Price Discovery under Disposition Sales

Transcription

1 Price Discovery under Disposition Sales Darwin Choi Hong Kong University of Science and Technology August 2015 Abstract This paper studies the impact of the disposition effect, which results in uninformed sales when investors realize their capital gains. Market makers should post tighter bid-ask spreads and quote prices that are less sensitive to sales for stocks trading at capital gains. However, I show that the price discovery process can be faster or slower under the disposition effect. This is because disposition traders contribute to more uninformed sales but not buys, and decrease the information content or noise of each sale. Using stocks with positive returns on merger and acquisition announcement dates, I find evidence that supports the predictions. Keywords: Uninformed Traders, The Disposition Effect, Liquidity, Price Impact, Price Discovery, Market Efficiency, Mergers and Acquisitions. JEL Classification: G11, G12, G14. Author Contact Information: Darwin Choi, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. Phone: , dchoi@ust.hk. I thank Nicholas Barberis, Tarun Chordia, Lauren Cohen, Sudipto Dasgupta, William Goetzmann, Alexander Ising, Michael Lemmon, Loriana Pelizzon, Rik Sen, Matthew Spiegel, Heather Tookes, and seminar participants at Asian Finance Association International Conference, European Finance Association Doctoral Tutorial, HKUST, University of Mannheim, and Yale School of Management for helpful comments. I acknowledge the Direct Allocation Grant (DAG12BM13-19) from HKUST. All errors are my own.

2 1 Introduction Uninformed trades are typically associated with more noise in the price discovery process. While the relationship holds if these trades are purely random, as is assumed in many theoretical models, I show that information can indeed be incorporated into prices more quickly if uninformed trades are asymmetric. In particular, I examine the impact of having more uninformed sales than buys under certain conditions, matching empirical findings of investor behavior. Evidence on the disposition effect, the tendency to hold losers and sell winners, is widely documented in the literature. 1 Odean (1998, 1999), Barber and Odean (2000, 2001, 2002), and Shumway and Wu (2005) show that individuals tend to exhibit the disposition effect and that stocks they sell outperform those they hold. Based on the assumption that the effect is not information-driven, I derive empirical implications by modifying Glosten and Milgrom s (1985) model. The implications are then tested using stocks with positive returns on merger and acquisition announcement dates, which help identify disposition sales. The model in this paper first shows that disposition sales decrease bid-ask spreads and decrease the price impact per sale. The finding is similar to an increase in noise trading activity in traditional microstructure models with anonymous trading, such as Kyle (1985), Glosten and Milgrom (1985), and Easley and O Hara (1987). However, the disposition effect creates opposing impacts on the speed of the price discovery process: while the price impact of each sale is weaker, there is a larger number of sales (relative to the number of buys). In the model, a small number of informed traders have perfect private knowledge of the stock s true value, buying on good news and selling on bad news. When informed traders buy on good news, the smaller price impact per sale reduces the noise of each uninformed 1 For example, Shefrin and Statman (1985), Odean (1998, 1999), Barber and Odean (2000, 2001, 2002), Dhar and Zhu (2006), and Barber, Odean, and Zhu (2009a) show the tendency of individual investors to hold losers and sell winners. Locke and Mann (2005) and Cici (2012) show that some professional traders and mutual fund managers hold onto losses longer than gains. Genesove and Mayer (2001) find that real estate owners are reluctant to realize losses. Weber and Camerer (1998) provide evidence of the disposition effect in experiments. Internationally, Chinese (Feng and Seasholes, 2005 and Shuwmay and Wu, 2005), Finnish (Grinblatt and Keloharju, 2000, 2001), and Israeli (Shapira and Venezia, 2001) investors are found to exhibit the disposition effect. 1

3 sale, but the larger number of uninformed sales hampers price discovery. On the other hand, when informed traders sell on bad news, the smaller price impact per sale lowers the information content of each sale, but the additional sales help bring down prices. I show that if the proportion of informed traders is high, good (bad) news travels quickly (slowly) when disposition traders realize their capital gains. Intuitively, the effect of reduced price impact outweighs the effect of increased uninformed sales if informed trading is more prevalent. The reverse is true when the proportion of informed trading is low, under some additional assumptions. To test the predictions, I use firms with sudden positive stock returns. If investors of these stocks have accumulated large unrealized capital gains, a sudden positive return would help identify the time when some disposition traders begin to realize their capital gains. Using a sample of over 2,300 firms that have positive returns on M&A announcement dates, I find evidence that an estimate of aggregate unrealized capital gains is positively related to liquidity improvements in quoted spread and depth, as well as to a decrease in price impact, proxied by the Amihud (2002) measure. Following Grinblatt and Han (2005), the unrealized capital gain variable is calculated using the difference between the stock price after announcement and the aggregate reference price; the latter is defined as the turnoverweighted average historical price. Noting that disposition trades are not directly observed in the aggregate data, I compare my results with a sample of firms that have negative returns on M&A announcement dates. Even though investors of these firms could still have unrealized capital gains, the volume of disposition sales should be low because some are reluctant to sell after the negative returns. Using the same regression specification, I do not observe a significant relationship between unrealized capital gains and changes in liquidity and price impact. The results are also robust after controlling for the deal value, offer premium, as well as merger arbitrage activity, which is proxied by the change in short interest in the acquirer s equity. 2

