SLIDE 1
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.
SLIDE 2 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.
SLIDE 3 3
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).
SLIDE 4 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
- nly 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
- f 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
- bservable 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
SLIDE 5 5 because any senior debt has to be valued simultaneously with convertible debt. This is why “structural” models are not popular among practitioners. In contrast to the “structural” models that model convertible bonds’ value in terms of firm value, recent papers use the so-called “reduced form” models that define convertible debt value as a function of the firm’s stock value. Nyborg (1996) argues that the main problem with modelling the convertible bond value as a function of firm value lies in the fact that the firm value is an unobservable value and is hard to estimate. Moreover, in this case it is necessary to model all senior debt simultaneously to obtain consistent results. Modelling convertible debt as a function of the firm’s stock price is more plausible. Models that take this approach are Tsiveriotis and Fernandes(1998), Takahashi, Kobayashi, and Nakagawa (2002), and Ayache, Forsyth and Vetzal (2003).2 Since convertible debt essentially consists of a straight debt part and a pure equity part, it is evident that only the debt part is subject to default risk. In order to account for this, different discounting rates may be used for the instances when convertible debt takes either the form of debt or equity. Tsiveriotis and Fernandes (1998) solve this problem by separating a convertible bond into a bond-like part and an equity-like part. They put down a system of partial differential equations consisting of two equations: one that governs the convertible bond value evolution, and one that governs the evolution of a hypothetical instrument, the cash part
- f the convertible bond. For the purpose of discounting they introduce a market observed credit
spread on straight bonds of the same firm. Thus, they discount the cash portion of CB using the default adjusted interest rate, while discounting the equity part with the risk free rate. Solving a system of two partial differential equations subject to boundary conditions provides the convertible bond’s price evolution.Yigitbasioglu (2001) extends the model of Tsiveriotis and Fernandes (TF) for pricing foreign exchange rate sensitive convertibes. As the extension of the TF model, the author proposes to add the exchange rate factor as an additional state variable. Models that value convertible bonds as a function of the stock price do not have an explicit bankruptcy feature. Ayache et al. (2003) develop a model that explicitly models the bankruptcy
- event. The authors assume a Poisson arrival process of default. They also explicitly model the
2 Another example is the model of McConnell and Schwartz (1986). Their model is developed for the pricing of a
specific type of convertible bond, i.e. a Liquid Yield Option Note (LYON). These LYONS are zero coupon convertible, callable and putable bonds with calling prices that increase over time.
SLIDE 6 6 behaviour of convertible bondholder in the event of bankruptcy. The authors claim that the model of Tsiveriotis and Fernandes is internally inconsistent. Takahashi, Kobayashi, and Nakagawa (2002), from now on TKN, develop a reduced form model for the pricing of convertible bonds. It is similar in spirit to the model of Duffie and Singleton (1999), but adds some extra features. TKN derive a formula for valuing pre-default value of defaultable securities assuming a diffusion path for the firm stock price. They discount this value using default adjusted discount rates. The probability of default is specified with a hazard rate, which is a declining function of the firm’s stock price. The stock price is assumed to depend on the probability of default itself. TKN choose the stock price as a variable influencing hazard rate since the stock price is the most observable of all firm market
- securities. This assumption makes sense since the higher the stock price, the lower are the
chances of default. In contrast to the extensive theoretical literature on convertible bond pricing, there is hardly 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 Wang (2002) use market data on single convertible bonds to verify their models. Ho and Pfeffer (1996) uses market data on seven convertible bonds to perform a sensitivity analysis of their two-factor binomial model. Carayannopoulos (1996) uses a sample of 30 US convertible bonds to test the Brennan- Schwartz model. TKN use data on four Japanese convertible bonds to compare their model to the models of Tsiveriotis-Fernandes (1998), Cheung and Nelken (1994), and Goldman-Sachs (1994). On the basis of absolute error ratio they conclude that their model had the best predictions of convertible bond prices. The remainder of this paper is structured as follows. In Section II we present the different convertible bond valuation models. Section III includes the data description. Section IV is devoted to the estimation of the parameters. The results of the estimation are presented in Section V. The paper concludes with Section VI where the summary and conclusions are presented.
- II. Convertible bond valuation models
- A. Model selection
SLIDE 7 7 A model comparison is possible if all the model input variables are either directly observable
- r can be estimated. Structural models that use non-directly observable variables, such as firm
value, are very difficult to estimate. Reduced form models, on the other hand, use directly
- bservable market variables and are much easier to estimate. This reason explains their
popularity among practitioners. The selection of models that were used for the comparison in
- ur study is based on model popularity as well as their sound theoretical underpinnings. In this
paper we compare the models of Tsiveriotis and Fernandes (1998) (from now on TF)3, Takahashi, Kobayashi and Nakagawa (2001) (from now on TKN), and Ayache, Vetzal and Forsyth (2003) (from now on AFV).
Convertible bonds give bondholders an option to convert the bond into a pre-specified number
- f stocks of the underlying company. Therefore, convertible bonds are hybrid securities that
consist of two parts: a bond part and a call option part. The valuation of such a combination of a fixed income instrument and a derivative security can be performed in two ways. First, a binomial (trinomial) tree can be built that mimics the behavior of the state variable. In our case, the underlying stock price rate plays the role of the state variable. After deriving the stock price path for the period of bond life, the value of the bond can be found for any stock price at any given point in time. Since it is always optimal for convertible bondholder to convert the bond into stocks as soon as the conversion value exceeds the bond value, the value of the convertible bond is the maximum of the conversion value and the straight bond value. In other words, the valuation of the convertible bond starts at the maturity date and continues backwards. At the maturity date, the conversion value is found for each stock price taken from the respective binomial tree. This value is then compared to the sum of the bond face value plus coupons and the maximum of two is taken. In such a way, the price of the convertible bond at the maturity date is obtained. Then, by discounting the value of the bond one period back with the appropriate discount rate, the bond value one period before maturity can be found. This value
3 Grimwood et. al (2002) in their review of convertible models argue that approach of Tsiveriotis-Fernandes based
- n the methodology of Jarrow-Turnbull (1995) is the most popular among practitioners.
SLIDE 8 8 is again compared to the conversion value and maximum is taken. In such a manner, the bond prices can be deduced for the entire duration of the bond. The second approach is based on the results of Ito’s lemma. If the stock price movement process can be assumed, then the derivative price movement process can be calculated using Ito’s lemma. An arbitrage argument can then be used to find the price of such a derivative. That price appears to be a solution to a partial differential equation (PDE). The solution to this equation can be found by imposing appropriate boundary conditions and by using numerical methods for solving the PDE.
