公司認股權證對股價之影響 - 政大學術集成

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(1)國立政治大學金融研究所博士論文指導教授：廖四郎博士. 政治大立公司認股權證對股價之影響. ‧ 國. 學 ‧. On Stock Return Processes and Conditional Heteroskedasticities with Warrant Introduction. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. 研究生：張瑞珍撰中華民國九十九年一月.

(2) Contents. Chapter 1. Introduction……………………………….….. p.4. Chapter 2. Literature Review……………………………. p.6. 2.1 Warrant derivatives. p.6. 2.2 GARCH model. p.9. 2.3 GARCH-in-mean (GARCH-M) model. p.10. 政治大 Methodology…………………………………. 立. p.10. 3.1.Theoretical Consideration on Testing the Effect of Warrants on the Underlying Stock Return Process. p.14. 2.4 Asymmetric GARCH. 學. ‧. ‧ 國. Chapter 3. p.14. p.15. 3.3.The GARCH-M Model with Volatility Modifications for Testing the Effect of Warrant Introduction. p.17. er. io. sit. y. Nat. 3.2.Stock Return Processes in the GARCH-M Model. al Data ………………………………………….. iv. p.24. Chapter 5. ngc Empirical Results…………………………….. p.27. n. Chapter 4. Ch. hi. e. Un. 5.1.Model 1: The Dilution-Adjusted GARCH-M Model. p.27. 5.2.Model 2: The Asymmetric Dilution-Adjusted GARCH-M Model. p.31. 5.3.Model 3: The Dilution-Adjusted GARCH-M Model with a Threshold for Exercise Prices. p.34. 5.4.Model 4: The Asymmetric Dilution-Adjusted GARCH-M Model with a Threshold for Exercise Prices. p.37. i.

(3) Discussion…………………………………….. p.40. 6.1 The Dilution Effect on Stock Return Processes. p.41. 6.2 A Plausible Explanation of Changes in the Stock Return Process. p.42. Chapter 7. Conclusions…………………………………... p.44. Appendix A. The Stock Return Process of Companies in Table Ⅰ. p.49. 立. 政治大. 學 ‧. ‧ 國 io. sit. y. Nat. n. al. er. Chapter 6. Ch. engchi. ii. i Un. v.

(4) Tables. Table Ⅰ. Summary of the 36 warrants issued from Jan. 1, 2001, to Dec. 31, 2004. p.26. Table Ⅱ. Maximum Likelihood Estimates Dilution-Adjusted GARCH-M Model). (The. p.29. Table Ⅲ. Summary Statistics for the Parameters of Introduction Dummy on Model 1. p.30. Table Ⅳ. Maximum Likelihood Estimates of Model 2 (The Asymmetric Dilution-Adjusted GARCH-M Model). Table Ⅴ. Summary Statistics for the Parameters of Introduction Dummy and Asymmetric Dummy on Model 2. p.33. Table Ⅵ. Maximum Likelihood Estimates on Model 3 (The Dilution-Adjusted GARCH-M Model with a Threshold for Exercise Prices). p.35. Table Ⅶ. Summary Statistics for the Parameters of Introduction Dummy on Model 3. 立. of. Model. 1. 政治大. 學. ‧. ‧ 國. Nat. n. al. p.36. er. io. sit. y. p.32. Ch. i Un. v. Table Ⅷ. Maximum Likelihood Estimates on Model 4 (The Asymmetric Dilution-Adjusted GARCH-M Model with a Threshold for Exercise Prices). p.37. Table Ⅸ. Summary Statistics for the Parameters of Introduction Dummy and Asymmetric Dummy on Model 4. p.39. engchi. iii.

(5) 論文摘要. 雖然許多研究已針對認股權證評價進行調整，但是其價格低估的問題仍無法解決。因此，本文將探討認股權證發行對股價報酬動態過程的影響。本文將證實是否認股權證發行將影響其標的股價之動態過程，倘若股價報酬的動態過程已反應了認股權證發行的潛在稀釋效果，則進行充分調整的股權稀釋模型將低估認股權證的價格。為了確認在評價認購權證時充分調整稀釋效果的必要性，本文將檢測權證發行對股票報酬過程的影響。本文利用延伸 Garch-M 模型，. 政治大酬的變異數顯著降低，該結果在釐清股權稀釋效果與不對稱效果之後，該稀釋立導出四個檢驗稀釋效果的模型。實證結果顯示，在發行認股權證之後，股價報. ‧. ‧ 國. 學. 效果依然顯著。. 關鍵字：認股權證，發行效果，稀釋效果，GARCH，資本結構. n. er. io. sit. y. Nat. al. Ch. engchi. i. i Un. v.

(6) Thesis Abstract. As the underestimation of warrants remains unsolved after many adjustments presented by previous researchers, we further investigate the impact of the warrant introduction on the underlying stock return processes. This research attempts to determine whether the introduction of warrants influences the return processes of underlying stocks. If the introduction creates a potential dilution effect in stock return processes, full dilution adjustment pricing models would lead to underestimation. To. 政治大 GARCH-M model has been extended to derive four models for testing the dilution 立 exam whether full dilution adjustments are required for warrant pricing, the. ‧ 國. 學. effect on stock return processes. Empirical results show that the volatilities of underlying stock return processes are significantly reduced following warrant. er. io. sit. y. Nat. effect.. ‧. introduction even after clarification and distinguishing dilution from asymmetric. Key Words: Warrant; Introduction Effect; Dilution Effect; GARCH; Capital Structure. n. al. Ch. engchi. ii. i Un. v.

(7) Chapter 1. Introduction Since Black and Scholes (1973) and Galai and Schneller (1978) priced warrants as an option on the stock of the underlying firm with some dilution modifications, the warrant pricing problem has become an important issue.1 Recently, Koziol (2006). 政治大 then analyzed their optimal立 exercise strategies for corporate warrants. As warrants are also found that the exercise behavior of warrant holders affects warrant values and. ‧ 國. 學. incorporated in many financial derivatives, it is important to accurately evaluate warrant prices. Numerous warrant pricing models are presented, following the option. ‧. pricing framework with some modifications, such as dilution adjustments. The most common and cost efficient method might be the Dilution-Adjusted Black-Scholes. y. Nat. sit. (DABS) model which is the Black-Scholes option pricing model with some dilution. n. al. er. io. adjustments. The model assumes that warrant listing increases both firm equity and. i Un. v. outstanding shares, and thus the dilution effect should be take into account.. Ch. engchi. Kremer and Roenfeldt (1993) and Hauser and Lauterbach (1997) suggested warrants are generally underpriced by the DABS model and Dilution-Adjusted Jump-Diffusion model. Crouhy and Galai (1991), Schulz and Trautmann (1994), Becchetti (1996) and Handley (2002) argued that stock price should already reflect the dilution effect during the life of the warrant. Handley (2002) also noted that the stock price of the underlying firm conditionally reflects dilution at any time following the announcement of warrant listing. No explicit adjustment for dilution is required. If. 1. The Black-Scholes warrant pricing model is presented by Black-Scholes (1973) and Galai and Schneller (1976), they showed that the Black-Sholes option pricing model can price warrants with some modifications and dilution adjustments. 4.

(8) the dilution effect is already a function of time, pricing errors of dilution-adjusted warrant pricing models will increase closer to expiration of the warrant. Therefore, this study investigates whether the underlying stock return process genuinely reflects the potential dilution effect during the life of the warrant. In general, warrant pricing models need to estimate the future volatility of underlying assets. If the introduction of warrants significantly lowers the return and volatility of the underlying stocks, the warrant price drops. In this case, the valuation of warrants using post-announcement stock return processes adjusting for dilution overcompensates, forcing underpricing of warrants. Although Crouhy and Galai. 政治大 potential dilution during the life of warrant, little attention has been drawn to 立 determine the effect of the warrant introduction on the underlying stock return. (1991) and Handley (2002) argued that the stock process should constantly reflect the. ‧ 國. 學. process.2 This is a closer examination on the life warrants instead of focusing on the. ‧. time of introduction. Since financial data generally exhibit time-varying variances and excess kurtoses, the GARCH model is extended to incorporate these. sit. y. Nat. characteristics.. er. io. Chapter 2 follows with a description of examination methodology for the. al. n. iv n C results; Chapter 5 summarizes and h discusses i U the conclusion is presented in e n gthe c hresults;. introduction effect; Chapter 3 describes the data set; Chapter 4 presents the empirical. Chapter 6.. 2. The literature focused on such impact was published by Alkeback and Hagelin (1998) and Becchetti (1996). Alkeback and Hagelin (1998) used the event study methodology to determine the effect on price, volatility and liquidity of the underlying stock at and around warrant introduction. Their results suggested that there is no real effect on the underlying stocks following the warrant introduction, thus there is no significant impact on the price or volatility. Becchetti (1996) analyzed the effect of bond plus equity warrant (WB) issues on underlying asset volatility, and the empirical results indicate that the underlying stock volatility decreased after the introduction of WB. 5.

(9) Chapter 2. Literature Review 2.1 Warrant derivatives Many previous researchers pay much attention to reduce the underestimation problem of warrant pricing. Several researchers concluded that this underestimation bias is improved by either considering the possibility of maturity extension by the. 治政 issuer, establishing that the equity return volatility is大 inversely related to the stock 立 price, including the flexibility for early exercise, or pricing the warrant with ‧ 國. 學. jump-diffusion model. However, the underestimation problem after these adjustments is only partially improved, not solved. Kremer and Roenfeldt (1993) suggested that. ‧. warrants are generally underpriced by the DABS model and the Dilution-Adjusted. Nat. sit. y. Jump-Diffusion model. Their empirical study indicated a large degree of underpricing. er. io. when DABS models are applied to samples of short maturity, and they also found that. al. iv n C underestimation is possibly resulted h from e n gnegligence c h i Uof maturity extension for some n. this pricing error increases as the maturity decreases. They argued that the. warrants. However, this concept is not the crucial reason of warrant underestimation.. Hauser and Lauterbach (1997) provided the evidence that the underestimation still significantly exists in DABS models with extensible maturity adjustments. Their examination on five warrant pricing models concluded the pricing errors of the DABS model are significant especially for out-of-the-money warrants, and some biases remain in the constant elasticity of variance (CEV) model despite its relatively good performance. Therefore, I suggest that some crucial factors of underestimation must still be ignored in these models. Crouhy and Galai (1991) argued that the stock price process should already reflect. 6.

