BASIC

PORTFOLIO

ANALYSIS

Fall 2002

Mean and Standard Deviation of Individual Securities Define: (1) ijR jth return on stock i

(2) iR expected return stock on i

(3) iσ standard deviation of return stock i

(4) M number of periods (5) N number of assets

M

ijRM

1jiR=Σ=

2

iR

ijRE

2

Mi

Rij

RM

1j2i

−=−

=Σ=σ

Note some use M-1.

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(

=−−=−−

=−=−

=−−=−

326i

66/382i 3/2

=

==

σ

σ

MEAN AND VARIANCE OF PORTFOLIOS

Two General Rules:

1. 2R1R2

RE1

REj2

Rj1

RE +=+=+

2. 1RCj1

CRE =

Two Asset Case (both risky) Define:

iX as the proportion in security i.

(1) Return on portfolio

ijRiXj2

R2

Xj1

R1

XpjR Σ=+=

(2) Mean return on portfolio

+=j2

R2

Xj1

R1

XEPR

+=j2

R2

XEj1

R1

XE

iRiX2R2X1R1X Σ=+=

(3) Variance = 2

pRpRE2p

−=σ

2)

2R

2X

1R

1X()

j2R

2X

j1R

1X(E2p

+−+=σ

2)

2R

j2R(

2X)

1R

j1R(

1XE

−+−=

−−+−+−= )2Rj2R)(1Rj1R(2X1X22)2Rj2R(22X2)1Rj1R(2

1XE

−−+−+−= )2Rj2R)(1Rj1R(E2X1X22)2Rj2R(E22X2)1Rj1R(E2

1X

122X1X22

222X2

121X σσσ ++=

−−= )2

Rj2

R)(1

Rj1

R(E12σ

Note:

(1) Measures joint movement

(2) Unrestricted to sign

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 jiijij σσρσ =

1ij1 +≤≤− ρ

Calculating ABσ

(12 - 10) (7 - 9) = -4 (10 - 10) (9 - 9) = 0 (8 - 10) (11 - 9) = -4

38 AB −=σ

38

38

AB38 ρ=−

1AB −=ρ

Calculating ACσ 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=ρ

Calculating ADσ 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

Three Security Case

1. Return on portfolio ijRiXj3R3Xj2R2Xj1R1XPjR Σ=++=

2. Mean return on portfolio )j3R3Xj2R2Xj1R1X(EPR ++=

3R3X2R2X1R1XpR ++=

3. Variance of return

2)3R3X2R2X1R1X()

j3R3X

j2R2X

j1R1X(E2

P

++−++=σ

2)3R

j3R(3X)2R

j2R(2X)1R

j1R(1XE2

P

−+−+−=σ

Terms Variance

2)1Rj1R(E21X −

2)2Rj2R(E2

2X −

2)3Rj3R(E2

3X −

Terms Covariance

−− )2

Rj2

R)(1

Rj1

R(E2X1X2

−− )3

Rj3

R)(1

Rj1

R(E3X1X2

−−)3

Rj3

R)(2

Rj2

R(E3X2X2

General Formulas:

Mean Return on Portfolio: iRiXpR Σ=

Variance of Return on Portfolio

ikkXiX N

ik1k

N

1i2i 2

iX N

1i2p σσσ

≠=Σ

=Σ+

=Σ=

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

−+=

ikN112

N1 σσ

−+=

ikik2iN

1 σσσ +−=

Efficient Set Theorem

(1). Holding PR constant minimize Pσ

(2). Holding Pσ constant maximize PR

Plotting Efficient Frontier

(two risky assets) R σ Proportion A 14 4 AX

B 8 2 )AX1(BX −=

Perfectly Positively Correlated Expected Return:

BR)AX1(ARAXpR −+=

)BRAR(AXBR −+=

BAAB)AX1(AX22B

2)AX1(2A

2AX2p σσρσσσ −+−+=

IF

1+=ρ

−+−+= 2B

2)A

X1(BA

)A

X1(A

X22A

2A

X2p σσσσσ

2

B)

AX1(

AAX

−+= σσ

or

B)AX1(AAXp σσσ −+=

)BA(AXBp σσσσ −+=

or

BA

BpAX

σσ

σσ

−

−=

Substituting into expected return equation:

−−

−+=

BR

AR

BA

BpBRpR

σσ

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+=

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

and

)BA(AXBp σσσσ ++−=

)BA(AXB σσσ +−+=

BA

Bpor BA

BpAX

σσ

σσ

σσ

σσ

+

+−

+

+=

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±=

with other ρ 's not a straight line

In standard definition proceeds full usable 1X 2X pR

+2 -1 20 +3 -2 26 +4 -3 32

Efficient Frontier with Riskless Asset

)FRAR(XFRARXFR)X1(cR −+=+−=

where X is fraction in risky portfolio A

−++−=FAAF

)X1(X22A

2X2F

2)X1(2c σσρσσσ

A

cX2A

2X2c σσ

σσ =⇒=

)FRAR(A

cFRcR −+=

σσ

cA

FR

AR

FRcR σσ

−+=

(1). Separation Theorem: Investors optimum choice of a risky portfolio is separate from his or her preferences. (2). Two Fund Theorem: An investor is not hurt by restriction to a choice of two funds. (3). Unambiguous objective function.