Certainly! Let’s break down the solution to this APV (Adjusted Present Value) problem in **two steps** with explanations and the final answer. Here’s how to approach it: --- ## **Step 1: Calculate the NPV as if All-Equity Financing** **Explanation:** First, ignore the debt, and calculate the NPV using the project's unlevered cash flows and the asset cost of capital (unlevered cost, 15%). ### **Given:** - Initial investment = $3.6 million - life = 5 years - Annual cash inflow = $1.15 million - Discount rate (all-equity) = 15% ### **Formula:** \[ NPV = - \text{Initial Investment} + \sum_{t=1}^{5} \frac{1.15 \text{ million}}{(1.15)^t} \] Calculate the present value (PV) of the annuity: \[ PV = 1.15 \times \left(1 - (1 + .15)^{-5}\right) / .15 \] \[ PV = 1.15 \times \left(1 - (1.15)^{-5}\right) / .15 \] \[ (1.15)^{-5} \approx .497 \] \[ 1 - .497 = .503 \] \[ PV = 1.15 \times .503 / .15 = 1.15 \times 3.353 = \$3.856 \text{ million} \] \[ NPV = 3.856 - 3.6 = \$.256 \text{ million} \] **End of Step 1:** **Unlevered NPV = $.256 million** --- ## **Step 2: Add the PV of the Interest Tax Shield (APV Adjustment)** **Explanation:** Now add the present value of the tax shield from the interest on debt. - Debt issued = $1.44 million - Interest rate = 10% - Tax rate = 35% - Debt is perpetual, so PV of tax shield = (Debt × Interest Rate × Tax Rate) / Interest Rate = Debt × Tax Rate \[ \text{PV of tax shield} = 1.44 \times .35 = \$.504 \text{ million} \] Now add this to the NPV from Step 1: \[ \text{APV} = .256 + .504 = \$.760 \text{ million} \] --- ## **Final Answer** \[ \boxed{APV = \$760,000} \] --- ### **Summary** 1. **Unlevered NPV:** $.256 million (using all-equity financing) 2. **PV of Interest Tax Shield:** $.504 million (from tax-deductible debt interest) 3. **APV (Adjusted Present Value):** $.760 million Let me know if you want to see the breakdown of any calculations!

Certainly! Let's solve the problem step by step and provide explanations along the way. --- ## **Step 1: Calculate the NPV with All-Equity Financing** **Explanation:** To begin, we will calculate the Net Present Value (NPV) of the project assuming it is entirely financed through equity. This involves using the unlevered cash flows and the cost of capital. ### **Given Data:** - Initial investment = $3.6 million - Life of the project = 5 years - Annual cash inflow = $1.15 million - Discount rate (cost of capital) = 15% ### **NPV Formula:** \[ NPV = - \text{Initial Investment} + \sum_{t=1}^{5} \frac{\text{Cash Flow}}{(1 + r)^t} \] ### **Calculating Present Value (PV) of Cash Flows:** Using the formula for the present value of an annuity: \[ PV = C \times \left(1 - (1 + r)^{-n}\right) / r \] where: - \( C = 1.15 \text{ million} \) - \( r = 0.15 \) - \( n = 5 \) ### **Calculating:** 1. Calculate \( (1 + 0.15)^{-5} \): \[ (1.15)^{-5} \approx 0.497 \] 2. Calculate the annuity factor: \[ 1 - 0.497 = 0.503 \] 3. Calculate PV: \[ PV = 1.15 \times \frac{0.503}{0.15} = 1.15 \times 3.353 \approx 3.856 \text{ million} \] ### **Final NPV Calculation:** \[ NPV = 3.856 - 3.6 = 0.256 \text{ million} \] **End of Step 1:** **Unlevered NPV = $0.256 million** --- ## **Step 2: Add the Present Value of the Interest Tax Shield** **Explanation:** Next, we will calculate the present value (PV) of the tax shield generated by the interest on the debt. This requires using the debt amount, interest rate, and tax rate. ### **Given Data for Debt:** - Debt issued = $1.44 million - Interest rate = 10% - Tax rate = 35% ### **PV of Tax Shield Formula:** \[ \text{PV of Tax Shield} = \text{Debt} \times \text{Tax Rate} \] ### **Calculating:** \[ \text{PV of Tax Shield} = 1.44 \times 0.35 = 0.504 \text{ million} \] ### **Final APV Calculation:** Now we combine the unlevered NPV and the PV of the tax shield: \[ \text{APV} = 0.256 + 0.504 = 0.760 \text{ million} \] --- ## **Final Answer** \[ \boxed{APV = \$760,000} \] --- ### **Summary** 1. **Unlevered NPV:** $0.256 million (calculated with all-equity financing) 2. **PV of Interest Tax Shield:** $0.504 million (calculated from the tax-deductible interest on debt) 3. **APV (Adjusted Present Value):** $0.760 million Let me know if you need further assistance or clarification!