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III. Methodology A. Trading Volume
The measures of liquidity are based on Amihud, Lauterbach, and Mendelson (2003).
First, we use the change of the stock’s trading volume to measure liquidity. We calculate the change in the relative trading volume in the stock as follows. Let DVOLj denote the change in the trading volume of stock j relative to the market volume,
(1) 𝐷𝑉𝑂𝐿𝑗 = 𝑙𝑜𝑔�𝑉𝑂𝐿𝑗𝐴∕ 𝑉𝑂𝐿𝑚𝐴� − 𝑙𝑜𝑔�𝑉𝑂𝐿𝑗𝐵∕ 𝑉𝑂𝐿𝐵𝑚�,
where VOL is the average daily volume (in monetary units) and j and m indicate stock j and the market, respectively. A indicates the period of 30 trading days following the expiration window, (days ＋1 to ＋30), B indicates the period of 30 trading days before the expiration window (days －30 to －1), and the expiration window consists of the one trading day straddling the expiration day.
B. Implicit Spread, Turnover, and Market Depth
The second measure of liquidity is based on Roll (1984), who shows that the implicit bid-ask spread which can be calculated using the autocovariance of stock returns, COVj＝COV (Rj,t, Rj,t－1), as
(2) 𝑆𝑃𝑅𝐸𝐴𝐷𝑗 = 2 ∙ �−𝐶𝑂𝑉𝑗.
We calculate the COVj of each stock j from daily returns over 30 trading days before (B) and 30 trading days after (A) the warrant expiration window. The change in the bid-ask spread,
(3) 𝐷𝑆𝑃𝐷𝑗 = 𝑆𝑃𝑅𝐸𝐴𝐷𝑗𝐴− 𝑆𝑃𝑅𝐸𝐴𝐷𝑗𝐵,
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can be used as an estimate of the change in liquidity.
The third measure of liquidity is the turnover rate of trading volume which can be calculated as following equation,
(4) 𝑇𝑅𝑄𝑗 = 𝑇𝑟𝑎𝑑𝑖𝑛𝑔 𝑉𝑜𝑙𝑢𝑚𝑒𝑗 𝑆h𝑎𝑟𝑒𝑠 𝑂𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔𝑗.
We calculate the TRQj of each stock j by using the same window on COVj. Then we define DTRQ as the following equation,
(5) 𝐷𝑇𝑅𝑄𝑗 = 𝑙𝑜𝑔 �𝑇𝑅𝑄𝑇𝑅𝑄𝑗𝐴
𝑗𝐵�,
which represents the change in the turnover rate of trading volume.
The fourth measure of liquidity is the standard deviation of returns of closed prices which is divided by the daily trading volume of stock j. We define it as below,
(6) 𝐷𝑠𝑡𝑑𝑗 = 𝑙𝑜𝑔 �𝑇𝑟𝑎𝑑𝑖𝑛𝑔 𝑉𝑜𝑙𝑢𝑚𝑒𝑠𝑡𝑑. �
𝐴− 𝑙𝑜𝑔 �𝑇𝑟𝑎𝑑𝑖𝑛𝑔 𝑉𝑜𝑙𝑢𝑚𝑒𝑠𝑡𝑑. �
𝐵, which represents the change in the market depth.
C. The Relation of Liquidity Changes to Fragmentation
Now we use the following regressions to test whether the consolidation of trading could increase liquidity by the warrant exercise. If so, the increase in liquidity should be positively related to the degree of fragmentation before the warrant expiration.
(7) 𝐷𝐿𝐼𝑄𝑈𝐼𝐷𝐼𝑇𝑌𝑗 = 𝛾0+ 𝛾1𝑉𝑂𝐿𝑅𝑊𝑆𝑗+ 𝑢𝑗
VOLRWSj denotes the ratio of the average daily trading volume of warrants to the average daily trading volume of the underlying stocks in the period of 30 trading days
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(days －30 to －1) before the warrants expiration. For DLIQUIDITYj, we use DVOLj, DSPDj, DTRQj, and Dstdj, the measures of the change in the stock liquidity at the warrant expiration.
