Derivatives Swaps Professor Andr Farber Solvay Business School
Derivatives Swaps Professor André Farber Solvay Business School Université Libre de Bruxelles Derivatives 05 Swaps
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. 08 January 2022 Derivatives 05 Swaps 2
Swaps: Introduction • Contract whereby parties are committed: – To exchange cash flows – At future dates • Two most common contracts: – Interest rate swaps – Currency swaps 08 January 2022 Derivatives 05 Swaps 3
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 + 3 t , t+ 4 t, . . . • Most common swap : 6 -month LIBOR 08 January 2022 Derivatives 05 Swaps 4
Interest Rate Swap: Example Objective A B • 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% Borrowing conditions Fix Var 5% Libor + 1% 4% Libor+ 0. 5% Fix Var Swap: 3. 80% Libor+1% Bank A Libor 08 January 2022 A free lunch ? 3. 70% B 4% Libor Derivatives 05 Swaps 5
Payoffs • Periodic payments (i=1, 2, . . , n) at time t+ t, t+2 t, . . t+i t, . . , T= t+n 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-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 08 January 2022 Derivatives 05 Swaps 6
IRS Decompositions • IRS: Cash Flows (Notional amount = 1, = t ) TIME 0 2 . . . Inflow r 0 r 1 . . . Outflow R R . . . (n-1) rn-2 R n rn-1 R • Decomposition 1: 2 bonds, Long Floating Rate, Short Fixed Rate TIME 0 2 … (n-1) n Inflow r 0 r 1 . . . rn-2 1+rn-1 Outflow R R . . . R 1+R • • • Decomposition 2: n FRAs TIME 0 2 Cash flow (r 0 - R) (r 1 -R) 08 January 2022 … … (n-1) (rn-2 -R) n (rn-1 - R) Derivatives 05 Swaps 7
Valuation of an IR swap • Since a long 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 08 January 2022 Derivatives 05 Swaps 8
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 08 January 2022 Derivatives 05 Swaps 9
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+ 100 rn-2 )/ (1+rn-2 ) = 100 and so on and on…. Vfloat 100 Time 08 January 2022 Derivatives 05 Swaps 10
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 08 January 2022 Derivatives 05 Swaps 11
(ii) IR Swap = Portfolio of n FRAs Value of FRA f. FRA, i = M DFi-1 - M (1+ R t) DFi 08 January 2022 Derivatives 05 Swaps 12
FRA Review Δt i i -1 Value of FRA f. FRA, i = M DFi-1 - M (1+ R t) DFi 08 January 2022 Derivatives 05 Swaps 13
(iii) Swap valuation based on forward rates Rewrite the value of a FRA as: 08 January 2022 Derivatives 05 Swaps 14
(iv) Swap valuation based on current swap rate As: 08 January 2022 Derivatives 05 Swaps 15
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% 08 January 2022 Derivatives 05 Swaps 16
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 • > 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 = S Vfra t 08 January 2022 Derivatives 05 Swaps 17
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