2.1 Swaps Lecture 2 2.2 Types of Rates • Treasury rates • LIBOR rates • Euribor rates 2.3 2.4 2.5 2.6 2.7 Zero Rates A zero rate (or spot rate), for maturity T, is the rate of interest earned on an investment that provides a payoff only at time T 2.8 Example Maturity (years) 0.5 Zero Rate (% cont comp) 5.0 1.0 5.8 1.5 6.4 2.0 6.8 2.9 Bond Pricing • To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate • In our example, the theoretical price of a twoyear bond providing a 6% coupon semiannually is 3e 0.05 0.5 3e 0.0581.0 3e 103e 0.0682.0 98.39 0.064 1.5 2.10 Bond Yield • The bond yield is the discount rate that makes the present value of the cash flows on the bond equal to the market price of the bond • Suppose that the market price of the bond in our example equals its theoretical price of 98.39 • The bond yield is given by solving 3e y 0.5 3e y 1.0 3e y 1.5 103e y 2.0 98.39 to get y=0.0676 or 6.76%. 2.11 Forward Rates The forward rate is the future zero rate implied by today’s term structure of interest rates Calculation of Forward Rates Zero Rate for Forward Rate an n -year Investment for n th Year Year (n ) (% per annum) (% per annum) 1 10.0 2 10.5 11.0 3 10.8 11.4 4 11.0 11.6 5 11.1 11.5 2.12 2.13 Formula for Forward Rates • Suppose that the zero rates for time periods T1 and T2 are R1 and R2 with both rates continuously compounded. • The forward rate for the period between times T1 and T2 is R2 T2 R1 T1 T2 T1 2.14 Duration • Duration of a bond that provides cash flow c i at time t i is c i e yt i ti i 1 B n where B is its price & y is its yield (continuously compounded) • This leads to B B Dy 2.15 Duration Matching • This involves hedging against interest rate risk by matching the durations of assets and liabilities • It provides protection against small parallel shifts in the zero curve 2.16 Nature of Swaps • A swap is an agreement to exchange cash flows at specified future times according to certain specified rules An Example of a “Plain Vanilla” Interest Rate Swap • An agreement by “Company B” to receive 6-month LIBOR & pay a fixed rate of 5% per annum every 6 months for 3 years on a notional principal of $100 million • Next slide illustrates cash flows 2.17 Cash Flows to Company B ---------Millions of Dollars--------LIBOR FLOATING FIXED Net Date Rate Cash Flow Cash Flow Cash Flow Mar.1, 1998 4.2% Sept. 1, 1998 4.8% +2.10 –2.50 –0.40 Mar.1, 1999 5.3% +2.40 –2.50 –0.10 Sept. 1, 1999 5.5% +2.65 –2.50 +0.15 Mar.1, 2000 5.6% +2.75 –2.50 +0.25 Sept. 1, 2000 5.9% +2.80 –2.50 +0.30 Mar.1, 2001 6.4% +2.95 –2.50 +0.45 2.18 2.19 Typical Uses of an Interest Rate Swap • Converting a liability from – fixed rate to floating rate – floating rate to fixed rate • Converting an investment from – fixed rate to floating rate – floating rate to fixed rate A and B Transform a Liability 5% 5.2% A B LIBOR+0.8% LIBOR A: from 5.2 fixed to floating ---> pays Libor+0.2% B: from floating Libor+0.8% to fixed ---> pays 5%+0.8% 2.20 2.21 A and B Transform an Asset 5% 4.7% A B LIBOR-0.25% LIBOR 2.22 The Comparative Advantage Argument • Company A wants to borrow floating • Company B wants to borrow fixed Fixed Floating Company A 10.00% 6-month LIBOR + 0.30% Company B 11.20% 6-month LIBOR + 1.00% 2.23 The Swap 9.95% 10% A B LIBOR+1% LIBOR A: from 10% fixed to floating ---> pays Libor+0.05% B: from floating Libor+1% to fixed ---> pays 9.95%+1% 2.24 Valuation of an Interest Rate Swap • Interest rate swaps can be valued as the difference between the value of a fixedrate bond & the value of a floating-rate bond 2.25 Valuation in Terms of Bonds • The fixed rate bond is valued in the usual way • The floating rate bond is valued by noting that it is worth par immediately after the next payment date Swapping a BTP 2.26 2.27 Credit Risk • A swap is worth zero to a company initially • At a future time its value is liable to be either positive or negative • The company has credit risk exposure only when its value is positive 2.28 Examples of Other Types of Swaps • • • • • • Amortizing & step-up swaps Extendible & puttable swaps Index amortizing swaps Equity swaps Commodity swaps Differential swaps