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### 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

- An investor will choose his optimal portfolio from the set of portfolios that offer

THE EFFICIENT SET THEOREM

- THE FEASIBLE SET
- DEFINITION: represents all portfolios that could be formed from a group of N securities

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

- Apply the efficient set theorem to the feasible set

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

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%

- Ark Shipping Company

- Consider two securities

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

CONCAVITY OF THE EFFICIENT SET

- USING THE FORMULA
we can derive the following:

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

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

- lie on a straight line connecting A and G

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

- all lie on two line segments

CONCAVITY OF THE EFFICIENT SET

- ACTUAL LOCATIONS OF THE PORTFOLIO
- What if correlation coefficient (r ij ) is zero?

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

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

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

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

THE MARKET MODEL

- ei, I CAN BE CONSIDERED A RANDOM VARIABLE
- DISTRIBUTION:
- MEAN = 0
- VARIANCE = sei

- DISTRIBUTION:

DIVERSIFICATION

- PORTFOLIO RISK
- TOTAL SECURITY RISK: s2i
- has two parts:
where = the market variance of index returns

= the unique variance of security i

returns

- has two parts:

- TOTAL SECURITY RISK: s2i

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

- Market Risk

- also has two parts: market and unique

DIVERSIFICATION

- Unique Risk
- mathematically can be expressed as

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