4 Finally, mergers and acquisitions provide a platform to examine how prices incorporate good and bad news. I use the target firm s stock price after the announcement as a probability estimate of the ultimate success or failure of the deal, similar to Samuelson and Rosenthal (1986). Target firms with known deal status and positive announcement returns are studied; deals that are ultimately completed (withdrawn) proxy for good (bad) news, and the stocks are likely influenced by the disposition effect. 2 I split the stocks into eight groups according to completed and withdrawn deals, high and low aggregate unrealized capital gains, and high and low bid-ask spreads after the announcement (which proxy for the proportion of informed traders). The results are consistent with the model s predictions. If the proportion of informed trading is high, good news travels more quickly over the first 20 trading days after the announcement in the group with strong disposition effect, relative to the group with weak disposition effect; bad news travels more slowly over the first 20 trading days after the announcement in the group with strong disposition effect, relative to the group with weak disposition effect. The relationship is reversed if the proportion of informed trading is low. A few other studies examine how the disposition effect influences asset prices. Grinblatt and Han (2005) show that the disposition effect creates a spread between stock prices and fundamental values, causing underreaction to information in equilibrium. Based on Grinblatt and Han s (2005) model, Goetzmann and Massa (2008) derive and test additional implications of the relationship between the disposition effect and stock return, volatility, and volume. Frazzini (2006) provides evidence that good (bad) earnings news travels slowly among stocks trading at capital gains (losses). All these papers argue that disposition investors, who sell at capital gains, create excess supply. In essence they only study the effect of the larger number of sales and ignore the uninformed nature of disposition sales. In contrast, this paper shows that the uninformed nature is empirically relevant, and the resulting decrease price impact per sale is important and can outweigh the effect of the number of 2 As in the model, good and bad news refers to the ultimate stock value, not the public news on the announcement date. Therefore, stocks that have bad news may have unrealized capital gains. 3

5 sales when informed trading is prevalent. The relationship between capital gains and the speed of the incorporation of news is opposite to Frazzini s (2006) results in this case. The paper is structured as follows. Section 2 incorporates the disposition effect into Glosten and Milgrom s (1985) model. Section 3 describes the sample of firms with merger and acquisition announcements. The empirical results are provided in Section 4. Section 5 concludes. 2 Glosten and Milgrom s (1985) Model with the Disposition Effect This section incorporates the disposition effect into a simplified version of Glosten and Milgrom s (1985) model. Although it only allows traders to buy or sell one unit of stock at a time, an advantage of the model is that it can study bid-ask spread and the price impact of trades as functions of information asymmetry. 3 The analysis generates empirical predictions on the impacts of uninformed disposition traders. The model examines trading in one risky stock. Suppose the stock s liquidation value at time T is V {V, V }, where V > V. At time t = 0, the prior probability of the high value V is δ 0 and of the low value V is 1 δ 0. The interest rate between times 0 and T is always zero. At each time t {1, 2,..., T 1}, one and only one trader arrives anonymously and randomly, according to the proportions of the traders. He trades according to his strategy. If he decides to trade, he submits a market order to either buy one unit of stock at the ask, A t, or sell one unit of stock at the bid, B t. A market maker determines the bid and ask prices, at which she is willing to trade with sellers and buyers, respectively. Not knowing the stock s true value, she revises the bid and ask prices after each transaction. 3 The focus of this paper is whether disposition traders affect prices and liquidity due to information asymmetry. However, in reality bid-ask spread and the price impact of trades could also depend on order processing costs and inventory holding costs. 4

6 Two types of traders participate in the market: informed traders (ITs) with a proportion of 1 µ and uninformed traders (UTs) with a proportion of µ, where µ (0, 1]. Only ITs have perfect knowledge of the stock s final value V, i.e., at all times they know whether V will be V or V. They can be interpreted as corporate insiders or anyone who has superior skills in processing public information. Specifically, informed traders receive an accurate private signal that reveals the stock s final value V, i.e., if V = V they receive a high signal, and if V = V they receive a low signal. Informed traders trading strategy is as follows: Buy with probability 1, if high signal is received and V > A t Buy with probability 0, otherwise; Sell with probability 1, if low signal is received and V < B t Sell with probability 0, otherwise; where A t and B t are the current ask and bid prices, respectively. In other words, ITs make arbitrage profits by trading until their signals are fully incorporated into prices. Uninformed traders (UTs) do not possess private information regarding the stock s value, but they will trade because of exogenous reasons or the disposition effect. Unlike traditional market microstruture models, UTs trades are asymmetric in this paper: they are more likely to sell than buy when there is an unrealized capital gain. This characteristic can be viewed as a special case of Glosten and Milgrom s (1985) framework. 4 It models the disposition effect and is consistent with empirical evidence of investor behavior. Let UTs trading strategies be the following: Buy with probability c; 4 Diamond and Verrecchia (1987) also incorporate asymmetric uninformed trades into a Gloston and Milgrom s (1985) framework. In their model, a restriction of short sales drives out some uninformed traders but does not affect informed traders. While the disposition effect is similar in spirit to a relaxation of this restriction, a key difference is that the short-sale restriction is exogenous, but the additional selling propensity arising from the disposition effect may depend on past prices. 5

