Feedback Effect and Capital Structure

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1 Feedback Effect and Capital Structure Minh Vo Metropolitan State University Abstract This paper develops a model of financing with informational feedback effect that jointly determines a firm s capital structure and cost of capital. We show that under some conditions capital structure affects traders incentives to produce information about the prospect of the firm which they then use to trade in its securities. On one hand, informed trades incorporate new information into security prices which in turn help the firm make more efficient operating decisions. On the other hand, they increase the cost of capital as uninformed investors demand extra compensation in the form of a liquidity premium in anticipation that they will lose to informed traders. The optimal capital structure which maximizes the value of the firm is determined by the trade-off between high operating efficiency and low cost of capital. When information is not imperative for the firm s operating decisions, Modigliani-Miller s irrelevance is maintained. However, when information is crucial for efficient operating decisions, the optimal capital structure is a balance between a high level of information revelation and a low cost of capital. The study can explain why many firms consistently hold low levels of debt and why firms with similar fundamentals may choose very different capital structures. JEL Classification: G14, G30, G3 Keywords: Feedback effect, Capital structure, Cost of capital 1

2 1 Introduction Financial markets play an important role in the economy. They are places where investors trade to profit from their information. In the trading process security prices aggregate all information from various sources and provide information that would be hard to generate otherwise. Although managers know a great deal about their own firms, to make good operation decisions they also need other things, for instance, information about consumers, suppliers, other firms in the industry, other industries, etc. Such information is dispersed among many agents in the economy and can only be aggregated efficiently via trades in financial markets. Security prices thus contain useful information that can help firms make efficient operation decisions. In literature, this is known as a feedback effect - prices not only reflect the cash flows generated by firms but also affect them (see Bond et al. (01) for a survey.) Feedback effects have been documented empirically in literature. Chen et al. (007) provide evidence that that managers learn new information from stock prices and incorporate it into corporate investment decisions. Luo (005) shows that merging companies tend to extract information from market reactions and take it into account when they close deals. Baker et al. (003) demonstrate that stock prices have a stronger impact on the investments of firms that rely on external equity to finance their marginal investments. Edmans et al. (01) show that financial markets have real effects by affecting the behavior of managers. On the theory side, financial economists incorporate feedback effects into models of financial markets to better understand the price formation process and its implications for the real economy. This includes Subrahmanyam and Titman (1999); Fulghieri and Lukin (001); Dow and Rahi (003); Foucault and Gehrig (008); Goldstein and Guembel (008); Chang and Yu (004, 010); Bond et al. (010), to name just a few. In this paper we study a model of financing with an informational feedback effect that jointly determines capital structure and cost of capital. Modigliani and Miller (1958) state that in perfect markets capital structure does not matter: firms with the same future cash flow distribution will have the same market value regardless of their financing methods. This

3 result rarely holds in practice as markets are not perfect. Frictions have been added to the standard theory to establish an interior financing optimum. However, researchers report that in reality firms choose capital structures that differ from the theoretically optimal ones. For instance, Graham (000) shows that firms are very conservative in using leverage. Furthermore, firms with similar fundamentals may choose very different capital structures. Recently, financial economists have tried a behavioral finance approach to explain how firms choose their capital structure. Malmendier et al. (011); Cronqvist et al. (01) suggest that leverage decisions are related to the personal characteristics of firms CEOs. Hackbarth (008, 009) use models with managerial traits to show that optimistic and/or overconfident managers may employ leverage aggressively and need not follow a pecking order. In this paper we incorporate informational feedback effect into a financing model in an attempt to answer the following research questions: (i) whether and to what extent a firm can use leverage to benefit from information production in financial market; (ii) whether there exists an optimal capital structure which maximizes the value of the firm; (iii) why firms with similar fundamentals could have very different capital structures. Our model features a firm facing a choice between a risky high growth strategy and a riskless conservative strategy, and then issuing debt and equity in primary markets to finance its business. The securities are then traded in secondary markets among uninformed investors who have liquidity needs and a speculator who can obtain costly information about the firm s profitability to trade for profit. The speculator chooses the optimal amount of information to produce to maximize profit. Trading causes some speculator s information to be incorporated into security prices. The firm then extracts information from prices for its operation decisions. The market makers set efficient security prices taking into account the fact that their prices will affect the firm s operations and therefore its cash flows. Thus, security prices both reflect and affect the firm s fundamentals - the so-called feedback effect. We show that when the firm has complete information or when information is not imperative for its decision making, the Modigliani-Miller s irrelevance is maintained. In this case, the speculator does not acquire 3

