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Derivatives Swaps Professor André Farber Solvay Business School Université Libre de Bruxelles

Derivatives Swaps

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Derivatives Swaps . Professor André Farber Solvay Business School Université Libre de Bruxelles. Interest Rate Derivatives. Forward rate agreement (FRA) : OTC contract that allows the user to "lock in" the current forward rate. - PowerPoint PPT Presentation

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Page 1: Derivatives Swaps

DerivativesSwaps

Professor André FarberSolvay Business SchoolUniversité Libre de Bruxelles

Page 2: Derivatives Swaps

Derivatives 05 Swaps |2April 22, 2023

Interest Rate Derivatives

• Forward rate agreement (FRA): OTC contract that allows the user to "lock in" the current forward rate.

• Treasury Bill futures: a futures contract on 90 days Treasury Bills

• Interest Rate Futures (IRF): exchange traded futures contract for which the underlying interest rate (Dollar LIBOR, Euribor,..) has a maturity of 3 months

• Government bonds futures: exchange traded futures contracts for which the underlying instrument is a government bond.

• Interest Rate swaps: OTC contract used to convert exposure from fixed to floating or vice versa.

Page 3: Derivatives Swaps

Derivatives 05 Swaps |3April 22, 2023

Swaps: Introduction

• Contract whereby parties are committed: – To exchange cash flows– At future dates

• Two most common contracts:– Interest rate swaps– Currency swaps

Page 4: Derivatives Swaps

Derivatives 05 Swaps |4April 22, 2023

Plain vanilla interest rate swap

• Contract by which– Buyer (long) committed to pay fixed rate R– Seller (short) committed to pay variable r (Ex:LIBOR)

• on notional amount M• No exchange of principal• at future dates set in advance • t + t, t + 2 t, t + 3t , t+ 4 t, ...

• Most common swap : 6-month LIBOR

Page 5: Derivatives Swaps

Derivatives 05 Swaps |5April 22, 2023

Interest Rate Swap: Example

Objective Borrowing conditions

Fix VarA Fix 5% Libor + 1%B Var 4% Libor+ 0.5%

Swap:

• Gains for each company• A BOutflow Libor+1% 4% 3.80% LiborInflow Libor 3.70%Total 4.80% Libor+0.3% Saving 0.20% 0.20%

A free lunch ?

A Bank BLibor Libor

4%Libor+1%3.80% 3.70%

Page 6: Derivatives Swaps

Derivatives 05 Swaps |6April 22, 2023

Payoffs

• Periodic payments (i=1, 2, ..,n) at time t+t, t+2t, ..t+it, ..,T= t+nt • Time of payment i: ti = t + i t

• Long position: Pays fix, receives floating

• Cash flow i (at time ti): Difference between • a floating rate (set at time ti-1=t+ (i-1) t) and • a fixed rate R • adjusted for the length of the period (t) and • multiplied by notional amount M• CFi = M (ri-1 - R) t

• where ri-1 is the floating rate at time ti-1

Page 7: Derivatives Swaps

Derivatives 05 Swaps |7April 22, 2023

IRS Decompositions

• IRS:Cash Flows (Notional amount = 1, = t )TIME 0 2 ... (n-1) n Inflow r0 r1 ... rn-2 rn-1

Outflow R R ... R R

• Decomposition 1: 2 bonds, Long Floating Rate, Short Fixed RateTIME 0 2 … (n-1) n Inflow r0 r1 ... rn-2 1+rn-1

Outflow R R ... R 1+R

• Decomposition 2: n FRAs• TIME 0 2 … (n-1) n • Cash flow (r0 - R) (r1 -R) … (rn-2 -R) (rn-1- R)

Page 8: Derivatives Swaps

Derivatives 05 Swaps |8April 22, 2023

Valuation of an IR swap

• Since a long position position of a swap is equivalent to:– a long position on a floating rate note– a short position on a fix rate note

• Value of swap ( Vswap ) equals:

