Swaps
Chapter 7
7.1
Goals of Chapter 7

Introduce interest rate (IR) swaps (利率交換)
–
–
–
–

Introduce currency swaps (貨幣交換)
–


Definition for swaps
An illustrative example for IR swaps
Discuss reasons for using IR swaps
Quotes and valuation of IR swaps
Payoffs, reasons for using currency swaps, and
the valuation of currency swaps
Credit risk of swaps
Other types of swaps
7.2
7.1 Interest Rate (IR) Swaps
7.3
Definition of Swaps

A swap is an agreement to exchange a series
of cash flows (CFs) at specified future time
points according to certain specified rules
– The first swap contracts were created in the early
1980s
– Swaps are traded in OTC markets
– Swaps now occupy an important position in OTC
derivatives markets
– The calculation of CFs depends on the future
values of an interest rate, an exchange rate, or
other market variables
7.4
Interest Rate Swap

The most common type of swap is a “plain
vanilla” IR swap
– One party agrees to pay CFs at a predetermined
fixed rate on a notional principal for several years
– The other party pay CFs at a floating rate on the
same notional principal for the same period of time
– The floating rate in most IR swaps depends on the
LIBORs with different maturities in major currencies
– An illustrative example: On Mar. 5 of 2013, Microsoft
(MS) agrees to receive 6-month LIBOR and pay a
fixed rate of 5% with Intel every 6 months for 3 years
7.5
on a notional principal of $100 million
Cash Flows of an Interest Rate
Swap
---------Millions of Dollars--------Floating
Fixed
Net
Cash Flow Cash Flow Cash Flow
Date
LIBOR
Rate
Mar.5, 2013
4.2%
Sept. 5, 2013
4.8%
+2.10
–2.50
–0.40
Mar.5, 2014
5.3%
+2.40
–2.50
–0.10
Sept. 5, 2014
5.5%
+2.65
–2.50
+0.15
Mar.5, 2015
5.6%
+2.75
–2.50
+0.25
Sept. 5, 2015
5.9%
+2.80
–2.50
+0.30
+2.95
–2.50
+0.45
Mar.5, 2016
※ For each reference period, the 6-month LIBOR in the beginning of the period
determine the payment amount at the end of the period
– According to this rule, there is no uncertainty about the first CF exchange
※ Note that it is not necessary to exchange the principal at any time point
– This is why the principal is termed the notional principal (名義本金或名目本金), or just
the notional
7.6
Cash Flows of an Interest Rate Swap
If the Principal was Exchanged
---------Millions of Dollars--------Floating
Fixed
Net
Cash Flow Cash Flow Cash Flow
Date
LIBOR
Rate
Mar.5, 2013
4.2%
Sept. 5, 2013
4.8%
+2.10
–2.50
–0.40
Mar.5, 2014
5.3%
+2.40
–2.50
–0.10
Sept. 5, 2014
5.5%
+2.65
–2.50
+0.15
Mar.5, 2015
5.6%
+2.75
–2.50
+0.25
Sept. 5, 2015
5.9%
+2.80
–2.50
+0.30
+102.95
–102.50
+0.45
Mar.5, 2016
※ If the principal were exchanged at the end of the life of the swap, the nature (or
said the net cash flows) of the deal would not be changed in any way
– Since only the net CF changes hands in practice for IR swaps, it is not necessary to
exchange the principal at any time point
※ The IR swap can be regarded as the exchange of a fixed-rate bond (with the
CFs in the 4th column) for a floating-rate bond (with the CFs in the 3rd column)
※ This characteristic helps to evaluate the value of an IR swap (introduced later)
7.7
Interest Rate Swap
– Day count conventions for IR swaps in the U.S.



Since the 6-month LIBOR is a U.S. money market rate, it is
quoted on an actual/360 basis
As for the fixed rate, it is usually quoted as actual/365
For the first CF exchange on Slide 7.6, because there are 184
days between Mar. 5, 2013 and Sep. 5, 2013, the accurate
CF amounts are
184
= $2.1467 mil. (for the floating-rate
360
184
= $2.5205 mil. (for the fixed-rate CF)
365
$100 mil.× 4.2% ×
$100 mil.× 5% ×

CF)
For clarity of exposition, this day count issue will be ignored in
the rest of this chapter
7.8
Interest Rate Swap

Reasons for using IR swaps
1.
Converting a liability from
fixed rate to floating rate
 floating rate to fixed rate
※The Intel and Microsoft example

5.2%
5%
MS
Intel
LIBOR + 0.1%
LIBOR
Original fixedrate debt of Intel
IR swap
Original floatingrate debt of MS
– The net borrowing rate for Intel’s liability is LIBOR + 0.2%
– The net borrowing rate for MS’s liability is 5.1%
7.9
Interest Rate Swap
2.
Converting an asset (or an investment) from
fixed rate to floating rate
 floating rate to fixed rate
※The Intel and Microsoft example

