Lecture 9 Basics on Swaps Agenda: 1. Introduction to Swaps ~ Definition: ~ Basic functions ~ Comparative advantage: 2. Swap quotes and LIBOR zero rate ~ Interest rate swap is combination of two bonds: ~ Determining the LIBOR zero curve (Swap zero curve) based on Swap rate:
1. Introduction to Swaps ~ Definition: A swap is an agreement between two companies to exchange cash flows in the future. It specifies the dates and the way to calculate cash flows. Futures contracts are a simple form of swaps. Whereas Futures contracts lead to the exchange of cash flows on just one future date, swaps lead to cash flow exchanges taking place on several future dates. Fixed for floating interest rates: A 5.0% 6-m LIBOR B Notional principal: $100 million Period: January 20011 through January 2014 Payment: every 6 months Firm B Date 6-m LIBOR Floating Fixed Net Jan-11 4.20% Jul-11 4.80% $2.10 $2.50 ($0.40) Jan-12 5.30% $2.40 $2.50 ($0.10) Jul-12 5.50% $2.65 $2.50 $0.15 Jan-13 5.60% $2.75 $2.50 $0.25 Jul-13 5.90% $2.80 $2.50 $0.30 Jan-14 6.40% $102.95 $102.50 $0.45 ~ The floating interest rates are determined in the beginning of each period. ~ Day count convention ~ A sells a floating-rate bond to B and purchases a fixed-rate bond from B. 1
~ Basic functions Without middleman Transform a liability: 5% 5.2% A B LIBOR+0.1% 6-m LIBOR Transform an asset: 5% LIBOR%-0.25% A B 4.7% 6-m LIBOR With middleman 4.985% 5.015% 5.2% A Broker B LIBOR+0.1% LIBOR LIBOR?? Market?? 5.2% A B LIBOR+0.1% Maker LIBOR LIBOR Day count convention: Floating: Actual / 360 Fixed: Actual / 365 Adjustment: LIBOR 365/360 or Fixed 360/365 2
~ Comparative advantage: Fixed rate bond Variable-rate bond Preference Quality company 9% LIBOR + ½% Variable Risky company 10 ½% LIBOR + 1% Fixed Quality company has a greater advantage in fixed-rate bonds. In order to save the borrowing cost, Quality company may issue fixed rate bond then arrange a floating-forfixed swap with Risky company. Quality Co. Variable-rate payments at LIBOR + ½ % Risky Co. Fixed-rate Payments at 9% Fixed-rate payments at 9 ½ % Variable-rate Payments at LIBOR +1% Investors in Fixedrate bonds issued by Quality Co. Investors in variable-rate bonds issued by Risky Co. Receive Pay Net cost Save Quality Co. 9 ½ % (9% + LIBOR+½%) LIBOR ½% Risky Co. LIBOR+½% (LIBOR+1% + 9 ½ % ) 10% ½% Without the SWAP, Quality Co. needs to pay LIBOR +½%, and Risky Co., 10 ½%. Through the SWAP, both of them can save ½% interest costs. 3
??? How to identify the cost saving opportunities through SWAP in financial markets???? How to calculate (arrange) the SWAP payments between two parties? Fixed rate bond Variable-rate bond Preference Quality company 9% LIBOR + ½% Variable Risky company 10 ½% LIBOR + 1% Fixed Difference 1 ½% ½% (gap: 1 ½% ½% =1%) As long as these two differences are not identical, there is a chance reduce borrowing costs through SWAP. The total cost saving is the gap between the two differences. If these two companies agree to equally share the gap (1%), then the payment arrangements can be solved by: Receive Pay Net cost Quality Co. R (9% + Q) = (LIBOR+½% ½%) Risky Co. Q (LIBOR+1% + R) = (10 ½% ½%) Where: R: the payments made by Risky Co. to Quality Co. Q: the payments made by Quality Co. to Risky Co. Solving these two equations, one set of solution is: R = 9 ½ % Q = LIBOR + ½ % 4
Comparative-advantage argument: The difference should be arbitraged away. The reason that spread differentials exist: Floating rate roll-over: The difference in fixed rate reflects that Risk company is more likely to default. 5
2. Swap quotes and LIBOR zero rate ~ Interest rate swap is combination of two bonds: A floating rate bond exchanges for a fixed rate bond. V swap = B fl -B fix Swap rate quotes: Maturity bid (%) offer (%) Swap rate (%) 2 6.03 6.06 6.045 3 6.21 6.24 6.225 4 6.35 6.39 6.37 5 6.47 6.51 6.49 7 6.65 6.68 6.665 10 6.83 6.87 6.85 Bid: pay fixed rate at bid rate and receive floating rate Offer: receive fixed rate at offer rate and pay floating rate Swap rate: average of bid and ask rates. If fixed rate = swap rate at initial date: V swap = 0 B fl = B fix =Par value=notional principal Because if the bond yield (market interest rate) = coupon rate, B fix = Par value If the bond variable coupon rates perfectly match LIBOR rates, B fl =Par value, because financial markets discount cash flows at LIBOR rates. Maturity LIBOR Par: Cash flow V B = PV of Cash flow 0 6.00% 100 100 1 7.00% 6 5.660377358 2 8.00% 7 6.171751014 3 9.00% 108 88.16787163 6
~ Determining the LIBOR zero curve (Swap zero curve) based on Swap rate: For short-term zero rate (up to 2~5 years), we use spot LIBOR and Eurodollar futures. For long-term zero rate, we rely on swap rates. Maturity S-T Zero rate (continuous) Swap rate (semiannual) Swap rate Interpolation (semiannual) L-T Zero rate Bootstrap (continuous) 0.5 5.5000% 1 5.7500% 1.5 5.9000% 2 6.0450%? 2.5 6.1350%? 3 6.2250% 3.5 6.2975% 4 6.3700% 5 6.4900% 7 6.6650% 10 6.8500% A 2-year bond with coupon rate 6.045% with 6.045% yield is priced par value. 3.0225e -5.5% 0.5 + 3.0225e -5.75% 1 + 3.0225e -5.9% 1.5 + 103.0225e -R 2 = 100 the 2-year zero rate =R = 5.9636% A 2.5-year bond with coupon rate 6.135% with 6.135% yield is priced par value. 3.0675e -5.5% 0.5 + ::: + 103.0675e -R 2.5 = 100 the 2.5-year zero rate =R = 6.0549% 7