Short Selling, Earnings Management, and Firm Value

Short Selling, Earnings Management, and Firm Value Jinzhi Lu October 23, 2018 Abstract This paper studies the interaction between short selling and earnings management (misreporting). I show informed short selling and uninformed short selling play different interaction roles. Ex post, informed short selling facilitates the detection of misreporting, whereas uninformed short selling distorts the detection of misreporting. Ex ante, short selling can affect the manager s choice of misreporting level. The main conclusions of this paper are as follows: (1) Whether uninformed short selling should be banned can depend on the trade-off between the ex-ante reduction in misreporting and the ex-post increase in distortion; (2) banning informed short selling may reduce misreporting, but always reduces firm value. I am grateful to Pingyang Gao for his guidance and support. I also appreciate helpful comments from Philip Berger, Miao Liu, Haresh Sapra, and the PhD students at the University of Chicago Booth School of Business. 1

1 Introduction Should short selling be encouraged or restricted? Some proponents of short selling argue that informed short selling leads to more informative stock prices that better guide firms real decisions. The mechanism behind their argument is the feedback effect. Feedback happens when outsiders of firms know information insiders don t know. Outsiders trade on their information, and therefore stock prices contain useful information that can guide insiders decisions (empirical evidence includes Durnev et al. (2004), Chen, Goldstein, and Jiang (2007), and Luo (2005)). Examples of theses proponents include Diamond and Verrecchia (1987), Leland (1992), and Goldstein and Guembel (2008). In these models, bad news is better reflected in stock prices if informed short selling is allowed. Managers learn from the stock prices and cancel negative NPV projects. Other proponents argue that informed short selling helps uncover accounting fraud. For example, in his account of short selling in Allied Capital, Inc., hedge fund manager David Einhorn argues short sellers help uncover financial reporting violations. Although opponents of short selling may not object to the role informed short sellers play, they argue that uninformed short sellers can reduce firm values through the self-fulfilling prophesy of short selling. For example, Goldstein and Guembel (2008) present a model in which the feedback effect generates equilibria in which uninformed speculators sell. Such uninformed short selling reduces the informativeness of stock price and leads to the cancellation of positive NPV projects, hence reducing firm value. Integrating the arguments of proponents and opponents, one might believe that informed short selling should be encouraged and uninformed short selling should be discouraged. However, such arguments focus only on the ex-post consequences of short selling. I argue that short selling also affects ex-ante actions. For example, consider the situation in which the shareholders of a firm are evaluating a manager s performance. The manager can inflate his 2

performance in such a way that the observed performance diverges from his true performance. Fortunately, a speculator exists who knows the manager s true performance and trades accordingly. The speculator s trading has two effects: (1) ex-post, it helps shareholders correctly evaluate the manager s performance; (2) ex-ante, it affects the manager s willingness to engage in performance inflation. I focus on the second channel. A common example of performance inflation is earnings management. In this paper, I develop a model to study the interaction between short selling and earnings management. Specifically, I consider a setting where the current shareholders of a firm are deciding whether to fire a manger. The manager s type can be good or bad. If no earnings management is present, a good manager produces a good report and a bad manager produces a bad report. However, if a bad manager can engage in costly manipulation to issue a good report, earnings reports are not a perfect signal of the manager s type. This scenario leaves some room for the shareholders to learn from the trading in the financial market. As in Goldstein and Guembel (2008), in the stock market, a speculator exists who may have private information about the manager s type and trades on this information. Noisy traders are also present in the market. As in Kyle (1985), the market maker can only observe the sum of order flows the noise traders and the speculator submit. Based on his information, the market maker sets a stock price that is equal to the expected value of the manager. Under this market structure, the speculator s trading orders can partially reveal his private information and provide useful feedback to the shareholders. My model predicts that banning short selling (either informed or uninformed) can increase or decrease the manager s choice of misreporting level. The intuition is as follows. First, note the shareholders are making the decision based on two information sources: the earnings report and stock price. Whereas a bad report reveals a bad manager perfectly, a good report can come from a bad or a good manager. Upon a good report, because the shareholders can 3

only observe the stock price, and not the speculator s type, and because a higher stock price indicates a better manager, the shareholders will unconditionally retain the manager if and only if the stock price is higher than a certain threshold. If short selling is banned, then ex post, the stock price will be higher, and issuing a good report will be more rewarding for the bad manager. However, shareholders rationally anticipate the higher stock price and will set the equilibrium threshold higher. A higher threshold implies less incentive to misreport. How the misreporting level changes depends on the trade-off between the higher stock price and the higher threshold. The trade-off can go both directions. In addition, I show uninformed short selling and informed short selling play fundamentally different roles in affecting firm value. First, I show banning uninformed short selling can reduce firm value because of the possible increase in misreporting. Second, I show banning informed short selling always reduces firm value, even though it can reduce misreporting. I discuss what tool policy makers can use to restrict short selling. If the ideal policy goal is to allow both informed and uninformed short sellers, a policy maker can achieve this goal through imposing no restrictions on short selling. On the other hand, based on my results, the ideal policy goal could also be to allow informed short sellers and ban uninformed ones. As Goldstein and Guembel (2008) suggested, a policy maker can achieve the goal through imposing short-selling costs. According to them, this policy can work because informed short sellers make more profit than uninformed short sellers. By setting a short-selling cost in the middle range, a policy maker is able to predict an equilibrium outcome in which the uninformed speculators do not trade, and the negatively unformed speculators sell. I formally prove this result in the paper. To sum up, the contribution of this study is threefold. First, it establishes a theory to explain how short selling affects misreporting and firm value. Second, it shows banning uninformed short selling can increase misreporting and reduce firm value. Third, it proves 4

