#### 5.1. Methodology

We analyze the PDR on the ex-date and daily abnormal returns for the days [−5, 5] relative to the ex-date to get a more holistic view. As common in the literature, we use closing prices for every event

i, and adjust a stock’s price drop on the ex-date by its expected return. The applied methodology follows

Elton et al. (

2005). The PDR is estimated following

Boyd and Jagannathan (

1994) and

McDonald (

2001), as stated in the following:

In an OLS analysis, we regress the adjusted drop from closing cum price,

${P}_{i,t-1}$, to closing ex price,

${P}_{i,t}$, on the dividend,

${D}_{i,t}$, of a stock. The slope coefficient

${\beta}_{i}$ states the PDR. We follow

MacKinlay (

1997) and use a market model specification with a 120-day window to estimate the expected return of a stock for the ex-date

t,

$\mathrm{E}\left[{R}_{i,t}\right]$. To estimate

${\widehat{\alpha}}_{i}$ and

${\widehat{\beta}}_{i}$, we regress a stock’s daily total returns on daily CDAX total returns over the days [−65, −6] and [6, 65] relative to the ex-date.

${\widehat{\beta}}_{i}$ is based on three market betas (lagged, matching and leading) and is eventually aggregated to arrive at a consistent

${\widehat{\beta}}_{i}$ for each stock (

Dimson 1979).

$\mathrm{E}\left[{R}_{i,t}\right]$ is calculated as follows:

where

${R}_{m,t}$ is the realized market return for the ex-date

t.

We set an 11-day event window around ex-dates, which represents the date itself plus five days before and five after an ex-date. We use Equation (5) to estimate the abnormal return of a stock for day

t in the event window,

${\mathit{AR}}_{i,t}$, defined as the return for day

t,

${R}_{i,t}$, minus the respective expected return.

We calculate the mean over all

N stocks to get the average abnormal return,

${AAR}_{t}$.

For the

${AAR}_{t}$, we apply

t-statistics by

Kolari and Pynnönen (

2010) to account for possible event-induced volatility and cross-sectional correlation. These are two-sided tests with expected value of zero.

To obtain a deeper understanding of the price behavior on ex-dates, we analyze the PDR in an additional regression analysis as shown in the following:

We regress the

${PDR}_{i}$, defined as the adjusted

14 price drop from closing cum price to closing ex price over the dividend payment, on the following explanatory variables. We take dividend yield,

${\mathit{DY}}_{i}$, as the dividend payment over the closing cum price, and use

${Risk}_{i}$, as the volatility of a stock’s daily return divided by the volatility of the CDAX daily return over the estimation period. Further we take the ex-date bid-ask spread,

${Spread}_{i}$, as well as the natural logarithm of the market capitalization,

${Cap}_{i}$, of a stock. Pooled OLS and panel regression with random effects are applied to estimate Equation (7).

#### 5.2. Results

This section presents the results of our price analysis. As outlined in

Section 3.1, we expect a PDR of unity for German stocks with tax-free dividend over the whole sample period.

Table 2 presents the PDR estimates following Equation (3). The slope coefficient of the dividend payment,

D, states the PDR on ex-dates. We find significant point estimates of 0.8698 for the full sample between 2002 and September 2019. Further, we find significant point estimates of 0.7937 between 2002 and 2008 (HI-system) and 0.9166 between 2009 and September 2019 (FT-system). The small number of observations over the years 2002 to 2008 reduces the validity of the result for the HI-system. However, the

t-statistics at the bottom of

Table 2 make clear that none of the three point estimates is significantly different to one.

Figure 3 depicts a kernel-density plot of the PDRs for the full sample between 2002 and September 2019. The plot shows a slightly higher mass of density for PDRs smaller than one, with its expected value of almost one.

Our results of the PDRs are in line with the theory as we expect a PDR of one, but the variation in the results is somewhat surprising. There are several potential reasons for this variation that we discuss in the following.

To adjust for price movements during the ex-date, we consider the expected return of a stock. To this end, we estimate three market betas (lagged, matching and leading) following

Dimson (

1979), to account for infrequent trading. We calculate the expected return again, but based on only one matching market beta. The results for Equation (3) are almost identical (see

Table A1 in the

Appendix A) to the results in

Table 2. Thus infrequent trading and the price adjustment do not affect the point estimates.

Market microstructure arguments seriously challenge tax-motivated reasoning, since they also hold in our case, in the absence of a discrepancy between dividend taxation and capital gains taxation. A reason for PDRs lower than one can be a bid-ask bounce in stock prices, where the closing cum price is close to the bid price and the closing price on the ex-date is closer to the ask price (

Frank and Jagannathan 1998). Analyses with bid and ask prices provide economically identical estimates to our results in

Table 2.