4 costly information since he cannot benefit from it. The cost of capital is thus zero and there is no feedback effect. When information extracted from security prices is crucial for the firm to make efficient operation decisions, the feedback effect does exist. However, whether capital structure matters depends on the level of leverage the firm employs. If the firm issues no bond or riskless bond its value is independent of capital structure. Although the firm still benefits from the feedback effect, it has no control over information production and cost of capital. However, if the firm issues risky bonds, capital structure does matter and there exists an optimal level of leverage which maximizes the value of the firm. Information produced in the market is negatively related to the leverage level. On one hand, when traders produce more information, they make security prices more informative which help the firm make efficient operation decisions. On the other hand, they increase the cost of capital as uninformed investors demand extra compensation in the form of a liquidity premium in anticipation that they will lose to informed traders. The optimal capital structure trades off the two effects on the firm s value. Our model predicts that under some conditions, high growth firms tend to use less debt than more established ones. Firms with similar fundamentals but different in information need can have very different capital structures. The study features a game theoretic model in which the speculator takes the firm s leverage as given and chooses the amount of costly information to produce to maximize his trading profit. Under some conditions, his information production is a function of the leverage. The firm then takes the speculator s information as given and chooses its capital structure to maximize its own value. In equilibrium, the information produced by the speculator is positively related to the market liquidity but negatively related to the level of debt. Intuitively, if the market is more liquid, the speculator can trade higher volumes without being detected and therefore get higher profit. He, thus, increases his information production. In addition, since equity is more information sensitive than debt, the speculator can benefit more when the firm issues more equity (less debt). Hence, he produces more information when the firm 4

5 uses less debt. The cost of capital rises with the speculator s profit and therefore with his information production. The optimal level of debt is increasing in the market liquidity and decreasing in the risk of the high growth strategy. Intuitively, when the market is highly liquid, the speculator increases his information acquisition and reveals more useful information via trades which helps the firm make efficient operation decisions. However, this also raises the firm s cost of capital since it is the profit of the speculator. To balance the two opposite effects, the firm needs to raise debt level. Other things equal, as the risk of the high growth strategy increases, the firm needs more information from the speculator to make efficient decisions and therefore it reduces debt. Thus, the model can explain why many firms consistently hold low levels of debt. The remainder of the paper is organized as follows. In Section we describe the model, derive the equilibrium and discuss the optimal information acquisition of the speculator. In Section 3, we derive the optimal capital structure and discuss the implications. Section 4 concludes the paper. The model.1 The economic setting There are four dates. Market participants include a firm (a manager), a speculator, a bond market maker, a stock market maker, and a number of liquidity traders in stock and bond markets. All of them are risk neutral. The interest rate is normalized to zero. At date 0, the manager chooses capital structure (the face value of the debt) F to maximize the value of the firm and then issues stocks and bonds to raise capital. We assume that the market makers and the speculator do not participate in the primary markets. 1 At date 1, the speculator acquires information about the firm and trades take place in the stock and bond secondary 1 This is a common assumption in literature to induce cost of capital. Since the market makers and the speculator are risk neutral and do not experience liquidity shocks, their participations in the primary markets would drive out liquidity traders and therefore eliminate the cost of capital. 5

6 Figure 1: The timeline markets. At date, after observing security prices, the manager makes an operation decision. At date 3, cash flows are realized and security holders are paid accordingly. The time line is summarized in Figure 1. In the following, we will provide more information about each market participant. The firm The firm owns a production technology which converts capital stochastically into cash flow θ at date 3. The value of the firm depends on the operation decision the manager makes at date. For simplicity, we assume that the manager chooses either conservative or growth investment strategy. The conservative strategy is riskless and yields a certain cash flow of θ c = M. The growth strategy, on the other hand, is risky. Its cash flow θ g is a function of a random variable µ which takes two values, H or L, with equal probability. More specifically, m + e θ g = m e if µ = H if µ = L, 6

7 where m > M > m e. Thus, the growth strategy yields higher expected cash flow. Given the firm s risk neutrality, the manager only chooses the conservative strategy when he knows µ = L with high probability. The term e can be considered as the payoff volatility (or risk) of the growth strategy. The level of debt F is assumed to be smaller than M. Thus, the debt is riskless if the manager chooses the conservative strategy. At date 1, the manager learns a signal x at no cost which reveals µ perfectly with probability f and is uninformative with probability 1 f. In this setting, f measures the quality of the firm s information. The speculator The speculator is able to acquire costly information to trade for profit. Thus, his problem is to choose the amount of information to maximize expected profits. At date 1, he expends resources to acquire signal y {h, l} such that prob ( µ = H y = h) = prob ( µ = L y = l) = 1 + α, where α [0, 1] at the cost C (α) = 1 cα. Thus, the more resources he spends, the higher quality is his signal. However, the cost of the signal increases at a faster rate than its precision. Liquidity traders Liquidity traders buy or sell securities for reasons exogenous to the model. At date 0, those investors buy the firm s securities. At date 1, they experience liquidity shocks and have to trade. The aggregate shares of liquidity trades in the stock (bond) market are denoted by n S (n B ) which takes values of σ S (σ B ) or σ S ( σ B ) with equal probability. Thus, σ S (σ B ) represents both the volatility of the liquidity trades and the stock (bond) market depth. In practice, the stock market is deeper and more volatile than the bond market; therefore, we assume that σ S > σ B and σ s σ B is approximately unchanged. 7

