one-that is, the probability that the project will still be available next period is high. This means that risk in acquiring more information is low: the project will most likely still exist after i is learned, and the decision whether to undertake it can then be made with perfect accuracy. At the other extreme, when q is close to zero, the project is likely to disappear while v is gathered. Unless the project is not worth undertaking using the information already known about it, there is no point in gathering more information about a project that will probably cease to exist. I the second stage therefore, the decision to gather more information will always be associated with larger values of g. In what follows, we use Qs, 8=0, 1, to generically denote the g-threshold above which more information will be gathered in the second stage after the decision-maker learns S=sE 0, 1 in the first stage. Similar reasoning leads to the conclusion that more information will be gathered only when p is large enough in the first stage. We define the p-threshold of the first stage, P, analogously. Thus the strategy of any decision-maker can be summarized by three information-gathering thresholds: P, Q1, and Qc 3.2 Updating, Firm Value, and First-Best Given that the risk of the project available to the firm is purely idiosyncratic and that the riskfree rate is zero, we know that the value of the firm to its well-diversified, risk-neutral shareholders is simply the expected value of its end-of-period cash Flows. The expected value of the risky project evolves as more information is gathered about it. Thus, depending on when the project undertaken, it will have a different value to the firm. Trivially, when the information about the risky project is perfect, the firm is worth one if i= 1(the project is undertaken)and 2 if i=0 (the project is dropped). After 8 is received however, the firm's value depends on the posterior probability of the risky project's success, and on what the manager does with the information. It is easy to verify that 1+a Pr{=1|8=1}= 1-a Pr{=1|8=0} so that E|=1=1+a E[|s=0]= a Notice that 1(To) gets closer to one(zero) as a increases: more weight is put on the information when its precision is large. This also translates into more extreme assessments of the risky projects
one—that is, the probability that the project will still be available next period is high. This means that risk in acquiring more information is low: the project will most likely still exist after ˜v is learned, and the decision whether to undertake it can then be made with perfect accuracy. At the other extreme, when ˜q is close to zero, the project is likely to disappear while ˜v is gathered. Unless the project is not worth undertaking using the information already known about it, there is no point in gathering more information about a project that will probably cease to exist. In the second stage therefore, the decision to gather more information will always be associated with larger values of ˜q. In what follows, we use Q¯s, s = 0, 1, to generically denote the ˜q-threshold above which more information will be gathered in the second stage after the decision-maker learns s˜ = s ∈ {0, 1} in the first stage. Similar reasoning leads to the conclusion that more information will be gathered only when ˜p is large enough in the first stage. We define the ˜p-threshold of the first stage, P¯, analogously. Thus the strategy of any decision-maker can be summarized by three information-gathering thresholds: P¯, Q¯1, and Q¯0. 3.2 Updating, Firm Value, and First-Best Given that the risk of the project available to the firm is purely idiosyncratic and that the riskfree rate is zero, we know that the value of the firm to its well-diversified, risk-neutral shareholders, is simply the expected value of its end-of-period cash flows. The expected value of the risky project evolves as more information is gathered about it. Thus, depending on when the project is undertaken, it will have a different value to the firm. Trivially, when the information about the risky project is perfect, the firm is worth one if ˜v = 1 (the project is undertaken) and 1 2 if ˜v = 0 (the project is dropped). After ˜s is received however, the firm’s value depends on the posterior probability of the risky project’s success, and on what the manager does with the information. It is easy to verify that Pr {v˜ = 1 | s˜ = 1} = 1 + a 2 ≡ π1, and Pr {v˜ = 1 | s˜ = 0} = 1 − a 2 ≡ π0, so that E[˜v | s˜ = 1] = 1 + a 2 > 1 2 , and (1) E[˜v | s˜ = 0] = 1 − a 2 < 1 2 . (2) Notice that π1 (π0) gets closer to one (zero) as a increases: more weight is put on the information when its precision is large. This also translates into more extreme assessments of the risky project’s 9
