CHAPTER 7 POWERPOINT PRESENTATION

CHAPTER SEVEN PORTFOLIO

ANALYSIS

THE EFFICIENT SET THEOREM



THE THEOREM

•

An investor will choose his optimal portfolio from the set of portfolios that offer



maximum expected returns for varying levels of risk, and



minimum risk for varying levels of returns

THE EFFICIENT SET THEOREM



THE FEASIBLE SET

•

DEFINITION: represents all portfolios that could be formed from a group of N

securities

THE EFFICIENT SET THEOREM

THE FEASIBLE SET

r

_P



_P

0

THE EFFICIENT SET THEOREM

 EFFICIENT SET THEOREM APPLIED TO THE FEASIBLE SET

•

Apply the efficient set theorem to the feasible set

the set of portfolios that meet first conditions of efficient set theorem must be identified

consider 2nd condition set offering minimum risk for varying levels of expected return lies on the “western”

boundary

remember both conditions: “northwest” set meets the requirements

THE EFFICIENT SET THEOREM



THE EFFICIENT SET

•

where the investor plots indifference

curves and chooses the one that is furthest

“northwest”

•

the point of tangency at point E

THE EFFICIENT SET THEOREM

THE OPTIMAL PORTFOLIO

E r

_P



_P

0

CONCAVITY OF THE EFFICIENT SET



WHY IS THE EFFICIENT SET CONCAVE?

•

BOUNDS ON THE LOCATION OF PORFOLIOS

•

^EXAMPLE:



Consider two securities

– Ark Shipping Company

• E(r) = 5%  = 20%

– Gold Jewelry Company

• E(r) = 15%  = 40%

CONCAVITY OF THE EFFICIENT SET



_P

r

_P

A

G

r_A = 5

_A=20 r_G=15

_G=40

CONCAVITY OF THE EFFICIENT SET



ALL POSSIBLE COMBINATIONS RELIE ON THE WEIGHTS (X

₁

, X

₂

)

X

2

= 1 - X

1

Consider 7 weighting combinations using the formula

2 2 1

1 1

r X

r X

r X

r ^N

i

i i

P

   



CONCAVITY OF THE EFFICIENT SET

Portfolio return A 5

B 6.7 C 8.3 D 10 E 11.7 F 13.3 G 15

CONCAVITY OF THE EFFICIENT SET



USING THE FORMULA

we can derive the following:

2 / 1

1 1 



 



 



 

N i

N j

ij j

i

P X X





CONCAVITY OF THE EFFICIENT SET

r

_P



_P=+1



_P=-1

A 5 20 20

B 6.710 23.33 C 8.3 0 26.67 D 10 10 30.00

E 11.7 20 33.33 F 13.3 30 36.67 G 15 40 40.00

CONCAVITY OF THE EFFICIENT SET



UPPER BOUNDS

•

lie on a straight line connecting A and G



i.e. all  must lie on or to the left of the straight line



which implies that diversification generally leads to risk reduction

CONCAVITY OF THE EFFICIENT SET



LOWER BOUNDS

•

all lie on two line segments



one connecting A to the vertical axis



the other connecting the vertical axis to point G

•

any portfolio of A and G cannot plot to the left of the two line segments

•

which implies that any portfolio lies within the boundary of the triangle

CONCAVITY OF THE EFFICIENT SET



G

upper bound

lower bound

r

_P



_P



CONCAVITY OF THE EFFICIENT SET



ACTUAL LOCATIONS OF THE PORTFOLIO

•

What if correlation coefficient (_ij ) is zero?

CONCAVITY OF THE EFFICIENT SET

RESULTS:



_B

= 17.94%



_B

= 18.81%



_B

= 22.36%



_B

= 27.60%



_B

= 33.37%

CONCAVITY OF THE EFFICIENT SET

ACTUAL PORTFOLIO LOCATIONS



C D  F

CONCAVITY OF THE EFFICIENT SET



IMPLICATION:

•

^If

^

ij

< 0 line curves more to left

•

^If

^

ij

= 0 line curves to left

•

^If

^

ij

> 0 line curves less to left

CONCAVITY OF THE EFFICIENT SET



KEY POINT

•

As long as -1 <



the portfolio line curves to the left and the northwest

portion is concave

•

i.e. the efficient set is concave

THE MARKET MODEL

 A RELATIONSHIP MAY EXIST BETWEEN A STOCK’S RETURN AN THE MARKET INDEX RETURN

where intercept term r_i = return on security

r_I = return on market index I

slope term

 random error term

iI I

i iI

i

r

r    

₁

 

THE MARKET MODEL



THE RANDOM ERROR TERMS 

_{i, I}

•

shows that the market model cannot explain perfectly

•

the difference between what the actual return value is and

•

what the model expects it to be is attributable to



i, I

THE MARKET MODEL





_{i, I}

CAN BE CONSIDERED A RANDOM

VARIABLE

• DISTRIBUTION:



MEAN = 0



VARIANCE =



_i

DIVERSIFICATION



PORTFOLIO RISK

•

TOTAL SECURITY RISK: _i



has two parts:

where = the market variance of index returns

= the unique variance of security i returns

2 2

2 2

i i

iI

i

  



  

2 2





_iI

2

i



DIVERSIFICATION



TOTAL PORTFOLIO RISK

•

also has two parts: market and unique



Market Risk

– diversification leads to an averaging of market risk



Unique Risk

– as a portfolio becomes more diversified, the smaller will be its unique risk

DIVERSIFICATION



Unique Risk

– mathematically can be expressed as



 

 



 ^N 

i

i

P ₁ N

2 2

2 1











 



   

 N N

N 2 2

2 2

1 ...

1



_



_



_

END OF CHAPTER 7