Interest Rate Swap
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Content
Interest rate swap
An interest rate swap (IRS) is a liquid financial derivative instrument in which two parties agree to exchange
interest rate cash flows, based on a specified notional
amount from a fixed rate to a floating rate (or vice versa)
or from one floating rate to another.[1] Interest rate swaps
can be used for both hedging and speculating.
1
B pays floating rate to A (B receives fixed rate)
Currently, A borrows from Market @ LIBOR +1.5%. B
borrows from Market @ 8.5%.
Consider the following swap in which Party A agrees
to pay Party B periodic fixed interest rate payments of
8.65% in exchange for periodic variable interest rate payments of LIBOR + 70 bps (0.70%) in the same currency. Note that there is no exchange of the principal
amounts and that the interest rates are on a “notional”
(i.e., imaginary) principal amount. Also note that interest
payments are settled in net; that is, Party A pays (LIBOR
+ 1.50%)+8.65% - (LIBOR+0.70%) = 9.45% net. The
fixed rate (8.65% in this example) is referred to as the
swap rate.[2]
Structure
At the point of initiation of the swap, the swap is priced so
that it has a net present value of zero. If one party wants
to pay 50 bps above the par swap rate, the other party has
to pay approximately 50bps over LIBOR to compensate
for this.
2 Types
Party A is currently paying floating rate, but wants to pay fixed
rate. Party B is currently paying fixed rate, but wants to pay floating rate. By entering into an interest rate swap, the net result is
that each party can swap their existing obligation for their desired
obligation.
In an interest rate swap, each counterparty agrees to pay
either a fixed or floating rate denominated in a particular
currency to the other counterparty. The fixed or floating
rate is multiplied by a notional principal amount (say, $1
million) and an accrual factor given by the appropriate
day count convention. When both legs are in the same
currency, this notional amount is typically not exchanged
between counterparties, but is used only for calculating
the size of cashflows to be exchanged. When the legs are
in different currencies, the respective notional amounts
are typically exchanged at the start and the end of the
swap, which is called cross currency interest rate swap.
Normally the parties do not swap payments directly, but rather
each sets up a separate swap with a financial intermediary such
as a bank. In return for matching the two parties together, the
bank takes a spread from the swap payments (in this case 0.30%
compared to the above example)
As OTC instruments, interest rate swaps can come in a
number of varieties and can be structured to meet the
specific needs of the counterparties. For example, the
legs of the swap could be in same or different currencies;
the notional of the swap could be amortized over time;
reset dates (or fixing dates) of the floating rate could be
irregular.
The most common interest rate swap involves counterparty A paying a fixed rate (the swap rate) to counterparty
B while receiving a floating rate indexed to a reference
rate like LIBOR, EURIBOR, or MIBOR. By market
convention, the counterparty paying the fixed rate is the
“payer” (while receiving the floating rate), and the counterparty receiving the fixed rate is the “receiver” (while
paying the floating rate).
The interbank market, however, only has a few standardized types which are listed below. Each currency has
A pays fixed rate to B (A receives floating rate)
1
2
2
TYPES
its own standard market conventions regarding the fre- receive JPY LIBOR + 35bps. With this, they have efquency of payments, the day count conventions and the fectively locked in a 35bps profit instead of running with
end-of-month rule.[3]
a current 40bps gain and index risk. The 5bps difference (w.r.t. the current rate difference) comes from the
swap cost which includes the market expectations of the
2.1 Fixed-for-floating rate swap, different future rate difference between these two indices and the
currencies
bid-offer spread, which is the swap commission for the
dealer.
For example, if a company has a $10 million fixed rate
Floating-for-floating rate swaps are also seen where both
loan at 5.3% paid monthly and a floating rate investment
sides reference the same index, but on different payment
of JPY 1.2 billion that returns JPY 1M Libor +50bps evdates, or use different business day conventions. This
ery month, and wants to lock in the profit in USD as they
can be vital for asset-liability management. An example
expect the JPY 1M Libor to go down or USDJPY to go up
would be swapping 3M LIBOR being paid with prior non(JPY depreciate against USD), then they may enter into a
business day convention, quarterly on JAJO (i.e., Jan,
fixed-for-floating swap in different currencies where the
Apr, Jul, Oct) 30, into FMAN (i.e., Feb, May, Aug, Nov)
company pays floating JPY 1M Libor+50bps and receives
28 modified following.
5.6% fixed rate, locking in 30bps profit against the interest rate and the FX exposure.
2.2
2.3
2.4
2.7 Fixed-for-fixed rate swap, different
currencies
Floating-for-fixed rate swap, same currency
Party P pays/receives fixed interest in currency A to reFixed-for-floating rate swap, same cur- ceive/pay fixed rate in currency B for a term of T years.
For example, you pay JPY 1.6% on a JPY notional of
rencies
1.2 billion and receive USD 5.36% on the USD equivalent notional of $10 million at an initial exchange rate of
Floating-for-fixed rate swap, different USDJPY 120.
currencies
2.5
Fixed-for-fixed rate swap, same cur- 2.8 Floating-for-floating rate swap, differrency
ent currencies
2.6
Floating-for-floating rate swap, same Party P pays/receives floating interest in currency A indexed to X to receive/pay floating rate in currency B incurrency
dexed to Y on a notional N at an initial exchange rate of
FX for a tenure of T years. The notional is usually exchanged at the start and at the end of the swap. This is
the most liquid type of swap with different currencies.
