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CHAPTER SEVEN 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 SET rP 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**

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 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 G rG=15 rA = 5 A sP sA=20 sG=40

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**CONCAVITY OF THE EFFICIENT SET**

ALL POSSIBLE COMBINATIONS RELIE ON THE WEIGHTS (X1 , X 2) X 2 = X 1 Consider 7 weighting combinations using the formula

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**CONCAVITY OF THE EFFICIENT SET**

Portfolio return A B C D E F G

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**CONCAVITY OF THE EFFICIENT SET**

USING THE FORMULA we can derive the following:

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**CONCAVITY OF THE EFFICIENT SET**

rP sP=+1 sP=-1 A B C D E F G

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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 upper bound A 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 = 17.94% sB = 18.81% sB = 22.36% sB = 27.60% sB = 33.37%

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**CONCAVITY OF THE EFFICIENT SET**

ACTUAL PORTFOLIO LOCATIONS F D E C B

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**CONCAVITY OF THE EFFICIENT SET**

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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**CONCAVITY OF THE EFFICIENT SET**

KEY POINT As long as < 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

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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:**

where = 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

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END OF CHAPTER 7

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Corporate Finance Portfolio Theory Prof. André Farber SOLVAY BUSINESS SCHOOL UNIVERSITÉ LIBRE DE BRUXELLES.

Corporate Finance Portfolio Theory Prof. André Farber SOLVAY BUSINESS SCHOOL UNIVERSITÉ LIBRE DE BRUXELLES.

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