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CHAPTER EIGHT
PORTFOLIO ANALYSIS
1
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
2
THE EFFICIENT SET
THEOREM
• THE FEASIBLE SET
– DEFINITION: represents all portfolios that
could be formed from a group of N securities
3
THE EFFICIENT SET
THEOREM
THE FEASIBLE SET
rP
0
sP
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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
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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
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THE EFFICIENT SET
THEOREM
THE OPTIMAL PORTFOLIO
rP
E
0
sP
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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% s = 20%
– Gold Jewelry Company
» E(r) = 15% s = 40%
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CONCAVITY OF THE
EFFICIENT SET
rP
rG=15
rA = 5
G
A
sA=20
sG=40
sP
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CONCAVITY OF THE
EFFICIENT SET
• ALL POSSIBLE COMBINATIONS RELIE
ON THE WEIGHTS (X1 , X 2)
X2= 1 - X1
Consider 7 weighting combinations
using the formula
rP 
N
X
i 1
i
ri  X 1 r1  X 2 r2
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CONCAVITY OF THE
EFFICIENT SET
Portfolio
A
B
C
D
E
F
G
return
5
6.7
8.3
10
11.7
13.3
15
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CONCAVITY OF THE
EFFICIENT SET
• USING THE FORMULA
1/ 2


s P   X i X js ij 
 i 1 j 1

N
N
we can derive the following:
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CONCAVITY OF THE
EFFICIENT SET
A
B
C
D
E
F
G
rP
sP=+1
5
6.7
8.3
10
11.7
13.3
15
20
10
0
10
20
30
40
sP=-1
20
23.33
26.67
30.00
33.33
36.67
40.00
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CONCAVITY OF THE
EFFICIENT SET
• UPPER BOUNDS
– lie on a straight line connecting A and G
• i.e. all s must lie on or to the left of the straight line
• which implies that diversification generally leads to
risk reduction
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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
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CONCAVITY OF THE
EFFICIENT SET
rP
G
lower bound
0
A
upper
bound
sP
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CONCAVITY OF THE
EFFICIENT SET
• ACTUAL LOCATIONS OF THE
PORTFOLIO
– What if correlation coefficient (r ij ) is zero?
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CONCAVITY OF THE
EFFICIENT SET
RESULTS:
sB
sB
sB
sB
sB
=
=
=
=
=
17.94%
18.81%
22.36%
27.60%
33.37%
18
CONCAVITY OF THE
EFFICIENT SET
ACTUAL PORTFOLIO LOCATIONS
C
D
E
F
B
19
CONCAVITY OF THE
EFFICIENT SET
• IMPLICATION:
– If rij <
0
– If rij = 0
– If rij > 0
line curves more to left
line curves to left
line curves less to left
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CONCAVITY OF THE
EFFICIENT SET
• KEY POINT
– As long as -1 < r< 1 , the portfolio line
curves to the left and the northwest portion is
concave
– i.e. the efficient set is concave
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THE MARKET MODEL
• A RELATIONSHIP MAY EXIST
BETWEEN A STOCK’S RETURN AN
THE MARKET INDEX RETURN
ri  a iI  b i1rI  e iI
where
aiI  intercept term
ri = return on security
rI = return on market index I
b iI  slope term
e iI  random error term
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THE MARKET MODEL
• THE RANDOM ERROR TERMS ei, 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
ei, I
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THE MARKET MODEL
 ei, I CAN BE CONSIDERED A RANDOM
VARIABLE
– DISTRIBUTION:
• MEAN = 0
• VARIANCE = sei
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DIVERSIFICATION
• PORTFOLIO RISK
TOTAL SECURITY RISK: s2i
• has two parts:
s  b s s
2
i
where
2
iI
b iI2s 2
s e2i
2
i
2
ei
= the market variance of index
returns
= the unique variance of security i
returns
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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
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DIVERSIFICATION
• Unique Risk
– mathematically can be expressed as
s
2
eP
2
 1 
2
 
s
ei

i 1  N 
N
1 s e21  s e22  ...  s e2N 



N 
N

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