For example, you pay floating USD 3M LIBOR on the
USD notional 10 million quarterly to receive JPY 3M TIBOR quarterly on a JPY notional 1.2 billion (at an initial
exchange rate of USDJPY 120) for 4 years; at the start
you receive the notional in USD and pay the notional in
JPY and at the end you pay back the same USD notional
(10 million) and receive back the same JPY notional (1.2
Floating-for-floating rate swaps are used to hedge against billion).
or speculate on the spread between the two indexes. For For example, consider a U.S. company operating in Japan
example, if a company has a floating rate loan at JPY 1M that needs JPY 10 billion to fund its Japanese growth. The
LIBOR and the company has an investment that returns easiest way to do this is to issue debt in Japan, but this
JPY 1M TIBOR + 30bps and currently the JPY 1M TI- may be expensive if the company is new in the Japanese
BOR = JPY 1M LIBOR + 10bps. At the moment, this market and lacking a good reputation among the Japanese
company has a net profit of 40bps. If the company thinks investors. Additionally, the company may not have the
JPY 1M TIBOR is going to come down (relative to the appropriate debt issuance program in Japan or may lack
LIBOR) or JPY 1M LIBOR is going to increase in the a sophisticated treasury operation in Japan. The company
future (relative to the TIBOR) and wants to insulate from could issue USD debt and convert to JPY on the FX marthis risk, they can enter into a float-float swap in same ket. This option solves the first problem, but it introduces
currency where they pay, say, JPY TIBOR + 30bps and two new risks:
Party P pays/receives floating interest in currency A indexed to X to receive/pay floating rate in currency A indexed to Y on a notional N for a tenure of T years. For
example, you pay JPY 1M LIBOR monthly to receive
JPY 1M TIBOR monthly on a notional JPY 1 billion for
three years or you pay EUR 3M EURIBOR quarterly to
receive EUR 6M EURIBOR semi-annually. The second
example, where the indexes are the same type but with
different tenors, are the most liquid and most commonly
traded same currency floating-for-floating swaps..
3.2
British local authorities
3
• FX risk: If this USDJPY spot goes up at the matu- The interest rate swap market in USD is closely linked to
rity of the debt, then when the company converts the the Eurodollar futures market which trades among others
JPY to USD to pay back its matured debt, it receives at the Chicago Mercantile Exchange.
less USD and suffers a loss.
• USD–JPY interest rate risk: If JPY rates come
down, the return on the investment in Japan may also 3.2 British local authorities
go down, introducing interest rate risk.
In June 1988 the Audit Commission was tipped off by
The FX risk can be hedged with long-dated FX forward someone working on the swaps desk of Goldman Sachs
contracts, but this introduces yet another risk where the that the London Borough of Hammersmith and Fulham
implied rate from the FX spot and the FX forward is a had a massive exposure to interest rate swaps. When the
fixed but the JPY investment returns a floating rate. Al- commission contacted the council, the chief executive
though there are several alternatives to hedge both expo- told them not to worry as “everybody knows that interest
sures effectively without introducing new risks, the easi- rates are going to fall"; the treasurer thought the interest
est and most cost-effective alternative is to use a floating- rate swaps were a “nice little earner”. The Commission’s
Controller, Howard Davies, realised that the council had
for-floating swap in different currencies.
put all of its positions on interest rates going down and
ordered an investigation.
2.9
Other variations
A number of other far less common variations are possible. Mostly tweaks are made to ensure that a bond is
hedged “perfectly”, so that all the interest payments received are exactly offset, which can lead to swaps where
the principal is paid on one or more legs, rather than just
interest (for example to hedge a coupon strip), or where
the balance of the swap is automatically adjusted to match
that of a prepaying bond like residential mortgage-backed
securities.
• Brazilian Swap
3
By January 1989 the Commission obtained legal opinions from two Queen’s Counsel. Although they did not
agree, the commission preferred the opinion which made
it ultra vires for councils to engage in interest rate swaps.
Moreover, interest rates had increased from 8% to 15%.
The auditor and the commission then went to court and
had the contracts declared illegal (appeals all the way up
to the House of Lords failed in Hazell v Hammersmith
and Fulham LBC); the five banks involved lost millions of
pounds. Many other local authorities had been engaging
in interest rate swaps in the 1980s.[4] This resulted in several cases in which the banks generally lost their claims
for compound interest on debts to councils, finalised in
Westdeutsche Landesbank Girozentrale v Islington London Borough Council.[5]
Uses
Interest rate swaps are used to hedge against or speculate
on changes in interest rates.
4 Valuation and pricing
Further information: Rational_pricing § Swaps
3.1
Speculation
The valuation of vanilla swaps was often done using the
so-called textbook formulas using a unique curve in each
currency. Some early literature described some incoherence introduced by that approach and multiple banks
were using different techniques to reduce them. It became even more apparent with the 2007–2012 global
financial crisis that the approach was not appropriate.
The now-standard pricing framework is the multi-curves
framework.
