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 set of portfolios meeting these two conditions is know as the efficient set(or the efficient frontier).

THE EFFICIENT SET THEOREM THE FEASIBLE SET DEFINITION: represents all portfolios that could be formed from a group of N securities All possible portfolios that could be formed from the n securities lie either on or within the boundary of the feasible set. The set will have an umbrella-type shape.

THE EFFICIENT SET THEOREM THE FEASIBLE SET rPrP PP 0

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 THEOREM Selection of the optimal portfolio the investor plots indifference curves on the same figure as the efficient set and then proceed to choose the portfolio that is on the indifference curve that is farthest northwest. The portfolio will correspond to the point at which an indifference curve is just tangent to the efficient set.

THE EFFICIENT SET THEOREM THE OPTIMAL PORTFOLIO E rPrP PP 0

THE EFFICIENT SET THEOREM Indifference curves for the risk-averse investor is positively sloped and convex. The efficient set is generally positively sloped and concave,meaning that if a straight line is drawn between any two points on the efficient set, the straight line will lie below the efficient set. There will be only one tangency point between the investor’s indifference curves and 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%  = 20% Gold Jewelry Company E(r) = 15%  = 40%

CONCAVITY OF THE EFFICIENT SET PP rPrP A G r A = 5  A =20 r G =15  G =40

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

BOUNDS ON THE LOCATION OF PORFOLIOS A B C D E F G X X

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 SET USING THE FORMULA we can derive the following:

CONCAVITY OF THE EFFICIENT SET r P  P=+1  P=-1 A52020 B C D E F G

CONCAVITY OF THE EFFICIENT SET For any given set of weights, the lower and upper bounds will occur when the correlation between the two securities is –1 and +1, respectively. UPPER BOUNDS lie on a straight line connecting A and G i.e. all  must lie on or to the left of the straight line which implies that diversification generally leads to risk reduction

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 SET  G upper bound lower bound rPrP PP 

CONCAVITY OF THE EFFICIENT SET SUMMARY For any given set of weights,the lower and upper bounds will occur when the correlation between the two securities is –1 and +1. Any portfolio consisting of securities a and g will lie within or on the boundary of the triangle, with its actual location depending on the magnitude of the correlation coefficient between the two securities.

CONCAVITY OF THE EFFICIENT SET ACTUAL LOCATIONS OF THE PORTFOLIO What if correlation coefficient (  ij ) is zero?

CONCAVITY OF THE EFFICIENT SET RESULTS:  B =17.94%  B =18.81%  B =22.36%  B =27.60%  B =33.37%

CONCAVITY OF THE EFFICIENT SET ACTUAL PORTFOLIO LOCATIONS  C D  F

CONCAVITY OF THE EFFICIENT SET The portfolio, consisting of two securities, lie on a line that is curved, or bowed, to the left. IMPLICATION: If  ij < 0line curves more to left If  ij = 0line curves to left If  ij > 0line curves less to left

CONCAVITY OF THE EFFICIENT SET KEY POINT As long as -1 <   the portfolio line curves to the left and the northwest portion is concave i.e. the efficient set is concave

THE MARKET MODEL A RELATIONSHIP MAY EXIST BETWEEN A STOCK ’ S RETURN AN THE MARKET INDEX RETURN where    intercept term r i = return on security r I = return on market index I    slope term    random error term

THE MARKET MODEL THE RANDOM ERROR TERMS  i, 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  i, I

THE MARKET MODEL  i, I CAN BE CONSIDERED A RANDOM VARIABLE DISTRIBUTION: MEAN = 0 VARIANCE =   i

THE MARKET MODEL Graphical representation of the market model: The vertical axis measures the return on the particular security The horizontal axis measures the return on the market index The line goes through the point on the vertical axis corresponding to the value of alpha. The line has a slope equal to beta.

THE MARKET MODEL Beta The slope in a security’s market model measures the sensitivity of the security’s returns to the market index’s returns

THE MARKET MODEL Beta Betas greater than 1 are more volatile than the market index and are known as aggressive stocks. Stocks with betas less than one are less volatile than the market index and are known as defensive stocks.

DIVERSIFICATION PORTFOLIO RISK TOTAL SECURITY RISK:   i has two parts: where = the market risk of security i = the unique variance of security i returns

DIVERSIFICATION PORTFOLIO RISK and return

DIVERSIFICATION PORTFOLIO RISK and return

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

DIVERSIFICATION Unique Risk mathematically can be expressed as

END OF CHAPTER 7