This section provides a theoretical analysis of the incentives for cum-ex trading. We derive stock-market equilibrium conditions to determine the expected price/drop ratio (PDR), i.e., the price drop on the ex-dividend date in relation to the dividend of a stock. On the ex-dividend date, the owner of the stock is no longer entitled to receive the current dividend. This causes a “technical” drop in the expected price of the stock at the ex-dividend date. Following the literature (Elton and Gruber 1970; Kalay 1982; McDonald 2001), we use a costly–arbitrage framework to derive conditions under which risk-neutral investors fail to find arbitrage opportunities—neither buying the stock cum-dividend and selling it ex-dividend (long–arbitrage strategy) nor shorting cum-dividend and closing the short position ex-dividend (short–arbitrage strategy) result in a profit.
Stock-market equilibrium
On the stock market, investors who consider selling the stock meet investors who consider buying the stock. Selling and buying on the stock market each come at transaction cost c. Taxation is taken into account by two tax rates, a dividend tax \(\tau _d\) and a tax on realized capital gains \(\tau _g\). In view of the tax conditions faced by a fully taxable institutional investor in Germany, which is the testing ground for the empirical analysis, we simplify the exposition and assume that both tax rates are equal to the withholding-tax rate \(\tau _w=\tau _g=\tau _d\).Footnote 13
Consider first an investor following a long-arbitrage strategy, who is buying a stock cum-dividend at the price \(P_{CUM}\) and selling the stock ex-dividend at the price \(P_{EX}\). The expected return is \((1-\tau _w){\cdot }(E[P_{EX}]-P_{CUM}-2c)+(1-\tau _w){\cdot }D\).Footnote 14 The long-arbitrage strategy yields an expected return that is smaller than or equal to zero if the expected PDR exceeds or equals a certain threshold, formally, \(\frac{P_{CUM}-E[P_{EX}]}{D}\ge 1-\frac{2c}{D}\).
An alternative strategy is short arbitrage. This consists of short selling a stock cum-dividend, such that the net dividend is forgone, and purchasing and returning the stock ex-dividend. This strategy yields an expected return of \((1-\tau _w){\cdot }(P_{CUM}-E[P_{EX}]-2c)-(1-\tau _w){\cdot }D\). The expected return is non-positive, if \(\frac{P_{CUM}-E[P_{EX}]}{D}\le 1+\frac{2c}{D}\).
The two inequalities allow us to derive a condition that ensures the absence of profitable arbitrage trading opportunities for common investors around ex-dividend dates. This holds if the PDR is within the interval
$$\begin{aligned} 1-\frac{2c}{D}\le \frac{P_{CUM}-E[P_{EX}]}{D} \le 1+\frac{2c}{D}. \end{aligned}$$
(1)
Cum-ex trading
In this subsection, we explore the incentives for cum-ex trading in a market equilibrium where the PDR is in accordance with inequality (1). In the first step, we analyze the profit opportunities of short seller and buyer separately in the stock-market equilibrium without illegitimate tax refund.Footnote 15 In the next step, we explore the profit opportunities if the buyer receives an illegitimate tax refund. In a third step, we explore the profit opportunities under collusion.
The short seller
The stock is sold cum-dividend at the price \(P_{CUM}\) and delivered at the ex-dividend date, when it is traded at price \(P_{EX}\). The transaction cost is 2c. The short seller is obliged to pay a compensation in the amount of the net-of-tax dividend \((1-\tau _w) \cdot D\) to the buyer. The short seller’s expected profit from the trade is:
$$\begin{aligned} \Pi _S=(P_{CUM}-E[P_{EX}]-2c)-(1-\tau _w) \cdot D \end{aligned}$$
(2)
The short seller holds the short position in the stock until the ex-dividend date. Hence, the larger the price drop, the more favorable is the price development from the short seller’s perspective. Given inequality (1), the maximum expected price drop in the stock-market equilibrium is \(P_{CUM}-E[P_{EX}]=D+2c\). In this case, the short seller earns a profit in the amount of \(\tau _w \cdot D\). More generally,
$$\begin{aligned} \tau _w \cdot D - 4c \le \Pi _S \le \tau _w \cdot D. \end{aligned}$$
(3)
Therefore, in the stock-market equilibrium described by inequality (1), provided the transaction cost is small, a trade would result in a positive profit for the short seller. Importantly, this holds only with taxable dividends, where \(\tau _w>0\). In the special case of tax-exempt dividends (\(\tau _w=0\)), the short seller never obtains a positive profit, as \(-4c\le \Pi _S \le 0\).