LIBOR – 0.2%
5%
MS
Intel
4.7%
LIBOR
Original floatingrate asset of Intel
IR swap
Original fixed-rate
asset of MS
– The net interest rate earned for Intel’s asset is 4.8%
– The net interest rate earned for MS’s asset is LIBOR – 0.3%
7.10
Interest Rate Swap

When a financial intuition is involved
– Usually two nonfinancial companies do not get in
touch directly to arrange a swap


It is unlikely for a company to find a trading counterparty
which needs the opposite position of the IR swap, i.e.,
another firm agrees with the principal and maturity but shows
a different preference for the floating or fixed IR
In practice, each of them deals with a financial institution (F.I.)
5.015%
4.985%
5.2%
F.I.
Intel
Original fixedrate debt of Intel
MS
LIBOR
LIBOR
IR swap
IR swap
LIBOR + 0.1%
Original floatingrate debt of MS
7.11
Interest Rate Swap
LIBOR – 0.2%
Original floatingrate asset of Intel


MS
F.I.
Intel

5.015%
4.985%
LIBOR
LIBOR
IR swap
IR swap
4.7%
Original fixed-rate
asset of MS
Note that the F.I. has two separate and offsetting IR swap
contracts, and it has to honor the both contracts even Intel
or MS defaults
In most cases, Intel and MS do not even know whether the
F.I. has entered into an offsetting swap with another firm
In practice, there are many F.I.’s as market markers (or say
dealers) in OTC markets and always preparing to trade IR
swaps without having an offsetting swap
– They can hedge their unoffset swap positions with Treasury
bonds, FRAs, or other IR derivatives
7.12
Quotes By a Swap Market Maker
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
※ The quotes of IR swaps are expressed as the rate for the fixed-rate side
– Bid rate: the fixed rate that the market maker pays for buying (receiving) a
series of CFs according to LIBOR
– Offer rate: the fixed rate the market marker earns for selling (paying) a series
of CFs according to LIBOR
– Swap Rate: the fixed rate such that the value of this swap is zero (introduced
later), and the bid-offer quotes usually center on the swap rate in practice
– The plain vanilla fixed-for-floating swaps are usually structured so that the
7.13
financial institution earns about 0.03% to 0.04% in the U.S.
Comparative Advantage Argument

The comparative advantage argument
explains the popularity of the IR swaps
– AAA Corp. prefers to borrow at a floating rate and
BBB Corp. prefers to borrow at a fixed rate
– The fixed or floating IRs they need to pay are
Fixed


Floating
AAA Corp.
4.00%
6-month LIBOR – 0.1%
BBB Corp.
5.20%
6-month LIBOR + 0.6%
A key feature is that the difference between the two
fixed rates (1.2%) is greater than the difference between
the two floating rates (0.7%)
AAA (BBB) Corp. has comparative advantage in
borrowing fixed-rate (floating-rate) debt
7.14
Comparative Advantage Argument

An ideal win-win solution with a swap
– AAA Corp. borrows fixed-rate funds at 4%
– BBB Corp. borrows floating-rate funds at LIBOR + 0.6%
– Both enter into a fixed-for-floating IR swap to obtain the IRs
they prefer
4.35%
4%
BBB
AAA
LIBOR + 0.6%
LIBOR
Borrow at a
fixed rate
IR swap
Borrow at a
floating rate
The net borrowing rate for AAA Corp. is LIBOR – 0.35%, which is
by 0.25% lower than LIBOR – 0.1% if it borrows at a floating rate
directly
 The net borrowing rate for BBB Corp. is 4.95%, which is by 0.25%
lower than 5.2% if it borrows at a fixed rate directly

7.15
Comparative Advantage Argument
– Suppose AAA and BBB cannot deal directly and a F.I. is involved
4%
BBB
F.I.
AAA
Borrow at a
fixed rate
4.37%
4.33%
LIBOR
LIBOR
IR swap
IR swap
LIBOR + 0.6%
Borrow at a
floating rate
The net interest rate for AAA Corp. is LIBOR – 0.33%, which is by
0.23% lower than LIBOR – 0.1% if it borrows at a floating rate directly
 The net interest rate for BBB Corp. is 4.97%, which is by 0.23% lower
than 5.2% if it borrows at a fixed rate directly
 The gain of the F.I. is 0.04%

– Note that in both cases, the total gains of all participants is 0.5% ,
which equals (1.2% – 0.7%), where 1.2% (0.7%) is the difference
between the fixed (floating) borrowing IRs for AAA and BBB Corp.
7.16
Criticism of the Comparative
Advantage Argument

The comparative advantage arises from the
unmatched maturities for different IR rates
– The 4% and 5.2% rates available to AAA and BBB
are, for example, 5-year rates
– The LIBOR – 0.1% and LIBOR + 0.6% rates are
available to AAA and BBB for only 6 months

The fixed IR level or the spread above or below the LIBOR
reflects the creditworthiness of AAA and BBB corporations
– Since the 6-month period is so short that the default prob. of
BBB is small, BBB can enjoy the comparative advantage on
borrowing at a floating rate
– In contrast, since lenders intend to cover the default
uncertainty for a longer period of time, the 5-year fixed
borrowing rate for BBB is relatively more expensive
7.17
Criticism of the Comparative
Advantage Argument

