TEXTBOOK INCONSISTENCIES IN GRAPHING VALUATION EQUATIONS: A FURTHER NOTE

In this note, the authors demonstrate that if a firm's leverage ratio is measured by debt-to-total value (debt-to-equity) in the pure capital structure rearrangement framework implicit in research reported by Modigliani and Miller (e.g., debt-to-equity exchanges), the market value of levered equity will be a concave (convex) function, and the market value of debt will be a convex (concave) function. These nonlinearities exist even without bankruptcy costs.