value, as shown in(1)and(2). The following lemma shows how the firms value evolves with the information gathering process of the firm. On average, after the manager learns that s=s in the first stage, the firm is won Second stage Lemma 3.1 Suppose that the manager adopts g-thresholds of Qs, 8=0, 1, for the E(Q)=2(1+)+(=3).- Note that Fs(Qs does not represent the optimal or maximum value of the firr stage. Instead it represents the value that is implied by a particular information gathering strategy chosen by the firms manager. Of course, the firms shareholders are ultimately interested in maximizing the initial value of the firm, which we calculate next. In the first stage, the manager either undertakes the project(when p< P) or acquires s(when p> P), which will be one or zero with equal probabilities. We can use the expected future values of the firm derived in Lemma 3. 1 to calculate the value of the firm at the outset Lemma 3. 2 Suppose that the manager adopts a p-threshold of P and g-thresholds of Qs, 8=0,1 The initial value of the firm is then given by FQ0)=折F(Q)+FQ)+1]-P where F1( and Fo() are as calculated in (3) This result shows how any capital budgeting strategy, which can be summarized with three thresholds (P, Q1, Qo, maps into a value for the firm. We start by characterizing the manager's set of decisions that will maximize this value. We refer to the value-maximizing strategy as the first- best strategy, and use a superscript"FB"to denote it. This strategy can be reached by assuming that the manager is a risk-neutral owner of the firm, as the personal objective of this owner is then precisely to maximize the firms value. Alternatively, it can be viewed as the solution to the following maximization problem F(P,Q1,Q0) {P,Q1,Qo}∈0,1 The following proposition solves for this first-best strategy and associated firm value Proposition 3.1(First-Best) The value of the firm is maximized with PB=0, Q0=0, and
value, as shown in (1) and (2). The following lemma shows how the firm’s value evolves with the information gathering process of the firm. Lemma 3.1 Suppose that the manager adopts q˜-thresholds of Q¯s, s = 0, 1, for the second stage. On average, after the manager learns that s˜ = s in the first stage, the firm is worth F¯s(Q¯s) ≡ 1 2 1 + 1 2 πs + πs − 1 2 Q¯s − 1 4 πsQ¯2 s. (3) Note that F¯s(Q¯s) does not represent the optimal or maximum value of the firm after the first stage. Instead it represents the value that is implied by a particular information gathering strategy chosen by the firm’s manager. Of course, the firm’s shareholders are ultimately interested in maximizing the initial value of the firm, which we calculate next. In the first stage, the manager either undertakes the project (when ˜p < P¯) or acquires ˜s (when ˜p ≥ P¯), which will be one or zero with equal probabilities. We can use the expected future values of the firm derived in Lemma 3.1 to calculate the value of the firm at the outset. Lemma 3.2 Suppose that the manager adopts a p˜-threshold of P¯ and q˜-thresholds of Q¯s, s = 0, 1. The initial value of the firm is then given by F¯(P, ¯ Q¯1, Q¯0) ≡ 1 4 F¯1(Q¯1) + F¯0(Q¯0)+1 − 1 4 P¯2, (4) where F¯1(·) and F¯0(·) are as calculated in (3). This result shows how any capital budgeting strategy, which can be summarized with three thresholds P, ¯ Q¯1, Q¯0 , maps into a value for the firm. We start by characterizing the manager’s set of decisions that will maximize this value. We refer to the value-maximizing strategy as the firstbest strategy, and use a superscript “FB” to denote it. This strategy can be reached by assuming that the manager is a risk-neutral owner of the firm, as the personal objective of this owner is then precisely to maximize the firm’s value. Alternatively, it can be viewed as the solution to the following maximization problem: P¯FB, Q¯FB 1 , Q¯FB 0 = argmax {P , ¯ Q¯1,Q¯0}∈[0,1]3 F¯(P, ¯ Q¯1, Q¯0). The following proposition solves for this first-best strategy and associated firm value. Proposition 3.1 (First-Best) The value of the firm is maximized with P¯FB = 0, Q¯FB 0 = 0, and Q¯FB 1 = π1 − 1 2 1 2π1 = 2a 1 + a . (5) 10
With this strategy, the initial value of the firm is 16+8(1+a) In the value-maximizing strategy, the firm always gathers information at the outset. This intuitive: the ex ante values of both the risky project and the safe investment are exactly 2; since the firm can always get this value of 2 later by dropping the risky project in favor of the safe investment, acquiring information is always optimal in the first stage. A similar argument applies when the outcome of the first stage of information gathering is 8=0. In that case, the risky project is worth o==<2. The firm benefits from acquiring more information, since the worst possible scenario after doing so is again 2 After s= l is observed however, perfect information is gathered only with some probability, depending on the outcome of g. In particular, the risky project is then worth T1=* a>).When the risky project is almost sure to disappear, i. e, g is close to zero, acquiring more information foolish: the perfect information will be useless if the project cannot