Valuation of a Leveraged Project: Apv, Fte and Wacc
Essay by Raphael Nadruz • June 11, 2017 • Study Guide • 1,053 Words (5 Pages) • 965 Views
Valuation of a Leveraged Project: APV, FTE and WACC
By JoΓ£o Amaro de Matos
Nova School of Business and Economics
Leveraged and Unleveraged Cash-flows in Perpetual Projects
Consider the case of a project generating a perpetual cash-flow X per period.
Projects with no Debt
Assume that the corporate tax rate is ππ. In the absence of debt, the cost of equity is π0 and at each period the shareholder receives the Unleveraged Cash Flow (ππΆπΉ) given by: ππΆπΉ=π(1βππ)
and the state receives πππ. In perpetuity, discounted at the rate π0, this provides shareholders with π, corresponding to the total value of the firm π0. π=π0=π(1βππ)π0.
Projects with Debt
Assume now that there is a perpetual debt B financing the project at a rate ππ. In that case the cash flow to the shareholder at each period will be the Leveraged Cash Flow (πΏπΆπΉ) given by: πΏπΆπΉ=(πβ πππ΅)(1βππ).
We thus have πΏπΆπΉ=ππΆπΉβ(1βππ)πππ΅ βππΆπΉ=πΏπΆπΉ+(1βππ)πππ΅
Also we notice that the state collects taxes in the amount (πβ πππ΅)ππ,
and the creditors receive πππ΅ as interest on debt.
Value of the Project (Proposition MM1)
It is important to notice two things. First the company pays less tax when they are leveraged as compared to an otherwise equivalent project with no debt. The difference, to be called tax-shield, is given by πππ₯ πβππππ=πππβ(πβ πππ΅)ππ=πππ΅ππ.
Second, these three cash-flows must be discounted at different rates, as they reflect different risks. Clearly the present value of debt should be obtained by discounting the cash-flow πππ΅ to creditors at the debt rate ππ in perpetuity. This provides the obvious equality π΅=πππ΅ππ.
Similarly, as the risk is the same as the debt risk, the present value of the tax shield is given by ππ(π‘ππ₯ π βππππ)=πππ΅ππππ=π΅ππ.
As the value of the firm is defined as π=π+π΅, and the leveraged firm pays less π΅ππ to the state than the unleveraged firm, this means that with debt the value of project is worth π=π0+ π΅ππ.
The Cost of Equity (Proposition MM2)
Since the project is now financed with debt, the cost of equity must change, as equity holders have now a riskier position in the sense that in a leveraged project the shareholders are only entitled to the residual claim after the creditors are paid. Let the cost of equity under a leveraged project be denoted by ππ . The value of equity must be given by the Leveraged Cash Flow discounted in perpetuity at that rate: π=πΏπΆπΉππ =(πβ πππ΅)(1βππ)ππ .
This also means that πΏπΆπΉ=πππ . As π=π+π΅=π0+ π΅ππ, we simply have to impose that (πβ πππ΅)(1βππ)ππ +π΅=π(1βππ)π0+π΅ππ.
Multiplying both sides by ππ π0 and simplifying leads to (ππ βπ0) π(1βππ)π0=ππ π΅(1βππ)βπππ΅(1βππ)=(ππ βππ)π΅(1βππ).
Noticing that π(1βππ)π0=π0=π+π΅β π΅ππ=π+π΅(1βππ),
and replacing this expression above we obtain (ππ βπ0)π=π΅(π0βππ)(1βππ),
leading to the well-known Modigliani-Miller expression for the cost of leveraged equity ππ =π0+π΅π(π0βππ)(1βππ).
Value of a Leveraged Project on a Perpetual Basis
The value of a perpetual leveraged project may be defined in different ways.
The Adjusted Present Value (APV)
We define the Adjusted Present Value (π΄ππ) of a project as the Value of the unleveraged project plus the side effects of debt. As such, in a perpetual project: π΄ππ=ππΆπΉπ0+ π΅ππ.
The Flow To EQUITY (FTE)
Alternatively we may define the value of the project as the Flow To Equity (πΉππΈ), namely as its definition of the sum of equity and debt: πΉππΈ=πΏπΆπΉππ +π΅=π+π΅.
The Equivalence between APV and FTE
We may easily show that both expressions lead to the same value (as they should).
First use πΏπΆπΉ=πππ . in the expression ππΆπΉ=πΏπΆπΉ+(1βππ)πππ΅ to get ππΆπΉ=πππ +(1βππ)πππ΅.
Using the Modigliani-Miller expression for ππ in this equation we obtain ππΆπΉ=ππ0+(1βππ)π0.
By replacing this in the expression of the APV, we immediately get π΄ππ=π+π΅=πΉππΈ.
The Weighted Average Cost of Capital (WACC)
The Weighted Average Cost of Capital is defined as a rate ππ€πππ such that the value of a project is given by the unleveraged cash-flow discounted at that rate.
In the perpetual context this means that
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