8 Case Probability Firm s information Speculator s information Prices 1 f/ H P (H) f/ L P (L) 3 1 f y P (y) Market makers Table 1: Information Structure The market makers observe aggregate order flows without knowing which order comes from which trader. They then set competitive prices to break even as in Kyle (1985). Since there is informational feedback effect from markets to the firm, the market makers take into account the impact of their prices on the value of the firm and vice versa. In other words, prices both reflect and affect the value of the firm. The information structure is summarized in Table 1. We observe that when the firm has complete information, security prices reflect its information. When it has no information, security prices are functions of the speculator s information.. The equilibrium We employ the notion of Bayesian-Nash equilibrium in this paper. Definition. A Bayesian-Nash equilibrium of the trading-investment game consists of (i) the firm s strategies on capital structure and investment, and belief given the market prices and its information, (ii) the speculator s trading strategies in the bond and equity markets, (iii) the market makers pricing rules and their beliefs given order flows, i.e. the posterior distribution of the firm s value given the aggregate order flows, such that: 1. The firm maximizes its value at date 0;. The speculator maximizes his expected profit; 3. The market makers set efficient security prices and break even; 4. The beliefs are consistent with Bayes rule wherever applicable. 8

9 We use the backward induction approach to derive the equilibriums. We will separately examine equilibriums when the firm has complete information (cases 1 and ) and when it has no information at all (case 3). Subscripts S and B will be used to denote stock and bond, respectively...1 Equilibrium when the firm has complete information Let P 0B (.), P 0S (.), π (.), and V 0 (.) be the bond and stock prices, the cost of capital, and the value of the firm, respectively, at time 0. Since the manager knows the realization of µ, he uses it to maximize the terminal cash flow. In particular, he chooses the growth strategy if µ = H and the conservative strategy if µ = L. Thus, the bond is riskless and so P 0B (H) = P 0B (L) = F. Given his investment strategy, the terminal cash flow is θ (H) = m + e and θ (L) = M. The stock price is P 0S (H) = θ (H) P 0B (H) = m + e F and P 0S (L) = θ (L) P 0B (L) = M F. Since security prices reflect all information, the speculator does not trade. Thus, the cost of capital is zero. Proposition 1 summarizes the above results. Proposition 1. If the firm has complete information, then : P 0B (.) = F, P 0S (H) = m + e F P 0S (L) = M F, π = 0, V 0 (H) = m + e, V 0 (L) = M Proposition 1 says that when the firm has full information, the value of the firm is not affected by the capital structure. The markets are of strong-form efficiency and the cost of capital is zero. 9

10 .. Equilibrium when the firm has no information In this case, by acquiring information the speculator has an information advantage over the manager and the market makers. He trades to exploit his information advantage. It should be noted that the speculator always trades stock since it is sensitive to information. However, he only trades risky bond since riskless bond is information insensitive and the interest rate is zero. More specifically, if F m e, the speculator trades only stock. If F > m e, he trades both stock and bond. In order to make money, he must disguise his information from the market makers who try to set efficient security prices given the aggregate order flows. He buys when he receives good news (y = h), sells when he gets bad news (y = l). To keep his information from being revealed completely to the market makers, he has to trade as if he were liquidity traders. Thus, his optimal strategy is buy σ S (σ B ) shares of stock ( risky bond) when y = h and sell σ S (σ B ) shares of stock (risky bond) when y = l. Formally, let d i (y), i = B, S be the trading strategy of the speculator in market i, then σ i d i (y) = σ i if y = h if y = l. Assuming the bond and stock market makers communicate with each other to incorporate all publicly available information into security prices, then markets are of semi-strong-form efficiency. Let X i, i = B, S be the aggregate order flow observed by the market maker in market i then X i = d i (y) + n i. Given the trading strategy of the speculator and the distribution of n i, X i can take 3 possible values: X i = σ i when the liquidity trades n i = σ i and y = h. X i = σ i when n i = σ i and y = l. 10

