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Derivatives Swaps

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.

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Derivatives Swaps

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  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. Derivatives 05 Swaps

  3. 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

  4. Plain vanilla interest rate swap • Contract by which • Buyer (long) committed to pay fixed rate R • Seller (short) committed to pay variabler (Ex:LIBOR) • on notional amountM • 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 Derivatives 05 Swaps

  5. 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 ? Interest Rate Swap: Example 3.80% 3.70% Libor+1% 4% Bank A B Libor Libor Derivatives 05 Swaps

  6. 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 Derivatives 05 Swaps

  7. 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 Rate TIME 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

  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 Derivatives 05 Swaps

  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 Derivatives 05 Swaps

  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 (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 100 Time Derivatives 05 Swaps

  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 Derivatives 05 Swaps

  12. (ii) IR Swap = Portfolio of n FRAs Value of FRA fFRA,i= MDFi-1 - M (1+ Rt) DFi Derivatives 05 Swaps

  13. FRA Review Δt i i -1 Value of FRA fFRA,i= MDFi-1 - M (1+ Rt) DFi Derivatives 05 Swaps

  14. (iii) Swap valuation based on forward rates Rewrite the value of a FRA as: Derivatives 05 Swaps

  15. (iv) Swap valuation based on current swap rate As: Derivatives 05 Swaps

  16. Swap Rate Calculation • Value of swap: fswap =Vfloat - Vfix = M - M [RSdi + dn] where dt = discount factor • Set R so that fswap = 0  R = (1-dn)/(Sdi) • 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

  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 • > 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 Derivatives 05 Swaps

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