banning informed short selling can reduce misreporting, but always reduces firm value. 1.1 Related literature The feedback effect assumption is implicit behind the debate on short selling, because if short selling does not affect firms real decisions, then at the firm level, no efficiency gain or loss will be associated with short selling. Many empirical studies support the feedback mechanism. For example, Durnev et al. (2004) document a robust cross-sectional positive association between stock return variation and investment efficiency, suggesting that a more informative stock price leads to better investment. Luo (2005) finds the market reaction to a merger and acquisition announcement affects the firm s later decision on the deal. Chen et al. (2007) find informed trading has a strong positive effect on the sensitivity of corporate investment to stock price. Durnev and Mangen (2009) find restatement in a firm s financial report is related to its competitors investment decisions. All this evidence points to the fact that outside information sources can guide firms real decisions. On the other hand, analytical studies in the feedback effect literature typically start with simplified assumptions on firms insiders. First, they often assume that insiders know nothing about firms future prospects. Second, the studies typically assume that no agency conflict exists between managers and shareholders. For example, in the models of Goldstein and Guembel (2008), Khanna and Mathews (2012), and Edmans et al. (2015), managers are uninformed and take actions to maximize firm value. Although insider information and agency conflicts are not the focus of these studies, exploring settings where they are incorporated remains interesting. Gao and Liang (2013) relax the first assumption and show disclosure by firm insiders lowers outsiders incentive to acquire information. Dow and Gorton (1997) relax the second assumption and explore the moral hazard problem where the existence of informed outsiders may lower managers incentives to acquire information. In my model, I relax the agency assumption to explore the interaction between short selling and earnings management. 5

My study is directly related to recent empirical studies such as Massa et al. (2015) and Fang et al. (2015). These two studies show short selling curbs earnings management. My model provides a theoretical framework to interpret their empirical results, and shows their results can arise from the trade-off of several forces. I organize the paper as follows. In section 2, I describe the model and define the equilibrium concept. In section 3, I explore the basic properties of the equilibrium and the conditions under which uninformed short selling exists. In section 4, I show the main results. First, I show banning uninformed short selling can increase misreporting and reduce firm value. Second, I show banning informed short selling always reduces firm value. Section 5 concludes. 2 Model The model introduces earnings management into a simplified Goldstein and Guembel (2008) setting. The current shareholders of a firm are deciding whether to fire a manager. Two signals about the type of the manager are generated: a public earnings report and a private signal that only a speculator learns. The speculator trades in the stock market, and therefore the stock price partially reflects the speculator s private information. The shareholders make the decision based on the earnings report and the stock price. All players are risk neutral. Details of the model are described below. 2.1 Setup The manager knows his ability θ {L, H}, which is equally likely ex ante. An H manager (good manager) increases the shareholder value by V, and an L manager (bad manager) destroys shareholder value by V. The manager receives an extra benefit of 1 from retaining the job. Without earnings management, an earnings report r(θ) = θ is disclosed. However, the manager can manipulate the report at a cost. In particular, a low-ability manager can issue an H report with probability β at a private cost of c(β) = 1 2 cβ2, where c is a parameter 6

that catches the magnitude of the cost. 1 2 Although the ex-ante mean of θ is 0, conditional on r = H, the expectation of θ is positive, as can be seen from E[θ r = H] = 1 β 1+β V 0. 3 In other words, r = H indicates a good manager on average. Hence, without looking at the stock price, the shareholders will retain the manager for sure when r = H and fire the manager for sure when r = L. The game has 3 additional players: a speculator, a noise trader, a market maker. I now describe their roles in detail. The speculator, who has no initial position in the firm s stock and will trade to maximize profit, receives a signal about the manager s type. With probability α, the speculator receives a perfect signal s = l if θ = L, s = h if θ = H. With probability 1 α, the speculator receives no information. For simplicity, I use the term positively informed to describe the speculator who gets s = h. Similarly, negatively informed and uninformed speculator describe the speculators who get s = l and s =, respectively. Trading occurs for only one period in which the speculator submits an endogenous order d { 1, 0, 1} (short-selling, no trade, buying) based on his information. Following Kyle (1985), the noise trader submits an exogenous order n f(n), where f is some distribution that has support (, ). I assume f(n) satisfies the following assumptions. Assumption 1 Suppose f(.) has the following properties: (1) (MLRP) f(n 1) f(n) is increasing in n. (2) (boundary condition) lim n f(n 1) f(n) = 0. 1 An alternative setup could be the following. The firm needs to decide whether to continue a project or not. The manager knows the quality of the project θ {L, H}, which is equally likely ex ante. An H project generates a future profit of V > 0, and an L project generates a future loss of V. The manager receives a rent of 1 if the firm invests in the project. Without earnings management, r(θ) = θ is disclosed. However, the manager can manipulate the report at a cost. In particular, if θ = L, the manager can issue an H report with probability β at a private cost of c(β) = 1 2 cβ2, where c is a parameter that catches the magnitude of the cost. 2 The main results of the paper still hold as long as the cost function is convex. 3 Hence, assuming E[θ] = 0 does not hurt the model s ability to illustrate how short sellers can destroy a good firm. 7

(3) (symmetry) f(n) = f( n), n. A typical example of f(.) is the normal distribution with mean 0. Only one round of trading occurs. The market maker observes the total order flow Q = n + d and sets the price based on the total order flow and manager s report: p(r, Q) = max(e[θ r, Q], 0). For simplicity, I assume the shareholders can also observe the total order flow Q. Because the shareholders are risk neutral, they will fire the manager if and only if E[θ r, Q] < 0. If the manager is retained, the speculator makes a profit of d(e[θ s, r] p(r, Q)). If the manager is fired, the speculator s profit is 0. Figure 1: information structure 2.2 Time line of the model The time line of the events is as follows: t=1, manager s ability θ {L, H} is realized, type L manager chooses management level β and pays a cost of c(β) = 1 2 cβ2. t=2, the speculator receives his private signal s. Manager s report r is generated. Everyone can observe the report r. The noise trader and the speculator submit trading orders d and n. 8