11 X i = 0 when n i = σ i and y = l, or n i = σ i and y = h. Thus, the market makers can infer precisely that y = h when they observe an X i = σ i, and y = l when they see an X i = σ i in which case, the speculator s information is revealed to the markets and he earns zero profit. However, when both X S and X B are zero, the market makers cannot infer y. As a result, the speculator s payoff is positive and security prices do not fully reflect y. Note that if the speculator deviates from this trading strategy, his signal is always revealed to the market makers. The manager inverts security prices to learn y and uses it to make investment decision. His investment strategy is as follows: 1. if y = h, follow growth strategy.. if y = l, follow conservative strategy. 3. If y is not revealed, follow growth strategy since the firm is risk neutral and the expected return of the growth strategy is higher than that of the conservative strategy. Proposition. If F m e the speculator does not trade bond. His optimal information acquisition α m, stock price P 0S, bond price P 0B, the cost of capital π, and the value of the firm V 0 are given by: α m = σ Se c, (1) π = σ S e 8c, () P 0S = 3m + M 4 P 0B = F, F + σ Se 8c σ S e 8c, V 0 = m [M (m α me)] π. (3) Proof. See Appendix. 11

12 Proposition indicates that when the firm has no information and chooses riskless debt, capital structure does not matter although the feedback effect still exists via stock market. Equation (1) shows that the speculator s optimal information acquisition is positively related to the stock market depth σ S and the risk of the growth strategy e but negatively related to the information cost c. A deeper stock market allows him to trade higher volume to exploit his information to a greater extent and thus get a high payoff. The riskier growth strategy makes information more valuable given the cost, so the speculator increases its acquisition. As a result, the cost of capital, which is the speculator s payoff from exploiting his information, is also positively related to the stock market depth and the risk of the growth strategy and negatively related to the information cost c. The value of the firm V 0 (equation (3)) consists of three terms. The first term is the value of the firm if there is no feedback effect nor cost of capital. The second term is the gain from the feedback effect. Recall that with the feedback effect, the manager chooses the conservative strategy if the stock price indicates that the speculator s signal is l. Thus, the gain is M (m α m e), where α m is the quality of his information. The gain is multiplied by 1 4 which is the probability that signal l is revealed. Note that the gain could be negative (or loss) depending on α m. In the extreme case, α m = 0, it is a loss since M < m by definition. The last term π is the cost of capital which is the discount liquidity traders require at the time of investing in the firm in anticipation that they might lose to the speculator at a later date. If the firm chooses risky debt (F > m e), the speculator trades both stock and bond since both are sensitive to information. The manager can thus extract information from the prices of both securities. Table presents all eight possibilities. Assuming all variables are independent, each possibility happens with probability 1 8. P 1i (y) is the price of security i at time 1 when y is revealed to the market makers and P 1i ( ) is the price of security i at time 1 when y is not revealed. We observe that with probability 3, the speculator s trades reveal 8 y = h or y = l; with probability 1, his trades do not reveal his information. 4 Proposition 3. If F > m e, the debt is risky and the speculator trades both stock and 1

13 y d B (y) d S (y) n B n S X B X S Price Firm s strategy h σ B σ S σ B σ S σ B σ S P 1i (h) Growth h σ B σ S σ B σ S σ B 0 P 1i (h) Growth h σ B σ S σ B σ S 0 σ S P 1i (h) Growth h σ B σ S σ B σ S 0 0 P 1i ( ) Growth l σ B σ S σ B σ S 0 0 P 1i ( ) Growth l σ B σ S σ B σ S 0 σ S P 1i (l) Conservative l σ B σ S σ B σ S σ B 0 P 1i (l) Conservative l σ B σ S σ B σ S σ B σ S P 1i (l) Conservative Table : Firm s investment strategy given security prices bond. Furthermore, if the aggregate order flows reveal the speculator s information then his payoff is zero and the security prices at time 1 are given by: P 1B (y = h) = F 1 + α + (m e) 1 α, P 1S (y = h) = (m + e F ) 1 + α, P 1B (y = l) = F, P 1S (y = l) = M F. Proof. See Appendix. With the feedback effect, the manager extracts information from security prices and uses it to make real investment decisions for the firm. In doing so, he is able to pursue growth with lower downside risk. A surprising consequence of the feedback effect is that the bond is riskless when the speculator receives unfavorable news (y = l) but risky when he receives favorable news (y = h), P 1B (y = l) P 1B (y = h). This is because if the manager gets an l signal from the markets, he follows the conservative strategy to avoid the bad outcome. When he gets an h signal, he engages in the high growth strategy; however, since the signal y might not be perfect, there is some chance that the investment fails; therefore the bond is risky. In the extreme case, if the speculator gets perfect signal, α = 1, the bond is always 13