D. The Effect of Warrant Exercise on Stock Values
As we explain in the introduction, we want to see how the exercise of warrant affects stock prices. First, we calculate the abnormal returns2
(8) 𝐴𝑅𝑗𝑡 = 𝑅𝑗𝑡− 𝑅𝑀𝑡,
,
where Rjt is the return on stock j on day t, and RMt is the return of market index on day t.
Then we calculate the two-day (days －1 and 0) cumulative abnormal returns,
(9) 𝐶𝐴𝑅𝑗 = 𝐴𝑅𝑗,−1+ 𝐴𝑅𝑗,0.
After we define CARj, we want to figure out whether CARj is an increasing function of the liquidity benefits from consolidation. We use the following regression,
(10) 𝐶𝐴𝑅𝑗 = 𝛿0+ 𝛿2𝐷𝐿𝐼𝑄𝑈𝐼𝐷𝐼𝑇𝑌𝑗+ 𝜈𝑗.
DLIQUIDITYj＝DVOLj, DCOVj, DTRQj, or Dstdj, or two of them.
Then considering the change in trading volume of the firm’s equity claims in excess of the increase in the number of shares following the warrant exercise,
(11) 𝐷𝑉𝑂𝐿𝑅𝐴𝑇𝑗 = 𝐷𝑉𝑂𝐿𝑗− 𝑙𝑜𝑔�1 + 𝑅𝐴𝑇𝑊𝑆𝑗�.
DVOLRATj is the difference between the relative change in the stock’s trading volume,
2 We could see Brown and Warner (1980, 1985) on this method of calculating the abnormal return. In our case, it is inappropriate to employ the conventional market model methodology and estimate the market model parameters in the period before the expiration. Because in some cases a warrant enters our sample of deep in-the-money warrants after a rise in the price of the underlying stock before the warrant expiration.
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DVOLj, and RATWSj, the relative increase indirectly in the number of shares of stock that results from the warrant exercise (despite investors can choose to receive stocks or cash when warrants expire). RATWSj is the ratio of the number of warrants to the number of shares of stock outstanding.
E. Investor Sentiment
We use the VIX of Taiwan which is calculated by Taiwan Futures Exchange as a proxy of investor sentiment. The VIX of Taiwan is used starting from December, 2006.
To match VIX, we have to give up parts of our samples are before December, 2006.
Following the criteria used by Bliss and Panigirtzoglou (2004), we use the mean of VIX in the all sample period to determinate whether the VIX is relatively high or low.
The mean value of VIX in the all sample period is 26.85. We calculate the mean value of VIX during the period of 30 trading days before and after the expiration dates following the window of (－30, －1) and (＋1, ＋30), respectively. If the mean of VIX is higher than 26.85, then we define it as the period of high VIX. Corresponsively, if the mean of VIX is lower than 26.85, it represents the period of low VIX.
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IV. Empirical Results A. Hypothesis
Our main test is whether stock prices are affected by the trading consolidation or not.
Given that the warrants exercise on the expiration dates have no information content (we include only warrants that are deep in-the-money prior to expiration), our null
hypothesis is that the ARs on the stock is zero. Against the null, the first alternative hypothesis is:
H1. Stock prices should rise if consolidation of trading is beneficial.
However, it could be argued that warrants offer investors additional tools to diversify their investment portfolios, upon warrants exercise, might reduce investors’ incentive and interest to invest in the stocks. Therefore, we consider another alternative
hypothesis:
H2. Stocks prices should fall because of the elimination of investment tools of
diversifying afforded by warrants.
In fact, the both effects may be present. The results will show us which effect is stronger.
Hypothesis H1 posits that the rise in stock prices is due to the increase in the stock liquidity upon the consolidation of trading after the warrants expiration. We exam this hypothesis by testing the relation between CAR and variables that reflect the liquidity benefits of trading consolidation. Our hypothesis is:
H3. CAR is an increasing function of liquidity benefits from consolidation of trading following the warrants expiration.