Only the total order flow Q = d + n is observed. t=3, Based on order flow Q and report r, the market maker sets price as the expected value of the manager: p(q, r) = max(e[θ Q, r], 0). The shareholders, who can also observe Q and r, will fire the manager if and only if p < 0. 2.3 Definition of the equilibrium The equilibrium concept that will be used is a Perfect Bayesian Equilibrium (PBE). It is defined as follows: (i) The speculator s trading strategy maximizes his expected profit, given his private signal s, the price-setting rule of the market maker, and the decision strategy of the shareholders. (ii) Type L manager s choice of earnings management level maximizes his expected payoff, given the strategy of the shareholders. (iii) A decision strategy the by the shareholders that maximizes their expected payoff, based on order flow Q and earnings report r, and given all other strategies. (iv) A price-setting rule p = E[θ Q, r] that allows the market maker to break even. (v) Players use Bayes rule to update their beliefs whenever possible. (vi) Each player s has a rational belief about other players strategies. 3 The Equilibrium The main goal of this paper is to study the interaction between short selling and earnings management and whether informed and uninformed short selling play different interaction roles. But when do informed and uninformed short selling exist? In this section, I investigate the basic properties of the equilibrium and derive conditions under which the uninformed speculators short sells. Such preparation work is needed for further analysis. 9

3.1 Basic properties of the equilibrium Lemma 1 below pins down the equilibrium strategies of the positively informed and the negatively informed speculators and states that informed short selling always exists in my model. It is extremely useful throughout the paper. Lemma 1 If no restriction is placed on trading, the positively informed speculator (s = h) plays d = 1 with probability 1, and the negatively informed speculator (s = l) plays d = 1 with probability 1. Lemma 1 is intuitive. The positively informed speculator wants to buy because he knows for sure the manager is of good type and his valuation of the firm is thus V. If his private information (s = h) is fully revealed through Q, then the price p will be set to V and he makes 0 profit. If his private information is not fully revealed through Q, p < V and he makes a positive profit. By not trading, he makes 0 profit, and by short selling, he makes negative profit. Therefore, the only possible action for type s = h is buying (d = 1). The intuition for the negatively informed speculator s action is similar. Now, I use β to denote the equilibrium level of earnings management. Proposition 1 below shows that in the equilibrium, the shareholders follow a threshold strategy. That is, upon seeing r = H, they will retain the manager if and only if the order flow Q is above some threshold q. Proposition 1 If no restriction is placed on trading, then in any equilibrium, (1) A threshold q 0 (q can be ) exists such that the manager will be retained if and only if r = H and q q. Besides, when β = 1, q = 0. 10

(2) The MM s equilibrium pricing function is p(q = q, r = H) = E[θ Q = q, r = H] > 0, if q > q ; p(q = q, r = H) = 0, if q q ; p(q = q, r = L) = 0, q. The intuition for proposition 1 is as follows. Note the shareholders are making the decision based on two information sources: earnings report and the order flow. Whereas a bad report reveals a bad manager perfectly, a good report can come from a bad or a good manager. Upon a good report (r = H), because the shareholders can only observe the order flow, not the speculator s type, and because a higher order flow indicates a better manager, the shareholders will unconditionally retain the manager if and only if the order flow is bigger than a certain threshold. Also note that q 0. The intuition is as follows. By the symmetry property of f(n), relative to r = H alone, Q > 0 and r = H implies a better manager, whereas Q < 0 and r = H implies a worse manager. Because E[θ r = H, Q = q ] = 0 < E[θ r = H], q must be negative. 3.2 Existence of uninformed short selling Lemma 2 presents some clues about the strategy of the uninformed speculator. Lemma 2 If no restriction is placed on trading, the uninformed speculator plays d = 1 with probability 0. Lemma 2 says the uninformed speculator will never buy in the equilibrium. The intuition is as follows. If he plays d = 1, he makes a positive profit when Q < 0. However, Q < 0 happens when n < 1, which happens with probability less than 1 2. When n > 1, which happens with probability larger than 1 2, he makes a negative profit. Therefore, overall, he loses if he buys. 11

Now I study the conditions for the existence of uninformed short selling. Let c 0 = 1 F (1). In lemma 6 of the appendix, I show c c 0 implies β = 1. To avoid this uninteresting case, I assume c > c 0 for the rest of the paper. Proposition 2 c > c 0 exists such that if c (c 0, c ), then β (0, 1) and the uninformed speculator plays d = 1 with probability 1 in the equilibrium. Proposition 2 is the main result of this section. I now explain the intuition in detail. If the total order flow turns out to be Q = q > 0, the uninformed speculator s short position is established at a price higher than his valuation and therefore he makes a positive profit. If the total order flow turns out to be q < Q = q < 0, the uninformed speculator s short position is established at a price lower than his valuation and therefore he makes a negative profit. If the total order flow turns out to be Q = q < q, the uninformed speculator s short position is established at a price of 0 and he makes 0 profit. Hence, the speculator s gain and loss are not symmetric, because the region for gain is q (0, + ), whereas the region for loss is q (q, 0). When Q < q, the manager will be fired and the uninformed speculator can guarantee 0 loss. If q is close enough to zero, which happens when β is close to 1 or the cost of earnings management is low enough (close to c 0 ), the gain should outweigh the loss and the speculator can lock in a strictly positive profit by short selling. In other words, the uninformed speculator short sells and hopes two things: (1) the total order flow is high; and (2) even if the total order flow is low, he won t lose much. In this way, he is guaranteed a positive expected profit on average. We can see that when the total order flow is low, the uninformed speculator really wants a good manager to be fired because his profit will otherwise be negative, instead of zero. 12