14 riskless. For stock, if α is big enough P 1S (y = h) > P 1S (y = l). Thus, in order to maximize the value of the firm, the manager needs to select the optimal capital structure which encourages the speculator to get more accurate signal. On one hand, more information helps the speculator get higher payoff from trades which translates to higher cost of capital since liquidity traders requires a discount in anticipation that they will lose to informed trades. On the other hand, when the speculator trades on his information, security prices become more informative which helps the manager make more efficient operation decisions. The optimal capital structure trades off the two effects. Proposition 4. If F > m e and if the aggregate order flows do not reveal the speculator s information, then the security prices at time 1 and the cost of capital π are given by: P 1B ( ) = 1 (m e + F ), P 1S ( ) = 1 (m + e F ), π ( ) = α [(σ S + σ B ) e + (σ S σ B ) (m F )]. (4) Proof. See Appendix. When order flows do not reveal the speculator s information, the risk-neutral manager selects the growth strategy since it has higher expected return. Security prices are the average of security values in both states µ = H and µ = L. Note that if µ = L, the firm goes bankrupt and its stock value is zero. Equation (4) relates the cost of capital to the quality of the speculator s information α, the risk of the growth strategy e, the market depth σ S + σ B and the capital structure. Other things equal, the cost of capital is increasing with the quality α of the speculator s information. Higher information quality leads to greater speculator s payoff and therefore higher cost of capital. 14

15 increasing with the volatility of the growth strategy e. Due to the fact that he is better informed than other market participants, higher volatility of the growth strategy allows the speculator to exploit his information to a greater extent for profit. increasing with the total market depth σ S + σ B. As the speculator must camouflage in liquidity traders to keep his information from being revealed, deep markets help him trade large volume without being detected, resulting in a higher cost of capital. Note that we assume σ S σ B is positive and unchanged. This is in contrast to the conventional wisdom that deep markets lead to lower cost of capital. decreasing with the level of debt F. This is because equity is more information sensitive than debt. Higher debt leads to lower equity available to trade in the market which reduces the speculator s profit. As a result, the cost of capital is lower. We next derive the optimal information acquisition α m of the speculator when the manager, who has no information about the growth strategy, chooses risky debt. Propositions 3 and 4 show that the speculator s payoff (or cost of capital) is positive if and only if his information is not revealed, which happens with probability 1 4 (Table.) Thus, his expected payoff is 1π ( ), with π ( ) is given in equation (4). Taking into account the cost of information, the 4 speculator solves the following profit maximizing problem: max α Π (α) = 1 4 π ( ) 1 cα (5) S.t. Π (α) 0. (6) Lemma 1. If F > m e, the profit-maximizing information acquisition α m of the speculator, his profit Π (α m ) and the cost of capital π (α m ) are given by: α m = 1 8c [(σ S + σ B ) e + (σ S σ B ) (m F )], (7) We distinguish between payoff and profit. Payoff is the gain from trading securities while profit is the payoff minus the cost of information acquisition. 15

16 Proof. See Appendix. Π (α m ) = 1 cα m = 1 18c [(σ S + σ B ) e + (σ S σ B ) (m F )]. (8) π (α m ) = cα m = 1 64c [(σ S + σ B ) e + (σ S σ B ) (m F )]. (9) Equation (7) shows that the speculator s optimal information acquisition depends on market depths (σ S and σ B ), the risk of the growth strategy e and the level of debt F. Other things equal, the speculator increases his information acquisition if the cost of the information c is low. the markets are more liquid. This is because high liquidity allows him to trade large volume without being detected which results in higher profit. Furthermore, he acquires more information if the liquidity difference between stock and bond market is larger. This can be explained by the fact that trading stock is more profitable than bond as stock is more sensitive to information. the volatility of the growth strategy increases. Higher volatility allows the speculator to exploit his information advantage to a greater extent for profit. the debt level of the firm decreases. Other things equal, less debt implies more stock available to trade in the market. Thus, the speculator s payoff is higher if he has more information. As a result, the manager can use debt as a control variable to affect the speculator s information acquisition. When he needs more information for operation decision making, he decreases the level of debt. This signals to the markets that the firm is risky. The speculator responds to the news by producing more information to trade. On one hand, the increase in the speculator s information acquisition helps the manager make more efficient operation decisions which result in higher value of the firm. On the other hand, more information acquisition leads to higher cost of capital which reduces the value of the firm. The manager thus needs to balance the two effects when deciding the capital structure of the firm. 16

17 Equations (8) and (9) show that at α m both the speculator s profit Π (α m ) and the cost of capital are increasing in α m but the cost of capital is twice as much as the speculator s profit. 3 Optimal capital structure According to Table, with probability 3, the speculator s signals h or l are revealed and 8 with probability 1 4 they are not, in which case the speculator gets positive payoff. Thus, the prices of bond and stock at date 0, P 0B and P 0S, are given by: P 0B = 3 8 P 1B (h) P 1B (l) P 1B ( ) π B, P 0S = 3 8 P 1S (h) P 0S (l) P 0S ( ) π S where π B and π S are the cost of debt and the cost of equity. Thus, the value of the firm at time 0 is: V 0 = P 0B + P 0S. Lemma. If F > m e, the value of the firm depends on its capital structure and is given by: V 0 = m [M (m α me)] cα m, (10) where α m = 1 8c [(σ S + σ B ) e + (σ S σ B ) (m F )]. Proof. See Appendix. Like the F m e case, the value of the firm (equation (10)) consists of three terms. The first term, m, is the value of the firm when the manager has no information and there is no feedback effect nor cost of capital. The second term is the gain due to the feedback effect which helps the manager avoid the low cash flow in the low state. The last term is the cost of capital. Unlike the F m e case, the amount of information acquired by the 17

