The Handbook of Hybrid Securities: Convertible Bonds, CoCo Bonds and Bail-In

10 Modeling Hybrids: Handling Credit

10.1 CREDIT SPREAD

10.1.1 Definition

A credit spread cs is an add-on to an existing risk-free rate r in order to discount risky cash flows. As such, this spread reflects the market’s view on the credit quality of a particular bond. In absence of liquidity risk, the theoretical value at time t of a zero-coupon bond with face value N expiring at time T and using a continuously compounding risk-free interest rate r is:

(10.1)

The same bond can have different credit spread definitions. This all depends on the choice of the risk-free reference curve. A corporate bond could yield 100 bps more than the yield offered on Treasury bonds. Using the Libor curve as reference, the spread could, for example, be only 50 bps. In this latter case the risk-free curve is the Libor curve. This spread is then called the Z-spread.¹ The Z-spread of the bond is the difference between the yield to maturity of the bond and the yield interpolated on the swap curve for the same maturity of the bond. The credit spread is an example of how a metric used in fixed-income mathematics fails to be reliable in the space of hybrid bonds.

Hybrid bonds have an embedded optionality, the existence of which questions the applicability of the credit spread to discount the cash flows. In the case of a convertible bond, it is the investor who has the option to convert the bond into ...