- C. Convertible bond valuation models
As mentioned before, we use the models of Tsiveriotis-Fernandes (TF), Takahashi-Kobayashi- Nakagawa (TKN), and Ayache-Forsyth-Vetzal (AFV) in our comparison. All of these models use approaches that allow for the use of actual market data estimation and testing. The TF-model The TF-model is a modified structural form model. This model discriminates between two parts of the convertible bonds: the bond-like or cash only part (COCB) and the equity-like part. The COCB is entitled to all cash payments and no equity flows that an optimally behaving
- wner of CB would receive. Therefore, the value of CB, V, is the sum of the COCB value, Σ,
and the equity value (V-Σ). The stock price is assumed to follow the continuous time process
t g
Sdw Sdt r dS δ + = , where rg is stock growth rate, δ is standard deviation of stock price, and w is Wiener process. Since the bond-like part is subject to default, the authors propose to discount it at a risky rate. The equity-like part is default-free and is discounted with the risk-free rate. CB valuation becomes a system of two coupled PDEs. For CB: ) ( ) ( ) ( 2 1
2 2
= + Σ + − Σ − − + + t f r r V r V SV r V V
c t s g ssδ
For COCB: ) ( ) ( 2 1
2 2
= + Σ + − Σ + Σ + Σ t f r r S r V
c t s g ssδ
SLIDE 9 9 S is the underlying stock price, , rc is the credit spread reflecting payoff default risk, r is the risk free interest rate; f(t) specifies different external cash flows for cash and equity (e.g. coupons
To find the value of convertible bond, it is necessary to solve the system of PDEs. At each point in time, the convertible prices should also satisfy boundary conditions. At the maturity: ) , max( ) , ( Coupon F aS T S V + = , ) , max( ) , ( F T S = Σ where a is the conversion ratio, F is the face value of the bond. At conversion points: ; aS t S V ≥ ) , ( = Σ if . Callability constraints: ; aS t S V ≤ ) , ( ) , Pr max( aS ice Call V ≤ = Σ if . Puttability constraints are ; if ice Call V Pr ≥ ice Put V Pr ≥ ice Put Pr = Σ ice Put V Pr ≤ . With the binomial apporach, it is first necessary to find the prices of the convertible bond at the maturity date, T. The grid points with stock prices where it is optimal to convert the bond is the "equity-like" part and is discounted at the risk-free rate, r. The points where the total of face value plus coupon value is higher than conversion value are discounted at the risky, r+rc. Working one period back, the convertible prices are calculated and the points found where the issuer can call the bond and holder can put the bond. The TKN model The TKN model is a reduced form model; it uses the stock price as a proxy for the firm value. The probability of default in this model is explained exogenously. In the TKN-model there is no discrimination between bond-like and equity-like parts. The probability of default is a decreasing function of the stock price. At every point the model adjusts the discount rate to account for the probability of default and the proportion of loss at the event of default. The discount rate in their model, rf+Lλ. , is the default-adjusted short rate, which is a function of risk-free rate rf, the default hazard rate λ(S,t) and a fractional loss of market value of the claim at default, L: The hazard rate is a decreasing function of the stock price ( , ) ( )
t b t
c S t S S λ λ θ = = + where θ≥0, c and b are constants. There are two state variables in this model: the firm stock price and the default event. The stock price is assumed to follow the
SLIDE 10
10 continuous time process
t
Sdw Sdt dS δ λ υ + + = ) ( According to Ito’s lemma and arbitrage arguments, the price of the convertible bond in such a case can be explained by the following partial PDE: ) ( ) ( ) ( 2 1
2 2
= + + − + + + t f V L r V SV V V
t s ss
λ λ υ δ At each point in time, the price of the convertible should be within the boundaries defined by the following conditions. ) , max( ) , ( F aS T S V = at maturity; for conversion before maturity; for calling; for putting, where aS t S V ≥ ) , ( ) , Pr max( aS ice Call V ≤ ice Put V Pr ≥ d r − ≡ ν , where d is stock dividend rate. Prices are first found for the terminal date: ) , max( ) , ( Coupon F aS T S V + = . They are then discounted at individual discount rates that depend on the stock price. It is necessary at each time step for every stock price value to check if it is optimal to call or put the bond as well as to convert it into stocks. The AFV-model The AFV-model is a modified reduced form model that assumes a Poisson default process. The authors of this model criticize the TF-model because it does not treat stock prices at default properly since it does not stipulate what happens to the price of distressed firm stock in the case of bankruptcy. The AFV-model boils down to solving the following equation ) , 1 ( max( = − − RF aS p MV η , V p r V SV d p r V V MV
t s ss
) ( ) ( 2 1
2 2
+ + − − + − − ≡ η δ subject to the boundary conditions , , where p is probability of default, η is the proportional fall in underlying stock value after default, R is proportion of bond face value recovered immediately after default. ) , Pr max( aS ice Call V ≤ ice Put V Pr ≥ Similar to the TKN model, the AFV model assumes the probability of default to be a decreasing function of stock price:
α
) (
0 S
S p ps = , where p0, S0 and α are constants for a given
SLIDE 11 11 firm; p0 is the probability of default when the stock price is S0. Parameters p0, S0., and α are calibrated from historical convertible bond data.
We use a sample of actively traded Canadian convertible bonds published by the Financial Post Market Data convertible debentures section. All bonds are traded on the Toronto Stock Exchange (TSX) in the period from January 1, 2005 to April 28, 2006. Information on historical bond prices is derived from the Globe Investor Gold database4 of Bell Globe Media Publishing Inc. We take detailed information on each issue including coupon rates, maturity dates, conversion conditions from the prospectuses available at the SEDAR5 (System of Electronic Document and Archive Retrieval) and Bloomberg databases. The data on existing debt and firm ratings are derived from the Dominion Bond Rating service website.6 The information on underlying stock dividends is taken from corresponding websites and from the Toronto stock exchange website.7 We use the sample of all active convertible bonds information that is published by the Financial Post newspaper. This sample consists of 97 issues of convertible and exchangeable bonds that are traded on the TSX as of the November, 1st 2005. The vast majority of the convertible bonds were issued by young growing firms many of which are income trusts8. We exclude all exchangeable bonds as well as bonds traded in currencies other than the Canadian
- dollar. After screening for the issues that have price series and prospectuses available as well
as information on underlying stocks and financial statements with dividend information we were left with 66 issues presented in Table I. The descriptive statistics of the convertible bond characteristics are presented in Table I. [Please insert Table II here] Since in the evaluation of the models risk-free interest rates are used, we use, as a proxy, the forward interest rate derived from the Bank of Canada zero coupon bond curve. The forward
4 See http://www.globeinvestorgold.com 5 See http://www.sedar.com 6 See http://www.dbrs.com 7 See http://www.tsx.com 8 Income trust is an investment trust that holds assets which are income producing. The income earnde on assets
is passed on to the unit holders.
