Download Interest Rate Swap

Interest Rate Swap: Definition
■ An Interest Rate Swap is a contract between two counterparties
consisting in exchanging interest flows
● At regular dates agreed in advance
● Calculated on an amount called notional
■ Plain vanilla swaps consist in exchanging floating to fixed interest flows
on a fixed notional amount, without any capital exchange at the
conclusion nor at maturity
■ The fixed rate, called the “swap rate” or “par swap rate”, must be set
contractually in order to have a fair contract for both parties
■ OTC market of swaps
● Swap are directly traded between banks, creating a very active market
■ Swaps are commonly used in ALM (in both banking and insurance)
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Interest Rate Swap: Example (1/2)
■ 4 years spot IRS between A & B
● Effective date: 16 February 2011
● Termination date: 16 February 2015
● Notional amount: €10,000,000
● Party A pays:
 Floating rate : 3 months Euribor (+ spread if A not a bank)
 Daycount: Act/360
 Payment dates: quarterly, on 16 May, 16 August, 16 November, 16
February each year, subject to adjustment in accordance with the
Business Day Convention
 Fixing dates: 2 business days in advance, reference Reuters Euribor01
2
Interest Rate Swap: Example (2/2)
■ B pays:
 Fixed rate : 3.2%
 Daycount: 30/360
 Payment dates: annually, on 16 February of each year, subject to
adjustment in accordance with the Business Day Convention.
A
Floating interests
B
Contract
conclusion
3
Interest Rate Swap: Characteristics
■ The fixed rate is set in advance so that the contract is fair for both parties
(value of the swap at conclusion = 0)
● One can show (see later) that this rate is equal to:
RS 
■ Fixed and floating legs can have different frequencies
1  PZC (0, Tn )
n
 (T  T
i 1
i
i 1
) PZC (0, Ti )
■ The swap market is OTC, but quotes (“swap rates”) for a series of plain
vanilla swaps are available in the market
● Maturities until 50 years
● Market quotes are available in real time (e.g. Bloomberg)
■ More complex contracts can be concluded in the OTC market, where e.g.
● The notional vary with time
● The contract is not spot but forward
● One or both legs are function of more than one reference rates (structured swaps)
■ A swap can be seen as a series of FRAs
● But in which each FRA considered separately is generally not valued at 0
4
Interest Rate Swap: Use in ALM (1/10)
■ A swap makes it possible to transform a fixed rate debt into a floating rate
debt on some period (or conversely)
■ It can also be useful in the case of firms having difficulties to fund their
activities with a long term debt at a fixed rate
● Such a firm will easily obtain short term credits, that it will renew on a
regular basis
● The debt appears for the firm as a floating rate debt
● The firm may find more convenient to enter a swap and finance itself on the
short term


The credit risk of a swap is not the same as credit risk of a bond, because there is not notional exchange at
maturity
 it can be more easy to obtain a swap than a long term debt
■ It also allows to lock for some period the cost of funding
■ In particular: In retail banking, a swap allows to lock in the return on a
portfolio of fixed rate mortgage loans
5
Interest Rate Swap: Use in ALM (2/10)
■ Let us consider a bank selling mortgage loans at a predefined fixed rate
● Suppose this bank funds its activity in the short term (as it is usually the
case)
● When loans are issued, the bank will conclude a swap in view of
guaranteeing its margin on the whole life of the loans
■ Example
● Mortgage loans portfolio with an average rate of 4.5% and length of 20
years, with a funding based on Euribor 6 months
● Suppose that in the market, the 20 years swap rate against Euribor 6
months is 3.5%
● If the bank enters a 20 years spot swap (with a decreasing notional in
order to reflect the amortizing schemes of the loans), then a margin of 1%
is guaranteed during 20 years
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Interest Rate Swap: Use in ALM (3/10)
Mortgage
loans
Clients
Funding
Bank 1
4.5%
Euribor 6M
3.5%
Swap
Bank 2
Financial
markets
Euribor 6M
 Guaranteed margin
of 1% for Bank 1
7
Interest Rate Swap: Use in ALM (4/10)
■ Hedging of life insurance contracts with guaranteed rate
● Covering assets: government bonds with average maturity 10 years
● When the bonds arrive at maturity, reinvestment risk for the company
 Risk
that the return on bonds at that time is insufficient w.r.t. the
guaranteed rate
 Conclusion of forward amortizing swaps on Euribor 1Y
■ Example: life insurance portfolio, guaranteed rate of 4.75%
● At t=0, investment in bonds, with coupon rate C0. In principle C0>4.75%.
 Maturity
of these bonds: T1
● At t=0: conclusion of a receiver forward swap, beginning in T1, ending in T2

The company has to pay Euribor 1Y, and receives a fixed rate K1 > 4.75%.

