Sesi 5 Portfolio Analysis

7/31/2019 Sesi 5 Portfolio Analysis

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CHAPTER 7

Investments

1

OPTIMAL RISKY

PORTFOLIOS

Ir. Toga Buana S. Lubis, MM, CFP

Portfolio Risk as a Function of theNumber of Stocks in the Portfolio2


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Risk Reduction with Diversification3

St. Deviation

Unique Risk

Number ofSecurities

Market Risk

Two-Security Portfolio: Return4

rp = W1r1 + W2r2W1 = Proportion of funds in Security 1

W2 = Proportion of funds in Security 2

r = Ex ected return on Security 1r2 = Expected return on Security 2

1==

n

1i

iw


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Two-Security Portfolio: Risk5

p = 1 1 2 2 1 2 1 2

12 = Variance of Security 1

22 = Variance of Security 2

Cov(r1

r2

) = Covariance of returns for

Security 1 and Security 2

Covariance6

Cov r r =

1,2 = Correlation coefficient of

returns

,

1 = Standard deviation of

returns for Security 1

2 = Standard deviation of

returns for Security 2


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Correlation Coefficients: Possible

Values7

ange o va ues or 1,2+ 1.0 > > -1.0

If = 1.0, the securities would be perfectly

positively correlated

If = - 1.0, the securities would be

perfectly negatively correlated

Portfolio Risk/Return Two Securities:Correlation Effects8

The relationship depends on correlation

coe c ent.

-1.0 < < +1.0

The smaller the correlation, the greater the risk

reduction potential.

. , no r s re uc on s poss e.


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Minimum-Variance Combination9

E r = .10 = .15Sec 1

22 - Cov(r1r2)

2E(r2) = .14 = .20Sec 212 = .2

1 2

1

+ - 2Cov(r1r2)

W2 = (1 - W1)

2 2


2

W1 =. - . . .

(.15)2 + (.2)2 - 2(.2)(.15)(.2)

=1 .

W2 = (1 - .6733) = .3267


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= =p . . . . .

p = [(.6733)2(.15)2 + (.3267)2(.2)2 +

1/2. . . . .

p = [.0171]1/2

= .1308


2

W1 =. - . . .

(.15)2 + (.2)2 - 2(.2)(.15)(-.3)

=1 .

W2 = (1 - .6087) = .3913


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Minimum -Variance: Return and Risk

with = -.313

= =p . . . . .

p = [(.6087)2(.15)2 + (.3913)2(.2)2 +

1/2. . . . -.

p= [.0102]1/2

= .1009

Three-Security Portfolio14

2p = W1

212 + W2

212

+ 2W W

rp = 1r1 + 2r2 + 3r3

Cov(r r )

+ W32

32

Cov(r1r3)+ 2W1W3

Cov(r2r3)+ 2W2W3


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Descriptive Statistics for Two Mutual

Funds15

Computation of Portfolio Variance fromthe Covariance Matrix16


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Expected Return and Standard Deviation

with Various Correlation Coefficients17

Portfolio Expected Return as a Functionof Investment Proportions18


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Portfolio Standard Deviation as a

Function of Investment Proportions19

Portfolio Expected Return as a functionof Standard Deviation20


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Correlation Effects21

The relationship depends on correlation coefficient.

-1.0 < < +1.0

The smaller the correlation, the greater the risk

reduction potential.

If = +1.0, no risk reduction is possible.

Determination of the Optimal OverallPortfolio22


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The Proportions of the Optimal Overall

Portfolio23

Extending Concepts to All Securities24

The optimal combinations result in lowest level

o r s or a g ven return.

The optimal trade-off is described as the

efficient frontier.

These portfolios are dominant.


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The Minimum-Variance Frontier of Risky

Assets25

Extending to Include Riskless Asset26

The optimal combination becomes linear.

A single combination of risky and riskless assets

will dominate.


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The Efficient Frontier of Risky Assets

with the Optimal CAL27

The Efficient Portfolio Set28


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Capital Allocation Lines (CAL) with

Various Portfolios from the Efficient Set29

Risk Reduction of Equally Weighted Portfolios inCorrelated and Uncorrelated Universes30


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CHAPTER 8

Investments

1

INDEX MODELS

Ir. Toga Buana S. Lubis, MM, CFP

Advantages of the Single Index Model2

Reduces the number of inputs for

vers cat on.

Easier for security analysts to specialize.


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Single Factor Model3

ri = E(Ri) + iF + e

i = index of a securities particular return to the

factor

F= some macro factor; in this case F is unanticipated

movement; F is commonly related to security

returns

Assumption:

a broad market index like the S&P500 is the commonfactor.

Single Index Model4

(ri - rf) = i + i(rm - rf) + ei

Risk Prem Market Risk Prem

or Index Risk Prem

i = the stocks expected return if the

markets excess return is zero (rm - rf) = 0i(rm - rf) = the component of return due to

movements in the market index

ei = firm specific component, not due to market

movements


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Risk Premium Format5

i = ri - rf

Rm = (rm - rf)

Risk premium

format

Ri = i + i(Rm) + ei

Components of Risk6

Market or systematic risk: risk related to the

macro econom c actor or mar et n ex.

Unsystematic or firm specific risk: risk not

related to the macro factor or market index.

Total risk = Systematic + Unsystematic


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Measuring Components of Risk7

i i m eiwhere:

i2 = total variancei2 m2 = systematic variance2(e

i) = unsystematic variance

Examining Percentage of Variance8

=

Systematic Risk/Total Risk = 2

i2 m2 / 2 = 2

i2 m2 / i2 m2 + 2(ei) = 2


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Index Model and Diversification9

eR ++=

1

1

1

1

N

i

PP

N

i

PP

N

N

=

=

=

=

)(

1

2222

1

PMP

i

PP

e

eN

e

p +=

= =

The Variance of a Portfolio with Risk CoefficientBeta in the Single-Factor Economy10


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Scatter Diagram of HP, S&P 500, and

Security Characteristic Line (SCL) for HP11

Regression Statistics for the SCL ofHewlett-Packard12


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Using the Single-Index Model with

Active Management13

The single-index model can

be extended to optimize

the portfolio with active

management

The portfolio consists of an

active portfolio and a

passive or index portfolio

A The weight of the active

portfolio is determined by

the information ratio

e A

Sharpe Ratio for the CombinedPortfolio14

+=22

e

ssA

MP

A


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Efficient Frontiers with the Index Model

and Full-Covariance Matrix15

Sesi 5 Portfolio Analysis

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