Similar to Goldstein and Guembel (2008), in my model, the uninformed speculator s shortselling strategy (d = 1) is manipulative in nature. This is because, based on the uninformed speculator s information set, E[θ r = H, s = ] = E[θ r = H] = 1 β 1+β V > 0. By short-selling, he pools with the negatively informed speculator, which leads to a higher probability that the shareholders fire a good manager. But why only short selling can destroy a good manager, not selling? In fact, the assumption that the speculator has no initial position is crucial for proposition 2. If the uninformed speculator has an initial position in the firm, his profit from selling is the difference between the price he receives if he sells today and the price he receives if he waits until the next period (after the decision on manager is made). If the uninformed speculator sells (d = 1) and this leads to the firing of the manager, today s price will be zero. On the other hand, if the uninformed speculator plays d = 0 instead of d = 1, then the price for the next period will more likely be positive because the manager is less likely to be fired. Hence, if the uninformed speculator has an initial position, he would prefer d = 0 (no trade), as destroying a good manager is not what he wants. 3.3 Introducing a short-selling cost into the model One natural way to study the effects of short selling is to compare the case where short selling is banned and the case in which short selling is allowed. However, using such a method, separating the effect of uninformed short selling from the effect of informed short selling would be hard. Is looking at their effects separately feasible? As Goldstein and Guembel (2008) suggested, imposing the short-selling cost can do the trick. I formalize the result in this subsection. Assumption 2 Suppose an additional player, a decision maker (DM), enters the game before t = 1. The DM aims to maximize expected firm value and can impose a short selling cost 13

κ 0. Proposition 3 Suppose c (c 0, c ), (1) if κ = 0, then d(s = h) = 1, d(s = l) = 1, and d(s = ) = 1 (buy-sell-sell, or BSS equilibrium). (2) κ 1 > 0 exists such that if κ = κ 1, then d(s = h) = 1, d(s = l) = 1, and d(s = ) = 0 (buy-sell-no trade, or BSN equilibrium). (3) κ 2 > 0 exists such that if κ = κ 2, then d(s = h) = 1, d(s = l) = 0, and d(s = ) = 0 (buy-no trade-no trade, or BNN equilibrium). The key intuition behind proposition 3 is that the negatively informed speculator makes more profit than the uninformed speculator. Hence, they can be separated by a short selling cost that is higher than the profit of the uninformed speculator but lower than the profit of the negatively informed short seller. Of course, if the short-selling cost is high enough, then both of them will be discouraged from short selling. Based on the definitions, the comparison between the BSS equilibrium and the BSN equilibrium reveals the role of uninformed short selling. The comparison between the BNN equilibrium and the BSN equilibrium reveals the role of informed short selling. From now on, I define the ban on uninformed short selling as the switch from κ = 0 to κ = κ 1. I define the ban on informed short selling as the switch from κ = κ 1 to κ = κ 2. 4 Main Results With the preparation work in the previous section, in this section, I return to the main research question of this paper: the interaction between short selling and earnings management. More specifically, first, I will investigate the effect of banning short selling on earnings management. 14

Second, I will investigate the efficiency of banning short selling (i.e., whether firm value increases or decreases due to the ban on short selling). 4.1 Uninformed short selling A natural way to study the effect of banning uninformed short selling is to compare the BSS equilibrium with the BSN equilibrium. Proposition 4 below replicates the main result of Goldstein and Guembel (2008). It shows that if the earnings management level β is exogenous, uninformed short selling reduces firm value. Proposition 4 If the misreporting level β is exogenous, the DM prefers κ = κ 1 to κ = 0; The intuition for proposition 4 is as follows. Because conditional on r = H, the manager is more likely to be good, the uninformed short speculators distort shareholders decision by trading in the wrong direction, which leads to a higher chance that a good manager is fired. Hence, from an ex-post perspective, banning uninformed short selling is always efficient. To sum up, if the level of earnings management β is exogenous, the firm value in the BSN equilibrium is the highest. However, if the level of earnings management β is endogenously determined, the conclusion of proposition 4 may no longer hold. In fact, the level of earnings management can be ex-ante affected through two channels. These two channels are discussed below. The first channel is the stock price (order-flow) channel. As we can see from proposition 1, in the equilibrium, conditional on r = H, the shareholders will retain the manager if and only if the total order flow is bigger than a certain threshold. If uninformed short selling is banned, then ex post, the order flow will be higher, and issuing a r = H report will be more rewarding for the type L manager. The second channel is the threshold channel. If uninformed short selling is banned, then 15

in the equilibrium, the threshold q will adjust accordingly. Because q can go up or down, the incentive for earnings management could be more or less. Based on these two channels, if uninformed short selling is banned, the level of earnings management can go up or down. Proposition 5, however, shows that if the cost of earnings management is sufficiently low and if uninformed short selling is banned, earnings management level will become higher and firm value will become lower. For notation purposes, I use β 1 and β 2 to denote the equilibrium earnings management level in the BSS and the BSN equilibrium, respectively. Proposition 5 c (c 0, c ] exists such that when c (c 0, c), (1) the decision maker prefers κ = 0 to κ = κ 1, and (2) 0 < β 1 < β 2 = 1. The intuition for proposition 5 is as follows. First, when the cost of earnings management (c) is low, namely when β is large and close to 1, the threshold q should be close to 0, and therefore the room for q to change is minimal. Therefore, banning uninformed short selling will mainly cause the stock price to become higher. That is to say, the stock price channel will dominate the threshold channel. As a result, if uninformed short selling is banned, the low-type manager understands he is more likely to be retained if he can issue a high report. In other words, the marginal benefit of engaging in earnings management becomes higher if uninformed short selling is banned. For this reason, ex-ante, a higher level of earnings management will be chosen. Second, although banning uninformed short selling benefits the shareholders ex-post decision, ex ante, the ban of uninformed short selling leads to a higher level of earnings management. When the cost of earnings management (c) is low, the choice of earnings management is more elastic and therefore the ex-ante effect dominates. As a result, when c is low, firm 16

