Answer 3 we see that v l v u 213 2 013 billion since

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ANSWER (3): We see that V L – V U = $2.13 – $2 = $0.13 billion . Since this value is just G L , we can show that G L for the personal tax view is greater than the G L for the corporate tax view. For the corporate tax view, we have: G L = TD = T(CPN / r d ) = 0.2(50 / 0.05) = $0.2 billion . Thus, the corporate tax view gives a value that is $0.2 – $0.13 = $0.07 billion or $70,000 million more . Most of the difference can be explained by the fact debtholders now pay taxes. Note that even without considering the impact of (1 – α ), we have: (1 – T d )CPN / r d = (1 – 0.85)$50M / 0.05 = $850 million. For the corporate tax view, this value would be $1 billion because debtholders do not pay personal taxes. 74. Consider the following definitions. The gain to leverage (G L ) equals leveraged firm value (V L ) minus unleveraged firm value (V U ). V L equals leveraged equity value (E L ) plus debt value (D). E L equals (1 – T e )(1 – T)( – CPN) / r L where T e is the applicable tax rate on equity income, T is the firm's applicable corporate tax rate, is the uncertain perpetual before-tax cash flow for unleveraged equity owners, CPN is the perpetual before-tax cash flow for debt owners, and r L is the endogenously determined cost of leveraged equity. V U is unleveraged equity value (E U ) and equals (1 – T e )(1 – T) / r where r is the exogenous cost of unleveraged equity. D is (1 – T d )CPN / r d where T d is the applicable tax rate on debt income, and r d is the cost of debt. Given these definitions answer the below questions. (1) Substitute the definitions of E L and D into the formula for V L and then substitute this expression into the definition for G L . After doing this, use algebra to see if you can get gain to leverage = G L = [1 – α r d / r L ]D – [1 – (r / r L )]E U where α = (1 – T e )(1 – T) / (1 – T d ). NOTE. The derivation is unimpeded regardless of whether r d is the exogenous riskless rate or is endogenously determined by the choice of D. (2) Why might this expression for gain to leverage be more valuable than other expressions? ANSWER (1): Substituting the definitions of E L and D into the formula for V L gives: V L = E L + D = [(1 – T e )(1 – T)( – CPN) / r L ] + D. Substituting this expression along with the definition V U = E U into the definition G L = V L – V U gives: G L = [(1 – T e )(1 – T)( – CPN) / r L ] + D – E U . Multiplying out the first component and rearranging: G L = D – [(1 – T e )(1 – T)CPN / r L ] – E U + [(1 – T e )(1 – T) / r L ]. Multiplying the second component by (1 – T d )r d / (1 – T d )r d = 1 and noting that this is equivalent to – {(1 – T e )(1 – T)r d / (1 – T d )r L }(1 – T d )CPN / r d which is – [(1 – T e )(1 – T)r d / (1 – T d )r L ]D, and factoring out D: G L = [1 – {(1 – T e )(1 – T)r d / (1 – T d )r L }D – E U + [(1 – T e )(1 – T) / r L ]. Multiplying the last component by r / r = 1 and noting this equals (r / r L )(1 – T e )(1 – T) / r which is (r / r L )E U , and factoring out E U : G L = [1 – {(1 – T e )(1 – T}r d / (1 – T d )r L }]D – [1 – (r / r L )]E U .
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