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Venture Capital Distributions
and
Stock Returns
Ziemowit Bednarek*
March 17, 2006
Abstract
This paper studies venture capital distributions and stock returns. We seek good predictors of the
distribution event timing. We collect the data on 2,486 distributions of firms between 1999 and 2002 and
find significant increase in the cumulative abnormal return before the distribution date (16% on average),
followed by sharp decline in performance (around 6% on average). Hazard functions for distribution events
are estimated with semi-parametric [Cox (1972)] and parametric models, using maximum likelihood
estimation method. The Cox model fits the data quite well, but parametric models offer a better fit.
Generalized gamma distribution has the highest Akaike Information Criterion. Stock return after the IPO,
selected Fama-French factors (HML, SMB, market excess return) as well as the lockout period have highly
significant influence on the occurrence of the distribution event. These findings are robust to different
analysis periods, ranging from one to five years.
*
Haas School of Business, UC Berkeley. This is a joint work with Elliot Geidt and Mark Seasholes (both
UC Berkeley). I thank Brandon Doerfler for providing the data and Mark Seasholes for valuable comments.
The event study section is largely based on the paper “Venture Capital Distributions and the Performance
of Stocks” by Elliot Geidt. All the results of event study were however replicated and the original code was
developed. The entire survival analysis section is an original work of the author and was prepared for the
purpose of the course project for Finance 621 at the Graduate School of Business, Stanford University, as
was this paper.
Introduction
This paper studies venture capital distributions and stock prices. We seek good predictors
of the distribution event timing. According to National Venture Capital Association
(www.nvca.org), the number of US venture capital investment deals increased from
1,433 in 1990 to 2,873 in 2004. The total amount of investment went up from $2.8 billion
to $21.0 billion over the same period. This growing industry primarily finances start-up
growth companies, many in the high-tech sector.
Our study is partly motivated by Gompers and Lerner (1997) paper (GL), which
documents stock price reactions to venture capital distributions. We use a dataset
obtained from Shott Capital Management, “…an asset management firm that specializes
in the management of post-venture distribution on behalf of limited partners”2. It includes
a larger number of distributions than the one used in GL. The event study section uses the
methodology developed in MacKinley (1997).
We find significant correlation between the anticipated venture capital
distribution and the stock abnormal return. The cumulative abnormal return over the
period of 50 days before the distribution is 16% on average. This is followed by a
subsequent drop of around 6% over the period of 0 to 2 days relative to the distribution
date. Changes in the abnormal returns around the distribution date are highly statistically
significant.
Survival analysis methods are then applied to estimate the hazard functions of the
distribution event. Our main findings follow:
2
Thanks to Elliot Geidt
1) An increase in the stock return in the period between the IPO and distribution
reduces the hazard rate. This decline is most noticeable for the one-year horizon.
These numbers are obtained through the semi-parametric estimation of the hazard
function using Cox proportional hazards model. Similar results are derived from
the parametric estimation.
2) An increase of the Fama-French HML factor causes a decline in the hazard
function of the same magnitude (largely negative coefficients). The Fama-French
SMB factor and market excess return have the opposite effect on the hazard
function. The hazard rate is largely positively correlated with these factors.
3) The end of the lockout period substantially reduces the hazard ratio.
All hazard functions, estimated with both semi-parametric and parametric estimation
methods, assume the possibility of multiple failures (the subject’s exit time is after the
first distribution).
The remainder of the paper is organized as follows: section 1 briefly describes the
data. Section 2 presents the results of the event study. Section 3 is divided into two
subsections that present results for the survival analysis using semi-parametric and
parametric models. We end with conclusions.
