Using APV to estimate a project's value

Question

Consider a project that requires an initial investment of $1,000,000 and has an expected cash-inflow of $85,000 a year in perpetuity. The opportunity cost of capital with an all- equity financing structure is 10%, and the project allows the firm to borrow at 7%. Assume the corporate tax rate is 35%). Use the APV to estimate the project’s value.

a.      Assume first that the project will be partly financed with $400,000 of debt and that the debt amount is to be fixed and perpetual

b.     Then assume that the initial borrowing will be increased or reduced in proportion to changes in the future market value of this project.

(Hint: Notice that the discount rate for the tax shields that you’ll in a and b is not the same. In the first case, the tax shields are fixed and constant while in the second case they will be moving in sync with the operating assets

Answer 1

(a) Project value with fixed debt

Step 1: Calculate the unlevered value of the project

The unlevered value of the project is the value of the project if it were financed entirely with equity. This is calculated by discounting the project's cash flows using the unlevered cost of capital:

Unlevered value = $85,000 / (10% - 0%)

= $850,000

Step 2: Calculate the present value of the tax shields

The present value of the tax shields is calculated by discounting the expected tax savings from the interest payments. The tax savings are calculated by multiplying the interest payments by the corporate tax rate:

Tax savings = $400,000 * 7% * 35%

= $98,000

The present value of the tax shields is calculated using the weighted average cost of capital (WACC), which is a weighted average of the cost of equity and the cost of debt:

WACC = (Cost of equity * Equity weight) + (Cost of debt * Debt weight)

WACC = (10% * 60%) + (7% * 40%)

= 9%

The present value of the tax shields is then calculated as follows:

Present value of tax shields = $98,000 / 9%

= $1,088,889

Step 3: Calculate the APV

The APV is calculated by adding the unlevered value of the project to the present value of the tax shields:

APV = Unlevered value + Present value of tax shields

APV = $850,000 + $1,088,889

= $1,938,889

Therefore, the APV of the project with fixed debt is $1,938,889.

(b) Project value with variable debt

Step 1: Calculate the unlevered value of the project

The unlevered value of the project is calculated the same way as in (a):

Unlevered value = $85,000 / (10% - 0%)

= $850,000

Step 2: Calculate the present value of the tax shields

The present value of the tax shields is calculated differently in this case because the debt level will vary depending on the market value of the project. To calculate the present value of the tax shields, we need to make some assumptions about how the debt level will change over time.

One assumption we can make is that the debt level will always be equal to 40% of the market value of the project. This means that the tax shields will be equal to 40% of the interest payments, which in turn will be equal to 7% of the debt level.

The present value of the tax shields can then be calculated using the WACC:

Present value of tax shields = $850,000 * 40% * 7% / 9%

= $280,000

Step 3: Calculate the APV

The APV is calculated the same way as in (a):

APV = Unlevered value + Present value of tax shields

APV = $850,000 + $280,000

= $1,130,000

Therefore, the APV of the project with variable debt is $1,130,000.

Conclusion

The APV of the project with fixed debt is higher than the APV of the project with variable debt. This is because the tax shields are more valuable when they are fixed and constant.