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B. The Analysis of Abnormal Returns
As mentioned in the part of introduction, first we want to see the abnormal returns (ARs) of stocks around the issued dates, the first trading dates, and the expiration dates of all warrants. We could see the statistic results in Table 2, 3 and 4.
In Table 2, the results show that almost all the means of daily ARs of stocks around the issued dates of all warrants are significantly positive. These results are contrary to the paper written by 張啟容 (1998), which discovered that there are negative ARs on the issued dates significantly. The paper uses the data in the sample period from 1997 to 1998 in Taiwan, and the author considered that the events of issuing warrants would convey somehow a kind of negative signal to investors. However, in our sample period, the phenomenon does not exist anymore. Conrad (1989) finds that options would have positive price effects on stocks beginning approximately three days before introduction.
And the effects are significantly until the day after issuing. Although warrants are not exactly like options, the results from Conrad (1989) are somewhat consistent with ours.
We think that issuers might buy portions of stocks to build their positions for hedge before they issue the warrants. Therefore the behavior of inventory buildup may lead the stock prices rising. This explanation is consistent with the study from Chan and Wei (2001).
Table 2
The Abnormal Returns of Stocks around the Issued Dates of All Warrants
t-3 t-2 t-1 t t+1 t+2 t+3
Mean 0.2790*** 0.2894*** 0.0850*** 0.0874*** 0.0682*** 0.0248 0.0313*
Std. 2.5503 2.5101 2.3336 2.2985 2.2936 2.2453 2.2504
t-value 13.19 13.90 4.39 4.58 3.58 1.33 1.68
N 14528
Day t is the issued dates of warrants. The abnormal returns (ARs) of stocks are the daily returns of underlying stocks minus the daily market returns. The sample comprises 14528 warrants over the period 2006-2010. ***, **, and * indicate significant level 1%, 5%, and 10%, respectively.
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In Table 3, it shows the results of the means of the ARs of stocks around the first trading dates of all warrants are almost significantly positive. On average, issuers would sell out their issuing positions in the first three days after the warrants listing (王佩甄 (2000)). However, there is no strong evidence to explain the ARs of stocks after the first trading dates of warrants. We infer a possible explanation is that since most of warrants are issued out-of-money (showed in Table 1), if issuers want to sell out their issuing warrants as soon as possible, they might somewhat play a role of market makers in the stock market to stimulate the stock prices to go up.
Table 3
The Abnormal Returns of Stocks around the First Trading Dates of All Warrants
t-3 t-2 t-1 t t+1 t+2 t+3
Mean 0.0225 0.0340* 0.0878*** 0.1348*** 0.1850*** 0.1674*** 0.1945***
Std. 2.2778 2.2349 2.2599 2.2508 2.2879 2.2986 2.2617
t-value 1.19 1.83 4.68 7.22 9.74 8.78 10.36
N 14521
Day t is the first trading dates of warrants. The abnormal returns (ARs) of stocks are the daily returns of underlying stocks minus the daily market returns. The sample comprises 14521 warrants over the period 2006-2010. ***, **, and * indicate significant level 1%, 5%, and 10%, respectively.
In Table 4, we could see that the days before expiration dates and the event dates have negative ARs of stocks. But the days after expiration dates have significantly positive ARs of stocks. We infer that investors might prefer to invest in the same equity claim. Since the warrants expired, investors have to rebalance their positions. The results could be explained that stock prices are improved by trading consolidation. We will discuss the relation between prices, liquidity, and trading consolidation in detail in the following section.
