Formula
Black-Scholes
S X = S − PV(x)
SX
ln[ PV(K ) ] σ T
d1 = +
σ T 2
d2 = d1 − σ T
C = S X N(d1) − PV(K )N(d2)
C = S0 N(d1) − Xe −pt N(d2)
Duration formula
ϵ
% ch ange = Durat ion ×
1+r
The equity cost of capital
r = rf + β(E[Rmkt ] − rf )
Asset or Unlevered cost of capital : E 12.8
E D
rU = rE + rD
E+D E+D
E + D: total market value of equity and debt
rE and rD : equity and debt costs of capital
Rearranged to get rE
D
rE = rU + (rU − rD )
E
Asset or Unlevered Beta : E 12.9
E D
βU = βE + βD
E+D E+D
Rearranged to get βE
D
βE = βU + (β − βD )
E U
Cash and Net Debt : 12.10
Net Debt = Debt - Excess Cash and Short Term Investment
, Taxes 12.11
E ective after-tax interest rate = r (1 − TC )
TC = corporate tax rate
Weighted Average cost of capital (WACC) 12.12 (after tax) levered
E D
rWACC = rE + rD(1 − TC )
E+D E+D
12.8 + 12.12 = 12.13
D
rWACC = rU − rDTC
E+D
Lecture 3
MM Proposition I
Market value of Equity = Market Value of Assets - Market Value of Debt and Other Liabilities
Leverage and the Equity Cost of Capital:
E+D=U=A
U = (unlevered equity)
Lecture 4
Cum-dividend
Pcum = Current Dividend + PV( Future Dividends )
The e ciency dividend tax-rate
(Pcum − Pex )(1 − Tg) = Div(1 − Td )
T_d = dividend tax rate
T_g = capital gains tax
After-tax loss = after-tax cash ow from the dividend
Share price drop (rearranged)
1 − Td Td − Tg
Pcum − Pex = Div( ) = Div(1 − ) = Div(1 − T*
d
)
1 − Tg 1 − Tg
Where T* is the e ected dividend tax rate
d
ff ffi ff fl
, Td − Tg
T*
d
=( ) this measures the additional that is instead tax paid by the investor
1 − Tg
per {dollar} of after-tax capital gains income received as dividends
Adjusting for investor taxes (rearrange) : 17.4
1 − Td
Pcum = Pex + Div0 × ( )
1 − Tg
Retain: 17.5
Div × (1 − Td )
Pretain =
r f × (1 − Ti )
T_i = investor tax rate
17.4 + 17.5
(1 − Tc )(1 − Tg)
Pretain = Pcum + ( ) = Pcum × (1 − T*
retain
)
1 − Ti
Where T*
retain
measures the e ective tax disadvantage of retaining cash:
(1 − TC )(1 − Tg)
T*
retain
= (1 − )
1 − Ti
Lecture 5
Interest Tax Shield: 15.1
Interest tax shield = corporate tax rate x interest payments
Interest Tax Shield and Firm Value: 15.2
V L = V U + PV(interest tax shield)
V L & V U = value of rm with and without leverage
r is between WACC and rm’s pre-tax WACC: 15.6
E D D
rWACC = rE + rD − rDTC
E+D E+D E+D
First two parts = Pre-tax Waco
Last bit = reduction due to interest tax shield
Lecture 7 & 8
Actuarially Fair Insurance Premium
fi fi ff
, CASH-and-CARRY and the pricing of currency forward
Cover-interest parity
1 + r$
F=S×
1 + r¥
(1 + r$)T
FT = S ×
(1 + r¥ )T
T = no. of years, the no-arbitrage forward rate for an exchange that will occur in T years
Interest Rate Risk Measurement: Duration
PV(Ct )
∑ P
Durat ion = ×t
T
Where
Ct = cash ow at date t
PV(Ct ) = present value evaluated at the bond’s yield
∑
P= PV(Ct ) = total present value at the cash ow
T
Duration and Interest Rate Sensitivity
e
Present change in value ≈ - Duration ×
1 + r /k
k = no. of compounding periods per year of the APR
e = coupon increase
Bond price
T
Ct + F
∑ (1 + r)t
P=
t=1
Duration and market values [a Portfolio] 30.8
A B
DA+B = DA + DB
A+B A+B
Immunising the Portfolio 30.10
Amount to Exchange = [ ΔPortfolio Duration X Portfolio Value ] / [ Δ Asset Duration ]