Data
We use data obtained from Shott Capital Management. The dataset includes
detailed information on the date of IPO, distribution date, number of shares distributed
and the distribution price. The stocks are identified by ticker. The original dataset
3
contains information on 2,486 distributions of 624 different companies financed by
venture capitalists (Table 1). On average one company has almost four distributions in
the sample period. For the purposes of survival analysis 2,104 distinct distributions are
used as 382 distributions were excluded because in some cases multiple distributions of
the same company were conducted on the same day. 43% of the distributions in the
sample were carried out in the first year after the IPO (Table 2). More than 80% of the
distributions occur in the first five years after the IPO date. We do not present here the
data on the value of distributed shares. From an intuitive point of view the value of the
distribution may have an effect on the stock price performance and this way also on the
distribution decision.
We include information on the lockout periods of 324 out of 624 distinct
companies in the sample. The data is hand-collected from information memoranda and
issuance prospectuses. Out of 324 lockout periods, 318 are 180 days. The other lockout
periods are 90, 120, 270 and 540 days.
All the distributions were conducted between 1999 and 2002. Some of the start-
ups were taken over by larger firms before they went public. In such cases our reported
IPO date is the IPO date of the acquiring company. For the purposes of this paper we
exclude such distributions (all above 5 years between IPO and distribution, accounting
for 17% of all distributions) from the survival analysis.
We also collect the time series of the selected Fama-French factors – HML, SMB
and market excess return from the Center for Research in Securities Prices (CRSP)
database. CRSP is also the source of information on the returns of distributed stock in the
period between the IPO and the distribution date.
4
Event study
This paper is mostly concerned with the survival analysis of the distribution
timing, therefore event study results are only briefly presented. Usual methodology for
event study is implemented, following MacKinley (1997). The event is defined as the
distribution of venture capital shares to the limited partners.
For each stock the abnormal return is calculated in the time frame -50 – + 50 days
around the distribution date and then mean abnormal return across all distributions is
reported for each day. We assume cross-sectional independence of returns. This allows us
to easily compute standard errors. All reported values are statistically significant at 10%.
Following MacKinley (1997), the abnormal return is defined as:
ARit = Rit − (α i + β i * Rmt ) (1)
We estimate betas for all distributed stocks using restricted least squares, setting
alphas to zero. We use Nasdaq value-weighted returns (including dividends) obtained
through CRSP as a proxy for market return.
Figures 1 and 2 in Appendix B present the results of the event study. There is a
significant increase in the stock value the days before the planned distribution, followed
by subsequent drop in value. Maximum cumulative abnormal return is achieved on the
day of distribution and equals roughly 16%. Highest daily abnormal return is attained
three days before the distribution date (this is likely due to the distribution being
announced, to be confirmed in future work) and it amounts to about 1.4%. Immediately
following the distribution day daily abnormal return attains its minimum value in the
analyzed time frame, roughly -4%. These findings confirm the results of GL (1997).
5
Survival analysis
In all the regressions for the semi-parametric and parametric methods of
estimations the distribution event constituted the failure of the dependent variable. The
covariates were: the abnormal stock return (referred to later as simply the stock return),
the Fama-French factor and the dummy variable for the lockout period (it was 1 when the
lockout period was imposed and 0 after it was released). The Fama-French factors used in
the analysis are HML, SMB and the excess return.
Cox proportional hazards model – semi-parametric analysis
The Cox (1972) proportional hazards model makes the following assumption
about the form of the hazard function for the ith subject in the data:
h(t | xi ) = h0 (t ) exp( xi β x ) , (2)
where β x denotes the set of coefficients estimated from the data. The nice feature of the
Cox model is that it makes no assumption about the baseline hazard function, h0 (t ) ,
which is not estimated by the model.
