Chapter 7 Investment Analysis and Portfolio Management. Frank K. Reilly & Keith C. Brown

Investment Analysis and

Portfolio Management

Frank K. Reilly & Keith C. Brown

Chapter 7 - An Introduction to

Portfolio Management

Questions to be answered:

1. What do we mean by risk aversion and what evidence indicates that investors are generally risk averse?

2. What are the basic assumptions behind the Markowitz portfolio theory?

3. What is meant by risk and what are some of the alternative measures of risk used in investments?

4. How do you compute the expected rate of return for an individual risky asset or a portfolio of assets?

5. How do you compute the standard deviation of rates of return for an individual risky asset?

6. What is meant by the covariance between rates of return and how do you compute covariance?

7. What is the relationship between covariance and correlation? 8. What is the formula for the standard deviation for a portfolio

of risky assets and how does it differ from the standard deviation of an individual risky asset?

9. Given the formula for the standard deviation of a portfolio, how and why do you diversify a portfolio?

10. What happens to the standard deviation of a portfolio when you change the correlation between the assets in the portfolio? 11. What is the risk-return efficient frontier?

12. Is it reasonable for alternative investors to select different portfolios from the portfolios on the efficient frontier?

13. What determines which portfolio on the efficient frontier is selected by an individual investor?

Background Assumptions

• investor memaksimumkan return pd tingkat

risiko tertentu.

• Portofolio melibatkan seluruh aset dan

kewajiban investor

• Hubungan antara return aset dlm portofolio

sangat penting

• Portofolio yg baik bukanlah kumpulan

Risk Aversion

(Benci Risiko)

• Dg satu pilihan antar dua aset dg return yg

sama, Investor umumnya memilih aset dengan

tingkat risiko lebih kecil

• Buktinya:

– Banyak investor membeli asuransi: kematian, kendaraan, kesehatan, dan ketidakpastian pendapatan.

• Pembeli mempertukarkan biaya yg pasti untuk risiko keugian yg tidak pasti

Not all investors are risk averse

• Preferensi Risiko: hrs dilakukan dengan

jumlah uang yg dikeluarkan-sedikit, untuk

memastikan kerugian yg besar

Definition of Risk

1. Ketidakpastian atas hasil mendatang, atau

2. Probabilitas dari hasil yg tidak diinginkan

Markowitz Portfolio Theory

• Mengkuantitatifkan risiko

• Menderivasi ukuran return harapan bg portofolio aset

dan risiko harapannya

• Menunjukkan bhw varian dari return mrp ukuran berarti

tentang risiko portofolio

• Menderivasi formula untuk menghitung varian

portfolio, yg menunjukkan bgm mendiversifikasi scr

efektif suatu portofolio

Assumptions of

Markowitz Portfolio Theory

1. Investor mempertimbangkan tiap alternatif investasi spt yg sdg disajikan dg distribusi probabilitas dr return ekspektasi slm

beberapa periode pemilikan investasi.

2. Investor meminimumkan utilitas ekpektasi satu-periode, dan kurve utilitasnya menunjukkan utilitas marjinal yg menurun dr kemakmuran (diminishing marginal utility of wealth).

3. Investor menestimasi risiko portofolio atas basis variabilitas return harapan.

4. Investor mendasarkan keputusan hanya pd return harapan dan risiko, sehingga kurve utilitasnya mrp fungsi dr return ekspektasi dan varian ekspektasi (atau deviasi standar) dr retun saja.

Markowitz Portfolio Theory

• Menggunakan 5 asumsi, aset tunggal atau

portofolio aset dianggap efisien jika:

– Tidak ada aset/portofolia aset yg menawarkan

return lbh tinggi dg risiko sama (atau lebih

rendah), atau

– Risiko lebih rendah dengan return sama (lbh

tinggi)

Alternative Measures of Risk

• Varian atau deviasi standar dari return harapan

• Kisaran return (Range of returns)

• Return di bawah harapan

– Semivarian – ukuran yg hanya

mempertimbangkan deviasi di bawah rerata

– Ukuran risiko ini mengasumsikan scr implisit bhw

investor ingin meminimumkan kurangnya return

yg lbh rendah dp tingkat target return

Expected Rates of Return

• Unt aset individual – jumlah dr retun

potensial dikalikan dg probabilitas return

• Untuk portofolio aset – rata-rata tertimbang

return harapan bg investasi individual dlm

portofolio

Computation of Expected Return for an

Individual Risky Investment

0.25

0.08

0.0200

0.25

0.10

0.0250

0.25

0.12

0.0300

0.25

0.14

0.0350

E(R) = 0.1100

Expected Return

(Percent)

Probability

Possible Rate of

Return (Percent)

Exhibit 7.1

Computation of the Expected Return for a

Portfolio of Risky Assets

0.20 0.10 0.0200 0.30 0.11 0.0330 0.30 0.12 0.0360 0.20 0.13 0.0260 E(R_{por i}) = 0.1150 Expected Portfolio Return (Wi X Ri) (Percent of Portfolio) Expected Security Return (Ri) Weight (Wi) Exhibit 7.2

i

asset

for

return

of

rate

expected

the

)