Note that the floating-rate loan will be reviewed (so as the
creditworthiness of the borrower) and rolled over every 6
months, so the true cost to borrow at a floating rate depends
on the (LIBOR + spread) in the future
– spread changes with the creditworthiness of BBB
– The net borrowing rate for BBB is NOT FIXED at (LIBOR + 0.6%)
+ 4.37% – LIBOR = 4.97% for 5 years, but is (LIBOR + spread) +
4.37% – LIBOR = 4.37% + spread dependent on its future
creditworthiness every 6 months
– In contrast, if BBB borrows at a fixed rate, the borrowing rate is
fixed at 5.2% for 5 years, regardless of its future creditworthiness
– As a result, BBB cannot achieve its goal with a swap perfectly if
its creditworthiness changes in the future
※ The above inference disprove that the comparative advantage
7.18
argument can explain the popularity of IR swaps
The Nature of Swap Rates

The n-year swap rate is a constant interest rate
corresponding to a credit risk for 2n
consecutive 6-month LIBOR loans to AA-rated
companies
– First, it is known that the 6-month LIBOR is a shortterm AA-rating borrowing rate
– Second, a F.I. can earn the n-year swap rate by
Lending for the first 6-month loan to a AA borrower and
relending it for successive 6-month periods to other AA
borrowers for n years, and
2. Entering into a IR swap to exchange the LIBOR income in
the above step for the constant CFs at the n-year swap rate7.19
1.
The Nature of Swap Rates
– Note that the n-year swap rates are lower than nyear AA-rating fixed borrowing rates
For the swap rate, the creditworthiness of the borrowers
are always AA for the whole n-year period
 For n-year AA-rating fixed borrowing rates, it is only
known that the initial creditworthiness of the borrower is
AA-rating at the beginning of the n-year period
※The lower credit risk for earning the swap rate leads to the
lower n-year swap rate than the n-year AA-rating fixed
borrowing rate
※Recall that the credit risks of AA-rating companies are
small in practice, it can be inferred that the swap rates are
closer to risk-free

7.20
Valuation of IR Swaps

There are two approaches to price IR swaps
1. Regard the value of an IR swap as the
difference between the values of a fixed-rate
bond and a floating-rate bond (see Slide 7.7)
2. Regard an IR swap as a portfolio of forward rate
agreements (FRAs) (For the Intel and MS 3year IR swap, it can be regarded separately as
5 FRAs)
7.21
Valuation of IR Swaps

Valuation in terms of bond prices
– For a swap where fixed CFs are received and
floating CFs are paid, its value can be expressed as
𝑉swap = 𝐵fix − 𝐵fl ,
where 𝐵fix and 𝐵fl denote the values of a fixed-rate
and floating-rate bonds
– The value of a fixed-rate bond (Bfix) can be derived
with the traditional discounted cash flow method
– The value of a floating-rate bond (Bfl) that pays 6month LIBOR is always equal to its PAR VALUE
immediately after the each payment date
7.22
Valuation of IR Swaps

Price Bfl (with the face value to be $100) with 1.5 years to
maturity in one possible scenario for the 6-month LIBOR
$100+$2
$3
$4
6%
4%
8%
t=0
t=0.5
t=1
$100 ( 
$100 ( 
$100 ( 
$3  $100
1  6%  0.5

$4  $100
1  8%  0.5
$3
1  6%  0.5


t=1.5
$2  $100
1  4 %  0 .5
$4
1  8%  0.5

)
$102
(1  8%  0.5)((1  4%  0.5)
$4
(1  6%  0.5)(1  8%  0.5)

)
$102
(1  6%  0.5)(1  8%  0.5)(1  4%  0.5)
)
※ Note that in any scenario for LIBOR and for different life time of
bonds, the Bfl is worth its par value on the issue date and on each
date immediately after the coupon payment date
7.23
Valuation of IR Swaps

Generalization for pricing Bfl (with the principal (or said
par value) L)
Value = L = PV
of L+k* at t*
Value = L
Value =
L+k*
0
Valuation
Date
t
t*
First Pmt
Date
(Floating
Pmt = k*)
Second
Pmt Date
Maturity
Date
※ Note that the value of a Bfl at any time point 𝑡 is the PV of (𝐿 + 𝑘 ∗ ) at 𝑡,
∗
i.e., 𝐿 + 𝑘 ∗ 𝑒 −𝑟(𝑡 −𝑡) , where 𝐿 + 𝑘 ∗ is the value of the Bfl on the next
payment date and 𝑟 is the continuously compounding zero rate
corresponding to the time to maturity of (𝑡 ∗ − 𝑡)
7.24
Valuation of IR Swaps

