1. Théorie Financière Structure financière et coût du capital
Professeur André Farber

2. Risk and return and capital budgeting
Objectives for this session:
Beta of a portfolio
Beta and leverage
Weighted average cost of capital
Modigliani Miller 1958
WACC with taxes
Tfin Intro to capital structure

3. Beta of a portfolio Consider the following portfolio Stock Value Beta
A nA  PA A
B nB  PB B
The value of the portfolio is:
V = nA  PA + nA  PB
The fractions invested in each stock are:
Xi = ( ni Pi) / V for i = A,B
The beta of the portfolio is the weighted average of the betas of the individual stocks
P = XA A + XB B
Example 1
Stock $  X
ATT 3,000 0,
GENENTECH 2,
V 5,000
P = 0.60   1.40
= 1.02
Tfin Intro to capital structure

4. Example Stock $  X ATT 3,000 0.76 0.60 Genetech 2,000 1.40 0.40
V 5,000
P = 0.60 * * 1.40 = 1.02
Tfin Intro to capital structure

5. Application 1: cost of capital for a division
Firm = collection of assets
Example: A company has two divisions
Value($ mio) 
Electrical
Chemical
V 600
firm = (100/600) * (0.50) + (500/600) * 0.90 = 0.83
Assume: rf = 5% rM - rf = 6%
Expected return on stocks: r = 5% + 6%  0.83 = 9.98 %
An adequate hurdle rate for capital budgeting decisions ? No
The firm should use required rate of returns based on project risks:
Electricity :  0.50 = 8% Chemical :  0.90 = 10.4%
Tfin Intro to capital structure

6. Application 2: leverage and beta
Consider an investor who borrows at the risk free rate to invest in the market portfolio
Assets $  X
Market portfolio 2,
Risk-free rate -1,
V 1,000
P = 2  1 + (-1)  = 2
Tfin Intro to capital structure

7. Expected Return Expected Return P 20% P M 14% 14% M 8% 8% 2 Sigma 1
Beta
Tfin Intro to capital structure

8. Cost of capital with debt
Up to now, the analysis has proceeded based on the assumption that investment decisions are independent of financing decisions.
Does
the value of a company change
the cost of capital change
if leverage changes ?
Tfin Intro to capital structure

9. An example CAPM holds – Risk-free rate = 5%, Market risk premium = 6%
Consider an all-equity firm:
Market value V 100
Beta 1
Cost of capital 11% (=5% + 6% * 1)
Now consider borrowing 20 to buy back shares.
Why such a move?
Debt is cheaper than equity
Replacing equity with debt should reduce the average cost of financing
What will be the final impact
On the value of the company? (Equity + Debt)?
On the weighted average cost of capital (WACC)?
Tfin Intro to capital structure

10. Weighted Average Cost of Capital
An average of:
The cost of equity requity
The cost of debt rdebt
Weighted by their relative market values (E/V and D/V)
Note: V = E + D
Tfin Intro to capital structure

11. Modigliani Miller (1958)
Assume perfect capital markets: not taxes, no transaction costs
Proposition I:
The market value of any firm is independent of its capital structure:
V = E+D = VU
Proposition II:
The weighted average cost of capital is independent of its capital structure
rwacc = rA
rA is the cost of capital of an all equity firm
Tfin Intro to capital structure

12. Using MM 58 Value of company: V = 100 Initial Final Equity 100 80
Debt
Total MM I
WACC = rA 11% 11% MM II
Cost of debt - 5% (assuming risk-free debt)
D/V
Cost of equity 11% % (to obtain rwacc = 11%)
E/V 100% 80%
Tfin Intro to capital structure

13. Cost of equity calculation
V (=VU ) = E + D
Value of equity rE
Value of all-equity firm rA rD
Value of debt
Tfin Intro to capital structure

14. Why is rwacc unchanged?
Consider someone owning a portfolio of all firm’s securities (debt and equity) with Xequity = E/V (80% in example ) and Xdebt = D/V (20%)
Expected return on portfolio = requity * Xequity + rdebt * Xdebt
This is equal to the WACC (see definition):
rportoflio = rwacc
But she/he would, in fact, own a fraction of the company. The expected return would be equal to the expected return of the unlevered (all equity) firm
rportoflio = rA
The weighted average cost of capital is thus equal to the cost of capital of an all equity firm
rwacc = rA
Tfin Intro to capital structure

15. What are MM I and MM II related?
Assumption: perpetuities (to simplify the presentation)
For a levered companies, earnings before interest and taxes will be split between interest payments and dividends payments
EBIT = Int + Div
Market value of equity: present value of future dividends discounted at the cost of equity
E = Div / requity
Market value of debt: present value of future interest discounted at the cost of debt
D = Int / rdebt
Tfin Intro to capital structure

16. Relationship between the value of company and WACC
From the definition of the WACC:
rwacc * V = requity * E + rdebt * D
As requity * E = Div and rdebt * D = Int
rwacc * V = EBIT
V = EBIT / rwacc
Market value of levered firm
EBIT is independent of leverage
If value of company varies with leverage, so does WACC in opposite direction
Tfin Intro to capital structure

