Solvay Business School Université Libre de Bruxelles

Advanced Corporate Finance: formulas

André Farber

Revised May 2006 Capital Structure and the Weighted Average Cost of Capital

MM 1958 Leverage and firm value MM I V = VURequired return to equityholders MM II: rE = rA + (rA – rD) (D/E) Beta Asset vs Beta Equity βE = βA + (βA – βD ) (D/E) Weighted average cost of capital : WACC = rE (E/V) + rD (D/V)

WACC = rA Taxes - Constant riskless debt

PV of tax shield: VTS = TCD Value of levered firm (MM I): V = VU + TCD Required return to equityholders (MM II): rE = rA + (rA – rD) (1 – TC) (D/E) Beta Asset vs Beta Equity βE = [1+(1-TC)D/E] βAWeighted average cost of capital WACC = rE (E/V) + rD (1-TC) (D/V)

WACC = rA – rA TC D/V Taxes – Debt proportional to value L = D/V

PV of tax shield: Value of levered firm (MM I): V = VU + VTS Required return to equityholders (MM II): rE = rA + (rA – rD) (D/E) Beta Asset vs Beta Equity βE = βA + (βA – βD ) (D/E) Weighted average cost of capital WACC = rE (E/V) + rD (1-TC) (D/V)

WACC = rA – rD TC D/V Taxes – Variable debt: the Capital Cash Flows Approach (Ruback)

Capital cash flow = Free cash flow unlevered + Tax shield Discount rate for capital cash flow = rA

European Option Pricing (non dividend paying stock)

Payoffs at maturity: Call: max(0, )TS K− Put: m ax(0, )TK S− Put-Call Parity (European options on non dividend stocks): ( )C PV K S P+ = + Binomial option pricing model

tΔ length of time step rf risk-free interest rate / time step

Today Up Down Underlying asset S uS dS

Derivative f fu fd

Gross returns in up and down states: tu eσ Δ= 1 td eu

σ− Δ= =

AdvFin 2006 Formulas V2 31/01/2007

Valuation formulas Synthetic option: f Delta S M= × +

( ) ( )(1

u d d u

)f

f f ufDelta MuS dS u d r

− −= =

− −df+

Risk neutral 1-period valuation formula: (1 )(1 )

u d

f

pf pfr

f+ −=

+

Risk-neutral probability of up: 1 fr d

pu d+ −

=−

(to calculate the true probability, replace rf by the expected return) State price: u u d df v f v f= +

1 1 1u d

f f

p pv vr r

−= =

+ +

Black-Scholes formulas (European option on a stock paying a constant dividend yield q)

Call option: 1 2( ) ( )qT rTC Se N d Ke N d− −= −

1

ln( )0.5

qT

rTSeKed T

Tσ

σ

−

−= + 2 1

ln( )0.5

qT

rTSeKed T

Tσ σ

σ

−

−= − = −d T

1( ) ( )qT CDelta Call e N dS

− ∂= =

∂

N(d2) = risk-neutral probability of exercising the call option

Put option: [ ] [ ]2 1 21 ( ) 1 ( ) ( ) (rT qT rT qTP Ke N d Se N d Ke N d Se N d− − − −= − − − = − − − 1)

Risky debt valuation Merton model

Market value of levered firm: V = VU Equity: Call option on the assets Debt: D = Risk-free debt – Put option D = Risk-free debt – PV(RNProba of default * RN expected loss given default) Beta asset vs beta equity: /E A EquityDelta V Eβ β= × × Beta asset vs beta debt: /D A DebtDelta V Dβ β= × ×

Leland model Market value of levered firm: V = VU + VTS - BC

Value of $1 if bankruptcy: ( ) 22

/r

B B Up V V σ=

PV of tax shield: [ ]/ (1 )C BVTS T C r p= − PV of bankruptcy costs: B BBC p Vα= Value of risky debt: ( / )(1 ) (1 )B BD C r p V pBα= − + − Endogeneous level of bankruptcy: 2(1 ) /( 0.5 )B CV T C r σ= − +

AdvFin 2006 Formulas V2 31/01/2007

Warrants and convertible bonds Warrants:

Exercise if: q(VT + mK) > mK Value of warrant at maturity WT = (1-q)CT

Zero-coupon bond with warrant / Convertible zero-coupon bond Exercise if: q(VT – F + mK) > mK Value of warrant at maturity mWT = q Max(0,VT – (nK+D))

Right issue Value of right: Right = [nnew/(nold+nnew)](Pcum – Psub)

AdvFin 2006 Formulas V2 31/01/2007