return market, beta, dan mathematika diversifikasi pertemuan 12 dan 13 matakuliah: f0892 - analisis...
DESCRIPTION
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.TRANSCRIPT
RETURN MARKET, BETA, DAN MATHEMATIKA DIVERSIFIKASI
Pertemuan 12 dan 13
Matakuliah : F0892 - Analisis KuantitatifTahun : 2009
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.
• Return market diperoleh dengan menghitung perubahan indeks per hari.
Bina Nusantara University 4
IHSGt+1 - IHSG1
- IHSG1
MATHEMATIKA DIVERSIFIKASI
Bina Nusantara University 5
6
Linear Combinations• Introduction• Return• Variance
7
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
8
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
9
Variance• Introduction• Two-security case• Minimum variance portfolio• Correlation and risk reduction• The n-security case
10
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
11
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
12
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
13
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
14
Two Security Case (cont’d)Example (cont’d)
What is the expected return and variance of this two-security portfolio?
15
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
16
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
17
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
18
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
19
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?
20
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
21
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
22
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
23
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
24
The n-Security Case (cont’d)• The equation includes the correlation coefficient
(or covariance) between all pairs of securities in the portfolio
25
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
26
Single-Index Model• Computational advantages• Portfolio statistics with the single-index model
27
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
28
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
29
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
30
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
31
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
32
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