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CHAPTER SEVEN

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

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  1. CHAPTER SEVEN PORTFOLIO ANALYSIS

  2. 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

  3. THE EFFICIENT SET THEOREM • THE FEASIBLE SET • DEFINITION: represents all portfolios that could be formed from a group of N securities

  4. THE EFFICIENT SET THEOREM THE FEASIBLE SET rP sP 0

  5. 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

  6. 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

  7. THE EFFICIENT SET THEOREM THE OPTIMAL PORTFOLIO rP E sP 0

  8. 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%

  9. CONCAVITY OF THE EFFICIENT SET rP G rG=15 rA = 5 A sP sA=20 sG=40

  10. 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

  11. CONCAVITY OF THE EFFICIENT SET Portfolioreturn A 5 B 6.7 C 8.3 D 10 E 11.7 F 13.3 G 15

  12. CONCAVITY OF THE EFFICIENT SET • USING THE FORMULA we can derive the following:

  13. 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

  14. 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

  15. 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

  16. CONCAVITY OF THE EFFICIENT SET rP G lower bound upper bound A sP 0

  17. CONCAVITY OF THE EFFICIENT SET • ACTUAL LOCATIONS OF THE PORTFOLIO • What if correlation coefficient (r ij ) is zero?

  18. CONCAVITY OF THE EFFICIENT SET RESULTS: sB = 17.94% sB = 18.81% sB = 22.36% sB = 27.60% sB = 33.37%

  19. CONCAVITY OF THE EFFICIENT SET ACTUAL PORTFOLIO LOCATIONS F D E C B

  20. 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

  21. 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

  22. 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

  23. 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

  24. THE MARKET MODEL • ei, I CAN BE CONSIDERED A RANDOM VARIABLE • DISTRIBUTION: • MEAN = 0 • VARIANCE = sei

  25. DIVERSIFICATION • PORTFOLIO RISK • TOTAL SECURITY RISK: s2i • has two parts: where = the market variance of index returns = the unique variance of security i returns

  26. 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

  27. DIVERSIFICATION • Unique Risk • mathematically can be expressed as

  28. END OF CHAPTER 7

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