Interest rate swaps are also used speculatively by hedge
funds or other investors who expect a change in interest rates or the relationships between them. Traditionally, fixed income investors who expected rates to fall
would purchase cash bonds, whose value increased as
rates fell. Today, investors with a similar view could enter a floating-for-fixed interest rate swap; as rates fall, investors would pay a lower floating rate in exchange for the
The present value of a plain vanilla (i.e., fixed rate for
same fixed rate.
Interest rate swaps are also popular for the arbitrage floating rate) swap can be computed by determining the
opportunities they provide.
Varying levels of present value (PV) of the fixed leg and the floating leg.
creditworthiness means that there is often a posi- The value of the fixed leg is given by the present value of
tive quality spread differential that allows both parties to the fixed coupon payments known at the start of the swap,
benefit from an interest rate swap.
i.e.
4
5
P Vfixed = N × C ×
n (
∑
)
δ˜i × P D (t˜i )
RISKS
Therefore, at the time the contract is entered into, there
is no advantage to either party, i.e.,
i=1
P Vfixed = P Vfloat
where C is the swap rate, n is the number of fixed payments, N is the notional amount, δ˜i is the accrual factor Thus, the swap requires no upfront payment from either
according to the day count convention for the fixed rate party.
period and P D (t˜i ) is the discount factor for the payment
During the life of the swap the same valuation technique
time t˜i .
is used, but since, over time, both the discounting factors
The value of the floating leg is given by the present value and the forward rates change, the PV of the swap will
of the floating coupon payments determined at the agreed deviate from its initial value. Therefore, the swap will
dates of each payment. However, at the start of the swap, be an asset to one party and a liability to the other. The
only the actual payment rates of the fixed leg are known in way these changes in value are reported is the subject of
the future, whereas the forward rates are unknown. The IAS 39 for jurisdictions following IFRS, and FAS 133 for
forward rate for each floating payment date is calculated U.S. GAAP. Swaps are marked to market by debt security
using the forward curves. The forward rate for the period traders to visualize their inventory at a certain time.
[t˜j−1 , tj ] with accrual factor δi is given by
Fj =
1
δj
(
)
P I (tj−1 )
−
1
P I (tj )
5 Risks
Interest rate swaps expose users to interest rate risk and
where I is the market index, such as USD LIBOR, and credit risk.
P I (tj ) is the discount factor associated to the relevant
forward curve. The value of the floating leg is given by
• Market risk: A typical swap consists of two legs—
the following:
one fixed, the other floating. The risks of these
two component will naturally differ. Newcomers to
market finance may think that the risky component
m
∑
is the floating leg, since the underlying interest rate
P Vfloat = N ×
(Fj × δj × P D (tj ))
floats, and hence, is unknown. This first impression
j=1
is wrong. The risky component is in fact the fixed
leg and it is very easy to see why this is so.[7]
where m is the number of floating payments, δi is the accrual factor according to the floating leg day count con(Comment: the above comment is not entirely accurate.
vention.
Normally people will assume a hypothetical notional exIn the event that
change at the end. After this hypothetical assumption, the
PD = PI ,
swap can be understood as a floating rate bond vs a fixed
rate bond. The risks on the floating bond side are small
this formula simplifies to
compared to the risks from the fixed rate bond side. HowP Vfloat = N × (1 − P D (t˜m ))
ever, without this hypothetical notional exchange at the
on the reset dates, since the summation in P Vfloat tele- end, purely the floating cash flow from the coupon payscopes to the first and last terms only. On reset dates, the ments will have higher risks than the cash flow from the
fixed coupon payments. Both understanding have their
value of an off-the-run swap (old issue) is given by
merit and fully understand these two views are important
P Vfixed − P Vfloat = N × (Beq − 1)
to see the risks, particularly for floating floating swaps)
where Beq is the value of hypothetical bond that mimics The discussion of pricing interest rate swaps illustrated
the fixed leg of the swap with a unit principal payable at an important point. Regardless of what happens to fuexpiry. On none reset dates the swap value becomes
ture Libor rates, the value of a rolling deposit or floating
P Vfixed − P Vfloat = N × (Beq − P D (tr ))
rate note (FRN) always equals the notional amount N at
the reset dates. Between the reset dates this value may
where tr is the nearest reset date. The fixed rate offered in
be different from N, but the discrepancy cannot be very
the swap is the rate which values the fixed rates payments
large since the δ will be 3 or 6 months. Interest rate flucat the same PV as the variable rate payments using today’s
tuations have minimal effect on the values of fixed inforward rates, i.e.:
struments with short maturities; in other words, the value
of the floating leg changes very little during the life of a
C = ∑n (NP×Vδ˜float×P D (t˜ )) [6]
swap.
i
i
i=1
5
On the other hand, the fixed leg of a swap is equivalent to
a coupon bond and fluctuations of the swap rate may have
major effects on the value of the future fixed payments.
• Credit risk on the swap comes into play if the swap
is in the money or not. If one of the parties is in
the money, then that party faces credit risk of possible default by another party. However, when the
swap is negotiated through an intermediary financial institution, usually the intermediary assumes the
default risk in exchange for a fixed percentage of
the transaction (the bid-ask spread). In an intermediated swap, the two parties are not typically even
aware of the identity of the second party to the transaction, making a quantification of the other party’s
credit risk not only irrelevant, but impossible.