We first test the data with the test of proportional hazards assumption, based on
Grambsch and Therneau (1994). Table 3 presents p-values of the χ 2 statistics for
different period horizons. We use the stock return, the Fama-French HML factor and
dummy variable for the lockout period (equal to 1 during the lockout period and 0
otherwise) as the set of explanatory variables. The proportional hazards assumption is not
violated for the one- and two-year horizons. Results are robust to the Fama-French SMB
and excess return factors. Grambsch and Therneau (1994) show that this test is equivalent
6
to a test of nonzero slope coefficient in a generalized linear regression of the scaled
Schoenfeld (1982) residuals on time. Figure 3 presents test of the proportional hazards
assumption with the scaled Schoenfeld residuals for the Fama-French HML factor on the
y-axis and time on the x-axis for the one-year horizon. The slope coefficient is roughly
zero, meaning we fail to reject the tested assumption. Longer analysis horizon (five
years) implies the violation of the proportional hazards assumption. Null hypothesis of
the existence of proportional hazards is rejected at 10% significance level for the five-
year horizon.
Table 4 summarizes the results of the Cox model regression with the stock return,
dummy for the lockout period and different Fama-French factors, for the one-year
horizon. Failure is defined as the distribution event. Multiple failures are allowed and
subjects exit the analysis on the date of the last recorded distribution.
An increase in the stock return causes significant drop in the hazard rate, from
1 − e −1.36 = 74% for the model with the HML to 1 − e −1.09 = 66% for the model with excess
return. Coefficients are significant at 5% significance level. Every one percentage point
increase in the stock return brings about 0.66pp-0.74pp decline in the hazard rate, ceteris
paribus. Intuitively, when the distributed stock gives higher return, venture capitalists are
less likely to distribute the shares to the limited partners.
The Fama-French HML factor has a large negative impact on the hazard rate. Its
hazard ratio is 1 − e −15.05 ≈ 0 , meaning that an increase in this factor by one percentage
point causes reduction in the hazard rate by one percentage point, ceteris paribus. The
effect of the other two Fama-French factors tested here, SMB and market excess return,
are of the opposite direction to HML. An increase in the market excess return by one
7
percentage point causes increase in the hazard rate by eleven percentage points, ceteris
paribus. The effect of SMB is disproportionately large, however the coefficient is highly
significant, with p-value close to zero.
Finally, the lockout period has no significant effect on the hazard function. In
order to confirm this finding, we perform logrank and Wilcoxon [Gehan (1965) and
Breslow (1970)] tests. The null hypothesis is that the survivor functions for different
groups of data are the same. Table 5 presents the outcome of the logrank and Wilcoxon
tests performed on the model with HML factor and for one-year horizon, for two possible
values of the lockout dummy variable. Both of these tests fail to reject the null
hypothesis. Results are robust for the other two Fama-French factors used in this analysis
as well as the two-year horizon.
Figure 4 depicts the smoothed hazard function for the Cox proportional hazards
model. There is a substantial increase in the hazard rate between 200 and 300 days after
the IPO. As was noted before, 99% of the companies had lockout periods of 180 days and
it seems like a significant increase in the hazard rate after the most prevalent lockout
period is justified by the sample. Also, this is consistent with the histogram of the time
between the IPO and distribution.
Although certain features of the Cox proportional hazards model may seem
appealing, it does not fit the data in the sample very well. Figure 5 shows empirical
Nelson-Aalen cumulative hazard measure versus partial Cox-Snell residuals. In case of
the perfect fit, Nelson-Aalen cumulative hazard curve should lie on the 45° line. In our
case these two lines do not lie very far apart, however the quality of the fit could be
better.
8
Parametric models estimation
Parametric methods of estimation of the hazard function offer a better fit for the
data in our sample. We estimate the hazard function fitting the exponential distribution,
Weibull distribution, log-normal distribution, log-logistic distribution and generalized
gamma distribution. We use maximum likelihood estimation method. The exponential
and Weibull distributions are fitted in the proportional hazards metric and all other
distributions in the accelerated time failure metric.
We estimate the hazard functions for the five distributions, using three horizons
defined above. Table 5 summarizes estimation results for fitting the distributions into the
data, with the stock return, the Fama-French HML factor and dummy for the lockout
period as explanatory variables. All coefficients are statistically significant, most of them
at 1% significance level. The last row of the table shows Akaike Information Criterion
[Akaike (1974)],
AIC = −2 LLF + 2(k + c) , (3)
where LLF is the value of log-likelihood function, k is the number of model covariates
and c the number of model-specific distributional parameters.