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Table 4
The Abnormal Returns of Stocks around the Expiration Dates of All Warrants
t-3 t-2 t-1 t t+1 t+2 t+3
Mean 0.0662*** 0.0821*** -0.0003 -0.0327* 0.0703*** -0.0026 -0.0112
Std. 2.1241 2.1791 2.1095 2.1442 2.0433 2.0965 2.0704
t-value 3.70 4.47 0.02 1.81 4.08 0.15 0.64
N 14102
Day t is the expiration dates of warrants. The abnormal returns (ARs) of stocks are the daily returns of underlying stocks minus the daily market returns. The sample comprises 14102 warrants over the period 2006-2010. ***, **, and * indicate significant level 1%, 5%, and 10%, respectively.
Further, we want to see if different level of investor sentiment would affect the ARs of stocks or not. In Table 5, we could see that in the period of high VIX, the ARs of stocks before t+2 are all as significantly positive as the results in Table 2. There is no apparent difference when investor sentiment is relatively high.
Table 5
The Abnormal Returns of Stocks During the Issued Dates of All Warrants in the Period of High VIX
t-3 t-2 t-1 t t+1 t+2 t+3
Mean 0.2729*** 0.2919*** 0.0666* 0.1023*** 0.0936*** 0.0348 -0.0385
Std. 2.8675 2.7640 2.6204 2.6161 2.6087 2.5563 2.5529
t-value 7.33 8.14 1.96 3.01 2.76 1.05 1.16
N 5935
Day t is the issued dates of warrants. The abnormal returns (ARs) of stocks are the daily returns of underlying stocks minus the daily market returns. The sample comprises 5935 warrants in the period of high VIX (we calculate the mean value of VIX during the period of 30 trading days before and after the issued dates following the window of (－30, ＋30)) exercise over 2006-2010. ***, **, and * indicate significant level 1%, 5%, and 10%, respectively.
Next, we could see the ARs of stocks after t-1 in Table 6 are all as significantly positive as the results in Table 3. Apparently, the ARs of stocks after the first trading dates would not be significantly affected by investor sentiment.
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Table 6
The Abnormal Returns of Stocks During the First Trading Dates of All Warrants in the Period of High VIX
t-3 t-2 t-1 t t+1 t+2 t+3
Mean -0.0325 -0.0407 0.0423 0.0680** 0.1873*** 0.1317*** 0.2030***
Std. 2.5696 2.5313 2.5539 2.5796 2.5830 2.5998 2.5245
t-value 0.97 1.24 1.27 2.03 5.58 3.90 6.19
N 5924
Day t is the first trading dates of warrants. The abnormal returns (ARs) of stocks are the daily returns of underlying stocks minus the daily market returns. The sample comprises 5924 warrants in the period of high VIX (we calculate the mean value of VIX during the period of 30 trading days before and after the first trading dates following the window of (－30, ＋30)) exercise over 2006-2010. ***, **, and * indicate significant level 1%, 5%, and 10%, respectively.
In Table 7, we could see the ARs of stocks on the day t are different from the results in Table 4. So in the period of high VIX, the ARs of stocks on the expiration dates are not significantly negative.
Table 7
The Abnormal Returns of Stocks During the Expiration Dates of All Warrants in the Period of High VIX
t-3 t-2 t-1 t t+1 t+2 t+3
Mean 0.1493*** 0.1407*** 0.0297 0.0095 0.1310*** 0.1020*** 0.0565
Std. 2.5419 2.5865 2.5210 2.5370 2.4491 2.5355 2.4600
t-value 3.78 3.50 0.76 0.24 3.44 2.59 1.48
N 4133
Day t is the expiration dates of warrants. The abnormal returns (ARs) of stocks are the daily returns of underlying stocks minus the daily market returns. The sample comprises 4133 warrants in the period of high VIX (we calculate the mean value of VIX during the period of 30 trading days before and after the expiration dates following the window of (－30, ＋30)) exercise over 2006-2010. ***, **, and * indicate significant level 1%, 5%, and 10%, respectively.
In Table 8, we could see that the ARs of stocks during the issued dates before t+2 are all as significantly positive as the results in Table 2 and Table 5. It suggests no matter in what levels of VIX, the existence of ARs of stocks is lasting.