17. MM II: another presentation
The equality rwacc = rA can be written as:
Expected return on equity is an increasing function of leverage:
requity
12.5%
Additional cost due to leverage
11% rA
rwacc 5%
rdebt
D/E
0.25
Tfin Intro to capital structure

18. Why does requity increases with leverage?
Because leverage increases the risk of equity.
To see this, back to the portfolio with both debt and equity.
Beta of portfolio: portfolio = equity * Xequity + debt * Xdebt
But also: portfolio = Asset
So: or
Tfin Intro to capital structure

19. Back to example Assume debt is riskless:
Tfin Intro to capital structure

20. Summary: the Beta-CAPM diagram
Executive Master in Finance – Modigliani Miller 1958
Summary: the Beta-CAPM diagram  
Beta L
βEquity U
βAsset r
rEquity
rAsset
rDebt=rf
D/E
The IS-LM diagram used by economist is the source of inspiration for this figure. For many years, I have tried to create a figure which would look as impressive as the IS-LM diagram. Here is the result.
The figure in the first quadrant shows the relationship between the debt-equity ratio and the beta equity. The figure is based on the assumption that the debt is riskless. In that case, the relationship is linear.
The security market line in illustrated in the second quadrant.
 The relationship between the debt-equity ratio and the cost of equity, the cost of debt and the WACC is in the third quadrant
 This is simply a -45% line.
This diagram can be used to show that investors can choose the level of leverage that they wish by rebalancing their portfolios.
If the company has no debt (point U), an investor can create leverage by borrowing at the risk-free interest to reach L, the risk and expected returns of a levered company.
If, on the hand, the company is levered (point L), an investor can allocate his money between the stock of the levered company and the risk free asset to undo the leverage and reach point U.  
rEquity
rDebt
D/E
WACC
Executive Master of Finance Modigliani Miller

21. Risky debt: Merton model
Limited liability: rules out negative equity
Market value of equity
Equity similar to a call option on the company
Company goes bankrupt if V<F
Market value of company V
Face value of debt F
Tfin Intro to capital structure

22. Risk debt decomposition
Risky debt = Risk-free debt - Put
Market value of debt
Bankruptcy
Face value of debt
Market value of company
Face value of debt
Tfin Intro to capital structure

23. Figure 21.9 Beta of Debt and Equity
The blue curve is the beta of equity and the red curve is the beta of debt as a function of the firm’s debt-to-equity ratio. The black line is the approximation derived in Chapter 14—the beta of equity when the beta of debt is assumed to be zero. The firm is assumed to hold five-year zero-coupon debt and reinvest all its earnings. (The firm’s beta of assets is 1, the risk-free interest rate is 3% per year, and the volatility of assets is 20% per year.)
Copyright © 2007 Pearson Addison-Wesley. All rights reserved.

24. Corporate Tax Shield Interest are tax deductible => tax shield
Tax shield = Interest payment × Corporate Tax Rate
= (rD × D) × TC
rD : cost of new debt
D : market value of debt
Value of levered firm
= Value if all-equity-financed + PV(Tax Shield)
PV(Tax Shield) - Assume permanent borrowing
VL=VU + TCD
Tfin Intro to capital structure

25. Cost of equity calculation
VU + T = E + D
Value of equity rA rE
Value of all-equity firm rD
Value of debt rD
Value of tax shield
Tfin Intro to capital structure

26. Cost of equity calculation (2)
General formula
If rTS = rD & VTS = TCD
Similar formulas for beta equity (replace r by β)
Tfin Intro to capital structure

27. WACC If g =0 (constant perpetuity case): VTS = TCD
wacc = rA(1-TC D/VL) This is the MM formula.
Tfin Intro to capital structure

28. Example A B Balance Sheet Total Assets 1,000 1,000
Book Equity 1,
Debt (8%)
Income Statement
EBIT
Interest
Taxable Income
Taxes (40%)
Net Income
Dividend
Total
Assume rA= 10%
(1) Value of all-equity-firm:
VU = 144 / 0.10 = 1,440
(2) PV(Tax Shield):
Tax Shield = 40 x 0.40 = 16
PV(TaxShield) = 16/0.08 = 200
(3) Value of levered company:
VL = 1, = 1,640
(5) Market value of equity:
EL = VL - D = 1, = 1,140
Tfin Intro to capital structure

29. What about cost of equity?
Proof:
But VU = EBIT(1-TC)/rA
and E = VU + TCD – D
Replace and solve
1) Cost of equity increases with leverage:
2) Beta of equity increases
In example:
rE = 10% +(10%-8%)(1-0.4)(500/1,140)
= 10.53% or
rE = DIV/E = 120/1,140 = 10.53%
Tfin Intro to capital structure

30. What about the weighted average cost of capital?
Weighted average cost of capital decreases with leverage
Weighted average cost of capital: discount rate used to calculate the market value of firm by discounting net operating profit less adjusted taxes (NOPLAT)
NOPLAT = Net Income + Interest + Tax Shield
= (EBIT-rDD)(1-TC) + rDD +TCrDD
= Net Income for all-equity-firm = EBIT(1-TC)
VL = NOPLAT / WACC
As:
In example: NOPLAT = 144
VL = 1,640
WACC = 10.53% x % x 0.60 x 0.31 = 8.78%
Tfin Intro to capital structure