An example for pricing IR swaps
– For the party to pay the six-month LIBOR and
receive fixed 8% (semi-annual compounding) on a
principal of $100 million
– Remaining life of the IR swap is 1.25 years
– LIBOR rates for 3-months, 9-months and 15months are 10%, 10.5%, and 11% (continuously
compounding)
– The 6-month LIBOR on the last payment date was
10.2% (semi-annual compounding)
7.25
Valuation of IR Swaps
Time 𝑩𝐟𝐢𝐱 CF
(yr)
𝑩𝐟𝐥 CF
Discount Factor
0.25
$4
$105.1
𝑒 −10%∙0.25 = 0.9753
$3.901
0.75
$4
𝑒 −10.5%∙0.75 = 0.9243
$3.697
1.25
$104
𝑒 −11%∙1.25 = 0.8715
$90.640
Total
PV of 𝑩𝐟𝐢𝐱
CF
$98.238
PV of 𝑩𝐟𝐥
CF
$102.505
$102.505
※ For per $100 principal
– The coupon payment of 𝐵fl after 3 months is 0.5 × 10.2% ×
$100 = $5.1
– The value of 𝐵fl today is $100 + $5.1 𝑒 −10%∙0.25 = $102.505
according to the formula on Slide 7.23
– 𝑉swap = 𝐵fix − 𝐵fl = $98.238 − $102.505 = −$4.267
7.26
Valuation of IR Swaps

Valuation in terms of FRAs
– Each exchange in an IR swap is an FRA


Note that for a newly issued IR swap, the first exchange
of payments is known when the swap is negotiated
For each of other exchanges, it can be regarded as a
FRA applied for a future period of 6 months
– Recall that for a FRA applied in (𝑇1 , 𝑇2 ], the payoff of the
lender at 𝑇2 is 𝐿(𝑅𝐾 − 𝑅𝑀 ) 𝑇2 − 𝑇1 (see Slide 4.35), where
𝐿 is the principal, 𝑅𝐾 is the fixed IR specified in the FRA
contract, and 𝑅𝑀 is the actual LIBOR in (𝑇1 , 𝑇2 ]
– Considering a pay-floating-receive-fixed IR swap with the
principal 𝐿, for each 6 months, the swap holder can receive
the net payoff of 𝐿 𝑅𝐾 − 𝑅𝑀 ×0.5, where 𝑅𝑀 is the actual 6month LIBOR for that period and 𝑅𝐾 is the fixed IR specified
in the swap contract
7.27
Valuation of IR Swaps
– Note that the value of any derivatives equals the
present value of it expected payoff
This approach has been used to price FRA on Slide 4.38
 To evaluate the expected payoff of an exchange in an IR
swap, the expectation of the future LIBOR is needed
𝑒 −𝑅2 𝑇2 𝐸 𝐿 𝑅𝐾 − 𝑅𝑀 𝑇2 − 𝑇1
= 𝑒 −𝑅2 𝑇2 𝐿 𝑅𝐾 − 𝐸[𝑅𝑀 ] 𝑇2 − 𝑇1
= 𝑒 −𝑅2 𝑇2 𝐿[𝑅𝐾 𝑇2 − 𝑇1 − 𝐸[𝑅𝑀 ] 𝑇2 − 𝑇1 ]
 It is known that the expected future LIBORs equal the
forward rates (𝑅𝐹 ) based on today’s term structure of IRs:
Value of an exchange = 𝑒 −𝑅2 𝑇2 𝐿[𝑅𝐾 𝑇2 − 𝑇1 − 𝑅𝐹 𝑇2 − 𝑇1 ]
= 𝑒 −𝑅2 𝑇2 𝐿(𝑅𝐾 − 𝑅𝐹 ) 𝑇2 − 𝑇1

※ The above formula is identical to the FRA pricing formula on Slide 4.38
7.28
Valuation of IR Swaps
– Consider the pay-floating-receive-fixed IR swap
example on Slide 7.24. For per $100 principal,
Time
(yr)
Fixed
CF
Expected
floating CF
Expected
net CF
Discount
factor
PV of expected
net CF
0.25
$4
$5.100
–$1.100
0.9753
–$1.073
0.75
$4
$5.522*
–$1.522
0.9243
–$1.407
1.25
$4
$6.051**
–$2.051
0.8715
–$1.787
–$4.267
Total
∗
𝑅𝐹 =
10.5%×0.75−10%×0.25
0.5
= 10.75% (cont. comp.) ⇒ 11.044% (semi-annual comp.)
Expected cash outflow at 𝑡 = 0.75 is $100 × 11.044% × 0.5 = $5.522
∗∗
𝑅𝐹 =
11%×1.25−10.5%×0.75
0.5
= 11.75% (cont. comp.) ⇒ 12.102% (semi-annual comp.)
Expected cash outflow at 𝑡 = 1.25 is $100 × 12.102% × 0.5 = $6.051
7.29
Valuation of IR Swaps

An IR swap is worth zero when it is first initiated
– When a swap contract is first negotiated, the swap
rate is determined such that the value of the swap is
zero initially

This feature is similar to set the delivery prices of futures
contracts to be the futures prices such that the futures
contracts are worth zero when they are initiated
– With the passage of time, the value of an IR swap
emerges and can be either positive or negative

