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CHAPTER EIGHT PORTFOLIO ANALYSIS 1 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 2 THE EFFICIENT SET THEOREM • THE FEASIBLE SET – DEFINITION: represents all portfolios that could be formed from a group of N securities 3 THE EFFICIENT SET THEOREM THE FEASIBLE SET rP 0 sP 4 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 5 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 6 THE EFFICIENT SET THEOREM THE OPTIMAL PORTFOLIO rP E 0 sP 7 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% 8 CONCAVITY OF THE EFFICIENT SET rP rG=15 rA = 5 G A sA=20 sG=40 sP 9 CONCAVITY OF THE EFFICIENT SET • ALL POSSIBLE COMBINATIONS RELIE ON THE WEIGHTS (X1 , X 2) X2= 1 - X1 Consider 7 weighting combinations using the formula rP N X i 1 i ri X 1 r1 X 2 r2 10 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 11 CONCAVITY OF THE EFFICIENT SET • USING THE FORMULA 1/ 2 s P X i X js ij i 1 j 1 N N we can derive the following: 12 CONCAVITY OF THE EFFICIENT SET A B C D E F G rP sP=+1 5 6.7 8.3 10 11.7 13.3 15 20 10 0 10 20 30 40 sP=-1 20 23.33 26.67 30.00 33.33 36.67 40.00 13 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 14 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 15 CONCAVITY OF THE EFFICIENT SET rP G lower bound 0 A upper bound sP 16 CONCAVITY OF THE EFFICIENT SET • ACTUAL LOCATIONS OF THE PORTFOLIO – What if correlation coefficient (r ij ) is zero? 17 CONCAVITY OF THE EFFICIENT SET RESULTS: sB sB sB sB sB = = = = = 17.94% 18.81% 22.36% 27.60% 33.37% 18 CONCAVITY OF THE EFFICIENT SET ACTUAL PORTFOLIO LOCATIONS C D E F B 19 CONCAVITY OF THE EFFICIENT SET • IMPLICATION: – If rij < 0 – If rij = 0 – If rij > 0 line curves more to left line curves to left line curves less to left 20 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 21 THE MARKET MODEL • A RELATIONSHIP MAY EXIST BETWEEN A STOCK’S RETURN AN THE MARKET INDEX RETURN ri a iI b i1rI e iI where aiI intercept term ri = return on security rI = return on market index I b iI slope term e iI random error term 22 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 23 THE MARKET MODEL ei, I CAN BE CONSIDERED A RANDOM VARIABLE – DISTRIBUTION: • MEAN = 0 • VARIANCE = sei 24 DIVERSIFICATION • PORTFOLIO RISK TOTAL SECURITY RISK: s2i • has two parts: s b s s 2 i where 2 iI b iI2s 2 s e2i 2 i 2 ei = the market variance of index returns = the unique variance of security i returns 25 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 26 DIVERSIFICATION • Unique Risk – mathematically can be expressed as s 2 eP 2 1 2 s ei i 1 N N 1 s e21 s e22 ... s e2N N N 27