SLIDE 12 12 rates used are 3-month forward rates for horizons from 3-months to 30 years. Zero-coupon bond data is available at the Bank of Canada website.9 For the purpose of defining the best convertible bond model the data is divided into two sub- samples: historical and forecasting. Using the historical sub-sample the model parameters are
- calibrated. Then weekly model prices are calculated for each convertible bond for the
forecasting period. The final forecast sub-period date of April, 28 2006 is stipulated by the availability of Bank of Canada data on risk-free rates. The beginning date of forecast sub- sample depends on the issuance date of the bond. If the bond was issued before January 1, 2004, then the starting date is defined as January, 1 2005. If the bond was issued before July 1, 2004 but after January 1, 2004, the starting date is chosen to be July 1, 2005. If the bond was issued after January 1, 2005, the starting date of the price comparison sample is January 1,
- 2006. The choice of starting date is stipulated by the need of historical sample data for the
estimation of the parameters of the model.10 In contrast to the estimation techniques that require the use of straight corporate bonds of the same firm for estimating model parameters, we use information inherent in the convertible bond prices for calibrating the parameters of the models we use. Many of the firms in our sample issue convertible debt instead of ordinary bonds in order to save the costs of interest in the absence of a high credit rating. These young and growing firms offer investors convertible bonds with lower coupons and conversion features as a sweetener. The majority of these firms do not have other publicly traded corporate bonds in their capital structure. Thus, using the method for convertible valuation that does not hinge on the presence of firm’s other corporate bonds promises to be valuable. The information contained in the prices of the convertible bonds may be helpful to calibrate parameters of the models in the absence of other forms of bonds for the firm. Using historical convertible prices we search for model parameters with the Levenberg-Marquardt algorithm (Marquardt, 1963) that minimizes the squared sum of residuals between convertible bond pricing model and market prices. Later, we use these parameters for forecasting the convertible
9 See http://www.bankofcanada.ca 10 Loncarski, ter Horst and Veld (2006a) found that during the first six months after issuing convertible bonds are
underpriced which provides possibility for convertible arbitrage. To avoid pricing biases we performed alternative pricing procedure which uses reduced historical samples where the first six months of data are
- dropped. The results of reduced sample estimation are similar to the ones using original data samples and are
available from authors upon request.
SLIDE 13 13 prices for our prediction sub-sample. Initial values and boundaries for parameters are provided based on the assumption of the corresponding model papers. In the TKN model θ≥0, in the AFV model p≥0, S(0)≥0, values for α range from -1.2 to -2. In their paper, Ayache et al. (2003) use a value of p(0)=0.02. Given the convertible valuation model, the Marquardt algorithm finds the theoretical convertible prices given the initial values for model parameters. In the next step, the algorithm changes the model parameters until the values that return the minimum squared deviations of the model prices from the observed market prices are found. The data needed for the estimation
- f the parameters and predicting the out-of-sample theoretical convertible bond prices consist
- f market prices of the convertible bonds, conversion prices, issuance, settlement and maturity
dates, coupon rates, number of coupons per year, market prices of the underlying stocks, volatilities of the underlying stocks, call schedules and call prices. Note that, together with convertible bond data, data on ordinary bonds can be used to calibrate the parameters. In this case the conversion price has to be specified as some unrealistically large number and call dates to be set for after maturity period. We expect that our parameter calibration approach will yield superior results compared to the approach where only corporate bond data are exploited, because our approach uses a wider set of market information.
The numerical routines used for calculating the model prices are written using the FORTRAN computing language as it allows for very fast computational processing. We use a Crank- Nicholson finite difference algorithm for solving the corresponding partial difference equations and the Marquardt iterative procedure for finding the values of parameters that produce the smallest deviations of model prices from the market prices. First data are used in order to evaluate the Tsiveriotis-Fernandes (TF) model.11 To be able to perform convertible price estimation the following data are needed: issue date of the bond, settlement date of the bond, risk-free rate, price of the underlying stock for the settlement date, maturity date, coupon rate, conversion ratio, dividend and call information, and the credit spread that reflects the credit rating of the issuer. The maturity dates, coupon rate, conversion
11 Note that this model can also be estimated using the “cbprice” function of Fixed Income Toolbox in Matlab.
SLIDE 14 14 ratio and underlying stock prices are derived from the Globe Investor database. The issue dates and calling information are taken from the convertible bond prospectuses available on the SEDAR website. The risk-free rates for the bonds of the same maturity as the convertible bonds are available on the website of the Bank of Canada.12 Dividend information is obtained from the companies’ financial statements available at their websites and the Toronto stock exchange website. The stock volatility is calculated using daily price series of the underlying stocks for the year 2005. The only input needed for calculating prices with the TF-model that can not be directly
- bserved from the market is the credit spread. In theory, this credit spread should be derived
from the ordinary non-convertible corporate bond of the same company that has identical characteristics as the convertible security except for the conversion possibility. Unfortunately, there is only few firms in our sample that have ordinary corporate bonds in their capital structure, which match the characteristics of the corresponding convertible bonds. Therefore, in practice, it is necessary to calculate the credit spread in other way. In this paper we use two alternative methods. The first alternative method is to use the average value of the credit spread for bonds that have the same credit ranking13. However, many of the bonds in our sample are issued by small firms, and therefore don’t have credit ratings assigned. For companies that do not have credit ratings assigned, we assume a BBB rating. The average credit spreads are then taken from the Canadian Corporate Bond Spread Charts published by RBC Capital Markets. The second alternative that we use is to produce the numerical routine which, given the market price series, derives the credit spreads that are implied by the data. It should be noted that in case of calibration, the numerical routines return the values of credit spreads that are lower than the ranking implied ones. Prices are also calculated for the Ayache-Forsyth-Vetzal (AFV) model. This model allows for a different behavior of stock prices in case the firm defaults on its corporate debt. The partial default version assumes that the price of the underlying stock is not affected by the firm’s default on its bonds. The total default model assumes that the stock price jumps to zero at the
12 See http:// www.bankofcanada.ca 13 In this study we call ‘TFS’ model the Tsiveriotis-Fernandes model where the credit spread is derived from the
corresponding firm credit ratings. We reserve the name ‘TF’ for the version of the Tsiveriotis-Fernandes model where credit spread is calibrated by the Marquardt algorithm.
SLIDE 15 15 moment that the default takes place. We assume that in case of default convertible bond holders recover no value. We use both the partial default and the total default models are used in our comparison. The last set of data comes from Takahashi-Kobayashi-Nakagawa (TKN) model. This model can be thought of as a version of the AFV total default model. The difference is in the functional form of the hazard function. Using FORTRAN routines we use the historical sub- samples of convertible bonds data in order to estimate the model parameters θ, c and b. Then, given these parameters we estimate the model prices for the forecast sub-sample.