In principle, K1 > 4.75% (otherwise the guaranteed rate is too high compared to the market!)
8
Interest Rate Swap: Use in ALM (5/10)
■ Example (continued)
● In T1: Investment of nominal values of the expired bonds in new bonds with same
maturity as the forward swap (if possible), and new coupon rate C1.
● From T1: The company receives a fixed coupon rate C1 and the fixed rate K1 of
the swap, but pays a floating rate in the swap, and has to put aside (in the
mathematical reserves) the guaranteed rate of 4.75% of the insurance contracts
 Some interest rate risk is still present, linked to the potential spread existing
between the floating rate (Euribor 1Y) paid and the fixed coupon C1 received
from the new bonds from time T1
 In T1, the company should conclude at that time a second payer (spot)
swap of maturity equal to the maturity T2 of the new bonds, paying a fixed
rate K2 (swap rate at T1 for the maturity T2)
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Interest Rate Swap: Use in ALM (6/10)
■ Example (continued)
● If the spread government – interbank is zero, then K2=C1 (see later)
 Coupons received from the new bonds allow exactly to pay the fixed leg of
this new swap
 Floating legs of both swaps (forward and spot) are compensating.
 Globally the margin of the company is guaranteed, and equal to K1 – 4.75%
● In practice, K2 is close to C1
● Remarks:
 The
hedging can be perfect if no surrender on the insurance contracts, or if very
stable surrender rates

Otherwise, some interest rate risk may subsist
 With
this strategy, the company will not benefit from an increase of interest rates in
the markets

The future revenue is locked, no benefit in case of future increase of rates in the market.
10
Interest Rate Swap: Use in ALM (7/10)
■ Illustration
Hedging
ASSETS
Investments
Conclusion of the
forward swap, beginning
in T1 and maturity T2
Investment in
bonds of
maturity T1,
coupon C0
T=0
LIABILITIES
Conclusion of the
spot payer swap
End of the
spot swap
Execution of the forward
receiver swap concluded in
T=0
End of the
forward swap
Investment in
bonds of
maturity T2,
coupon C1
T=T1
End of bonds
investment
T=T2
11
Selling the
insurance
contracts
End of
insurance
contracts
11
Interest Rate Swap: Use in ALM (8/10)
■ Illustration
Hedging
ASSETS
Investments
Conclusion of the
forward swap, beginning
in T1 and maturity T2
Investment in
bonds of
maturity T1,
coupon C0
T=0
LIABILITIES
Conclusion of the
spot payer swap
End of the
spot swap
Execution of the
forward receiver
swap
End of the
forward swap
Investment in
bonds of
maturity T2,
coupon C1
End of bonds
investment
T=T1
T=T2
12
Selling the
insurance
contracts
End of
insurance
contracts
Receives C0,
Receives C1 + K1 + EUR1Y,
Pays 4.75%
Pays
4.75% + K2 + EUR1Y
12
Interest Rate Swap: Use in ALM (9/10)
Remarks
■ We will see later that swaps allow to increase the durations of assets
●  Usage in ALM in order to better immunise liabilities with very long
duration.
■ Forward swaps play the role of forward bonds. However bonds are
always spot contracts, forward bonds are not common.
■ Comparison bank – insurance
● In banking
 Fixed
rate mortgages correspond to assets with a fixed revenue (fixed rate) while the
funding of the bank is generally on the short term (floating rate)
 Banks typically have assets with an important duration, more than the funding
liabilities.
● In insurance (life)
 Revenues
of the company are based on assets with a duration usually shorter than
the duration of liabilities (that can have a very long duration).
 Sometimes the duration of liabilities is longer than the longest bond we can find in the
market.
 Inverted situation with respect to banks
13
Interest Rate Swap: Use in ALM (10/10)
Remarks
■ Other derivatives can be used: interest rates options like caps, floors,
swaptions (cf. later)
● They allow to hedge the company against adverse interest rate
movements, but to benefit from good movements
● But these are options: they have a price!
■ Products directly linked to swap rates: CMS floor or cap
● CMS floor: contract in which the company will receive on a regular basis
the difference between some threshold (e.g. 4.75%) and the swap rate for
a fixed maturity (e.g. 10 years)
 This
corresponds to guarantee the return of a bonds portfolio frequently rebalanced
and of maturity always close to 10 years (and not managed in a buy and hold policy)
14
Interest Rate Swap: Valuation
■ The fixed rate of the swap can be easily determined by equating the
market value of the fixed and floating legs
■ We have to determine K such that both legs are equal
■ Let us suppose that the notional is equal to 1
■ The fixed leg can be seen as an instrument with deterministic cash-
flows. It is a collection of zero-coupon bonds and its value is hence:
n
Fixed leg   K (Ti  Ti 1 ) PZC (0, Ti )
i 1
15
Interest Rate Swap: Valuation
■ The floating leg is an instrument with unknown cash-flows, delivering the
Euribor rate fixed at the beginning of each period
■ Each cash-flow can be seen as the floating leg of a FRA
■ Let us consider the floating cash-flow paid at maturity Ti (called a “floater”)
■ We have seen that the value of a FRA is zero provided the fixed rate is
equal to the forward rate Ki , given by:
 PZC (0, Ti 1 ) 
1

