Some Useful Formulas
1
Present Value (Chapter 4)
The discounted value of T future cash flows
PV =
2
C1
C2
CT
+
+...+
=
1 + r (1 + r )2
(1 + r )T
t =1
C
r
C
[1 − 1/(1 + r )T ]
r
Growing Perpetuity (Chapter 4)
The value of a perpetuity that grows at rate g, where the first payment is C
PV =
6
C
Annuity (Chapter 4)
The value of C received each year for T years
PV =
5
T
∑ (1 +t r )t
Perpetuity (Chapter 4)
The value of C received each year, for ever
PV =
4
C
t =1
Net Present Value (Chapter 4)
Present value minus initial costs
NPV = PV − Cost
C0 = −Cost
NPV = C0 +
3
T
∑ (1 +t r )t
C
r−g
Growing Annuity (Chapter 4)
The value of a T-period annuity that grows at the rate g, where the first payment is C
T
 1
1 + g 
1

PV = C
−
×
 
 r − g r − g  1 + r  
7
Measures of Risk for Individual Assets (Chapter 10)
Varr( RA ) = σ A2
SD( RA ) = σ A
Cov
v( RA , RB ) = σ AB
Corr( RA , RB ) = ρAB
8
=E
Expected
xpected value
value of
of (RA − ‰ A)2
= Var(RA )
= Expected value of [(RA − ‰ A)( RB − ‰B )]
= Cov( RA RB )/σAσ B
Expected Return on a Portfolio of Two Assets (Chapter 10)
‰ p = X A ‰A + XB ‰B
9
Variance of a Portfolio of Two Assets (Chapter 10)
σ 2p = X A2 × σ A2 + 2 XA XB × σ AB + XB2 × σ B2
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Some Useful Formulas
10
Beta of a Security (Chapter 10)
βA =
11
A33
Cov(RA , RM)
σ 2 ( RM)
Capital Asset Pricing Model (Chapter 10)
‰ A = RF + β A × ( ‰M − RF)
12
k-Factor Model (Chapter 11)
Ri = RF + βi1 F1 + βi 2 F2 + . . . + βik Fk + mj
13
Leverage and the Cost of Equity (Chapter 15)
Before tax:
RS = R0 +
B
( R0 − RB )
S
After tax:
RS = R0 +
14
B
(1 − TC )( R0 − RB )
S
Value of the Firm under Corporate Taxes (Chapter 15)
VL = VU + TC B
15
Weighted Average Cost of Capital (Chapter 15)
 S 
 B 
 S + B  RS +  S + B  RB (1 − t C )
16
Equity Beta (Chapter 17)
No-tax case: βUnlevered firm =
Corporate tax case: βUnlevered firm =
17
Equity
× βEquity
Debt + Equity
Equity
× βEquity
Equity + (1 − t C ) Debt
Black-Scholes Model (Chapter 22)
C = SN( d1 ) − Ee − Rt N(d2 )
where d1 = [ln ( S /E ) + ( R +
1
2
σ 2 )t ]/ σ 2t
d2 = d1 − σ 2t
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