Swaps Chapter 7 7.1. Goals of Chapter 7 7.2 Introduce interest rate (IR) swaps ( 利率交換 ) – Definition for swaps – An illustrative example for IR swaps

Swaps

Chapter 7

7.1

Goals of Chapter 7

7.2

Introduce interest rate (IR) swaps (利率交換 ) – Definition for swaps– An illustrative example for IR swaps– Discuss reasons for using IR swaps– Quotes and valuation of IR swaps

Introduce currency swaps (貨幣交換 )– Payoffs, reasons for using currency swaps, and

the valuation of currency swaps Credit risk of swaps Other types of swaps

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 on a notional principal of $100 million 7.5

7.6

---------Millions of Dollars---------

LIBOR Floating Fixed Net

Date Rate Cash Flow Cash Flow Cash Flow

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

Mar.5, 2016 +2.95 –2.50 +0.45

Cash Flows of an Interest Rate Swap

※ 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

7.7

Cash Flows of an Interest Rate Swap If the Principal was Exchanged

※ 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)

---------Millions of Dollars---------

LIBOR Floating Fixed Net

Date Rate Cash Flow Cash Flow Cash Flow

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

Mar.5, 2016 +102.95 –102.50 +0.45

7.8

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

(for the floating-rate CF) (for the fixed-rate CF)

For clarity of exposition, this day count issue will be ignored in the rest of this chapter

Reasons for using IR swaps1. Converting a liability from

fixed rate to floating rate floating rate to fixed rate

※ The Intel and Microsoft example

7.9

Interest Rate Swap

Intel MS

LIBOR

5% LIBOR + 0.1%5.2%

{ ¿ { ¿ { ¿

Original fixed-rate debt of Intel

Original floating-rate debt of MS

IR swap

– The net borrowing rate for Intel’s liability is LIBOR + 0.2%– The net borrowing rate for MS’s liability is 5.1%

2. Converting an asset (or an investment) from fixed rate to floating rate floating rate to fixed rate

※ The Intel and Microsoft example

7.10

Interest Rate Swap

– 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%

Intel MS

LIBOR

5% 4.7%LIBOR – 0.2%

{ ¿ { ¿ { ¿

Original floating-rate asset of Intel

Original fixed-rate asset of MS

IR swap

7.11

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.)

IntelMS

LIBOR

5.015% LIBOR + 0.1%5.2%

{ ¿ { ¿ { ¿Original fixed-rate debt of Intel

IR swap Original floating-rate debt of MS

F.I.

LIBOR{ ¿

IR swap

4.985%

7.12

Interest Rate Swap

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

IntelMS

LIBOR

5.015% 4.7%LIBOR – 0.2%

{ ¿ { ¿ { ¿

Original floating-rate asset of Intel

IR swap Original fixed-rate asset of MS

F.I.

LIBOR{ ¿IR swap

4.985%

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

7.13

※ 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

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

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

Fixed Floating

AAA Corp. 4.00% 6-month LIBOR – 0.1%

BBB Corp. 5.20% 6-month LIBOR + 0.6%


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

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

AAA BBB

LIBOR

4.35%LIBOR + 0.6%4%

{ ¿ { ¿ { ¿

Borrow at a fixed rate

Borrow at a floating rate

IR swap


– Suppose AAA and BBB cannot deal directly and a F.I. is involved

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

AAABBB

LIBOR

4.37% LIBOR + 0.6%4%

{ ¿ { ¿ { ¿

Borrow at a fixed rate

IR swap Borrow at a floating rate

F.I.

LIBOR{ ¿IR swap

4.33%

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 + ) in the future– 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 + ) + 4.37% – LIBOR = 4.37% + 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 argument can explain the popularity of IR swaps 7.18

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 short-

term AA-rating borrowing rate– Second, a F.I. can earn the n-year swap rate by

1. 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

The Nature of Swap Rates

– Note that the n-year swap rates are lower than n-year 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 swaps1. 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 3-year IR swap, it can be regarded separately as 5 FRAs)

7.21

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

,

where and 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 6-month LIBOR is always equal to its PAR VALUE immediately after the each payment date

7.22


Price Bfl (with the face value to be $100) with 1.5 years to maturity in one possible scenario for the 6-month LIBOR

7.23


6% 8% 4%

$3 $4$100+$2

t=0 t=0.5 t=1 t=1.5

))5.0%41)((5.0%81(

102$

5.0%81

4$

5.0%81

100$4$( 100$

))5.0%41)(5.0%81)(5.0%61(

102$

)5.0%81)(5.0%61(

4$

5.0%61

3$

5.0%61

100$3$( 100$

)5.0%41

100$2$( 100$

※ 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

Generalization for pricing Bfl (with the principal (or said par value) L)

※ Note that the value of a Bfl at any time point is the PV of at , i.e., , where is the value of the Bfl on the next payment date and is the continuously compounding zero rate corresponding to the time to maturity of 7.24


0 t*

Valuation Date

First PmtDate

(Floating Pmt = k*)

SecondPmt Date

Maturity Date

Value = L

Value = L+k*

Value = L = PV of L+k* at t*

t

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 15-

months 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

※ For per $100 principal– The coupon payment of after 3 months is

– The value of today is according to the formula on Slide 7.23


7.26

Time (yr)

