An Empirical Comparison of Convertible Bond Valuation Models

Beginning of article

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.

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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. Therefore, the Black and Scholes (1973) methodology can be used for valuing convertible bonds. The models of Brennan and Schwartz (1977) and Ingersoll (1977) apply the structural approach to the valuation of convertible bonds. In these models, the interest rates are assumed to be nonstochastic. Brennan and Schwartz (1980) correct this by incorporating stochastic interest rates. However, they conclude that for a reasonable range of interest rates, the errors from the nonstochastic interest rate model are small. For practical purposes, it is preferable to use the simpler model with nonstochastic rates.

Nyborg (1996) argues that one of the main problems inherent in the implementation of structural form models is that the convertible bond value is assumed to be a function of the firm value, a variable not directly observable. To circumvent this problem, some authors model the price of convertible bonds as a function of the stock price, a variable directly observable in the market. The model of McConnell and Schwartz (1986) is such an example. In this model, they price Liquid Yield Option Notes (LYONs), which are zero coupon convertible bonds callable by the issuer and putable by the bondholder. However, one of the main drawbacks of their model is the absence of a bankruptcy feature.

The second approach used for the valuation of defaultable corporate obligations is the reduced form approach. In contrast to the structural approach where default is an endogenous event tied to the firm's value and capital structure, in the reduced form models default is an exogenous event. In the reduced form approach, the default risk of a firm and its value are not explicitly related; at any point in time, the probability of default is defined by a Poisson arrival process and is described by a hazard function. The application of this approach for valuing defaultable nonconvertible bonds can be found in the models of Jarrow and Turnbull (1995), Duffle and Singleton (1999), and Madan and Unal (2000). (3) The attractiveness of this approach is that the convertible bond value can be modeled as a function of the stock price. The models of Tsiveriotis and Fernandes (1998), Takahashi, Kobayashi, and Nakagawa (2001), and Ayache, Forsyth, and Vetzal (2003) use the reduced form approach for valuing convertible bonds. (4)

In contrast to the extensive theoretical literature on convertible bond pricing, there is very little empirical literature on this topic. Some researchers use market data to verify the degree of accuracy of their own models. Cheung and Nelken (1994) and Hung and Wang (2002) use market data on single convertible bonds to verify their models. Ho and Pfeffer (1996) use market data on seven convertible bonds to perform a sensitivity analysis of their two-factor multinomial model. King (1986) uses a sample of 103 US convertible bonds and finds that the average predicted prices of the Brennan and Schwartz (1980) model with nonstochastic interest rates are not significantly different from the mean market prices. Carayannopoulos (1996) uses the stochastic interest rate variant of the Brennan and Schwartz (1980) model. For a sample of 30 US convertible bonds, he finds a significant overpricing of deep-in-the-money convertible bonds. Takahashi et al. (2001) use data on four Japanese convertible bonds to compare their model to the models of Tsiveriotis and Fernandes (1998), Cheung and Nelken (1994), and Goldman-Sachs (1994). On the basis of the mean absolute deviation, which is calculated as the difference between the model and the market price expressed as a percentage of the market price, they conclude that their model produces the best predictions of convertible bond prices. Ammann, Kind, and Wilde (2003) use 18 months of daily French market data and the Tsiveriotis and Fernandes (1998) model to find that, on average, market prices of French convertibles are 3% lower than the model-predicted prices.

Most companies that issue convertible bonds do not have straight bonds outstanding. For this reason, we cannot use straight bond parameters, such as the credit rating, when calculating model prices for convertible bonds. Moreover, other model parameters, such as the underlying state variable volatility, the dividend yield, and the default rate, are often not directly observable. Therefore, we use an innovative technique that allows for the calculation of model prices even when the values of the parameters are not observable. We divide our sample into two parts: 1) a historical sample and 2) a forecasting sample. Instead of using the values of the parameters inferred from the plain debt data or underlying stock market data, we use the information contained in the historical convertible bond prices to estimate the necessary parameters. This approach allows for forecasting the convertible bond prices using the convertible bond price series only. The data from the historical sample are used to calibrate model parameters. We then calculate model prices for the forecasting period and compare these to market prices.

The estimation procedure becomes very complicated if all the features of convertible bond contracts are taken into account. Lau and Kwok (2004) demonstrate that the dimension of the valuation procedure increases rapidly if the soft-call feature is accommodated. They also determine that the calling period essentially increases the optimal call price at which the issuers should call the bond. This effective call price can be viewed as the original call price multiplied by a call price adjustment, (1+ [pi]), in which [pi] is the excess calling cost defined as the difference between the critical call price and the published call price. In our study, we only account for call and call notice features and we do not price the soft-call feature. The call notice period is featured by including and calibrating the excess calling cost parameter. Therefore, the results of this study are subject to the fact that some convertible bond contract features are ignored. (5)

First, we estimate the Brennan and Schwartz (1980) model. This seminal model for the valuation of convertible bonds has a very sound theoretical background as it explains the economic mechanisms behind the default event connecting the bankruptcy with the capital structure of the firm. However, since it is a structural form model, it requires a simultaneous estimation of all the other defaultable assets. This fact seriously complicates the estimation of the model. To eliminate this complication, we estimate this model using a subsample of firms with a simple capital structure defined as a capital structure that only consists of equity, risk-free straight debt, and convertible debt. The assumption of a simple capital structure substantially simplifies the estimation process. However, it also reduces the domain of applicability for this model.