Our comparison of the convertible bond pricing models is based on the indicators that show the ability of a model to generate prices that are close to market prices. We assume that markets are efficient and that they price convertible bonds efficiently. There are several indicators that we will base our decision upon. These are based on describing the proximity of the predicted (model) price and market prices. The mispricing is both calculated relative to market and to model prices. The first indicator is the “mean deviation”. This is calculated as the average deviation of the model price from the market price. This deviation is both calculated relative to market prices and to model prices. The second indicator is the “mean absolute deviation”. This is calculated as follows:
) Pr Pr Pr ( ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ice Market ice Market ice Model abs average MAD p
where the subscript m indicates that the error is calculated relative to the observed market price. The mean absolute deviation relative to theoretical price predicted by the model is calculated as:
) Pr Pr Pr ( ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ice Model ice Market ice Model abs average MADm
Another indicator of model fit is the “mean squared error”, which is calculated as follows:
2
Pr Pr Pr ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ice Market ice Market ice Model average MSE p
2
Pr Pr Pr ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ice Model ice Market ice Model average MSEm
This indicator also can be relative either to model or market price.
SLIDE 16 16 The “mean absolute deviation” and “mean deviation” scores treat all errors with the same
- weight. There is no additional penalty for the instances when the model price is far from the
market price. On the other hand, mean squared error score gives larger weights for large deviations. Tables IV and V provide rankings of the models based on the indicators mentioned. While Table IV shows the mispricing relative to the market prices, Table V provides mispricing results relative to the model predicted prices. [Please insert Tables IV and V here] Based on the results in Table IV it can be concluded that the TF-model has the best predictive
- power. This model shows the lowest values for all mispricing scores. The average value of the
mispricing error is 2.3%. This result is quite impressive given that this model requires the lowest number of parameters, i.e. it just requires the value of the credit spread added to the risk-free interest rate. On average, T-F model slightly overprices or underprices the convertible bonds depending on the way the credit spread is derived. The average overpricing of the T-F model with firm ranking-derived credit spread (T-F “spread” model) is only 0.07% while for the T-F model with the model-implied spread the average underpricing is a mere 0.05%. Moreover, the pricing errors are not significantly different form zero so we can conclude that the T-F model on average prices the convertibles correctly. This results contrast the one of Amman et al. (2003) who finds that on average T-F model overprices the French market convertible bonds by 3%. The second best model is the TKN-model. Based on the mispricing score, it performs worse than the TF- model, having on average a 3.2% difference between the model price and actual market price. The TKN model on average overprices the convertible bonds by 0.37 %. Nevertheless, the average pricing error is not statistically different from zero. Both the partial and the total default AFV-models show the largest mispricing, around 4% on
- average. The average under or overpricing of the AFV model depends on the assumption of
what happens to the stock price after default. The total default AFV model, the one where stock price goes to zero in the event of default, on average underprices the convertibles relative to the market by 2.42%. The partial default AFV model on average overprices the convertibles by 2.23%. Both average pricing errors are not significantly different form zero.
SLIDE 17 17 It is also interesting to see whether there are convertible bond characteristics that affect the mispricing in a systematic way. In order to check for any such regularities we perform a regression analysis in which the mispricing errors are regressed on characteristics of the convertible security. These characteristics include for example moneyness (ratio of current market price to the conversion price = SX), annual volatility of the underlying stock (VOLAT), convertible bond coupon rate (COUPON) and remaining time to maturity of the convertible security (TMAT). The results of this regression analysis are included in Tables VI and Table VII From these tables it can be seen that TF-model, which should appeal most to the practitioners because of its precise pricing ability, tends to overprice the bonds that are deep in the money regardless of the way the credit spread is obtained. This result is in line with the results of Amman et al. (2003), King (1986), and Carayannopoulos (1996) who report positive relationship between overpricing and moneyness. All of the models used in our study except for the AFV with total default report the same positive relationship between overpricing and
- moneyness. As can be seen from Table VIII and Table IX14, the mispricing, expressed as the
percentage absolute deviation of the model price from the market price, statistically depends on the degree of the convertible bond moneyness. Convertible securities that are deep in the money with conversion values exceeding the nominal values are more heavily mispriced as compared to the convertible securities that are at or out of the money. Since the pricing error is defined as market price less model price, the positive coefficient of stock volatility in the TFS model implies that model price decreases relative to the market price as volatility of the stock increases. Moreover, from the Table VIII we can see that the TFS model predicts heavier mispricing for convertible bonds with highly volatile underlying stocks. The same time TF model reveals no statistically significant effect of volatility on the pricing
- deviations. All other models in the study predict heavier mispricing for the bonds with more
volatile underlying stocks. All the models in our study to the exclusion of AFV with partial default overprice bonds that are close to their maturities. For partial default AFV model this effect is not statistically
- significant. The result is the same as of Amman et al. (2003) who find larger underpricing for
14 Tables 6 and 7 report the regression results where the dependant variable is the actual value of the pricing error
as a percentage of convertible price. Regressions of the actual values of the pricing errors help to find the variables that explain the ‘direction’ of mispricing, i.e. whether the convertibles are under/overpriced. Tables 8 and 9 show the results of the regressions where the dependant variable is the absolute value. The results in these tables help to find the variables that explain the ‘precision’ of the models.
SLIDE 18 18 bonds with longer term to maturity. In the TF model time to maturity has no significant effect
- n the value of average absolute mispricing. The average absolute deviation of the TFS model
price from the market price increases with the time to maturity. This is different from the result
- f King (1986) who finds heavier mispricing for the convertibles close to maturity.
The effect of the convertible bond coupon rate does not have statistically significant effect on the model pricing in both versions of the Tsiveriotis-Fernandes model. The same time in both versions of AVF model convertibles with higher coupon rates are priced more accurately that the ones with lower coupon rates while the TKN prices lower coupon bonds more accurately.
