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Recurrent Survival Analysis of Sequential Conversions of Convertible Bond
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Lie-Jane Kao
Department of Finance and Banking, KaiNan University, Taoyuan,Taiwan
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Li-Shya Chen
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Department of Statistics, Cheng-Chi University,Taiwan
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Cheng-Few Lee
Department of Finance & Economics, Rutgers University, NJ, USA
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12
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Instead of a block conversion assumed in previous literature, this study explores the
relationship between the hazard rate of sequential conversion over the lifespan of a
convertible bond and individual bond’s time-dependent characteristics, including the
difference between the equity’s price and conversion price, the dividend, and the risk-free
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interest rate. The histories of 130 convertible bonds’ conversions and attributes listed on
the Taiwan Stock Exchange outstanding from January 2004 to December 2009 are collected.
By considering individual bond’s sequential conversions as recurrent events and a recurrent
survival analysis technique (Anderson and Gill, 1982; Prentice, Williams, and Peterson,
1981), an extension of the Cox proportional hazard model, is adopted. Our analysis shows
that the difference between the equity’s price and conversion price has a positive effect on
the hazard rate of conversion, while dividend and risk-free interest rate have negative effects.
It is hoped the results can provide a cornerstone for the pricing of convertible bonds based on
the observation of sequential conversion in real life instead of a block conversion assumed in
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previous literature.
KeyWords. Block Conversion, Hazard Rate, Sequential Conversion, Recurrent Event,
Recurrent Survival Analysis.
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1.Introduction
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A convertible bond is a security that bundles straight debt with an option for the
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bondholder to convert the bond for a preset number of shares of equity stock.
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considered as a vehicle to mitigate the asset substitution problem by curbing shareholders’
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incentive for risk.
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bondholders in a one-period setting, shareholders often find it is more difficult to shift
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investment opportunities to riskier projects and to transfer wealth to their own benefits.
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Mayers (1998, 2002) argued that convertible debts ensure future investment opportunities are
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made only if profitable, thus it can serve as a sequential financing vehicle to control the
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over-investment problem.
It is often
In Green (1984), it was argued that in the presence of convertible
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Nevertheless, the risk-mitigating effect of a convertible bond can be mitigated due to
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the decision to convert the bonds prior to maturity by the bondholders or due to the decision
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of a call by the shareholders (Francois et al., 2006).
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will never be optimal to convert a convertible bond strategically prior to the maturity or the
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call under the assumptions of a perfect market, constant conversion terms and no dividend
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payments (Ingersoll, 1977).
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dividends are paid and adverse changes in the conversion terms are not allowed, the optimal
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conversion strategy is to convert a convertible bond either immediately prior to a dividend
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payment date or at the maturity.
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bondholders ensures the risk-mitigating effect of a convertible bond.
From the bondholders’ viewpoint, it
In another paper by Brennan and Schwartz (1977, 1980), if
In either situation, the optimal conversion strategy of the
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In literature, the determinants of a bondholder’s decision to convert and the timings of
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conversions are inclusive. It is argued that the expected probability of a conversion at
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maturity or the expected time to conversion is determined by the initial design features of a
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convertible bond (Lewis et al., 2003; Lee et al., 2009). According to Lewis et al. (2003), the
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expected probability of a conversion at maturity is positively correlated with the differences
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between the stock and conversion prices as well as the risk-free interest rate at the time of
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convertible issue, but negatively correlated with the firm’s dividend yield for the year before
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the convertible issue date.
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separation of control and ownership rights are prone to issue more debt-like convertibles,
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which have lower expected probabilities of a conversion at maturity, to avoid expropriation
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of minority shareholders’ wealth by controlling shareholders.
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addresses the strategic dimension of conversion by bondholders (Emanuel, 1983;
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Constantinides, 1984; Asquith & Mullins, 1991; Bühler and Koziol, 2004; Francois et al., 2006;
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Sirbu and Shreve, 2006).
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value is sufficiently high and the downside protection premium offered by the bond is
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sufficiently low, the cash flow differential of dividends over coupon interest can be great
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enough to engender strategic conversion. In Sirbu and Shreve (2006), by considering the
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conversion-call strategy as a two-person, zero-sum game, it was shown that if the coupon
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rate is below the dividend rate times the call price, then conversion should precede call and
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strategic conversion should be more prevalent for bonds with greater conversion values. On
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the other hand, if the dividend rate times the call price is below the coupon rate, call should
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precede conversion.
In Lee et al. (2009), it is documented that firms with higher
There are also literature
In Asquith & Mullins (1991), it is argued that when the conversion
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All the aforementioned literature implicitly or explicitly assumed a block conversion in
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which all the convertible bonds are completely converted at the same time or not at all.
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Instead of a block conversion, however, one often observes sequential conversions by
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competitive bondholders over time instead of a block conversion. Constantinides (1984)
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showed that in the absence of straight bonds, a competitive equilibrium exists so that
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sequential conversions can be optimal in that the price of a divisible convertible bond equals
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that of a block one. In Bühler and Koziol (2004), by including additional straight bonds in a
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firm’s capital structure, it was shown sequential strategic conversions is in general optimal
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rather than a block conversion.
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In Francois et al. (2006), by allowing the risk shifting due to
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investment opportunity occurring randomly after capital structure was previously designed,
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the decisions to shift risk and to convert strategically are considered as a non-cooperative
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game between shareholders and convertible bondholders. When the game is sequential, it
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was shown that shareholders substitute assets to shift risk and bondholders strategically
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convert arises as an attainable Nash equilibrium which can prevail sequentially over time.
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This study explores the relationship between the intensity of sequential conversion over
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the lifespan of a convertible bond and the determinants of sequential conversions of
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individual bond, including the conversion price, the equity’s price, the dividend, and the
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bond’s coupon rate.
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and attributes listed on the Taiwan Stock Exchange outstanding from January 2004 to
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December 2009 are collected. Compared to previous studies in which a block conversion is