7 Sell with probability c + g; where c > 0 is a constant and g > 0. 5 c and g are defined such that 2c + g 1 (UTs can buy, sell, or decide not to trade when they arrive at the market). g can be a binary variable where it is a positive constant if there is an unrealized capital gain, and zero otherwise. Alternatively, g can be a more complicated function such that it is increasing in the unrealized capital gain; this will capture disposition traders increasing tendency to sell when the capital gain is larger. 6 The unrealized capital gain is defined relative to an aggregate reference price, R t. Section 4.1 provides a way to estimate R t based on past prices and turnover, similar to Grinblatt and Han (2005). 7 A single market maker clears the market by acting as an intermediary. She knows the high and low values (V and V ) and their prior probabilities, but does not know whether V = V or V. She is able to observe historical prices and order flows, as well as the proportion and trading rule of each trader-type. As in Glosten and Milgrom (1985), Kyle (1985), and Easley and O Hara (1987), the market maker faces Bertrand competition and earns the lowest non-negative expected profits in every transaction. At each time t {1, 2,..., T 1}, before the trader arrives she sets the bid and ask prices B t and A t, making them known to the market. These represent the prices at which she is willing to buy and sell one unit of stock from the trader at time t, whose type is not identified. The market maker is risk-neutral, has 5 One concern is that UTs will deplete their holdings when the probability of selling is higher than that of buying. Therefore, the model should be applied to situations where there are temporarily more sales than buys, rather than to an extended period of time. 6 Grinblatt and Han (2005) argue that if the disposition effect is caused by prospect theory (Kahneman and Tversky, 1979) and mental accounting (Thaler, 1980), then investors tend to sell winners and buy losers. I restrict g to be positive in this paper to examine the higher propensity to sell winners, matching main empirical evidence of the disposition effect and the empirical results of this paper. The tendency to hold or buy losers can be incorporated into this model if g is negative. Besides, Pasquariello (2014) studies market quality when speculators display their preferences consistent with prospect theory. The focus is different because speculators in his paper can acquire private information. For a more in-depth discussion on the disposition effect and prospect theory, see also Kyle, Ou-Yang, and Xiong (2006), Barberis and Xiong (2009), Kaustia (2010), Henderson (2012), and Li and Yang (2013). 7 Realistically, the unrealized capital gains and reference prices are different for different investors. If g is increasing in the unrealized capital gain in the model (and therefore decreasing in R t ), one can interpret it as a larger number of heterogeneous disposition traders who have unrealized capital gains. 6

8 unlimited supply of the stock, and does not face any inventory or transaction costs. Bid-ask spreads compensate her for the adverse selection costs due to informed traders. 2.1 Bid and Ask Prices At each time t, an informed trader arrives with probability 1 µ or an uninformed trader arrives with probability µ. Each time when a trader buys, the market maker s expected profit in this transaction is given by: Pr(Uninformed Buy) E(Profit Uninformed Buy) Pr(Informed Buy) E(Loss Informed Buy) = µc (A t E t 1 ) (1 µ)δ t 1 (V A t ) φ(a t, µ, c, δ t 1, V, V ). (1) The market maker s expected stock value, E t 1, is determined using all available information to her at time t 1 (after the trade at time t 1 but before the trade at time t): E t 1 = δ t 1 V + (1 δ t 1 )V, (2) where δ t 1 is the probability of V = V at time t 1, conditional on market maker s information. 8 As will be shown later, the probabilities and expected value are revised after each trade at time t. 8 For simplicity, I do not state the conditioning event in the notations when there is no confusion. 7

9 When an anonymous trader sells, the market maker s expected profit is equal to: Pr(Uninformed Sale) E(Profit Uninformed Sale) Pr(Informed Sale) E(Loss Informed Sale) = µ(c + g) (E t 1 B t ) (1 µ)(1 δ t 1 ) (B t V ) ψ(b t, µ, c, g, δ t 1, V, V ). (3) The market maker faces Bertrand competition and sets prices to earn the lowest nonnegative expected profit. Therefore, the lowest possible ask price and the highest possible bid price are picked, i.e., A t = inf{a φ(a, µ, c, δ t 1, V, V ) 0}; (4) B t = sup{b ψ(b, µ, c, g, δ t 1, V, V ) 0}; (5) A t E t B t ; (6) A t > 0, B t > 0. (7) Such bid and ask prices may not exist. For example, in the event that g is a linear function in B t, equation (5) becomes a quadratic equation and the discriminant could be smaller than zero. As in Glosten and Milgrom (1985), the market will shut down in this case as there is not enough revenue from uninformed traders to compensate for adverse selection costs. Throughout this paper, I focus on cases where the market maker can find real prices that solve equations (4) (7). 9 9 Glosten and Milgrom (1985, P. 81) show that A t > E t > B t always holds in the presence of informed traders. The proof is omitted in this paper to preserve space. 8

