Ebook The Pricing of Finite Maturity Corporate Coupon Bonds with Rating-Based Covenants
Since Merton’s (1974)[27] seminal paper which pioneered the so called structural approach to pricing corporate bonds, we have seen an increasing growth of the literature in this field. At an early stage, an important contribution was provided by Black and Cox (1976)[5] who, in defining default as a trigger event that may happen at any moment of a bond’s life instead of occurring only at the maturity, relaxed one of the simplifying assumptions present in Merton’s model, and established a feature common to almost all structural models published thereafter. In this type of modeling exercise, default is triggered when the value of the firm’s assets reaches some specified value, the barrier level. The way in which this barrier level is set, either endogenously or exogenously, has been a distinguishing factor between different models. One way is to consider that the barrier level is determined by the shareholders, in order to maximize the equity value (e.g. Black and Cox (1976)[5], Leland (1994)[20], Leland and Toft (1996)[22], Goldstein, Ju and Leland (2001)[14], Ericsson and Reneby (2003)[10]). The alternative takes into consideration a barrier level set exogenously, reflecting the presence of some kind of covenant in the bond indenture (e.g Black and Cox (1976)[5], Kim, Ramaswamy and Sundaresan (1993)[19], Longstaff and Schwartz (1995)[23], Briys and de Varenne (1997)[6], Ericsson and Reneby (1998)[9], Schobel (1999)[29], Hsu, Sa-Requejo and Santa-Clara (2003)[15], Hui, Lo and Tsang (2003)[17], Taurén (1999)[30], Collin-Dufresne and Goldstein (2001)[7], Ju and Ou-Yang (2004)[18], Huang et al. (2003)[16]).
Besides different extensions concerning the interest rate process, recovery values, debt structure, bond characteristics, definition of barriers and dynamic capital structure, to name a few, the research in this field has also focused on aspects related to the asset substitution problem (Mello and Parsons (1992)[26], Leland (1998)[21], Ericsson (2000)[8], Bhanot and Mello (2006)[4]), the equity/bond holders strategic behavior (Anderson and Sundaresan [1], Anderson, Sundaresan and Tychon [2], Mella-Barral and Peraudin (1997)[25], Mella-Barral (1999)[24], Fan and Sundaresan (2000)[11]), and bankruptcy codes (François and Morrelec (2004)[12], Moraux (2002)[28], Galai, Raviv and Wiener (2003)[13] and Yu (2003)[31]).
The structural model proposed here aims to price corporate bonds whose indenture incorporates a rating trigger based covenant, which links the pay-offs to bondholders with the credit rating of the firm. As put by Bhanot and Mello (2006)[4]: ”a ”rating trigger clause” in a corporate bond indenture requires a firm to prepay its debt or to change the coupon rate on its debt if the firm’s credit rating reaches a specified level.” Although the Bhanot and Mello (2006) framework does take into consideration this kind of bond, and considers two types of rating trigger covenants (partial amortization of the debt’s principal or an accrued coupon rate), their model only applies to perpetual debt. By contrast, the present paper proposes a framework capable of dealing with finite maturity bonds.
The finite maturity case was addressed by Bhanot (2003)[3], whose model assumes the existence of two possible credit events: namely, a downgrade in the credit rating of the firm and bankruptcy, which implies the liquidation of the firm. These two events were modeled through the specification of two barriers levels, VB1 and VB2 respectively (with VB1 > VB2), established exogenously. However, there are two limitations in Bhanot (2003) that we will try to overcome in the current paper. In the first place, albeit the purpose of Bhanot (2003) is to price bonds with rating based covenants, the model does not explicitly assume any kind of change in bondholder pay-offs, when the covenant is triggered. In other words, when the value of the firm’s assets reaches the first barrier (VB1), the rating change only affects some parameter values associated with the diffusion process governing the value dynamics of the firm’s assets. Additionally, even if the previous remark is not taken into consideration, the price formula developed by Bhanot (2003) assumes that the payment to bondholders at maturity (admitting that, in the mean time, the firm has not entered in bankruptcy and so has not been liquidated, which is equivalent to admitting that the second barrier VB2 as not been reached) always corresponds the bond principal. This final cash-flow only makes sense in a scenario where the value of the firm’s assets is enough to cover it. Otherwise, if the value of these assets is insufficient to cover the face value of the bond, at most the bondholders will only receive the value of the assets, since the equity holders will not be willing to pay the difference. In this sense, we may say that Bhanot (2003) overvalues the bondholders expected cash-flows, resulting in an overpricing of the bond.
Using the same base structure of Bhanot’s model, which defines both credit events (rating change and bankruptcy) through the barrier levels, VB1 and VB2, we propose to obtain a bond pricing formula that takes into account those two remarks. Besides obtaining the bond value before the rating change takes place, we also derive value expressions at the moment of the rating change and immediately after that. The same is done for equity, bankruptcy costs, tax benefits and leveraged firm value.
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