The log-logistic distribution has the lowest score of 192, however the generalized
gamma distribution with AIC of 193 attains the highest value of likelihood function.
Also, this distribution is the most general of the five distributions. Based on this criterion
we choose the generalized gamma distribution in the accelerated failure time metric to fit
9
the data and to test the properties of the hazard functions. In generalized gamma
regression model we assume that
τ j ~ gamma( β , κ , σ ) ,
where τ j is the failure time. Cumulative distribution function of the failure time is:
F (τ ) = I (γ , u ) , if κ > 0
F (τ ) = Φ ( z ) , if κ = 0
F (τ ) = 1 − I (γ , u ) , if κ < 0 ,
ln(τ ) − β
where γ = κ −2 , z0 = sign(κ ) , u = γe γ z0
, Φ() is the standard normal cumulative
σ
distribution function and I (a, x) is the incomplete gamma function,
1 x − v a −1
Γ(a ) ∫
I (a, x ) = e v dv . (4)
0
We run several maximum likelihood estimations with different combinations of
the explanatory variables and time horizons. The explanatory variables sets tested are the
stock return, dummy for the lockout period and one of the three Fama-French factors
(HML, SMB and market excess return). Tables 7 through 9 present results of the
estimation for the three different time horizons. They show the coefficients obtained from
the above described regressions. Most of the coefficients are significant at 1%
significance level.
Hazard rates are not directly computable from the coefficients and so we interpret
the coefficients after plotting the hazard functions for different values of the explanatory
variables.
All the results are robust across the three tested analysis horizons. An increase in
the stock return between the IPO date and the distribution date reduced the hazard rate
10
substantially (Figure 7). This is consistent with the results of the semi-parametric
analysis. The Fama-French HML factor has the similar effect on the hazard rate as the
stock return. Its high values substantially reduced the hazard rate. Coefficients of the
stock return and the HML factor are highly significant.
The other two tested Fama-French factors, SMB (Figure 8) and market excess
return, have similar effect on the hazard rate. Increase in their values brings about
substantial increase in the hazard rate. Again this is consistent with the results of the
semi-parametric model estimation. The coefficients are significant except for the case of
the two-year horizon, where SMB coefficient is not significant.
As expected, the hazard rate is significantly lower in the lockout period,
increasing substantially after the lockout obligation expires (Figure 6). Coefficients of the
lockout period dummy variable are significant, except for the two-year horizon model.
Figure 9 shows the test of goodness-of-fit with the partial Cox-Snell residuals on
the x-axis and Nelson-Aalen cumulative hazard on the y-axis. It seems like the gamma
regression yields a better fit than the Cox model, especially for small values of the partial
Cox-Snell residuals.
Results of the parametric estimation seem to be in most part very intuitive. Higher
stock return in the period between the IPO and distribution date may cause the venture
capitalists to hold the shares longer before passing them through to the limited partners.
A very sharp drop in the abnormal stock return around the distribution date (Figure 1),
amounting to around 4% on average, is a good indicator of the distribution event. This
result and highly significant coefficients should be however treated with caution. The
event study presented earlier indicates the run-up in the stock price before the shares
11
distribution as the market anticipates this event. We expect that the cumulative abnormal
return over the time frame of for example 20 days before the distribution should have a
very high predictive power. We intend to include this covariate in the future work on this
paper.
The lockout period is complied with and this is easily noticeable in Figure 6.
After the lockout obligation is released (dummy variable equal to 0), the risk of
distributing the shares is substantially higher than with the lockout obligation in place
(dummy variable equal to 1).