(10) the dilution effect during the life of the warrant, and they also emphasized that “imposition of an exogenous adjustment coefficient results in double-counting of dilution effect.” In addition, the above point of view is also supported by several studies (Schulz and Trautmann, 1994; Handley, 2002). From their perspectives, there is no need to make any specific modification for the dilution effect. Schulz and Trautmann (1994) argued that using the option pricing model with the dilution adjustment to value warrant price is an inadequate application. They deliberately analyzed the non-stationarity of stock volatility, and then stated that “setting the stock volatility changeable over the life of the outstanding warrant, there is no. 政治大 the stock price as the state variable.” Their empirical analysis also supports that even 立 without dilution adjustment the option-like warrant can be correctly priced. In line. dilution-related pricing bias of the American constant variance diffusion model with. ‧ 國. 學. with Schulz and Trautmann (1994), Handley (2002) noted that the stock price of the. ‧. underlying firm conditionally reflects dilution at any time following the announcement of warrant listing, and no explicit adjustment for dilution is required.. Nat. sit. y. Moreover, the empirical results of Hauser and Lauterbach (1997) showed that the. er. io. absolute pricing errors in all of the five models are significantly negatively correlated. al. n. iv n C expiration decreases and as stock h prices e nfall. h i U this study wonders whether g cTherefore,. with the time to maturity. It indicates that pricing errors increase as the time to. the underlying stock return process really reflects some potential dilution effect during the warrant life. Since the dilution effect is embedded in time, pricing errors of dilution-adjusted warrant pricing models will increase as the time to expiration. On one hand, some researchers argue that the exogenous adjustment for dilution effect creates problems in double-counting. On the other hand, numerous researchers evaluate warrants with dilution adjustments and emphasize the importance of dilution adjustment. These contrasting views lead to the necessity for an investigation of the warrant introduction effect on the underlying stock return process. Conceptually, if the behavior of stock return during the life of warrants already reflects the potential. 7.

(11) dilution, the warrant pricing model with any exogenous adjustment for dilution effect would underestimate warrant prices. In general, warrant pricing models need to estimate the future volatility of underlying assets. If the warrant introduction itself makes the return and volatility of the underlying stock significantly lower, the warrant price will decrease with lower volatility. In this case, the valuation of warrants using post-announcement stock return processes with the dilution adjustment will result in overcompensation and make warrants underpriced. Therefore, this study would like to examine the impact of warrant introduction to determine whether the warrant introduction affects the underlying stock return process.. 政治大 and the underlying stock price conditionally reflects dilution at any time following the 立 announcement of warrant listing, little attention has been drawn to determine the Although several researchers suggested that the dilution adjustment is unnecessary. ‧ 國. 學. effect of the warrant introduction on the underlying stock return process. The. ‧. literature focused on such impact was published by Alkeback and Hagelin (1998).3 They used the event study methodology to determine the effect on price, volatility. Nat. sit. y. and liquidity of the underlying stock at and around warrant introduction.. er. io. Unfortunately, their conclusion did not support that the stock price already reflects the. al. n. iv n C h e n gintroduction, underlying stocks following the warrant c h i U thus there is no significant. dilution of warrant listing. Their results suggested that there is no real effect on their. impact on the price or volatility. Based on their results that the warrant introduction has no real influence on its underlying stock, option pricing models should modify the potential equity dilution to price warrants. This conclusion seems to contradict to the studies mentioned above (Crouhy and Galai, 1991; Schulz and Trautmann, 1994; Handley, 2002). Consequently, this ambiguity justifies the need for further empirical research. Because Crouhy and Galai (1991) and Handley (2002) argued that the stock. 3. Another literature proposed by Becchetti (1996) analyzed the effect of bond plus equity warrant (WB) issues on underlying asset volatility, and the empirical results indicate that the underlying stock volatility decreased after the introduction of WB. 8.

(12) process should constantly reflect the potential dilution during the warrant life, this study is a closer examination on the warrants life instead of focusing on the time of introduction. Since financial data generally exhibit time-varying variances and excess kurtoses, the GARCH model is extended to incorporate these characteristics.. 2.2 GARCH model The ARCH process proposed by Engle (1982) explicitly recognizes the difference between the unconditional and the conditional variance. They stated that the conditional variance can change over time as a function of past errors. They also. 政治大 Ω 立 at time t − 1 . For extending the ARCH class of models. specify the past error being a random variable with mean and variance conditionally on the information set. t −1. ‧ 國. 學. to allow for more flexible lag structure, the GARCH(p, q) process (Generalized Autoregressive Conditional Heteroskedasticity) introduced by Bollerslev (1986) is. (. ‧. then given by:. ). y. sit. io. σ = α 0 + ∑ α u + ∑aβl iσ t2−i , q. i =1. 2 i t −i. p. n. 2 t. er. ut = Z tσ t ,. Nat. ut Ω t-1 ∼ N 0, σ t2 ,. i =1. Ch. engchi. i Un. v. where p ≥ 0, q > 0, α 0 > 0, α i ≥ 0, for i = 1,..., q, β i ≥ 0, for i = 1,..., p, ut is the. error or innovation term to denote a real valued stochastic process in normal distribution with mean zero, E {ut Ωt −1} = 0 , and conditional variance σ t2 ,. σ t2 = E {ut2 Ωt −1} , and Ωt is the information set ( σ -field) of all information available at time t. ut is decomposed to be Z tσ t , where Z t is a sequence of independent, identically distributed random variable ( i.i.d ) with zero mean and unit variance. For p = q = 0 , the process reduced to simply white noise. As p = 0, the conditional variance is specified as a linear function of past sample variances only, and the process is the so-called ARCH(q) process. The extension of GARCH process well characterized asset return dynamics by allowing lagged conditional variances to 9.

(13) enter as well. The GARCH(p, q) process is stationary if and only if. q. p. i =1. i =1. ∑ α i + ∑ βi < 1.. 2.3 GARCH-in-mean (GARCH-M) model. Engle, Lilien and Robins (1987) extended Engle's (1982) ARCH model to add the heteroskedasticity term into the mean equation and is called the ARCH-in-mean (ARCH-M) process. The standard GARCH-M process proposed by Engle and Bollerslev (1986) for stock excess return, xt , is. xt = a0 + a1σ t2 + ut ,. σ t2 = b0 + b1σ t2−1 + g1ut2−1 ,. 立. where ut is the innovation with. 政治大 E ( u ) = 0 and E ( u ) = σ t −1. 2 t. t −1. t. 2 t. .. ‧ 國. 學. Thus, GARCH-M model is an extension of GARCH model for allowing the conditional variance to be a determinant of the mean.. ‧. 2.4 Asymmetric GARCH model. y. Nat. sit. Black (1976) and Christie (1982) suggested that the decreasing in stock prices. n. al. er. io. increases the leverage and makes the stock return more volatile. Thus, the standard. i Un. v. GARCH-M model is not rich enough to depict the asymmetric effect of conditional. Ch. engchi. volatility. To incorporate the asymmetric innovation impacts on conditional volatilities, several modified GARCH model are proposed.. 2.4.1 GJR GARCH model. Based on the standard GARCH-M process proposed by Bollerslev (1986), Glosten. et al. (1993) generalized the GARCH-M model by allowing asymmetric innovations. The two asymmetric GARCH-M models they proposed are: Model 1: The positive and negative innovations have different impacts on conditional variance. xt = a0 + a1σ t2 + ut , 10.

(14) σ t2 = b0 + b1σ t2−1 + g1ut2−1 + g 2ut2−1 I t −1 , where xt is stock excess return, ut is the innovation with Et −1 ( ut ) = 0 and. Et −1 ( ut2 ) = σ t2 , and I t −1 is a dummy variable. When ut is negative, I t −1 is 1; otherwise, I t −1 is zero.. Model 2: As several seasons amplify the conditional volatility, Glosten et al. (1993) divided the conditional volatility into the fundamental volatility (exhibit no seasonal patterns) and the seasonal effect volatility.. xt = a0 + a1σ t2 + ut. 政治大 I ,. ut = (1 + λ1 Jant + λ2 Dect )ηt ,. 立. σ t2 = b0 + b1σ t2−1 + g1ηt2−1 + g 2ηt2−1. t −1. ‧ 國. 學. where ut is the innovation and ηt is the fundamental part of the excess return innovation. Thus, the innovation ut is a scale multiple of the fundamental. ‧. innovation ηt . σ t2 denotes the conditional variance of the fundamental innovation,. sit. io. n. al. er. asymmetric effect.. y. Nat. Et −1 ⎡⎣ηt2 ⎤⎦ . As mentioned in model 1, I t −1 is a dummy variable to illustrate the. 2.4.2 EGARCH model. Ch. engchi. i Un. v. The exponential general autoregressive conditional heteroskedastic (EGARCH) model was proposed by Nelson (1991). They made the ln (σ t2 ) linear in some function of time and lagged of innovations and the construction leaves no need of parameter restrictions. The conditional variance equation is defined as p. q. k =1. k =1. ln σ t2 = wt + ∑ β k ln σ t2− k + ∑ α k g ( Z t − k ) ,. (. g ( Zt ) = θ Zt + λ Zt − E ( Zt. )) ,. where σ t2 is the conditional variance, Z t ∼ i.i.d . with zero mean and unit variance, and wt , β k , and α k are real and nonstochastic parameters. Since exponentiation. 11.

