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RETURN MARKET, BETA, DAN MATHEMATIKA DIVERSIFIKASI
Pertemuan 12 dan 13
Matakuliah : F0892 - Analisis KuantitatifTahun : 2009
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RETURN MARKET• Return market : ialah return dari seluruh usaha
yang ada di suatu wilayah tertentu. • Karena sukar menghitung return seluruh usaha
dalam wilayah tertentu maka bisa diwakilkan dengan menghitung return dari seluruh saham yang tercatat di bursa. (Di Indonesia ialah Bursa Efek Indonesia).
• Yang digunakan ialah indeks dapat IHSG, LQ 45, atau Kompas 100.
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• Return market diperoleh dengan menghitung perubahan indeks per hari.
Bina Nusantara University 4
IHSGt+1 - IHSG1
- IHSG1
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MATHEMATIKA DIVERSIFIKASI
Bina Nusantara University 5
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Linear Combinations• Introduction• Return• Variance
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Introduction• A portfolio’s performance is the result of the
performance of its components– The return realized on a portfolio is a linear combination
of the returns on the individual investments
– The variance of the portfolio is not a linear combination of component variances
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Return• The expected return of a portfolio is a weighted
average of the expected returns of the components:
1
1
( ) ( )
where proportion of portfolio invested in security and
1
n
p i ii
i
n
ii
E R x E R
xi
x
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Variance• Introduction• Two-security case• Minimum variance portfolio• Correlation and risk reduction• The n-security case
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Introduction• Understanding portfolio variance is the essence of
understanding the mathematics of diversification– The variance of a linear combination of random
variables is not a weighted average of the component variances
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Introduction (cont’d)• For an n-security portfolio, the portfolio variance
is:2
1 1
where proportion of total investment in Security correlation coefficient between
Security and Security
n n
p i j ij i ji j
i
ij
x x
x i
i j
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Two-Security Case• For a two-security portfolio containing Stock A
and Stock B, the variance is:2 2 2 2 2 2p A A B B A B AB A Bx x x x
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Two Security Case (cont’d)Example
Assume the following statistics for Stock A and Stock B:
Stock A Stock B
Expected return .015 .020Variance .050 .060Standard deviation .224 .245Weight 40% 60%Correlation coefficient .50
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Two Security Case (cont’d)Example (cont’d)
What is the expected return and variance of this two-security portfolio?
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Two Security Case (cont’d)Example (cont’d)
Solution: The expected return of this two-security portfolio is:
1
( ) ( )
( ) ( )
0.4(0.015) 0.6(0.020)
0.018 1.80%
n
p i ii
A A B B
E R x E R
x E R x E R
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Two Security Case (cont’d)Example (cont’d)
Solution (cont’d): The variance of this two-security portfolio is:
2 2 2 2 2
2 2
2
(.4) (.05) (.6) (.06) 2(.4)(.6)(.5)(.224)(.245).0080 .0216 .0132.0428
p A A B B A B AB A Bx x x x
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Minimum Variance Portfolio• The minimum variance portfolio is the particular
combination of securities that will result in the least possible variance
• Solving for the minimum variance portfolio requires basic calculus
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Minimum Variance Portfolio (cont’d)
• For a two-security minimum variance portfolio, the proportions invested in stocks A and B are:
2
2 2 2
1
B A B ABA
A B A B AB
B A
x
x x
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Minimum Variance Portfolio (cont’d)
Example (cont’d)
Assume the same statistics for Stocks A and B as in the previous example. What are the weights of the minimum variance portfolio in this case?
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Minimum Variance Portfolio (cont’d)
Example (cont’d)
Solution: The weights of the minimum variance portfolios in this case are:
2
2 2
.06 (.224)(.245)(.5) 59.07%2 .05 .06 2(.224)(.245)(.5)
1 1 .5907 40.93%
B A B ABA
A B A B AB
B A
x
x x
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Minimum Variance Portfolio (cont’d)
Example (cont’d)
0
0,2
0,4
0,6
0,8
1
1,2
0 0,01 0,02 0,03 0,04 0,05 0,06
Wei
ght A
Portfolio Variance
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Correlation and Risk Reduction• Portfolio risk decreases as the correlation
coefficient in the returns of two securities decreases
• Risk reduction is greatest when the securities are perfectly negatively correlated
• If the securities are perfectly positively correlated, there is no risk reduction
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The n-Security Case• For an n-security portfolio, the variance is:
2
1 1
where proportion of total investment in Security correlation coefficient between
Security and Security
n n
p i j ij i ji j
i
ij
x x
x i
i j
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The n-Security Case (cont’d)• The equation includes the correlation coefficient
(or covariance) between all pairs of securities in the portfolio
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The n-Security Case (cont’d)• A covariance matrix is a tabular presentation of
the pairwise combinations of all portfolio components– The required number of covariances to compute a
portfolio variance is (n2 – n)/2
– Any portfolio construction technique using the full covariance matrix is called a Markowitz model
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Single-Index Model• Computational advantages• Portfolio statistics with the single-index model
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Computational Advantages• The single-index model compares all securities
to a single benchmark– An alternative to comparing a security to each of the
others
– By observing how two independent securities behave relative to a third value, we learn something about how the securities are likely to behave relative to each other
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Computational Advantages (cont’d)• A single index drastically reduces the number of
computations needed to determine portfolio variance– A security’s beta is an example:
2
2
( , )
where return on the market index
variance of the market returns
return on Security
i mi
m
m
m
i
COV R R
R
R i
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Portfolio Statistics With the Single-Index Model
• Beta of a portfolio:
• Variance of a portfolio:1
n
p i ii
x
2 2 2 2
2 2
p p m ep
p m
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Portfolio Statistics With the Single-Index Model (cont’d)
• Variance of a portfolio component:
• Covariance of two portfolio components:
2 2 2 2i i m ei
2AB A B m
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Multi-Index Model• A multi-index model considers independent
variables other than the performance of an overall market index– Of particular interest are industry effects
• Factors associated with a particular line of business
• E.g., the performance of grocery stores vs. steel companies in a recession
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Multi-Index Model (cont’d)• The general form of a multi-index model:
1 1 2 2 ...where constant
return on the market index
return on an industry index
Security 's beta for industry index
Security 's market beta
retur
i i im m i i in n
i
m
j
ij
im
i
R a I I I Ia
I
I
i j
i
R
n on Security i