Ebook The Valuation of Convertible Bonds With Credit Risk
The market for convertible bonds has been expanding rapidly. In the U.S., over $105 billion of new convertibles were issued in 2001, as compared with just over $60 billion in 2000. As of early in 2002, there were about $270 billion of convertibles outstanding, more than double the level of five years previously, and the global market for convertibles exceeded $500 billion.1 Moreover, in the past couple of decades there has been considerable innovation in the contractual features of convertibles. Examples include liquid yield option notes (McConnell and Schwartz, 1986), mandatory convertibles (Arzac, 1997), “death spiral” convertibles (Hillion and Vermaelen, 2001), and cross currency convertibles (Yigitbasioglu, 2001). It is now common for convertibles to feature exotic and complicated features, such as trigger prices and “soft call” provisions. These preclude the issuer from exercising its call option unless the firm’s stock price is either above some specified level, has remained above a level for a specified period of time (e.g. 30 days), or has been above a level for some specified fraction of time (e.g. 20 out of the last 30 days).
The modern academic literature on the valuation of convertibles began with the papers of Ingersoll (1977) and Brennan and Schwartz (1977, 1980). These authors build on the “structural” approach for valuing risky non-convertible debt (e.g. Merton, 1974; Black and Cox, 1976; Longstaff and Schwartz, 1995). In this approach, the basic underlying state variable is the value of the issuing firm. The firm’s debt and equity are claims contingent on the firm’s value, and options on its debt and equity are compound options on this variable. In general terms, default occurs when the firm’s value becomes sufficiently low that it is unable to meet its financial obligations.2 An overview of this type of model is provided in Nyborg (1996). While in principle this is an attractive framework, it is subject to the same criticisms that have been applied to the valuation of risky debt by Jarrow and Turnbull (1995). In particular, because the value of the firm is not a traded asset, parameter estimation is difficult. Also, any other liabilities which are more senior than the convertible must be simultaneously valued.
To circumvent these problems, some authors have proposed models of convertible bonds where the basic underlying factor is the issuing firm’s stock price (augmented in some cases with additional random variables such as an interest rate). As this is a traded asset, parameter estimation is simplified (compared to the structural approach). More over, there is no need to estimate the values of all other more senior claims. An early example of this approach is McConnell and Schwartz (1986). The basic problem here is that the model ignores the possibility of bankruptcy. McConnell and Schwartz address this in an ad hoc manner by simply using a risky discount rate rather than the risk free rate in their valuation equation. More recent papers which similarly include a risky discount rate in a somewhat arbitrary fashion are those of Cheung and Nelken (1994) and Ho and Pfeffer (1996).
An additional complication which arises in the case of a convertible bond (as opposed to risky debt) is that different components of the instrument are subject to different default risks. This is noted by Tsiveriotis and Fernandes (1998), who argue that “the equity upside has zero default risk since the issuer can always deliver its own stock [whereas] coupon and principal payments and any put provisions ...depend on the issuer’s timely access to the required cash amounts, and thus introduce credit risk” (p. 95). To handle this, Tsiveriotis and Fernandes propose splitting convertible bonds into two components: a “cash-only” part, which is subject to credit risk, and an equity part, which is not. This leads to a pair of coupled partial differential equations that can be solved to value convertibles. A simple description of this model in the binomial context may be found in Hull (2003). Yigitbasioglu (2001) extends this framework by adding an interest rate factor and, in the case of cross-currency convertibles, a foreign exchange risk factor.
Recently, an alternative to the structural approach has emerged. This is known as the “reduced-form” approach. It is based on developments in the literature on the pricing of risky debt (see, e.g. Jarrow and Turnbull, 1995; Duffie and Singleton, 1999; Madan and Unal, 2000). In contrast to the structural approach, in this setting default is exogenous, the “consequence of a single jump loss event that drives the equity value to zero and requires cash outlays that cannot be externally financed” (Madan and Unal, 2000, p. 44). The probability of default over the next short time interval is determined by a specified hazard rate. When default occurs, some portion of the bond (either its market value immediately prior to default, or its par value, or the market value of a default-free bond with the same terms) is assumed to be recovered. Authors who have used this approach in the convertible bond context include Davis and Lischka (1999), Takahashi et al. (2001), Hung and Wang (2002), and Andersen and Buffum (2003). As in models such as that of Tsiveriotis and Fernandes (1998), the basic underlying state variable is the firm’s stock price (though some of the authors of these papers also consider additional factors such as stochastic interest rates or hazard rates).
While this approach is quite appealing, the assumption that the stock price instantly jumps to zero in the event of a default is highly questionable. While it may be a reasonable approximation in some circumstances, it is clearly not in others. For instance, Clark and Weinstein (1983) report that shares in firms filing for bankruptcy in the U.S. had average cumulative abnormal returns of -65% during the three years prior to a bankruptcy announcement, and had abnormal returns of about -30% around the announcement. Beneish and Press (1995) find average cumulative abnormal returns of -62% for the three hundred trading days prior to a Chapter 11 filing, and a drop of 30% upon the filing announcement. The corresponding figures for a debt service default are -39% leading up to the announcement and -10% at the announcement. This clearly indicates that the assumption of an instantaneous jump to zero is extreme. In most cases, default is better characterized as involving a gradual erosion of the stock price prior to the event, followed by a significant (but much less than 100%) decline upon the announcement, even in the most severe case of a bankruptcy filing.
However, as we shall see below, in some models it is at least implicitly assumed that a default has no impact on the firm’s stock price. This may also be viewed as unsatisfactory. To address this, we propose a model where the firm’s stock price drops by a specified percentage (between 0% and 100%) upon a default. This effectively extends the reduced form approach which, in the case of risky debt, specifies a fractional loss in market value for a bond, to the case of convertibles by similarly specifying a fractional decline in the issuing firm’s stock price.
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