value goes down if uninformed short selling is banned. Therefore, the decision maker prefers κ = 0 over κ = κ 1. In fact, proposition 5 is equivalent to saying that uninformed short selling can deter earnings management. To understand this deterring effect, first note that ex-post, both uninformed short selling and informed short selling drive down stock price. Hence, when shareholders see a low stock price, they are not able to tell whom it comes from and have to fire the manager unconditionally if the fraction of informed speculators is large enough. The low-type manager anticipates the shareholders action and knows that even if he issues a high report, his chance of being retained becomes lower because the uninformed speculator drives down the stock price. Hence, in this case, the natural consequence of allowing uninformed short selling is less earnings management. 4.1.1 An Example One might be concerned proposition 5 relies on corner solution (β 2 = 1). Here I provide an analytically tractable example that does not rely on corner solution but delivers the same intuition as proposition 5. Suppose the distribution of noise trading (f(n)) is as follows: Noisy trading n -2-1 0 1 2 1 2 1 2 1 Probability 9 9 3 9 9 Noe the random variable n is now discrete, but its distribution f(n) still satisfies the MLRP property and the symmetry property. The random variable n can be decomposed as n = n 1 + n 2, where n 1 and n 2 are i.i.d. and take the value { 1, 0, 1} with probability 1 3 each. In other words, we can think of noise trading as submitted by two independent noise traders who buy, sell, and don t trade with equal probability. For notation purposes, I still use β 1 and β 2 for the level of earnings management in the BSS and the BSN equilibrium, respectively. Similar to proposition 2, proposition 6 shows that when c is sufficiently small, the BSS 17

equilibrium is the only possible outcome. Proposition 6 If κ = 0, the uninformed speculator short sells with probability 1 in any PBE if and only if c (0, 1 3 2α ). Further, β (0, 1) for c ( 1 3, 1 3 2α ). Proposition 7 illustrates that uninformed short selling can deter earnings management and save firm value. Proposition 7 Suppose c ( 1 3, 1 3 2α ), then (1) 0 < β 1 < β 2 < 1. (2) If α ( 1 4 (9 33), 1) and c ( 3 α 3, 1 3 2α ), the decision maker prefers κ = κ 1 to κ = 0. (3) Otherwise, if c ( 1 3, 3 α 3 ), the decision maker prefers κ = 0 to κ = κ 1. The intuition for proposition 7 is similar to that for proposition 5. Ex ante, uninformed short selling deters earnings management, as can be seen from β 1 < β 2. Ex post, uninformed short selling distorts shareholders decisions. Hence, whether uninformed short selling should be banned depends on the severity of the earnings management problem. When c is low, indicating a severe problem, uninformed short selling should be allowed as a weapon against earnings management. 4.2 Informed short selling This subsection compares the BSN equilibrium with the BNN equilibrium. Such comparison reveals the role of informed short selling. 4.2.1 Can banning informed short selling reduce misreporting? As one would expect, informed short selling can also affect β through two channels. The stock price (order-flow) channel is identical to the stock price channel in section 4.1. That is, banning informed short selling raises order flow and makes earnings management more rewarding. 18

The threshold channel, however, differs from the threshold channel in section 4.1. Here, if informed short selling is banned, the shareholders rationally understand that even a high stock price (order flow) may contain considerable amount of bad news. Hence, they tend to raise the equilibrium threshold. A higher threshold implies less incentive to misreport. Hence, depending on the trade-off between the two channels, the earnings management level can go up or down. Proposition 8 below gives an example in which banning informed short selling reduces the earnings management level. Proposition 8 Consider the same example in section 4.1.1. If c ( 1 3, 1 3 α ), then the switch from κ = κ 1 (the BSN equilibrium) to κ = κ 2 (the BNN equilibrium) will (1) raise the threshold from q = 0 to q = 1. (2) reduce the earnings management level from β = ( 2 3 1 3 α)/c to β = 1 3c. To understand proposition 8, first note q = 1 is a very high threshold, requiring the shareholders to have a very pessimistic belief about the manager even if r = H. Only when the shareholders suspect a high level of β, will they update their belief pessimistically. Hence, to sustain a high q in a rational expectation equilibrium, β has to be in the high range and therefore c has to be low, which explains the c < 1 3 α condition. But why does the earnings management level drop after informed short selling is banned? To understand this, remember that upon r = H, the shareholders will retain the manager if and only if the total order flow Q is higher than the threshold q. Also remember the manager will manage earnings only when his type is bad. Conditional on the manager being bad, the speculator can only be negatively informed or uninformed. The negatively informed speculator s order flow increases from 1 to 0 after informed short selling is banned. However, the threshold also increases by 1 (from q = 0 to q = 1). These two effects exactly offset each other. Hence, if the speculator is negatively informed, the ban on informed short selling will not change the bad manager s incentive to engage in earnings management. Nevertheless, 19

the threshold change is independent of the speculator s type. After informed short selling is banned, the uninformed speculator s order flow remains at 0 while the threshold still increases. Hence, if the speculator is uninformed, the ban on informed short selling will lower the bad manager s incentive to engage in earnings management. As a result, because the bad manager understands the speculator is either negatively informed or uninformed, overall, he will have less incentive to engage in earnings management. 4.2.2 Can banning informed short selling increase firm value? One might expect that because banning informed short selling can reduce the earnings management level, it can also increase firm value. However, the proposition below gives us the result that banning informed short selling always reduces firm value. Proposition 9 If α > 0, the firm value in the BSN equilibrium is strictly higher than the firm value in the BNN equilibrium. Proposition 9 tells us that although banning informed short selling may help discipline ex-ante misreporting, it actually distorts the ex-post detection of misreporting so much that firm value always decreases. Based on the results so far, from the ex-post perspective, both allowing uninformed short selling and banning informed short selling reduces firm value because both of them reduce the informativeness of stock price. From the ex-ante perspective, both allowing uninformed short selling and banning informed short selling can deter earnings management. However, when the ex-post and the ex-ante effects are both considered, banning informed short selling always reduces firm value, whereas allowing uninformed short selling can increase firm value. Why is the difference between the efficiency roles of informed and uninformed short selling so big? 20

