Matakuliah
Tahun
: F0892 - Analisis Kuantitatif
: 2009
VARIANS DAN STANDAR DEVIASI
PORTFOLIO
Pertemuan 10
EXPECTED RETURN SINGLE SECURITY
Expected return =
Where Ri is return and wi is the weighting of
component asset i.
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3
VARIANS PORTFOLIO
• Portfolio variance =
•
where i≠j.
Alternatively the expression can be written as:
where ρij = 1 for i=j.
Portfolio volatility=
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4
• For a two asset portfolio:– Portfolio return=
– Portfolio variance:
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• The formula to compute the standard deviation of a portfolio of
N securities is :
 N
 p 
 i 1
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N

j 1
 1/2
W iW j ij 

.
6
• The standard deviation is the square root of the variance.
• Note that  1
ii
• Hence,
 ii ii  i  i   2i
• What is the correlation coefficient between a riskfree security
and a stock? Zero!
• Therefore, the covariance between a riskfree security and a
stock is also zero!
• What is the formula for the standard deviation of a portfolio with
two securities (say, security 1 and security 2)? Note that X2 =1X1,  11  11  1  1   21 and  22   22  2  2   22 .
Hence,
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2
 p 
 i1
2

j 1
 1/2
W iW j ij 

W 1W 1 11W 1W 2 12W 2W 1 21W 2W 2 22 
W  W  2W 1W 2 12 
2
1

2
1
2
2
1/2
2
2
 W  (1 W 1)  2W 1(1 W 1) 12
2
1
2
1
2
2
2

1/2
.
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Example
You estimated that the standard deviations of General Electric
and General Motors stocks are 18% and 31%, respectively.
Also, you estimated that the correlation coefficient between
the return of the two securities is 0.35. What is the standard
deviation of a portfolio (including the two securities) with GE
having a weight of 40%?
The formula in the last slide for the standard deviation of a
portfolio with two securities is

 p W  (1 W 1)  2W 1(1 W 1) 12
2 2
1 1
2
2
2

1/2
.
Note that the covariance between the returns of GE and GM is
0.01953
[







0
.
35
*
0
.
18
*
0
.
31
].
GEGM
GEGM
GE
GM
Hence,
= 0,01953

 p  ( 0 .4 ) ( 0 .1 8 )  ( 0 .6 ) ( 0 .3 1 )  2 ( 0 .4 ) ( 0 .6 ) ( 0 .0 1 9 5 3 )
2
2
2
2
 2 2 .1 7 % .
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
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