(15) ensures positivity, these parameters no need for positive sign restrictions. When 0 < zt < ∞ , g ( zt ) is linear in zt with slope θ + λ ; while −∞ < zt < 0 , g ( zt ) is linear in zt with slope θ − λ . Therefore, the construction of g ( zt ) makes the conditional variance process can respond asymmetrically to positive and negative shocks on stock price. Following Black (1976) many researchers have found evidence that the change in variance tomorrow is conditionally correlated with excess returns today. The feature of σ t2 is determined by positivity or negativity of unanticipated excess returns and thus σ t2 is correlated with the algebraic sign of Z t . The term Z t − E ( Z t. 政治大 ln h is positive(negative) with the magnitude of Z larger(smaller) 立 expected value, E Z . Moreover, if λ = 0 and θ < 0 , the innovation in. ). in. EGARCH depicts this asymmetric effect. As θ = 0 and λ > 0 , The innovation in 2 t +1. t. ( ). positive when returns innovations, Z t , are negative.. ln σ t2 is. 學. ‧ 國. t. then its. ‧. 2.4.3 TGARCH model. Nat. sit. y. Zakoian (1994) proposed the Threshold GARCH (TGARCH) model which is the. er. io. counterpart of the GJR GARCH model to create the asymmetric impact response. al. n. iv n C can obtain standard hdeviation e n g c hh i U. curve. Instead of modeling conditional volatility, the conditional standard deviation is modeled.. Since. the. t. negative. values,. Rabemananjara and Zakoïan (1993) proposed a more general model with the release of non-negative constraints on parameters, but it has not wide accepted. The general TARCH model is specified as. ut = σ t Z t ,. σ t = α 0 + ∑ (α u − α u q. i =1. + + i t −i. − − i t −i. ) +∑ β σ p. j =1. j. t− j. ,. where Z t ∼ i.i.d . is independent of ut with zero mean and unit variance,. (. ). (. ). ut+−i = max ut −i , 0 , and ut−−i = min ut −i , 0 . There are some non-negative constraints. 12.

(16) on the parameters α 0 , α i+ , α i− , and β j , where i, j is from 1 to q and p , respectively. For simplification, I take TGARCH(1,1) for illustration: ut = σ t Z t ,. σ t = α 0 + α1+ ut+−1 − α1−ut−−1 + β1 σ t −1 . While α1+ ≠ α1− , the impact response curve is asymmetric, i.e. the distribution of Z t is asymmetric. For example, if α1+ < α1− , the negative innovation is more volatile than positive one.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 13. i Un. v.

(17) Chapter 3. Methodology. 3.1 Theoretical Consideration on Testing the Effect of Warrants on the Underlying Stock Return Process. Some prior studies stated that warrant introduction may change the underlying stock return process, but others left the stock return process unaffected. If the stock. 治政 return process is unaffected by warrant introduction,大 the dilution effect should be 立 taken into account for the valuation of warrants. Otherwise, if the stock return process. ‧ 國. 學. has changed following warrant introduction and already accounts for potential. ‧. dilution impact, there would be no need for specific dilution adjustment. This. y. Nat. controversy leads us to investigate the introduction of warrants and its relationship to. io. sit. the underlying stock return process.. n. al. er. When a firm issues warrants, its equity increases with cash inflow associated with. Ch. i Un. v. the warrant premium. Furthermore, if the underlying stock price is higher than the. engchi. exercise price during the life of the warrants, the firm takes more equity when warrants are exercised and new stocks are offered. Hence the capital structure of the firm has changed after warrant listing. According to proposition II of the capital structure theory (Modigliani and Miller, 1958), the risks to shareholders increase with higher leverage and the expected return on equity is positively related. In other words, the higher leveraged shareholders have better return in good times and worse returns in bad times than lower leveraged shareholders. This means that the higher the leverage is, the greater the risks; therefore, shareholders would ask for more expected return as risk premium. As the warrant introduction decreases the debt-equity ratio 14.

(18) and lowers the leverage of the firm, the company is exposed to less risk (systematic, market, and default risk). Meanwhile, the shareholders of the lower leveraged firm take less risk and get less risk premium. We argue that the underlying stock return process changes and has lower volatility and expected return after warrant introduction.. 3.2 Stock Return Processes in the GARCH-M Model. Time series models of stock prices must measure the time-varying volatilities, because financial data consistently exhibits different degrees of variation at different. 政治大. points of time. Following the constant-elasticity-of-variance model by Cox and Ross. 立. stochastic. volatility.. The. generalized. 學. with. ‧ 國. (1976) and Beckers (1980), Hull and White (1987) presented an option pricing model autoregressive. conditional. heteroskedasticity (GARCH) process of Bollerslev (1986) and its extensions. ‧. characterized asset return dynamics by different conditional volatility models. As the. y. Nat. sit. GARCH model captures the volatility clustering and evolution in financial assets,. n. al. er. io. there are many researchers, such as Heston (1993) and Duan (1995), producing. i Un. v. reasonable option pricing using the GARCH model. Many extended asymmetric. Ch. engchi. GARCH models adequately represent shocks and the leverage effect of different volatility response. Lauterbach and Schultz (1990) noted the constant variance assumption of the dilution adjusted Black-Scholes models appears to cause biases in model prices for almost all warrants over the entire sample period. Alberg et al. (2008) also concluded that the asymmetric GARCH model with fat-tailed densities improves overall estimation for measuring conditional variance. There is no reason for using the constant volatility to investigate the impact of warrant introduction on underlying stocks. While the family of GARCH models has become an important empirical method for modeling financial time series data, the GARCH model is extended to. 15.

(19) verify whether the warrant introduction has any effect. As investors require compensation for holding risky assets, the expected return of a risky asset increases with higher variance. When an asset becomes riskier, its conditional volatility increases as does expected rate of return. The relationship between mean and variance of the excess returns follows the framework proposed by Engle, Lilien, and Robins (1987), the GARCH in the mean model (GARCH-M model), to allow for impacts of conditional variances on the conditional expected returns. The dynamics of the stock return modeled by GARCH -M process of order (p,q) are. 政治大. 立. ln( Sit / Si ( t −1) ) ≡ Rit ,. ‧ 國. p. 學. Rit = r + λiσ it − (1/ 2)σ it2 + σ it Z it , q. s =1. ‧. σ it2 = ci + ∑ α i (t − s )σ i2(t − s ) + ∑ β i (t − s )ui2(t − s ) , s =1. y. sit. is the return of stock i at time t with a conditional mean. er. Rit. io. where. Nat. uit ≡ σ it Z it , with Z it Ωt −1 ~ N (0,1) ,. E ( Rit Ωt −1 ) = r + λiσ it − (1/ 2)σ it2 , and conditional deviation N (0, σ it2 ) . r represents. n. al. Ch. engchi. i Un. v. the risk-free rate of return, and λi is the price of risk of stock i. uit , or σ it Z it , is the difference between ex ante and ex post returns of stock i at time t, and Z it , conditional on information Ωt −1 at time t-1, represents a sequence of independent, identically and normally distributed random variables with zero mean and unit variance. The coefficients α i and β i should satisfy some regularity conditions to ensure unconditional volatility σ it2 is finite. Hence, the conditional expected return of stock i at time t is E ( Rit Ωt −1 ) = r + λiσ it − (1/ 2)σ it2 ,. (1). 16.

(20) and its conditional volatility is Var ( Rit Ωt −1 ) = σ it2 ,. (2). where Ωt −1 denotes the information set at time t-1. Stock return process can be represented by a conditional expected return term plus a conditional volatility term. Moreover, as Eq. (1) shows, the conditional expected return is the risk-free rate with a scaled multiple of conditional volatility to compensate for risk. Thus the GARCH-M model is extended to allow the risk premium serial correlated with the volatility process σ it2 .. 治政大 for Testing the Effect of 3.3 The GARCH-M Model with Volatility Modifications 立 Warrant Introduction ‧ 國. 學. The general market perspective is that investors demand compensation for holding. ‧. risky assets. In line with market perspective, the GARCH-M model allows the conditional expected stock return to change proportionally with serial correlations of. y. Nat. io. sit. volatilities based on the following assumption: higher volatility accompanies higher. n. al. er. expected stock return because investors demand a risk premium. Low risk premium is. Ch. i Un. v. expected for lower volatility. According to this setup of the GARCH-M model, we. engchi. only verify the volatility difference to evaluate the effect of warrant introduction on the stock return process. It is reasonable that when warrant introduction decreases the firm’s debt-equity ratio, leverage and risk exposure, the shareholders of the lower leveraged firm take less risk and receive a smaller premium. In order to test the impact of the underlying stock return process and to determine whether the introduction itself reflects some potential dilution effects, we introduce dummy variables into stock return process and modify the GARCH-M model with Gaussian innovation by incorporating the impact of warrant introduction. Following this framework, four extensions of the model are. 17.

(21) derived with different conditional volatility settings. The four modified models are divided into two groups: Model 1 and Model 2 in a one dummy variable framework, and Model 3 and Model 4 in a two dummy variables framework.. A. One Dummy Variable Framework. A dummy variable is added into the conditional volatility of stock return processes to incorporate the effect of warrant introduction. The prime form is presented in Model 1, and the extended form clarifying the ambiguity with asymmetric effect is displayed in Model 2.. 立. 政治大. Model 1: The Dilution-Adjusted GARCH-M Model. ‧ 國. 學. To investigate whether the stock return process is influenced by warrant introduction, a dummy variable is introduced into the conditional volatility of the. ‧. stock return process. Then, the conditional standard deviation is changed to. io. sit. y. Nat. σ itD = σ it (1 − δ i I it ) ,. (3). n. al. er. where σ itD is the standard deviation including the introduction of stock i at time t,. Ch. i Un. v. and σ it is the fundamental standard deviation without any volatility dilution effect. engchi. from warrant introduction. I it is the dummy or indicator variable of stock i at time t. I it is equal to 1 if observations recorded after warrant introduction. Otherwise, I it. is zero. δ i is the parameter for warrant introduction. If δ i is positive, conditional volatility is reduced after warrant introduction. As Eq. (3) shows, we assume that the standard deviation of stock return after warrant introduction is divided into two parts. One part is the fundamental standard deviation before listing, and the other part is a scaled multiple of the fundamental standard deviation after warrant introduction. The conditional standard deviation in Eq. (3) can also be shown as. 18.