10 2.2 The Price Impact of Trades Whenever there are informed buys (sales), the market maker should immediately set prices equal to V (V ). (The two events are mutually exclusive since all informed traders receive the correct signal and trade in the same direction.) However, since all trades are anonymous, the market maker is not certain about the direction of informed trades. She can only estimate from past transactions and other information. From her perspective, each buy increases the likelihood of informed buys and decreases the likelihood of informed sales. The conditional probability of V = V after a buy transaction at time t (i.e., δ t Buy t ) is given by the Bayes rule: δ t Buy t = Pr{V = V Buy t } = Pr{Buy t V = V } Pr{V = V }, (8) Pr{Buy t } where Buy t denotes a buy transaction at time t. The prior probability Pr{V = V } is calculated using the market maker s information before the trade at time t, so Pr{V = V } = δ t 1 in equation (8). Note that the conditional probability of V = V after a buy at time t (i.e., Pr{V = V Buy t }) is 1 Pr{V = V Buy t } = 1 δ t Buy t. One can obtain the following recursive equation in δ t Buy t : δ t Buy t = Pr{Buy t V = V } 1 δ t Buy t Pr{Buy t V = V } Pr{V = V } Pr{V = V } = 1 µ + µc µc δ t 1 1 δ t 1. (9) In the first term on the right hand side, the numerator (denominator) is the probability of a buy at time t given that the stock s true value is high (low). (Note that ITs never buy on a low signal.) Similarly, each sale increases the likelihood of informed sales and decreases the likeli- 9

11 hood of informed buys. The following equation gives the updating rule of δ t Sell t (i.e., the probability of V = V conditional on a sell transaction at time t): δ t Sell t = Pr{Sell t V = V } 1 δ t Sell t Pr{Sell t V = V } Pr{V = V } Pr{V = V } µ(c + g) = 1 µ + µ(c + g) δ t 1. (10) 1 δ t 1 Uninformed traders exhibit the disposition effect and are more likely to sell when there is an unrealized capital gain, independent of the true value. As such, g appears in both the numerator and denominator (referring to probabilities of selling in cases of high and low values, respectively). In the event of no trading, which occurs when an uninformed trader arrives and decides not to trade (if 2c + g < 1), there is no new information and δ t = δ t 1. The stock s conditional expected value, E t, is updated using the conditional probabilities and equation (2). δ t and E t are always revised upward (downward) after a buy (sale) at time t when informed traders are present. This is consistent with other microstructure models with anonymous trading (e.g., Glosten and Milgrom, 1985 and Easley and O Hara, 1987). After each trade, the market maker uses equations (4) (7) to determine new bid and ask prices for the next period, B t+1 and A t The Impact of the Disposition Effect This section is concluded by introducing three propositions regarding prices under the influence of the disposition effect. Proposition 1 The bid-ask spread is tighter when the selling propensity g is larger. 10

12 Proof: In the Appendix. Proposition 2 When the selling propensity g is larger, the downward revision of the stock s conditional expected value after each sale is smaller. Proof: In the Appendix. Propositions 1 and 2 are intuitive: when the selling propensity g is larger, the market maker has anticipated the extra uninformed disposition trades. Each sale is less likely to be informed from the market maker s perspective, resulting in a tighter bid-ask spread and a reduced price impact per sale. However, there is a larger number of sales than buys when disposition traders sell stocks to realize their capital gains. The overall impact of the disposition effect on the price discovery process is not clear. To understand the overall impact, I make two additional assumptions in order to derive closed-form solutions: (i) g is a (positive) constant, and (ii) 2c + g = 1. These assumptions refer to situations where uninformed traders always buy or sell when they arrive at the market maker, and they have a constantly higher propensity to sell than buy. 10 Let b be the number of buys and s be the number of sells that occur during times 1 to t (t = b + s as trades always occur). Also let Pr{Buy t V = V } Pr{Buy V = V } = 1 µ + µc p, and Pr{Buy t V = V } Pr{Buy V = V } = µc q. Then equations (9) and (10) can be written as δ t 1 δ t = ( p q )b ( 1 p 1 q )s δ 0 1 δ 0. (11) Note that if assumption (i) is violated, then Pr{Sell t V = V } and Pr{Sell t V = V } are not constants; if assumption (ii) is violated, then Pr{Buy V = V } + Pr{Sell V = V } 1 and 10 Another implicit assumption is that the stock is trading at a capital gain even if V = V. This is still consistent with the disposition effect if the reference price is lower than V. 11

13 Pr{Buy V = V } + Pr{Sell V = V } 1. In neither cases does equation (11) hold. I will discuss the relaxation of the assumptions later. Take log and divide by b + s on both sides, 1 b + s log( δ t ) = b 1 δ t b + s log(p q ) + s b + s log(1 p 1 q ) + 1 b + s log( δ 0 ). (12) 1 δ 0 This equation can help us understand the rate at which δt 1 δ t converges. The rate captures how quickly prices reveal the true value of V. Proposition 3 shows that the price discovery process can be faster or slower under the disposition effect. Proposition 3 Assume that g is a positive constant and 2c + g = 1. If V = V, prices converge to the true value more quickly when g is larger; if V = V, prices can converge to the true value more quickly or more slowly when g is larger. In both cases of V, the convergence speed also depends on the value of µ. Proof: Again b + s = t. As t goes to infinity, the last term on the right hand side of equation (12) converges to zero. If V = V, b b+s to Pr{Sell V = V } 1 p. Similarly, if V = V, s b+s converges to Pr{Buy V = V } p and s b b+s converges to Pr{Sell V = V } 1 q. Equation (12) becomes 1 t log( δ t ) p log( p 1 δ t q 1 t log( δ t ) q log( p 1 δ t q converges b+s converges to Pr{Buy V = V } q and p ) + (1 p) log(1 ), if V = V. (13) 1 q p ) + (1 q) log(1 ), if V = V. (14) 1 q Therefore, δ t 1 δ t converges exponentially at rates [p log( p 1 p ) + (1 p) log( )] and [q log( p ) + q 1 q q (1 q) log( 1 p )], under V = V and V = V, respectively. It can be shown that the rate is 1 q positive in the former case and negative in the latter case. 11 So δt 1 δ t converges to positive 11 The rates are, respectively, the Kullback-Leibler divergence of q from p and the negative of the Kullback- Leibler divergence of p from q. The Kullback-Leibler divergence is a measure of the difference between two 12

