BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B...

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BASIC PORTFOLIO ANALYSIS Fall 200

Transcript of BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B...

Page 1: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

BASIC

PORTFOLIO

ANALYSIS

Fall 2000

Page 2: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

Mean and Standard Deviation of Individual Securities

Define:

(1)ij

R jth return on stock i

(2)i

R expected return stock on i

(3)i

σ standard deviation of return stock i

(4) M number of periods

(5) N number of assets

Page 3: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

Mij

RM

1jiR

=Σ=

2

iR

ijRE

2

M

iR

ijRM

1j2i

−=−

=Σ=σ

Note some use M-1.

Page 4: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With
Page 5: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

Example:

MONTH Return

Dec 5%

Nov -2%

Oct 3%

Sept 2%

Aug -1%

July -1%6%

%16%6

6ij

R

iR ==Σ=

42)11(

42)11(

12)12(

42)13(

92)12(

162)15(

=−−

=−−

=−

=−

=−−=−

326

i

66/382i

3/2

=

==

σ

σ

Page 6: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

MEAN AND VARIANCE OF PORTFOLIOS

Two General Rules:

1. 2

R1

R2

RE1

REj2

Rj1

RE +=+=+

2. 1

RCj1

CRE =

Two Asset Case (both risky)

Define:

iX as the proportion in security i.

(1) Return on portfolio

ijR

iX

j2R

2X

j1R

1X

pjR Σ=+=

Page 7: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

(2) Mean return on portfolio

+=j2

R2

Xj1

R1

XEPR

+=j2

R2

XEj1

R1

XE

iR

iX

2R

2X

1R

1X Σ=+=

Page 8: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

(3) Variance = 2

pRpRE2p

−=σ

2)

2R

2X

1R

1X()

j2R

2X

j1R

1X(E2

p

+−+=σ

2)

2R

j2R(

2X)

1R

j1R(

1XE

−+−=

−−+−+−= )2

Rj2

R)(1

Rj1

R(2

X1

X22)2

Rj2

R(22

X2)1

Rj1

R(21

XE

Page 9: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

−−+−+−= )2

Rj2

R)(1

Rj1

R(E2X1X22)2Rj2R(E22X2)1Rj1R(E2

1X

122X

1X22

222

X21

21

X σσσ ++=

−−= )2

Rj2

R)(1

Rj1

R(E12

σ

Note:

(1) Measures joint movement

(2) Unrestricted to sign

Page 10: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

Example (assume equally likely) 6A

Return

Condition A B C Rainfall D

Good 12 7 8 Heavy 8

Average 10 9 6 Average 6

Poor 8 11 4 Light 4

r 10 9 6 6

σ 8/3 8/3 8/3 8/3

Useful

jiijijσσρσ =

1ij

1 +≤≤− ρ

Page 11: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With
Page 12: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

Calculating AB

σ

(12 - 10) (7 - 9) = -4

(10 - 10) (9 - 9) = 0

(8 - 10) (11 - 9) = -4

38

AB−=σ

38

38

AB38 ρ=−

1AB

−=ρ

Page 13: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

CalculatingAC

σ 6C

(12 - 10) (8 - 6) = 4

(10 - 10) (6 - 6) = 0

(8 - 10) ( 4 - 6) = 4 8

38

AB=σ

38

38

AB38 ρ=

1AB

Page 14: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

Calculating AD

σ 6D

(12 - 10) (8 - 6) = +4

(12 - 10) (6 - 6) = 0

(12 - 10) (4 - 6) = -4

(10 - 10) (8 - 6) = 0

(10 - 10) (6 - 6) = 0

(10 - 10) (4 - 6) = 0

(8 - 10) (8 - 6) = -4

(8 - 10) (6 - 6) = 0

(8 - 10) (4 - 6) = +4

0AD

0AD

=

=

ρ

σ

Page 15: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

Three Security Case

1. Return on portfolio

ijR

iX

j3R

3X

j2R

2X

j1R

1X

PjR Σ=++=

2. Mean return on portfolio

)j3

R3

Xj2

R2

Xj1

R1

X(EPR ++=

3R

3X

2R

2X

1R

1XpR ++=

3. Variance of return

2)

3R

3X

2R

2X

1R

1X()

j3R

3X

j2R

2X

j1R

1X(E2

P

++−++=σ

2)

3R

j3R(

3X)

2R

j2R(

2X)

1R

j1R(

1XE2

P

−+−+−=σ

Page 16: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

Terms Variance

2)1

Rj1

R(E21

X −

2)2

Rj2

R(E22

X −

2)3

Rj3

R(E23

X −

Terms Covariance

−− )2

Rj2

R)(1

Rj1

R(E2

X1

X2

−− )3

Rj3

R)(1

Rj1

R(E3

X1

X2

−−)3

Rj3

R)(2

Rj2

R(E3

X2

X2

Page 17: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

General Formulas:

Mean Return on Portfolio:

iR

iXpR Σ=

Variance of Return on Portfolio

ikkX

iX

N

ik1k

N

1i2i

2i

X N

1i2p σσσ

≠=Σ

=Σ+

=Σ=

Page 18: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

The Effect of Diversification

Assume random selection and equal amount in each security.

