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**Unformatted text preview: **Proofs for capital budgeting with leverage 1 Notation : D Dollar value of debt E Dollar value of equity V Firm value V Firm value under all-equity financing r D Cost of debt capital r E Cost of equity capital r Cost of capital if the firm is all-equity financed t C Corporate tax rate Proof of unlevering and levering formulas We first prove the unlevering equation r = (1- t C ) D (1- t C ) D + E r D + E (1- t C ) D + E r E . (1) By the MM propositions with taxes, we know that V = V + t C D. (2) This says that the firm is a portfolio consisting of its unlevered value and the value of the tax shield. We can therefore use portfolio theory to compute the expected rate of return for the levered firm. Call this rate r L . Then r L = V V + t C D r + t C D V + t C D r D . Note that we have assumed that debt is perpetual and that the risk of the tax shield is the same as the risk of the debt....

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