14 infinity when V takes the high value and to zero when V takes the low value, revealing the true value of V. I first analyze the the case where V takes the high value. The convergence rate is r(v, µ, c) p log( p p ) + (1 p) log(1 q 1 q ) = (1 µ + µc) log( 1 µ + µc µc ) + (µ µc) log( µ µc ). (15) 1 µc The partial derivative with respect to c is r(v, µ, c) c = 1 µ + µc c + µ(µ µc) 1 µc + µ log( 1 µ + µc µc 1 µc ). (16) µ µc As c = 1 g, r = 1 r r. Figure 1a shows that 2 g 2 c g and g (µ (0, 1], g (0, 0.5]). Therefore, the rate at which with g in the model. Also from Figure 1a, when µ is lower, r g when g is higher (or equivalently, c is lower) as well. When V takes the low value, the convergence rate is is always positive for relevant ranges of µ δt 1 δ t converges always increases is more positive. This happens r(v, µ, c) q log( p p ) + (1 q) log(1 q 1 q ) = µc log( 1 µ + µc µc ) + (1 µc) log( µ µc ). (17) 1 µc The partial derivative with respect to c is r(v, µ, c) c = µ 2 c µ(1 µc) + µ log( 1 µ + µc 1 µ + µc µ µc µc 1 µc ). (18) µ µc Again r = 1 r. g 2 c r The partial derivative g can be positive or negative in the range µ (0, 1], g (0, 0.5], as shown in Figure 1b. Recall that r(v, µ, c) is always negative probability distributions and is positive as long as p q. 13

15 and δ 0 1 δ 0 is converging to zero, so a more negative r(v, µ, c) signals a faster convergence; a positive (negative) r g (quickly) when g is larger. From Figure 1b, r g as well as when g is low (high). means that information is incorporated into prices more slowly is positive (negative) when µ is low (high), With additional assumptions, Proposition 3 refers to an increase in g and a decrease in c (holding µ constant), which make uninformed trades more asymmetric towards sales. This is closer to empirical findings on the disposition effect but deviates from traditional market microstructure models. While price impact is lower for each sale, there is a larger number of sales. The price discovery process can be faster or slower, depending on a number of parameters. The assumption of 2c + g = 1 makes changes in c and g go in opposition directions. Does Proposition 3 still hold if c and g are independent and if g is a decreasing function of the reference price, R t? Note that if the reference price is a function of past prices (for example, based on investors purchase price), the bid and ask prices in the model become path dependent and closed-form solutions are not available. I run a simulation where c is a constant and g increases with the capital gain ( Bt Rt R t ), and R t is updated after each trade: R t = w t B t + (1 w t )R t 1. I vary R 0 to study the impact of the disposition effect: if R 0 is low, g takes a high value and the disposition effect is stronger. The following parameters are used in the simulation: V = $70, V = $30, δ 0 = 0.5, c = The proportion of uninformed traders, µ, is either 0.8 or The reference price at time 0, R 0, is either $30 or $70. The additional selling propensity, g, is increasing with the capital gain, consistent with the disposition effect. Specifically, 0, if B t R t g = max{0.05, 0.05 Bt Rt R t }, if B t > R t. 14

16 R t is updated by: R t = 0.01B t R t 1. To study price discovery, I study the number of trades needed for prices to hit a certain level. (Note that the number of trades is different from time t in the model, because when 2c + g < 1 there can be no trades at some particular times.) This level is determined by the odds ratio, δ t 1 δ t. If the true value of the stock is $70, then the ratio needs to be higher than a number, odds; if the true value of the stock is $30, then the ratio needs to be lower than 1. I set odds = 50 and report the average number of trades when the odds ratio first hits odds the level. The simulation is run 1,000 times for each case of µ, R 0, and the true value of the stock. Table 1 shows the results. 12 Most findings in Proposition 3 still hold: the speed at which information is revealed by prices is faster under stronger disposition effect when µ is low for V = V and when µ is high for V = V ; it is slower under stronger disposition effect when µ is low for V = V. Unlike Proposition 3, however, in the simulation the speed can also be slower under stronger disposition effect when V = V ; this happens when µ is high. A reason for this difference between Proposition 3 and the simulation is that good news always travels quickly when c is lower. In the simulation, g is increased but c is held constant, while g and c go in opposite directions under the assumptions of Proposition 3. Absent the effect of c, the region where r g is weakly positive (when µ is high) in Figure 1a would become negative in the simulation. Table 2 summarizes the relationship between the speed of price discovery, disposition effect, and informed trading in the simulation. In contrast to our traditional understanding of uninformed trades, disposition sales may make prices less noisy. Increased number of sales and decreased price impact per sale create opposing effects on the speed of price adjustment. If informed traders are buying (when V = V ), the false signal coming from the increased number of sales reduces the speed 12 The simulation results hold for a wide range of different parameter values as well. 15

