Théorie Financière Structure financière et coût du capital Professeur André Farber

August 23, 2004 Tfin Risk and return (2) |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

August 23, 2004 Tfin Risk and return (2) |3 Beta of a portfolio Consider the following portfolio StockValueBeta An A  P A  A Bn B  P B  B The value of the portfolio is: V = n A  P A + n A  P B The fractions invested in each stock are: X i = ( n i P i ) / V for i = A,B The beta of the portfolio is the weighted average of the betas of the individual stocks  P = X A  A + X B  B

August 23, 2004 Tfin Risk and return (2) |4 Example Stock$  X ATT3, Genetech2, V5,000  P = 0.60 * * 1.40 = 1.02

August 23, 2004 Tfin Risk and return (2) |5 Application 1: cost of capital for a division Firm = collection of assets Example: A company has two divisions Value($ mio)  Electrical Chemical V600  firm = (100/600) * (0.50) + (500/600) * 0.90 = 0.83 Assume: r f = 5%r M - r f = 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%

August 23, 2004 Tfin Risk and return (2) |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 portfolio2,00012 Risk-free rate-1, V1,000  P = 2  1 + (-1)  0 = 2

August 23, 2004 Tfin Risk and return (2) |7 8% 14% 1 2 Beta M P 8% 14% 20% Sigma Expected Return M P

August 23, 2004 Tfin Risk and return (2) |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 ?

August 23, 2004 Tfin Risk and return (2) |9 An example CAPM holds – Risk-free rate = 5%, Market risk premium = 6% Consider an all-equity firm: Market value V100 Beta1 Cost of capital11% (=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)?

August 23, 2004 Tfin Risk and return (2) |10 Weighted Average Cost of Capital An average of: The cost of equity r equity The cost of debt r debt Weighted by their relative market values (E/V and D/V) Note: V = E + D

August 23, 2004 Tfin Risk and return (2) |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 = V U Proposition II: The weighted average cost of capital is independent of its capital structure r wacc = r A r A is the cost of capital of an all equity firm

August 23, 2004 Tfin Risk and return (2) |12 Using MM 58 Value of company: V = 100 InitialFinal Equity Debt 0 20 Total MM I WACC = r A 11%11% MM II Cost of debt-5% (assuming risk-free debt) D/V00.20 Cost of equity11%12.50% (to obtain r wacc = 11%) E/V100%80%

August 23, 2004 Tfin Risk and return (2) |13 Cost of equity calculation Value of all-equity firm Value of equity Value of debt V (=V U ) = E + D rErE rDrD rArA

August 23, 2004 Tfin Risk and return (2) |14 Why is r wacc unchanged? Consider someone owning a portfolio of all firm’s securities (debt and equity) with X equity = E/V (80% in example ) and X debt = D/V (20%) Expected return on portfolio = r equity * X equity + r debt * X debt This is equal to the WACC (see definition): r portoflio = r wacc 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 r portoflio = r A The weighted average cost of capital is thus equal to the cost of capital of an all equity firm r wacc = r A

August 23, 2004 Tfin Risk and return (2) |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 / r equity Market value of debt: present value of future interest discounted at the cost of debt D = Int / r debt

August 23, 2004 Tfin Risk and return (2) |16 Relationship between the value of company and WACC From the definition of the WACC: r wacc * V = r equity * E + r debt * D As r equity * E = Div and r debt * D = Int r wacc * V = EBIT V = EBIT / r wacc Market value of levered firm EBIT is independent of leverage If value of company varies with leverage, so does WACC in opposite direction

August 23, 2004 Tfin Risk and return (2) |17 MM II: another presentation The equality r wacc = r A can be written as: Expected return on equity is an increasing function of leverage: rArA D/E r equity 11% r debt 5% % r wacc Additional cost due to leverage

August 23, 2004 Tfin Risk and return (2) |18 Why does r equity 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 * X equity +  debt * X debt But also:  portfolio =  Asset So: or

August 23, 2004 Tfin Risk and return (2) |19 Back to example Assume debt is riskless:

August 23, 2004 Tfin Risk and return (2) |20 Corporate Tax Shield Interest are tax deductible => tax shield Tax shield = Interest payment × Corporate Tax Rate = (r D × D) × T C r D : 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 V L =V U + T C D

August 23, 2004 Tfin Risk and return (2) |21 Cost of equity calculation Value of all-equity firm Value of tax shield Value of equity Value of debt V U + T = E + D rErE rDrD rArA rDrD

August 23, 2004 Tfin Risk and return (2) |22 Cost of equity calculation (2) If r TS = r D & VTS = T C D Similar formulas for beta equity (replace r by β) General formula

August 23, 2004 Tfin Risk and return (2) |23 WACC

August 23, 2004 Tfin Risk and return (2) |24 Example A B Balance Sheet Total Assets 1,000 1,000 Book Equity 1, Debt (8%) Income Statement EBIT Interest 0 40 Taxable Income Taxes (40%) Net Income Dividend Interest 0 40 Total Assume r A = 10% (1) Value of all-equity-firm: V U = 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: V L = 1, = 1,640 (5) Market value of equity: E L = V L - D = 1, = 1,140

August 23, 2004 Tfin Risk and return (2) |25 What about cost of equity? 1) Cost of equity increases with leverage: 2) Beta of equity increases Proof: But V U = EBIT(1-T C )/r A and E = V U + T C D – D Replace and solve In example: r E = 10% +(10%-8%)(1-0.4)(500/1,140) = 10.53% or r E = DIV/E = 120/1,140 = 10.53%

August 23, 2004 Tfin Risk and return (2) |26 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-r D D)(1-T C ) + r D D +T C r D D = Net Income for all-equity-firm = EBIT(1-T C ) VL = NOPLAT / WACC As: In example: NOPLAT = 144 V L = 1,640 WACC = 10.53% x % x 0.60 x 0.31 = 8.78%