1 AN EMPIRICAL COMPARISON OF CONVERTIBLE BOND VALUATION MODELS - PDF document

1 AN EMPIRICAL COMPARISON OF CONVERTIBLE BOND VALUATION MODELS Robert Jones, Chris Veld, and Yuriy Zabolotnyuk * November, 21 2006 JEL-codes: C12, C63, G13. Keywords: convertible bonds, credit risk, Ayache-Forsyth-Vetzal model, Tsviveriotis- Fernandes model, Takahashi-Kobayashi-Nakagawa model, convertible arbitrage * Robert Jones is Professor of Economics at the Department of Economics, Simon Fraser University, Burnaby, Canada (e-mail: r.jones@sfu.ca). Chris Veld is Professor of Finance in the Department of Accounting and Finance, University of Stirling, United Kingdom and Adjunct Professor of Finance, Faculty of Business Administration, Simon Fraser University, Burnaby, Canada (e-mail: c.h.veld@stir.ac.uk). Yuriy Zabolotnyuk is PhD student in the Faculty of Business Administration, Simon Fraser University, Burnaby. Corresponding author: Yuriy Zabolotnyuk, Faculty of Business Administration, Simon Fraser University, 8888 University Drive, Burnaby, BC, Canada V5A 1S6, tel: 604-291-5565, fax: 604-291-4920, e-mail: yzabolot@sfu.ca Chris Veld gratefully recognizes the financial support of the Social Sciences and Humanities Research Council of Canada. The usual disclaimer applies.

2 AN EMPIRICAL COMPARISON OF CONVERTIBLE BOND VALUATION MODELS Abstract This paper compares three theoretical models for the valuation of convertible bonds using a sample of Canadian convertible bonds for the period from January 2005 to April 2006. Model prices are calculated using numerical methods and are then compared to market prices. The absolute deviation of the model price from the market price is 2.3% for the Tsiveriotis- Fernandes (1998) model, 3.2% for the Takahashi-Kobayashi-Nakagawa (2001) model and 4.1% for the total default and the partial default Ayache-Forsyth-Vetzal (2003) models. For this and other measures of fit, the Tsiveriotis-Fernandes model, which also requires the lowest number of parameters, outperforms the other two models. Pricing errors are related to bond characteristics such as moneyness, volatility of the underlying stock, coupon rate, and time to maturity. However, these relationships are different between models.

3 I. Introduction Exchange-listed companies frequently attract capital by issuing convertible bonds. These are bonds that, at the option of the holder, can be exchanged into shares of common stock of the issuing company. Convertible bonds possess characteristics of both equity and debt. They resemble debt because they pay a fixed coupon interest. On the other hand, they resemble equity, because part of the price that is paid for them is for the option to exchange the bonds into shares. During the period from 1990 to 2003 there were no less than 7208 issues of convertible bonds. A large part of these bonds were issued in the United States (2166), but they were also relatively popular in countries such as Japan (1632), South Korea (827), Canada (280), and Australia (235). 1 An important problem that comes with convertible bonds is that they are difficult to value. This is caused by the fact that the exercise of the conversion right means that the bond has 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 practice, most convertible bonds are callable. 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. This callability provision adds to the difficulties in pricing convertible bonds. Even though both academics and practitioners agree that convertible bond valuation is an important topic there is not much empirical literature on this topic. This paper aims to fill this gap in the literature by empirically comparing three different convertible bond valuation models for a large sample of Canadian convertible bonds. The modern literature on the pricing of convertible bonds is generally considered to start with Merton (1974) who is the first to create a model that uses the so-called “structural approach” for valuing convertible bonds. He specifies the default process as an endogenous process depending on the capital structure of the firm. Therefore, default occurs as soon as the diffusion process for firm value takes it below the debt value. Ingersoll (1977a) is the first to develop a model for pricing convertible bonds and preferred stocks that is based on the Black- Scholes (1973) model. In his paper, Ingersoll (1977a) determines optimal conversion and call policies. He establishes that the optimal policy for the issuer is to call bond when its call price is equal to conversion value. In order to check the optimal policy, it is only necessary to know 1 See Loncarski, ter Horst and Veld (2006a).

4 the call price, the conversion terms and the current stock price. Brennan and Schwartz (1977 ) extend the Black-Scholes (1973) option pricing model to price convertible bonds. They add discrete bond coupons, underlying stock dividends, positive firm default probability, call, and convertibility features. In this model, the firm assets consist of common stock and convertible debt. The risk free rate is assumed to be constant and known. They find that it is optimal to convert the bond only either right before the dividend payment date, before an adverse change in the conversion terms, or at the maturity date. At the same time, the firm needs to call bond only if the value of the bond if not called is equal to the call price. By using Ito’s lemma, the authors are able to show that market value of a convertible bond must follow some partial differential equation (PDE). By adding the bond value boundary conditions the authors solve the PDE for the convertible bond value using an explicit difference scheme. In a follow-up paper, Brennan and Schwartz (1980) add additional senior debt to the assets of the firm. They also make the interest rate assumptions more realistic by introducing stochastic interest rates. In their model, the value of a convertible bond depends on two variables: the value of the firm and the interest rate. The bond price depends on the firm value since it defines the probability of default and the stock price, while the interest rate influences the convertible bond price through the cash flow discounting rate. The authors assume for the interest rate to follow Ito’s process. They derive a PDE for valuing convertible bonds. They solve the equation subject to call, conversion, bankruptcy and conversion boundary conditions obtaining closed form solution with an implicit difference scheme. An interesting result they get shows that values of the bonds only slightly differ for stochastic interest rates, suggesting the use of the constant interest rates model for practical purposes. Longstaff and Schwartz (1995) argue that the assumption of bankruptcy that happens when firm’s assets go down to zero is unrealistic. They point out that usually firms become bankrupt before all their assets disappear. This bankruptcy feature improves the model forecasts. “Structural” models like those of Ingersoll (1977) and Brennan and Schwartz (1977, 1980) have one inherent problem that complicates the calculation of convertible bond prices. These models determine convertible stock prices as function of a firm value, variable not direclty observable in the market. For a convertible price calculation, a firm value has to be calculated first, an arduous task given complex corporate capital structure. Estimation complications arise