CF CF Discount Factor PV of CF PV of CF

0.25 $4 $105.1 $3.901 $102.505

0.75 $4 $3.697

1.25 $104 $90.640

Total $98.238 $102.505

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 , the payoff of the lender at

is (see Slide 4.35), where is the principal, is the fixed IR specified in the FRA contract, and is the actual LIBOR in

– Considering a pay-floating-receive-fixed IR swap with the principal , for each 6 months, the swap holder can receive the net payoff of , where is the actual 6-month LIBOR for that period and is the fixed IR specified in the swap contract

7.27


– 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

=

= It is known that the expected future LIBORs equal the

forward rates ) based on today’s term structure of IRs:

Value of an exchange =

=7.28


※ The above formula is identical to the FRA pricing formula on Slide 4.38

– Consider the pay-floating-receive-fixed IR swap example on Slide 7.24. For per $100 principal,

7.29


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

Total –$4.267

(cont. comp.) 11.044% (semi-annual comp.) Expected cash outflow at is (cont. comp.) 12.102% (semi-annual comp.) Expected cash outflow at is


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


– 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)– : the quotes of spot LIBOR (given different ) provided

by financial institutions are used– in (sometimes ): the quotes of Eurodollar futures

are used to derive the LIBOR zero rates Suppose the zero rate for is known With the convexity adjustment, the forward rates () for can

be derived from the futures rates implied from the quotes of Eurodollars futures

Finally, we can deduce through

7.32

Determine LIBOR Zero Curve with Eurodollar Futures and Swaps

– For longer : 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 and thus

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

should be solved first before solving LIBOR rates for longer 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 three-

month 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


7.35

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)

– 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


7.36

– 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 three-month periods

– In Dec. 2011, due to the concern of the crisis in Greece, this 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– months: the quotes of OIS rates provided by financial

institutions are used– For a longer (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%, 3.5%, 4%, and 4.5% with continuous compounding 7.37

Determine Zero Curve Using OISs

This OIS contracts implies that

Solve for the 1.25-year OIS zero rate to be 3.9798%

– For a is so long such that the quotes of OIS rates are not available or unreliable, e.g., 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) spread to derive the OIS rate zero curve 7.38

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) every year for 5 years 7.40

7.41

Currency Swap

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

IBM BP

Sterling 5%

Dollar 6%

IBM BP

$15 mil.

£10 mil.

IBM BP

£10 mil.

$15 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

Typical uses of a currency swap is to – Convert a liability in one currency to a liability in

another currency

– Convert an investment in one currency to an investment in another currency

7.42

Currency Swap

IBM BP

Sterling 5%

Dollar 6%Dollar 6% Sterling 5%

IBM BP

Sterling 5%

Dollar 6%

Sterling 5%

Dollar 6%

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

※GE has a comparative advantage in the USD market, whereas QA has a comparative advantage in the AUD market

Comparative Advantage Arguments for Currency Swaps

7.43

USD AUD

General Electric 5.0% 7.6%

Qantas Airways 7.0% 8.0%

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

※ GE pays 6.9% in AUD (0.7% better off) and QA pays 6.3% in USD (0.7% better off)


7.44

GEQA

AUD 8.0%

USD 6.3% AUD 8.0%USD 5.0%F.I.

AUD 6.9%

USD 5.0%

– Different ways to arrange the currency swaps1. QA bears some foreign exchange risk

2. GE bears some foreign exchange risk

※ 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

GEQA

AUD 6.9%

USD 5.2% AUD 8.0%USD 5.0%F.I.

AUD 6.9%

USD 5.0%

GEQA

AUD 8.0%

USD 6.3% AUD 8.0%USD 5.0%F.I.

AUD 8.0%

USD 6.1%

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

,

where is the domestic bond defined by the remaining USD CFs, is the bond defined by the remaining foreign-currency CFs, and 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 7.46

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


※ (million $)


Time (yr)

Cash Flows of (million $)

PV of (million $)

Cash flows of (million yen)

PV of (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 1,230.55

7.48


– 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 and the specified trading price is (expressed as domestic dollars / per foreign dollar)

– Payoff of a FX forwards to purchase the foreign currency at is , where is the domestic-dollar price of the foreign currency (or the FX rate) at

– Rewrite the payoff to be , which can be reinterpreted as buying units of a foreign currency (worth at ) with units of domestic dollars at

7.49


– The value of a FX forward is the PV of its expected payoff

– The forward (or futures) price of the foreign currency provide the unbiased approximation for based on the information of IRs today

– Since the forward price of the foreign currency is (introduced on Slides 5.23-5.24), we can obtain

– Thus, the value of a FX forward is

– The forward FX rates of Japanese Yen, , in the example on Slide 7.47 are

7.50

Time (yr) 1 2 3

Forward FX rate ($/per Yen)


– Take the first exchange in the currency swap for example: it can be regarded as a FX forward to purchase million with $0.8 million

– Value of the first exchange

(million $)

7.51

Time (yr)

Dollar CF (mil. $)

Yen CF (mil. Yen)

Expected future 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

7.3 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

7.56

※ 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

7.4 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


– 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


– 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 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 fixed-for-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 swap7.61


– 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


– 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 pre-

agreed 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

Documents

Swaps Chapter 7 7.1. Goals of Chapter 7 7.2 Introduce interest rate (IR) swaps ( 利率交換 ) – Definition for swaps – An illustrative example for IR swaps