18 speculator α m is a function of the firm s capital structure. Thus, the manager can use F as a control variable to maximize the value of the firm by affecting the information acquisition of the speculator. More specifically, the manager solves the following optimization problem: { max m + 3 } F 8 [M (m α me)] cαm st. α m = 1 8c [(σ S + σ B ) e + (σ S σ B ) (m F )]. Proposition 5. The optimal capital structure which maximizes the value of the firm is given by: F = m 3 (σ S + σ B ) e. (11) (σ S σ B ) At this capital structure, the speculator s information acquisition is: α m (F ) = 3e 16c. (1) Proof. See Appendix. Equation (11) says that the optimal level of debt F is a function of the expected return of the growth strategy m, market depths σ S and σ B, and the risk of the growth strategy e. Other things equal, If the markets are sufficiently deep (σ S + σ B > 3/), the optimal level of debt is positively related to risk (e). In deep markets, high risk causes the speculator to produce too much information which can result in too high cost of capital and reduce the firm s value. The manager therefore increases leverage to discourage too much information production. However, if the markets are not very deep (σ S + σ B < 3/), high risk alone may not motivate the speculator to produce enough information the firm needs. The manager therefore decreases leverage to encourage more information production. As a result, the optimal level of debt is inversely related to risk. 18

19 The above effects are amplified further when the stock-bond market difference (σ S σ B ) increases. Although stock and bond give the firm the same feedback effect benefits, the stock s cost of capital is higher than that of the bond because the stock is more information sensitive. When the stock market is too deep compared to the bond market, the manager increases debt to deter the speculator from obtaining too much information for trading which could cause excessive cost of capital. The debt level is positively correlated with m, the expected return of the risky strategy. Often m is correlated with the firm size. This implies that large firms tend to use more debt than the smaller ones. Equation (1) says that at the optimal capital structure, the amount of information produced by the speculator increases with the risk of the growth strategy and decreases with the cost of producing information. Corollary. A sufficient condition for the firm to choose risky debt is M > m 5e 3c. Proof. See appendix. Under a feedback effect, M is the cash flow the firm gets when the security prices accurately signal that µ = L. Thus, the gain from feedback effect is higher if M is larger, other things equal. However, as shown in Lemma, in order to control the informativeness of the security prices, the manager must use risky debt. Large expected gain from the feedback effect provides more incentive for the firm to employ risky debt. So far, we have investigated two extreme cases and shown that if the firm has complete information, the Modigliani-Miller irrelevance is maintained. However, if it has no information and uses risky debt, there exists an optimal level of debt, given in equation (11), which maximizes the value of the firm. For cases in between, we assign some probability f the 19

20 firm has complete information and some probability 1 f it has no information at all. Since capital structure does not matter in the former, (11) is also an optimal level of debt when the firm has partial information. 4 Conclusions This paper develops a game theoretic model of financing with an informational feedback effect to investigate the impact of a firm s capital structure policy on the informational efficiency of its security prices. We show that when the firm has complete information or when information is not critical for its efficient operating decisions, the Modigliani-Miller irrelevance is maintained. The debt is riskless and the speculator does not acquire information to trade. However, when the firm has incomplete information and information is imperative to make efficient operating decisions, capital structure does matter and the firm can use it to control the information production in the market. On one hand, the more information the speculator acquires for trade, the more informative are the security prices which in turn help the firm make efficient operating decisions. On the other hand, more information acquisition by the speculator leads to higher cost of capital which decreases the value of the firm. The optimal capital structure trades off the two opposite effects on the firm. Our results indicate that the speculator s information acquisition is positively related to the market liquidity but negatively related to the debt level of the firm. There exists an optimal level of debt which is positively related to the firm size and the market liquidity but negatively related to the risk of the business. 0

21 A Appendix A.1 Proof of Proposition Proof. If F m e, the debt is riskless, P 0B = P 1B (.) = F and the speculator only trades stock since bond is insensitive to his information. With probability 1, the stock aggregate order flow reveals y = h, the manager chooses 4 the growth strategy and the speculator s payoff is zero. Thus, P 1S (y = h G) = (m + e F ) p ( µ = H y = h) + (m e F ) p ( µ = H y = l) = (m + e F ) 1 + α = m + αe F. + (m e F ) 1 α With probability 1, the stock aggregate order flow reveals y = l, the manager chooses 4 the conservative strategy and the speculator s payoff is zero. Thus, P 1S (y = l) = M F. With probability 1, the stock aggregate order flow does not reveal y, the manager chooses the growth strategy. Let P 1S (y =. G) denote the stock price when the firm follows the growth strategy, then P 1S (y = ) = 1 [P 1S (y = h G) + P 1S (y = l G)]. P S (y = l G) = (m + e F ) p ( µ = H y = l) + (m e F ) p ( µ = L y = l) = m F eα. = P S (y = ) = m F. 1

