## Further Development of the Formulas

In their original derivation of the relationship between ku and WACC, Modigliani and Miller assume a company that does not grow and whose debt is risk free.1 Their analysis leads to Equation 1.

Suppose we relax the assumption that growth is zero. Assuming constant growth (g ) into perpetuity, the relationship between ku and WACC becomes:

If you go one more step and assume discrete cash flows without constant growth, the relationship between ku and WACC becomes:

where,

PVT = Present value of future interest tax shields

If you accept that the tax benefit of interest expense should be discounted at the risk free rate or the cost of debt, you should use either Equation 3 or 4 to relate ku and WACC.

Working with Equation 2, where the tax benefit of interest is discounted at ku, and relaxing the no-growth assumption, results in the following relationship between ku and WACC:

As you can see, this formula is the same as the original one. The same formula also holds for discrete cash flows with variable growth rates.

Exhibits A.1, A.2, and A.3 summarize the formulas for various sets of assumptions.

1 F. Modigliani and M. Miller, ''The Cost of Capital, Corporation Finance, and the Theory of Investment," American Economic Review, June 1958, pp. 261-297.

Wi4CC -ku-k

Wi4CC -ku-k

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