(22) , if I it = 0, ⎧σ σ itD = ⎨ it ⎩σ it (1 − δ i ), if I it = 1.. Fundamental conditional volatility is defined as the function of the square of the fundamental standard deviation:. σ it2 = β 0i + β1iσ i2(t −1) + β 2i ui2(t −1) .. (4). If the stock return process already reflects some potential dilution effect during the life of warrants, δ i should be positive showing the lower volatility. Where the volatility of stock returns increases during the life of warrants, δ i should be negative. The modified GARCH(1,1)-M model incorporating a dilution-adjusted dummy can be written as. 政治大. 立. ‧ 國. ( ). Rit = r + λiσ itD − (1/ 2) σ itD. 學. ln( Sit / Si ( t −1) ) ≡ Rit ,. 2. + uitD ,. (5). ‧. y. Nat. where uitD ≡ σ itD Z it = (1 − δ i I it ) σ it Z it , with Z it Ωt −1 ~ N (0,1) . To ensure the positive. io. sit. value of conditional volatility, we need to set β 0i > 0 , β1i ≥ 0 , and β 2i ≥ 0 . The. er. sum of β1i and β 2i should be less than one to ensure unconditional variance of Rit. n. al. Ch. is finite. The conditional expected return therefore is. ( ). E ( Rit Ωt −1 ) = r + λiσ itD − (1/ 2) σ itD. 2. engchi. i Un. v. 2. = r + λiσ it (1 − δ i I it ) − (1/ 2) ⎡⎣σ it (1 − δ i I it ) ⎤⎦ , (6). and the conditional volatility is. ( ). Var ( Rit Ωt −1 ) = Vart −1 uitD = (σ itD ) 2 = (1 − δ i I it ) 2 σ it2 .. (7). Consequently, Model 1 makes an allowance for the measurement of the warrant introduction effect. In Eqs. (6) and (7), if δ i is significantly positive (negative), the conditional volatility decreases (increases) with the introduction of warrants, and therefore the expected return decreases (increases) with lower (higher) conditional volatility. 19.

(23) From Eqs. (3) and (4), 2. 2. 2. ⎡⎣σ itD /(1 − δ i I it ) ⎤⎦ = β 0i + β1i ⎡⎣σ iD( t −1) /(1 − δ i I i ( t −1) ) ⎤⎦ + β 2i ⎡⎣uiD( t −1) /(1 − δ i I i ( t −1) ) ⎤⎦. (. = ⎡⎣1/(1 − δ i I i ( t −1) ) 2 ⎤⎦ ⎡(1 − δ i I i ( t −1) ) 2 β 0i + β1i σ iD( t −1) ⎢⎣. ). 2. (. + β 2i u. D i ( t −1). ). 2. ⎤. ⎥⎦. (8). Rearranging (8), the following specification of the conditional volatility function was obtained:. (σ ). D 2 it. 2 2 2 = ⎡⎣(1 − δ i I it ) /(1 − δ i I i ( t −1) ) ⎤⎦ ⎡(1 − δ i I i ( t −1) ) 2 β 0i + β1i (σ iD( t −1) ) + β 2i ( uiD( t −1) ) ⎤ . (9) ⎣⎢ ⎦⎥. It can also be expressed as. (σ ). 立. 政治大. (10). 學. ‧ 國. D 2 it. ⎧ β + β (σ D ) 2 + β ( u D ) 2 , if I it = I i (t −1) = 0, i ( t −1) i ( t −1) 1i 2i ⎪ 0i 2 2 ⎪ = ⎨(1 − δ i ) 2 ⎡ β 0i + β1i (σ iD(t −1) ) + β 2i ( uiD( t −1) ) ⎤ , if I it = 1, I i (t −1) = 0, ⎢⎣ ⎥⎦ ⎪ ⎪ ⎡(1 − δ ) 2 β ⎤ + β σ D 2 + β u D 2 , if I = I i it i ( t −1) = 1. 0i ⎦ 1i ( i ( t −1) ) 2 i ( i ( t −1) ) ⎩⎣. Eq. (10) shows that the conditional volatility after warrant listing (σ itD ) would be a 2. ‧. scaled multiple of β 0i + β1i (σ iD( t −1) ) + β 2i ( uiD( t −1) ) , or of the constant term β 0i . If δ i 2. sit. y. Nat. 2. n. al. er. io. is significantly positive (negative), the conditional volatility will decrease (increase). i Un. v. with the warrant listing. Thus Model 1 depicts the changes in conditional volatility of warrant introduction.. Ch. engchi. Model 2: The Asymmetric Dilution-Adjusted GARCH-M Model. As the stock price decreases from negative shocks, the equity value of a firm gets smaller relative to its debt, and its stocks become riskier with the higher financial leverage. This asymmetric phenomenon is referred to as the leverage effect. It is important to distinguish the volatility change due to the dilution effect from this asymmetric leverage effect in order to avoid ambiguity. Corresponding with Engle and Ng (1993), Glosten et al. (1993), and Christoffersen and Jacobs (2004), the 20.

(24) conditional variance equation in Model 1 (the Dilution-Adjusted GARCH Model) is modified to contain the asymmetric effect as follows:. ( ). Rit = r + λiσ itD − (1/ 2) σ itD. 2. + uitD ,. uitD ≡ σ itD Z it = (1 − δ i I it ) σ it Z it , with Z it Ωt −1 ~ N (0,1) ,. σ itD = (1 − δ i I it )σ it ,. (. ). 2. σ it2 = β 0i + β1iσ i2(t −1) + β 2i ui (t −1) − li ui (t −1) .. (11). The parameters β1i , β 2i , and li should satisfy some regularity conditions to ensure. 政治大 l >0, negative return shocks increase volatility more than positive shocks. Therefore, 立. that the unconditional volatility of stock return process is finite. In Eq. (11), where i. and warrant effects is. (σ ) D it. 2. (. 2 = ⎡⎣(1 − δ i I it ) /(1 − δ i I i ( t −1) ) ⎤⎦ ⎡(1 − δ i I i (t −1) ) 2 β 0i + β1i σ iD( t −1) ⎢⎣. ). (. ). 2 + β 2i uiD( t −1) − li uiD( t −1) ⎤ .(12) ⎥⎦. Nat. sit. y. 2. ‧. ‧ 國. 學. the conditional volatility function for stock return process accounting for asymmetric. n. al. er. io. B. Two Dummy Variables Framework. Ch. i Un. v. In general, the dilution effect is due to the possible exercise of warrants; higher. engchi. stock prices lead to a greater possibility of exercising them. When stock prices are high enough, the dilution effects may already partly or entirely be reflected in the conditional volatility for representing the possibility of wealth transferring from stock holders to warrant holders. This argument is supported by Hauser and Lauterbach (1997). Hence, a threshold dummy variable is added to judge if the stock price is higher than the exercise price, the two dummy variable framework makes the conditional volatility unchanged after warrant introduction until stock price exceeds the exercise price. Similar to the one dummy variable framework, the original form is presented in 21.

(25) Model 3. Model 4 shows the extended form clarifying the ambiguity of asymmetric effect.. Model 3: The Dilution-Adjusted GARCH-M Model with a Threshold for Exercise Prices. The threshold dummy variable helps to identify whether the relationship between stock price and exercise price affects the volatility. Since Eq. (10) indicates conditional volatility scales down after warrant introduction, for simplicity we redefine the conditional volatility function with warrant introduction as. σ = (1 − δ i I i (t −1) ) ( β 0i + β1iσ 2 it. 2. 2 i ( t −1). 立. 政治大 +β u ). 2 2 i i ( t −1). ‧ 國. 學. Then, by adding a threshold dummy variable Di ( t −1) to identify the relation between exercise price and stock price, the conditional volatility can be modified as. ‧. ln( Sit / Si ( t −1) ) ≡ Rit ,. io. uit ≡ σ it Z it , with Z it Ωt −1 ~ N (0,1) ,. n. σ it2 = (1 − δ i I i (t −1) Di ( t −1) ) 2 ( β 0i a+ lβ1iσ i2( t −1) + β 2i ui2( t −1) ) ,. Ch. engchi. er. sit. y. Nat. Rit = r + λiσ it − (1/ 2)σ it2 + uit ,. i Un. v. (13). where Di ( t −1) is the threshold dummy variable of stock i at time t-1, when stock price is higher than exercise price, Si (t −1) > k , Di ( t −1) is 1. Otherwise, Di ( t −1) is 0. Regular conditions should satisfied by δ i , β 0i , β 1i , and β 2i to ensure conditional volatility is always positive and unconditional volatility is finite. Eq. (13) shows that the warrant introduction leaves the conditional volatility unchanged until the firm’s stock price is higher than the exercise price. We can also express it as ⎧ β 0i + β1iσ i2(t −1) + β 2i ui2( t −1) , if I i ( t −1) = 0, ⎪ ⎪ σ it2 = ⎨ β 0i + β1iσ i2(t −1) + β 2i ui2( t −1) , if I i ( t −1) = 1, Di ( t −1) = 0, ⎪ 2 2 2 ⎪⎩(1 − δ i ) ( β 0i + β1iσ i ( t −1) + β 2i ui ( t −1) ), if I i ( t −1) = 1, Di ( t −1) = 1, 22.

(26) The conditional expected return of stock i at time t is E ( Rit Ωt −1 ) = r + λiσ it − (1/ 2)σ it2 , and its conditional volatility is. ( ). Vart −1 ( Rit Ωt −1 ) = Vart −1 uit = σ it2 = (1 − δ i I i ( t −1) Di ( t −1) ) 2 ( β 0i + β1iσ i2(t −1) + β 2i ui2( t −1) ) .. Model 4: The Asymmetric Dilution-Adjusted GARCH-M Model with a Threshold for Exercise Prices. In order to distinguish the dilution effect of warrant introduction from the. 政治大. asymmetric leverage effect, as in Model 2, the conditional volatility function in Model 3 is transformed to. 立. ‧ 國. 學. σ it2 = (1 − δ i I i (t −1) Di (t −1) ) 2 ( β 0i + β1iσ i2( t −1) + β 2i ( ui ( t −1) − li ui ( t −1) ) 2 ) .. (14). ‧. With li >0 , negative shocks increase volatility more than positive shocks and li is the parameter for asymmetric effect.. n. er. io. sit. y. Nat. al. Ch. engchi. 23. i Un. v.