One party’s gains are the other party’s losses, so two
parties of a swap have opposite points of view on the swap
value
7.30
Valuation of IR Swaps
– Although the swap is zero initially, it does not mean
that the value of each individual FRA is zero initially


The initial zero value of a swap actually means that the
sum of the values of all FRAs underlying the swap is zero
For a pay-fixed-receive-floating swap on the issue date,
– If the zero curve is upward sloping  forward rates ↑ with T

The forward rates with shorter time to maturities < the swap rate
 negative values for FRAs with shorter time to maturities
 The forward rates with longer time to maturities > the swap rate
 positive values for FRAs with longer time to maturities
– If the zero curve is downward sloping  forward rates ↓ with T

The forward rates with shorter time to maturities > the swap rate
 positive values for FRAs with shorter time to maturities
 The forward rates with longer time to maturities < the swap rate
 negative values for FRAs with longer time to maturities
7.31
Determine LIBOR Zero Curve with
Eurodollar Futures and Swaps

Construct the LIBOR zero curve
(Note that derivatives traders commonly use LIBOR as
proxies for risk-free rates when trading derivatives)
– 𝑇 < 1: the quotes of spot LIBOR (given different 𝑇)
provided by financial institutions are used
– 𝑇 in [1,2] (sometimes [1,5]): the quotes of Eurodollar
futures are used to derive the LIBOR zero rates



Suppose the zero rate 𝑅1 for 𝑇1 is known
With the convexity adjustment, the forward rates (𝑅𝐹 ) for
[𝑇1 , 𝑇2 ] can be derived from the futures rates implied from
the quotes of Eurodollars futures
Finally, we can deduce 𝑅2 through
𝑅𝐹 =
𝑅2 𝑇2 −𝑅1 𝑇1
𝑇2 −𝑇1
⇒ 𝑅2 =
𝑅𝐹 𝑇2 −𝑇1 +𝑅1 𝑇1
𝑇2
7.32
Determine LIBOR Zero Curve with
Eurodollar Futures and Swaps
– For longer 𝑇: the quotes of swap rates are used to
derive the LIBOR zero rate
Consider a pay-floating-receive-fixed IR swap with the swap
rate of 5%, principal of $100, and 2 years to maturity
 Suppose the 6-month, 12-month, and 18-month LIBOR zero
rates are 4%, 4.5%, and 4.8% with cont. compounding
 Since the initial value of a swap is zero, then 𝑉swap = 𝐵fix −
𝐵fl = 𝐵fix − $100 = 0 and thus
$2.5𝑒 −4%∙0.5 + $2.5𝑒 −4.5%∙1 + $2.5𝑒 −4.8%∙1.5 + $102.5𝑒 −𝑅∙2 = $100
 Solve for the 2-year LIBOR zero rate to be 4.953%
 The above equation also demonstrates a swap rate equals a
par yield
※Similar to the bootstrap method, LIBOR rates for shorter 𝑇
should be solved first before solving LIBOR rates for longer 𝑇 7.33

Overnight Indexed Swaps (OISs)

An OIS is a swap where a fixed rate for a period
is exchanged for the geometric average of the
overnight rates during the period
– The fixed rate is referred to as the OIS rate, which is
determined such that an OIS is worth zero initially
– OISs tend to have short lives (≤ 3 months)
– Longer-term OISs are typically divided into threemonth sub-periods

At the end of each sub-period, the net of the actual geometric
average of the overnight rates during the sub-period and the
fixed OIS rate will be exchange
– Should the 3-month OIS rate equal the 3-month
LIBOR rate?
7.34
Overnight Indexed Swaps (OISs)
1. Borrow $100 in the overnight market for 3 months (92 days for
example), rolling the interest and principal on the loan forward each
night (pay the geometric average of the overnight rates)
2. Enter into an OIS to convert the geometric average of the overnight
rates to the 3-month OIS rate
3. Lend the borrowed $100 to another AA-rated financial institution for
three months at LIBOR
※ Final payoff = $100 × (92/365) (LIBOR − OIS rate) > 0
– Thus, the OIS rate is lower than the LIBOR



This is because OIS rate is a continually refreshed overnight
rate (always lend or borrow daily with AA financial institutions)
To earn 3-month LIBORs, the bank will bear the default risk of
its trading counterparty, which is rated AA initially
The OIS rate is even closer to the risk-free interest rate
7.35
Overnight Indexed Swaps (OISs)
– In practice, many derivatives dealers choose to use
OIS rates for discounting collateralized transactions
(less risky) and use LIBORs for discounting
noncollateralized transactions (more risky)
– The (LIBOR – OIS) spread


Defined as the 3-month LIBOR rate over the 3-month OIS rate
Can be used to measure the degree of stress in financial
markets
– In normal market condition, this spread is about 10 basis points
– In Oct. 2009, this spread spiked to an all-time high of 364 basis
points because banks are reluctant to lend to each other for threemonth periods
– In Dec. 2011, due to the concern of the crisis in Greece, this
7.36
spread rose to 50 basis points
Determine Zero Curve Using OISs