6
Market size
On its December 2014 statistics release, the Bank for International Settlements reported that interest rate swaps
were the largest component of the global OTC derivative
market representing 60% of it, with the notional amount
outstanding in OTC interest rate swaps of $381 trillion,
and the gross market value of $14 trillion.[8]
Interest rate swaps can be traded as an index through the
FTSE MTIRS Index.
7
See also
• Swap rate
[4] Duncan Campbell-Smith, “Follow the Money: The Audit Commission, Public Money, and the Management of
Public Services 1983-2008”, Allen Lane, 2008, chapter 6
passim.
[5] [1996] UKHL 12, [1996] AC 669
[6] “Understanding interest rate swap math & pricing” (PDF).
California Debt and Investment Advisory Commission.
January 2007. Retrieved 2007-09-27.
[7] http://chicagofed.org/webpages/publications/
understanding_derivatives/index.cfm
[8] “OTC derivatives statistics at end-December 2014”
(PDF). Bank for International Settlements.
• Pricing and Hedging Swaps, Miron P. & Swannell
P., Euromoney books 1991
Early literature on the incoherence of the one curve pricing approach.
• Interest rate parity, money market basis swaps and
cross-currency basis swaps, Tuckman B. and Porfirio P., Fixed income liquid markets research,
Lehman Brothers, 2003.
• Cross currency swap valuation, Boenkost W. and
Schmidt W., Working Paper 2, HfB - Business
School of Finance & Management, 2004. SSRN
preprint.
• The Irony in the Derivatives Discounting, Henrard
M., Wilmott Magazine, pp. 92–98, July 2007.
SSRN preprint.
Multi-curves framework:
• Interest rate cap and floor
• Equity swap
• Total return swap
• Inflation derivative
• Eurodollar
• Constant maturity swap
• FTSE MTIRS Index
8
References
[1] “Interest Rate Swap”. Glossary. ISDA.
• A multi-quality model of interest rates, Kijima M.,
Tanaka K., and Wong T., Quantitative Finance,
pages 133-145, 2009.
• Two Curves, One Price: Pricing & Hedging Interest
Rate Derivatives Decoupling Forwarding and Discounting Yield Curves, Bianchetti M., Risk Magazine, August 2010. SSRN preprint.
• The Irony in the Derivatives Discounting Part II: The
Crisis, Henrard M., Wilmott Journal, Vol. 2, pp.
301–316, 2010. SSRN preprint.
9 External links
[2] "Interest Rate Swap" by Fiona Maclachlan, The Wolfram
Demonstrations Project.
• Understanding Derivatives: Markets and Infrastructure Federal Reserve Bank of Chicago, Financial
Markets Group
[3] "Interest Rate Instruments and Market Conventions
Guide" Quantitative Research, OpenGamma, 2012.
• Bank for International Settlements - Semiannual
OTC derivatives statistics
6
9
• Glossary - Interest rate swap glossary
• Investopedia - Spreadlock - An interest rate swap future (not an option)
• Basic Fixed Income Derivative Hedging - Article on
Financial-edu.com.
• Hussman Funds - Freight Trains and Steep Curves
• Interest Rate Swap Calculator
• Historical LIBOR Swaps data
• “All about money rates in the world: Real estate interest rates”, WorldwideInterestRates.com
EXTERNAL LINKS
7
10
10.1
Text and image sources, contributors, and licenses
Text
• Interest rate swap Source: https://en.wikipedia.org/wiki/Interest_rate_swap?oldid=710329676 Contributors: SimonP, Maury Markowitz,
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Wp.duan, Wynandbez, Swapsbroker, Sun Creator, Dekisugi, DumZiBoT, Mkipnis, Addbot, Theobaldr, Amkdude2, Kiril Simeonovski,
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10.2
Images
• File:Vanilla_interest_rate_swap.png Source: https://upload.wikimedia.org/wikipedia/commons/1/1c/Vanilla_interest_rate_swap.png
License: CC BY-SA 3.0 Contributors: Own work Original artist: Suicup
• File:Vanilla_interest_rate_swap_with_bank.png Source: https://upload.wikimedia.org/wikipedia/commons/8/81/Vanilla_interest_
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10.3
Content license
• Creative Commons Attribution-Share Alike 3.0
An interest rate swap (IRS) is a liquid financial derivative instrument in which two parties agree to exchange
interest rate cash flows, based on a specified notional
amount from a fixed rate to a floating rate (or vice versa)
or from one floating rate to another.[1] Interest rate swaps
can be used for both hedging and speculating.
1
B pays floating rate to A (B receives fixed rate)
Currently, A borrows from Market @ LIBOR +1.5%. B
borrows from Market @ 8.5%.
Consider the following swap in which Party A agrees
to pay Party B periodic fixed interest rate payments of
8.65% in exchange for periodic variable interest rate payments of LIBOR + 70 bps (0.70%) in the same currency. Note that there is no exchange of the principal
amounts and that the interest rates are on a “notional”
(i.e., imaginary) principal amount. Also note that interest
payments are settled in net; that is, Party A pays (LIBOR
+ 1.50%)+8.65% - (LIBOR+0.70%) = 9.45% net. The
fixed rate (8.65% in this example) is referred to as the
swap rate.[2]
Structure
At the point of initiation of the swap, the swap is priced so
that it has a net present value of zero. If one party wants
to pay 50 bps above the par swap rate, the other party has
to pay approximately 50bps over LIBOR to compensate
for this.
