How to get the same valuation using APV and WACC methods?

Let's try to get the same numbers for a valuation using APV (Adjusted Present Value) and WACC (Weighted Average Cost of Capital) methodology! What's most important is the insight behind the formulas.

APV METHOD:

1) When we calculate APV we calculate first the firm value all equity financed and then we calculate the value of the tax shield provided by the debt.

NPV = EBIT * (1-Tc) / ra
TAX_SHIELD = DV * rd * Tc / rd (we must consider that the cost of opportunity of the tax shield is the same as the debt)

Where:
DV = Debt Value
ra = Cost of Assets
rd = Cost fo Debt
Tc = Tax Rate

FV = NPV + TAX_SHIELD
(FV = Firm Value)

2) To get Equity Value (EV) we subtract the debt value EV = FV - DV


WACC METHOD:

1) Using WACC we have first to find the cost of equity (re).

re = ra + (ra - rd)*(1-Tc)*D/E

D = % of debt (debt value over firm value)
E = (1-D)

I'll explain in the end why do we have to use this formula.

2) Then we calculate the WACC

WACC = re * EV/FV + rd * DV/FV *(1-Tc)

3) Then we calculate the firm value:

FV = FCFF / WACC

Where:
FCFF = EBIT * (1-Tc) (Free cash flow to the firm)

4) We can also calculate equity value (EV):

EV = FCFE / re

Where:
FCFE = (EBIT - DV*rd) * (1-Tc)
re is the same: re = ra + (ra-rd)*(1-Tc)*D/E

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To understand the equation for re, we have to think about the firm balance sheet.

We have to remind that the Firm Value is composed not only by its ASSETS, but also by the value of the TAX SHIELD. So

So the value of the assets is:

ASSETS = FV - rd*DV*Tc / rd (assuming that the cost of opportunity of the tax shield is the sam as the debt)

To find the return on equity we need to calculate the return on assets less the cost of debt, divided by the equity value.

re = ( ra*ASSETS - rd*DV*(1-Tc) ) / EV

Where: EV = FV - DV

Note that I'm considering here that the shareholder gets the return on assets plus the value of the tax shield. When we substitute ASSETS on the equation for re, we get exactly the equation for re a I showed before.

If we valuate a project using both methods with the same premises considered here, we will get the same firm value and project value.