All the tested Fama-French factors have a significant impact on the timing of the
distribution of shares to the limited partners, for all analysis horizons, except for the two-
year model with the SMB factor. An increase in the excess return of high book-to-market
over low book-to-market firms reduces the hazard rate substantially. An increase in the
excess return of small companies over big companies, as well as increase in the market
excess return both cause substantial increase in the hazard rate. In other words HML
factor has a similar effect on the hazard rate as the stock return and the other two factors
have the opposite effect.
These results are in most part very intuitive once we take account of the
characteristics of firms in the sample. A vast majority are small, growth high-tech
companies. We expect that at the time of the IPO, usually five to six years after the
investment, their stock is small and book-to-market ratio is low. Therefore a decrease in
the HML factor should increase the hazard rate substantially, following the logic of the
earlier argument about the stock return. Along the same lines we expect that as the stocks
in the sample are small, a decrease in the SMB factor should have a similar effect as a
12
decrease in the stock return. This should cause a significant increase in the hazard rate.
However we observe the opposite effect of the SMB factor on the hazard function. This
puzzle could probably be better explained if we looked at the financial statements of
companies in the sample for a period of time long before the IPO and immediately
afterwards. Perhaps at the time of the IPO the stocks have characteristics of big stocks
with low book-to-market value.
Conclusions
This paper documents the stock price performance in response to the venture
capital distribution. We find that in the period between the IPO and distribution,
cumulative abnormal stock returns increase on average. This is followed by drop in
abnormal returns after the distribution event. Price decline is substantial and statistically
significant. These findings confirm results of the GL (1997) paper.
We conduct the survival analysis with multiple failures, allowing for more than
one stock distribution of the same company. Semi-parametric estimation done through
the Cox proportional hazards model allows us to conclude that both the stock return and
the tested Fama-French factors can be good predictors of the distribution event. However,
the overall fit of the model as measured with the partial Cox-Snell residuals is not
satisfactory.
Parametric methods of estimation of the hazard function resulted in a better fit to
the data. Out of the five tested regression models, we choose the generalized gamma
distribution as the one with the highest likelihood function and a very high Akaike
13
Information Criterion value. Again, we conclude that both the stock return and the tested
Fama-French factors can be good predictors of the distribution event. In case of the
gamma regression also the dummy variable for the lockout period is highly significant
and could add some predictive power.
We do not claim that the tested set of covariates is optimal in terms of explaining
the timing of venture capital distributions. We did not test so far the Fama-French
momentum factor. One could also include to the set of explanatory variables for example
the cumulative abnormal return before the distribution, the Fama-French portfolios’
returns or value of distributed shares. Also, the hazard functions could be estimated
simultaneously for the three tested horizons. Significance of the estimates could be then
tested based on some form of the joint Wald test.
Obtained results are in most part intuitive. This refers to the influence of the stock
return, the HML factor and the lockout period obligation on the hazard rate. Effect of the
Fama-French SMB factor on the hazard rate would be easy to interpret if companies in
the sample were big firms. However the nature of venture capital is to finance small start-
ups. One possible explanation would be a substantial growth rate of the company before
the IPO. This question offers interesting opportunities for the future research of venture
capital distributions.
14
References
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and Automatic Control AC-19: 716-723
Breslow, N.E., 1970, A generalized Kruskal-Wallis test for comparing k samples subject
to unequal patterns of censorship, Biometrika 57: 579-594
Cleves, M.A., W.W. Gould, and R.G. Gutierrez, 2004, An introduction to survival
analysis using Stata®, Stata Corporation
Cox, D.R, 1972, Regression models and life tables (with discussion), Journal of the
Royal Statistical Society, Series B 34: 187-220
Cox, D.R, and E. J. Snell, 1968, A general definition of residuals (with discussion),
Journal of the Royal Statistical Society, Series B 30: 248-275
Fama, Eugene F., and Keneth R. French, 1996, Size and Book-to-Market Factors in
Earnings and Returns, Journal of Finance 50, 131-156
Gehan, E.A., 1965, A generalized Wilcoxon test for comparing arbitrarily singly
censored data, Biometrika 52: 203-223
Geidt, E., 2004, “Venture Capital Distributions and the Performance of Stocks”,
University of California, Berkeley working paper
Gompers, Paul A., and Josh Lerner, 1997, Venture Capital Distributions: Short-Run and
Long-Run Reactions, Journal of Finance 53, 2161-2183
Grambsch, P.M., and T.M. Therneau, 1994, Proportional hazards tests and diagnostics
based on weighted residuals, Biometrika 81: 515-526
Mackinlay, A. Craig, 1997, Event Studies in Economics and Finance, Journal of
Economic Literature, Vol. 35, No. 1, 13-39
Schoenfeld, D., 1982, Partial residuals for the proportional hazards regression model,
Biometrika 69: 239-241
15
Appendix A – Tables
Table 1. Data description.
