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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

CHAPTER SEVEN

PORTFOLIO ANALYSIS

the efficient set theorem
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 theorem1
THE EFFICIENT SET THEOREM
  • THE FEASIBLE SET
    • DEFINITION: represents all portfolios that could be formed from a group of N securities
the efficient set theorem2
THE EFFICIENT SET THEOREM

THE FEASIBLE SET

rP

sP

0

the efficient set theorem3
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
the efficient set theorem4
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
the efficient set theorem5
THE EFFICIENT SET THEOREM

THE OPTIMAL PORTFOLIO

rP

E

sP

0

concavity of the efficient set
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%
concavity of the efficient set1
CONCAVITY OF THE EFFICIENT SET

rP

G

rG=15

rA = 5

A

sP

sA=20

sG=40

concavity of the efficient set2
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

concavity of the efficient set3
CONCAVITY OF THE EFFICIENT SET

Portfolioreturn

A 5

B 6.7

C 8.3

D 10

E 11.7

F 13.3

G 15

concavity of the efficient set4
CONCAVITY OF THE EFFICIENT SET
  • USING THE FORMULA

we can derive the following:

concavity of the efficient set5
CONCAVITY OF THE EFFICIENT SET

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

concavity of the efficient set6
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
concavity of the efficient set7
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
concavity of the efficient set8
CONCAVITY OF THE EFFICIENT SET

rP

G

lower bound

upper bound

A

sP

0

concavity of the efficient set9
CONCAVITY OF THE EFFICIENT SET
  • ACTUAL LOCATIONS OF THE PORTFOLIO
    • What if correlation coefficient (r ij ) is zero?
concavity of the efficient set10
CONCAVITY OF THE EFFICIENT SET

RESULTS:

sB = 17.94%

sB = 18.81%

sB = 22.36%

sB = 27.60%

sB = 33.37%

concavity of the efficient set11
CONCAVITY OF THE EFFICIENT SET

ACTUAL PORTFOLIO LOCATIONS

F

D

E

C

B

concavity of the efficient set12
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
concavity of the efficient set13
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
the market model
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

the market model1
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
the market model2
THE MARKET MODEL
  • ei, I CAN BE CONSIDERED A RANDOM VARIABLE
    • DISTRIBUTION:
      • MEAN = 0
      • VARIANCE = sei
diversification
DIVERSIFICATION
  • PORTFOLIO RISK
    • TOTAL SECURITY RISK: s2i
      • has two parts:

where = the market variance of index returns

= the unique variance of security i

returns

diversification1
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
diversification2
DIVERSIFICATION
  • Unique Risk
    • mathematically can be expressed as
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