MBF2263 Portfolio Management Lecture 6: Portfolio Analysis · CONCAVITY OF THE EFFICIENT SET •KEY POINT –As long as -1 < r< 1 , the portfolio line curves to the left and the northwest

MBF2263 Portfolio Management

Lecture 6: 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 FEASIBLE SET

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

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THE FEASIBLE SET

rP

sP0

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• 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

– 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 OPTIMAL PORTFOLIO

E

rP

sP0

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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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sP

rP

A

G

rA = 5

sA=20

rG=15

sG=40

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

X 2 = 1 - X 1

Consider 7 weighting combinations

using the formula

2211

1

rXrXrXrN

i

iiP

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Portfolio return

A 5

B 6.7

C 8.3

D 10

E 11.7

F 13.3

G 15

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• USING THE FORMULA

we can derive the following:

2/1

1 1

N

i

N

j

ijjiP XX ss

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rP sP=+1 sP=-1

A 5 20 20

B 6.7 10 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

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• 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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• 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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A

G

upper bound

lower bound

rP

sP0

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• ACTUAL LOCATIONS OF THE PORTFOLIO

– What if correlation coefficient (r ij ) is zero?

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RESULTS:

sB = 17.94%

sB = 18.81%

sB = 22.36%

sB = 27.60%

sB = 33.37%

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ACTUAL PORTFOLIO LOCATIONS

B

CD E

F

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• IMPLICATION:

– If rij < 0 line curves more to left

– If rij = 0 line curves to left

– If rij > 0 line curves less to left

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• 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

where aiI intercept term

ri = return on security

rI = return on market index I

b iI slope term

e iI random error term

iIIiiIi rr eba 1

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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 RISKTOTAL SECURITY RISK: s2

i• has two parts:

where = the market variance of index returns

= the unique variance of security ireturns

2222

iiiIi essbs

22sbiI

2

ies

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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

i

iPN1

2

2

2 1ee ss

NN

N

22

2

2

1 ...1 eee sss

Documents

MBF2263 Portfolio Management Lecture 6: Portfolio Analysis · CONCAVITY OF THE EFFICIENT SET •KEY POINT –As long as -1 < r< 1 , the portfolio line curves to the left and the northwest