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

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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