15 Our estimates of the PDRs are thus not systematically biased by bid-ask bounces.

Both tax related literature as well as short-term arbitrage literature find a positive correlation between the price drop on ex-dates and the dividend yield of a stock (e.g., (

Elton and Gruber 1970;

Kalay 1982)). Moreover, the noise in the PDR estimation is higher with stocks showing a relatively low dividend yield. To increase the granularity of our analysis, we split our sample into dividend-yield tertiles and estimate the PDRs.

Table 3 provides our results. For all three tertiles, the estimated PDRs are approximately one. The medium tertile even shows a PDR larger than one with a point estimate of 1.0409 and a significant and negative

$\alpha $. The other tertiles also show negative, but insignificant,

$\alpha $ estimates. Our results indicate no correlation between the price drop on ex-dates and the dividend yield of German stocks with tax-free dividend. This is in line with our expectations. Due to tax equivalence, there are no tax clienteles systematically affecting ex-date prices. The results also suggest that short-term arbitrageurs exercise no systematic impact. The

$\alpha $ estimates show, as expected, an inverse trend with increasing dividend-yield. Following

Kalay (

1982) and

McDonald (

2001),

$\alpha $ can be interpreted as twice the proportional transaction costs in a costly arbitrage model for sufficiently large dividends. If dividends are small compared to transaction costs, arbitrage can become unprofitable, because costs exceed potential gains. Hence, the varying

$\alpha $ in

Table 3 provides evidence that transaction costs may affect ex-date prices.

In the following, we deepen our analysis and regress the

PDRs, as stated in Equation (7), on the variables

DY,

Risk, as well as

Spread and

Cap as proxies for transaction costs.

Table 4 shows the results for two models. In model (1), we use pooled OLS estimators with adjusted standard errors following

Newey and West (

1987). Model (2) is a panel regression with random effects

16 and cluster-robust standard errors (clustered at firm level). Our results in model (1) prove to be robust, as the estimators for model (2) are similar. We do not find a significant relationship between the

PDR on ex-dates and dividend yield,

DY, nor between

PDR on ex-dates and

Risk. This result also supports tax-motivated reasoning. The insignificant coefficients for

DY and

Risk are consistent with our results in

Table 3. The coefficients of

Spread and

Cap are also insignificant. This is somewhat surprising as we expect larger, liquid traded stocks to be more efficiently priced with PDRs closer to one. However, transaction costs do not explain the variation in the PDRs either. Our results show no statistically significant evidence for short-term arbitrageurs trading on ex-dates.

In the end, we can conclude that ex-date prices of German stocks with tax-free dividend show an expected pattern. Our results are most consistent with the tax clientele theory. Further, we find no systematic impact from short-term arbitrageurs, nor do technical market microstructure arguments cause PDRs to significantly deviate from one. However, the variation in the PDRs remains partially unexplained.

Finally, the analysis of abnormal returns around ex-dates gives further insights. In

Table 5, we see almost no significant average abnormal returns (

AAR_{t}) in the five days prior to the ex-date and insignificant or significant and negative

AAR_{t} after the ex-date. On the ex-date itself, we find significant and positive

AAR_{t} since 2009. This result stands in contrast to the PDR estimation of one. There is no obvious reason from a tax point of view. The literature on the German stock market by

Lasfer (

2008) and

Haesner and Schanz (

2013) among others, reports comparable outcomes for the ex-date. Stale prices, as suggested by

McDonald (

2001), are not a valid explanation, as we control for this. Our analysis based on bid or ask prices reveals economically identical results for the ex-date. However, the estimation of the AAR differs from the methodology of the PDR estimation, because we implicitly consider transaction costs with the

$\alpha $ in Equation (3).

Another potential, not technical, explanation may be a bias in investors’ behavior.

Hartzmark and Solomon (

2019) show that investors suffer a free dividend fallacy, when investors do not (fully) link an ex-dates price decline to the dividend payment. Investors may be attracted by a reduced price trying to optimally time their purchases. But, in the light of a PDR estimation of one, the technical reasoning outlined above appears to be more likely.

Since there is no tax-related trading, the insignificant

AAR_{t} prior to the ex-date indicate that short-term arbitrage does not significantly affect prices of German stocks with tax-free dividend either. The significant and negative

AAR_{t} after the ex-date are consistent with the declining price pressure of dividend-seeking investors. Dividend-seeking investors can purchase stocks before our event window, so we do not necessarily expect positive

AAR_{t} over the five days in advance of the ex-date (

Kreidl and Scholz 2020). However, the

AAR_{t} on the ex-date do contrast with the otherwise consistent outcomes of our price analysis.