Valuing corporate debt: The effect of cross-holdings of stock and debt

We have developed a simple approach to valuing risky corporate debt when corporations own securities issued by other corporations. We assume that corporate debt can be valued as an option on corporate business asset value, and derive payoff functions when there exist cross-holdings of stock or debt between two firms. Next we show that payoff functions with multiple cross-holdings can be solved by the contraction principle. The payoff functions which we derive provide a number of insights about the risk structure of company cross-holdings. First, the Modigliani–Miller theorem can obtain when there exist cross-holdings between firms. Second, by establishing cross-shareholdings each of stock holders distributes a part of its payoff values to the bond holder of the other's firm, so that both firms can decrease credit risks by cross-shareholdings. In the numerical examples, we show that the correlation in firms can be a critical condition for reducing credit risk by cross-holdings of stock using Monte Carlo simulation. Moreover, we show we can calculate the default spread easily when complicated cross-holdings exist in many firms by a simulation that includes the contraction principle method. By using our method, we can reevaluate which cross-held shares are beneficial or disadvantageous for the cross-holding firms.