To answer this question, we need to go back to the mechanism for uninformed short selling. Uninformed short sellers can profit from destroying firm value, because decision makers cannot distinguish them from informed short sellers. If a sufficient fraction of informed speculators is in the market, a low stock price implies a sufficient degree of negative information. As a consequence, even if a project can be good conditional on the speculator being uninformed, it will still be canceled. However, if no one is informed in the market and this information is common knowledge, uninformed speculators cannot profit from short selling, because a low stock price would no longer suggest negative information. Hence, uninformed short selling can distort ex-post decision making only when informed short selling provides a sufficient amount of useful information. Therefore, the distorting role of uninformed short-selling is secondary to the information-providing role of informed short selling, which explains why banning informed short selling always reduces firm value whereas allowing uninformed short selling can increase firm value. 5 Conclusions Although the ex-post role of short selling has been widely studied, the ex-ante role has not. In this paper, I link the ex-ante role of short selling to manager s misreporting problem, or earnings management. Specifically, I introduce managers misreporting incentive into the Goldstein and Guembel (2008) model. Similar to their results, the feedback effect generates the equilibrium whereby the uninformed speculator short sells. Such short-selling strategy decreases ex-post firm value because it increases the fraction of good managers fired (type I error). Hence, ex post, uninformed short-selling reduces firm value. On the ex-ante roles of short selling, my model predicts banning short selling can increase or reduce misreporting. Given that misreporting reduces firm value, I conjecture and prove 21

that banning uninformed short selling can reduce firm value. However, informed short selling are fundamentally different from uninformed short selling. I show that even though banning informed short selling can reduce misreporting, it always reduces firm value. This conclusion is equivalent to claiming the primary role of informed short selling is to convey bad news to the decision makers in the firms. References [1] Chen, Qi, Itay Goldstein, and Wei Jiang. Price informativeness and investment sensitivity to stock price. Review of Financial Studies 20.3 (2007): 619-650. [2] Diamond, Douglas W., and Robert E. Verrecchia. Constraints on short-selling and asset price adjustment to private information. Journal of Financial Economics 18.2 (1987): 277-311. [3] Dow, James, and Gary Gorton. Stock market efficiency and economic efficiency: is there a connection? Journal of Finance Vol. 52, No. 3: 1087-1129. [4] Durnev, Art, and Claudine Mangen. Corporate investments: Learning from restatements. Journal of Accounting Research 47.3 (2009): 679-720. [5] Durnev, Art, Randall Morck, and Bernard Yeung. Valueenhancing capital budgeting and firmspecific stock return variation. The Journal of Finance 59.1 (2004): 65-105. [6] Edmans, Alex, Jiang Wei, and Itay Goldstein. Feedback Effects, Asymmetric Trading, and the Limits to Arbitrage. The American Economic Review 105.12 (2015): 3766-3797. [7] Fang, Vivian W., Allen H. Huang, and Jonathan M. Karpoff. Short selling and earnings management: A controlled experiment. The Journal of Finance (2015). 22

[8] Gao, Pingyang, and Pierre Jinghong Liang. Informational Feedback, Adverse Selection, and Optimal Disclosure Policy. Journal of Accounting Research 51.5 (2013): 1133-1158. [9] Goldstein, Itay, and Alexander Guembel. Manipulation and the allocational role of prices. The Review of Economic Studies 75.1 (2008): 133-164. [10] Khanna, Naveen, and Richmond D. Mathews. Doing battle with short sellers: The conflicted role of blockholders in bear raids. Journal of Financial Economics 106.2 (2012): 229-246. [11] Kyle, Albert S. Continuous auctions and insider trading. Econometrica: Journal of the Econometric Society (1985): 1315-1335. [12] Luo, Yuanzhi. Do insiders learn from outsiders? Evidence from mergers and acquisitions. The Journal of Finance 60.4 (2005): 1951-1982. [13] Massa, Massimo, Bohui Zhang, and Hong Zhang. The Invisible Hand of Short Selling: Does Short Selling Discipline Earnings Management?. Review of Financial Studies 28.6 (2015): 1701-1736. A Appendix A.1 proof of proposition 1 Lemma 3 (monotone likelihood ratio property) In any PBE, the ratio Pr(Q=q θ=h) Pr(Q=q θ=l) increasing in q. Further, Pr(Q=q θ=h) Pr(Q=q θ=l) q=0 = 1. is Proof: By lemma 1, the positively informed speculator buys and the negatively informed speculator short sells. Suppose that in equilibrium, the uninformed speculator buys, doesn t 23

trade, and short sells with probability µ 1, µ 2, and µ 3, respectively. Hence, Pr(Q = q θ = H) Pr(Q = q θ = L) = αf(q 1) + (1 α)(µ 1f(q 1) + µ 2 f(q) + µ 3 f(q + 1) αf(q + 1) + (1 α)(µ 1 f(q 1) + µ 2 f(q) + µ 3 f(q + 1) = (α + (1 α)µ 1)f(q 1)/f(q + 1) + (1 α)µ 2 f(q)/f(q + 1) + (1 α)µ 3 (1 α)µ 1 f(q 1)/f(q + 1) + (1 α)µ 2 f(q)/f(q + 1) + α + (1 α)µ 3 α(f(q 1)/f(q + 1) 1) = 1 + (1 α)µ 1 f(q 1)/f(q + 1) + (1 α)µ 2 f(q)/f(q + 1) + α + (1 α)µ 3 α = 1 + (1 α)µ 1 + (1 α)µ 2 f(q)/f(q 1) + (α + (1 α)µ 3 )f(q + 1)/f(q 1) α. (1 α)µ 1 f(q 1)/f(q + 1) + (1 α)µ 2 f(q)/f(q + 1) + α + (1 α)µ 3 By the MLRP property, both the second and the third term is increasing in q, and we conclude d Pr(Q=q θ=h) Pr(Q=q θ=l) dq > 0. Q.E.D. Now, upon observing r = H and Q = q, the shareholders will retain the manager if and only if E[θ Q = q, r = H] 0 Pr(θ = H Q = q, r = H)V Pr(θ = L Q = q, r = H)V 0 Pr(Q = q θ = H) Pr(Q = q θ = H) + β Pr(Q = q θ = L) V β Pr(Q = q θ = L) Pr(Q = q θ = H) + β Pr(Q = q θ = L) V 0 Pr(Q = q θ = H) Pr(Q = q θ = L) β From the proof of lemma 2 we know Pr(Q = q θ = H) lim q 0 Pr(Q = q θ = L) = 1 lim q Pr(Q = q θ = H) Pr(Q = q θ = L) = (1 α)µ 3 α + (1 α)µ 3. Pr(Q=q θ=h) If lim q Pr(Q=q θ=l) < β, q (, 0] exists such that Pr(Q=q θ=h) Pr(Q=q θ=l) β if and only if q q. The following equation determines the cut-off q : Pr(Q = q θ = H) Pr(Q = q θ = L) = β. (1) If lim q Pr(Q=q θ=h) Pr(Q=q θ=l) β, the manager will always be retained, which means q =. Besides, by lemma 4, we know that when β = 1, only q = 0 satisfies (1). Q.E.D. 24

