CHAPTER 2. THE MODIGLIANI-MILLER THEOREM Note that the space of commodity bundles that can be spanned by trading uity and securities is exogenous. but only because we have assumed the firms choice of production plan is exogenous. In other words, there is no financial innovation. This assumption is crucial for the MM theorem 2.3 Default 2.3.1 Default with complete markets For simplicity we assume there is a single firm i= l with a single feasible production plan y(w)>0, and a single security with payoffs z(w)=1 Limited liability raises the possibility of default and risky debt. Let i(aj,w denote the return to risky debt and i(ai, w) the return to equity in a firm with risky debt. Then 4y,)=1((-a)i)+3)<0 and y(w)+aj2(w) if y(w)+aj2(w)20 if ya)+ <0. If there are complete markets, the value of the risky debt is q=p·2(a) and the value of equity is P·叭0j The value of the firm to the original shareholders V=+ P·(ay)+ayP·(a) So default doesn't add value to the firm Assume that there is a single type of firm consisting of a continuum of identical firms. These firms choose different levels of risky debt. The number of securities may be great enough to span the entire commodity space R For example, suppose y(w)=w and choose ai=-w+1 for w=1, ., 926 CHAPTER 2. THE MODIGLIANI-MILLER THEOREM Note that the space of commodity bundles that can be spanned by trading equity and securities is exogenous, but only because we have assumed the firm’s choice of production plan is exogenous. In other words, there is no financial innovation. This assumption is crucial for the MM theorem. 2.3 Default 2.3.1 Default with complete markets For simplicity we assume there is a single firm j = 1 with a single feasible production plan y(ω) > 0, and a single security with payoffs z(ω)=1. Limited liability raises the possibility of default and risky debt. Let zˆ(αj , ω) denote the return to risky debt and yˆ(αj , ω) the return to equity in a firm with risky debt. Then zˆ(αj , ω) = ½ z(ω) if y(ω) + αjz(ω) ≥ 0 y(ω)/(−αj2) if y(ω) + αjz(ω) < 0. and yˆ(αj , ω) = ½ y(ω) + αjz(ω) if y(ω) + αjz(ω) ≥ 0 0 if y(ω) + αjz(ω) < 0. If there are complete markets, the value of the risky debt is qˆ = p · zˆ(αj ) and the value of equity is vˆ = p · yˆ(αj ). The value of the firm to the original shareholders is Vˆ = ˆv + ˆqαj = p · yˆ(αj ) + αjp · zˆ(αj ) = p · y. So default doesn’t add value to the firm. Assume that there is a single type of firm j consisting of a continuum of identical firms. These firms choose different levels of risky debt. The number of securities may be great enough to span the entire commodity space RΩ. For example, suppose y(ω) = ω and choose αω j = −ω + 1 for ω = 1, ..., |Ω|