- VI. Summary and conclusions
In this paper we compare the price prediction ability of three convertible bond pricing models using actual market data from the Toronto Stock Exchange. The sample consists of 66 Canadian convertible bonds and spans for the period from January 1, 2005 to April 28, 2006 The models that are compared in this study are those of Tsiveriotis and Fernandes (1998), Takahashi et al. (2001), and Ayache et al. (2003). The results of the study show that the model of Tsiveriotis and Fernandes appears to be the most accurate. This is surprising, since this model uses the lowest number of parameters of all three models. On average, the absolute deviation of the model predicted price from the market price is 2.3% for the Tsiveriotis-Fernandes model, 3.2% for the Takahashi et al. (2001) model, and 4.1% for the total default and 4% for the partial default Ayache et al. (2003) models. It should be mentioned that the Ayache model is the most general model of all three models requiring the largest number of parameters for estimation. Takahashi model is nested within the Ayache model as a special case of total default model with modified hazard rate function. The approach for choosing the best model on the base of the deviation of its predicted prices relative to the market price assumes that market prices are efficient and correct and that arbitrage is impossible. A regression analysis shows that there are certain characteristics of convertible bonds that influence the models’ pricing errors with statistical significance. In all of the models except for the partial default Ayache model, convertible bond moneyness, that is the extent to which stock price exceeds the conversion price, has negative effect on the both relative and absolute values
SLIDE 19 19
- f pricing errors. In the case of a partial default Ayache model, the “moneyness effect” is not
clear, since the significance of the coefficient depends on whether the pricing error is defined relative to the market or model predicted price. Time remaining to maturity of the convertible bonds has similar negative effect on the relative pricing errors though the effect is statistically insignificant for Ayache model with partial default. The impact of time to maturity on absolute values of errors is equivocal. The effects of the underlying stock volatility and the convertible bond coupon rate are unclear. Coupon rates have insignificant effects for both versions of Tsiveriotis-Fernandes model, negative significant effect for Ayache model with total default and positive significant effects for partial default Ayache and Takahashi et al. models. Underlying stock volatility has significant positive effect on the absolute value of models’ deviation from market observed prices for all models except total default Ayache model. The effect of volatility on the relative value of pricing error is ambiguous. It ranges from insignificant positive for the Tsiveriotis-Fernandes model with model-implied spread and Ayache total default model to positive and significant for rating-derived Tsiveriotis-Fernandes model and partial default Ayache models.
SLIDE 20 20
Table I Characteristics of the Convertible Bonds Used in the Study
This table shows the main characteristics of the convertible securities used in the study. The sample consists of 66 Canadian convertible bonds traded on the T
- ronto Stock Exchange . Conversion ratio shows the number of stocks that can be obtained in the case of conversion for each 100 dollars of bond face value.
Underlying stock volatility is expressed as annualized standard deviation. In call schedule the first number refers to call price per 100 dollars of face value, second number refers to the starting date of calling at this price. Calling continues until the next call date (if any) or until maturity date if not specified
- therwise. Credit spreads are derived from corporate credit rating using 2005 Royal Bank of Canada relative value curves for Canadian corporate bonds.
Issuer/Symbol Issue Date Issue Size, millions dollars Maturity Date Coupon Rate, % Conversion Ratio, per 100$ of face value Underlying Stock Volatility Credit Spread, basis points Call schedule
Advantage Energy AVN.DB.A Jul-03 30 Aug-08 9 5.9 0.21 45 105 - 08/01/06, 102.5 -08/01/07 Advantage Energy AVN.DB.B Dec-03 60 Feb-09 8.25 6.1 0.21 65 105 - 02/01/07, 102.5 -02/01/08 Advantage Energy AVN.DB.C Jul-03 75 Oct-09 7.5 4.9 0.21 65 105 - 10/01/07, 102.5 -10/01/08 Advantage Energy AVN.DB.D Jan-05 50 Dec-11 7.75 4.8 0.21 75 105 - 12/01/07, 102.5 -12/01/08 Agricore United AU.DB Nov-02 100 Nov-07 9 13.3 0.30 65 100 - 12/01/05 Alamos Gold AGI.DB Jan-05 50 Feb-10 5.5 18.9 0.48 65 100 - 02/15/08 Alexis Nihon AN.DB Aug-04 55 Jun-14 6.2 7.3 0.14 128 100 - 06/30/08 Algonquin Power APF.DB Jul-04 85 Jul-11 6.65 9.4 0.17 69 100 - 07/31/07 Baytex Energy BTE.DB Jun-05 100 Dec-10 6.4 6.8 0.26 115 105 - 12/31/08, 102.5 -12/31/09 Bonavista Energy BNP.DB Jan-04 100 Jun-09 7.5 4.4 0.26 83 105 - 02/01/07, 102.5 -02/01/08 Bonavista Energy BNP.DB.A Dec-04 135 Jul-10 6.75 3.5 0.26 105 105 - 12/31/07, 102.5 -12/31/08 Boyd Group BYD.DB Sep-03 4 Sep-08 8 11.6 0.60 45 105 - 09/30/04, 102.5 -09/30/05 Calloway REIT CWT.DB Apr-04 55 Jun-14 6 5.9 0.22 88 100 - 06/30/08 Cameco Corp CCO.DB Sep-03 200 Oct-13 5 4.6 0.40 82 100 - 10/01/08 Can Hotel Inc. HOT.DB Feb-02 55 Sep-07 8.5 10.4 0.15 65 100 - 03/01/05 Can Hotel Inc. HOT.DB.A Nov-04 60 Nov-14 6 8.5 0.15 128 100 - 11/30/08 Chemtrade CHE.DB Dec-02 41 Dec-07 10 6.9 0.23 45 105 - 12/31/05, 102.5 -12/31/06 Cineplex Galaxy CGX.DB Jul-05 105 Dec-12 6 5.3 0.27 75 100 - 12/31/08 Clean Power CLE.DB Jun-04 55 Dec-10 6.75 9.8 0.34 69 100 - 06/30/07 Clublink LNK.DB Apr-98 80 May-08 6 5.0 0.18 65 100 - 03/15/03 Cominar CUF.DB Sep-04 100 May-14 6.3 5.8 0.15 88 100 - 06/30/08 Creststreet Power CRS.DB Jan-05 27 Mar-10 7 10.0 0.19 65 100 - 03/15/08 Daylight Energy DAY.DB Oct-04 80 Dec-09 8.5 10.5 0.21 65 105 - 12/01/07, 102.5 -12/01/08 Dundee REIT D.DB May-04 75 Jun-14 6.5 4.0 0.17 117 100 - 06/30/08 Dundee REIT D.DB.A Apr-05 100 Mar-15 5.7 3.3 0.17 117 100 - 03/31/09 Esprit Energy EEE.DB Jul-05 100 Dec-10 6.5 7.2 0.23 65 105 - 12/31/08, 102.5 -12/31/09 Fort Chicago Energy FCE.DB.A Jan-03 135 Jun-08 7.5 11.1 0.22 45 100 - 01/31/06 Fort Chicago Energy FCE.DB.B Oct-03 63 Dec-10 6.75 9.4 0.22 55 100 - 12/31/06 Gerdau AmeriSteel