A.2 proof of lemma 2 Suppose that in equilibrium, the uninformed speculator buys, doesn t trade, and short sells with probability µ 1, µ 2, and µ 3, respectively. The goal is to show µ 3 = 0. I use proof by contradiction. Suppose otherwise that µ 3 > 0, then the uninformed speculator s profit from buying is π d=1 = = V V + q + (E[θ r = H] E[θ Q = q, r = H])f(q 1)dq 1 Pr(Q = q θ = H)/ Pr(Q = q θ = L) f(q 1)dq q Pr(Q = q θ = H)/ Pr(Q = q θ = L)/β + β + Pr(Q = q θ = H)/ Pr(Q = q θ = L) + 1 1 Pr(Q = q θ = H)/ Pr(Q = q θ = L) f(q 1)dq. Pr(Q = q θ = H)/ Pr(Q = q θ = L)/β + β + Pr(Q = q θ = H)/ Pr(Q = q θ = L) + 1 (2) + The last step is due to q 0 and Pr(Q = q θ = H)/ Pr(Q = q θ = L) < 1 when q < 0. Now I decompose (2) into the following two components. RHS of (2) = V V + 0 0 1 Pr(Q = q θ = H)/ Pr(Q = q θ = L) f(q 1)dq+ Pr(Q = q θ = H)/ Pr(Q = q θ = L)/β + β + Pr(Q = q θ = H)/ Pr(Q = q θ = L) + 1 1 Pr(Q = q θ = H)/ Pr(Q = q θ = L) f(q 1)dq Pr(Q = q θ = H)/ Pr(Q = q θ = L)/β + β + Pr(Q = q θ = H)/ Pr(Q = q θ = L) + 1 When q > 0, f(q 1) > f(q + 1). Therefore, the denominator in the first term is less than the denominator in the second term. Therefore, RHS of (2) < 0 and π d=1 RHS of (2) < 0, which is a contradiction because, by deviating to d = 0, the uninformed speculator can increase his profit to 0. Hence, we must have µ 3 = 0. Q.E.D. A.3 proof of proposition 2 A.3.1 step 1: preparation I use µ 1, µ 2 to denote the probability that the uninformed speculator plays d = 0 and d = 1. Therefore, in equilibrium, conditional on r = H, the probability that a type L manager will 25

be retained is (I use the notation γ) γ(q, µ 1, µ 2 ) = µ 1 (1 F (q )) + µ 2 ((1 α)(1 F (q + 1))) + α(1 F (q + 1)). (3) Further, I use π d= 1, π d=0 to denote the uninformed speculator s profit in the equilibrium if he plays d = 1 and d = 0, respectively. Note π d=0 = 0. Lemma 4 (1) Any pure strategy equilibrium with interior earnings management level β is determined by the following conditions: cβ = γ(q, µ 1, µ 2 ), Pr(Q = q θ = H) Pr(Q = q θ = L) = β, µ 1 = 1 if π d=0 = 0 π d= 1, and µ 2 = 1 if π d= 1 π d=0 = 0. (2) Any mixed-strategy equilibrium with interior earnings management level β is determined by the following conditions: cβ = γ(q, µ 1, µ 2 ), Pr(Q = q θ = H) Pr(Q = q θ = L) = β, and µ 1 = µ 2 > 0 if π d=0 = π d= 1 = 0. Proof: Type L manager s problem is max β βγ(ˆq) 1 2 cβ2 FOC cβ = γ(ˆq, µ 1, µ 2 ), where ˆq is the manager s guess of q. Applying the equilibrium condition q = ˆq, the FOC becomes cβ = γ(q, µ 1, µ 2 ). (4) 26

In addition, the threshold level q is determined by Pr(Q = q θ = H) Pr(Q = q θ = L) = β. (5) The equilibrium is thus determined by (4), (5), and the non-deviation conditions. Lemma 5 In any PBE, E[θ Q = q, r = H] E[θ r = H] > 0 if and only if q > 0. Further, when q > 0, E[θ Q = q, r = H] E[θ r = H] increases in β. Hence, when q > 0, we could write g(q, β) E[θ Q = q, r = H] E[θ r = H] with g 1 > 0 and g 2 > 0. Proof: To start, note Pr(Q = q θ = H) E[θ Q = q, r = H] = Pr(Q = q θ = H) + β Pr(Q = q θ = L) V β Pr(Q = q θ = L) Pr(Q = q θ = H) + β Pr(Q = q θ = L) V = Pr(Q = q θ = H)/ Pr(Q = q θ = L) Pr(Q = q θ = H)/ Pr(Q = q θ = L) + β V β Pr(Q = q θ = H)/ Pr(Q = q θ = L) + β V. Hence, E[θ Q = q, r = H] increases in Pr(Q=q θ=h) Pr(Q=q θ=l), which increases in q by lemma 3. As a result, E[θ Q = q, r = H] E[θ r = H] increases in q. Because when q = 0, Pr(Q=q θ=h) Pr(Q=q θ=l) = 1 and E[θ Q = q, r = H] = 1 β 1+β V = E[θ r = H]. Therefore, E[θ Q = q, r = H] > E[θ r = H] if and only of q > 0. It remains to show E[θ Q = q, r = H] E[θ r = H] increases in β when q > 0. Pr(Q = q θ = H)/ Pr(Q = q θ = L) β E[θ Q = q, r = H] E[θ r = H] = Pr(Q = q θ = H)/ Pr(Q = q θ = L) + β V 1 β 1 + β V. β(pr(q = q θ = H)/ Pr(Q = q θ = L) 1) = (Pr(Q = q θ = H)/ Pr(Q = q θ = L) + β)(1 + β) Pr(Q = q θ = H)/ Pr(Q = q θ = L) 1 = Pr(Q = q θ = H)/ Pr(Q = q θ = L)/β + β + Pr(Q = q θ = H)/ Pr(Q = q θ = L) + 1. When q > 0, Pr(Q = q θ = H)/ Pr(Q = q θ = L) > 1 and that Pr(Q = q θ = H)/ Pr(Q = q θ = L)/β + β decreases in β. Therefore, E[θ Q = q, r = H] E[θ r = H] increases in β when q > 0. This completes the proof of lemma 6. Lemma 6 Let c 0 = γ(0, 0, 1) = 1 F (1). If c c 0, β = 1, then µ 1 = 0 and µ 2 = 1. 27