(27) Chapter 4. Data. Data listed on Hong Kong Exchanges and Clearing Limited (HKEx) is used for demonstration. Hong Kong is one of the world’s three most actively traded warrant. 政治大 In general, equity warrants have a long expiration period. In the 立. markets. Top six exchanges represent almost 90% of the aggregate warrant turnover around the world.. 4. interest of observing total trading period of a warrant, unexpired warrants are. ‧ 國. 學. excluded; empirical analysis utilizes expired warrant data issued from Jan. 1, 2001, to. ‧. Dec. 30, 2004. Total observations of each underlying stock return includes its entire. y. Nat. warrant trading life, the sample period, and the same amount of time before warrant. er. io. sit. introduction. The time prior warrant introduction is referred to as the control period. Observations for each stock include the sample and control periods.. n. al. Ch. i Un. v. All official daily closing prices of stocks after capital action adjustments are. engchi. obtained from the Datastream. The data for exercise provisions and other descriptions of the warrants are collected from the annual Fact Book published by HKEx. There were 82 new equity warrants listed on HKEx during 2001-2004. An equal amount of time as warrant lifetime is required for the control group prior its introduction. A. 4. In 2005, Hong Kong Exchanges and Clearing Limited (HKEx) published a brief comparison of the Hong Kong warrant market with oversea counterparts in terms of the number issued and turnover. It showed that Hong Kong was ranked number two in terms of annual turnover of listed warrants among world stock exchanges in 2003, just behind Deutsche Börse (DB) of Germany. Clarification of the double counts problem in Germany, Hong Kong became the world’s most actively traded warrant market by turnover value in 2003. 24.

(28) considerable number of stocks are excluded, which include those with another warrant listing during the observation, or those with warrants shortly after an Initial Public Offering (IPO) making the control period too short for comparison.5 To avoid the complication of different exchange rates, the warrants traded in currencies other than Hong Kong dollars are also excluded. Lastly, the study also excludes a few coding error warrants or stocks no longer in the public equity market. After elimination, the final sample includes 36 warrants issued from 2001 to 2004 with 37748 observations. Table Ⅰ summarizes the 36 warrants sorted by listing date. Since almost all. 政治大. subscription periods, except a warrant issued by Regal Hotels Intl. HDG. in 2004,. 立. start prior to the listing day of warrants, the starting date of subscription periods is. ‧ 國. 學. considered the same as the date warrants are introduced. The warrants in Table Ⅰ different. lengths. of. time. and. range. from. deep-in-the-money. ‧. cover. to. deep-out-of-the-money. Exercise prices of warrants are drawn and plotted with daily. y. Nat. sit. returns of the underlying stocks in Appendix A to show the basic patterns of stock. n. al. er. io. returns and to see the relationship between stock and exercise prices.6 As indicated in. Ch. i Un. v. Appendix A, stock returns display smaller volatilities after warrant introduction.. 5. engchi. Riche Multi-Media HDG. was the only exception, because it went public on Feb. 15, 2000. The control period is briefly unavailable from 19990617-2000214.. 6. Each stock is plotted and it is found that most stocks appear to have smaller volatilities after warrant introduction. Because of page limitation it is only possible to show the stock return process for some sample companies in Appendix A. 25.

(29) Table Ⅰ Summary of the 36 warrants issued from Jan. 1, 2001, to Dec. 31, 2004 Subscription period. Date in Control Group. Total Date in Sample. 20010112-20030111 20010209-20020208 20010611-20030610 20010614-20040617 20010619-20060619 20010703-20030630 20010711-20040630 20010903-20031231 20011102-20041101 20020226-20040226 20020301-20030829 20020410-20040409 20020507-20030506 20020529-20050430 20020531-20031129 20020617-20050616 20020621-20040630 20020709-20030708 20020723-20030723 20020829-20031231 20020902-20050901 20020926-20040325 20030805-20050804 20030903-20060302 20031013-20050412 20031208-20041207 20031205-20041206 20031202-20051202 20040114-20070113 20040524-20050523 20040602-20060531 20040528-20070531 20040618-20050623 20040628-20050627 20050202-20070726 20041104-20061103. 19990112-20010111 20000209-20010208 19990611-20010610 19980614-20010613 19960619-20010618 19990701-20010702 19980713-20010710 19990501-20010902 19981102-20011101 20000226-20020225 20000901-20020228 20000410-20020409 20010507-20020506 19990627-20020528 20001129-20020530 20000214-20020616 20000621-20020620 20010709-20020708 20010723-20020722 20010401-20020828 19990902-20020901 20000926-20020925 20010805-20030804 20010302-20030902 20020413-20031012 20021208-20031207 20021205-20031204 20011202-20031201 20010114-20070113 20030524-20040523 20020603-2040601 20010528-20040527 20030618-20040617 20030628-20040627 20020726-20050201 20021104-20041103. 19990112-20030111 20000209-20020208 19990611-20030610 19980614-20040617 19960619-20060619 19990701-20030630 19980713-20040630 19990501-20031231 19981102-20041101 20000226-20040226 20000901-20030829 20000410-20040409 20010507-20030506 19990627-20050430 20001129-20031129 20000214-20050616 20000621-20040630 20010709-20030708 20010723-20030723 20010401-20031231 19990902-20050901 20000926-20040325 20010805-20050804 20010302-20060302 20020413-20050412 20021208-20041207 20021205-20041206 20011202-20051202 20010114-20070113 20030524-20050523 20020603-20060531 20010528-20070531 20030618-20050623 20030628-20050627 20020726-20070726 20021104-20061103. 政治大. engchi. 26. sit. y. ‧. Ch. 學. n. al. er. 立. io. 20010115 20010214 20010615 20010618 20010622 20010703 20010711 20010903 20011102 20020226 20020305 20020412 20020507 20020603 20020605 20020620 20020625 20020709 20020725 20020829 20020902 20020926 20030807 20030905 20031016 20031210 20031205 20031205 20040114 20040524 20040602 20040601 20040624 20040630 20040804 20041104. Introduction date 20010112 20010209 20010611 20010614 20010619 20010703 20010711 20010903 20011102 20020226 20020301 20020410 20020507 20020529 20020531 20020617 20020621 20020709 20020723 20020829 20020902 20020926 20030805 20030903 20031013 20031208 20031205 20031202 20040114 20040524 20040602 20040528 20040618 20040628 20050202 20041104. Nat. (1) SUN HUNG KAI & CO. (2) GOLD PEAK INDS. (3) COSMOS MACHINERY ENTS. (4) LUKS GROUP (5) LEI SHING HONG (6) CHINA TRAVEL INTL.INVS. (7) KITH HOLDINGS (8) KINGBOARD CHEMICALS HDG. (9) CITY TELECOM (10) HAIER ELECTRONICS GP. (11) PAUL Y ENGR.GP. (12) ASIA ALUMINUM HOLDINGS (13) FAR EAST PHARM.TECH. (14) HOP HING HOLDINGS (15) SINOLINK WORLDWIDE HDG. (16) RICHE MULTI-MEDIA HDG. (17) HARMONY ASSET (18) PREMIUM LAND (19) SOUTH CHINA HDG. (20) CHINA STRATEGIC HDG. (21) ALCO HOLDINGS (22) PACIFIC ANDES INTL.HDG. (23) PEACE MARK HDG. (24) SOUNDWILL HOLDINGS (25) HERITAGE INTL.HDG. (26) EFORCE HOLDINGS (27) ALLIED PROPERTIES (28) KENFAIR INTL.HDG. (29) QUALITY HLTHCR.ASIA (30) PLAYMATES HOLDINGS (31) CHINA TRAVEL INTL.INVS. (32) GLOBAL BIO-CHEM TECH.GP. (33) U-RIGHT INTL.HDG. (34) RONTEX INTL.HDG. (35) REGAL HOTELS INTL.HDG. (36) MAN YUE INTL.HDG.. Listing date. ‧ 國. Sample Sequence Number & Company. i n U. v. Number of Valid Observations 1043 522 1042 1568 2608 1042 1557 1217 1565 1043 780 1044 521 1524 782 1392 1050 521 522 717 1565 912 1043 1304 781 521 522 952 1564 520 1042 1568 526 520 1304 1044. Exercise price per unit (HK$) 3 2.2 0.4 0.9 3 1.22 2.2 5.8 0.11 0.52 0.4 0.77 2.62 0.27 1 3.6 0.08 0.22 0.42 0.16 0.98 0.85 0.65 2 0.17 0.28 2.5 0.69 2.5 1.42 1.508 9.8 0.2 0.102 0.25 0.48.

(30) Chapter 5. Empirical Results. Using maximum likelihood estimation procedure, I run the regression respectively for each stock in each model. In this chapter, I report the empirical results for the four. 政治大 extended GARCH-M model 立incorporating a dummy variable for warrant introduction, dilution-adjusted GARCH-M models. First, I present the estimates of Model 1, an. ‧ 國. 學. to determine the estimated relationship between volatility and warrant introduction. Second, in order to distinguish the volatility fluctuation of dilution effect from. ‧. asymmetric leverage effect, an asymmetric effect variable is added into Model 2.. sit. y. Nat. Third, different from previous models, I consider the level of stock prices by. er. io. including a threshold dummy variable in Model 3. This framework separates the. al. dilution effect from other introduction effects and makes the conditional volatility. n. iv n C U than exercise prices. Finally, unchanged after introduction until stock h e nprices h ihigher g care by adding an asymmetric effect variable in Model 3, Model 4 is obtained.. 5.1 Model 1: The Dilution-Adjusted GARCH-M Model. Table Ⅱ shows the empirical results of Model 1 by maximizing its log-likelihood function. All parameters are estimated simultaneously on the daily returns of total observations for each sample. The full sample period is from Jan. 1, 2001, to Dec. 30, 2004. Assuming the risk-free rate of return, r, is a constant 5% annual rate as shown in Christoffersen and Jacobs (2004) and the daily return rate is 0.000137. The second to the sixth column of TableⅡ provide the parameter estimates for r, λ, β 0 , β1 , and 27.

