Questions:
What is the main difference between WACC-based valuation and APV valuation?

My company is considering an acquisition of another company for $100 million. Our current best estimate is that this new company will generate cash flows of $1 million dollars per year, and this will grow indefinitely at a rate of 4% per year. Our cost of capital is 5% and the interest rate on the loan is 8%. 

To afford this acquisition, we will fund 25% of the acquisition through debt. What is the APV? Assume the corporate tax rate is 50% and we will keep our debt-to-equity ratio constant.

Reference answers:
The main difference lies in how they try to account for the interest rate tax shield. The WACC method tries to roll this into its estimate for the required rate of return. APV instead tries to first compute the unlevered value and then directly incorporate the value of the interest rate tax shield.

First, we need to compute the unlevered value of this company. Since it is going to generate a constantly growing stream of cash flows, we can use:

PV = C / (g - r) = 1e6 / (5% - 4%) = $100,000,000

This means that the unlevered value of the acquisition is $100 million.

We can do the same calculation to estimate the value of the tax shield. SInce we are funding 25% of the $100 million in debt, that means that we have $25 million. In the first year, we will be required to pay 0.08 * 25 = $2 million dollars. Since the tax rate is 50% that means that we have a tax shield of 2 * 0.5 = $1 million in the first year. However, since we are keeping the debt-to-equity ratio constant, we need to value not only the $1 million dollars of tax shield that this acquisition will support, but the tax shield from all of the additional debt as well. Since this value is growing as a perpetuity, this will grow as a perpetuity as well. This means that its value is also:

PV = C / (g - r) = 1e6 / (5% - 4%) = $100,000,000

Therefore,

APV = Unlevered Present Value + Present Value of Interest Rate Tax Shield = 100 + 100 = $200 Million.