4.2. THE RISK SHIFTING PROBLEM Expected Value of equity =0.02 x 15,000= 300 otice that the bondholders are worse off by 900 and the equityholders are better off by 300. The NPV of the project was -600 so this is the majority of the drop in value with the other 300 coming from the transfer to equity. Thus even though its a lousy project, it's worth doing as far as the equityholders are concerned. The conclusion is that the stockholders of levered firms gain when business risk increases and this leads to an incentive to take risk 4.2.2 A Formal Model of Risk Shifting Let a denote the set of actions available to the manager with generic element a. Typically, A is either a finite set or an interval of real numbers. Let s denote a set of states with generic element s. For simplicity, we assume that the set S is finite. The probability of the state s conditional on the action a is denoted by p(a, s). The revenue in state s is denoted by R(s)20 The manager's utility depends on both the action chosen and the con- sumption he derives from his share of the revenue. The shareholder's utility depends only on his consumption. We maintain the following assumptions about preferences · The agent' s utility function u:A×R+→ R is additively separable (a,c)=U(c)-v(a) Further, the function U: R+- R is C and satisfies U(c>0 and U"(c)≤0. The principal's utility function V: R-R is C and satisfies V(c>0 nd"(c)≤0 Notice that the manager's consumption is assumed to be non-negative This is interpreted as a liquidity constraint or limited liability. Pcr Risk shifting iects, other things being equal. We can think of this as a ccurs when the manager has a convex reward schedule and fers riskier proj case where the principal is a bondholder and the agent is the managers of the firm acting in the shareholders'interest who have issued debt to finance the risky venture4.2. THE RISK SHIFTING PROBLEM 5 Expected Value of equity = 0.02 x 15,000 = 300 Notice that the bondholders are worse off by 900 and the equityholders are better off by 300. The NPV of the project was -600 so this is the majority of the drop in value with the other 300 coming from the transfer to equity. Thus even though its a lousy project, it’s worth doing as far as the equityholders are concerned. The conclusion is that the stockholders of levered firms gain when business risk increases and this leads to an incentive to take risks. 4.2.2 A Formal Model of Risk Shifting Let A denote the set of actions available to the manager with generic element a. Typically, A is either a finite set or an interval of real numbers. Let S denote a set of states with generic element s. For simplicity, we assume that the set S is finite. The probability of the state s conditional on the action a is denoted by p(a, s). The revenue in state s is denoted by R(s) ≥ 0. The manager’s utility depends on both the action chosen and the con￾sumption he derives from his share of the revenue. The shareholder’s utility depends only on his consumption. We maintain the following assumptions about preferences: • The agent’s utility function u : A × R+ → R is additively separable: u(a, c) = U(c) − ψ(a). Further, the function U : R+ → R is C2 and satisfies U0 (c) > 0 and U00(c) ≤ 0. • The principal’s utility function V : R → R is C2 and satisfies V 0 (c) > 0 and V 00(c) ≤ 0. Notice that the manager’s consumption is assumed to be non-negative. This is interpreted as a liquidity constraint or limited liability. Risk shifting occurs when the manager has a convex reward schedule and prefers riskier projects, other things being equal. We can think of this as a case where the principal is a bondholder and the agent is the managers of the firm acting in the shareholders’ interest who have issued debt to finance the risky venture