(31) β 2 as in standard GARCH-M models. Note that the t-statistics of the parameters β1 and β 2 are strongly significant in most samples, indicating the volatility clustering in stock returns and justifying the suitability of GARCH models. The second to the last column in Table Ⅱ provide the estimates for parameter δ , the introduction effect dummy variable. This table shows that out of the 36 stocks, δ of 30 stocks is estimated to be significant. Thus the dummy variable identifies some of the introduction effect. The positive (negative) estimate of parameter δ indicates that stock return volatility decreases (increases) with warrant introduction. Empirical results of Model. 政治大. 1 are summarized in Table Ⅲ. In the full sample, δ is significantly different from. 立. zero of 30 samples at the 5% level with a high rejection rate of 0.8333. Considering. ‧ 國. 學. the dilution effect of volatility, only significant rejection with positive δ was selected. There is still a high rejection rate of 0.75. This shows that stock returns changed. ‧. significantly in the volatility following warrant introduction, and in most cases the. Nat. sit. y. volatility is diluted. Of note is that two of the three significantly negative δ are. n. al. er. io. from the warrants issued with deep-in-the-money.7 This is reasonable, considering. i Un. v. the dilution effect or wealth transferring from stock holders to warrant holders already. Ch. engchi. reflected in stock prices after the announcement of deep-in-the-money warrant listing. Two distinctive samples are exclude evaluating the rejection rate to 0.7941 in the last column of Table Ⅲ.. 7. The two deep-in-the-money issued warrants are Harmony Asset and Heritage Intl. HDG. When the warrants were introduced, the stock price was 102 times of the exercise price for the former and 45 times for the latter. 28.

(32) Table Ⅱ Maximum Likelihood Estimates of Model 1 (The Dilution-Adjusted GARCH-M Model). ( ). ln( Sit / Si ( t −1) ) ≡ Rit = r + λiσ itD − (1/ 2) σ itD. 2. + uitD ;. σ itD = (1 − δ i I it )σ it , after warrant introduction I it = 1 , and otherwise Iit = 0 ; uitD ≡ σ itD Z it = (1 − δ i I it ) σ it Z it , and Zit Ωt -1 ~ N (0,1) ; σ it2 = β 0i + β1iσ i2( t −1) + β 2i ui2( t −1) ; Parameter estimates on daily return of the 36 samples are obtained by maximizing the log-likelihood function of Model 1. t-statistics are reported below with each estimate in parenthesis. For the risk free rate, r, we assume a constant 5% annual rate as Christoffersen and Jacobs (2004) and a daily return rate of 0.000137. 0.0168 (0.6339) -0.0211 (-0.5003) 0.0093 (0.2949) 0.0046 (0.1747) -0.0223 (-1.1790) 0.0269 (0.9123) 0.0270 (1.0873) 0.0698 (2.3294) 0.0039 (0.1623) -0.0045 (-0.1572) -0.0239 (-0.6848) 0.0299 (1.0127) 0.0816 (1.8943) -0.0024 (-0.0990) 0.0748 (2.4722) 0.0039 (0.1520) -0.0186 (-0.6176) -0.0074 (-0.1680) -0.0507 (-1.1136) -0.0062 (-0.1591) 0.0364 (1.5663). 0.000137. (4) LUKS GROUP. 立. 0.000137. (5) LEI SHING HONG. (8) KINGBOARD CHEMICALS HDG.. 0.000137 0.000137 0.000137. (14) HOP HING HOLDINGS (15) SINOLINK WORLDWIDE HDG. (16) RICHE MULTI-MEDIA HDG. (17) HARMONY ASSET (18) PREMIUM LAND (19) SOUTH CHINA HDG. (20) CHINA STRATEGIC HDG.. al. 0.000137. n. (13) FAR EAST PHARM.TECH.. (21) ALCO HOLDINGS. 0.000137. io. (12) ASIA ALUMINUM HOLDINGS. 0.000137. 0.000137 0.000137 0.000137 0.000137 0.000137 0.000137 0.000137 0.000137. 0.0001 0.8342 (3.6755) (36.63169) 0.0003 0.4503 (2.7038) (3.1437) 0.0018 0.0824 (4.8435) (0.5070) 0.0002 0.7732 (5.3455) (24.9845) 0.0000 0.8606 (10.0033) (72.1045) 0.0001 0.8861 (4.3261) (55.4358) 0.0000 0.7837 (7.4119) (58.4544) 0.0002 0.6953 (6.8508) (18.9458) 0.0009 0.4628 (5.9158) (7.7147) 0.0003 0.7813 (6.5184) (43.9579) 0.0004 0.6809 (2.2438) (16.1904) 0.0002 0.6137 (5.2357) (13.7912) 0.0001 0.8128 (2.8863) (16.5678) 0.0007 0.4697 (8.6732) (8.4908) 0.0012 0.0477 (7.4394) (0.5470) 0.0009 0.4013 (7.1140) (7.8823) 0.0010 0.0841 (6.3178) (0.7505) 0.0001 0.9273 (3.2188) (57.0207) 0.0014 0.1294 (4.0017) (0.7338) 0.0013 0.5281 (3.5240) (4.4298) 0.0006 0.1552 (5.7972) (1.3493). 政治大. Ch. n engchi U. 29. δ. Log-likelihood. 0.1333 (6.4567) 0.2117 (3.8583) 0.1216 (3.2339) 0.1396 (6.4198) 0.0442 (6.1469) 0.0757 (5.8586) 0.1178 (7.7641) 0.1487 (4.0174) 0.3221 (7.8890) 0.2187 (8.6624) 0.3191 (2.8533) 0.2447 (6.5021) 0.1129 (3.3450) 0.1993 (4.7449) 0.2148 (4.1506) 0.4375 (7.2431) 0.1686 (4.0526) 0.0621 (4.0048) 0.0883 (2.0028) 0.0811 (3.1474) 0.2219 (6.7015). 0.1458 (1.9713) 0.5127 (12.2628) -0.0173 (-0.3606) 0.5257 (17.3588) -0.0080 (-0.2710) 0.4193 (8.1958) 0.0871 (6.5784) 0.0774 (2.4755) 0.4585 (16.1411) 0.4750 (8.1851) 0.4597 (3.8207) 0.1555 (2.4854) 0.3096 (4.1645) -0.0166 (-0.4132) 0.2496 (4.9954) 0.5930 (26.7244) -0.9801 (-10.3083) 0.3158 (3.5635) 0.0113 (0.1778) 0.2703 (6.2371) 0.2890 (9.3450). 2100.00. ‧. (11) (PAUL Y ENGR.GP.. 0.000137. Nat. (10) HAIER ELECTRONICS GP.. 0.000137. β2. β1. 學. ‧ 國. (6) CHINA TRAVEL INTL.INVS.. β0. y. 0.000137. (3) COSMOS MACHINERY ENTS.. (9) CITY TELECOM. 0.000137 0.000137. (2) GOLD PEAK INDS.. (7) KITH HOLDINGS. λ. sit. (1) SUN HUNG KAI & CO.. r. er. Sample Sequence Number & Company. iv. 1316.51 1709.69 3393.68 6794.03 2195.42 4012.59 2591.97 2795.17 1628.98 1354.26 2225.89 1079.74 2757.61 1518.49 2893.33 1648.27 909.28 928.72 1145.77 3561.59.

(33) Table Ⅱ--Continued Sample Sequence Number & Company. r. (22) PACIFIC ANDES INTL.HDG. 0.000137 0.000137. (23) PEACE MARK HDG. (24) SOUNDWILL HOLDINGS. 0.000137. (25) HERITAGE INTL.HDG.. 0.000137. (26) EFORCE HOLDINGS. 0.000137. (27) ALLIED PROPERTIES. 0.000137. (28) KENFAIR INTL.HDG.. 0.000137. (29) QUALITY HLTHCR.ASIA. 0.000137. (30) PLAYMATES HOLDINGS. 0.000137. (31) CHINA TRAVEL INTL.INVS. (32) GLOBAL BIO-CHEM TECH.GP.. 0.000137. 立. 0.000137. (34) RONTEX INTL.HDG.. 0.000137. β2. δ. Log-likelihood. 0.0926 (2.8343) 0.0693 (2.2703) 0.0015 (0.0542) -0.0427 (-1.2863) 0.0151 (0.3640) 0.0098 (0.2252) 0.1619 (4.4383). 0.0003 (2.1831) 0.0002 (2.5029) 0.0001 (2.7824) 0.0013 (6.3435) 0.0000 (5.4483) 0.0000 (3.2980) 0.0003 (4.3350). 0.6889 (5.9560) 0.8196 (17.1138) 0.8272 (35.3832) 0.4093 (4.7453) 0.9342 (306.3754) 0.8253 (26.6715) 0.5375 (6.4191). 0.1186 (3.2141) 0.1013 (4.1854) 0.1458 (7.7712) 0.1732 (4.6709) 0.0400 (5.2904) 0.1385 (4.7910) 0.1679 (4.1766). 0.3576 (8.4637) 0.4441 (10.3882) 0.2208 (2.9173) -0.3116 (-4.2578) -0.7632 (-3.9230) -0.0175 (-0.1351) 0.4181 (11.5656). 1914.96. -0.0071 (-0.2790) 0.0733 (1.6716) 0.0171 (0.5627) 0.0337 (1.3606) 0.0085 (0.2223) 0.1187 (2.6326) 0.0271 (0.9246) 0.0842 (3.0419). 0.0005 (3.8410) 0.0002 (2.3254) 0.0002 (3.4921) 0.0003 (4.5613) 0.0002 (1.7952) 0.0004 (3.3083) 0.0000 (2.2226) 0.0013 (8.6249). 0.5702 (5.5810) 0.7472 (8.5180) 0.4820 (3.9419) 0.6346 (10.4745) 0.7337 (5.6383) 0.7429 (12.3208) 0.8685 (31.6399) 0.1285 (1.6670). 0.1056 (4.0888) 0.1364 (2.8212) 0.1565 (3.7418) 0.1576 (5.3635) 0.0001 (0.0044) 0.1317 (3.6998) 0.1006 (5.7975) 0.2603 (4.8883). 0.4574 (17.5870) 0.1199 (1.3420) 0.1142 (2.3113) 0.1052 (2.4446) 0.1992 (2.2694) 0.3412 (5.9941) 0.1514 (2.1506) 0.1716 (4.1396). 政治大. 2161.25 2324.41 1086.87 975.72 1301.84 2256.12. 3304.88 1032.70 2560.85 3142.89 1146.09 935.89 2776.86 1897.10. sit. y. Nat. Table Ⅲ. 0.000137. β1. ‧. (36) MAN YUE INTL.HDG.. 0.000137. β0. 學. ‧ 國. (33) U-RIGHT INTL.HDG.. (35) REGAL HOTELS INTL.HDG.. 0.000137. λ. n. al. er. io. Summary Statistics for the Parameters of Introduction Dummy on Model 1 This table shows the number and percentage of stocks with significant changes in volatility after warrant introduction on model 1. Rejections of the null hypothesis are reported at the 5% level. Summary A reports the number and percentage of stocks with significant changes in volatility after warrant introduction. Then, we only select the rejections with positive parameter, i.e. their volatility is significantly diluted, in summary B. The second column shows the results of the total samples, while the sub-samples without deep-in-the-money issued warrants are reported in the last column. Samples without deep-in-the-money Sample Full sample issued warrants Number of samples 36 34 H 0 : δ =0. Ch. engchi. Summary A Number of rejections Rate of rejection Summary B Number of rejections with positive parameters Rate of rejection with positive parameters. 30. i Un. v. 30 0.8333. 28 0.8235. 27 0.75. 27 0.7941.