Apr-97 125 Apr-07 6.5 3.8 0.44 65 100 - 04/30/02 Harvest Energy HTE.DB Jan-04 50 May-09 9 7.1 0.31 90 105 - 05/31/07, 102.5 -05/31/08 Harvest Energy HTE.DB.B Jul-05 75 Dec-10 6.5 3.2 0.31 90 105 - 12/31/08, 102.5 -12/31/09
SLIDE 21
21
InnVest INN.DB.A Mar-04 58 Apr-11 6.25 8.0 0.19 105 100 - 04/15/08 Inter Pipeline IPL.DB Nov-02 100 Dec-07 10 16.7 0.22 45 100 - 12/31/05 IPC US REIT IUR.DB.U Nov-04 40 Nov-14 6 10.5 0.21 90 100 - 11/30/08 IPC US REIT IUR.DB.V Sep-05 60 Sep-10 5.75 9.1 0.21 65 100 - 09/30/08 Keyera KEY.DB Jun-04 100 Jun-11 6.75 8.3 0.24 65 100 - 06/30/07 Legacy Hotels LGY.DB Feb-02 150 Apr-07 7.75 11.4 0.20 65 100 - 04/01/04 Magellan Aerospace MAL.DB Dec-02 55 Jan-08 8.5 22.2 0.44 45 100 - 01/31/06 MDC Partners MDZ.DB Jun-05 45 Jun-10 8 7.1 0.34 83 100 - 06/30/08 Morguard Real Estate MRT.DB.A Jul-02 75 Nov-07 8.25 10.0 0.13 45 100 - 11/01/05 NAV Energy NVG.DB May-04 50 Jun-09 8.75 9.1 0.27 65 105 - 06/30/07, 102.5 -06/30/08 Northland Power NPI.DB Aug-04 65 Jun-11 6.5 8.0 0.25 69 100 - 06/30/07 Paramount Energy PMT.DB Aug-04 48 Sep-09 8 7.0 0.22 65 105 - 09/30/07, 102.5 -09/30/08 Paramount Energy PMT.DB.A Apr-05 100 Jun-10 6.25 5.2 0.22 65 105 - 06/30/08, 102.5 -06/30/09 Pembina PIF.DB.A Dec-01 75 Jun-07 7.5 9.5 0.24 45 100 - 06/30/05 Pembina PIF.DB.B Jun-03 220 Dec-10 7.35 8.0 0.24 55 100 - 06/30/06 Primaris REIT PMZ.DB Jun-04 50 Jun-14 6.75 8.2 0.20 117 100 - 06/30/08 Primewest Energy PWI.DB.A Aug-04 150 Sep-09 7.5 3.8 0.24 83 105 - 09/30/07, 102.5 -09/30/08 Primewest Energy PWI.DB.B Aug-04 100 Dec-11 7.75 3.8 0.24 105 105 - 12/31/07, 102.5 -12/31/08 Progress Energy PGX.DB Jan-05 100 May-10 6.75 6.7 0.26 65 105 - 12/31/07, 102.5 -12/31/08 Provident Energy PVE.DB.A Sep-03 75 Dec-08 8.75 9.1 0.21 65 100 - 01/01/07 Provident Energy PVE.DB.B Jul-04 50 Jul-09 8 8.3 0.21 83 100 - 07/31/07 Provident Energy PVE.DB.C Feb-05 100 Aug-12 6.5 7.3 0.21 105 100 - 08/31/08 Retirement Res REIT RRR.DB.B Jul-03 150 Jan-11 8.25 8.1 0.25 95 100 - 07/31/07 Retirement Res REIT RRR.DB.C Apr-05 200 Mar-15 5.5 8.8 0.25 95 100 - 03/31/09 Retrocom Mid-Market RMM.DB Jul-05 20 Jul-12 7.5 12.1 0.29 75 100 - 08/31/09 Rogers Sugar RSI.DB.A Mar-05 50 Jun-12 6 18.9 0.24 75 100 - 06/29/08 Royal Host Real Estate RYL.DB Feb-02 40 Mar-07 9.25 14.3 0.19 65 100 - 03/01/05 Summit Real Estate SMU.DB Feb-04 100 Mar-14 6.25 4.7 0.21 88 100 - 03/31/08 Superior Propane SPF.DB Jan-01 100 Jul-07 8 6.3 0.27 45 100 - 02/01/04 Superior Propane SPF.DB.A Dec-02 250 Nov-08 8 5.0 0.27 45 100 - 11/01/05 Superior Propane SPF.DB.B Jun-05 175 Dec-12 5.75 2.8 0.27 69 100 - 07/01/08 Superior Propane SPF.DB.C Oct-05 75 Oct-15 5.85 3.2 0.27 88 100 - 10/31/08 Taylor NGL TAY.DB Mar-05 50 Sep-10 5.85 9.7 0.27 75 100 - 09/10/08
SLIDE 22 22
Table II Descriptive Statistics of the Convertible Bond Sample Used
This table presents the descriptive statistics of the sample of 66 convertible bonds used in
- ur study of comparing convertible bond pricing models. All the bonds in the sample are
traded on the Toronto Stock Exchange. VOLAT refers to annualized standard deviation of the underlying stocks; TMAT refers to the time remaining to maturity of the bonds as of December, 1 2005. COUPON refers to the convertible bond coupon rates. S/X refers to the ratio of average stock price during the forecast period to the conversion price. VOLAT COUPON TMAT S/X MIN 0.13 0.05 1.25 0.3 MAX 0.6 0.1 9.92 2.46 AVERAGE 0.25 0.07 4.92 1.06 MEDIAN 0.23 0.07 4.72 1.02
0.08 0.01 2.4 0.35
Table III Descriptive Statistics of Model Errors
This table provides the descriptive statistics for the deviations of model prices from the observed market prices expressed as a percentage of reference prices. ERRP column is for the errors as a percentage of the market price, ERRM is for the errors as a percentage of the model prices. TF refers to the Tsiveriotis- Fernandes model with price fitted credit spread, TF-S refers to the Tsiveriotis-Fernandes model with firm credit ratings implied credit spread. Ayache-Forsyth- Vetzal model has two versions: partial default model (AFV-P) in which the stock price is unchanged at the event of default and total default model (AFV-T) where stock price goes to zero at the moment of default. TKN refers to Takahashi-Kobayashi-Nakagawa model. Models TF TF-S AFV-P AFV-T TKN ERRP ERRM ERRP ERRM ERRP ERRM ERRP ERRM ERRP ERRM min
- 24.10
- 19.42
- 24.10
- 19.42
- 26.88
- 21.18
- 24.10
- 19.42
- 47.18
- 32.05
max 25.36 33.98 25.32 33.91 41.83 71.92 34.19 51.95 32.30 47.71 median 0.23 0.23 0.08 0.08
- 0.78
- 0.78
- 0.29
- 0.29
- 0.17
- 0.17
Table IV Average Pricing Errors as Percentage of Market Prices
Table shows the average sample model pricing errors as a percentage of the convertible security market price. Errors are calculated as convertible market price minus its model predicted price. Average error, AVERROR refers to the average of pricing errors for the whole sample. ERRP refers to the average error compared to the market price. 'Market' mean squared error (MSEP) is calculated as T-statistics are reported in brackets.