17 of news transmission, but the weakened signal per sale resulting from the decreased price impact improves the price adjustment speed. The opposite is true in the low value case. If informed traders are selling, disposition trades help incorporate the true signal into prices, but each sale is less able to correct the noise from uninformed buys. Depending on which of the opposing effects is stronger, prices can be more or less noisy when there are more disposition traders. The simulation suggests that the effect of a larger number of sales is more important when µ is high (i.e., when uninformed traders are more prevalent) and less important when µ is low, resulting in the patterns in the price discovery processes. I will describe the data in the next section. The objective is to test the model using firms that experience a sudden increase in stock prices. These firms will be affected by an influx of uninformed disposition traders, which corresponds to a larger value of g in the model. Note that a larger value of g (which refers to a representative investor in the model) does not necessarily mean that all disposition traders are more likely to sell because of larger capital gains. It can also mean that more investors, who have different reference prices, are in the positive gain territory and are therefore more prone to the disposition effect. 3 Data The proportions of the traders and the trading propensities are difficult to specify in the model. Empirical studies by Grinblatt and Han (2005) and Frazzini (2006) use stocks and mutual funds data to develop proxies for the disposition effect. Using individual accounts from a discount brokerage house, Goetzmann and Massa (2008) construct proxies for the proportion of disposition traders. None of the studies provides a direct estimate of the proportions of different traders and the trading propensities. I use information events that are likely to trigger disposition traders to sell shares. However, not all events are suitable: Chae (2005) shows that some uninformed traders react 16

18 to adverse selection costs and trade less actively prior to scheduled news announcements. They defer trading until the information asymmetry is resolved on the news announcement date. Chordia, Roll, and Subrahmanyam (2001) document increases in trading activity before scheduled macroeconomic announcements. As such, the difference before and after scheduled events is likely to be affected by other factors. On the other hand, Chae (2005) argues that the reduction in uninformed trading activity is not observed if the events are unscheduled, such as merger and acquisition announcements. I use M&A announcements and focus of firms with positive stock returns on the announcement dates. 13 In these firms, investors experience an increase in the unrealized capital gains around the announcement date. The impact of the disposition effect is identifiable by comparing the periods before and after the announcement date. The merger and acquisition announcements are obtained from SDC Mergers & Acquisitions database. Deals that are classified as rumors, recapitalizations, repurchases, or spinoffs are excluded. I study NYSE/AMEX/Nasdaq firms that make announcements between March 1993 and September Since I calculate liquidity and price impact measures using data on the 20 trading days after announcements, I exclude deals that are withdrawn or completed within a month, and firms that make another M&A announcement within a month. Prices and quotes data are obtained from CRSP and TAQ. Short interest data come from Compustat. The final sample consists of 2,391 firms with a positive stock return on the announcement date (if the announcement is made on a holiday, the return on the next trading day is used). Panel A of Table 3 shows the summary statistics of firm and deal characteristics. The mean and median proposed deal value are $244 million and $149 million, respectively. The mean (median) market cap before the announcement is $521 million ($158 million). Average 13 This does not mean that all investors of the sample firms experience unrealized capital gains. If the sample firms are sorted into four portfolios of equal size, based on the estimated aggregate unrealized capital gains (calculated from equations (19) and (20) in Section 4.1), the portfolio that experiences the largest unrealized capital gains has an average of 37.7%, while the portfolio that experiences the smallest unrealized capital gains has an average of 68.6%. 17

19 equity return on the announcement date is 14.5%. About 35% are acquirers (the rest are targets) and 24% are NYSE firms (the rest are AMEX/Nasdaq firms). Panels B and C show the targets and acquirers in the sample separately. Targets are smaller in market cap and experience a higher positive return. The average offer premium is 18.19%, while the average change in acquirer s short interest as a percentage of shares outstanding is 0.44%. Panel D of Table 3 shows the statistics of 1,607 firms that have negative stock returns on announcement dates. This group of firms serves as a comparison in the analysis in Section 4: while investors can still have unrealized capital gains, the gains are lower after the announcement. The disposition effect is therefore weaker after the announcement and its impact is less detectable in the data. Average return on the announcement date is 5.0%. Compared to the sample in Panel A, the proposed deal value is similar, while market cap is larger. Acquirers are more common to have negative announcement returns. 4 Empirical Results 4.1 Unrealized Capital Gain, Liquidity, and Price Impact Measures Grinblatt and Han (2005) argue that the reference price in estimating aggregate unrealized capital gains should be related to past turnover. They use five years of weekly data to calculate the reference price in week t 1: R t 1 = 1 k 260 n 1 (V t 1 n n=1 τ=1 [1 V t 1 n+τ ])P t 1 n, (19) where V t is the turnover in week t, P t is the stock price at the end of week t (adjusted for stock splits), and k is a constant that makes the weights on past prices sum to one. The intuition is that the turnover V t 1 n approximates the likelihood of the stock being purchased 18