be undertaken after that information is learned. If on the other hand the project will most likely continue to exist I IS close to one, acquiring i makes sense, as a better informed decision can be made without much risk of losing the project. This is the tradeoff that leads the firm to undertake the project only when q>Q1. It is easy to show that Qi is increasing in T1 and a. Thus a firm with a high-ability manager is less willing to gather perfect information: the manager's imperfect information is already ery informative, and so there is no point in further risking losing the project to competition. Thus a high-ability manager can make accurate decisions quickly There are a few things to notice about the first-best value of the firm, as calculated in(6). First this value is also increasing in a: the firm directly benefits from the ability of its manager. Second FFB is not only greater than but bounded away from it: the initial value of the firm, regardless of the ability of its manager, exceeds the present value of its initial opportunities, namely the risky project and the safe investment. This is because, even when a= 0, the firm can always choose to gather information for two stages, and make its decision about the risky project afterwards. O average,after the second stage, the project will still exist with probability Elp the project has a net value of 2 half the time (payoff of i=l with an initial investment of 2), the value created from information gathering without skill is 4 2. 2=16. This is why the initial value of the firm is 2+16= i6 when the manager is unskilled
With this strategy, the initial value of the firm is F¯FB = 9 16 + a2 8(1 + a) . (6) In the value-maximizing strategy, the firm always gathers information at the outset. This is intuitive: the ex ante values of both the risky project and the safe investment are exactly 1 2 ; since the firm can always get this value of 1 2 later by dropping the risky project in favor of the safe investment, acquiring information is always optimal in the first stage. A similar argument applies when the outcome of the first stage of information gathering is ˜s = 0. In that case, the risky project is worth π0 = 1−a 2 < 1 2 . The firm benefits from acquiring more information, since the worst possible scenario after doing so is again 1 2 . After ˜s = 1 is observed however, perfect information is gathered only with some probability, depending on the outcome of ˜q. In particular, the risky project is then worth π1 = 1+a 2 > 1 2 . When the risky project is almost sure to disappear, i.e., ˜q is close to zero, acquiring more information is foolish: the perfect information will be useless if the project cannot be undertaken after that information is learned. If on the other hand the project will most likely continue to exist, i.e., ˜q is close to one, acquiring ˜v makes sense, as a better informed decision can be made without much risk of losing the project. This is the tradeoff that leads the firm to undertake the project only when q˜ ≥ Q¯FB 1 . It is easy to show that Q¯FB 1 is increasing in π1 and a. Thus a firm with a high-ability manager is less willing to gather perfect information: the manager’s imperfect information is already very informative, and so there is no point in further risking losing the project to competition. Thus a high-ability manager can make accurate decisions quickly. There are a few things to notice about the first-best value of the firm, as calculated in (6). First, this value is also increasing in a: the firm directly benefits from the ability of its manager. Second, F¯FB is not only greater than 1 2 but bounded away from it: the initial value of the firm, regardless of the ability of its manager, exceeds the present value of its initial opportunities, namely the risky project and the safe investment. This is because, even when a = 0, the firm can always choose to gather information for two stages, and make its decision about the risky project afterwards. On average, after the second stage, the project will still exist with probability E[˜p] · E[˜q] = 1 4 . Since the project has a net value of 1 2 half the time (payoff of ˜v = 1 with an initial investment of 1 2 ), the value created from information gathering without skill is 1 4 · 1 2 · 1 2 = 1 16 . This is why the initial value of the firm is 1 2 + 1 16 = 9 16 when the manager is unskilled. 11