2 Types
Party A is currently paying floating rate, but wants to pay fixed
rate. Party B is currently paying fixed rate, but wants to pay floating rate. By entering into an interest rate swap, the net result is
that each party can swap their existing obligation for their desired
obligation.
In an interest rate swap, each counterparty agrees to pay
either a fixed or floating rate denominated in a particular
currency to the other counterparty. The fixed or floating
rate is multiplied by a notional principal amount (say, $1
million) and an accrual factor given by the appropriate
day count convention. When both legs are in the same
currency, this notional amount is typically not exchanged
between counterparties, but is used only for calculating
the size of cashflows to be exchanged. When the legs are
in different currencies, the respective notional amounts
are typically exchanged at the start and the end of the
swap, which is called cross currency interest rate swap.
Normally the parties do not swap payments directly, but rather
each sets up a separate swap with a financial intermediary such
as a bank. In return for matching the two parties together, the
bank takes a spread from the swap payments (in this case 0.30%
compared to the above example)
As OTC instruments, interest rate swaps can come in a
number of varieties and can be structured to meet the
specific needs of the counterparties. For example, the
legs of the swap could be in same or different currencies;
the notional of the swap could be amortized over time;
reset dates (or fixing dates) of the floating rate could be
irregular.
The most common interest rate swap involves counterparty A paying a fixed rate (the swap rate) to counterparty
B while receiving a floating rate indexed to a reference
rate like LIBOR, EURIBOR, or MIBOR. By market
convention, the counterparty paying the fixed rate is the
“payer” (while receiving the floating rate), and the counterparty receiving the fixed rate is the “receiver” (while
paying the floating rate).
The interbank market, however, only has a few standardized types which are listed below. Each currency has
A pays fixed rate to B (A receives floating rate)
1
2
2
TYPES
its own standard market conventions regarding the fre- receive JPY LIBOR + 35bps. With this, they have efquency of payments, the day count conventions and the fectively locked in a 35bps profit instead of running with
end-of-month rule.[3]
a current 40bps gain and index risk. The 5bps difference (w.r.t. the current rate difference) comes from the
swap cost which includes the market expectations of the
2.1 Fixed-for-floating rate swap, different future rate difference between these two indices and the
currencies
bid-offer spread, which is the swap commission for the
dealer.
For example, if a company has a $10 million fixed rate
Floating-for-floating rate swaps are also seen where both
loan at 5.3% paid monthly and a floating rate investment
sides reference the same index, but on different payment
of JPY 1.2 billion that returns JPY 1M Libor +50bps evdates, or use different business day conventions. This
ery month, and wants to lock in the profit in USD as they
can be vital for asset-liability management. An example
expect the JPY 1M Libor to go down or USDJPY to go up
would be swapping 3M LIBOR being paid with prior non(JPY depreciate against USD), then they may enter into a
business day convention, quarterly on JAJO (i.e., Jan,
fixed-for-floating swap in different currencies where the
Apr, Jul, Oct) 30, into FMAN (i.e., Feb, May, Aug, Nov)
company pays floating JPY 1M Libor+50bps and receives
28 modified following.
5.6% fixed rate, locking in 30bps profit against the interest rate and the FX exposure.
2.2
2.3
2.4
2.7 Fixed-for-fixed rate swap, different
currencies
Floating-for-fixed rate swap, same currency
Party P pays/receives fixed interest in currency A to reFixed-for-floating rate swap, same cur- ceive/pay fixed rate in currency B for a term of T years.
For example, you pay JPY 1.6% on a JPY notional of
rencies
1.2 billion and receive USD 5.36% on the USD equivalent notional of $10 million at an initial exchange rate of
Floating-for-fixed rate swap, different USDJPY 120.
currencies
2.5
Fixed-for-fixed rate swap, same cur- 2.8 Floating-for-floating rate swap, differrency
ent currencies
2.6
Floating-for-floating rate swap, same Party P pays/receives floating interest in currency A indexed to X to receive/pay floating rate in currency B incurrency
dexed to Y on a notional N at an initial exchange rate of
FX for a tenure of T years. The notional is usually exchanged at the start and at the end of the swap. This is
the most liquid type of swap with different currencies.
For example, you pay floating USD 3M LIBOR on the
USD notional 10 million quarterly to receive JPY 3M TIBOR quarterly on a JPY notional 1.2 billion (at an initial
exchange rate of USDJPY 120) for 4 years; at the start
you receive the notional in USD and pay the notional in
JPY and at the end you pay back the same USD notional
(10 million) and receive back the same JPY notional (1.2
Floating-for-floating rate swaps are used to hedge against billion).
