Interest Rate Swap: Definition - Personal Homepageshomepages.ulb.ac.be/~cazizieh/Statf508_files/swaps.pdfThe fixed rate is set in advance so that the contract is fair for both parties

■ 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)

Interest Rate Swap: Definition

1

■ 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

Interest Rate Swap: Example (1/2)

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.

Interest Rate Swap: Example (2/2)

A

B

Floating interests

Contract

conclusion

3

■ 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:

■ Fixed and floating legs can have different frequencies

■ 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

Interest Rate Swap: Characteristics

4

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i

iZCii

nZCS

TPTT

TPR

1

1 ),0()(

),0(1

■ 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

Interest Rate Swap: Use in ALM (1/10)

5

■ 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


6


Clients Bank 1 Financial

markets 4.5% Euribor 6M

Bank 2

Euribor 6M 3.5%

Mortgage loans Funding

Swap

Guaranteed margin of 1% for Bank 1

7

■ 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

9

■ 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)


9

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

11

■ Illustration


Execution of the forward

receiver swap concluded in

T=0

Selling the

insurance

contracts

ASSETS

LIABILITIES

Conclusion of the

forward swap, beginning

in T1 and maturity T2

Investment in

bonds of

maturity T1,

coupon C0

Investment in

bonds of

maturity T2,

coupon C1

Conclusion of the

spot payer swap

Hedging

Investments

T=0 T=T1 T=T2

End of the

spot swap

End of bonds

investment

End of

insurance

contracts

End of the

forward swap

11

12

■ Illustration


Selling the

insurance

contracts

ASSETS

LIABILITIES

Conclusion of the

forward swap, beginning

in T1 and maturity T2

Investment in

bonds of

maturity T1,

coupon C0

Investment in

bonds of

maturity T2,

coupon C1

Hedging

Investments

T=0 T=T1 T=T2

End of the

spot swap

End of bonds

investment

End of

insurance

contracts

End of the

forward swap

Receives C0,

Pays 4.75%

Receives C1 + K1 + EUR1Y,

Pays 4.75% + K2 + EUR1Y

Execution of the

forward receiver

swap

Conclusion of the

spot payer swap

12

■ 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


Remarks

13

■ 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)


Remarks

14

■ 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:

Interest Rate Swap: Valuation

),0()( leg Fixed1

1

n

i

iZCii TPTTK

15

■ 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:


),,0(1),0(

),0(

)(

11

1

1

ii

iZC

iZC

ii

i TTFTP

TP

TTK

16

■ 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. :

■ The value of the floating leg of the swap is the sum of these floating cash-flows (floaters):


),0(),0(

),0()(1),0(

),0(

)(

1),0()(

1

11

1

1

iZCiZC

iZCii

iZC

iZC

ii

iZCiii

TPTP

TPTTTP

TP

TTTPTTK

n

1i

1 ),0(1),0(),0( swap theof leg floating nZCiZCiZC TPTPTP

= value of a floater

17

■ The fixed rate of the swap is determined by equating the market value of the fixed and floating legs:

■ Which leads to:


01 ),0()( 1

1 ),T(PTPTTK nZC

n

i

iZCii

n

i

iZCii

nZC

TPTT

TPK

1

1 ),0()(

),0(1

(1)

18

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:

■ By summing on the different maturities and equating to the fixed leg value, we get another expression for the swap rate K:

■ The swap rate appears as a weighted mean (convex combination) of forward rates


n

i

iZCii

n

i

iZCiiii

TPTT

TPTTTTF

K

1

1

1

11

),0()(

),0())(,;0(

(2)

),0())(,;0(),0()( 111 iZCiiiiiZCiii TPTTTTFTPTTK

19

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:

■ 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 )

Interest Rate Swap: Valuation – Remarks

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11 ),0())(,;0(),0(

1),0())(,;0(),0(1

11

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20

■ This can also be directly deduced from the value of each floater

Remark 2

■ 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:

Interest Rate Swap: Valuation – Remarks

1

)0(1)0(

leg floating)0(

)0( bond rate floating1

nZCnZC

nZC

n

i

inZC

,TP,TP

,TP

floaters,TP

1),0(),0()(

),0()(

),0(1

1 nominal andn maturity K,coupon with bond a of price

1

1

1

1

n

i

nZCiZCiin

i

iZCii

nZC TPTPTTK

TPTT

TPK

21

■ 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

Yield Curve: Remarks

22 Basic Financial Tools

■ By the same reasoning as for getting the par swap rate, the market value of a (receiver) swap at t is obtained as:

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

Interest Rate Swaps : Valuation after conclusion

n

ki

iZCiiiiS

kZCkkkkS

TtPTTTTtFR

TtPTTTTLR

1

11

11

).,()))(,;((

),()))(,(( at t valueSwap


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

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)


31/12/2012

(Valuation Date)

31/3/2013:

Next refixing, next

payment

30/9/2013:

Last payment

30/9/2012:

Last refixing:

floating 6M at 6%


■ Hypothesis: Current yield curve:

● R(0.25) = 5%, R(0.75) = 7% (annual compounding).

■ Fixed leg value:

■ Floating leg value:

● Forward Libor rate (= simple rate) at 31/12/2012 for the period going from 31/3/2013 to 30/9/2013:


93.919,96)07.01(

000,50

)05.01(

000,50leg Fixed

75.025.0

1)05.01(

)07.01(5.0)9/30 ,3/31 ;12/31(

25.0

75.0

F


%86.71)05.01(

)07.01(2)9/30 ,3/31 ;12/31(

25.0

75.0

F

■ Floating leg value:

●So that the floating leg is equal to:

■ Swap value = fixed leg – floating leg

= 96,919.93 – 66,990.86 = 29,929.07


86.990,66)07.01(

5.0598,78

)05.01(

5.0000,60leg Floating

75.025.0


Documents

Interest Rate Swap: Definition - Personal Homepageshomepages.ulb.ac.be/~cazizieh/Statf508_files/swaps.pdfThe fixed rate is set in advance so that the contract is fair for both parties