Translation between levered and unlevered equity return for explicit cash flows and debt financing

What is it about?

Contemporary textbooks propose formulas for the translation between required returns on unlevered and levered equity based on Modigliani and Miller (1963) or Miles and Ezzell (1980). These formulas are derived under strict assumptions, one of which is constant leverage. Modigliani and Miller additionally assume a stationary perpetual annuity, and Miles and Ezzell assume a Martingale process. At the same time, textbooks contain examples and exercises where these formulas are applied to cash flows that do not comply with these assumptions. This leads to an incorrect calculation of the value of the equity or firm. In addition, the valuation results will become inconsistent (different) when applying different discounted-cash-flow models at the same time. The primary purpose of this paper is to derive the correct translational formula between the unlevered and levered returns on equity for the specific case where cash flows have a finite lifetime and the flow to debt is prespecified. While the parameters included in the conventional formulas are the equity-to-firm value ratio, the tax rate and the return on debt, we find that the new formula also depends on the value of interest payments relative to the value of debt, which is different for different types of loans. A secondary purpose is to promote an understanding of the importance of the type of stochasticity of cash flows for translation formulas. Furthermore, we describe the general derivation of the discount rate in the free-cash-flow method and the translation formula between unlevered and levered returns on equity.

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Photo by Ehud Neuhaus on Unsplash

Photo by Ehud Neuhaus on Unsplash

Why is it important?

To know the correct translation between the return on unlevered equity and the return on levered equity is important for a consistent valuation of the firm by the equity method, the free-cash flow method and the adjusted-present-value method. Furthermore, this translation formula shows the factors that influence the required return on equity whenever financial leverage (debt ratio) is changed.