E (r ) 
Expected return of an asset:
1
T

n t 1
T
Covariance between two assets:
 xy 
1
 (r  E (r )) * ((r  E (r ))
xt
x
yt
y
 x y
 2p  w12 12  w22 22  2 w1 w2 1 2 12
 c2  y 2 2p
Portfolio variance (1 risky asset and 1 risk-free)
Slope of CAL (reward-to-variability ratio): S 
[ E (r p )  r f ]
p
Optimal fraction to be invested in the risky asset
Weight of asset 1 in the minimum variance portfolio
Equation of CAL: E ( rc )  r f 
y* 
w1 
w1 
Weight of asset A in the portfolio that maximizes utility with
only 2 risky assets
2
E (rM )  r f  A M
Contribution of an individual asset to the risk of market
n
portfolio: Cov ( r , r )   w Cov ( r , r )
I M
k
I k
k 1
wA 
c
[ E ( rp )  r f ]  r f  S  c
p
[ E (r p )  r f ]
A 2p
 22  cov( r1 , r2 )
 12   22  2 cov( r1 , r2 )
Weight of asset 1 in the optimal risky portfolio
Third prediction of CAPM:
 xy
Portfolio expected return: E (rp )  w1 E (r` )  w2 E (r2 )
1
A 2
2
Portfolio variance (2 risky assets)
Utility: U  E (r ) 

1
( rt  E ( r )) 2
( n  1) t 1
Correlation between two assets:  
T
n t 1
2 
Variance of an asset:
rt
[ E (r1 )  r f ] 22  [ E (r2 )  r f ] 1 2 12
[ E ( r1 )  r f ] 22  [ E ( r2 )  r f ] 12  [ E ( r1 )  r f  E ( r2 )  r f ] 1 2 12


A     2   
E (rA )  E (rB )  A  B2   A B  AB
CAPM:
2
A
2
B
A
B
AB
E (rI )  r f   [ E (rM )  r f ]
CAPM beta :  
 MI
 M2
n
P 
Portfolio beta:
w 
i
Jensen’s alpha:   ractual  (r f   [ E (rM )  r f ])
i
i 1
Third property of mean-variance efficient portfolios:
Cov ( ri , rP )  Cov ( rQ , rP )
E ( ri )  E ( rQ )  [ E ( rP )  E ( rQ )]
 P2  Cov ( rQ , rP )
Zero-beta CAPM:
Single-factor model: ri  E (ri )   i F  ei
Single-index model: Ri   i   i Rm  ei
Decomposition of the variance of an asset (based on singleindex model)
Covariance of an asset (based on single-index model)
2
var( ri )   2 M
  2 (ei )
Estimation of beta:  
t-statistic: t 
n
n 1
*
cov( rM , rA )
E (ri )  E ( rZ ( M ) )  [ E ( rM )  E ( rZ ( M ) )]
Cov ( Ri , R j )   i  j  M2
Estimation of alpha:   rA   rM
R-squared:
st .error
R2 
 2 M2
 A2
Firm-specific risk of an equally-weighted portfolio:
2
 M2
 M2
  0
1
 2 (e P )   
n
Cov( ri , rM )
n
  ( e)
2
No arbitrage condition:
 1
 2 ( ei )
E ( rU ) - r f
U
 A2

E ( rV )  r f
V
i 1
APT: E (rV )  r f  V [ E (rM ) - r f ]
Put-call parity condition: P  S  C  Xe  rt
C 0  S 0 N ( d 1 )  Xe  rT N ( d 2 )
Bond equivalent yield:
BEY  n * YTM
ln( S 0 / X )  ( r   / 2)T
Effective annual yield:
EAY  (1  YTM ) n  1
2
d1 
Black-Scholes formula:
 T
d 2  d1   T
P
Bond price:
FV
(1  i )
n

t
 (1  i)
t 1
Duration of a corporate bond:
Forward rate:
(1  f n ) 
C
1 y
y
CY 
C
P
(1  y)
Duration of level perpetuity:
y
t
(1  y )  T (c  y )
c[(1  y ) T  1]  y
Duration of level annuity:
1 y
y

T
(1  y ) T  1
(1  y n ) n
(1  y n 1 ) n 1
CFt
T
Duration:

Current yield:

(1  y )t
D
t * wt , where wt 
bond price
t 1
Modified duration: D *  D
(1  y )
Change in the price of a bond as a function of change in duration:
 (1  y ) 
P
 D  

P
 1 y 
Change in the price of a bond as a function of change in modified duration:
P
  D *  y
P