valuation of ir swaps
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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
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 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– Therefore, there is no uncertainty about the first CF exchange
※ The principal is also known as the notional principal (名義本金或名目本金 ), or just the notional– Only net CFs change hands not necessary to exchange the principal at
any time point
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 CFs) of the deal would not be changed in any way
※ An 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 IR swaps (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
– 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 is ignored in the rest of this chapter
Interest Rate Swap
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 swaps and it has to honor the both contracts even if 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 swap 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 6.0453 years 6.21 6.24 6.2254 years 6.35 6.39 6.3705 years 6.47 6.51 6.4907 years 6.65 6.68 6.665
10 years 6.83 6.87 6.850
7.13
※ 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%
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
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
Comparative Advantage Argument
– 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 low, 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– With the IR swaps on Slide 7.16, 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 an IR swap perfectly if its creditworthiness changes in the future
※ The above inference disproves that the comparative advantage argument can fully 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 is
always AA for the whole n-year period For n-year AA-rating fixed lending rates, it is only known
that the initial creditworthiness of the borrower is AA-rating at the beginning of the n-year period
※ Credit risk comparison: n-year swap < n-year AA-rating lending
※ IR level comparison: n-year swap rate < n-year AA-rating lending rate
※ Furthermore, since the credit risks of AA-rating companies are very minor in practice, it can be inferred that the swap rates are even 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 (excluding the first exchange)
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 equal to its par value immediately after each coupon payment date (if it is discounted semiannually) 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
7.23
Valuation of IR Swaps
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 a Bfl (with the principal (or said par value) L) at any time point
※ A Bfl is worth 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
Valuation of IR Swaps
0 t*
Valuation Date
First PmtDate
(Floating CF = k*)
SecondPmt Date
Maturity Date
Value = L
Value = L+k*
Value = L = PV of L+k* at t*
t
LastPmt Date
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)
Valuation of IR Swaps
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
Valuation of IR Swaps
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 for
– 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
Valuation of IR Swaps
– The value of any derivatives equals the present value of it expected payoff This approach has been used to price FRAs 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
Valuation of IR Swaps
※ 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 7.25. For per $100 principal,
7.29
Valuation of IR Swaps
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
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 viewpoints on its 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 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 times to maturity < the swap rate negative values for corresponding FRAs The forward rates with longer times to maturity > the swap rate positive values for corresponding FRAs
– If the zero curve is downward sloping forward rates ↓ with T The forward rates with shorter times to maturity > the swap rate positive values for corresponding FRAs The forward rates with longer times to maturity < the swap rate negative values for corresponding FRAs 7.31
Determine LIBOR Zero Curve with Eurodollar Futures and Swaps
Construct the LIBOR zero curve(It is important since traders commonly use LIBORs as proxies for risk-free rates when trading derivatives)– : the quotes of spot LIBOR (given different )
provided by financial institutions are used– in (or in ): the quotes of Eurodollar futures are
used to derive 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 that swap rate = 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
Overnight Indexed Swaps (OISs)
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
※ Payoff = $100 (92/365) (LIBOR OIS rate) in practice– OIS rates are lower than LIBORs
OIS rates correspond to continually refreshed overnight rates (always lend or borrow daily with AA financial institutions)
To earn 3-month LIBORs, the bank bears the default risk of its trading counterparty, which is rated AA initially
OIS rates are even closer to risk-free interest rates
Overnight Indexed Swaps (OISs)
7.36
– In practice, many derivatives dealers choose to use OIS zero rates for discounting collateralized transactions (less risky) and use LIBOR zero rates 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 are used– For a longer (assume there are periodic settlements
(usually every 3 months) in OIS contracts) The OIS rate defines a par yield bond if the daily
discounting frequency is considered 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
The 1.25-year OIS zero rate is 3.9798% by solving
– For a being so long such that the quotes of OIS rates are not available or unreliable, e.g., years Note that LIBOR-based 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., for the 5-year time point, the corresponding (LIBOR – OIS) zero-rate 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
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)
Comparative Advantage Arguments for Currency Swaps
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 adopted in practice because the firms prefer to eliminate the foreign exchange risk with currency swaps thoroughly
Comparative Advantage Arguments for Currency Swaps
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
Valuation of Currency Swaps
※ (million $)
Valuation of Currency Swaps
Time (yr)
Cash Flows of (million $)
PV of (million $)
(discounted at 9%)
Cash flows of (million yen)
PV of (million yen)
(discounted at 4%)
1 0.8 (=10×8%) 0.7311 60 (=1,200×5%)
57.65
2 0.8 (=10×8%) 0.6682 60 (=1,200×5%)
55.39
3 0.8 (=10×8%) 0.6107 60 (=1,200×5%)
53.22
3 10 7.6338 1,200 1,064.30
Total 9.6439 1,230.55
7.48
Valuation of Currency Swaps
– Valuation in terms of forward contracts Each exchange of payments (including the first one) in a
fixed-for-fixed currency swap is a foreign exchange (FX) forward contract– FX forwards is an agreement to trade an amount of a foreign
currency at the specified price on a predetermined future date– FX forwards are similar to the foreign currency futures
contracts (introduced in Ch. 5) except that FX forwards are traded in OTC markets and thus there is no daily settlement requirement
– The principal is foreign dollars and the specified trading price is (expressed as domestic dollars / per foreign dollar)
– Payoff of a FX forwards to purchase the foreign currency at is , where is the domestic-dollar price of the foreign currency (or the FX rate) at 7.49
Valuation of Currency Swaps
– The value of a FX forward is the PV of its expected payoff
– The forward (or futures) price of the foreign currency provide the unbiased approximation for based on the information of IRs today
– Since the forward price of the foreign currency is (introduced on Slide 5.23), we obtain
– Thus, the value of a FX forward is
– The forward FX rates of Japanese Yen, , in the example on Slide 7.47 are
7.50
Time (yr) 1 2 3Forward FX rate ($/per Yen)
Valuation of Currency Swaps
– 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. $)
(discounted at 9%)
1 –0.8 60 0.009557 0.5734 –0.2266 –0.20712 –0.8 60 0.010047 0.6028 –0.1972 –0.16473 –0.8 60 0.010562 0.6337 –0.1663 –0.12693 –10 1,200 0.010562 12.6746 +2.6746 +2.0417
Total +1.5430
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 his
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/bond 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 default losses on a currency swap are greater than those 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
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 amount reduces (or increases) in a predetermined way
– Deferred (延遲 ) 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) 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 Interests on one or both sides are 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
immediately 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 (累積 ) swaps The interests on one side accrue only when the floating
reference rate is in a certain range (The other side pays in a standard way)
For example, only when the LIBOR rate is between [1%,2%], the interests on that day is accumulated– For every 6 months, the number of days where the specified
condition is met determines the exchanging CF 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 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 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 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
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