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assumed, our approach considers individual bond’s sequential conversions as recurrent
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events and recurrent survival analysis technique (Anderson and Gill, 1982; Prentice,
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Williams, and Peterson, 1981), an extension of the Cox proportional hazard model, is
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adopted. To our knowledge, this is the first study that uses the recurrent survival technique in
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studying the conversion behaviors of bondholders in a dynamic setting.
To be able to do so, the histories of 130 convertible bonds’ conversions
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The paper is organized as follows. Section 2 gives a review for the works in the survival
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analysis of recurrent events. Section 3 develops the model. An empirical study is given in
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Section 4, and Section 5 concludes.
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1. Recurrent Survival Analysis: Variance-correction versus Frailty
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Survival analyses of recurrent events that occur repeatedly over time for a given subject
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are common in the study of health science (Prentice et al., 1981; Andersen and Gill, 1982;
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Wei et al., 1989, 1990, 1997; Kelly and Lim, 2000; Cook and Lawless, 2002; Duchateau et. al.,
105
2003; Schaubel and Cai, 2005).
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international conflicts (Beck et. al., 1998), presidential nominations (McCarty and Razaghian,
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1999).
4
They also occur in the scope of political science, e.g.
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In the study of recurrent events, event dependency within an individual is a common
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theme. In addition, sometimes recurrent events within the same individual involve different
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types of categories and the ordering of the events is important. The traditional Cox’s
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proportional hazard model (Cox PH), which deals with a subject’s first failure, is therefore
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biased in the recurrent events context. Two extensions of Cox PH model, namely, the
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variance-correction models and frailty models were developed to account for this deficiency
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(Box-Steffensmeier et al., 2006).
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Variations within the variance-correction models depend on whether recurrent events
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within the same individual are treated as identical or not.
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different categories and the order of the recurrent events is important, variance-correction
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models are developed using a stratified Cox model with different risk sets defined for each
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strata. Three widely used variance-correction models are: the conditional risk set (PWP)
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model (Prentice, Williams, Peterson, 1981), the Andersen-Gill (AG) model (Anderson and
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Gill, 1982), and the marginal risk set (WLW) model (Wei, Lin, Weissfeld, 1989).
If recurrent events involve
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In the Andersen-Gill (AG) model, which is based on a non-homogeneous Poisson
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counting processes, all the recurrent events within an individual are treated as identical
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(Andersen and Gill, 1982). Unless time-dependent covariates that capture the dependence of
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recurrent events within an individual are included, the recurrent events on the same
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individual are independent and identical under the AG model. In cases recurrent events
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within the same individual are of different categories and the order of the events is important,
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either the conditional risk set model or the marginal risk set model should be adopted.
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the conditional risk set model by Prentice, et al. (1981), the risk set of the kth recurrent event
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are restricted to the individuals who have experienced the first k-1 recurrent events, and the
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baseline hazard functions are allowed to vary with k, k=1,2,…, K. In the marginal risk set
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model proposed by Wei, et al. (1989), the risk set of the kth recurrent event contains all the
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subjects that had not experienced the kth recurrent event or censored.
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In
Similar to the
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conditional risk set model, the baseline hazard function and the regression parameters may
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vary with the order of recurrent event k in the marginal risk set model.
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In contrast to variance-correction models, the frailty models use unobservable random
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covariates to account for the dependence among the recurrent events within an individual,
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which explain the heterogeneity across individuals. The simplest form of a frailty model is
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the Andersen-Gill model with a random covariate. An estimation procedure for the
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regression parameters, the variance of the frailty, and the underlying hazard function is
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proposed by Nielsen et al. (1992). Whereas more advanced frailty model is proposed by
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Clayton and Cuzick (1985), which assumes that the unobserved random covariate is from a
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gamma distribution with mean one and an unknown variance. For the situation that the order
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of recurrent events is important, the conditional frailty model that combines a random
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covariate
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Box-Steffensmeier et al., 2002, 2006).
with
event-based
stratification
is
proposed
(Kelly
and
Lim,
2000;
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2. Model Development: Sequential Conversions as Recurrent Events
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Suppose the time horizon is divided into L time intervals (t0, t1], (t1, t2], … , (tL-1, tL].
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Consider a set of N convertible bonds For each bond, one or more of the following events
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are observed: (1) A sequence of conversions, (2) The call announcement by the firm, (3) The
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maturity of the bond, and (4) The exercise of put options by the bondholders.
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of this paper is to explore the behavior of sequential conversions till the maturity date of the
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bonds, the events of a call and exercises of put options will be ignored.
.
As the theme
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To model the hazard of a conversion for bond i at time t, 1iN, the determinants of a
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conversion in literature including the conversion price, the stock’s price, the dividend, the
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bond’s coupon rate, and the risk-free interest rate are considered (Constantinides, 1984;
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Asquith & Mullins, 1991; Bühler and Koziol, 2004; Francois et al., 2006). As the coupon
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159
rates of the sample bonds are all zeros, we consider the following three time-dependent
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covariates at time t: (1) The difference between the highest stock price and conversion price
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X1i(t), (2) The dividend rate X2i(t), and (3) The risk-free interest rate X3i(t).
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dividend effect is traced back to four months prior to the distribution date of the dividends.
Note the
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If one considers the sequence of conversions within a bond as identical events and the
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order of conversions is not important, one might adopt the variance corrected AG model by
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Anderson and Gill (1982) due to its efficiency and precision in giving the most reliable
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estimates of the covariates’ effects (Therneau and Grambsch, 2000).
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proportional hazard (Cox PH) model, the hazard function of bond i for its kth conversion at
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time t has the proportional form
ik (t )  0 (t ) exp 1 X 1i (t )   2 X 2i (t )   3 X 3i (t )
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Like the Cox’s
(1)
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where 0(．) is the baseline hazard function.
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same individual are treated as independent observations, and therefore the partial likelihood
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can be expressed as
In the AG model, the recurrent events of the
 ik
N K i 1

