Essay About Valuation Of A Leveraged Project And 𝑟0

Valuation of a Leveraged Project: Apv, Fte and Wacc
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Valuation of a Leveraged Project: APV, FTE and WACC
By João 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 𝑇𝑐. In the absence of debt, the cost of equity is 𝑟0 and at each period the shareholder receives the Unleveraged Cash Flow (𝑈𝐶𝐹) given by: 𝑈𝐶𝐹=𝑋(1−𝑇𝑐)

and the state receives 𝑋𝑇𝑐. In perpetuity, discounted at the rate 𝑟0, this provides shareholders with 𝑆, corresponding to the total value of the firm 𝑉0. 𝑆=𝑉0=𝑋(1−𝑇𝑐)𝑟0.

Projects with Debt
Assume now that there is a perpetual debt B financing the project at a rate 𝑟𝑏. In that case the cash flow to the shareholder at each period will be the Leveraged Cash Flow (𝐿𝐶𝐹) given by: 𝐿𝐶𝐹=(𝑋− 𝑟𝑏𝐵)(1−𝑇𝑐).

We thus have 𝐿𝐶𝐹=𝑈𝐶𝐹−(1−𝑇𝑐)𝑟𝑏𝐵 ⇒𝑈𝐶𝐹=𝐿𝐶𝐹+(1−𝑇𝑐)𝑟𝑏𝐵
Also we notice that the state collects taxes in the amount (𝑋− 𝑟𝑏𝐵)𝑇𝑐,
and the creditors receive 𝑟𝑏𝐵 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 𝑇𝑎𝑥 𝑆ℎ𝑖𝑒𝑙𝑑=𝑋𝑇𝑐−(𝑋− 𝑟𝑏𝐵)𝑇𝑐=𝑟𝑏𝐵𝑇𝑐.

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 𝑟𝑏𝐵 to creditors at the debt rate 𝑟𝑏 in perpetuity. This provides the obvious equality 𝐵=𝑟𝑏𝐵𝑟𝑏.

Similarly, as the risk is the same as the debt risk, the present value of the tax shield is given by 𝑃𝑉(𝑡𝑎𝑥 𝑠ℎ𝑖𝑒𝑙𝑑)=𝑟𝑏𝐵𝑇𝑐𝑟𝑏=𝐵𝑇𝑐.
As the value of the firm is defined as 𝑉=𝑆+𝐵, and the leveraged firm pays less 𝐵𝑇𝑐 to the state than the unleveraged firm, this means that with debt the value of project is worth 𝑉=𝑉0+ 𝐵𝑇𝑐.

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 𝑟𝑠. The value of equity must be given by the Leveraged Cash Flow discounted in perpetuity at that rate: 𝑆=𝐿𝐶𝐹𝑟𝑠=(𝑋− 𝑟𝑏𝐵)(1−𝑇𝑐)𝑟𝑠.

This also means that 𝐿𝐶𝐹=𝑆𝑟𝑠. As 𝑉=𝑆+𝐵=𝑉0+ 𝐵𝑇𝑐, we simply have to impose that (𝑋− 𝑟𝑏𝐵)(1−𝑇𝑐)𝑟𝑠+𝐵=𝑋(1−𝑇𝑐)𝑟0+𝐵𝑇𝑐.
Multiplying both sides by 𝑟𝑠𝑟0 and simplifying leads to (𝑟𝑠−𝑟0) 𝑋(1−𝑇𝑐)𝑟0=𝑟𝑠𝐵(1−𝑇𝑐)−𝑟𝑏𝐵(1−𝑇𝑐)=(𝑟𝑠−𝑟𝑏)𝐵(1−𝑇𝑐).
Noticing that 𝑋(1−𝑇𝑐)𝑟0=𝑉0=𝑆+𝐵− 𝐵𝑇𝑐=𝑆+𝐵(1−𝑇𝑐),
and replacing this expression above we obtain (𝑟𝑠−𝑟0)𝑆=𝐵(𝑟0−𝑟𝑏)(1−𝑇𝑐),
leading to the well-known Modigliani-Miller expression for the cost of leveraged equity 𝑟𝑠=𝑟0+𝐵𝑆(𝑟0−𝑟𝑏)(1−𝑇𝑐).
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 (𝐴𝑃𝑉) of a project as the Value of the unleveraged project plus the side effects of debt. As such, in a perpetual project: 𝐴𝑃𝑉=𝑈𝐶𝐹𝑟0+ 𝐵𝑇𝑐.

The Flow To EQUITY (FTE)
Alternatively we may define the value of the project as the Flow To Equity (𝐹𝑇𝐸), namely as its definition of the sum of equity and debt: 𝐹𝑇𝐸=𝐿𝐶𝐹𝑟𝑠+𝐵=𝑆+𝐵.

The Equivalence between APV and FTE
We may easily show that both expressions lead to the same value (as they should).
First use 𝐿𝐶𝐹=𝑆𝑟𝑠. in the expression 𝑈𝐶𝐹=𝐿𝐶𝐹+(1−𝑇𝑐)𝑟𝑏𝐵 to get 𝑈𝐶𝐹=𝑆𝑟𝑠+(1−𝑇𝑐)𝑟𝑏𝐵.
Using the Modigliani-Miller expression for 𝑟𝑠 in this equation we obtain 𝑈𝐶𝐹=𝑆𝑟0+(1−𝑇𝑐)𝑟0.
By replacing this in the expression of the APV, we immediately get 𝐴𝑃𝑉=𝑆+𝐵=𝐹𝑇𝐸.
The Weighted Average Cost of Capital (WACC)
The Weighted Average Cost of Capital is defined as a rate 𝑟𝑤𝑎𝑐𝑐 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