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

DerivativesSwaps

Professor André Farber

Solvay Business School

Université Libre de Bruxelles

Derivatives 05 Swaps |2April 18, 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.

Derivatives 05 Swaps |3April 18, 2023

Swaps: Introduction

• Contract whereby parties are committed:

– To exchange cash flows

– At future dates

• Two most common contracts:

– Interest rate swaps

– Currency swaps

Derivatives 05 Swaps |4April 18, 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

Derivatives 05 Swaps |5April 18, 2023

Interest Rate Swap: Example

Objective Borrowing conditions

Fix Var

A Fix 5% Libor + 1%

B Var 4% Libor+ 0.5%

Swap:

• Gains for each company

• A B

Outflow Libor+1% 4%

3.80% Libor

Inflow 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%

Derivatives 05 Swaps |6April 18, 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

Derivatives 05 Swaps |7April 18, 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)

Derivatives 05 Swaps |8April 18, 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

Derivatives 05 Swaps |9April 18, 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

Derivatives 05 Swaps |10April 18, 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

Derivatives 05 Swaps |11April 18, 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

Derivatives 05 Swaps |12April 18, 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

Derivatives 05 Swaps |13April 18, 2023

FRA Review

i -1 i

Δt

1

1

( )

(1 )i

i

r R tM

r t

1

1

(1 ) (1 )

(1 )i

i

r t R tM

r t

M (1 )M R t

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

Derivatives 05 Swaps |14April 18, 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 DF

DF

Rewrite the value of a FRA as:

1

ˆ( )n

swap i it

f M R R t DF

Derivatives 05 Swaps |15April 18, 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

Derivatives 05 Swaps |16April 18, 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 3

Spot 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%

Derivatives 05 Swaps |17April 18, 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