20 in week t 1 n, while the product of (1 V t 1 n τ ) approximates the likelihood of the stock not being sold between weeks t n and t 1. The aggregate unrealized capital gain on the announcement date 0 is calculated using the reference price estimate: CapitalGain = P 0 R 1 R 1, (20) where P 0 is the stock price on the announcement date (after market close) and R 1 is the closest end-of-week reference price prior to the announcement date. 14 Note that CapitalGain has a lower bound of 100% but no theoretical upper bound. As shown in Panel A of Table 4, the mean (median) CapitalGain is 1.1% (13.8%). I use time-weighted average dollar spread and percentage spread to examine liquidity. These are defined as the difference between bid and ask quotes, expressed as dollars and percentage of bid-ask midpoint, respectively. Although the model in Section 2 does not analyze trading size, quoted depth is also included as liquidity proxies: dollar depth is the sum of bid and ask depths in dollars, while percentage depth is the sum of depths divided by the number of shares outstanding. The Amihud (2002) measure, defined as the average log ratio of daily absolute return to dollar volume, is a proxy for the price impact of trades. 15 The summary statistics, calculated using the 20 trading days ending 20 trading days prior to the announcement, are shown in Panel A of Table 4. In the main analysis that follows, I calculate changes in the measures from before to after the announcement date. See Figure 2 for a timeline. Pre-period is the 20 trading 14 I follow Grinblatt and Han (2005) and use weekly data to compute the reference price. As a robustness check, I also calculate two other reference prices: using daily data and using the arithmetic average price in the past 52 weeks. The results using these other measures are similar and are not reported. 15 Although the model generates predictions on bid prices and the price impact of sales, in the empirical analysis I do not separately examine bid and ask, or buys and sales. This is because I cannot identify disposition sales based on transaction data. While previous studies employ Lee and Ready s (1991) algorithm to classify transactions into buyer- and seller-initiated trades, disposition traders may not be initiating the trades by submitting market orders. For example, Linnainmaa (2010) analyzes Finnish trading records and shows that the disposition effect can be due to investors use of limit orders. 19

21 days ending 20 trading days prior to the announcement [ 40, 21]; post-period is the 20 trading days starting the trading day after the announcement [+1,+20]. 16 Throughout the paper I define changes ( ) as the post-period average log measures minus the pre-period average log measures. This is comparing a period that is more likely to be affected by the disposition effect to a period that is less. Negative changes in spreads and positive changes in depths correspond to liquidity improvements, while a negative change in log Amihud (2002) measure suggests a decrease in price impact. The statistics of the changes ( ) are reported in Panel B of Table 4. The mean absolute change varies from 0.31 (a decrease in $Spread) to 1.32 (a decrease in Amihud), and are all 1% significant. They suggest an improvement in liquidity and a decrease in price impact after the announcement. However, not all the changes are attributed to the disposition effect: M&A announcements are likely to resolve some information uncertainty. In unreported analysis, liquidity improvements and a decrease in price impact are also observed in the sample with negative announcement date returns. The unrealized capital gain measure in equation (20) depends on past turnover and the return on the announcement date. Table 5 reports correlations of between CapitalGain and log T urnover (in the pre-period ) and between CapitalGain and AnnouncementReturn; suggesting that the variables are quite different. 17 Nevertheless, in the regression analysis I control for the other two variables when studying the effect of CapitalGain. Most liquidity and price impact changes have correlations with CapitalGain that are 1% statistically significant: Amihud (correlation = 0.101), $Spread ( 0.053), %Spread ( 0.092), and $Depth (0.070). 16 The results are qualitatively similar if the pre-period is defined using [ 20, 1] instead. 17 In Schwert s (1996) sample of target firms in , he also finds that the pre-announcement runup and the post-announcement increase in the target s stock price are generally uncorrelated. 20

22 4.2 Liquidity and Price Impact: Regression Analysis This section examines the impact of the aggregate unrealized capital gains and controls for other firm characteristics that may affect liquidity or price impact. The baseline regression specification is as follows: P riceimpact i or Liquidity i = β 1 CapitalGain i + β 2 AnnouncementReturn i + β 3 log T urnover i + β 4 MarketCap i + β 5 V olatility i + β 6 Acquirer i + β 7 NY SE i t=1993 β 8t Y eardummy i + ɛ i. (21) The dependent variables are log changes ( ) in price impact or liquidity. The main independent variable of interest is CapitalGain, defined in (20). Two major controls are AnnouncementReturn, the stock return on the announcement date, and log T urnover, which is the average log (dollar volume divided by number of shares outstanding), calculated using data in the pre-period. As noted before, these two variables are related to CapitalGain. 18 But they could represent different trading strategies: T urnover may capture trading activity based on volume or liquidity (level) of the stock, and AnnouncementReturn may proxy for contrarian strategies based on price run-ups (but not unrealized capital gains), for example, by investors that do not hold the stock. In the presence of these two controls, a significant coefficient of CapitalGain is evidence of the impact of the disposition effect. The other controls are: M arketcap and V olatility, the changes (from pre-period to post-period) in market capitalization and stock return volatility, respectively; Acquirer and NY SE, which are dummy variables indicating that the firm is an acquirer and an NYSE firm, respectively. Note that M arketcap is also capturing the change in stock returns, if there is no change in the number of shares outstanding. 19 The year fixed-effects are omitted 18 In unreported analysis, I also try the following: 1. Calculate T urnover using past five years of data, matching the period of CapitalGain. 2. Control for the average return in the past five years, instead of just controlling for AnnouncementReturn. The results are unaffected. 19 Another potential control variable is the change in institutional holdings, as individual investors may 21

