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Valuation of a Leveraged Project: Apv, Fte and Wacc

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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 𝑇𝑐. In the absence of debt, the cost of equity is π‘Ÿ0 and at each period the shareholder receives the Unleveraged Cash Flow (π‘ˆπΆπΉ) given by: π‘ˆπΆπΉ=𝑋(1βˆ’π‘‡π‘)

and the state receives 𝑋𝑇𝑐. In perpetuity, discounted at the rate π‘Ÿ0, this provides shareholders with 𝑆, corresponding to the total value of the firm 𝑉0. 𝑆=𝑉0=𝑋(1βˆ’π‘‡π‘)π‘Ÿ0.

Projects with Debt

Assume now that there is a perpetual debt B financing the project at a rate π‘Ÿπ‘. In that case the cash flow to the shareholder at each period will be the Leveraged Cash Flow (𝐿𝐢𝐹) given by: 𝐿𝐢𝐹=(π‘‹βˆ’ π‘Ÿπ‘π΅)(1βˆ’π‘‡π‘).

We thus have 𝐿𝐢𝐹=π‘ˆπΆπΉβˆ’(1βˆ’π‘‡π‘)π‘Ÿπ‘π΅ β‡’π‘ˆπΆπΉ=𝐿𝐢𝐹+(1βˆ’π‘‡π‘)π‘Ÿπ‘π΅

Also we notice that the state collects taxes in the amount (π‘‹βˆ’ π‘Ÿπ‘π΅)𝑇𝑐,

and the creditors receive π‘Ÿπ‘π΅ 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 π‘‡π‘Žπ‘₯ π‘†β„Žπ‘–π‘’π‘™π‘‘=π‘‹π‘‡π‘βˆ’(π‘‹βˆ’ π‘Ÿπ‘π΅)𝑇𝑐=π‘Ÿπ‘π΅π‘‡π‘.

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 π‘Ÿπ‘π΅ to creditors at the debt rate π‘Ÿπ‘ in perpetuity. This provides the obvious equality 𝐡=π‘Ÿπ‘π΅π‘Ÿπ‘.

Similarly, as the risk is the same as the debt risk, the present value of the tax shield is given by 𝑃𝑉(π‘‘π‘Žπ‘₯ π‘ β„Žπ‘–π‘’π‘™π‘‘)=π‘Ÿπ‘π΅π‘‡π‘π‘Ÿπ‘=𝐡𝑇𝑐.

As the value of the firm is defined as 𝑉=𝑆+𝐡, and the leveraged firm pays less 𝐡𝑇𝑐 to the state than the unleveraged firm, this means that with debt the value of project is worth 𝑉=𝑉0+ 𝐡𝑇𝑐.

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 π‘Ÿπ‘ . The value of equity must be given by the Leveraged Cash Flow discounted in perpetuity at that rate: 𝑆=πΏπΆπΉπ‘Ÿπ‘ =(π‘‹βˆ’ π‘Ÿπ‘π΅)(1βˆ’π‘‡π‘)π‘Ÿπ‘ .

This also means that 𝐿𝐢𝐹=π‘†π‘Ÿπ‘ . As 𝑉=𝑆+𝐡=𝑉0+ 𝐡𝑇𝑐, we simply have to impose that (π‘‹βˆ’ π‘Ÿπ‘π΅)(1βˆ’π‘‡π‘)π‘Ÿπ‘ +𝐡=𝑋(1βˆ’π‘‡π‘)π‘Ÿ0+𝐡𝑇𝑐.

Multiplying both sides by π‘Ÿπ‘ π‘Ÿ0 and simplifying leads to (π‘Ÿπ‘ βˆ’π‘Ÿ0) 𝑋(1βˆ’π‘‡π‘)π‘Ÿ0=π‘Ÿπ‘ π΅(1βˆ’π‘‡π‘)βˆ’π‘Ÿπ‘π΅(1βˆ’π‘‡π‘)=(π‘Ÿπ‘ βˆ’π‘Ÿπ‘)𝐡(1βˆ’π‘‡π‘).

Noticing that 𝑋(1βˆ’π‘‡π‘)π‘Ÿ0=𝑉0=𝑆+π΅βˆ’ 𝐡𝑇𝑐=𝑆+𝐡(1βˆ’π‘‡π‘),

and replacing this expression above we obtain (π‘Ÿπ‘ βˆ’π‘Ÿ0)𝑆=𝐡(π‘Ÿ0βˆ’π‘Ÿπ‘)(1βˆ’π‘‡π‘),

leading to the well-known Modigliani-Miller expression for the cost of leveraged equity π‘Ÿπ‘ =π‘Ÿ0+𝐡𝑆(π‘Ÿ0βˆ’π‘Ÿπ‘)(1βˆ’π‘‡π‘).

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=π‘ˆπΆπΉπ‘Ÿ0+ 𝐡𝑇𝑐.

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=πΏπΆπΉπ‘Ÿπ‘ +𝐡=𝑆+𝐡.

The Equivalence between APV and FTE

We may easily show that both expressions lead to the same value (as they should).

First use 𝐿𝐢𝐹=π‘†π‘Ÿπ‘ . in the expression π‘ˆπΆπΉ=𝐿𝐢𝐹+(1βˆ’π‘‡π‘)π‘Ÿπ‘π΅ to get π‘ˆπΆπΉ=π‘†π‘Ÿπ‘ +(1βˆ’π‘‡π‘)π‘Ÿπ‘π΅.

Using the Modigliani-Miller expression for π‘Ÿπ‘  in this equation we obtain π‘ˆπΆπΉ=π‘†π‘Ÿ0+(1βˆ’π‘‡π‘)π‘Ÿ0.

By replacing this in the expression of the APV, we immediately get 𝐴𝑃𝑉=𝑆+𝐡=𝐹𝑇𝐸.

The Weighted Average Cost of Capital (WACC)

The Weighted Average Cost of Capital is defined as a rate π‘Ÿπ‘€π‘Žπ‘π‘ 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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