– Value of FR note Vfloat - Value of fixed rate bond Vfix

Vswap = Vfloat - Vfix

• Fix rate R set so that Vswap = 0

Page 9: Derivatives Swaps

Derivatives 05 Swaps |9April 22, 2023

Valuation

• (i) IR Swap = Long floating rate note + Short fixed rate note

• (ii) IR Swap = Portfolio of n FRAs

• (iii) Swap valuation based on forward rates (for given swap rate R):

• (iv) Swap valuation based on current swap rate for same maturity

Page 10: Derivatives Swaps

Derivatives 05 Swaps |10April 22, 2023

Valuation of a floating rate note

• The value of a floating rate note is equal to its face value at each payment date (ex interest).

• Assume face value = 100• At time n: Vfloat, n = 100

• At time n-1: Vfloat,n-1 = 100 (1+rn-1)/ (1+rn-1) = 100

• At time n-2: Vfloat,n-2 = (Vfloat,n-1+ 100rn-2)/ (1+rn-2) = 100

• and so on and on….

Vfloat

Time

100

Page 11: Derivatives Swaps

Derivatives 05 Swaps |11April 22, 2023

IR Swap = Long floating rate note + Short fixed rate note

Value of swap = fswap = Vfloat - Vfix

1

( )n

Swap i nt

f M M R t DF DF

Fixed rate R set initially to achieve fswap = 0

Page 12: Derivatives Swaps

Derivatives 05 Swaps |12April 22, 2023

(ii) IR Swap = Portfolio of n FRAs

Value of FRA fFRA,i = M DFi-1 - M (1+ R t) DFi

, 11 1 1

(1 )n n n

swap FRA i i i i ni i i

f f M DF M R t DF M M R t DF DF

, 11 1

(1 )n n

swap FRA i i ii i

f f M DF M R t DF

Page 13: Derivatives Swaps

Derivatives 05 Swaps |13April 22, 2023

FRA Review

i -1 iΔt

1

1

( )(1 )

i

i

r R tMr t

1

1

(1 ) (1 )(1 )

i

i

r t R tMr t

M (1 )M R t

Value of FRA fFRA,i = M DFi-1 - M (1+ R t) DFi

Page 14: Derivatives Swaps

Derivatives 05 Swaps |14April 22, 2023

(iii) Swap valuation based on forward rates

1,

ˆ(1 ) ( )iFRA i i i i

i

DFf M R t DF M R R t DFDF

Rewrite the value of a FRA as:

1

ˆ( )n

swap i it

f M R R t DF

Page 15: Derivatives Swaps

Derivatives 05 Swaps |15April 22, 2023

(iv) Swap valuation based on current swap rate

1

( )n

swap swap ii

f M R R t DF

1

n

swap i ni

M R t DF M M DF

As:

1

( )n

Swap i n float fixt

f M M R t DF DF V V

Page 16: Derivatives Swaps

Derivatives 05 Swaps |16April 22, 2023

Swap Rate Calculation

• Value of swap: fswap =Vfloat - Vfix = M - M [R di + dn]

where dt = discount factor

• Set R so that fswap = 0 R = (1-dn)/(di)• Example 3-year swap - Notional principal = 100

Spot rates (continuous)Maturity 1 2 3Spot rate 4.00% 4.50% 5.00%Discount factor 0.961 0.914 0.861

R = (1- 0.861)/(0.961 + 0.914 + 0.861) = 5.09%

Page 17: Derivatives Swaps

Derivatives 05 Swaps |17April 22, 2023

Swap: portfolio of FRAs

• Consider cash flow i : M (ri-1 - R) t– Same as for FRA with settlement date at i-1

• Value of cash flow i = M di-1- M(1+ Rt) di

• Reminder: Vfra = 0 if Rfra = forward rate Fi-1,I

• Vfra t-1

• > 0 If swap rate R > fwd rate Ft-1,t

• = 0 If swap rate R = fwd rate Ft-1,t

• <0 If swap rate R < fwd rate Ft-1,t

• => SWAP VALUE = Vfra t