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

June 17, 2015 Derivatives 05 Swaps |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.

June 17, 2015 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

June 17, 2015 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 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

June 17, 2015 Derivatives 05 Swaps |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 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 B Libor 4%Libor+1% 3.80%3.70%

June 17, 2015 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: t i = t + i  t 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)  t) and a fixed rate R adjusted for the length of the period (  t) and multiplied by notional amount M CF i = M  (r i-1 - R)   t where r i-1 is the floating rate at time t i-1

June 17, 2015 Derivatives 05 Swaps |7 IRS Decompositions IRS:Cash Flows (Notional amount = 1,  =  t ) TIME0  2 ...(n-1)  n  Inflowr 0  r 1 ...r n-2  r n-1  OutflowR  R ...R  R  Decomposition 1: 2 bonds, Long Floating Rate, Short Fixed Rate TIME0  2  …(n-1)  n  Inflowr 0  r 1 ...r n-2  1+r n-1  OutflowR  R ...R  1+R  Decomposition 2: n FRAs TIME0  2  …(n-1)  n  Cash flow(r 0 - R)  (r 1 -R)  … (r n-2 -R)  (r n-1 - R)

June 17, 2015 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 ( 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

June 17, 2015 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

June 17, 2015 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: V float, n = 100 At time n-1: V float,n-1 = 100 (1+r n-1  )/ (1+r n-1  ) = 100 At time n-2: V float,n-2 = (V float,n r n-2  )/ (1+r n-2  ) = 100 and so on and on…. V float Time 100

June 17, 2015 Derivatives 05 Swaps |11 IR Swap = Long floating rate note + Short fixed rate note Value of swap = f swap = V float - V fix Fixed rate R set initially to achieve f swap = 0

June 17, 2015 Derivatives 05 Swaps |12 (ii) IR Swap = Portfolio of n FRAs Value of FRA f FRA,i = M  DF i-1 - M  (1+ R  t)  DF i

June 17, 2015 Derivatives 05 Swaps |13 FRA Review i -1 i ΔtΔt Value of FRA f FRA,i = M  DF i-1 - M  (1+ R  t)  DF i

June 17, 2015 Derivatives 05 Swaps |14 (iii) Swap valuation based on forward rates Rewrite the value of a FRA as:

June 17, 2015 Derivatives 05 Swaps |15 (iv) Swap valuation based on current swap rate As:

June 17, 2015 Derivatives 05 Swaps |16 Swap Rate Calculation Value of swap: f swap =V float - V fix = M - M [R   d i + d n ] where d t = discount factor Set R so that f swap = 0  R = (1-d n )/(   d i ) Example 3-year swap - Notional principal = 100 Spot rates (continuous) Maturity Spot rate 4.00% 4.50% 5.00% Discount factor R = ( )/( ) = 5.09%

June 17, 2015 Derivatives 05 Swaps |17 Swap: portfolio of FRAs Consider cash flow i : M (r i-1 - R)  t –Same as for FRA with settlement date at i-1 Value of cash flow i = M d i-1 - M(1+ R  t) d i Reminder: V fra = 0 if R fra = forward rate F i-1,I V fra t-1 > 0If swap rate R > fwd rate F t-1,t = 0If swap rate R = fwd rate F t-1,t <0If swap rate R < fwd rate F t-1,t => SWAP VALUE =  V fra t