Welcome Debt Policy , Corporate Taxes , and Discount Rates ∗ †

Debt Policy, Corporate Taxes, and Discount Rates This paper studies the valuation of assets with debt tax shields when debt policy is a general time-dependent function of the asset’s unlevered cash flows, value, and history. In a continuous-time setting, it shows that the value of a project’s debt tax shield satisfies a partial differential equation, which simplifies to an easily solved ordinary differential equation for most plausible debt policies. A large class of cases exhibits closed-form solutions for the value of a levered asset, the value of its tax shield, and the appropriate cost of capital for discounting unlevered cash flows so as to account for the value of the tax shield. Perhaps the most popular application of financial theory is capital budgeting. Virtually every student of finance starts his education in the field by learning how to discount future cash flows. By the end of a first course, the student has developed the basic tools to implement a discounted cash flow analysis in a real world setting. Because the real world setting must account for the relative advantage of debt financing, arising from the debt interest tax subsidy, students of finance generally learn that such subsidies can be accounted for by discounting unlevered cash flows (also referred to as “free cash flows”) at a tax-adjusted weighted average cost of capital (or WACC). Such tax adjustments to the discount rate generate a value for levered assets that exceed the value they would have if they were not levered with debt financing. Despite the central importance of this topic, research on how to do a proper valuation for capital budgeting purposes is sparse and largely ancient, particularly when it comes to debt tax shields. An intrinsic difficulty associated with the valuation of debt tax shields is identifying the risk of the tax deductions arising from the stream of future debt interest expenses. The rate at which one discounts the future stream of interest-related tax shields, and hence the value of those tax shields, has eluded prior research, except for the simplest of cases. These cases impose stringent restrictions on the cash flow process and debt policy to circumvent the complex issue of risk and valuation. Among these are the models of Modigliani and Miller (1958) and Miles and Ezzell (1985). The Modigliani and Miller debt policy is one where the debt level is constant and debt is both perpetual and default-free. This debt policy implies that one can discount the stream of future interest-based tax shields at the risk-free rate. If the tax rate is constant, as they assume, the debt tax shield’s present value is necessarily proportional to the present value of the debt because the cash flow stream from debt and the tax shield are proportional to one another. Here, since the constant of proportionality is the corporate tax rate, the present value of the debt tax shield is the product of the corporate tax rate and the present value of the debt. Modigliani and Miller (1958) also use this model to develop formulas for discount rates that account for the value of the tax shield when cash flows have no tendency to grow. The interesting case studied by Miles and Ezzell focuses on the dynamic issuance of perpetual risk-free debt. This case assumes: 1) the unlevered cash flow realization at each date follows a random walk with no drift, which is paid out upon its realization (hence there is no expected growth), 2) the unlevered cash flow stream is valued by applying a constant discount rate, and 3) the debt-to-asset ratio is constant. Under these assumptions, the cash flow from each date’s tax shield is of the same risk as the one period lagged unlevered cash flow. As Grinblatt and Titman (1997, 2002) and Brealey and Meyers (2000) point out, Despite attempts to introduce the Adjusted Present Value method into the classroom, the Weighted Average Cost of Capital approach still vastly dominates the practitioner landscape. For example, Graham and Harvey (2001) observe that of twelve capital budgeting techniques, many of which are long out of favor with finance academics, the Adjusted Present Value method is the least-used method.