TF TF-S AFV partial AFV total TKN AVERROR, %
0.84
(-0.01) (-0.03) (-0.33) (-0.08) (-0.08) ERRP,% 0.06
2.42
(-0.02) (-0.02) (-0.35) (-0.37) (-0.06) abs(ERRP),% 2.29 2.29 4.08 4.08 3.19 (-0.75) (-0.78) (-0.75) (-0.76)
MSEP 14.63 13.93 46.32 32.53 33.99 (-0.32) (-0.31) (-0.43) (-0.23)
⎜ ⎛
2
100 * ⎟ ⎠ ⎞ ⎝ − MarketP ModelP MarketP
SLIDE 23 23
Table V Average Pricing Errors as Percentage of Model Predicted Prices
Table shows the average sample model pricing errors as a percentage of the model predicted price. Errors are calculated as convertible market price minus its model predicted price. Average error, AVERROR refers to the average of pricing errors for the whole sample. ERRM refers to the error compared to the model price. 'Model' mean squared error (MSEM) is calculated as T-statistics are reported underneath the values.
TF-S TF AFV partial AFV total TKN AVERROR,%
0.84
(-0.01) (-0.03) (-0.33) (-0.08) (-0.08) ERRM,% 0.2 0.06
1.31
(-0.05) (-0.02) (-0.29) (-0.17) (-0.01) abs (ERRM), % 2.26 2.25 3.87 4.04 3.18 (-0.79) (-0.83) (-0.75) (-0.6) (-0.65) MSEM 13.33 12.53 41.69 63.82 34.25 (-0.33) (-0.32) (-0.3) (-0.23) (-0.27)
2
100 * ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ModelP ModelP MarketP
Table VI Regression Results for the Model Errors Relative to the Market Prices.
Table shows regression results of the model predicted errors (calculated as the model price minus the market price) on explanatory variables such as ratio of market stock price to conversion price (S/X), time to maturity of convertible security (TMAT), volatility of the underlying stock (VOLAT), and convertible bond coupon rate. Dependent variable is expressed as the model error divided by observed market price. Sample consists of 66 Canadian convertible bonds traded at the Toronto Stock Exchange. Start of the sample period depends on the issuing date of the bond and starts January 1st 2005, July 1st 2006 or January 1st 2006. Sample period ends on April, 28th 2006. TF refers to the Tsiveriotis-Fernandes model with price fitted credit spread, TF-S refers to the Tsiveriotis-Fernandes model with firm credit ratings implied credit spread. The Ayache-Forsyth-Vetzal model has two versions: partial default model (AFV-P) in which the stock price is unchanged at the event of default and total default model (AFV-T) where stock price goes to zero at the moment
- f default. TKN refers to the Takahashi-Kobayashi-Nakagawa model.
Model intercept S/X VOLAT COUPON TMAT R2
TF 3.58
0.47
0.1 (10.65) (-12.46) (0.50) (-0.98) (-9.28) TF-S 2.37
2.63 0.03
0.08 (7.11) (-11.93) (2.85) (0.03) (-7.38) AFV-T 8.81
4.23
0.11 (11.17) (-15.29) (1.94) (-2.44) (-6.65) AFV-P
0.83 13.39 12.71
0.04 (-11.66) (3.17) (8.10) (4.56) (-0.21) TKN 1.55
3.89
0.01 (2.87) (-2.44) (-2.85) (2.23) (-2.46)
SLIDE 24 24
Table VII Regression Results for the Model Error Relative to the Models' prices
This table shows regression results of the model predicted errors (calculated as the model price minus the market price) on explanatory variables such as ratio of market stock price to conversion price (S/X), time to maturity of convertible security (TMAT), volatility of the underlying stock (VOLAT), and convertible bond coupon rate. Dependent variable is expressed as the model error divided by the price predicted by model. The sample consists of 66 Canadian convertible bonds traded at the Toronto Stock Exchange. Start of the sample period depends on the issuing date of the bond and starts January 1st 2005, July 1st 2006 or January 1st 2006. Sample period ends on April, 28th 2006. TF refers to the Tsiveriotis-Fernandes model with price fitted credit spread, TF-S refers to the Tsiveriotis-Fernandes model with firm credit ratings implied credit spread. The Ayache- Forsyth-Vetzal model has two versions: partial default model (AFV-P) in which the stock price is unchanged at the event of default and total default model (AFV-T) where stock price goes to zero at the moment of default. TKN refers to the Takahashi-Kobayashi-Nakagawa model.
Model intercept S/X VOLAT COUPON TMAT R2
TF
3.57
0.8
0.11 (11.17) (-13.23) (0.90) (-0.69) (-9.32)
TF-S
2.22
3.16 0.49
0.09 (7.06) (-12.45) (3.61) (0.48) (-7.18)
AFV-T
11.7
2.42
0.19 (16.67) (-21.93) (1.25) (-3.07) (-8.68)
AFV-P
0.32 16.65 11.42
0.05 (-11.44) (1.28) (10.50) (4.27) (-0.63)
TKN
1.56
4.81
0.01 (2.87) (-4.23) (-1.34) (2.74) (-1.54)
Table VIII Regression Results for the Absolute Model Error Relative to the Market Prices
Table shows regression results of the absolute values of the models’ predicted errors (calculated as the model price minus the market price) on explanatory variables such as ratio of market stock price to conversion price (S/X), time to maturity of convertible security (TMAT), volatility of the underlying stock (VOLAT), and convertible bond coupon rate. Dependent variable is expressed as the absolute model error divided by observed market price. Sample consists of 66 Canadian convertible bonds traded at the Toronto Stock Exchange. Start of the sample period depends on the issuing date of the bond and starts January 1st 2005, July 1st 2006 or January 1st 2006. Sample period ends on April, 28th 2006. TF refers to the Tsiveriotis-Fernandes model with price fitted credit spread, TF-S refers to the Tsiveriotis-Fernandes model with firm credit ratings implied credit spread. The Ayache-Forsyth-Vetzal model has two versions: partial default model (AFV-P) in which the stock price is unchanged at the event of default and total default model (AFV-T) where stock price goes to zero at the moment of default. TKN refers to the Takahashi-Kobayashi-Nakagawa model.
