CHAPTER SEVEN PORTFOLIO ANALYSIS THE EFFICIENT SET THEOREM

CHAPTER SEVEN PORTFOLIO ANALYSIS

THE EFFICIENT SET THEOREM n THEOREM • An investor will choose his optimal portfolio from the set of portfolios that offer 3 maximum expected returns for varying levels of risk, and 3 minimum risk for varying levels of returns

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

THE EFFICIENT SET THEOREM THE FEASIBLE SET r. P 0 s. P

THE EFFICIENT SET THEOREM n EFFICIENT SET THEOREM APPLIED TO THE FEASIBLE SET • Apply the efficient set theorem to the feasible set 3 the set of portfolios that meet first conditions of efficient set theorem must be identified 3 consider 2 nd condition set offering minimum risk for varying levels of expected return lies on the “western” boundary 3 remember both conditions: “northwest” set meets the requirements

THE EFFICIENT SET THEOREM n 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 THEOREM THE OPTIMAL PORTFOLIO r. P E 0 s. P

CONCAVITY OF THE EFFICIENT SET n WHY IS THE EFFICIENT SET CONCAVE? • BOUNDS ON THE LOCATION OF PORFOLIOS • EXAMPLE: 3 Consider two securities – – Ark Shipping Company • E(r) = 5% s = 20% Gold Jewelry Company • E(r) = 15% s = 40%

CONCAVITY OF THE EFFICIENT SET r. P G r. G=15 r. A = 5 A s. A=20 s. G=40 s. P

CONCAVITY OF THE EFFICIENT SET n ALL POSSIBLE COMBINATIONS RELIE ON THE WEIGHTS (X 1 , X 2) X 2= 1 - X 1 Consider 7 weighting combinations using the formula

CONCAVITY OF THE EFFICIENT SET Portfolio A B C D E F G return 5 6. 7 8. 3 10 11. 7 13. 3 15

CONCAVITY OF THE EFFICIENT SET n USING THE FORMULA we can derive the following:

CONCAVITY OF THE EFFICIENT SET A B C D E F G r. P s. P=+1 s. P=-1 5 6. 7 8. 3 10 11. 7 13. 3 15 20 10 20 30 40 20 23. 33 26. 67 30. 00 33. 33 36. 67 40. 00

CONCAVITY OF THE EFFICIENT SET n UPPER BOUNDS • lie on a straight line connecting A and G 3 i. e. all s must lie on or to the left of the straight line 3 which implies that diversification generally leads to risk reduction

CONCAVITY OF THE EFFICIENT SET n LOWER BOUNDS • all lie on two line segments 3 one connecting A to the vertical axis 3 the other connecting the vertical axis to G point • 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 SET r. P G lower bound 0 A upper bound s. P

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

CONCAVITY OF THE EFFICIENT SET RESULTS: s. B s. B = = = 17. 94% 18. 81% 22. 36% 27. 60% 33. 37%

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

CONCAVITY OF THE EFFICIENT SET n 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 SET n 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 n A RELATIONSHIP MAY EXIST BETWEEN A STOCK’S RETURN AN THE MARKET INDEX RETURN where ai. I = intercept term ri = return on security r. I = return on market index I b i. I = slope term e i. I = random error term

THE MARKET MODEL n 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 MODEL n ei, I CAN BE CONSIDERED A RANDOM VARIABLE • DISTRIBUTION: 3 MEAN = 0 3 VARIANCE = sei

DIVERSIFICATION n PORTFOLIO RISK • TOTAL SECURITY RISK: 3 has two parts: where s 2 i = the market variance of index returns = the unique variance of security i returns

DIVERSIFICATION n TOTAL PORTFOLIO RISK • also has two parts: 3 Market Risk market and unique – diversification leads to an averaging of market risk – as a portfolio becomes more diversified, the smaller will be its unique risk 3 Unique Risk

DIVERSIFICATION 3 Unique Risk – mathematically can be expressed as

END OF CHAPTER 7