Derivatives Swaps

1
DerivativesSwaps

• Professor André Farber
• Solvay Business School
• Université Libre de Bruxelles

2
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.

3
Swaps Introduction

• Contract whereby parties are committed
• To exchange cash flows
• At future dates
• Two most common contracts
• Interest rate swaps
• Currency swaps

4
Plain vanilla interest rate swap

• Contract by which
• Buyer (long) committed to pay fixed rate R
• Seller (short) committed to pay variable r
  (ExLIBOR)
• on notional amount M
• No exchange of principal
• at future dates set in advance
• t ?t, t 2 ?t, t 3?t , t 4 ?t, ...
• Most common swap 6-month LIBOR

5
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 Libor1 4
• 3.80 Libor
• Inflow Libor 3.70
• Total 4.80 Libor0.3
• Saving 0.20 0.20
• A free lunch ?

3.80
3.70
4
Libor1
Bank
A
B
Libor
Libor
6
Payoffs

• Periodic payments (i1, 2, ..,n) at time t?t,
  t2?t, ..ti?t, ..,T tn?t
• 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-1t (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

7
IRS Decompositions

• IRSCash 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 Rate
• TIME 0 ? 2? (n-1)? n ?
• Inflow r0 ? r1 ? ... rn-2 ? 1rn-1 ?
• Outflow R ? R ? ... R ? 1R ?
• Decomposition 2 n FRAs
• TIME 0 ? 2? (n-1)? n ?
• Cash flow (r0 - R)? (r1 -R)? (rn-2
  -R)? (rn-1- R)

8
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

9
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

10
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 (1rn-1?)/
  (1rn-1?) 100
• At time n-2 Vfloat,n-2 (Vfloat,n-1 100rn-2?)/
  (1rn-2?) 100
• and so on and on.

Vfloat
100
Time
11
IR Swap Long floating rate note
Short fixed rate note
Value of swap fswap Vfloat - Vfix
Fixed rate R set initially to achieve fswap 0
12
(ii) IR Swap Portfolio of n FRAs
Value of FRA fFRA,i M ? DFi-1 - M ? (1 R ?t) ?
DFi
13
FRA Review
?t
i -1
i
Value of FRA fFRA,i M ? DFi-1 - M ? (1 R ?t) ?
DFi
14
(iii) Swap valuation based on forward rates
Rewrite the value of a FRA as
15
(iv) Swap valuation based on current swap rate
As
16
Swap Rate Calculation

• Value of swap fswap Vfloat - Vfix M - M R
  S di dn
• where dt discount factor
• Set R so that fswap 0 ? R (1-dn)/(S 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

17
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 R?t) di
• Reminder Vfra 0 if Rfra forward rate Fi-1,I
• Vfra t-1
• gt 0 If swap rate R gt fwd rate Ft-1,t
• 0 If swap rate R fwd rate Ft-1,t
• lt0 If swap rate R lt fwd rate Ft-1,t
• gt SWAP VALUE S Vfra t