E(R

i

asset

in

portfolio

the

of

percent

the

W

:

where

R

W

)

E(R

i i 1 i por

=

=

=

∑

= n i i i

Variance (Standard Deviation) of

Returns for an Individual Investment

• Deviasi standar adl akar pangkat dua dari

varian

• Varian adl ukuran tentang variasi return yg

mungkin terjadi R

_i

, dr return harapan

Variance (Standard Deviation) of

Returns for an Individual Investment

∑

=

=

n

i

1

i

2

i

i

2

P

)]

E(R

-R

[

)

(

Variance

σ

Notasi

P

_i

= probabilitas dr return yg mungkin diterima (possible

Variance (Standard Deviation) of

Returns for an Individual Investment

∑

=

=

n

i

1

i

2

i

i

-

E(R

)]

P

R

[

)

(

σ

Deviasi Standar

Variance (Standard Deviation) of

Returns for an Individual Investment

Possible Rate Expected

of Return (Ri) Return E(Ri) Ri - E(Ri) [Ri - E(Ri)] 2 Pi [Ri - E(Ri)] 2 Pi 0.08 0.11 0.03 0.0009 0.25 0.000225 0.10 0.11 0.01 0.0001 0.25 0.000025 0.12 0.11 0.01 0.0001 0.25 0.000025 0.14 0.11 0.03 0.0009 0.25 0.000225 0.000500 Exhibit 7.3 Varian ( 2_{) = .0050} Deviasi Standar ( ) = .02236

σ

σ

Variance (Standard Deviation) of Returns for a

Portfolio

Penghitungan return bulanan:

Exhibit 7.4

Closing Closing

Date Price Dividend Return (%) Price Dividend Return (%)

Dec.00 60.938 45.688 Jan.01 58.000 -4.82% 48.200 5.50% Feb.01 53.030 -8.57% 42.500 -11.83% Mar.01 45.160 0.18 -14.50% 43.100 0.04 1.51% Apr.01 46.190 2.28% 47.100 9.28% May.01 47.400 2.62% 49.290 4.65% Jun.01 45.000 0.18 -4.68% 47.240 0.04 -4.08% Jul.01 44.600 -0.89% 50.370 6.63% Aug.01 48.670 9.13% 45.950 0.04 -8.70% Sep.01 46.850 0.18 -3.37% 38.370 -16.50% Oct.01 47.880 2.20% 38.230 -0.36% Nov.01 46.960 0.18 -1.55% 46.650 0.05 22.16%

Covariance of Returns

• Ukuran tentang derajat dimana dua variabel

berubah bersama (“move together”) retalif

pada nilai rerata individualnya

• Unt dua aset, i dan j, kovarian return

ditentukan sbg:

Covariance and Correlation

• Koefisien korelasi dihitung dg menstandarisasi (membagi) kovarian dg angka deviasi standar individual

• Koefisien Korelasi berubah2 dari -1 to +1

it i ij

R

of

deviation

standard

the

R

of

deviation

standard

the

returns

of

t

coefficien

n

correlatio

the

r

:

where

Cov

r

=

=

=

j i ij ij

σ

σ

σ

Correlation Coefficient

• Koefisien korelasi berubah-ubah hanya dlm

kisaran +1 s/d -1.

• Nilai +1 akan mengindikasikan hubungan

positif sempurna

–

bhw return dua aset bergerak bersama dlm pola linier sempurna.

• Nilai –1 akan mengindikasikan hubungan

negatif sempurnal

– Bhw return dua aset memiliki persentasi perubahan sama, tetap dg arah kebalikan

Portfolio Standard Deviation

Formula

σ σ σ σ σ σ ij 2 i i port n 1 i n 1 i ij j n 1 i i 2 i 2 i port r Cov where j, and i assets for return of rates e between th covariance the Cov i asset for return of rates of variance the portfolio in the value of proportion by the determined are weights where portfolio, in the assets individual the of weights the W portfolio the of deviation standard the : where Cov w w w = = = = = + =

∑

∑∑

= = =

Portfolio Standard Deviation

Calculation

• Beberapa aset dr portofolio bisa digambarkan dg

dua karakteristik:

– Return harapan

– Deviasi standar harapan dari return

• Korelasi diukur dg kovarian, yg berpengauh pd

deviasi standar portofolio

• Korelasi rendah mengurangi risiko portofolio

namun tak mempengaruhi return harapan

Combining Stocks with Different Returns and

Risk

Case Correlation Coefficient Covariance

a +1.00 .0070

b +0.50 .0035

c 0.00 .0000

d -0.50 -.0035

W

)

