The Correct Discount Rate for the Tax Shield: The N-period Case

Using no-arbitrage arguments in an M & M world, we show that in the N-period case, the appropriate discount rate for the tax shield is rho, the return to unlevered equity. We make no assumption about the appropriate discount rate for the tax shield. Instead, the appropriate discount rate for the tax shield is deduced from the no-arbitrage arguments. Furthermore, it is shown that the appropriate discount rate for the tax shield does not depend on whether the value of the debt is a fixed amount or is a fixed proportion of the levered value of the firm. The analysis begins at the end of the penultimate period N-1. First, we assume that the value of the levered cash flow is higher than the sum of the value of the unlevered cash flow and the value of the tax shield. It is shown that the inequality cannot hold because arbitrage opportunities exist. The equality only holds if the discount rate for the tax shield is rho, the return to unlevered equity. Second, we assume that the value of the levered cash flow is lower than the sum of the value of the unlevered cash flow and the value of the tax shield. Again, it is shown that the inequality cannot hold because arbitrage opportunities exist. The equality only holds if the discount rate for the tax shield is rho, the return to unlevered equity. Using an iterative process, the argument can be extended period by period backwards to period zero. In conclusion, in the N-period case, the appropriate discount rate for the tax shield is rho, the return to unlevered equity.