Distributions 2,486
Unique companies 624
Average number of distributions per company 3.98
Table 2. Histogram of the time between the IPO and the distribution date.
Time between IPO and distribution date Number of Cum. number of Percentage Cumulative
distributions distributions percentage
0 – 90 days 29 29 1.17% 1.17%
90 – 180 days 185 214 7.44% 8.61%
180 – 270 days 511 725 20.56% 29.16%
270 – 360 days 348 1,073 14.00% 43.16%
1 – 2 years 542 1,615 21.80% 64.96%
2 – 5 years 446 2,061 17.94% 82.90%
> 5 years 425 2,486 17.10% 100.00%
Sum 2,486 x 100.00% x
Table 3. Test of proportional hazards assumption for different analysis horizons. The test fails to reject the assumption of
proportional hazards for the one- and two-year horizons. P-value of the global test for the five-year horizon allows to reject the
hypothesis of the proportional hazards at 10% significance level.
-------------------------------------------------------------
FF factor in| 1 year 2 years 5 years
regression |
------------+------------------------------------------------
ret | 0.6795 0.3200 0.0199
hml | 0.9297 0.2304 0.8769
lock | 0.6119 0.3996 0.6847
------------+------------------------------------------------
global test | 0.9257 0.4703 0.0914
-------------------------------------------------------------
16
Table 4. The Cox model coefficients for regressions with different Fama-French factors, one-year horizon, proportional
hazards metric. We use Wald test to find the p-values of the coefficients.
-----------------------------------------------------------
Variable | HML SMB FFer
-------------+---------------------------------------------
ret | -1.36** -1.17** -1.09**
hml | -15.05***
lock | 0.39 0.41 0.40
smb | 17.28***
FFer | 7.04**
-----------------------------------------------------------
legend: * p<.1; ** p<.05; *** p<.01; FFer – FF excess return
Table 5. Log-rank test and Wilcoxon test for equality of survivor functions in regression with the HML factor and one-year
horizon, for different values of the lockout dummy variable. P-values above 10% give no evidence to reject the null hypothesis of
the equality of the survivor functions.
Log-rank test for equality of survivor Wilcoxon (Breslow) test for equality of
functions survivor functions
| Events Events | Events Events Sum of
lock | observed expected lock | observed expected ranks
------+------------------------- ------+--------------------------------------
0 | 592 597.78 0 | 592 597.78 -1841
1 | 157 151.22 1 | 157 151.22 1841
------+------------------------- ------+--------------------------------------
Total | 749 749.00 Total | 749 749.00 0
chi2(1) = 2.37 chi2(1) = 2.65
Pr>chi2 = 0.1239 Pr>chi2 = 0.1036
Table 6. Comparison of parametric models for the regression with the HML factor, one-year horizon. We present the
coefficients from the regressions. All the tested models yield significant coefficients. Akaike Information Criterion is presented in the
last row. We use Wald test to find the significance level of the coefficients. Exponential and Weibull models are on the proportional
hazards metric, all the other model on the accelerated failure time metric.