Similar to the method for constructing the LIBOR
zero curve, we can derive zero curve using OIS
quotes
– 𝑇 < 3 months: the quotes of OIS rates provided by
financial institutions are used
– For a longer 𝑇 (when there are periodic settlements
(usually every 3 months) in OIS contracts)



The OIS rate approximately defines a par yield bond
For a 1.25-year OIS contract with the OIS rate to be 4%, it
can be regarded as a bond paying a quarterly coupon at a
rate of 4% per annum and sold at par
Suppose the 3-, 6-, 9-, and 12-month OIS zero rates are 3%,
7.37
3.5%, 4%, and 4.5% with continuous compounding
Determine Zero Curve Using OISs


This OIS contracts implies that
$1𝑒 −3%∙0.25 + $1𝑒 −3.5%∙0.5 + $1𝑒 −4%∙0.75 + $1𝑒 −4.5%∙1
+$101𝑒 −𝑅∙1.25 = $100
Solve for the 1.25-year OIS zero rate to be 3.9798%
– For a 𝑇 is so long such that the quotes of OIS rates
are not available or unreliable, e.g., 𝑇 > 5 years



Note that LIBOR IR swaps are traded for longer maturities
than OIS
Assume the (LIBOR – OIS) spread is constant and as it is for
the longest OIS maturity for which there is reliable data, e.g.,
the 5-year OIS contract and the corresponding (LIBOR –
OIS) spread is 20 basis points
Use the LIBOR zero curve minus the constant (LIBOR – OIS)
7.38
spread to derive the OIS rate zero curve
7.2 Currency Swaps
7.39
Currency Swap

Currency swap is another popular type of swaps
– It involves exchanging principal and interest
payments in one currency for principal and interest
payments in another currency


Different from IR swaps, the principal amounts (in different
currencies) are exchanged at the beginning and at the end
of the life of a currency swap
The principal amounts are chosen to be approximately
equivalent using the exchange rate at the swap’s initiation
– An example of a currency swap: IBM pays 5% on a
principal of £10,000,000 and receive 6% on a
principal of $15,000,000 from British Petroleum (BP)
7.40
every year for 5 years
Currency Swap
£10 mil.
Year
Dollar CF
for IBM
(millions)
Sterling CF
for IBM
(millions)
2013
–15.00
+10.0
2014
+0.90
–0.5
2015
+0.90
–0.5
2016
+0.90
–0.5
2017
+0.90
–0.5
2018
+15.90
–10.5
BP
IBM
$15 mil.
Dollar 6%
BP
IBM
Sterling 5%
$15 mil.
BP
IBM
£10 mil.
※ A currency swap can be regarded as two
concurrent loans denominated in different
currencies
※ The values of $15 mil. and £10 mil. are set to be
equivalent initially ⇒ Two parties lend equivalent
amount of loans to each other ⇒ The net value
of the currency swap is zero initially
7.41
Currency Swap

Typical uses of a currency swap is to
– Convert a liability in one currency to a liability in
another currency
Dollar 6%
Dollar 6%
BP
IBM
Sterling 5%
Sterling 5%
– Convert an investment in one currency to an
investment in another currency
Dollar 6%
BP
IBM
Sterling 5%
Dollar 6%
Sterling 5%
7.42
Comparative Advantage
Arguments for Currency Swaps

The comparative advantage argument
explains the popularity of the currency swaps
– General Electric (GE) prefers to borrow AUD and
Qantas Airways (QA) prefers to borrow USD
– The USD and AUD borrowing IRs they face are
USD
AUD
General Electric
5.0%
7.6%
Qantas Airways
7.0%
8.0%
※GE has a comparative advantage in the USD
market, whereas QA has a comparative advantage
in the AUD market
7.43
Comparative Advantage
Arguments for Currency Swaps

Exploit the comparative advantage with
currency swaps
– Suppose that GE intends to borrow 20 mil. AUD
and QA intends to borrow 15 mil. USD, and the
current exchange rate is 0.75USD per AUD
– GE borrows USD, QA borrows AUD, and they use
currency swaps to transform GE’s USD loan into
a AUD loan and QA’s AUD loan into a USD loan
USD 6.3%
USD 5.0%
USD 5.0%
QA
F.I.
GE
AUD 6.9%
AUD 8.0%
AUD 8.0%
※GE pays 6.9% in AUD (0.7% better off) and QA pays 6.3%
in USD (0.7% better off)
7.44
Comparative Advantage
Arguments for Currency Swaps
– Different ways to arrange the currency swaps
1. QA bears some foreign exchange risk
USD 5.2%
USD 5.0%
USD 5.0%
QA
F.I.
GE
AUD 8.0%
AUD 6.9%
AUD 6.9%
2. GE bears some foreign exchange risk
USD 6.3%
USD 6.1%
USD 5.0%
QA
F.I.
GE
AUD 8.0%
AUD 8.0%
AUD 8.0%
※ These two alternatives are unlikely to be used in practice
because the firms prefer to eliminate the foreign exchange
risk with currency swaps thoroughly
7.45
Valuation of Currency Swaps

