1. Corporate Financial Policy 2004-2005 WACC
Professor André Farber
Solvay Business School
Université Libre de Bruxelles
In a previous lecture, we have analyzed the impact of taxes on the value of a company in a simplified setting: constant level of debt, perpetuities.
We obtained the following results:
1.The value of a levered company is equal to the value of the unlevered company plus the value of the tax shield equal to market value of debt multiplied by the corporate tax rate
V = VU + TCD
2. The cost of equity and the beta of equity are increasing functions of the debt-to-equiy ratio:
rE = rA + (rA – rD)(1 – TC)(D/E)
βE = β A + (β A – βD)(1 – TC)(D/E)
3.Two formulas can be used to calculate the WACC
a. the textbook formula
WACC = rE (1 – L) + rD(1 – TC) L
b. a formula derived from the value of the tax shield
WACC = rA(1 – TC L) with L = D/V
The objective of this lecture is to generalize these results.

2. How to value a levered company?
Corporate Financial Policy 3. WACC
How to value a levered company?
Value of levered company: V = VU + VTS = E + D
In general, WACC changes over time
Expected payoff = Free cash flow unlevered + Interest Tax Shield + Expected value
Expected return for debt and equity investors
Rearrange:
We first show that it is always possible to express the value of a levered company as the present value of the unlevered free cash flow discounted with the WACC. This is not however very useful because the WACC might vary over time as D/V and rE vary.
Practioners use a constant WACC. The problem is to understand under what condition this practice is correct.
Solve:
Cofipo WACC

3. Corporate Financial Policy 3. WACC
Comments
In general, the WACC changes over time. But to be useful, we should have a constant WACC to use as the discount rate. This can be obtained by restricting the financing policy.
2 possible financing rules:
Rule 1: Debt fixed
Borrow a fraction of initial project value
Interest tax shields are constant. They are discounted at the cost of debt.
Rule 2: Debt rebalanced
Adjust the debt in each future period to keep it at a constant fraction of future project value.
Interest tax shields vary. They are discounted at the opportunity cost of capital (except, possibly, for next tax shield –cf Miles and Ezzel)
Two possible financing rules under which the WACC is constant have been identified in the litterature.
Rule 1. Debt fixed + constant perpetual free cash flow.
This is the situation analyzed by Modigliani and Miller. In this approach, D is a constant (and thus VTS = TCD). VU is also a constant because of the assumption of a constant perpetuity
Rule 2. Debt rebalanced
The level of debt is adjusted to keep the ratio D/V constant. This financing rule was introduced by Miles and Ezzel in It can be used to value any set of future cash flow.
Cofipo WACC

4. Corporate Financial Policy 3. WACC
A general framework
V = VU + VTS = E + D
Value of equity rA rE
Value of all-equity firm rD
Value of debt
rTS
Value of tax shield
I propose here a general framework to understand the relationship between the value of the tax shield, the cost of equity and the weighted average cost of capital.
Suppose that you have a model that gives the value of the tax shield. Let rTS be the discount rate used to calculate this value.
As the balance sheet balances, the expected return on the liability side is equal to the expected return on the asset side:
This equality leads to formulas for the expected return on equity rE (and the beta of equity) and the weighted average cost of capital WACC
Cofipo WACC

5. Cost of equity calculation
Corporate Financial Policy 3. WACC
Cost of equity calculation
If rTS = rD (MM) and VTS = TCD
We derive here the expression for the expected return on equity. The MM formula turns out to be one particular case of this general formula.
Similar formulas for beta equity (replace r by β)
Cofipo WACC

6. Corporate Financial Policy 3. WACC
If rTS = rD and VTS = TC D (MM)
We now derive a general formula for the WACC. We show that the MM formula is a special case of the general formula.
Cofipo WACC

7. Rule 1: Debt fixed (Modigliani Miller)
Corporate Financial Policy 3. WACC
Rule 1: Debt fixed (Modigliani Miller)
Assumption: constant perpetuities FCFt = EBIT(1-TC) = rA VU
D constant.
Define: L = D/V
In the fixed debt financing rule analyzed by MM:
D is a constant
→VTS = TCD is also a constant
VU is a constant since EBIT is a constant perpetuity
→V is a constant
Cofipo WACC

8. Rule 2a: Debt rebalanced (Miles Ezzel)
Corporate Financial Policy 3. WACC
Rule 2a: Debt rebalanced (Miles Ezzel)
Assumption: any cash flows Debt rebalanced Dt/Vt = L ( a constant)
We now turn to rule 2. The level of debt is rebalanced to keep the ratio D/V constant. Consider a finite set of future cash flows FCF1,…,FCFT
The value at time T-1 is: VT-1 = FCFT/(1+rA) + rDTCLVT-1/(1+rD)
= VU,T-1+αVT-1
with α = rDTCL/(1+rD)
Therefore: VT-1 = VU,T-1/(1 – α) = FCFT /[(1- α)(1+rA)]
and VT-1 = VU,T-1 + VTST-1 = VU,T-1 + [α/(1 – α)]VU,T-1
This means that VT-1 is proportional to VU,T-1 and hence has the same risk.
At time T-2: VT-2 = FCFT-1/(1+rA) + α VT-2 + VT-1/(1+rA)
The subtle point (and the key insight of Miles and Ezzel) is to have proven that the discount rate to use for VT-1 is rA.
Some tedious algebra leads to:
VT-2 = FCFT-1 /[(1- α)(1+rA)] + FCFT /[(1- α)(1+rA)]²
This expression shows that the weighted average cost of capital is:
WACC = [(1- α)(1+rA)] = rA – rDTCL(1+rA)/(1+rD)
and
VT-2 = VU,T-2 + VTST-2 = VU,T-2 + [α/(1 – α)]VU,T-1 + VTST-1/ [(1 – α)(1+rA)]
This last expression leads to the value of the tax shield.
VTSt = [α/(1 – α)]VU,t +VTSt+1/[(1 – α)(1+rA)]
Not a straightfoward expression, I agree.
Cofipo WACC