3.3 The Effect of risk a version The first-best outcome maximizes the current value of the firm to its risk-neutral shareholders to attain it, the firms manager must not care about risk when making his capital budgeting decisions However, capital budgeting decisions will be made by agents whose human capital is tied to the firm, e.g., the CEO of the firm. As pointed out by Jensen and Meckling(1976), this agent's risl aversion is likely to affect his decisions. Compensation contracts can be used to reduce these agency costs by realigning the objective of the firm's manager with those of the shareholders. We discuss these in section 5. For now, we concentrate on the problem of a risk-averse manager whose utility depends only upon the value of the firm. We show how the risk aversion of this decision-maker will affect his capital budgeting decisions and in turn the value of his firm To keep the analysis of this and later sections tractable, we model risk aversion as a utility cost r>0 that is incurred by the firm's manager when his firm is worth nothing, that is when the firm undertakes the risky project and this projects turns out to be bad. This cost effectively makes the firms manager risk-averse: the three potential outcomes of the capital budgeting decisions, 10,2,1, will respectively yield T, 2, 1) in utility to the manager, making his utility function a convex function of the firm's end-of-period value. Note that, with this three-outcome specification suming more traditional utility functions will not change any of our results. Our specification however, has the advantage that it allows us to solve for most results in the paper analytically Note also that the cost r can alternatively be interpreted as the negative reputation that a manager acquires after running a firm down to the ground-the cost of getting fired, for example The risk-averse manager has the same information technology as the risk-neutral manager of section 3. 2. His decisions only departs from first-best due to the fact that he is more reluctant t undertake the risky project with less than perfect information. Indeed, the risk-averse manager ffers more than the risk-neutral manager when the firm loses all of its value. Looking at Propo- sition 3. 1, we see that it is only optimal for the risky project to be undertaken without perfect information when it is known that s= 1; this happen when q < Qi. This is where the risk-averse manager's decisions will fail to maximize firm value. as the following result shows Proposition 3.2 Suppose that the firm is managed by a single risk-averse individual with risk aversion r>0. The information acquisition strategy of this manager is given by a p-threshold of P(r)≡0,andq- thresholds of c0(r)≡0and a-(1-a)r 1+a
3.3 The Effect of Risk Aversion The first-best outcome maximizes the current value of the firm to its risk-neutral shareholders. To attain it, the firm’s manager must not care about risk when making his capital budgeting decisions. However, capital budgeting decisions will be made by agents whose human capital is tied to the firm, e.g., the CEO of the firm. As pointed out by Jensen and Meckling (1976), this agent’s risk aversion is likely to affect his decisions. Compensation contracts can be used to reduce these agency costs by realigning the objective of the firm’s manager with those of the shareholders. We discuss these in section 5. For now, we concentrate on the problem of a risk-averse manager whose utility depends only upon the value of the firm. We show how the risk aversion of this decision-maker will affect his capital budgeting decisions and in turn the value of his firm. To keep the analysis of this and later sections tractable, we model risk aversion as a utility cost r ≥ 0 that is incurred by the firm’s manager when his firm is worth nothing, that is when the firm undertakes the risky project and this projects turns out to be bad. This cost effectively makes the firm’s manager risk-averse: the three potential outcomes of the capital budgeting decisions, 0, 1 2 , 1 , will respectively yield −r, 1 2 , 1 in utility to the manager, making his utility function a convex function of the firm’s end-of-period value. Note that, with this three-outcome specification, assuming more traditional utility functions will not change any of our results. Our specification, however, has the advantage that it allows us to solve for most results in the paper analytically. Note also that the cost r can alternatively be interpreted as the negative reputation that a manager acquires after running a firm down to the ground—the cost of getting fired, for example. The risk-averse manager has the same information technology as the risk-neutral manager of section 3.2. His decisions only departs from first-best due to the fact that he is more reluctant to undertake the risky project with less than perfect information. Indeed, the risk-averse manager suffers more than the risk-neutral manager when the firm loses all of its value. Looking at Proposition 3.1, we see that it is only optimal for the risky project to be undertaken without perfect information when it is known that ˜s = 1; this happen when ˜q < Q¯FB 1 . This is where the risk-averse manager’s decisions will fail to maximize firm value, as the following result shows. Proposition 3.2 Suppose that the firm is managed by a single risk-averse individual with risk aversion r ≥ 0. The information acquisition strategy of this manager is given by a p˜-threshold of P¯(r) ≡ 0, and q˜-thresholds of Q¯0(r) ≡ 0 and Q¯1(r) ≡ 2 a − (1 − a)r 1 + a < Q¯FB 1 . (7) 12