(34) 5.2 Model 2: The Asymmetric Dilution-Adjusted GARCH-M Model. In Model 2, the asymmetric effect was incorporated by estimating the parameter l to allow different positive and negative shocks to act on the conditional volatility. If the parameter l is not significantly different from zero, the asymmetric effect is necessarily zero and Model 2 is reduced to Model 1. Table Ⅳ shows that δ. is still significantly different from zero for most. samples—although the asymmetric effect is included as an explanatory variable in conditional volatility. Therefore, the statistically significant changes of following. 政治大. warrant introduction are not referred to as an omission of asymmetric phenomenon.. 立. Table Ⅴ is a summary of Table Ⅳ . The rejection rate of the parameter for. ‧ 國. 學. asymmetric effect l, in the full sample is 0.5556, for approximately half of the samples. If we test whether the negative shocks have a larger effect on the volatility. ‧. than positive shocks, l > 0, the rejection rate is only 0.3611. The result suggests the. y. Nat. sit. impact of parameter l on asymmetric effect is unstable. Therefore, the results of. al. n. volatility.. er. io. Model 2 indicate that the samples are not significantly asymmetric in the conditional. Ch. engchi. 31. i Un. v.

(35) Table Ⅳ Maximum Likelihood Estimates of Model 2 (The Asymmetric Dilution-Adjusted GARCH-M Model). ( ). ln( Sit / Si ( t −1) ) ≡ Rit = r + λiσ itD − (1/ 2) σ itD. 2. + uitD ;. σ itD = (1 − δ i I it )σ it , after warrant introduction I it = 1 , and otherwise Iit = 0 ; uitD ≡ σ itD Z it = (1 − δ i I it ) σ it Z it , and Zit Ωt -1 ~ N (0,1) ;. (. ). 2. σ it2 = β0i + β1iσ i2(t −1) + β 2i uiD(t −1) − li uiD(t −1) ; with li > 0, negative return shocks increase volatility more than positive shocks, thus including asymmetric effects. Parameter estimates of daily return of the 36 samples are obtained by maximizing the log-likelihood function of Model 2. t-statistics are reported below with each estimate in parentheses. For the risk free rate, r, we assume a constant 5% annual rate as in Christoffersen and Jacobs (2004) and a daily return rate of 0.000137. Sample Sequence Number & Company. β2. δ. l. Log-likelihood. 0.000137. -0.0118 (-0.3638) -0.0310 (-0.7463) 0.0100 (0.3235) 0.0237 (0.9866) -0.0380 (-1.9581) 0.0075 (0.2231) 0.0235 (1.0136) 0.0629 (2.5414) 0.0240 (0.9376) 0.0038 (0.1332) -0.0232 (-0.3131) 0.0078 (0.2509) 0.0589 (1.3782) -0.0073 (-0.2868) 0.0713 (1.9183) 0.0341 (1.2950) -0.0194 (-0.6361) -0.0386 (-0.9197) -0.0571 (-1.3364) -0.0260 (-0.7058) 0.0582 (2.2938) 0.0945 (2.8348) 0.0628 (2.0667). 0.0001 (3.1632) 0.0003 (2.2942) 0.0018 (4.1179) 0.0002 (5.0937) 0.0000 (8.8693) 0.0001 (4.1834) 0.0000 (5.4910) 0.0002 (10.9709) 0.0010 (5.2717) 0.0003 (6.4329) 0.0004 (1.8018) 0.0002 (5.1629) 0.0002 (3.0398) 0.0007 (9.4466) 0.0013 (7.7146) 0.0006 (6.2602) 0.0000 (3.9277) 0.0001 (3.1520) 0.0016 (2.2646) 0.0013 (4.4358) 0.0004 (3.6289) 0.0003 (2.2505) 0.0001 (2.4913). 0.8978 (44.1669) 0.5253 (3.1098) 0.0890 (0.4586) 0.7663 (21.8356) 0.8576 (57.6971) 0.8947 (58.4510) 0.7845 (29.6822) 0.6496 (79.3989) 0.4325 (6.1907) 0.7905 (48.4760) 0.6816 (18.2798) 0.5990 (11.9818) 0.8182 (14.0649) 0.4292 (8.4153) 0.0379 (0.4608) 0.4411 (8.1004) 0.9501 (99.5937) 0.9330 (60.3619) 0.0001 (0.0003) 0.5378 (5.6523) 0.4340 (3.5086) 0.6872 (6.1373) 0.8241 (18.3335). 0.0901 (5.3298) 0.1659 (2.6411) 0.1205 (3.2851) 0.1355 (6.3657) 0.0204 (3.7004) 0.0739 (5.3719) 0.1182 (6.7298) 0.1592 (6.6040) 0.3064 (7.0230) 0.1889 (8.7920) 0.3147 (3.4939) 0.2153 (5.3175) 0.0658 (1.6985) 0.2335 (4.5820) 0.2192 (4.8432) 0.4127 (6.6998) 0.0308 (4.7926) 0.0547 (3.8231) 0.0381 (1.5491) 0.0884 (2.9163) 0.1512 (4.3946) 0.1188 (3.1139) 0.1005 (4.1880). 0.4145 (4.0584) 0.5227 (12.3852) -0.0174 (-0.3452) 0.5226 (17.1389) -0.0423 (-1.3469) 0.4594 (8.0349) 0.0898 (5.0346) 0.0898 (2.6884) 0.4655 (15.6944) 0.4494 (7.7943) 0.4641 (3.2069) 0.1552 (2.7753) 0.3421 (5.7964) -0.0283 (-0.6733) 0.2534 (5.5678) 0.5326 (21.2252) -0.5308 (-3.9046) 0.3799 (4.6202) 0.0413 (0.4351) 0.2769 (6.5869) 0.2877 (8.9913) 0.3592 (8.5117) 0.4401 (9.8544). 0.3669 (3.7944) 0.1645 (1.1653) -0.0220 (-0.1769) -0.2282 (-3.4573) 1.0000 (5.4447) 0.2367 (2.8562) 0.0371 (0.7067) 0.0710 (11.8745) -0.1628 (-2.6376) -0.0532 (-1.4159) -0.0058 (-0.0294) 0.1899 (2.3548) 0.5271 (1.5718) 0.1350 (1.5663) 0.0519 (0.5071) -0.1664 (-2.7267) -0.0785 (-0.7970) 0.4172 (2.8390) 1.0000 (2.2223) 0.3854 (3.3652) -0.3482 (-3.3369) -0.0275 (-0.2903) 0.1314 (1.4606). 2110.40. 0.000137. Nat. (10) HAIER ELECTRONICS GP.. 0.000137. (13) FAR EAST PHARM.TECH. (14) HOP HING HOLDINGS (15) SINOLINK WORLDWIDE HDG. (16) RICHE MULTI-MEDIA HDG. (17) HARMONY ASSET (18) PREMIUM LAND (19) SOUTH CHINA HDG. (20) CHINA STRATEGIC HDG. (21) ALCO HOLDINGS (22) PACIFIC ANDES INTL.HDG.. 0.000137. al. n. (12) ASIA ALUMINUM HOLDINGS. io. (11) PAUL Y ENGR.GP.. 0.000137. 0.000137 0.000137 0.000137 0.000137 0.000137 0.000137 0.000137 0.000137 0.000137 0.000137 0.000137. 0.000137. Ch. engchi. 32. y. 0.000137. (8) KINGBOARD CHEMICALS HDG. (9) CITY TELECOM. 0.000137. ‧. ‧ 國. (6) CHINA TRAVEL INTL.INVS.. 學. 0.000137. 政治大. sit. 立. 0.000137. (4) LUKS GROUP. (23) PEACE MARK HDG.. β1. 0.000137. (3) COSMOS MACHINERY ENTS.. (7) KITH HOLDINGS. β0. 0.000137. (2) GOLD PEAK INDS.. (5) LEI SHING HONG. λ. er. (1) SUN HUNG KAI & CO.. r. i Un. v. 1317.17 1709.71 3399.97 6839.72 2199.80 4012.69 2592.46 2799.63 1629.33 1354.27 2229.31 1084.42 2758.79 1518.60 2857.97 1676.67 914.49 931.92 1150.30 3569.01 1914.99 2162.40.

