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. 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 2
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 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 + Δt, t + 2 Δt, t + 3Δt, t+ 4 Δt,... Most common swap : 6-month LIBOR Derivatives 05 Swaps 4
Interest Rate Swap: Example Objective Borrowing conditions Fix Var A Fix 5% Libor + 1% B Var 4% Libor+ 0.5% Swap: Libor+1% 3.80% 3.70% A Bank B 4% 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? Libor Libor Derivatives 05 Swaps 5
Payoffs Periodic payments (i=1, 2,..,n) at time t+δt, t+2δt,..t+iδt,..,t= t+nδt Time of payment i: t i = t + i Δt Long position: Pays fix, receives floating Cash flow i (at time ti): Difference between a floating rate (set at time t i-1 =t+ (i-1) Δt) and a fixed rate R adjusted for the length of the period (Δt) and multiplied by notional amount M CF i = M (r i-1 - R) Δt where r i-1 is the floating rate at time t i-1 Derivatives 05 Swaps 6
IRS Decompositions IRS:Cash Flows (Notional amount = 1, τ= Δt ) TIME 0 τ 2τ... (n-1)τ n τ Inflow r 0 τ r 1 τ... r n-2 τ r n-1 τ Outflow R τ R τ... R τ R τ Decomposition 1: 2 bonds, Long Floating Rate, Short Fixed Rate TIME 0 τ 2τ (n-1)τ n τ Inflow r 0 τ r 1 τ... r n-2 τ 1+r n-1 τ Outflow R τ R τ... R τ 1+R τ Decomposition 2: n FRAs TIME 0 τ 2τ (n-1)τ n τ Cash flow (r 0 - R)τ (r 1 -R)τ (r n-2 -R)τ (r n-1 - R) Derivatives 05 Swaps 7
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 ( V swap ) equals: Value of FR note V float - Value of fixed rate bond V fix V swap = V float - V fix Fix rate R set so that Vswap = 0 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 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: V float, n = 100 At time n-1: V float,n-1 = 100 (1+r n-1 τ)/ (1+r n-1 τ) = 100 At time n-2: V float,n-2 = (V float,n-1 + 100r n-2 τ)/ (1+r n-2 τ) = 100 and so on and on. V float 100 Time Derivatives 05 Swaps 10
IR Swap = Long floating rate note + Short fixed rate note n f = M M ( R Δ t) DF + DF Swap i n t= 1 Value of swap = f swap = V float - V fix Fixed rate R set initially to achieve f swap = 0 Derivatives 05 Swaps 11
(ii) IR Swap = Portfolio of n FRAs n n [ (1 ) ] = = + Δ f f M DF M R t DF swap FRA, i i 1 i i= 1 i= 1 Value of FRA f FRA,i = M DF i-1 - M (1+ R Δt) DF i n n n f = f, = [ M DF 1 M(1 + RΔ t) DF] = M M RΔ t DF + DF swap FRA i i i i n i= 1 i= 1 i= 1 Derivatives 05 Swaps 12
FRA Review i -1 Δt i M ( r R) Δt i 1 (1 + r Δt) i 1 M (1 + ri 1Δt) (1 + RΔt) (1 + r Δt) i 1 M M(1 + RΔt) Value of FRA f FRA,i = M DF i-1 - M (1+ R Δt) DF i Derivatives 05 Swaps 13
(iii) Swap valuation based on forward rates n f = M ( Rˆ R) Δt DF swap i i t= 1 Rewrite the value of a FRA as: f DF = M (1 + RΔ t) DF = M ( Rˆ R) Δt DF i 1 FRA, i i i i DF i Derivatives 05 Swaps 14
(iv) Swap valuation based on current swap rate n f = M ( R R) Δt DF swap swap i i= 1 As: n M R Δ t DF = M M DF i= 1 swap i n n f = M M ( R Δ t) DF + DF = V V Swap i n float fix t= 1 Derivatives 05 Swaps 15
Swap Rate Calculation Value of swap: f swap =V float - V fix = M - M [R Σ d i + d n ] where d t = discount factor Set R so that f swap = 0 R = (1-d n )/(Σ d i ) 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 16
Swap: portfolio of FRAs Consider cash flow i : M (r i-1 - R) Δt Same as for FRA with settlement date at i-1 Value of cash flow i = M d i-1 - M(1+ RΔt) d i Reminder: V fra = 0 if R fra = forward rate F i-1,i V fra t-1 > 0 If swap rate R > fwd rate F t-1,t = 0 If swap rate R = fwd rate F t-1,t <0 If swap rate R < fwd rate F t-1,t => SWAP VALUE = Σ V fra t Derivatives 05 Swaps 17