or speculate on the spread between the two indexes. For For example, consider a U.S. company operating in Japan
example, if a company has a floating rate loan at JPY 1M that needs JPY 10 billion to fund its Japanese growth. The
LIBOR and the company has an investment that returns easiest way to do this is to issue debt in Japan, but this
JPY 1M TIBOR + 30bps and currently the JPY 1M TI- may be expensive if the company is new in the Japanese
BOR = JPY 1M LIBOR + 10bps. At the moment, this market and lacking a good reputation among the Japanese
company has a net profit of 40bps. If the company thinks investors. Additionally, the company may not have the
JPY 1M TIBOR is going to come down (relative to the appropriate debt issuance program in Japan or may lack
LIBOR) or JPY 1M LIBOR is going to increase in the a sophisticated treasury operation in Japan. The company
future (relative to the TIBOR) and wants to insulate from could issue USD debt and convert to JPY on the FX marthis risk, they can enter into a float-float swap in same ket. This option solves the first problem, but it introduces
currency where they pay, say, JPY TIBOR + 30bps and two new risks:
Party P pays/receives floating interest in currency A indexed to X to receive/pay floating rate in currency A indexed to Y on a notional N for a tenure of T years. For
example, you pay JPY 1M LIBOR monthly to receive
JPY 1M TIBOR monthly on a notional JPY 1 billion for
three years or you pay EUR 3M EURIBOR quarterly to
receive EUR 6M EURIBOR semi-annually. The second
example, where the indexes are the same type but with
different tenors, are the most liquid and most commonly
traded same currency floating-for-floating swaps..
3.2
British local authorities
3
• FX risk: If this USDJPY spot goes up at the matu- The interest rate swap market in USD is closely linked to
rity of the debt, then when the company converts the the Eurodollar futures market which trades among others
JPY to USD to pay back its matured debt, it receives at the Chicago Mercantile Exchange.
less USD and suffers a loss.
• USD–JPY interest rate risk: If JPY rates come
down, the return on the investment in Japan may also 3.2 British local authorities
go down, introducing interest rate risk.
In June 1988 the Audit Commission was tipped off by
The FX risk can be hedged with long-dated FX forward someone working on the swaps desk of Goldman Sachs
contracts, but this introduces yet another risk where the that the London Borough of Hammersmith and Fulham
implied rate from the FX spot and the FX forward is a had a massive exposure to interest rate swaps. When the
fixed but the JPY investment returns a floating rate. Al- commission contacted the council, the chief executive
though there are several alternatives to hedge both expo- told them not to worry as “everybody knows that interest
sures effectively without introducing new risks, the easi- rates are going to fall"; the treasurer thought the interest
est and most cost-effective alternative is to use a floating- rate swaps were a “nice little earner”. The Commission’s
Controller, Howard Davies, realised that the council had
for-floating swap in different currencies.
put all of its positions on interest rates going down and
ordered an investigation.
2.9
Other variations
A number of other far less common variations are possible. Mostly tweaks are made to ensure that a bond is
hedged “perfectly”, so that all the interest payments received are exactly offset, which can lead to swaps where
the principal is paid on one or more legs, rather than just
interest (for example to hedge a coupon strip), or where
the balance of the swap is automatically adjusted to match
that of a prepaying bond like residential mortgage-backed
securities.
• Brazilian Swap
3
By January 1989 the Commission obtained legal opinions from two Queen’s Counsel. Although they did not
agree, the commission preferred the opinion which made
it ultra vires for councils to engage in interest rate swaps.
Moreover, interest rates had increased from 8% to 15%.
The auditor and the commission then went to court and
had the contracts declared illegal (appeals all the way up
to the House of Lords failed in Hazell v Hammersmith
and Fulham LBC); the five banks involved lost millions of
pounds. Many other local authorities had been engaging
in interest rate swaps in the 1980s.[4] This resulted in several cases in which the banks generally lost their claims
for compound interest on debts to councils, finalised in
Westdeutsche Landesbank Girozentrale v Islington London Borough Council.[5]
Uses
Interest rate swaps are used to hedge against or speculate
on changes in interest rates.
4 Valuation and pricing
Further information: Rational_pricing § Swaps
3.1
Speculation
The valuation of vanilla swaps was often done using the
so-called textbook formulas using a unique curve in each
currency. Some early literature described some incoherence introduced by that approach and multiple banks
were using different techniques to reduce them. It became even more apparent with the 2007–2012 global
financial crisis that the approach was not appropriate.
The now-standard pricing framework is the multi-curves
framework.
Interest rate swaps are also used speculatively by hedge
funds or other investors who expect a change in interest rates or the relationships between them. Traditionally, fixed income investors who expected rates to fall
would purchase cash bonds, whose value increased as
rates fell. Today, investors with a similar view could enter a floating-for-fixed interest rate swap; as rates fall, investors would pay a lower floating rate in exchange for the
The present value of a plain vanilla (i.e., fixed rate for
same fixed rate.
Interest rate swaps are also popular for the arbitrage floating rate) swap can be computed by determining the
opportunities they provide.
Varying levels of present value (PV) of the fixed leg and the floating leg.
creditworthiness means that there is often a posi- The value of the fixed leg is given by the present value of
tive quality spread differential that allows both parties to the fixed coupon payments known at the start of the swap,
benefit from an interest rate swap.
i.e.