Proof: Use proof by contradiction. Suppose otherwise that β < 1. Then cβ c 0 β < c 0 = 1 F (1) γ(q, µ 1, µ 2 ), which suggests the marginal benefit of earnings management exceeds the marginal cost. Therefore, β = 1, which is a contradiction. Hence we must have β = 1 and q = 0. By lemma 4, when q = 0, the uninformed speculator earns a strictly positive profit from short selling. If he does not trade, he earns zero. If he buys, he makes a strictly negative profit. Hence he must short sell with probability 1. Lemma 7 δ > 0 exists such that when c (c 0, c 0 + δ), the equilibrium β, µ 1, and µ 2 are all functions of c. We could thus write β = β(c), µ 1 = µ 1 (c), µ 2 = µ 2 (c), and they are all continuous functions. Proof: The implicit function theorem. A.3.2 step 2: the uninformed speculator short sells with probability 1 Suppose otherwise that the uninformed speculator does not trade with strictly positive probability. Let s consider the deviation from no trading to short selling. By making this deviation, the uninformed speculator s profit is: π d= 1 = = + q 0 + 0 0 = (E d=0 [θ Q = q, r = H] E[θ r = H])f(q + 1)dq q (E d=0 [θ Q = q, r = H] E[θ r = H])f(q + 1)dq+ (E d=0 [θ Q = q, r = H] E[θ r = H])f(q + 1)dq q (g(q, β))f(q + 1)dq + + 0 (g(q, β))f(q + 1)dq. Now, because lim c c0 β = 1, we know for any ɛ 1 > 0, δ 1 > 0 exists such that when c (c 0, c 0 + δ 1 ), β > 1 ɛ 1. By lemma 5, the integrand in the second integral is always positive 28

and takes its smallest value when β = 1 ɛ 1. That is, when q > 0, g(q, β) g(q, 1 ɛ 1 ) > 0. Therefore, we know + 0 g(q, β)f(q + 1)dq + 0 g(q, 1 ɛ 1 )f(q + 1)dq M 2. We know when c c 0, β 1 and q 0. Hence, for ɛ (0, 1), δ 2 > 0 exists such that c 0 < c < c 0 + δ 2 implies 0 Now for c 0 < c < c 0 + δ 2, we have π d= 1 > q (E[θ r = H] E(θ Q = q, r = H))f(q + 1)dq ɛm 2. 0 q (E[θ r = H] E(θ Q = q, r = H))f(q + 1)dq + M 2 0 0. q (E[θ r = H] E(θ Q = q, r = H))f(q + 1)dq + ɛm 2 Take c = min(c 0 + δ, c 0 + δ 1, c 0 + δ 2 ). We know when c (c 0, c ), short selling guarantees strictly positive expected profit for the uninformed speculator. He will thus deviate to short selling with probability 1 and this completes step 2. Q.E.D. A.4 proof of proposition 3 By proposition 2, when κ = 0 and when c (c 0, c ), the uninformed speculator short sells with probability 1. Let κ 1 = + 0 V f(q + 1)dq. In any equilibrium, the uninformed speculator s profit from short-selling is: π s= + d= 1 = (E(θ Q = q, r = H) E[θ r = H])f(q + 1)dq < q + 0 + 0 (E(θ Q = q, r = H) E[θ r = H])f(q + 1)dq V f(q + 1)dq = κ. 29

Hence when κ = κ 1, the uninformed speculator will not short sell. Besides, for the negatively informed speculator, his profit from short selling is + d= 1 = (E(θ Q = q, r = H) + V )f(q + 1)dq π s=l > q + 0 + 0 (E(θ Q = q, r = H) + V )f(q + 1)dq V f(q + 1)dq = κ. Therefore, when κ = κ 1 the negatively informed speculator still short sells with probability 1. Finally, let κ 2 = 2V. It can be easily seen that when κ κ 2, both the uninformed and the negatively informed speculator play d = 0. A.5 proof of proposition 4 I use q 1 and q 2 to denote the equilibrium threshold q in the BSS and the BSN equilibrium, respectively. If β is exogenous, then by equation (1), q 1 and q 2 are determined by α f(q 1 1) + (1 α) = β. f(q 1 + 1) (6) αf(q 2 1) + (1 α)f(q 2 ) = β. αf(q 2 + 1) + (1 α)f(q 2 ) (7) I use v 1 (q) and v 2 (q) to denote the ex-ante firm value as a function of the threshold q, under the BSS and the BSN, respectively. We have v 1 (q) = 1 2 V ((1 α)(1 F (q + 1)) + α(1 F (q 1))) 1 }{{} 2 V β(1 F (q + 1)) }{{} the probability that the good manager is retained the probability that the bad manager is retained (8) v 2 (q) = 1 2 V ((1 α)(1 F (q )) + α(1 F (q 1))) 1 }{{} 2 V β((1 α)(1 F (q )) + α(1 F (q + 1))). }{{} the probability that the good manager is retained the probability that the bad manager is retained (9) I want to show v 2 (q 2 ) v 1 (q 1 ). I first prove a lemma. Lemma 8 v 1 (q 1 ) v 2 (q 2 ), v 2 (q 2 ) v 1 (q 1 ). 30