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# CHAPTER SEVEN - PowerPoint PPT 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.

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### CHAPTER SEVEN

PORTFOLIO ANALYSIS

• 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 FEASIBLE SET

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

THE FEASIBLE SET

rP

sP

0

• 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

• where the investor plots indifference curves and chooses the one that is furthest “northwest”

• the point of tangency at point E

THE OPTIMAL PORTFOLIO

rP

E

sP

0

• 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%

rP

G

rG=15

rA = 5

A

sP

sA=20

sG=40

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

X 2 = 1 - X 1

Consider 7 weighting combinations

using the formula

Portfolioreturn

A 5

B 6.7

C 8.3

D 10

E 11.7

F 13.3

G 15

• USING THE FORMULA

we can derive the following:

rPsP=+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

• 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

• 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

rP

G

lower bound

upper bound

A

sP

0

• ACTUAL LOCATIONS OF THE PORTFOLIO

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

RESULTS:

sB = 17.94%

sB = 18.81%

sB = 22.36%

sB = 27.60%

sB = 33.37%

ACTUAL PORTFOLIO LOCATIONS

F

D

E

C

B

• IMPLICATION:

• If rij < 0 line curves more to left

• If rij = 0 line curves to left

• If rij > 0 line curves less to left

• 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

• 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

• 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

• ei, I CAN BE CONSIDERED A RANDOM VARIABLE

• DISTRIBUTION:

• MEAN = 0

• VARIANCE = sei

• PORTFOLIO RISK

• TOTAL SECURITY RISK: s2i

• has two parts:

where = the market variance of index returns

= the unique variance of security i

returns

• 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

• Unique Risk

• mathematically can be expressed as