Ki 
 1  F (0, Ti 1 , Ti )
(Ti  Ti 1 )  PZC (0, Ti )

16
Interest Rate Swap: Valuation
■ Now, the value at t=0 of the floating part of the FRA for maturity Ti is
hence equal to the value of its fixed part, i.e. :
K i  (Ti  Ti 1 ) PZC (0, Ti ) 
 PZC (0, Ti 1 ) 
1

 1(Ti  Ti 1 ) PZC (0, Ti )
(Ti  Ti 1 )  PZC (0, Ti )

 PZC (0, Ti 1 )  PZC (0, Ti )
= value of a floater
■ The value of the floating leg of the swap is the sum of these floating
cash-flows (floaters):
n
floating leg of the swap   PZC (0, Ti 1 )  PZC (0, Ti )  1  PZC (0, Tn )
i 1
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Interest Rate Swap: Valuation
■ The fixed rate of the swap is determined by equating the market value of
the fixed and floating legs:
n
 K (T
i 1
i
 Ti 1 ) PZC (0, Ti )  1  PZC ( 0 ,Tn )
■ Which leads to:
K
1  PZC (0, Tn )
n
 (Ti  Ti 1 ) PZC (0, Ti )
(1)
i 1
18
Interest Rate Swap: Valuation
Alternative expression for the swap rate K:
● We can express the swap rate in function of forward rates for the different expiry
and maturities Ti
● For this, we go back to the valuation of the different floaters
● Each floater is worth:
K i (Ti  Ti 1 ) PZC (0, Ti )  F (0; Ti 1 , Ti )(Ti  Ti 1 ) PZC (0, Ti )
■ By summing on the different maturities and equating to the fixed leg
value, we get another expression for the swap rate K:
n
K
 F (0; T
i 1
i 1
, Ti )(Ti  Ti 1 ) PZC (0, Ti )
n
 (T
i 1
i
 Ti 1 ) PZC (0, Ti )
(2)
■ The swap rate appears as a weighted mean (convex combination) of
forward rates
19
Interest Rate Swap: Valuation – Remarks
Remark 1:
■ A floating rate bond can be seen as the floating leg of a swap plus the
capital redeemed at maturity. Its value can hence be easily obtained as:
n
PZC (0, Tn )   F (0; Ti 1 , Ti )(Ti  Ti 1 ) PZC (0, Ti )
i 1
■ By equating (1) and (2) we see that a floating rate bond is always
quoting at par at each re-fixing date (after coupon payment )
n
PZC (0, Tn )   F (0; Ti 1 , Ti )(Ti  Ti 1 ) PZC (0, Ti )  1
i 1
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Interest Rate Swap: Valuation – Remarks
■ This can also be directly deduced from the value of each floater
n
floating rate bond  PZC (0 ,Tn )   floatersi
i 1
 PZC (0 ,Tn )  floating leg
 PZC (0 ,Tn )  1  PZC (0 ,Tn )
Remark 2
1
■ The par swap rate appears as the coupon rate (or yield to maturity) of a
bond quoting at par.
■ This is directly deduced from equation (1) above:
K
1  PZC (0, Tn )
n
  K (Ti  Ti 1 ) PZC (0, Ti )  PZC (0, Tn )  1
i 1