The buyer’s profit without tax refund
The buyer purchases the stock cum-dividend at the price \(P_{CUM}\) and receives the stock ex-dividend at a value of \(P_{EX}\). For simplicity, we assume that the buyer immediately sells the stock and realizes the price \(P_{EX}\). In addition, the buyer has a transaction cost of 2c. As a compensation for forgoing the net dividend, the buyer receives \((1-\tau _w) \cdot D\) from the short seller. The buyer’s expected profit is:
$$\begin{aligned} \Pi _B=(1-\tau _w) \cdot D-(P_{CUM}-E[P_{EX}]+2c) \end{aligned}$$
(4)
Since the buyer holds a long position in the stock until the ex-dividend date, the smaller the price drop, the more favorable is the price development from the buyer’s perspective. Given inequality (1), the minimal expected price drop in the stock-market equilibrium is \(P_{CUM}-E[P_{EX}]=D-2c\). In this case, the buyer expects a profit of \(-\tau _w \cdot D\). More generally,
$$\begin{aligned} -\tau _w \cdot D - 4c \le \Pi _B \le -\tau _w \cdot D. \end{aligned}$$
(5)
Therefore, in the stock-market equilibrium, the buyer incurs a loss. Even in the special case of a tax-exempt dividend with \(\tau _w=0\), the buyer expects a profit of \(-4c\le \Pi _B\le 0\). As such trading never results in a positive profit for the buyer, the buyer has no incentive to participate. Furthermore, the maximum profit for the short seller is equal to the buyer’s minimal loss. Therefore, the short seller cannot compensate the buyer without making a loss. More generally, regardless of the PDR, \(\Pi _S+\Pi _B\le 0\).
The buyer’s profit with illegitimate tax refund
The buyer’s incentive to participate in a trade may change, if the buyer receives an illegitimate refundable tax credit of \(\tau _w \cdot D\). We assume the buyer has a cost of non-compliance \(\kappa _B \cdot D\; \text{ with }\; \kappa _B \ge 0\). This captures efforts to reap the tax credit or to get a refund, the risk of denial and possible prosecution because of tax evasion. The buyer’s expected profit with an illegitimate tax refund is:
$$\begin{aligned} \Pi _{B}^n=\tau _w \cdot D+(1-\tau _w) \cdot D - (P_{CUM}-E[P_{EX}]+2c)-\kappa _B \cdot D \end{aligned}$$
(6)
Still, the smaller the price drop, the larger the profits are. Inserting the minimal expected price drop according to inequality (1) yields a maximum profit for the buyer with illegitimate refund in the amount of \(-\kappa _B \cdot D.\) More generally,
$$\begin{aligned} \Pi _{B}^n \le -\kappa _B \cdot D. \end{aligned}$$
(7)
Hence, in the standard arbitrage equilibrium, even if an illegitimate tax refund is obtained, the trading between short seller and buyer would result in a loss for the buyer, if there is some cost associated with claiming the tax refund. If there are no costs and no uncertainty associated with the illegitimate tax refund, the buyer’s profit is zero.
Collusion
In the above setting, while the short seller would make a profit, even a buyer that receives an illegitimate tax refund has no gain and incurs the cost and risk associated with the refund. Given this asymmetric distribution of profits, mutually profitable cum-ex trading requires that the short seller shares some part of the profit with the buyer. To explore the possibilities for profitable collusion, we sum the agents’ expected profit functions, Eqs. (2) and (6). As the buyer obtains an illegitimate tax credit, sharing profits requires short seller and buyer to collude and, thus, to commit joint tax fraud. Because joint fraud may be fraught with difficulties due to potential conflict between participants, moral concerns or mistakes (Kleven et al. 2016), or because of different legal consequences in case of detection, the cost of non-compliance is likely to increase under collusion.Footnote 16
We, therefore, subtract the term \(\kappa _C \cdot D,\; \text{ with } \; \kappa _C \ge 0\), and derive the expected total profit from collusion:
$$\begin{aligned} \Pi _C \; = \; \Pi _{B}^n+\Pi _S-\kappa _C \cdot D \; = \; \tau _w \cdot D-(\kappa _B+\kappa _C) \cdot D-4c \end{aligned}$$
(8)
This equation shows that the joint profit is positive only as long as \(\tau _w>(\kappa _B+\kappa _C)+\frac{4c}{D}\). Hence, the two parties have an incentive to collude, if the tax refund is larger than the total costs of collusion, non-compliance and transaction. Importantly, this requires that \(\tau _w>0\). There is no profitable collusion with stocks whose dividends are tax exempt with \(\tau _w=0\).Footnote 17
Empirical implications
The theoretical discussion enables us to make predictions as to how cum-ex trades may affect the stock market in an arbitrage equilibrium.