fl fl
Black-Scholes
S X = S − PV(x)
SX
ln[ PV(K ) ] σ T
d1 = +
σ T 2
d2 = d1 − σ T
C = S X N(d1) − PV(K )N(d2)
C = S0 N(d1) − Xe −pt N(d2)
Duration formula
ϵ
% ch ange = Durat ion ×
1+r
The equity cost of capital
r = rf + β(E[Rmkt ] − rf )
Asset or Unlevered cost of capital : E 12.8
E D
rU = rE + rD
E+D E+D
E + D: total market value of equity and debt
rE and rD : equity and debt costs of capital
Rearranged to get rE
D
rE = rU + (rU − rD )
E
Asset or Unlevered Beta : E 12.9
E D
βU = βE + βD
E+D E+D
Rearranged to get βE
D
βE = βU + (β − βD )
E U
Cash and Net Debt : 12.10
Net Debt = Debt - Excess Cash and Short Term Investment
, Taxes 12.11
E ective after-tax interest rate = r (1 − TC )
TC = corporate tax rate
Weighted Average cost of capital (WACC) 12.12 (after tax) levered
E D
rWACC = rE + rD(1 − TC )
E+D E+D
12.8 + 12.12 = 12.13
D
rWACC = rU − rDTC
E+D
Lecture 3
MM Proposition I
Market value of Equity = Market Value of Assets - Market Value of Debt and Other Liabilities
Leverage and the Equity Cost of Capital:
E+D=U=A
U = (unlevered equity)
Lecture 4
Cum-dividend
Pcum = Current Dividend + PV( Future Dividends )
The e ciency dividend tax-rate
(Pcum − Pex )(1 − Tg) = Div(1 − Td )
T_d = dividend tax rate
T_g = capital gains tax
After-tax loss = after-tax cash ow from the dividend
Share price drop (rearranged)
1 − Td Td − Tg
Pcum − Pex = Div( ) = Div(1 − ) = Div(1 − T*
d
)
1 − Tg 1 − Tg
Where T* is the e ected dividend tax rate
d
ff ffi ff fl
, Td − Tg
T*
d
=( ) this measures the additional that is instead tax paid by the investor
1 − Tg
per {dollar} of after-tax capital gains income received as dividends
Adjusting for investor taxes (rearrange) : 17.4
1 − Td
Pcum = Pex + Div0 × ( )
1 − Tg
Retain: 17.5
Div × (1 − Td )
Pretain =
r f × (1 − Ti )
T_i = investor tax rate
17.4 + 17.5
(1 − Tc )(1 − Tg)
Pretain = Pcum + ( ) = Pcum × (1 − T*
retain
)
1 − Ti
Where T*
retain
measures the e ective tax disadvantage of retaining cash:
(1 − TC )(1 − Tg)
T*
retain
= (1 − )
1 − Ti
Lecture 5
Interest Tax Shield: 15.1
Interest tax shield = corporate tax rate x interest payments
Interest Tax Shield and Firm Value: 15.2
V L = V U + PV(interest tax shield)
V L & V U = value of rm with and without leverage
r is between WACC and rm’s pre-tax WACC: 15.6
E D D
rWACC = rE + rD − rDTC
E+D E+D E+D
First two parts = Pre-tax Waco
Last bit = reduction due to interest tax shield
Lecture 7 & 8
Actuarially Fair Insurance Premium
fi fi ff
, CASH-and-CARRY and the pricing of currency forward
Cover-interest parity
1 + r$
F=S×
1 + r¥
(1 + r$)T
FT = S ×
(1 + r¥ )T
T = no. of years, the no-arbitrage forward rate for an exchange that will occur in T years
Interest Rate Risk Measurement: Duration
PV(Ct )
∑ P
Durat ion = ×t
T
Where
Ct = cash ow at date t
PV(Ct ) = present value evaluated at the bond’s yield
∑
P= PV(Ct ) = total present value at the cash ow
T
Duration and Interest Rate Sensitivity
e
Present change in value ≈ - Duration ×
1 + r /k
k = no. of compounding periods per year of the APR
e = coupon increase
Bond price
T
Ct + F
∑ (1 + r)t
P=
t=1
Duration and market values [a Portfolio] 30.8
A B
DA+B = DA + DB
A+B A+B
Immunising the Portfolio 30.10
Amount to Exchange = [ ΔPortfolio Duration X Portfolio Value ] / [ Δ Asset Duration ]
fl fl