Like IR swaps, currency swaps can be valued
either as the difference between 2 bonds or as
a portfolio of forward contracts
– Valuation in terms of bond prices


For a receive-dollar-pay-foreign-currency currency swap,
then
𝑉swap = 𝐵𝐷 − 𝑆0 𝐵𝐹 ,
where 𝐵𝐷 is the domestic bond defined by the remaining
USD CFs, 𝐵𝐹 is the bond defined by the remaining foreigncurrency CFs, and 𝑆0 is the spot exchange rate (expressed
as dollars for per unit of foreign currency)
In contrast, for a pay-dollar-receive-foreign-currency
currency swap, then 𝑉swap = 𝑆0 𝐵𝐹 − 𝐵𝐷
7.46
Valuation of Currency Swaps

An example for pricing currency swaps
– All Japanese LIBOR zero rates are 4% (foreign
IR)
– All USD LIBOR zero rates are 9% (domestic IR)
– A currency swap is to received 5% in yen and pay
8% 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 dollar
7.47
Valuation of Currency Swaps
Time
(yr)
Cash Flows of
𝑩𝑫 (million $)
PV of 𝑩𝑫
(million $)
Cash flows of
𝑩𝑭 (million yen)
PV of 𝑩𝑭
(million yen)
1
0.8 (=10×8%)
0.7311
60 (=1,200×5%)
57.65
2
0.8 (=10×8%)
0.6682
60 (=1,200×5%)
55.39
3
0.8 (=10×8%)
0.6107
60 (=1,200×5%)
53.22
3
10
7.6338
1,200
1,064.30
Total
9.6439
※𝑉swap = 𝑆0 𝐵𝐹 − 𝐵𝐷 =
1,230.55
−
110
1,230.55
9.6439 = 1.543 (million $)
7.48
Valuation of Currency Swaps
– Valuation in terms of forward contracts

Each exchange of payments in a fixed-for-fixed currency
swap is a foreign exchange (FX) forward contract
– FX forwards is an agreement to trade an amount of a foreign
currency at the specified price on a predetermined future date
– FX forwards are similar to the foreign currency futures contracts
(introduced in Ch. 5) except that FX forwards are traded in OTC
markets and thus there is no daily settlement requirement
– Suppose the principal is 𝐿 and the specified trading price is 𝐾
(expressed as domestic dollars / per foreign dollar)
– Payoff of a FX forwards to purchase the foreign currency at 𝑇 is
𝐿(𝑆𝑇 − 𝐾), where 𝑆𝑇 is the domestic-dollar price of the foreign
currency (or the FX rate) at 𝑇
– Rewrite the payoff to be 𝐿𝑆𝑇 − 𝐿𝐾, which can be reinterpreted
as buying 𝐿 units of a foreign currency (worth 𝑆𝑇 at 𝑇) with 𝐿𝐾
7.49
units of domestic dollars at 𝑇
Valuation of Currency Swaps
– The value of a FX forward is the PV of its expected payoff
𝑒 −𝑟𝑇 𝐸 𝐿𝑆𝑇 − 𝐿𝐾 = 𝑒 −𝑟𝑇 (𝐿𝐸 𝑆𝑇 − 𝐿𝐾)
– The forward (or futures) price of the foreign currency provide
the unbiased approximation for 𝐸 𝑆𝑇 based on the information
of IRs today
– Since the forward price of the foreign currency is 𝐹0 =
𝑆0 𝑒 (𝑟−𝑟𝑓)𝑇 (introduced on Slides 5.23-5.24), we can obtain
𝐸 𝑆𝑇 = 𝐹0 = 𝑆0 𝑒 𝑟−𝑟𝑓 𝑇
– Thus, the value of a FX forward is
𝑒 −𝑟𝑇 (𝐿𝐹0 − 𝐿𝐾)
– The forward FX rates of Japanese Yen, 𝐹0 , in the example on
Slide 7.47 are
Time (yr)
Forward FX rate
($/per Yen)
1
2
3
1 (9%−4%)∙1
𝑒
110
= 0.009557
1 (9%−4%)∙2
𝑒
110
= 0.010047
1 (9%−4%)∙3
𝑒
110
= 0.010562
7.50
Valuation of Currency Swaps
– Take the first exchange in the currency swap for example: it can
be regarded as a FX forward to purchase ￥60 million with $0.8
million
– Value of the first exchange = 𝑒 −9%∙1 ￥60 ∙ 𝐸 𝑆1 − $0.8
= 𝑒 −9%∙1 ￥60 ∙ 0.009557 − $0.8
= −0.2071 (million $)
Time Dollar CF Yen CF Expected future
(yr)
(mil. $)
(mil. Yen)
FX rate =
forward FX rate
Yen CF
in dollar
(mil. $)
Net CF
(mil. $)
PV of net
CF (mil. $)
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
1,200
0.010562
12.6746
+2.6746
+2.0417
Total
+1.5430
7.51
7.3 Credit Risk of Swaps
7.52
Credit Risk