First, in order to obtain an illegitimate tax certificate through the settlement process, the cum-ex trade needs to be in accordance with stock-exchange rules. Depending on the trading venue, the short sale would have to take place on specific days before the ex-dividend date of a stock. This implies that transaction volumes of stocks used for cum-ex trades are greater on these days than otherwise.
Second, unlike the short sale of stocks before the ex-dividend date, the transaction volumes reported on the stock market at the ex-dividend date or later may be unaffected. To see this, note that at or after the ex-dividend date there is no need to use specific modes of transaction to ensure a dividend settlement. Traders may therefore choose among alternative options, including repurchase transactions or securities lending, to ensure that the stock can be delivered ex-dividend. In fact, in the typical arrangement, after all transactions associated with a cum-ex trade are completed, the final owner of the stock is identical to the original owner (Special Investigation Committee 2017, 75). In order to ensure a full circle of transactions, cum-ex trades involve the participation of at least three agents: the cum-ex seller, the cum-ex buyer and the original owner. To deliver the stocks after the ex-dividend date, the cum-ex seller obtains the stock from the original owner. The cum-ex buyer, however, transfers the stock back to the original owner. To conceal transactions, as is noted by the Special Investigation Committee of the German Federal Parliament, cum-ex trades might involve further intermediaries (Special Investigation Committee 2017, 440). However, by including the original owner, the agents can ensure that it is not required to access the market to clear positions. This precludes price effects from the fabricated purchases and, importantly, helps to avoid major risks associated with a change in prices.
Third, the circular trading is a possible structure for cum-ex trades, because these trades generate profit from an illegitimate tax rebate and not from a change in ownership. A change in the original ownership of a stock is not required and would only increase risk. But since actual market demand and supply of the stock are not changed, cum-ex trades will not affect the market prices of stocks around ex-dividend dates.
Further predictions could be made if additional assumptions are imposed regarding the cost of non-compliance for the buyer \(\kappa _B\) and the cost of collusion \(\kappa _C\). If costs differ between stocks or dividend events, for instance, cum-ex trades would tend to focus on those stocks and events where these costs are low and/or where dividends and tax refunds are high. Moreover, as it may become increasingly difficult to conceal the tax fraud, these costs may be increasing in the absolute volume of tax refunds or in the volume of tax refunds relative to total withholding taxes collected at a stock’s dividend date. As a consequence, there might be limits to cum-ex trading. In an equilibrium, in which no further profitable collusion is possible, cum-ex trades are then likely to focus on stocks of large companies or on dividend dates where several corporations pay dividends. However, detailed information is not available in the German case as to how withholding taxes, remittances and tax refunds actually have been monitored and checked by the tax authorities. We can only speculate about the precise determinants of differences in the costs of non-compliance and collusion. If these costs differ between stocks and events—and since also the potential profits differ—the effects of cum-ex trades on the transaction volumes are likely different. This calls for an analysis of the heterogeneity of transaction effects among stocks and events. This is important in particular for providing estimates about the magnitude of cum-ex trading. As potential observable dimensions behind differential effects on transaction volumes, the dividend yield and the market capitalization come into mind. The former determines the amount of refunded taxes and, hence, the potential profits of cum-ex trading. The latter is likely important for the probability of detection as stocks with higher levels of market capitalization often show higher transaction volumes, anyway.
An implication of the collusive nature of cum-ex trades is that transactions are likely to avoid the anonymous regular order book. As noted above, cum-ex trades are often implemented as circular trades, which implies that there is no need to buy stocks from unrelated third parties. However, the choice of the trading platform is not unrestricted, since the short sale needs to trigger the official settlement procedure in order to generate a tax certificate. One potential option to meet these criteria is to use “midpoint” orders, which are entered in a closed order book and executed at the midpoint of the bid/ask price in the regular (open) order book in accordance with the principle of volume/time priority. This means that even large orders can be executed without moving market prices. Another form that is possibly suited for cum-ex trading are “block trades” where cum-ex seller and buyer coordinate their offers and trade large orders at market-compliant conditions. However, it is also likely and has been noted in the proceedings of the Special Investigation Committee, that cum-ex trades involve “over-the-counter” (OTC) transactions (e.g., Special Investigation Committee 2017, 447).