International Investments
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Formula sheet
Portfolio risk and return
E (r )   p ( s)r ( s)
s
 C  y P
E ( rP )  r f
SP 
P
 2   p ( s )[ r ( s )  E ( r )]2
s
Cov ( rX , rY )   Pr( s )[ rX ( s )  E ( rX )][ rY ( s )  E ( rY )]
s

(X, Y)
 
i

Cov(rX , rY )
 XY
Cov(ri, rM )
 M2
E (ri )  rf   i[ E (rM )  rf ]
k  rf   [ E (rM )  rf ]
E (rc)  yE (rp)  (1  y)rf  rf  y[ E (rp)  rf ]
y* 
E ( rp )  rf
A P2
U  E (r )  0.5 A 2
E (rP)  wXE (rX )  wYE (rY )
 P2  wX2  X2  wY2 Y2  2w w Cov(r , r )
X
Y
X
Y
Cov(rX , rY )   XY X  Y
[ E ( rX )  rf ] Y2  [ E ( rY )  rf ]Cov( rX , rY )
wX 
[ E ( rX )  rf ] Y2  [ E ( rY )  rf ] X2  [ E ( rX )  rf  E ( rY )  rf ]Cov( rX , rY )
wMin ( X ) 
 Y2  Cov (r , r )
 X2   Y2  2Cov ( r , r )
X
Y
X
Y
Formula sheet
International Investments
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Formula sheet
Stock and bond valuation
D1
D2
DH  PH
V0 

 ... 
2
1  k (1  k )
(1  k ) H
D1
kg
P0 
E ( r )  Dividend yield  Capital gains yield 
D1 P1  P 0 D1


g
P0
P0
P0
E1
 PVGO
k
g  ROE  b
P0 
T
Coupon Par value

t
(1  r )T
t 1 (1  r )
V0 
T
D   t  wt
t 1
wt 
CFt /(1  y )t
Bond price
D* 
D
1 y
P
  D * y
P
Convexity 
T
1
CFt
[
(t 2  t )]

2
t
P  (1  y ) t 1 (1  y )
P
1
  D * y   Convexity  ( y ) 2
P
2
P / P
Effective duration  
r
Asset duration  w  duration1  (1  w)  duration 2
Durationperpetuity 
Durationannuity 
1 y
y
1 y
T

y
(1  y )T  1
Durationcoupon bond 
1  y (1  y )  T (c  y )

y
c[(1  y )T  1]  y
Durationcoupon bond at par 
1 y
1
[1 
]
y
(1  y )T
International Investments
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The index model
 i2   i2 M2   2
( ei )
Cov ( ri, rj )   i  j M2
1 2
 
( eP ) n ( e )
2
Options
P  C  S 0  PV ( X )  PV ( Dividends )
H
CU  CD
uS 0  dS 0
pupward ( risk neutral ) 
1  r   d
ud
C 0  S 0 N ( d 1)  Xe  rT N ( d 2)
d1 
ln( S 0 / X )  ( r   2 / 2)T
 T
d 2  d1  T
Term structure, futures and forwards
(1  yn ) n
(1  rn ) 
(1  yn  1) n 1
(1  yn ) n
1  fn 
(1  yn  1) n 1
F 0  S 0 (1  rf )  D  S 0 (1  rf  d )
F (T 2 )  F (T 1)(1  rf  d ) (T 2 T 1)
Performance measures and performance contribution
E ( rP )  rf
Sharpe ' s measure 
P
E ( rP )  rf
Treynor ' s measure 
P
P
Information ratio 
 ( eP )
Jensen ' s measure   P  E (rP )  [rf   P[ E (rM )  rf )
n
n
n
i 1
i 1
i 1
rP  rB   wPirPi   wBirBi   ( wPirPi  wBirBi )
Formula sheet
International Investments
Cumulative normal distribution
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Formula sheet
International Investments
Finite Geometric series:
Sn = a. [1 – (k)n] / [1 - k], k=quotient
Infinite Geometric series: (n goes to infinity)
Sn = a. [1] / [1 - k], k=quotient (<1)
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Formula sheet