ik (Tik )

L(  )     N
K j  1 I (T


T

T
)

(
T
)


i 1 k 1 
j ,l 1
ik
j ,l
jl ik 
 j 1 l 1
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(2)
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where =(1, 2, 3) are the coefficients in (1), Tik is the time of the kth conversion of the ith
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bond, 1kKi, while Ti , K i 1 is the termination time of the ith bond, 1iN. The indicator I()
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is one if Tik  (T j ,l 1 , T j ,l ] and zero otherwise, while the indicator δik equals to 1 if bond i is
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converted at Tik and zero otherwise.
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random effect or frailty to account for the heterogeneity among bonds and the dependence
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between sequential conversions within an individual bond. The hazard function (1) for the
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AG frailty model becomes
ik (t )  0 (t ) exp 1 X 1i (t )   2 X 2i (t )   3 X 3i (t )   i 
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The above AG model can be extended to include a
(3)
where  i is the frailty of bond i, 1iN. Note it is assumed the frailties  1 , …,  N share a
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common mean one and a common variance  2 .
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However, if one considers the sequence of conversions within a bond as heterogeneous
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events and the order of conversions is important, the conditional risk set approach by
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Prentice Williams, and Peterson (1981), i.e., the PWP approach, in which the observations
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are stratified by the order of conversions, should be used.
In the PWP approach, different
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baseline hazards will be employed for different stratum.
Let C=max{K1,…,KN} be the
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maxima number of conversions within a bond among the N sample bonds, and thus there are
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C stratum.
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function of bond i for its kth conversion at time t has the usual proportional hazard form
Let 0k(t) be the baseline hazard at time t for the kth stratum, 1kC, the hazard
*ik (t )  0k (t ) exp1 X 1i (t )   2 X 2i (t )  3 X 3i (t )
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(4)
Since a bond is not at risk for the kth conversion until its (k-1)th
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where 1iN, 1kC.
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conversion has been occurred, only the bonds whose (k-1)th and kth conversion occurs prior to
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and after Tik, respectively, are considered to be at risk, where Tik is the kth conversion time for
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the ith bond.


