CHAPTER 4. AGENCY PROBLEMS IN CORPORATE FINANCE There is a fixed cost of financing a project which we normalize to zero and a finite number of projects a= 1,. A identified with the probabilit distributions p(a, s). The contract between the shareholder and the manager requires the shareholders to pay the cost of the investment and pay the manager w(s) contingent on the outcome R(s). The manager has a utility function U(c) and chooses the project a to maximize his expected utility Spla, sU(w(s)). Because the manager has limited liability and no personal resources,(s)≥0 We assume that the principal and agent both know all the parameters of the model, the cost function v(a), the possible outcomes R(s), the agent's utility function U(), etc. There is asymmetric information about the choice of project, which gives rise to an incentive problem. Both the principal and the agent observe the contract (a, w() that specifies the manager's remuneration and the project that should be chosen and they both observe the realization s. However, only the manager observes the actual choice of p roject Casting this in the form of a principal-agent problem, the principal is assumed to choose the contract (a, w()) to maximize his expected return SpLa, s)V(R(s-w(s), subject to an incentive constraint(IC) and an individual rationality or participation constraint (IR) pla, S (IC)∑。p(a,s)U((s)≥∑,P(b,s)U((s),b p(a,s)U((s)≥ Of course, there is a participation constraint for the principal as well. If the solution to this problem does not give the principal a return greater than his opportunity cost, it may not be optimal for him to invest in the project at Suppose that the principal is risk neutral and the agent strictly risk averse Then the obvious solution is to offer the agent a fixed wage w(s)=w such that u(w)=i. The agent will be indifferent between all projects so it will be optimal for him to choose the project that maximizes the principals payoff, namely, the project a that maximizes expected revenue. But note that it is also optimal for him to choose any other project so we have not found a very robust method of implementing the efficient project Suppose that the agent is risk neutral and the principal strictly risk averse Then optimal risk sharing would require that the agent bear all the risk assuming that this is consistent with the budget constraint. Recall that we
6 CHAPTER 4. AGENCY PROBLEMS IN CORPORATE FINANCE There is a fixed cost of financing a project which we normalize to zero and a finite number of projects a = 1, ..., A identified with the probability distributions p(a, s). The contract between the shareholder and the manager requires the shareholders to pay the cost of the investment and pay the manager w(s) contingent on the outcome R(s). The manager has a utility P function U(c) and chooses the project a to maximize his expected utility s p(a, s)U(w(s)). Because the manager has limited liability and no personal resources, w(s) ≥ 0. We assume that the principal and agent both know all the parameters of the model, the cost function ψ(a), the possible outcomes R(s), the agent’s utility function U(·), etc. There is asymmetric information about the choice of project, which gives rise to an incentive problem. Both the principal and the agent observe the contract (a, w(·)) that specifies the manager’s remuneration and the project that should be chosen and they both observe the realization s. However, only the manager observes the actual choice of project. Casting this in the form of a principal-agent problem, the principal is assumed to choose the contract P (a, w(·)) to maximize his expected return s p(a, s)V (R(s) − w(s)), subject to an incentive constraint (IC) and an individual rationality or participation constraint (IR): max(a,w(·)) P s p(a, s)V (R(s) − w(s)) (IC) P s p(a, s)U(w(s)) ≥ P s p(b, s)U(w(s)), ∀b (IR) P s p(a, s)U(w(s)) ≥ u¯ Of course, there is a participation constraint for the principal as well. If the solution to this problem does not give the principal a return greater than his opportunity cost, it may not be optimal for him to invest in the project at all. Suppose that the principal is risk neutral and the agent strictly risk averse. Then the obvious solution is to offer the agent a fixed wage w(s)= ¯w such that u( ¯w)=¯u. The agent will be indifferent between all projects so it will be optimal for him to choose the project that maximizes the principal’s payoff, namely, the project a that maximizes expected revenue. But note that it is also optimal for him to choose any other project so we have not found a very robust method of implementing the efficient project. Suppose that the agent is risk neutral and the principal strictly risk averse. Then optimal risk sharing would require that the agent bear all the risk, assuming that this is consistent with the budget constraint. Recall that we