(36) Table Ⅳ--Continued Sample Sequence Number & Company (24) SOUNDWILL HOLDINGS. 0.000137. (28) KENFAIR INTL.HDG. (29) QUALITY HLTHCR.ASIA (30) PLAYMATES HOLDINGS. (35) REGAL HOTELS INTL.HDG. (36) MAN YUE INTL.HDG.. 0.000137 0.000137 0.000137 0.000137 0.000137. 立. 0.000137. 0.000137. β1. δ. l. Log-likelihood. 0.2564 (4.0790) -0.3077 (-4.0884) -0.1100 (-0.7129) -0.0619 (-0.4356) 0.4232 (11.8455) 0.4136 (17.7927) 0.1179 (1.2768) 0.1124 (2.3429) 0.1105 (2.3587) 0.0981 (1.4738) 0.3447 (5.8723) 0.1338 (2.1134) 0.1716 (4.2036). -0.1370 (-1.7506) -0.1614 (-1.2582) 0.9990 (3.2973) -0.2385 (-2.2419) -0.2125 (-2.6747) 0.2993 (2.3865) 0.2562 (2.3765) 0.1825 (2.0690) 0.2141 (3.0722) -1.0006 (-3.0989) -0.1183 (-1.0452) -0.0484 (-1.3260) -0.0178 (-0.2136). 2327.75. β2. 0.0001 0.8347 0.1279 (2.0275) (22.0565) (6.5801) 0.0013 0.4308 0.1493 (5.1751) (4.2785) (3.7128) 0.0000 0.9702 0.0132 (3.8857) (125.0793) (2.1960) 0.0000 0.8284 0.1207 (3.3749) (25.8350) (3.7979) 0.0003 0.5852 0.1714 (4.2018) (8.1479) (4.6682) 0.0012 0.0774 0.1214 (7.1804) (0.6838) (4.1889) 0.0002 0.6666 0.1900 (2.1381) (4.9581) (2.4071) 0.0002 0.5482 0.1441 (3.4554) (5.1288) (3.3473) 0.0003 0.6527 0.1466 (4.6165) (11.1047) (5.2264) 0.0001 0.8387 0.0176 (6.0693) (36.3930) (2.0188) 0.0004 0.7456 0.1213 (3.0733) (11.2735) (3.3128) 0.0000 0.8746 0.0964 (19.5568) (90.3966) (93.2648) 0.0013 0.1297 0.2588 (9.7844) (1.8274) (4.7991). 政治大. 學. Table Ⅴ. 0.000137. ‧. ‧ 國. (34) RONTEX INTL.HDG.. 0.0193 (0.5968) -0.0318 (-0.8691) 0.0063 (0.1542) 0.0391 (0.8268) 0.0164 (0.5322) -0.0180 (-0.7160) 0.0581 (1.2869) 0.0080 (0.2557) 0.0167 (0.6528) 0.0152 (0.3369) 0.1267 (3.0281) 0.0315 (0.9830) 0.0854 (2.8312). 0.000137. (27) ALLIED PROPERTIES. (33) U-RIGHT INTL.HDG.. 0.000137. 0.000137. (26) EFORCE HOLDINGS. (31) CHINA TRAVEL INTL.INVS. (32) GLOBAL BIO-CHEM TECH.GP.. λ. 0.000137. (25) HERITAGE INTL.HDG.. β0. r. 1087.92 984.58 1304.48 2259.01 3305.60 1035.06 2562.85 3148.25 1149.45 936.48 2777.19 1897.12. n. al. er. io. sit. y. Nat. Summary Statistics for the Parameters of Introduction Dummy and Asymmetric Dummy on Model 2 The table shows the number and percentage of stocks with significant changes in volatility after warrant introduction of model 2. Rejections of the null hypothesis, H 0 , are reported at the 5% level. Summary A reports the number and percentage of stocks with significant changes in volatility after warrant introduction. Then, we only select the rejections with positive parameter, i.e. their volatility is significantly diluted, in summary B. The second column shows the results of the total samples, while the sub-samples without deep-in-the-money issued warrants are reported in the last column. Summary C and Summary D show the results of parameter l for determining the asymmetric effect on conditional volatility. Samples without deep-in-the-money Sample Full sample issued warrants Number of samples 36 34 H 0 : δ =0. Ch. engchi. Summary A Number of rejections Rate of rejection Summary B Number of rejections with positive parameters Rate of rejection with positive parameters H 0 : l=0 Summary C Number of rejections Rate of rejection Summary D Number of rejections with positive parameters Rate of rejection with positive parameters. 33. i Un. v. 28 0.7778. 26 0.7647. 26 0.7222. 26 0.7647. 20 0.5556. 20 0.5882. 13 0.3611. 13 0.3824.

(37) 5.3 Model 3: The Dilution-Adjusted GARCH-M Model with a Threshold for Exercise Prices. To distinguish dilution from other introduction effects, a threshold for excise prices in stock is chosen. As discussed in Section 2, the conditional volatility is affected by the compound dummy variable which synthesizes the warrant introduction dummy and the threshold dummy in stock prices. The conditional volatility is not changed after warrant introduction until stock prices exceed exercise prices. Never and almost never in-the-money samples make the threshold dummy variable. 政治大 the total number of samples 立in Model 3 is 28. Table Ⅵ documents the results for. equal to zero all the time, and thus those samples are excluded.8 After the elimination,. ‧ 國. 學. Model 3. It shows that the parameter δ of the compound dummy variable is sill statistically significant in most samples. The results are briefly summarized in Table. ‧. Ⅶ. Of the samples, 23 out of 28 are rejected at a rate of 0.8214. By excluding the two. sit. y. Nat. deep-in-the-money issued warrants, as in Model 1, the rejection rate goes up to. io. er. 0.8462. It is interesting that all significant parameters of the compound dummy. al. variable are positive except a deep-in-the-money warrant, Harmony Asset, whose. n. iv n C issued price is 102 times of the exercise h e nprice. g c hIf iweUexclude the deep-in-the-money warrants, all significant parameters of the compound dummy variable are positive. The interpretation of dilution effect after warrant introduction becomes much clearer. Comparing Model 3 to Model 1, we also find maximum log-likelihood estimation is improved in 13 samples. As shown in Table Ⅵ, Model 3 performs better than Model 1 in 13 samples. Although Model 3 simplifies the introduction effect of model. 8. Omitted samples: 1, 2, 3, 13, 15, 19, 27 and 32 are in Table Ⅰ. Six of these eight samples (1, 2, 13, 15, 27 and 32) are never in the money, and the other two samples (number 3 and 19) are almost never in the money.. 34.

(38) 1, the additional information of the relationship between stock and exercise price makes Model 3 perform better in almost half of the samples.. Table Ⅵ Maximum Likelihood Estimates on Model 3 (The Dilution-Adjusted GARCH-M Model with a Threshold for Exercise Prices) ln( Sit / Si ( t −1) ) ≡ Rit = r + λiσ it − (1/ 2)σ it2 + uit ;. uit ≡ σ it Z it , and Zit Ωt -1 ~ N (0,1) ;. σ it2 = (1 − δ i I i ( t −1) Di ( t −1) ) 2 ( β 0i + β1iσ i2( t −1) + β 2i ui2( t −1) ) ; where Dit is the dummy variable of stock i at time t, when the stock price is higher than the exercise price, Sit > k , Dit is 1; otherwise when the stock price is lower than the exercise price, Sit < k , Dit is 0. The samples which are never or almost never in-the-money are excluded to avoid the parameter δ from being zero all the time. Then parameter estimates on daily return of the 28 samples are obtained by maximizing the log-likelihood function of Model 3. t-statistics are reported below with each estimate in parentheses. For the risk free rate, r, we assume a constant 5% annual rate and a daily return rate of 0.000137.. 0.000137 0.000137 0.000137. (10) HAIER ELECTRONICS GP. (11) PAUL Y ENGR.GP. (12) ASIA ALUMINUM HOLDINGS (14) HOP HING HOLDINGS (16) RICHE MULTI-MEDIA HDG. (17) HARMONY ASSET (18) PREMIUM LAND (20) CHINA STRATEGIC HDG. (21) ALCO HOLDINGS (22) PACIFIC ANDES INTL.HDG.. 0.000137. al. n. (9) CITY TELECOM. 0.000137. io. (8) KINGBOARD CHEMICALS HDG.. 0.000137. Ch 0.000137 0.000137 0.000137 0.000137 0.000137 0.000137 0.000137 0.000137 0.000137. δ. Log-likelihood. 0.0387 (3.2219) 0.0159 (3.4818) 0.0238 (2.8814) -0.0027 (-0.4228) 0.0113 (2.2596) 0.1967 (6.4308) 0.0138 (2.1640) 0.0462 (2.3302) -0.0038 (-0.2286) 0.5402 (2.7019) 0.0533 (5.2890) -0.0898 (-4.6742) 0.0203 (2.0036) 0.1165 (2.6930) 0.1030 (3.6511) 0.0920 (2.6727). 3405.79. β2. β1. 0.0000 0.8861 0.1047 (3.1441) (52.3392) (6.3179) 0.0000 0.8642 0.0454 (9.8800) (70.1245) (5.8496) 0.0000 0.9212 0.0669 (2.7803) (57.5371) (4.4813) 0.0000 0.7714 0.1186 (37.3025) (67.7593) (9.1087) 0.0001 0.8645 0.0872 (3.2887) (70.9211) (9.2804) 0.0007 0.5029 0.3586 (4.9997) (7.2806) (6.3968) 0.0001 0.8259 0.1740 (4.0449) (40.3910) (8.5750) 0.0001 0.6787 0.3136 (5.0045) (18.4627) (5.6150) 0.0002 0.6216 0.2297 (4.9713) (12.9609) (5.8895) 0.0007 0.4589 0.2092 (9.6859) (9.1672) (5.4729) 0.0000 0.9005 0.0899 (4.2005) (250.7212) (11.3608) 0.0003 0.7242 0.0769 (6.8691) (19.4284) (5.2711) 0.0001 0.9025 0.0890 (2.1421) (24.8821) (2.5494) 0.0011 0.5974 0.0793 (2.4701) (3.9870) (2.4580) 0.0002 0.5762 0.1962 (3.8903 (7.1888) (5.9105) 0.0002 0.7721 0.1046 (2.4882) (10.6788) (3.3879). y. 0.0086 (0.3918) -0.0223 (-1.0939) 0.0188 (0.5820) 0.0263 (1.6880) 0.0762 (6.2165) 0.0091 (0.3822) 0.0021 (0.0305) -0.0167 (-0.5283) 0.0285 (0.9409) -0.0035 (-0.1465) 0.0059 (0.2176) -0.0239 (-0.7446) -0.0057 (-0.1285) -0.0088 (-0.2515) 0.0379 (1.5961) 0.0927 (2.8743). Nat. (7) KITH HOLDINGS. 0.000137. β0. sit. (6) CHINA TRAVEL INTL.INVS.. λ. ‧. (5) LEI SHING HONG. ‧ 國. (4) LUKS GROUP. r. 學. Sample Sequence Number & Company. er. 立. 政治大. engchi. 35. i Un. v. 6801.18 2200.39 4012.34 2597.29 2783.77 1622.10 1350.67 2223.27 2758.01 2893.41 1678.05 899.03 1143.43 3556.49 1909.80.