P$10\mathrm{.}8$ Incorrect Valuation Assumptions: A privately held company finances itself with long$-$term debt and preferred stock using the capital structure shown below. The company's current free cash flow is $$100,$ and it expects to generate a series of cash flows that will grow at $3\%$ per year in perpetuity. The company plans to maintain its current capital structure strategy of $50\%$ debt and $20\%$ preferred stock in perpetuity, refinancing the company on an ongoing basis.

Chapter $10$ I Levering and Unlevering the Cost of Capital and Beta

Initial debt to firm value $.........................................................................................$ Initial preferred stock to firm value $..........................................................................................$

Preferred stock cost of capital $.........................................................................................$

Tax rate on all income. $...........................................................$

The company's chief financial officer $($CFO$)$ measured the company's equity cost of capital using the following formula.

$r_E=r_{UA}+(r_{UA}-r_D)(1-T_{DNT})\frac{V_D}{V_E}+(r_{UA}-r_{PS})\frac{V_{PS}}{V_E}$

The CFO valued the company and its equity using the WACC valuation method. The CFO wants to check the WACC valuation by using the APV valuation method but is unsure how to use this valuation method to check the WACC valuation. The CFO calculated a $$15,152$ firm value using the WACC valuation method.

a$.$ Reproduce the CFO's equity cost of capital, weighted average cost of capital, and WACC valuation and compare it to an APV valuation. Are the valuations consistent?

b$.$ Use the WACC valuation method to value the firm and the equity correctly by discounting the tax shields at the unlevered cost of capital. Use the APV valuation method to value the firm, and use the Equity Free Cash Flow valuation method to value the company's equity. Are the valuations consistent?