Portfolio Analysis

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PORTFOLIO ANALYSIS

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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

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THE EFFICIENT SET THEOREM

• THE FEASIBLE SET– DEFINITION: represents all portfolios that

could be formed from a group of N securities

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THE EFFICIENT SET THEOREM

THE FEASIBLE SETrP

P0

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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

E

rP

P0

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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% = 20%– Gold Jewelry Company

» E(r) = 15% = 40%

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CONCAVITY OF THE EFFICIENT SET

P

rP

A

G

rA = 5

A=20

rG=15

G=40

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CONCAVITY OF THE EFFICIENT SET

• ALL POSSIBLE COMBINATIONS RELIE ON THE WEIGHTS (X1 , X 2)

X 2 = 1 - X 1

Consider 7 weighting combinations

using the formula

22111

rXrXrXrN

iiiP

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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

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CONCAVITY OF THE EFFICIENT SET

• USING THE FORMULA

we can derive the following:

2/1

1 1

N

i

N

jijjiP XX

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CONCAVITY OF THE EFFICIENT SET

rP P=+1 P=-1

A 5 20 20

B 6.7 10 23.33C 8.3 0 26.67D 10 10 30.00E 11.7 20 33.33F 13.3 30 36.67G 15 40 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 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

G

upper bound

lower bound

rP

P

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CONCAVITY OF THE EFFICIENT SET

• ACTUAL LOCATIONS OF THE PORTFOLIO– What if correlation coefficient (ij ) is zero?

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CONCAVITY OF THE EFFICIENT SET

RESULTS:

B = 17.94%

B = 18.81%

B = 22.36%

B = 27.60%

B = 33.37%

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CONCAVITY OF THE EFFICIENT SET

ACTUAL PORTFOLIO LOCATIONS

CD

F

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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

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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

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THE MARKET MODEL

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

where intercept term

ri = return on security

rI = return on market index I

slope term

random error term

iIIiiIi rr 1

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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

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THE MARKET MODEL

i, I CAN BE CONSIDERED A RANDOM

VARIABLE

– DISTRIBUTION:• MEAN = 0

• VARIANCE = i

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DIVERSIFICATION• PORTFOLIO RISK

TOTAL SECURITY RISK: i

• has two parts:

where = the market variance of index returns

= the unique variance of security i returns

2222iiiIi

22 iI2i

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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

N

iiP N1

22

2 1

NNN

222

21 ...1