---------------------------------------------------------------------------------------
Variable | Exp Weibull Lognorm Loglog Gamma
-------------+-------------------------------------------------------------------------
_t |
ret | -1.98*** -1.95*** 1.02*** 0.84*** 1.23***
hml | -15.34*** -15.81*** 8.12*** 7.29*** 7.14***
lock | -1.52*** -0.32** 0.28*** 0.26** 0.55**
_cons | -4.40*** -11.55*** 5.10*** 5.08*** 4.74***
-------------+-------------------------------------------------------------------------
ln_p _cons | 0.76***
-------------+-------------------------------------------------------------------------
ln_sig _cons | -0.79*** -0.81***
-------------+-------------------------------------------------------------------------
ln_gam _cons | -1.40***
-------------+-------------------------------------------------------------------------
kappa _cons | -0.45***
-------------+-------------------------------------------------------------------------
aic | 417.92 314.14 206.45 192.85 193.46
-------------+-------------------------------------------------------------------------
legend: * p<.1; ** p<.05; *** p<.01
17
Table 7. Comparison of the generalized gamma models for regressions with different Fama-French factors, one-year horizon,
accelerated failure time metric. In tables 7 through 9 we use the Wald test to find the p-values of the coefficients.
-----------------------------------------------------------
Variable | HML SMB FFer
-------------+---------------------------------------------
_t |
ret | 1.23*** 1.04*** 1.22***
hml | 7.14***
lock | 0.55** 0.51** 0.63**
smb | -5.56*
FFer | -3.97**
_cons | 4.74*** 4.78*** 4.65***
-------------+---------------------------------------------
ln_sig _cons | -0.81*** -0.81*** -0.81***
-------------+---------------------------------------------
kappa _cons | -0.45*** -0.43*** -0.50***
-----------------------------------------------------------
legend: * p<.1; ** p<.05; *** p<.01; FFer – FF excess return
Table 8. Comparison of the generalized gamma models for regressions with different Fama-French factors, two-year horizon,
accelerated failure time metric.
-----------------------------------------------------------
Variable | HML SMB FFer
-------------+---------------------------------------------
_t |
ret | 1.71*** 1.42*** 1.66***
hml | 6.77**
lock | 1.73 1.55 1.84
smb | -2.61
FFer | -4.20**
_cons | 3.54*** 3.71*** 3.42**
-------------+---------------------------------------------
ln_sig _cons | -0.78*** -0.78*** -0.79***
-------------+---------------------------------------------
kappa _cons | -0.67*** -0.66*** -0.68***
-----------------------------------------------------------
legend: * p<.1; ** p<.05; *** p<.01; FFer – FF excess return
Table 9. Comparison of the generalized gamma models for regressions with different Fama-French factors, five-year horizon,
accelerated failure time metric.
-----------------------------------------------------------
Variable | HML SMB FFer
-------------+---------------------------------------------
_t |
ret | 1.48*** 1.26*** 1.43***
hml | 6.90**
lock | 0.97*** 0.92*** 0.98***
smb | -4.30
FFer | -4.17**
_cons | 4.31*** 4.36*** 4.29***
-------------+---------------------------------------------
ln_sig _cons | -0.78*** -0.78*** -0.79***
-------------+---------------------------------------------
kappa _cons | -0.60*** -0.59*** -0.60***
-----------------------------------------------------------
legend: * p<.1; ** p<.05; *** p<.01; FFer – FF excess return
18
Appendix B – Figures
Figure 1. Abnormal returns in the time frame of 50 days around the distribution date. Distribution date is denoted by 0. Market
model as in MacKinley(1997) is estimated in the period from -300 to -51 days relative to the distribution date. Expected stock return is
computed around the distribution date, using out-of-sample estimates.
Abnormal returns (+- 50 days)
2.00%
1.00%
0.00%
-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50
Return
-1.00%
-2.00%
-3.00%
-4.00%
Days
Figure 2. Cumulative abnormal returns in the time window of 50 days around the distribution date. Distribution date is denoted
by 0. Market model as in MacKinley(1997) is estimated in the period from -300 to -51 days relative to the distribution date. Expected
stock return is computed around the distribution date, using out-of-sample estimates.
