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Lecture 7: Financing Based on Market Values II Discounted Cash Flow, Section 2.4.4 – 2.4.5 © 2004, Lutz Kruschwitz and Andreas Löffler

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2.4.4 Miles-Ezzell: the problem Up to now we know three procedures to evaluate (or ). In case of financing based on market values these procedures coincide, otherwise not. But what can we tell about the relation between WACC and the unlevered cost of capital, i.e. ? This topic of adjustment formulas. And this time we need the assumption of weak autoregressive cash flows.

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2.4.4 Miles-Ezzell adjustment Theorem 2.11 (Miles-Ezzell 1980): If cash flows of the unlevered firm are weak autoregressive, the levered firm is financed based on market values and WACC is deterministic, then and is deterministic.

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2.4.4 Remarks The original article of Miles–Ezzell required a constant leverage ratio : we do not! This adjustment formula finally shows why WACC might be useful (remember apples and oranges?). The assumption of weak autoregressive cash flows is necessary (we come back to this). The proof is not an easy task. What is the connection to the Modigliani–Miller (1963) adjustment formula? Later...

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2.4.4 Proof (part I) We start from fundamental theorem for levered firm with its tax advantage (see (2.9))

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2.4.4 Proof (part I) Using rule 3 (linearity)

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2.4.4 Proof (part I) Rearranging terms

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2.4.4 Proof (part I) On the ground of riskless debt we get

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2.4.4 Proof (part I) Using rule 5 (expectation of realized quantities)

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2.4.4 Proof (part I) Rearranging terms

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2.4.4 Proof (part I) Employing the definition of leverage ratio

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2.4.4 Proof (part I) Rearranging terms

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2.4.4 Proof (part I) Rearranging terms

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2.4.4 Proof (part I) And finally have a nice recursive equation!

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2.4.4 Proof (part II) We start with the recursive equation

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2.4.4 Proof (part II) Since cash flows of the unlevered firm are weak auto- regressive, cost of capital can be used as discount rates

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2.4.4 Proof (part II) This must satisfy the following recursion equation

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2.4.4 Proof (part II) Rearranging terms

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2.4.4 Proof (part II) Using the definition of WACC type 2

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2.4.4 Proof (part II) q.e.d.

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2.4.4 Modigliani-Miller: the problem This equation is the first (1963) adjustment formula. Two assumptions are necessary: constant amount of debt (autonomous financing!) and infinite lifetime. Claims a nice formula: Theorem 2.12 (Modigliani-Miller): If WACC type 2 (WACC) and the unlevered firm´s cost of equity are deterministic, then the firm is financed based on market values. This is a contradiction to autonomous financing, hence the Modigliani-Miller formula is not applicable!

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2.4.4 Contradiction to Modigliani-Miller Summary: In order to apply MoMi two costs of capital must be deterministic:, otherwise the (nice) formula does not make sense. But then the firm is financed based on market values (see above theorem). This contradicts the assumption of MoMi (constant debt)! Hence, there is no connection to Miles–Ezzell, since the nice formula is not applicable under any circumstances!

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2.4.4 Modigliani-Miller again Is the original paper Modigliani-Miller (1963) flawed? No, since for both (as for Miles-Ezzell as well) costs of capital were not conditional expected returns! Although you can use MoMi theorem (theorem 2.5), you cannot use MoMi adjustment formula. Put differently: «discount rates» and «expected returns» cover different economic concepts.

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2.4.4 Proof Use Mo-Mi theorem (theorem 2.5) and the fact that cashflow are weak auto-regressive

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2.4.4 Proof

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Everything except is deterministic. QED

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2.4.5 The finite example Consider financing based on market values WACC results from Miles-Ezzell adjustment (theorem 2.11)

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2.4.5 The finite example The firm value is

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2.4.5 The infinite example The leverage ratio is l = 20% constant through lifetime. With Miles-Ezzell adjustment the WACC is And the firm value is

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Summary The connection between costs of capital of a levered firm and an unlevered firm are given by adjustment formulas. In particular a relation between WACC and can be proven, hence WACC is indeed useful. The adjustment by Miles–Ezzell is applicable (but this requires financing based on market values!). The adjustment by Modigliani–Miller is not applicable, although the Modigliani–Miller theorem can be used.

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