9. Corporate Financial Policy 3. WACC
Miles-Ezzel: example
Base case NPV = =
Data
Investment 300
Pre-tax CF
Year
Year
Year
Year
Year
rA 10%
rD 5%
TC 40%
L 25%
Using Miles-Ezzel formula
WACC = 10% x 0.40 x 5% x 1.10/1.05 = 9.48%
APV = = 44.85
Initial debt: D0 = 0.25 V0 = (0.25)(344.55)=86.21
Debt rebalanced each year:
Year Vt Dt
Using MM formula:
WACC = 10%( x 0.25) = 9%
APV = = 49.21
Debt: D = 0.25 V = (0.25)(349.21) = 87.30
No rebalancing
Cofipo WACC

10. Corporate Financial Policy 3. WACC
Miles-Ezzel: example
Table 1
Table 2
Here you have the details of the example. All calculations are performed starting from the final year and moving back in time.
Table 1 shows the two possible decompositions of the value of the company: V = VU + VTS V = E + D
where Vt = (FCFt+1+Vt+1)/(1+WACC)
VU,t = (FCFt+1+Vt+1)/(1+rA)
Table 2 verifies the relationship:
It is based on the following equations:
Divt = FCFt + TCrDDt-1 – rDDt-1 + (Dt – Dt-1)
Intt = rDDt-1
rTS,t = (TC rDDt + VTSt+1 – VTSt)/VTSt
rE,t = (Divt+1 + Et+1 – Et)/Et
Note that both VTS/V and rTS vary of time. At time T-1 (year 4 in our example), the tax shield is riskless. As time to the final year increases, the risk of the tax shield increases.
Cofipo WACC

11. Rule 2b: Debt rebalanced (Harris & Pringle)
Corporate Financial Policy 3. WACC
Rule 2b: Debt rebalanced (Harris & Pringle)
Any free cash flows – debt rebalanced continously Dt = L Vt
The risk of the tax shield is equal to the risk of the unlevered firm
rTS = rA
Rule 2b is a variant of the Miles-Ezzel method proposed by Harris and Pringle in As in the Miles-Ezzel setting, the level debt is rebalanced to keep the ratio Dt/Vt constant. However, this is done continuously. The only difference with the Miles-Ezzel analysis is that the tax shield of the following year is discounted with rA (the tax shield is the same risk as the cash flow of the unlevered firm). The rest of the development is identical to Miles-Ezzel.
Consider again a finite set of future cash flows FCF1,…,FCFT
The value at time T-1 is: VT-1 = FCFT/(1+rA) + rDTCLVT-1/(1+rA)
= VU,T-1+αVT-1
with α = rDTCL/(1+rA) (instead of α = rDTCL/(1+rD) in Miles-Ezzel)
This leads to:
VT-2 = FCFT-1 /[(1- α)(1+rA)] + FCFT /[(1- α)(1+rA)]²
The weighted average cost of capital is:
WACC = [(1- α)(1+rA)] = rA – rDTCL
and the value of the tax shield is
VTSt = VU,t + [α/(1 – α)]VU,t + VTSt+1/ [(1 – α)(1+rA)]
Cofipo WACC

12. Harris-Pringle: example
Corporate Financial Policy 3. WACC
Harris-Pringle: example
Here we illustrate the Harris-Pringle method with the data used in the Miles-Ezzel example.
The only noteworthy difference is that rTS is now a constant equal to rA
Cofipo WACC

13. Corporate Financial Policy 3. WACC
Summary of Formulas
Modigliani Miller
Miles Ezzel
Harris-Pringle
Operating CF
Perpetuity
Finite or Perpetual
Finite of Perpetual
Debt level
Certain
Uncertain
First tax shield
WACC
L = D/V
rE(E/V) + rD(1-TC)(D/V)
rA (1 – TC L)
rA – rD TC L
Cost of equity
rA+(rA –rD)(1-TC)(D/E)
rA+(rA –rD) (D/E)
Beta equity
βA+(βA – βD) (1-TC) (D/E)
βA +( βA – βD) (D/E)
Source: Taggart – Consistent Valuation and Cost of Capital Expressions With Corporate and Personal Taxes Financial Management Autumn 1991
Cofipo WACC

14. Constant perpetual growth
Corporate Financial Policy 3. WACC
Constant perpetual growth
Which formula to use if unlevered free cash flows growth at a constant rate?
In a provocative paper, Fernandez (2004) challenges the solutions currently proposed in the leading corporate finance textbooks. Fernandez claims that the value of tax shields is not the present value of tax shields. He argues that the correct value of tax shields is equal to the difference between the present value of taxes paid by the unlevered company and the present value of taxes paid by the levered company. These present values should be calculated using different discount rates.
The example shows that substantial valuation differences appear depending on the method.
A lively debate is currently taking place to clarify the issue.
Cofipo WACC

15. Varying debt levels
How to proceed if none of the financing rules applies?
Two important instances:
(i) debt policy defined as an amount of borrowing instead of as a target percentage of value
(ii) the amount of debt changes over time
Use the Capital Cash Flow method suggested by Ruback
(Ruback, Richard A Note on Capital Cash Flow Valuation, Harvard Business School, , January 1995)
Cofipo WACC

16. Capital Cash Flow Valuation
Assumptions:
CAPM holds
PV(Tax Shield) as risky as operating assets
Capital cash flow =FCF unlevered +Tax shield
Cofipo WACC

17. Capital Cash Flow Valuation: Example
Cofipo WACC