Risky Bonds, Floaters and Swaps
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We turn from risk-free government debt to the more general case of risky bonds. These are usually valued using discount rates that are based on the risk-free rate plus some spread to reflect the higher risk. We start by discussing the concept of duration, a necessary input to our analysis. We discuss two types of spread: First, the Z-spread—a fixed spread above the zero curve or the term structure of spot interest rates; Second, the nominal spread—a fixed premium above the yield to maturity of a government coupon bond that is approximately similar to the bond being valued. We then turn to estimating the spread for a particular bond using a market approach—finding the spread for quoted bonds with a similar credit rating. This leads us to estimating a credit rating when not available—a synthetic credit rating, by comparing standard financial ratios for an issuer, with published averages for issuers with various ratings. We discuss how to use the Jarrow-Lando-Turnbull model to handle future changes in credit rating based on a transition matrix. We turn next to the valuation of floating-rate notes or floaters. We show that a floater with discount rates matching its coupon rates has a value at par. We use this fact to derive floater values under more general conditions. We turn next to a vanilla interest rate swap—an instrument where a fixed rate is exchanged for a floating rate, or vice-versa. We show how a vanilla interest rate swap can be valued using a replicating portfolio that includes a long position in a fixed-rate instrument paired with a short position in a floating-rate instrument, or vice-versa. Finally, we discuss complex instruments that have embedded options—convertible bonds, callable bonds, and puttable bonds. We discuss the Goldman-Sachs lattice model to value convertible bonds. We discuss how to use the Black-Derman-Toy lattice to value callable and puttable bonds.