22 The speculator s expected payoff, which is also the cost of capital, is: π = 1 σ S [P S (y = h G) P S (y = ) + P S (y = ) P S (y = l G)] = σ Seα. The speculator solves the following profit maximization problem to determine the amount of information α to acquire: max α σ S eα 1 cα The first order condition is: σ S e cα m = 0 α m = σ Se c. Given the optimal amount of information acquired α m, the speculator s maximum payoff from his optimal information acquisition is: π m = σ Seα m = σ S e 4c which is also the cost of capital. Subtracting the cost of information from the payoff, we get the maximum profit of the speculator: σ S e 4c 1 cα m = σ S e 8c. The stock price at time 1 is given by: P 1S = 1 4 (m + eα m F ) (M F ) + 1 (m F ) = 3 4 m M F + σ Se 8c.

23 The value of the firm at time 0: V 0 = P 1S + P 1B π m = 3m + M 4 + σ Se 8c σ S e 4c = m [M (m α me)] π m A. Proof of Proposition 3 Proof. If aggregate order flows reveal y = h, the manager follows the growth strategy. As a result, P 1B (y = h) = F.p ( µ = H y = h) + (m e) p ( µ = L y = h), = F 1 + α + (m e) 1 α, P 1S (y = h) = (m + e F ) p ( µ = H y = h) + 0p ( µ = L y = h) = (m + e F ) 1 + α. If aggregate order flows indicate y = l, the manager engages in the conservative strategy which results in a sure cash flow of M greater than F. Thus, the debt is riskless. P 1B (y = l) = F, P 1S (y = l) = M F. Since security prices fully reflect the speculator s information, his payoff is zero. 3

24 A.3 Proof of Proposition 4 Proof. The manager follows the growth strategy when he is uncertain about the signal y of the speculator. Knowing that, the bond market maker sets the bond price as follows: P B ( ) = 1 [P B (y = h G) + P B (y = l G)], where P B (. G)denotes the bond price conditioned on the manager following the growth strategy. Similarly, P B (y = h G) = F 1 + α P B (y = l G) = F 1 α P B ( ) = m e + F + (m e) 1 α, + (m e) 1 + α. P S ( ) = 1 [P S (y = h G) + P S (y = l G)].. Since P S (y = h G) = (m + e F ) p ( µ = H y = h) + 0.p ( µ = L y = h) = (m + e F ) 1 + α, P S (y = l G) = (m + e F ) p ( µ = H y = l) + 0.p ( µ = L y = l) = (m + e F ) 1 α. P S ( ) = m + e F Given his trading strategy, the speculator s expected payoffs in each market are given by: { 1 π B ( ) = σ B [P B (y = h) P B ( )] + 1 } [P B ( ) P B (y = l G)] = σ B [P B (y = h) P B (y = l G)].. 4

25 π B ( ) = σ [ B F 1 + α + (m e) 1 α F 1 α (m e) 1 + α ] = σ Bα [F (m e)]. { 1 π S ( ) = σ S [P S (y = h) P S ( )] + 1 } [P S ( ) P S (y = l G)] = σ S [P S (y = h) P S (y = l G)]. π S ( ) = σ S [ (m + e F ) 1 + α (m + e F ) 1 α ] = σ Sα (m + e F ). Thus, the total payoff of the speculator is: π ( ) = π B ( ) + π S ( ) = α [(σ S + σ B ) e + (σ S σ B ) (m F )]. A.4 Proof of Lemma 1 Proof. From the speculator s payoff and profit equations (4) and (5): π ( ) = α [(σ S + σ B ) e + (σ S σ B ) (m F )], (13) Π (α) = 1 4 π ( ) 1 cα = α 8 [(σ S + σ B ) e + (σ S σ B ) (m F )] 1 cα. (14) 5

26 The speculator solves the following optimization problem: max Π (α) α The first order condition is: dπ dα = 1 8 [(σ S + σ B ) e + (σ S σ B ) (m F )] cα = 0, α m = 1 8c [(σ S + σ B ) e + (σ S σ B ) (m F )]. (15) The second order condition is: d Π = c < 0. dα Substituting α m into (13) and (14), we get: π (α m ) = 1 64c [(σ S + σ B ) e + (σ S σ B ) (m F )], Π (α m ) = 1 18c [(σ S + σ B ) e + (σ S σ B ) (m F )]. A.5 Proof of Lemma Proof. From Propositions 3 and 4, we have: P 1B (h) = F 1 + α m + (m e) 1 α m, P 1S (h) = (m + e F ) 1 + α m, P 1B (l) = F, P 1S (l) = M F, 6