4
5
P Vfixed = N × C ×
n (
∑
)
δ˜i × P D (t˜i )
RISKS
Therefore, at the time the contract is entered into, there
is no advantage to either party, i.e.,
i=1
P Vfixed = P Vfloat
where C is the swap rate, n is the number of fixed payments, N is the notional amount, δ˜i is the accrual factor Thus, the swap requires no upfront payment from either
according to the day count convention for the fixed rate party.
period and P D (t˜i ) is the discount factor for the payment
During the life of the swap the same valuation technique
time t˜i .
is used, but since, over time, both the discounting factors
The value of the floating leg is given by the present value and the forward rates change, the PV of the swap will
of the floating coupon payments determined at the agreed deviate from its initial value. Therefore, the swap will
dates of each payment. However, at the start of the swap, be an asset to one party and a liability to the other. The
only the actual payment rates of the fixed leg are known in way these changes in value are reported is the subject of
the future, whereas the forward rates are unknown. The IAS 39 for jurisdictions following IFRS, and FAS 133 for
forward rate for each floating payment date is calculated U.S. GAAP. Swaps are marked to market by debt security
using the forward curves. The forward rate for the period traders to visualize their inventory at a certain time.
[t˜j−1 , tj ] with accrual factor δi is given by
Fj =
1
δj
(
)
P I (tj−1 )
−
1
P I (tj )
5 Risks
Interest rate swaps expose users to interest rate risk and
where I is the market index, such as USD LIBOR, and credit risk.
P I (tj ) is the discount factor associated to the relevant
forward curve. The value of the floating leg is given by
• Market risk: A typical swap consists of two legs—
the following:
one fixed, the other floating. The risks of these
two component will naturally differ. Newcomers to
market finance may think that the risky component
m
∑
is the floating leg, since the underlying interest rate
P Vfloat = N ×
(Fj × δj × P D (tj ))
floats, and hence, is unknown. This first impression
j=1
is wrong. The risky component is in fact the fixed
leg and it is very easy to see why this is so.[7]
where m is the number of floating payments, δi is the accrual factor according to the floating leg day count con(Comment: the above comment is not entirely accurate.
vention.
Normally people will assume a hypothetical notional exIn the event that
change at the end. After this hypothetical assumption, the
PD = PI ,
swap can be understood as a floating rate bond vs a fixed
rate bond. The risks on the floating bond side are small
this formula simplifies to
compared to the risks from the fixed rate bond side. HowP Vfloat = N × (1 − P D (t˜m ))
ever, without this hypothetical notional exchange at the
on the reset dates, since the summation in P Vfloat tele- end, purely the floating cash flow from the coupon payscopes to the first and last terms only. On reset dates, the ments will have higher risks than the cash flow from the
fixed coupon payments. Both understanding have their
value of an off-the-run swap (old issue) is given by
merit and fully understand these two views are important
P Vfixed − P Vfloat = N × (Beq − 1)
to see the risks, particularly for floating floating swaps)
where Beq is the value of hypothetical bond that mimics The discussion of pricing interest rate swaps illustrated
the fixed leg of the swap with a unit principal payable at an important point. Regardless of what happens to fuexpiry. On none reset dates the swap value becomes
ture Libor rates, the value of a rolling deposit or floating
P Vfixed − P Vfloat = N × (Beq − P D (tr ))
rate note (FRN) always equals the notional amount N at
the reset dates. Between the reset dates this value may
where tr is the nearest reset date. The fixed rate offered in
be different from N, but the discrepancy cannot be very
the swap is the rate which values the fixed rates payments
large since the δ will be 3 or 6 months. Interest rate flucat the same PV as the variable rate payments using today’s
tuations have minimal effect on the values of fixed inforward rates, i.e.:
struments with short maturities; in other words, the value
of the floating leg changes very little during the life of a
C = ∑n (NP×Vδ˜float×P D (t˜ )) [6]
swap.
i
i
i=1
5
On the other hand, the fixed leg of a swap is equivalent to
a coupon bond and fluctuations of the swap rate may have
major effects on the value of the future fixed payments.
• Credit risk on the swap comes into play if the swap
is in the money or not. If one of the parties is in
the money, then that party faces credit risk of possible default by another party. However, when the
swap is negotiated through an intermediary financial institution, usually the intermediary assumes the
default risk in exchange for a fixed percentage of
the transaction (the bid-ask spread). In an intermediated swap, the two parties are not typically even
aware of the identity of the second party to the transaction, making a quantification of the other party’s
credit risk not only irrelevant, but impossible.
6
Market size
On its December 2014 statistics release, the Bank for International Settlements reported that interest rate swaps
were the largest component of the global OTC derivative
market representing 60% of it, with the notional amount
outstanding in OTC interest rate swaps of $381 trillion,
and the gross market value of $14 trillion.[8]
Interest rate swaps can be traded as an index through the
FTSE MTIRS Index.
7
See also
• Swap rate
[4] Duncan Campbell-Smith, “Follow the Money: The Audit Commission, Public Money, and the Management of
Public Services 1983-2008”, Allen Lane, 2008, chapter 6
passim.
[5] [1996] UKHL 12, [1996] AC 669
[6] “Understanding interest rate swap math & pricing” (PDF).
California Debt and Investment Advisory Commission.
January 2007. Retrieved 2007-09-27.
[7] http://chicagofed.org/webpages/publications/
understanding_derivatives/index.cfm
[8] “OTC derivatives statistics at end-December 2014”
(PDF). Bank for International Settlements.
• Pricing and Hedging Swaps, Miron P. & Swannell
P., Euromoney books 1991
Early literature on the incoherence of the one curve pricing approach.