With this strategy, the initial value of the firm is F(r)=FB_(1-a)2 Since Q1(r) decreases with r, we see that risk aversion makes the manager acquire more infor mation,as q will exceed the threshold more often. In fact, any value of r larger than i-a makes Q1(r)smaller than zero, and so would result in the manager always acquiring more information in the second stage. To avoid such an extreme effect of risk aversion and the corner solution that it introduces, we restrict our attention to values of r smaller than 1-a for the rest of the paper As before, the firm's value F(r) is increasing in its manager's ability a. However, the firm's value is strictly decreasing in r. The loss of firm value results from the fact that acquiring perfect information in the second stage comes with a probability that the project will be lost to competition In other words, for the risk-averse manager, the tradeoff between perfect and imperfect information is tilted towards the larger risk reduction that perfect information offers. The manager's utility gain from reducing risk does not transfer to the firms shareholders 4 The Role of Overconfidence and Optimism Firm value is negatively affected by risk aversion. In this section, we show how managerial over confidence and optimism may help restore firm value. In some cases, as we show, the first-best outcome may even be restored 4.1 Definitions and Updating Following the work of Daniel, Hirshleifer and Subrahmanyam(1998), Odean(1998) and Odean(2001), we refer to overconfidence as the perception that private information is more precise and more reliable than it really is. In particular, we assume that the overconfident manager thinks that a is equal to A E [ a, 1, the difference A-a E [0, 1-a] measuring the degree of verconfidence. Optimism, on the other hand, refers to the manager's ex ante view of the project An optimistic manager thinks that the project is better than it really is. As in Malmendier and Tate(2001)and Heaton(2002), we assume that the manager thinks that the probability of a good outcome for the risky project(i=1)is not 2, but BE [2, 1, where B-2[ 0, 2] measures the A Note that this has no impact on any of our results, which only get stronger as r increases. This assumpt only allows us to analyze the effects of risk aversion without worrying about the fact that solutions have a differ analytical form for different ranges of
With this strategy, the initial value of the firm is F¯(r) = F¯FB − (1 − a)2 8(1 + a) r2. (8) Since Q¯1(r) decreases with r, we see that risk aversion makes the manager acquire more information, as ˜q will exceed the threshold more often. In fact, any value of r larger than a 1−a makes Q¯1(r) smaller than zero, and so would result in the manager always acquiring more information in the second stage. To avoid such an extreme effect of risk aversion and the corner solution that it introduces, we restrict our attention to values of r smaller than a 1−a for the rest of the paper.4 As before, the firm’s value F¯(r) is increasing in its manager’s ability a. However, the firm’s value is strictly decreasing in r. The loss of firm value results from the fact that acquiring perfect information in the second stage comes with a probability that the project will be lost to competition. In other words, for the risk-averse manager, the tradeoff between perfect and imperfect information is tilted towards the larger risk reduction that perfect information offers. The manager’s utility gain from reducing risk does not transfer to the firm’s shareholders. 4 The Role of Overconfidence and Optimism Firm value is negatively affected by risk aversion. In this section, we show how managerial overconfidence and optimism may help restore firm value. In some cases, as we show, the first-best outcome may even be restored. 4.1 Definitions and Updating Following the work of Daniel, Hirshleifer and Subrahmanyam (1998), Odean (1998), and Gervais and Odean (2001), we refer to overconfidence as the perception that private information is more precise and more reliable than it really is. In particular, we assume that the overconfident manager thinks that a is equal to A ∈ [a, 1], the difference A − a ∈ [0, 1 − a] measuring the degree of overconfidence. Optimism, on the other hand, refers to the manager’s ex ante view of the project. An optimistic manager thinks that the project is better than it really is. As in Malmendier and Tate (2001) and Heaton (2002), we assume that the manager thinks that the probability of a good outcome for the risky project (˜v = 1) is not 1 2 , but B ∈ 1 2 , 1 , where B − 1 2 ∈ 0, 1 2 measures the 4Note that this has no impact on any of our results, which only get stronger as r increases. This assumption only allows us to analyze the effects of risk aversion without worrying about the fact that solutions have a different analytical form for different ranges of r. 13