4.2. THE RISK SHIFTING PROBLEM assume the agent's consumption is non-negative(limited liability). In the first best, we have seen that when the manager is risk neutral there is a number r>0 such that w(s)= maxR(s-T, 01, s, and the return to the principal is R(s)-(s)=min(R(s), r1,Vs With this payment structure, the entrepreneur chooses a to maximize his expected return ∑pa,s)m(s)=∑p(,)max{(s)-r,0 Suppose that the principal is restricted to offering an incentive scheme of this form. Then the(constrained) principal-agent problem max(a,r(a, s)V(minT, R(s))) r≥0 ∑、p(a,s)max{f(s)-r,0}≥∑,p(a,s)max{(s)-r,0},vs max R(s)-r,0} For any probability vector p=(p1, .,ps)let P()=∑ d A=∑P(R(a+1)-Ra) A distribution p is a mean-preserving spread of p if it satisfies one of the following equivalent conditions Proposition1 Suppose that∑、p,R(s)=∑。p,R(s). The following cona tion ≤∑。=04 (ii) for any non-decreasing function f: S-R with non-increasing difference∑spf(s)≤∑。Psf(s) (iii) p is obtained from p in a finite sequence of transformations at each step of which the probability of one outcome is reduced and the probability mass is redistributed to a higher and lower outcome in a
4.2. THE RISK SHIFTING PROBLEM 7 assume the agent’s consumption is non-negative (limited liability). In the first best, we have seen that when the manager is risk neutral there is a number r > 0 such that w(s) = max{R(s) − r, 0}, ∀s, and the return to the principal is R(s) − w(s) = min{R(s), r}, ∀s. With this payment structure, the entrepreneur chooses a to maximize his expected return X s p(a, s)w(s) = X s p(a, s) max{R(s) − r, 0}. Suppose that the principal is restricted to offering an incentive scheme of this form. Then the (constrained) principal-agent problem is max(a,r) P s p(a, s)V (min{r, R(s)}) s.t. r ≥ 0 (IC) P s p(a, s) max{R(s) − r, 0} ≥ P s p(a, s) max{R(s) − r, 0}, ∀s (IR) P s p(a, s) max{R(s) − r, 0} ≥ u¯ For any probability vector p = (p1, ..., pS) let P(s) = Xs σ=0 pσ and As = Xs σ=0 Pσ(R(σ + 1) − R(σ)). A distribution p0 is a mean-preserving spread of p if it satisfies one of the following equivalent conditions: Proposition 1 Suppose that P s psR(s) = P s p0 sR(s). The following conditions are equivalent: (i) Ps σ=0 Aσ ≤ Ps σ=0 A0 σ; (ii) for any non-decreasing function f : S → R with non-increasing differences P s p0 sf (s) ≤ P s psf (s) ; (iii) p0 is obtained from p in a finite sequence of transformations at each step of which the probability of one outcome is reduced and the probability mass is redistributed to a higher and lower outcome in a mean-preserving way
8 CHAPTER 4. AGENCY PROBLEMS IN CORPORATE FINANCE Suppose that p(, ) is a mean-preserving spread of P(b, ) Then, for any number r ∑p(a,s)max{R()-r,0≥∑p(,smax{B(s)-r,0 This follows immediately from the proposition and the fact that the functi f(s)= max(R(s)-r, 0 has non-decreasing differences. In other words the entrepreneur has a preference for risk(a preference for mean-preserving The principal on the other hand wants to maximize > pla, s)V(min(, R(s))) At any solution of the principal-agent problem, the participation constraint should be satisfied with equality: s P(a, s)w(s)= sp( a snax r,0=i. Thus, in the absence of the incentive problem the principal seeks to maximize ∑pa,sV(min{,R(s)) subject to p(a, s)max(R(s)-r,Of For example, if the principal is risk neutral he would always prefer a project with a higher expected value. But once the incentive constraint is imposed the risk shifting preferences of the agent have to be taken into account 4.3 Debt Overhang The risk taking or asset substitution problem is not the only one. Myers (1977)pointed out that firms rather than accepting negative NPV projects have an incentive to forego positive NPv projects. This incentive results from a debt overhang problem. The reason this arises is that equityholders with existing debt have to share the rewards of new projects with bondholders To see how this works consider another simple example 4.3.1 A Simple Example of Debt Overhang The firm has no cash and has debt of S10.000 The firm does nothing
8 CHAPTER 4. AGENCY PROBLEMS IN CORPORATE FINANCE Suppose that p(a, ·) is a mean-preserving spread of p(b, ·). Then, for any number r, X s p(a, s) max{R(s) − r, 0} ≥ X s p(b, s) max{R(s) − r, 0}. This follows immediately from the proposition and the fact that the function f(s) = max{R(s) − r, 0} has non-decreasing differences. In other words, the entrepreneur has a preference for risk (a preference for mean-preserving spreads). The principal on the other hand wants to maximize P s p(a, s)V (min{r, R(s)}). At any solution of the principal-agent problem, the participation constraint should be satisfied with equality: P s p(a, s)w(s) = P s p(a, s) max{R(s) − r, 0} = ¯u. Thus, in the absence of the incentive problem the principal seeks to maximize X s p(a, s)V (min{r, R(s)}) subject to X s p(a, s) max{R(s) − r, 0} = ¯u. For example, if the principal is risk neutral he would always prefer a project with a higher expected value. But once the incentive constraint is imposed, the risk shifting preferences of the agent have to be taken into account. 4.3 Debt Overhang The risk taking or asset substitution problem is not the only one. Myers (1977) pointed out that firms rather than accepting negative NPV projects have an incentive to forego positive NPV projects. This incentive results from a debt overhang problem. The reason this arises is that equityholders with existing debt have to share the rewards of new projects with bondholders. To see how this works consider another simple example. 4.3.1 A Simple Example of Debt Overhang The firm has no cash and has debt of $10,000. The firm does nothing: