 = -0.21
((60 * 0.0100) - 0.8)e-0.09x2 = -0.16
((60 * 0.0106) - 0.8)e-0.09x3 = -0.13
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.37
Example Valued as FRAs
(continued)
The final exchange of principal involves
receiving 1,200 million yen for $10 million,
The value of the forward contract
corresponding to this transaction is
((1,200 * 0.0106) - 10)e-0.09x3 = 2.04
Hence,the total value of the swap is
2.04 -0.13 - 0.16 - 0.21 = 1.54 million
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.38
Swaps & Forwards
A swap can be regarded as a convenient
way of packaging forward contracts
The ―plain vanilla‖ interest rate swap in our
example consisted of 6 FRAs (page 133)
The ―fixed for fixed‖ currency swap in our
example consisted of a cash transaction
& 5 forward contracts (ex.5.4)
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.39
Valued as Forward Contracts
The value of the swap is the sum of the values
of the forward contracts underlying the swap
Both swaps and forwards are normally
“at-the-money” initially
– This means that it costs NOTHING to enter into
a forward or swap
– It does NOT mean that each forward contract
underlying a swap is,at-the-money” initially
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.40
Credit Risk (page 143)
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
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.41
Examples of Other Types of
Swaps
Amortizing & step-up swaps
(本金分期减少方式互换与本金逐步增加的互换 )
Extendible & puttable swaps (可延长与可赎回互换 )
Index amortizing rate swaps (指数递减比率互换 )
Swaption,Options on swaps (互换权 )
Equity swaps (股权的互换 )
Commodity swaps (商品的互换 )
Differential swaps (差异互换 )
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,Shanghai Normal University
5.42
Assignments
5.1,5.2,5.3,5.4,5.8,5.9,5.10,5.11,5.12,
5.15
Assignment Questions