27 P 1B ( ) = 1 (m e + F ), P 1S ( ) = 1 (m + e F ). Therefore, V 0 = 3 8 [P 1B (h) + P 1S (h)] [P 1B (l) + P 1S (l)] [P 1B ( ) + P 1S ( )] π (α m ) = 5 8 m M α me π (α m ) = m [M (m α me)] cα m, where α m is given by equation (15). A.6 Proof of Proposition 5 Proof. The firm solves the following optimization problem: max V 0 = m + 3 F 8 [M (m α me)] cαm St. α m = 1 8c [(σ S + σ B ) e + (σ S σ B ) (m F )]. The first order condition is: dv 0 df = dv ( ) ( 0 dα m 3 dα m df = 8 e cα σ ) S σ B = 0 8c α m = 3 16c e, 1 8c [(σ S + σ B ) e + (σ S σ B ) (m F )] = 3e 16c F = m 3 (σ S + σ B ) e. (σ S σ B ) 7

28 The second order condition is: d V 0 df = (σ S σ B ) 3c < 0. Thus, V 0 (F ) is a maximum and ) (M m + 3e c 3 e V 0 (F ) = m m + 3M = 8 + 9e 16 c. 16c 16 c A.7 Proof of Corollary Proof. From Proportions and 5: V 0 (F m e) = 3m + M 4 = 3m + M 4 3m + M 4 + σ Se 8c σ S e 4c + e 8c σ S (1 σ S ) + e 8 c. The inequality in the third line is because σ S (1 σ S ) achieves the maximum of 1 8 when σ S = 1 4. V0 (F > m e) = 5m + 3M 8 + 9e 16 c. Thus, a sufficient condition for V 0 (F > m e) V 0 (F m e) is 5m + 3M 8 + 9e 16 c 3m + M 4 M m 5e 3c. + e 8 c 8

29 References Baker, M., J. C. Stein, and J. Wurgler (003). When does the market matter? Stock prices and the investment of equity-dependent firms. Quarterly Journal of Economics 118 (3), Bond, P., A. Edman, and I. Goldstein (01). The real effects of financial markets. The Annual Review of Financial Economics 4, Bond, P., I. Goldstein, and E. S. Prescott (010). Market-based corrective actions. Review of Financial Studies 3 (), Chang, C. and X. Yu (004). Investment opportunities, liquidity premium, and conglomerate mergers. The Journal of Business 77 (1), Chang, C. and X. Yu (010). Informational efficiency and liquidity premium as the determinants of capital structure. Journal of Financial and Quantitative Analysis 45 (), Chen, Q., I. Goldstein, and W. Jiang (007). Price informativeness and investment sensitivity to stock price. Review of Financial Studies 0 (3), Cronqvist, H., A. K. Makhija, and S. E. Yonker (01). Behavioral consistency in corporate finance: CEO personal and corporate leverage. Journal of financial economics 103 (1), Dow, J. and R. Rahi (003). Informed trading, investment, and welfare. The Journal of Business 76 (3), Edmans, A., I. Goldstein, and W. Jiang (01). The real effects of financial markets: The impact of prices on takeovers. The Journal of Finance 67 (3), Foucault, T. and T. Gehrig (008). Stock price informativeness, cross-listings, and investment decisions. Journal of Financial Economics 88 (1),

30 Fulghieri, P. and D. Lukin (001). Information production, dilution costs, and optimal security design. Journal of Financial Economics 61 (1), 3 4. Goldstein, I. and A. Guembel (008). Manipulation and the allocational role of prices. The Review of Economic Studies 75 (1), Graham, J. R. (000). How big are the tax benefits of debt? The Journal of Finance 55 (5), Hackbarth, D. (008). Managerial traits and capital structure decisions. Journal of Financial and Quantitative Analysis 43 (04), Hackbarth, D. (009). Determinants of corporate borrowing: A behavioral perspective. Journal of Corporate Finance 15 (4), Kyle, A. S. (1985). Continuous auctions and insider trading. Econometrica, Luo, Y. (005). Do insiders learn from outsiders? Evidence from mergers and acquisitions. The Journal of Finance 60 (4), Malmendier, U., G. Tate, and J. Yan (011). Overconfidence and early-life experiences: the effect of managerial traits on corporate financial policies. The Journal of finance 66 (5), Modigliani, F. and M. H. Miller (1958). The cost of capital, corporation finance and the theory of investment. The American economic review, Subrahmanyam, A. and S. Titman (1999). The going-public decision and the development of financial markets. The Journal of Finance 54 (3),

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