(
T

T
)
P
(
0
,
T
)
 i i 1 ZC i
n
i 1
price of a bond with coupon K, maturity n and nominal 1
21
Yield Curve: Remarks
■ Alternative to a ZC yield curve: yields of bonds quoting at par (i.e. price
= nominal).
● In general this is only the case at issuance
● In this case, there is a bijection between maturity and yield to maturity
● We have seen that this is the equivalent of par swap rates
 Different types of rates:
● Par swap rate for a given maturity
● Zero-coupon rate for a given maturity
● Coupon rate of an existing bond of a given maturity
● Yield to maturity of an existing bond of a given maturity
■ All these rates are different in general
Basic Financial Tools
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Interest Rate Swaps : Valuation after conclusion
■ By the same reasoning as for getting the par swap rate, the market value
of a (receiver) swap at t is obtained as:
Swap value at t  ( RS  L(Tk 1 , Tk ))(Tk  Tk 1 ) PZC (t , Tk )

n
 (R
i  k 1
S
 F (t ; Ti 1 , Ti ))(Ti  Ti 1 ) PZC (t , Ti ).
where Tk-1 = refixing date of the floating rate for the next floating payment
(Tk-1 < t < Tk)
■ We supposed here that floating and fixed legs have the same frequency.
Otherwise separate fixed and floating legs in the sums above
Basic Financial Tools
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Interest Rate Swaps : Valuation after conclusion
Example
Let us consider the following swap:
■ Notional = € 1,000,000
■ Maturity: 30 September 2013
■ Conclusion: 30 September 2003
■ Receiver swap
■ Floating rate: 6-months Libor, fixed on March 31st and September 30th
■ Fixed rate: 10%, received semi-annually
Basic Financial Tools
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Interest Rate Swaps : Valuation after conclusion
Example
We want to value the swap on December 31, 2012.
■ Swap residual maturity : 9 months
■ Next payment: on March 31st based on the last fixing of Libor 6M at 6%.
■ Next fixing: on March 31st (last fixing for this swap)
30/9/2012:
31/12/2012
31/3/2013:
30/9/2013:
Last refixing:
floating 6M at 6%
(Valuation Date)
Next refixing, next
payment
Last payment
Basic Financial Tools
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Interest Rate Swaps : Valuation after conclusion
Example
■ Hypothesis: Current yield curve:
● R(0.25) = 5%, R(0.75) = 7% (annual compounding).
■ Fixed leg value:
50,000
50,000
Fixed leg 

 96,919.93
0.25
0.75
(1  0.05)
(1  0.07)
■ Floating leg value:
● Forward Libor rate (= simple rate) at 31/12/2012 for the period going from
31/3/2013 to 30/9/2013:
(1  0.07) 0.75
F (31 / 12; 31 / 3, 30 / 9)  0.5 
1
0.25
(1  0.05)
 (1  0.07) 0.75

 F (31 / 12; 31 / 3, 30 / 9)  2  
 1  7.86%
0.25
 (1  0.05)

Basic Financial Tools
26
Interest Rate Swaps : Valuation after conclusion
Example
■ Floating leg value:
● So that the floating leg is equal to:
Floating leg 
60,000  0.5 78,598  0.5

 66,990.86
0.25
0.75
(1  0.05)
(1  0.07)
■ Swap value = fixed leg – floating leg
= 96,919.93 – 66,990.86 = 29,929.07
Basic Financial Tools
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