23 in the tables. The standard errors of all coefficient estimates are heteroskedasticity-consistent and clustered at both year-month and 1-digit SIC codes. Table 6 shows that CapitalGain are negatively related to changes in the Amihud price impact measure, dollar spread, and percentage spread; and positively linked to changes in dollar depth and percentage depth. These all support Propositions 1 and 2, which link liquidity and price impact with the selling propensity, g. Investors with higher unrealized capital gains are more likely to sell (captured by a higher g in the model and identified by positive announcement returns in the data), providing liquidity and lowering price impact. The coefficients of CapitalGain are all statistically significant. Increasing the aggregate unrealized capital gains by one standard deviation is associated with the following log changes: Amihud ( = 0.19), $Spread ( 0.05), %Spread ( 0.05), $Depth (0.03), and %Depth (0.03). 20 The coefficients of AnnouncementReturn generally carry the same sign as CapitalGain, and are significant in two out of five regressions. Therefore, the impact of the announcement return, which comes from the price change from day 1 to day 0, is weaker than that of CapitalGain, which is calculated based on prices in the past five years. One interesting note is that the relationships of the dependent variables with log T urnover (in the pre-period) are opposite to those with CapitalGain. There are also likely other factors influencing liquidity and price impact, ranging from changes in market cap and volatility to fixed-effects of being an acquirer and an NYSE firm. The next two tables attempt to further establish the link between liquidity or price impact changes and the disposition effect. Table 7 repeats regressions (21) using firms with negative announcement returns. Although some of these stocks could still have unrealized capital gains, the disposition effect be more prone to the disposition effect. However, institutional holdings for this sample period are only available at the quarterly level, which does not match the pre- and post-periods definitions. Nevertheless, if I also control for the change in institutional holdings, defined using the quarter-end before and after the announcement, the results are unaffected. 20 There are potentially other indirect impacts. For example, Massa and Xu (2013) argue that the target firm s liquidity can affect the M&A transaction itself and the resulting merged company. 22

24 is weaker after the announcement than before, as uninformed investors are generally less willing to sell (i.e., the selling propensity is lower after the announcement because the unrealized capital gain, if any, is smaller). The unrealized capital gains (measured up to the announcement date) do not suggest higher disposition sales in the post-period, and so there should not be any systematic relationship between the post- minus pre- changes and the CapitalGain variable. Consistent with the predictions, the relationships between four of the five dependent variables and CapitalGain are insignificant. The one that is statistically significant ( Amihud) has the opposite sign to Table 6. It is also possible that trading activity of other informed or uninformed traders leads to an increase or decrease in information asymmetry, and that the activity depends on certain deal characteristics. A notable strategy in M&A announcements is merger arbitrage, which typically involves short selling the acquirer s stock and buying the target s stock. For each firm in the sample, I use the change in short interest ratio in the acquirer ( AcquirerShortInterest) around the announcement to proxy for this arbitrage activity. Short interest data are available monthly prior to 2006 and semi-monthly beginning in AcquirerShortInterest is defined as the short interest in the acquirer after the announcement minus that before the announcement, divided by the total number of shares outstanding before the announcement. I also control for other deal characteristics: log DealV alue, which is the (log) proposed deal value at the announcement, and OfferPremium, which is the percentage of the price offered to the target over the stock price prior to announcement. These characteristics may affect investors who try to trade based on their estimate of the deal s final outcome. 21 The three variables are added to the baseline regressions, and the results are in Table 8. The sample is smaller because of missing data in the variables. In the presence of additional controls, the relationships between the dependent variables and CapitalGain are still significant in all five regressions. AcquirerShortInterest and OfferPremium are 21 In addition, investors of target firms may be selling because they will no longer have a pure play that they initially invested in, but this factor should have a relatively small impact because the deal is still not completed in the post-period. 23

25 insignificant or marginally significant in all regressions, while log DealV alue are significant only in the regressions of depths. 4.3 The Price Discovery Process Using target firms with positive announcement date returns, I check if the speed of the price discovery process is related to the disposition effect, as stated in Section 2. The advantage of using target firms is that the stock price reflects market s expectation of the successful probability of the M&A deal. The M&A environment is close to the binary outcomes of the true value V in Section 2: the deal is completed or withdrawn. Similar to Samuelson and Rosenthal (1986), who study prices as predictors of tender offer outcomes, I compute probability estimates from the target s stock price in the first 20 trading days after the announcement. Table 9 first confirms that the previous findings still hold for the subsample. I repeat regression (21) (Table 6) with the 1,123 target firms that have positive announcement date returns and known status of the deal in SDC ( Completed or Withdrawn ). The results are even stronger than the main sample, perhaps because target firms on average experience larger announcement date returns than acquirers (Table 3). Additional assumptions are required to compute the probability estimates. In the context of the model in Section 2, the ultimate price needs to take either the high or the low value (V or V ), and the current price is the expected value of this ultimate price (equation (2)). For the high value, I assume that it is the final price paid by the acquirer for each share of the target if the deal is successful, or the proposed price if the deal is withdrawn. For the low value, I assume that it is the target s average stock price in the 20 trading days ending 20 trading days before the announcement (the pre-period ). 22 From these estimates and 22 Samuelson and Rosenthal (1986) construct a more complicated estimate for the low value based on unsuccessful offers. I do not follow their methodology because their sample (tender offers in ) is different and there are not many withdrawn deals in my sample. 24