Cumulative abnormal returns (+- 50 days)
16.00%
12.00%
Return
8.00%
4.00%
0.00%
-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50
Days
19
Figure 3. Test of the proportional hazards assumption with the scaled Schoenfeld residuals for the one-year horizon. We plot
the residuals for the Fama-French HML factor. Zero slope coefficient of the fitted smooth function of time indicates that the
proportional hazards assumption is not violated.
Test of PH Assumption
600 400
scaled Schoenfeld - hml
-200 0 200
-400
0 100 200 300 400
Time
bandwidth = .8
Figure 4. Smoothed hazard function estimated with the Cox proportional hazards model. Covariates are the stock return, Fama-
French HML factor and dummy for the lockout period. Horizon is one-year. Approximately 99% of the distributions have the lockout
period equal to 180 days and the hazard rate attains its maximum after that time.
Cox proportional hazards regression
.015
Smoothed hazard function
.005 0.01
0 100 200 300 400
analysis time
20
Figure 5. Goodness-of-fit evaluation for the Cox proportional hazards model. Partial Cox-Snell residuals are on the x-axis. Fit of
the model is considered good if Nelson-Aalen cumulative hazard lies close the 45° line. The Cox proportional hazards model does not
offer a perfect fit for the data. The fit is much better for smaller values of the partial Cox-Snell residuals.
Nelson-Aalen cum. hazard vs. Cox-Snell residuals
Goodness-of-fit for Cox model (HML) for the one year horizon
1.5
1
.5
0
0 .1 .2 .3 .4
partial Cox-Snell residual
Nelson-Aalen cumulative hazard partial Cox-Snell residual
Figure 6. Smoothed hazard function for the generalized gamma regression model with the Fama-French HML factor and one-
year horizon. The figure plots two hazard functions with the estimated values of parameters, for two different values of the lockout
dummy. When the lockout period dummy is equal to 1 (time between the IPO and distribution is less than the lockout period), the
hazard rates are much lower than when there is no lockout period in place.
Hazard function for gamma regression with one year horizon
.015 .01
Hazard function
.005 0
0 100 200 300 400
analysis time
lock=0 lock=1
21
Figure 7. Smoothed hazard function for the generalized gamma regression model with Fama-French HML factor and one-
year horizon. The figure plots two hazard functions with the estimated values of parameters, for two different values of the stock
return. Higher stock return in the period between the IPO date and distribution causes significant drop in the hazard function.
Hazard function for gamma regression with one year horizon
.01
.004 .006 .008
Hazard function
.002
0
0 100 200 300 400
analysis time
ret=0.01 ret=0.1
Figure 8. Smoothed hazard function for the generalized gamma regression model with Fama-French SMB factor and one-year
horizon. The figure plots two hazard functions with the estimated values of parameters, for two different values of the SMB factor.
Higher SMB in the period between the IPO date and distribution causes significant increase in the hazard function.
Hazard function for gamma regression with one year horizon
.02 .015
Hazard function
.01 .005
0
0 100 200 300 400
analysis time
smb=0.01 smb=0.1
22
Figure 9. Goodness-of-fit evaluation for the generalized gamma regression model. Partial Cox-Snell residuals are on the x-axis.
Fit of the model is considered good if Nelson-Aalen cumulative hazard lies close the 45° line. The generalized gamma regression
model offers a better fit for the data than the Cox model. Again, fit is much better for the smaller values of partial Cox-Snell residuals.
Nelson-Aalen cum. hazard vs. Cox-Snell residuals
Goodness-of-fit for the gamma model (HML) for the one year horizon
.25
.2
.15
.1
.05
0
0 .02 .04 .06 .08 .1
partial Cox-Snell residual
Nelson-Aalen cumulative hazard partial Cox-Snell residual
23
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