Contracts such as swaps or forwards that are
private arrangements between two parties
entail credit risk
– A swap is worth zero to both counterparties initially
– At a future time point, its value is possible to be
either positive or negative
– A swap trader has credit risk exposure only when
the value of its swap or forward position is positive

The trading counterparty has the chance not to honor its
losses, which results in the credit risk
– Potential losses from defaults on a swap are much
less than the potential losses from defaults on a
loan with the same principal
7.53
Credit Risk

Since the value of a swap is the difference between two
concurrent bonds, the value of a swap is usually a small
fraction of its notional principal
– Potential losses from defaults on a currency swap
are greater on an IR swap because the principal
amounts in different currencies are exchanged at
the end of the life of a currency swap
– Credit vs. Market risks


Credit risk arises from the possibility of a default, but the
market risk arises from the changes of the market
variables, such as IR and FX rates
Market risks can be hedged by entering into offsetting
contracts, but credit risks are more difficult to hedge
7.54
Credit Risk

Credit default swaps (信用違約交換) (CDSs)
– Invented by JPMorgan in 1997
– An insurance policy with payoffs depending on the
occurrence of the default event of a corporate bond
or loan
– CDSs can shift the default risk from the protection
buyer to the protection seller (see the next slide)
– CDSs allow financial institutions to hedge credit risks
in the same way that they have hedged market risks
– The total size of outstanding CDS contracts reaches
a peak of $63 trillion before the 2008-2009 credit
crisis (US GDP is about $14 trillion per year)
7.55
Credit Risk
※ When the default event occurs, the protection seller should compensate the protection
buyer any losses on principal in the default event
※ For the protection buyer, CDS provides insurance against the possibility that a
borrower (the reference entity (參考實體)) might not pay
※ For the protection seller, CDS provides a way to earn profits by bearing default risk
without ever holding the credit instrument physically
7.56
7.4 Other Types of Swaps
7.57
Other Types of Swaps

Variations on IR swaps
– The tenors (i.e., the payment frequency) for the
floating- and fixed-rate sides could be different
– Other floating rates, like the commercial paper rate,
could be used
– Amortizing (攤銷) (or step up) swaps

The principal reduces (or increases) in a predetermined
way
– Deferred swaps

Also known as the forward-start swap, where the parties
do not begin to exchange interest payments until some
future date
7.58
Other Types of Swaps
– Constant maturity swaps (CMS swaps)


An agreement to exchange a LIBOR + spread (or a fixed rate)
for a fixed-maturity swap rate (constant maturity side)
For example, exchange the 6-month LIBOR + 0.1% (or 5.5%)
for 10-year swap rates every 6 months for the next 5 years
– Constant maturity Treasury swaps

E.g., exchange 6-month LIBOR + 0.15% for the 2-year
Treasury par yield every 6 months for the next 3 years
– Par yield: a coupon rate that causes the bond price to equal its
face value
– Compounding swaps

Interest on one or both sides is compounded to the end of
the life of the swap and thus there is only one payment at the
end of the life of the swap
7.59
Other Types of Swaps
– LIBOR-in-arrears (遞延) swaps


The LIBOR observed on the payment date is used to
calculate the payment on that date
Note that for standard IR swaps, the 6-month LIBOR
prevailing six months ago determines the current floating
payment
– Accrual (孳生) swaps


The interest on one side accrues only when the floating
reference rate is in a certain range
For example, when the LIBOR rates are between [1%,2%],
the floating interest payments should be exchanged
7.60
Other Types of Swaps

Other currency swaps
– Fixed-for-floating currency swaps



A LIBOR in one currency is exchanged for a fixed rate in
another currency
A combination of a fixed-for-floating IR swap and a fixedfor-fixed currency swap
It is also known as a cross-currency interest rate swap
– Floating-for-floating currency swaps


A LIBOR in one currency is exchanged for a LIBOR in
another currency
A combination of two fixed-for-floating IR swaps and a
fixed-for-fixed currency swap
7.61
Other Types of Swaps
– Differential Swaps




For example, for an amount of notional principal in USD,
exchange LIBOR in USD with LIBOR in yen
Note that the theoretically LIBOR in yen should be applied
to the principal in yen rather than the principal in USD
It is also known as quanto swap
Other types of swaps
– Equity swaps



Exchange the total return (dividends plus capital gains)
realized on an equity index for either a fixed or a floating IR
Used by portfolio managers to purchase a series of equity
index returns with a fixed or floating IR
Useful to escape from the capital controls of some nations 7.62
Other Types of Swaps
– Commodity swaps


An agreement where a floating (or market or spot) price
based on an underlying commodity is exchanged for a
fixed price for a following period
It can be decomposed into a series of forward contracts on
a commodity with different maturity dates and identical
delivery price
– Volatility swaps


At the end of each reference period, one side pays a preagreed volatility, e.g., 20%, and the other side pays the
actual volatilities of the underlying variable, e.g., 23%, in
the past reference period
Both volatilities are multiplied by the same notional
principal in calculating payments
7.63