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Table 8
The Abnormal Returns of Stocks During the Issued Dates of All Warrants in the Period of Low VIX
t-3 t-2 t-1 t t+1 t+2 t+3
Mean 0.2832*** 0.2877*** 0.0977*** 0.0771*** 0.0507** 0.0179 0.0796***
Std. 2.3061 2.3188 2.1130 2.0506 2.0478 2.0026 2.0139
t-value 11.39 11.50 4.29 3.48 2.29 0.83 3.66
N 8593
Day t is the issued dates of warrants. The abnormal returns (ARs) of stocks are the daily returns of underlying stocks minus the daily market returns. The sample comprises 8593 warrants in the period of low VIX (we calculate the mean value of VIX during the period of 30 trading days before and after the issued dates following the window of (－30, ＋30)) exercise over 2006-2010. ***, **, and * indicate significant level 1%, 5%, and 10%, respectively.
Then we could see the results showed in Table 9, comparing with Table 3 and Table 6, the ARs of stocks are all as significantly positive as the results above. It proves again that the investor sentiment did not visibly affect the ARs of stocks during the first trading dates of warrants.
Table 9
The Abnormal Returns of Stocks During the First Trading Dates of All Warrants in the Period of Low VIX
t-3 t-2 t-1 t t+1 t+2 t+3
Mean 0.0604*** 0.0854*** 0.1192*** 0.1808*** 0.1834*** 0.1920*** 0.1887***
Std. 2.0521 2.0033 2.0324 1.9918 2.0602 2.0655 2.0614
t-value 2.73 3.95 5.44 8.42 8.25 8.62 8.49
N 8597
Day t is the first trading dates of warrants. The abnormal returns (ARs) of stocks are the daily returns of underlying stocks minus the daily market returns. The sample comprises 8597 warrants in the period of low VIX (we calculate the mean value of VIX during the period of 30 trading days before and after the first trading dates following the window of (－30, ＋30)) exercise over 2006-2010. ***, **, and * indicate significant level 1%, 5%, and 10%, respectively.
In Table 10, we could see that the ARs of stocks are almost as same as the results in Table 4. According to our results, we could infer that the effects of investor sentiment play a role in the ARs of stocks during the expiration dates of warrants. As we exam the
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variances of the ARs of stocks during the issued dates and the first trading dates in different levels of VIX, there is no visibly change among our statistic results.
Table 10
The Abnormal Returns of Stocks During the Expiration Dates of All Warrants in the Period of Low VIX
t-3 t-2 t-1 t t+1 t+2 t+3
Mean 0.0318* 0.0578*** -0.0128 -0.0502** 0.0451** -0.0459** -0.0393**
Std. 1.9236 1.9855 1.9132 1.9582 1.8486 1.8832 1.8849
t-value 1.65 2.90 0.67 2.56 2.44 2.43 2.08
N 9969
Day t is the expiration dates of warrants. The abnormal returns (ARs) of stocks are the daily returns of underlying stocks minus the daily market returns. The sample comprises 9969 warrants in the period of low VIX (we calculate the mean value of VIX during the period of 30 trading days before and after the expiration dates following the window of (－30, ＋30)) exercise over 2006-2010. ***, **, and * indicate significant level 1%, 5%, and 10%, respectively.
In this paper, we mainly focus on the warrants which are deep in-the-money. The exercise of those type of warrants is quite certain, it suggests that we can eliminate time value and information cost inside the warrants. In Table 11, comparing with the results in Table 4, the ARs of stocks on the expiration dates of deep in-the-money warrants are more significantly negative than the results of all warrants.
Table 11
The Abnormal Returns of Stocks During the Expiration Dates of All Deep In-the-money Warrants
t-3 t-2 t-1 t t+1 t+2 t+3
Mean 0.0211 0.0033 0.0283 -0.1327*** 0.1483*** 0.0399 0.0633*
Std. 2.0357 2.0373 2.0163 2.1160 2.0136 2.0608 2.1040
t-value 0.65 0.10 0.84 3.74 4.39 1.15 1.79
N 3549
Day t is the expiration dates of warrants. The abnormal returns (ARs) of stocks are the daily returns of underlying stocks minus the daily market returns. The sample comprises 3549 deep in-the-money warrants over the period 2006-2010. ***, **, and * indicate significant level 1%, 5%, and 10%, respectively.