*ik (Tik )

L(  )     N
*

I
(
T

T

T
)

(
T
)

i 1k 1 
j ,k 1
ik
jk
jk
ik 
 j 1
N
197
198
Hence, the partial likelihood can be formulated as
 ik
C
(5)
A frailty term could also be included in (4), and the hazard becomes
*ik t   0k (t ) exp1 X1,i (t )  2 X 2,i (t )  3 X 3,i (t )  i 
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(6)
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where  i is the frailty of bond i, 1iN. Note it is assumed the frailties  1 , …,  N share a
201
common mean one and a common variance  2 .
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3. Data
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From the list on the Taiwan Stock Exchange, convertible bond that were outstanding
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from January 2004 to December 2009 and terminated before December 31, 2009 are eligible
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for our study.
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Among the eligible convertible bond issues, we select our sample bond
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issues based on the following criteria:
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(1) Exclude the listed companies in the financial sector due to the comparability of
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financial indicators;
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(2) If two or more convertible bonds are issued by the same company, then only one of
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them is randomly selected.
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We thus have 170 eligible issues of convertible bonds. As the main purpose of the
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analysis is to develop a hazard model that exhibits the dynamic features of the sequential
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conversion phenomenon observed empirically. Therefore, the bond issues that were traded
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with more shares for call or put are excluded from the sample. Further analysis shows 18
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bond issues were traded with more shares for call, while 22 were traded with more shares for
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put. After removing them, there are 130 bonds issues that are traded mainly for conversion.
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For each sample bond, the history of conversions during January 2004 to December 2009 is
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collected
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http://www.gretai.org.tw/ch/bond_trading_info/bonds_info/monthly/cbnote.php. While the
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covariates’ histories and the risk-free interest rates during January 2004 to December 2009
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are collected from Taiwan Economic Journal (TEJ).
from
the
website
of
Taiwan’s
GreTai
Securities
Markets
at
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From January 2004 to December 2009, all monthly records of the 130 eligible convertible
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bonds are of the counting process format, i.e., (ID, 1, 2, Status, X1, X2, X3, …), where 1 and
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2 denote the times when the monthly interval started and ended, respectively. Status
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indicates whether conversion occurring at the interval specified by [1, 2), and X1, X2, X3, …
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denote the covariates. As pointed by Therneau and Grambsch (2000) and Allison (2010),
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counting process format helps to handle datasets with recurrent events and time-dependent
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covariates.
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Among the 130 eligible convertible bonds, observations with their differences between
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the highest stock price and the conversion price exceeding $200 are considered as outliers.
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By removing these outliers, our final sample consists of N=128 bonds and there is a total of
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128
 ni  4497 observations for the empirical study. Table 1 gives the summary statistics of the
i 1
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three covariates X1, X2 and X3 for the 128 sample bonds.
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represents the standardized differences between the highest stock price and the conversion
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price.
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4. Empirical Analysis
Note that the covariate X1
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The lifespans or durations in months since publication of the N=130 sample bonds are
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summarized in Table 2, in which the frequency and the cumulative percent frequency
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distributions of the lifespan are given.
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lasted no more than 40 months: 12% of them had been traded fast and terminated within 15
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months and one third of them terminated within 25 months. Only 16.9% of the sample bonds
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had stayed to the end of the sample period.
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percentages of conversions been made from January 2004 to December 2009 for a total of 60
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months. Among the 130 bonds being studied, since their issuances, 6% converted before the
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4th month, 41% converted before the 10th month, 67% converted before the 14th month. The
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last conversion occurs at the 32th month.
From Table 2, about two third of the sample bonds
Table 3 provides frequencies and cumulative
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In Table 4, the AG model with and without the frailty term are estimated, respectively.
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Panel I shows the estimation result for the AG model without frailty; while Panel II shows
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the results with the frailty effect. For all cases, all the three covariate X1, X2, and X3 are
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significant at =0.10 level.
