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

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
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

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 SETrP 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 = X 1
Consider 7 weighting combinations
using the formula

11. CONCAVITY OF THE EFFICIENT SET
Portfolio return A B C D E F G

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

13. CONCAVITY OF THE EFFICIENT SET
rP sP=+1 sP=-1 A B C D E F G

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 SETrP G
lower bound
upper bound A sP

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 < 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