Model intercept S/X VOLAT COUPON TMAT R2
TF 2.11
3.66
0.01 (7.49) (-4.21) (4.68) (-0.30) (-0.38) TF-S 0.83
5.79 0.83 0.08 0.02 (3.08) (-2.45) (7.69) (0.94) (3.00) AFV-T 7.48
2.98
0.03 (10.05) (-8.52) (1.45) (-2.03) (-2.08) AFV-P 4.86
15.54
0.11 (9.50) (-13.55) (11.57) (-4.28) (-2.08) TKN 2.66
11.51 0.60 0.12 0.08 (6.08) (-12.50) (9.49) (0.43) (2.85)
SLIDE 25 25
Table IX Regression Results for the Absolute Model Error Relative to the Model Prices
Table shows regression results of the absolute values of the models’ predicted errors (calculated as the model price minus the market price) on explanatory variables such as the ratio of the market stock price to the conversion price (S/X), the time to maturity of the convertible security (TMAT), the volatility of the underlying stock (VOLAT), and convertible bond coupon rate. Dependent variable is expressed as the absolute model error divided by the price predicted by model. Sample consists of 66 Canadian convertible bonds traded at the Toronto Stock Exchange. Start of the sample period depends on the issuing date of the bond and starts January 1st 2005, July 1st 2006 or January 1st 2006. Sample period ends on April, 28th 2006. TF refers to the Tsiveriotis-Fernandes model with price fitted credit spread, TF-S refers to the Tsiveriotis-Fernandes model with firm credit ratings implied credit
- spread. The Ayache-Forsyth-Vetzal model has two versions: partial default model (AFV-P) in which the stock price is unchanged at the event of default
and total default model (AFV-T) where stock price goes to zero at the moment of default. TKN refers to the Takahashi-Kobayashi-Nakagawa model.
Model intercept S/X VOLAT COUPON TMAT R2
TF
2.48
3.31
0.02 (9.43) (-5.83) (4.53) (-0.14) (-1.69)
TF-S
1.07
5.66 1.14 0.05 0.03 (4.27) (-3.93) (8.13) (1.40) (2.24)
AFV-T
10.62
0.53
0.09 (16.17) (-14.45) (0.29) (-3.06) (-4.41)
AFV-P
4.07
17.43
0.13 (8.48) (-14.32) (13.84) (-3.7) (-2.49)
TKN
2.57
11.69 2.07 0.13 0.08 (5.86) (-13.17) (9.60) (1.45) (3.20)
SLIDE 26 26 REFERENCES
Ammann, M., A. Kind, and C. Wilde, 2003, Are convertible bonds underpriced? An analysis of the French
- market. Journal of Banking and Finance, 27, 635-653.
Asquith, P., and D.W. Mullins, 1991, Convertible debt: corporate call policy and voluntary conversion, The Journal of Finance 46, 1273-1289. Asquith, P., 1995, Convertible bonds are not called late, The Journal of Finance 50, 1275-1289. Ayache, E., P.A. Forsyth, and K.R. Vetzal, 2003, The valuation of convertible bonds with credit risk, The Journal
Baumol, W.J., B.G. Malkiel, and R.E. Quandt, 1966, Valuation of convertible securities, Quarterly Journal of Economics 80, 48-59. Black, F., and J. Cox, 1976, Valuing corporate securities: some effects of bond indenture provisions, The Journal
Black, F. and M. Scholes, 1973, The pricing of options and corporate liabilities, The Journal of Political Economy 81, 637-654. Brennan, M.J. and E.S. Schwartz, 1980, Analyzing convertible bonds, Journal of Financial and Quantitative Analysis 15, 907-929. Brennan, M.J. and E.S. Schwartz, 1977, Convertible bonds: valuation and optimal strategies for call and conversion, The Journal of Finance 32, 1699-1715. Brigham, E.F., 1966, An analysis of convertible debentures, The Journal of Finance 21, 35-54. Carayannopoulos, P., 1996, Valuing convertible bonds under the assumption of stochastic interest rates: an empirical investigation, Quarterly Journal of Business and Economics 35 (3), 17–31. Cheung, W. and I. Nelken, 1994, Costing the converts, RISK, July, 47-49. Crank J. and P. Nicolson, 1947, A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type, Proceedings of the Cambridge Philosophical Society 43, 50–64. Duffie, D., and K.J. Singleton, 1999, Modeling term structure of defaultable bonds, Review of Financial Studies 12, 687-720. Grimwood, R. and S. Hodges, 2002, The valuation of convertible bonds: A Study of Alternative Pricing Models, Working paper, Warwick University. Ho, T. and D. Pfeffer, 1996, Convertible bonds: model, value attribution and analytics, Financial Analysts Journal 52, 35-44. Hung, M.W. and J.Y. Wang, 2002, Pricing convertible bonds subject to default risk”, The Journal of Derivatives, 75-87. Ingersoll, J., 1977a, A contingent claims valuation of convertible securities, Journal of Financial Economics 4, 289-322. Ingersoll, J., 1977b, An examination of corporate call policies on convertible securities, The Journal of Finance 32, 463-478.
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Jarrow, R. and S. Turnbull, 1995, Pricing derivatives on financial securities subject to credit risk, The Journal of Finance 50, 53-85. King, R., 1986, Convertible bond valuation: An empirical test, Journal of Financial Research 9 (1), 53–69. Levenberg, K, 1944, A method for the solution of certain problems in least squares, Quarterly Journal on Applied Mathematics 2, 164-168. Loncarski, I., J. ter Horst, and C. Veld, 2006a, Why do companies issue convertible bonds? A review of theory and empirical evidence, In: L. Renneboog (editor), Advances in Corporate Finance and Asset Pricing, Elsevier, Amsterdam, 311-339. Loncarski, I., J. ter Horst, and C. Veld, 2006b, The convertible arbitrage strategy analyzed, Working Paper, Tilburg University and Simon Fraser University, August. Longstaff, F. and E. Schwartz, 1995, A simple approach to valuing risky fixed and floating rate debt, The Journal
Madan, D. and H. Unal, 2000, A two-factor hazard rate model for pricing risky debt and the term structure of credit spreads, Journal of Financial and Quantitative Analysis 35, 43-65. Marquardt, D, 1963, An algorithm for least-squares estimation of nonlinear parameters, SIAM Journal of Applied Mathematics 11, 431-441. McConnell, J. and E. Schwartz , 1986, LYON Taming, The Journal of Finance 50, 53-85. Merton, R., 1974, On the pricing of corporate debt: the risk structure of interest rates”, The Journal of Finance 29, 449-470. Nyborg, K., 1996, The use and pricing of convertible bonds, Applied Mathematical Finance 3, 167-190. Takahashi, A., T. Kobayashi and N. Nakagawa, 2001, Pricing convertible bonds with default risk, Journal of Fixed Income 11, 20-29. Tsiveriotis, K. and C. Fernandes, 1998, Valuing convertible bonds with credit risk, Journal of Fixed Income 8, 95- 102 Yigitbasioglu, Ali Bora, 2002, Pricing convertible with interest rate, equity, credit and FX risk, ISMA Centre Finance Discussion Paper No. 2001-14