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between the highest stock price and the conversion price X1 has a positive effect on the
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hazard rate of conversion, while the dividend X2 and the risk-free interest rate X3 have
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negative effects on the hazard rate of conversion.
Among the three covariates, the standardized difference
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For the PWP model, as the maximum number of conversions within an individual bond
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C=32, to avoid the problem of over stratification, we consider two stratification schemes
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with four strata each.
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The first scheme classifies observations of the 1st and 2nd conversion
258
as the first strata, observations after the 2nd up to the 5th as the second strata, observations
259
after the 5th conversion up to the 10th conversion as the third strata, while the remaining
260
observations are classified as the fourth strata. The second scheme classifies observations
261
up to the 5th conversion as the first strata, observations after the 5th conversion up to the 10th
262
conversion as the second strata, observations after the 10th conversion up to the 15th
263
conversion as the third strata, while the remaining observations are classified as the fourth
264
strata.
265
In Table 5, the PWP models under the two aforementioned stratification schemes are
266
estimated.
267
term, respectively, under the first stratification scheme are given. Estimates under the
268
second stratification scheme are given in Panels III and IV for the cases with and without the
269
frailty term, respectively.
270
with those of the AG models: all the three covariate X1, X2, and X3 are significant at =0.10
271
level.
272
price and the conversion price X1 has a positive effect on the hazard rate of conversion, while
273
the dividend X2 and the risk-free interest rate X3 have negative effects on the hazard rate of
274
conversion.
275
In Panels I and II, estimation results of PWP model with and without the frailty
In all cases, the results of the PWP models basically coincide
Among the three covariates, the standardized difference between the highest stock
The positive effect of X1, i.e., the standardized difference between the highest stock price
276
and the conversion price, coincides with that of Lewis et al. (2003).
277
of negative dividend effect 2 is in line with Lewis et al. (2003), in which the expected
278
probability of a conversion at maturity is negatively correlated with the dividend of the fiscal
279
year before the convertible issue. On the other hand, the estimated negative dividend effect
280
2 contradicts with those in Asquith & Mullins (1991) and Sirbu and Shreve (2006). As the
281
coupon rates are all zeros in our sample bonds, the cash flow differential of dividends over
282
coupon interest is not of the main concerns by the bondholders.
283
dividend yield diminishes the growth of future stock price, thus results in negative effect on
11
In addition, the result
Instead, the higher
284
the hazard rate of conversion.
285
portfolio consisting of an straight discount bond plus an option entitling the bondholders to
286
purchase the same fraction of equity with an exercise price determined by the principle of the
287
bond (Ingersoll, 1977), higher risk-free interest rate results not only lower straight bond value,
288
but also diminishes the value of the option to purchase the equity.
289
expect the negative risk-free interest rate effect 3. This contradicts the result that the
290
expected probability of a conversion at maturity is positively correlated with the risk-free
291
interest rate in Lewis et al. (2003).
In addition, as a convertible bond can be considered as a
Therefore, one can
292
293
5. Conclusion
294
From an empirical perspective, bondholders convert part of the outstanding convertibles
295
as long as it was feasible to convert is more prevalent than a block conversion. Not only that,
296
from a theoretical perspective, sequential conversion can also be optimal if one investor
297
holds all convertibles, or if competitive convertible bond holders act strategically (Emanuel,
298
1983; Constantinides, 1984). As the conversion strategy impacts the number of outstanding
299
stocks, their value, and the firm’s capital structure, thus the value of the convertible bonds
300
depends on the conversion strategy followed by the investors. However, most literature
301
implicitly or explicitly assumes a block conversion, i.e., convertible bonds are converted at
302
the same time or not at all to price a convertible bond.
303
technique to explore the relationship between the hazard of sequential conversion and three
304
determinants of a conversion, namely, the highest stock price and the conversion price, the
305
dividend, and the risk-free interest rate in a dynamic setting. It is hoped that the results can
306
provide a cornerstone for the pricing of a convertible bond based on the observation of
307
sequential conversion in real life.
308
12
This study uses a recurrent survival
309
310
Table 1. Summary Statistics of Covariates
Excluding Two Outliers
Variable
X1
X2
X3
311
312
Total obs.
4497
4497
4497
Min.
-1.477
0.000
0.862
1st Q.
-0.289
0.000
1.795
Median
-0.189
0.000
1.911
3rd Q.
0.0163
0.000
2.229
Max
Mean
25.47 10.437
7.494 0.088
2.649 1.985
Std
40.486
0.474
0.307
Note. X1 is the standardized difference between the conversion price and stock price; X2 is the dividend;
X3 is the risk-free interest rate.
313
13
314
Table 2. Distribution of Lifespans
Duration in
Months
1-5
6-10
11-15
16-20
21-25
26-30
31-35
36-40
41-45
46-50
51-55
56-60
315
316
Frequecy
1
3
11
11
17
10
15
17
12
4
2
25
Cumulative
Frquency%
0.78%
3.13%
11.72%
20.31%
33.59%
41.41%
53.13%
66.41%
75.78%
78.91%
80.47%
100.00%
Note. The two outliers of the N=130 sample bonds
are excluded.
317
14
318
319
Table 3. Distribution of Conversion Times
Duration in
Months
1-2
3-4
5-6
7-8
9-10
11-12
13-14
15-16
17-18
19-20
21-22
23-24
25-26
27-28
31-32
320
321
Frequecy
2
6
7
20
17
12
21
11
9
9
5
2
4
1
1
Cumulative
Frequency%
1.57%
6.30%
11.81%
27.56%
40.94%
50.39%
66.93%
75.59%
82.68%
89.76%
93.70%
95.28%
98.43%
99.21%
100.00%
Note. The two outliers of the N=130 sample bonds
are excluded.
322
323
324
325
326
327
15
328
329
Table 4. Analysis of AG Model
I. AG model without frailty
j
s.e.
X1
0.4376
0.0276
2value
d.f.
251.8
1
0.00000
p-value
X2
-0.0517
0.0312
2.75
1
0.09710
X3
-0.3144
0.1268
6.15
1
0.01321
3
0.00000
LRT
239.2
II. AG model with frailty
j
s.e.
2value
d.f.
X1
0.4339
0.0285
232.2
1
0.00000
X2
-0.0764
0.0322
5.61
1
0.01800
X3
-0.2924
0.1338
4.78
1
0.02900