• Interest rate parity, money market basis swaps and
cross-currency basis swaps, Tuckman B. and Porfirio P., Fixed income liquid markets research,
Lehman Brothers, 2003.
• Cross currency swap valuation, Boenkost W. and
Schmidt W., Working Paper 2, HfB - Business
School of Finance & Management, 2004. SSRN
preprint.
• The Irony in the Derivatives Discounting, Henrard
M., Wilmott Magazine, pp. 92–98, July 2007.
SSRN preprint.
Multi-curves framework:
• Interest rate cap and floor
• Equity swap
• Total return swap
• Inflation derivative
• Eurodollar
• Constant maturity swap
• FTSE MTIRS Index
8
References
[1] “Interest Rate Swap”. Glossary. ISDA.
• A multi-quality model of interest rates, Kijima M.,
Tanaka K., and Wong T., Quantitative Finance,
pages 133-145, 2009.
• Two Curves, One Price: Pricing & Hedging Interest
Rate Derivatives Decoupling Forwarding and Discounting Yield Curves, Bianchetti M., Risk Magazine, August 2010. SSRN preprint.
• The Irony in the Derivatives Discounting Part II: The
Crisis, Henrard M., Wilmott Journal, Vol. 2, pp.
301–316, 2010. SSRN preprint.
9 External links
[2] "Interest Rate Swap" by Fiona Maclachlan, The Wolfram
Demonstrations Project.
• Understanding Derivatives: Markets and Infrastructure Federal Reserve Bank of Chicago, Financial
Markets Group
[3] "Interest Rate Instruments and Market Conventions
Guide" Quantitative Research, OpenGamma, 2012.
• Bank for International Settlements - Semiannual
OTC derivatives statistics
6
9
• Glossary - Interest rate swap glossary
• Investopedia - Spreadlock - An interest rate swap future (not an option)
• Basic Fixed Income Derivative Hedging - Article on
Financial-edu.com.
• Hussman Funds - Freight Trains and Steep Curves
• Interest Rate Swap Calculator
• Historical LIBOR Swaps data
• “All about money rates in the world: Real estate interest rates”, WorldwideInterestRates.com
EXTERNAL LINKS
7
10
10.1
Text and image sources, contributors, and licenses
Text
• Interest rate swap Source: https://en.wikipedia.org/wiki/Interest_rate_swap?oldid=710329676 Contributors: SimonP, Maury Markowitz,
Ram-Man, Edward, Michael Hardy, Karada, Pcb21, Rl, Ehn, Peter Damian (original account), Donreed, Justanyone, Zigger, Dratman,
RScheiber, Vladan~enwiki, Sam Hocevar, Fenice, MBisanz, Cje~enwiki, Jerryseinfeld, Leifern, PaulHanson, Wikidea, Jberkes, Ercolev, Krexwall, Timrichardson, Wikiklrsc, BD2412, SLi, Gurch, Andrew G Ross, Antiuser, YurikBot, Grafen, GraemeL, Fred2028,
ArielGold, DocendoDiscimus, Minnesota1, Veinor, CarbonCopy, OrphanBot, KaiserbBot, Radagast83, Jeremy norbury, TheChieftain,
SashatoBot, Korovioff, Es330td, JDAWiseman, Hu12, CmdrObot, Cydebot, Peripitus, Future Perfect at Sunrise, KeithWright, AlekseyP, Legis, Thijs!bot, SvenAERTS, Ste4k, Msankowski, Alphachimpbot, Zidane tribal, Barek, Drdariush, Davidmanheim, Igirisujin,
Pauly04, STBot, Tgeairn, Poddrick, Idioma-bot, Sam Blacketer, Pleasantville, JohnSorrellAu, McTavidge, Dagroup, Purple Aubergine,
Altasoul, Gloomy Coder, Brianga, Gherrington, WRK, Int21h, Pjleahy, Finnancier, Desx2501, EoGuy, Harfo91, Enthusiast01, Raisaahab,
Wp.duan, Wynandbez, Swapsbroker, Sun Creator, Dekisugi, DumZiBoT, Mkipnis, Addbot, Theobaldr, Amkdude2, Kiril Simeonovski,
Gail, Xqbot, FrescoBot, Unstable-equilibrium, LucienBOT, Diroussel, Haeinous, Anatoly.karpov, VernoWhitney,
, Amit1law, Peterjleahy, Dewritech, Lancastle, ZéroBot, Liquidmetalrob, Vega47, Gwen-chan, Peteravel, Ashaykakde, Fancitron, Chinacat2002, BattyBot,
Luvoneanother, Quantresearch, Kkumaresan26, Brianrisk and Anonymous: 219
10.2
Images
• File:Vanilla_interest_rate_swap.png Source: https://upload.wikimedia.org/wikipedia/commons/1/1c/Vanilla_interest_rate_swap.png
License: CC BY-SA 3.0 Contributors: Own work Original artist: Suicup
• File:Vanilla_interest_rate_swap_with_bank.png Source: https://upload.wikimedia.org/wikipedia/commons/8/81/Vanilla_interest_
rate_swap_with_bank.png License: CC BY-SA 3.0 Contributors: Own work Original artist: Suicup
10.3
Content license
• Creative Commons Attribution-Share Alike 3.0
