An Empirical Comparison of Convertible Bond Valuation Models
Zabolotnyuk, Yuriy, Jones, Robert, Veld, Chris, Financial Management
This paper empirically compares three convertible bond valuation models. We use an innovative approach where all model parameters are estimated by the Marquardt algorithm using a subsample of convertible bond prices'. The model parameters are then used for out-of-sample forecasts of convertible bond prices. The mean absolute deviation is 1.86% for the Ayache-Forsyth-Vetzal model, 1.94% for the Tsiveriotis-Fernandes model, and 3.73% for the Brennan-Schwartz model. For this and other measures of fit, the Ayache-Forsyth-Vetzal and Tsiveriotis-Fernandes models outperform the Brennan-Schwartz model.
Exchange-listed companies frequently attract capital by issuing convertible bonds. During the period from 1990 to 2003, there were globally more than 7,000 issues of convertible bonds. (1) An important issue with convertible bonds is that they are difficult to value. This is due to the fact that the exercise of the conversion right requires the bond to be redeemed in order to acquire the shares. For this reason, a conversion right is, in fact, a call option with a stochastic exercise price. In addition, most convertible bonds are callable in practice. (2) This means that the issuing firm has the right to pay a specific amount, the call price, to redeem the bond before the maturity date. In some convertible bond contracts the call notice period is specified, thereby requiring the firm to announce the calling date well before the redemption can be performed. Often, the call notice period is combined with a soft call feature where the bond can only be called if the underlying stock price stays above a certain prespecified level for a prespecified period. All these features complicate the valuation process for convertible bonds.
Despite the importance of convertible bond valuation for both academic and practical purposes, there is not much empirical literature on this topic. This paper aims to fill this gap by empirically comparing three different convertible bond valuation models for a large sample of Canadian convertible bonds.
Convertible bonds are issued by corporate issuers and, as such, are subject to the possibility of default. There are two main approaches for valuing securities with default risk. The first approach, called structural approach, assumes that default is an endogenous event and bankruptcy happens when the value of the firm's assets reaches some low threshold level. This approach was pioneered by Merton (1974) who assumes that the firm value follows a stochastic diffusion process and default happens as soon as the firm value falls below the face value of the debt. However, as pointed out by Longstaff and Schwartz (1995), a default usually happens well before the firm depletes all of its assets. Valuation of the multiple debt issues in Merton (1974) is subject to strict absolute priority where any senior debt has to be valued before any subordinated debt is considered. This creates additional computational difficulties for valuing defaultable debt of firms with multiple debt issues. Moreover, the credit spreads implied by the approach of Merton (1974) are much smaller than those observed in financial markets.
In contrast, a default in the Longstaff and Schwartz (1995) model happens before the firm exhausts all of its assets and as soon as the firm value reaches some predefined level common for all issues of debt. The values of the credit spreads predicted by their model are comparable to the market observed spreads. The common default threshold for all securities allows valuation of multiple debt issues. To obtain more realistic credit spreads, particularly for short maturity issues, Zhou (2001) develops a structural approach model where both diffusion and jumps are allowed in the asset value process. The addition of a jump process allows for the possibility of instantaneous default caused by a sudden drop in firm value.
In the structural approach, debt is viewed as an option on the value of the assets of the firm, and an option embedded in the convertible bond can be viewed as a compound option on the value of firm assets. …