0.1120
--
166.7
72.9
0.00000
75.8
0.00000
2
LRT
487.0
p-value
330
Note. LRT is the likelihood ratio of H0: 1=2=3 vs.
331
332
H1: Not all js are zeros.  2 is the common variance of the
N frailties  1 , …,  N .
333
334
16
335
336
Table 5. Analysis of PWP Model
I. First stratification scheme without frailty
j
se
2value
d.f.
p-value
X1
0.3667
0.0287
163.70
1
0.00000
X2
-0.0635
0.0318
3.99
1
0.04570
X3
-0.2893
0.1269
5.20
1
0.02260
3
0.00000
d.f.
p-value
LRT
161.5
II. First stratification scheme with frailty
j
se
2value
X1
0.3867
0.0294
172.69
1
0.00000
X2
-0.0708
0.0324
4.79
1
0.02900
X3
-0.2354
0.1338
3.10
1
0.07800
0.0706
--
57.5
0.00016
60.4
0.00000

2
LRT
104.13
313.0
III. Second stratification scheme without frailty
j
se
2value
d.f.
p-value
X1
0.4110
0.0285
207.79
1
0.00000
X2
-0.0651
0.0315
4.26
1
0.03900
X3
-0.3747
0.1285
8.51
1
0.00353
3
0.00000
d.f.
p-value
LRT
204.1
IV. Second stratification scheme with frailty
j
se
2value
X1
0.4362
0.0294
220.92
1
0.00000
X2
-0.0796
0.0328
5.89
1
0.01500
X3
-0.2755
0.1377
4.00
1
0.04500
 2
0.1750
260.65
85.6
0.00000
88.5
0.00000
LRT
-482.0
337
Note. LRT is the likelihood ratio of H0: 1=2=3 vs. H1: Not all js are
338
zeros.  2 is the common variance of the N frailties  1 , …,  N .
339
340
341
17
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
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