N1

iX =

ik

2

N1N

ik1k

N

1i 2

i

2

N1N

1i2P σσσ

≠=Σ

=Σ+

=Σ=

−≠=Σ

=Σ−+

=Σ=

ik1N1

N1N

ik1k

N

1iN1N

N

2i

N

1iN1 σ

σ

ikN1N2

iN1 σσ

−+=

Page 19: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

ikN112

N1 σσ

−+=

ikik2iN

1 σσσ +−=

Page 20: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With
Page 21: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With
Page 22: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

Efficient Set Theorem

(1). Holding PR constant minimize Pσ

(2). Holding Pσ constant maximize PR

Page 23: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With
Page 24: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

Plotting Efficient Frontier

(two risky assets)

R σ Proportion

A 14 4A

X

B 8 2 )A

X1(B

X −=

Page 25: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

Perfectly Positively Correlated

Expected Return:

BR)

AX1(

AR

AXpR −+=

)B

RA

R(A

XB

R −+=

BAAB)

AX1(

AX22

B2)

AX1(2

A2A

X2p σσρσσσ −+−+=

IF

Page 26: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

1+=ρ

−+−+= 2

B2)

AX1(

BA)

AX1(

AX22

A2A

X2p σσσσσ

2

B)

AX1(

AAX

−+= σσ

or

Page 27: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

B)

AX1(

AAXp σσσ −+=

)BA

(A

XBp σσσσ −+=

or

BA

BpA

Xσσ

σσ

−=

Substituting into expected return equation:

Page 28: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

−−

−+=

BR

AR

BA

BpB

RpRσσ

σσ

−+

−−=

BA

BR

AR

pBA

BR

AR

BBR

σσσ

σσσ

This is, of course, a straight line. With the example:

−−+

−−−=

24814

p2481428pR σ

p32pR σ+=

Page 29: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With
Page 30: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

Perfect Negative Correlation

If 1−=ρ

−+−−= 2B

2)A

X1(BAA

X1A

X22A

2A

X2p σσσσσ

This can come from either

−−=B

)A

X1(AA

Xp σσσ

or

−+−=B

)A

X1(AA

Xp σσσ

Page 31: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

and

)BA

(A

XBp σσσσ ++−=

)BA

(A

XB

σσσ +−+=

BA

Bpor

BA

BpAX

σσ

σσ

σσ

σσ

+

+−

+

+=

Page 32: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

Substituting into expected return:

)BRAR(

BA

BpBRpR −

+

++=

σσ

σσ

or

)BRAR(

BA

BpBRpR −

+

+−+=

σσ

σσ

+

−±

+

−+=

BA

BR

AR

pBA

BR

AR

BBRpR

σσσ

σσσ

+−±

+−+=

24814

p2481428 σ

p10 σ±=

Page 33: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

with other ρ 's not a straight line

Page 34: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With
Page 35: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With
Page 36: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

In standard definition proceeds full usable

1X

2X pR

+2 -1 20

+3 -2 26

+4 -3 32

Page 37: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With
Page 38: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

Efficient Frontier with Riskless Asset

)F

RA

R(XF

RA

RXF

R)X1(cR −+=+−=

where X is fraction in risky portfolio A

−++−=

FAAF)X1(X22

A2X2

F2)X1(2

c σσρσσσ

A

cX2A

2X2c σ

σσσ =⇒=

)F

RA

R(

A

cF

RcR −+=σ

σ

cA

FR

AR

FRcR σσ

−+=

Page 39: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With

(1). Separation Theorem:

Investors optimum choice of a risky portfolio is separatefrom his or her preferences.

(2). Two Fund Theorem:

An investor is not hurt by restriction to a choice oftwo funds.

(3). Unambiguous objective function.

Page 40: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With
Page 41: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With
Page 42: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With
Page 43: BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B B R A R p A B B R A R B B R s s s s s s This is, of course, a straight line. With