Matakuliah
Tahun
: F0892 - Analisis Kuantitatif
: 2009
RETURN MARKET, BETA, DAN
MATHEMATIKA DIVERSIFIKASI
Pertemuan 12 dan 13
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.
IHSGt+1 - IHSG1
- IHSG1
Bina Nusantara University
4
MATHEMATIKA DIVERSIFIKASI
Bina Nusantara University
5
Linear Combinations
• Introduction
• Return
• Variance
6
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
7
Return
• The expected return of a portfolio is a weighted average
of the expected returns of the components:
n
E ( R p )    xi E ( Ri ) 
i 1
where xi  proportion of portfolio
invested in security i and
n
x
i 1
i
1
8
Variance
•
•
•
•
•
Introduction
Two-security case
Minimum variance portfolio
Correlation and risk reduction
The n-security case
9
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
10
Introduction (cont’d)
• For an n-security portfolio, the portfolio variance is:
n
n
   xi x j ij i j
2
p
i 1 j 1
where xi  proportion of total investment in Security i
ij  correlation coefficient between
Security i and Security j
11
Two-Security Case
• For a two-security portfolio containing Stock A and Stock
B, the variance is:
  x   x   2 xA xB  AB A B
2
p
2
A
2
A
2
B
2
B
12
Two Security Case (cont’d)
Example
Assume the following statistics for Stock A and Stock
B:
Stock A
Stock B
Expected return
Variance
Standard deviation
Weight
Correlation coefficient
.015
.050
.224
40%
.020
.060
.245
60%
.50
13
Two Security Case (cont’d)
Example (cont’d)
What is the expected return and variance of this twosecurity portfolio?
14
Two Security Case (cont’d)
Example (cont’d)
Solution: The expected return of this two-security
portfolio is:
n
E ( R p )    xi E ( Ri ) 
i 1
  x A E ( RA )    xB E ( RB ) 
  0.4(0.015)    0.6(0.020) 
 0.018  1.80%
15
Two Security Case (cont’d)
Example (cont’d)
Solution (cont’d): The variance of this two-security
portfolio is:
 2p  xA2 A2  xB2 B2  2 xA xB  AB A B
 (.4) (.05)  (.6) (.06)  2(.4)(.6)(.5)(.224)(.245)
2
2
 .0080  .0216  .0132
 .0428
16
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
17
Minimum Variance Portfolio (cont’d)
• For a two-security minimum variance portfolio, the
proportions invested in stocks A and B are:
 B2   A B  AB
xA  2
2
 A   B  2 A B  AB
xB  1  x A
18
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?
19
Minimum Variance Portfolio (cont’d)
Example (cont’d)
Solution: The weights of the minimum variance
portfolios in this case are:
 B2   A B  AB
.06  (.224)(.245)(.5)
xA  2

 59.07%
2
 A   B  2 A B  AB .05  .06  2(.224)(.245)(.5)
xB  1  xA  1  .5907  40.93%
20
Minimum Variance Portfolio (cont’d)
Example (cont’d)
1,2
1
Weight A
0,8
0,6
0,4
0,2
0
0
0,01
0,02
0,03
0,04
0,05
0,06
Portfolio Variance
21
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
22
The n-Security Case
• For an n-security portfolio, the variance is:
n
n
   xi x j ij i j
2
p
i 1 j 1
where xi  proportion of total investment in Security i
ij  correlation coefficient between
Security i and Security j
23
The n-Security Case (cont’d)
• The equation includes the correlation coefficient (or
covariance) between all pairs of securities in the portfolio
24
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
25
Single-Index Model
• Computational advantages
• Portfolio statistics with the single-index model
26
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
27
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:
i 
COV ( Ri , Rm )
 m2
where Rm  return on the market index
 m2  variance of the market returns
Ri  return on Security i
28
Portfolio Statistics With the Single-Index Model
• Beta of a portfolio:
n
 p   xi  i
i 1
• Variance of a portfolio:
 2p   p2 m2   ep2
 
2
p
2
m
29
Portfolio Statistics With the Single-Index Model
(cont’d)
• Variance of a portfolio component:
    
2
i
2
i
2
m
2
ei
• Covariance of two portfolio components:
 AB   A  B
2
m
30
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
31
Multi-Index Model (cont’d)
• The general form of a multi-index model:
Ri  ai   im I m   i1 I1   i 2 I 2  ...   in I n
where ai  constant
I m  return on the market index
I j  return on an industry index
 ij  Security i's beta for industry index j
 im  Security i's market beta
Ri  return on Security i
32