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In Table 12, the ARs of stocks on the expiration dates are not as significantly negative as the results in Table 11. A possible explanation is that investors would certainly exercise their call warrants in the expiration dates, so the issuers need no more stocks to hedge their short positions of warrants. They would sell their present holding positions in the stock market.
Table 12
The Abnormal Returns of Stocks During the Expiration Dates of All Non-Deep In-the-money Warrants
t-3 t-2 t-1 t t+1 t+2 t+3
Mean 0.0810*** 0.1086*** -0.0100 0.0010 0.0440** -0.0169 -0.0363*
Std. 2.1529 2.2242 2.1400 2.1527 2.0526 2.1083 2.0585
t-value 3.87 5.01 0.48 0.05 2.20 0.82 1.81
N 10553
Day t is the expiration dates of warrants. The abnormal returns (ARs) of stocks are the daily returns of underlying stocks minus the daily market returns. The sample comprises 10553 non-deep in-the-money warrants over the period 2006-2010. ***, **, and * indicate significant level 1%, 5%, and 10%, respectively.
In the following sections, we discuss further how the warrant market affects the stock market in different levels of investor sentiment.
C. Liquidity Change and Cumulative Abnormal Returns
The results in Table 13 show that the mean DVOL is －0.0309 with t＝10.66, highly significant. The coefficient of VOLRWS is significantly positive. It means if the average daily trading volume of warrants relative to the average daily trading volume of the stock before the (days －30 to －1) warrant expired raise, the trading volume of stocks relative to the market volume would also raise. Even though the mean of DVOL is negative, the increase in average stock volume is significantly induced by the decrease of numbers of living warrants.
The mean DSPD is positive but insignificant. We could see the coefficient of
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VOLRWS is significantly negative. It is consistent with the results of DVOL which means
that the greater the trading volume of warrants is, the greater the trading volume of the underlying stock is. And it also improved the liquidity of the stock market since the bid-ask spread decreased.
Then we could see the mean DTRQ is －0.0493 with t＝16.22, highly significant.
However, the coefficient of VOLRWS is significantly positive which represents that the liquidity of the stock market is indeed improved.
Finally, the average Dstd is significantly positive. It means that the market depth of the stock market is poorer after the warrants expired, but the coefficient of VOLRWS is insignificant.
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Changes in The stock liquidity following the warrant Exercise Regression Estimation Results
Dependent Variable Mean Constant VOLRWS R2
DVOL -0.0309*** indicate stock j and the market, respectively. A indicates the period of 30 trading days after the warrant expiration, days ＋1 to ＋30, and B indicates the period of 30 days before expiration, days －30 to －1.
𝐷𝑆𝑃𝐷𝑗= 𝑆𝑃𝑅𝐸𝐴𝐷𝑗𝐴− 𝑆𝑃𝑅𝐸𝐴𝐷𝑗𝐵. Roll (1984) proposes that 𝑆𝑃𝑅𝐸𝐴𝐷𝑗= 2 ∙ �−𝐶𝑂𝑉𝑗, where COV is the ratio of the average daily trading volume of warrants to that of the stock in the period of 30 trading days (days －30 to －1) before the warrant expired. The sample comprises 3549 warrants exercise over the period 2006-2010.
The estimated models are
(7) 𝐷𝐿𝐼𝑄𝑈𝐼𝐷𝐼𝑇𝑌𝑗= 𝛾0+ 𝛾1𝑉𝑂𝐿𝑅𝑊𝑆𝑗+ 𝑢𝑗,
Where DLIQUIDITYj＝DVOLj, DSPDj, DTRQj, or Dstdj.
Where DLIQUIDITYj＝DVOLj, DSPDj, DTRQj, or Dstdj.