Théorie Financière

Structure financière et coût du capital

Professeur André Farber

Structure financière |2April 18, 2023

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 ?

Structure financière |3April 18, 2023

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

WACC = rA

• rA is the cost of capital of an all equity firm

Structure financière |4April 18, 2023

Weighted Average Cost of Capital (no taxes)

• 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

equity debt

E DWACC r r

V V

Structure financière |5April 18, 2023

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)?

Structure financière |6April 18, 2023

Using MM 58

• Value of company: V = 100

• Initial Final

• Equity 100 80

• Debt 0 20

• Total 100 100 MM I

• WACC = rA 11% 11% MM II

• Cost of debt - 5% (assuming risk-free debt)

• D/V 0 0.20

• Cost of equity 11% 12.50% (to obtain rwacc = 11%)

• E/V 100% 80%

Structure financière |7April 18, 2023

Cost of equity calculation

Value of all-equity firm

Value of equity

Value of debt

V (=VU ) = E + D

rE

rD

rA

LD

LEA V

Dr

V

Err

Structure financière |8April 18, 2023

Why is WACC 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 = 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

rportoflio = rA

• The weighted average cost of capital is thus equal to the cost of capital of an all equity firm

WACC = rA

Structure financière |9April 18, 2023

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

Structure financière |10April 18, 2023

Relationship between the value of company and WACC

• From the definition of the WACC:

WACC * V = requity * E + rdebt * D

• As requity * E = Div and rdebt * D = Int

rwacc * V = EBIT

V = EBIT / WACC

Market value of levered firm

EBIT is independent of leverage

If value of company varies with leverage, so does WACC in opposite direction

Structure financière |11April 18, 2023

MM II: another presentation

• The equality WACC = rA can be written as:

• Expected return on equity is an increasing function of leverage:

E

Drrrr debtAAequity )(

rA

D/E

requity

11%

rdebt

5%

0.25

12.5%

rwacc

Additional cost due to leverage

Structure financière |12April 18, 2023

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

DE

D

DE

EDebtEquityAsset

E

DDebtAssetAssetEquity )(

Structure financière |13April 18, 2023

Back to example

• Assume debt is riskless:

25.1)80

201(1 Equity

E

V

E

DAssetAssetEquity )1(

%50.1225.1%6%5)( EquityFMFEquity rrrr

Structure financière |14April 18, 2023

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

DTr

DrTTaxShieldPV C

D

DC

)(

Structure financière |15April 18, 2023

Cost of equity calculation

Value of all-equity firm

Value of tax shield

Value of equity

Value of debt

VU + T = E + D

rE

rD

rA

rD

LD

LE

L

CD

L

UA V

Dr

V

Er

V

DTr

V

Vr

Structure financière |16April 18, 2023

Cost of equity calculation (2)

DrErVTSrVTSVr

V

Dr

V

Er

V

VTSr

V

VTSVr

DATSLA

LD

LA

LTS

L

LA

)(

E

DTrrrr CDAAE )1)((

E

VTSrr

E

Drrrr TSADAAE )()(

If rTS = rD & VTS = TCD

Similar formulas for beta equity (replace r by β)

General formula

E

DTCDAAE )1)((

Structure financière |17April 18, 2023

WACC

AL

UA

LCD

LE

rV

Vr

V

DTr

V

Erwacc

)1(

Structure financière |18April 18, 2023

Example

A B

Balance SheetTotal Assets 1,000 1,000Book Equity 1,000 500Debt (8%) 0 500

Income StatementEBIT 240 240Interest 0 40Taxable Income 240 200Taxes (40%) 96 80Net Income 144 120Dividend 144 120Interest 0 40Total 144 160

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 = 16PV(TaxShield) = 16/0.08 = 200

(3) Value of levered company:VL = 1,440 + 200 = 1,640

(5) Market value of equity:EL = VL - D = 1,640 - 500 = 1,140

Structure financière |19April 18, 2023

What about cost of equity?

1) Cost of equity increases with leverage:

2) Beta of equity increases

E

DTrrrr CDAAE )1()(

])1(1[E

DTCAE

Proof:

But VU = EBIT(1-TC)/rA

and E = VU + TCD – D

Replace and solve

E

CD

r

TDrEBITE

)1()(

In example:rE = 10% +(10%-8%)(1-0.4)(500/1,140) = 10.53%orrE = DIV/E = 120/1,140 = 10.53%

Structure financière |20April 18, 2023

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: )1()1( CCDE TEBITDTrEr

LCD

LE V

DTr

V

ErWACC )1(

In example: NOPLAT = 144VL = 1,640WACC = 10.53% x 0.69 + 8% x 0.60 x 0.31 = 8.78%