In order to be able to value the convertible bonds of firms with a nonsimple capital structure, we need to rely on the reduced form approach as it is not dependent on the capital structure of a firm. For this reason, we use two other convertible bond valuation models. The first is the Tsiveriotis and Fernandes (TF) (1998) model. The second is the model of Ayache, Forsyth, and Vetzal (AFV) (2003) model.

We find that, using the full sample of 64 bonds, the mean absolute deviation (MAD) of the model price from the market price, expressed as a percentage of the market price, is 1.86% for the AFV (2003) model. This deviation is 1.94% for the TF (1998) model. For the subsample of 17 firms that have a simple capital structure, the Brennan and Schwartz (BS) (1980) model has an MAD of 3.73%. The corresponding results of the AFV (2003) and TF (1998) models for the subsample used by the BS (1980) model are 2.16% and 2.17%, respectively. The BS (1980) model illustrates the smallest range of pricing errors. For the TF (1998) and AFV (2003) models, we find a negative correlation between moneyness and the absolute values of the pricing errors while this relationship is positive for the BS (1980) model. This means that the AFV (2003) and TF (1998) models misprice convertibles that have in-the-money conversion options less than convertibles with conversion options that are at-the-money or out-of-the-money. We find a positive association for the reduced form models between the absolute values of pricing errors and the volatility of the returns of the underlying stocks. The effect of volatility on the absolute deviations in the BS (1980) model is statistically insignificant.

The remainder of this paper is structured as follows. In Section I, we present different convertible bond valuation models. Section II includes the data description. Section III is devoted to the estimation of the parameters. The results of the estimation are presented in Section IV. The paper wraps up with Section V where the summary and conclusions are presented.

I. Convertible Bond Valuation Models

A. Model Selection

A comparison of valuation models is possible if all the model input variables are either directly observable or can be estimated. Structural models that use nondirectly observable variables, such as firm value, are very difficult to estimate. Their estimation becomes easier if a simple capital structure of the firm is assumed. Alternatively, reduced form models use directly observable market variables and are much easier to estimate. This explains their popularity among practitioners.

The selection of models that are used for the comparison in our study is based on popularity with practitioners as well as their sound theoretical underpinnings. In this paper, we compare the models of Brennan and Schwartz (BS) (1980), Tsiveriotis and Fernandes (TF) (1998), and Ayache, Forsyth, and Vetzal (AFV) (2003). (6)

B. Convertible Bond Valuation Models

1. The BS (1980) Model

Brennan and Schwartz (1977, 1980) develop a structural-type approach for valuing convertibles where the convertible bond value is modeled in terms of the firm value. The main assumptions of their approach are: 1) the firm value, W, is the central state variable, the risk-adjusted return on which is the risk-free rate at each instant; 2) the dilution effect resulting from conversion must be handled consistently; 3) the effect of all cash flows on the evolution of the firm value must be accounted for; 4) assets must be sufficient to fund all assumed recoveries in default; and 5) the share price process is endogenously determined by all of this.

The firm value, W, is assumed to be governed by the stochastic process d W = (r WD - (W) - r[B.sub.s] - [cB.sub.e] dt + [sigma] WdW in which r is the instantaneous risk-free interest rate, [B.sub.s] is the par value of senior straight bonds outstanding, [B.sub.c] is the par value of convertible subordinated bonds outstanding, c is the annualized continuous coupon rate on convertible bonds, D(W) = d max{0, W - [B.sub.s] - [B.sub.c]} is the total continuous dividend payout on shares, d is the constant dividend yield on the book value of equity, and [sigma] is the constant proportional volatility of the asset value. The stochastic process above is applied when the firm is not in default. Following Brennan and Schwartz (1980), we further assume a constant default boundary prior to convertible debt maturity at the firm asset level W [equivalent to] [B.sub.s] + [rho][B.sub.c], where [rho] denotes the convertible bond early recovery rate as a fraction of par. (7) This early default boundary implies W is just sufficient to fund full recovery on the senior straight debt, recovery on the convertibles, and zero recovery on equity at the time of default. (8)

The assumptions described earlier imply that due to standard arbitrage arguments, the value of the entire convertible bond issue, V, has to follow the partial differential equation (PDE) 1/2 [V.sub.WW][[sigma].sup.2] [W.sup.2] + (rW - D(W) - [rB.sub.s] - [rB.sub.c]) [V.sub.W] + [cB.sub.c] + [V.sub.t] - rV =0 where the subscripts indicate partial differentiation.

Boundary conditions characterize the convertible bond value at maturity, at early default point W, at the rational early conversion level W*, and at the rational early call-level W if applicable. These conditions are as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

V(W, t) [greater than or equal to] C(W) for all W, t(voluntary conversion),

V(W, t) = [rho][B.sub.c] (early default),

V(W, t) = max{[P.sub.c], C(W)} for V [greater than or equal to] (1 + [pi])[P.sub.c] for all t [greater than or equal to] [T.sub.c] (early call).

In the above, T is the maturity date of the convertible bond, [P.sub.c] is the early call price of the bond, [T.sub.c] is the first call date of the bond, C(W) is the conversion value of the bond given W, and [pi] is the excess value required for an early call. (9) The excess value [pi] required for early call is introduced to accommodate the presence of a call notice period. Note that upon notice of early call, bondholders exercise the conversion right …