E(R

Asset

_{i i}

σ

2i

σ

_i 1 .10 .50 .0049 .07 2 .20 .50 .0100 .10

Combining Stocks with Different

Returns and Risk

• Aset mungkin berbeda dlm return harapan

dan deviasi standar individual

• Korelasi negatif menurunkan risiko

portofolio

• Mengkombinasikan dua aset dg korelasi

-1.0 menurunkan deviasi standar portofolio

menjadi nol hanya jika deviasi standar

Constant Correlation with Changing Weights

Case W1 W 2 E(R_i) f 0.00 1.00 0.20 g 0.20 0.80 0.18 h 0.40 0.60 0.16 i 0.50 0.50 0.15 j 0.60 0.40 0.14

)

E(R

Asset

_i 1 .10 r_ij = 0.00 2 .20

Constant Correlation with Changing Weights

Case

W

1

W

2

E(R

i

)

E(

port

)

f

0.00

1.00

0.20

0.1000

g

0.20

0.80

0.18

0.0812

h

0.40

0.60

0.16

0.0662

i

0.50

0.50

0.15

0.0610

j

0.60

0.40

0.14

0.0580

k

0.80

0.20

0.12

0.0595

l

1.00

0.00

0.10

0.0700

Portfolio Risk-Return Plots for Different

Weights

-0.05 0.10 0.15 0.20 E(R) R_ij = +1.00 1 2

With two perfectly

correlated assets, it is only possible to create a two asset portfolio with risk-return along a line between either

Portfolio Risk-Return Plots for Different

Weights

-0.05 0.10 0.15 0.20 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 E(R) R_ij = 0.00 R_ij = +1.00 f g h i j k 1 2 With uncorrelated assets it is possible to create a two asset

portfolio with lower risk than either single asset

Portfolio Risk-Return Plots for Different

Weights

-0.05 0.10 0.15 0.20 E(R) R_ij = 0.00 R_ij = +1.00 R_ij = +0.50 f g h i j k 1 2

With correlated assets it is possible to create a two asset portfolio between the first two curves

Portfolio Risk-Return Plots for Different

Weights

-0.05 0.10 0.15 0.20 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 E(R) R_ij = 0.00 R_ij = +1.00 R_ij = -0.50 R_ij = +0.50 f g h i j k 1 2 With negatively correlated assets it is possible to create a two asset

portfolio with much lower risk than

Portfolio Risk-Return Plots for

Different Weights

-0.05 0.10 0.15 0.20 E(R) R_ij = 0.00 R_ij = +1.00 R_ij = -1.00 R_ij = +0.50 f g h i j k 1 2

With perfectly negatively correlated assets it is

possible to create a two asset portfolio with almost no risk

R_ij = -0.50

Estimation Issues

• Hasil alokasi portofolio tergantung pd input

statistikal yg akurat

• Estimasi dari

– Return harapan

– Deviasi Standar

– Koefisien Korelasi

• Di antara seluruh pasangan aset

• Dg 100 aset, 4,950 estimasi korelasi

Estimation Issues

• Dg asumsi bhw return saham dpt digambarkan

dg model pasar tunggal (single market model),

jumlah korelasi yg diperlukan mengurangi

jumlah aset

• Single index market model:

• bi = koefisien slope yg menghubungkan return

sekuritas-i dg return agregrat pasar saham

• Rm = Return pasar saham agregat

i m i i i

a

b

R

R

=

+

+

ε

Estimation Issues

• Jika semua sekuritas berhubungan sama dg

pasar dan a b

_i

diderivasi untuk tiap sekuritas

(each one), dpt ditunjukkan bhw koefisien

korelasi antara dua sekuritas i dan j

ditunjukkan (given):

market

stock

aggregate

for the

returns

of

variance

the

where

b

b

r

2 m i 2 m j i ij

=

=

σ

σ

σ

σ

j

The Efficient Frontier

• The efficient frontier menyatakan bhw

– set portofolio dg return maksimum unt tiap level

risiko tertentu, atau

– Isiko minimum untuk tiap tingkat return

• Frontier lbh tepat untuk portfolio investasi dp

sekuritas individual

– Kecual unt aset dg return tertinggi dan aset risiko

terendah

Efficient Frontier

for Alternative Portfolios

Efficient Frontier A B C Exhibit 7.15 E(R)

The Efficient Frontier

and Investor Utility

• Kurve utilitas investor menunjukkan saling tukar (trade-offs) yg diinginkan investor antara return dan risiko

• Slope kurve efficient frontier turun scr tetap (steadily) ketika kit bergerak naik (upward)

• Dua interaksi tsb akan menentukan portofolio tertentu yg dipilih oleh investorr individual

• Portofolio optimal memiliki utilitas tertinggi bag investor tertentu • Port optimal terletak pd titik tangen antara efficient frontier dan

Selecting an Optimal Risky

Portfolio

)

E(

σ

)

E(R

_port X Y U₃ U₂ U₁ U_3’ U_2’ U_1’ Exhibit 7.16

The Internet

Investments Online

www.pionlie.com

www.investmentnews.com

www.micropal.com

www.riskview.com

www.altivest.com

Future topics

Chapter 8

• Capital Market Theory

• Capital Asset Pricing Model

• Beta

• Expected Return and Risk

• Arbitrage Pricing Theory