chapter seven portfolio analysis. the efficient set theorem n the theorem an investor will choose...
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THE EFFICIENT SET THEOREM THE THEOREM
•An investor will choose his optimal portfolio from the set of portfolios that offermaximum expected returns for varying
levels of risk, andminimum risk for varying levels of
returns
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 setthe set of portfolios that meet first conditions of
efficient set theorem must be identifiedconsider 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 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% = 20%
– Gold Jewelry Company• E(r) = 15% = 40%
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
22111
rXrXrXrN
iiiP
CONCAVITY OF THE EFFICIENT SET USING THE FORMULA
we can derive the following:
2/1
1 1
N
i
N
jijjiP XX
CONCAVITY OF THE EFFICIENT SET
rP P=+1 P=-1
A 5 20 20B6.7 10 23.33C8.3 0 26.67D10 10 30.00E 11.7 20 33.33F 13.3 30 36.67G15 40 40.00
CONCAVITY OF THE EFFICIENT SET UPPER BOUNDS
•lie on a straight line connecting A and Gi.e. all must lie on or to the left of the
straight linewhich implies that diversification
generally leads to risk reduction
CONCAVITY OF THE EFFICIENT SET LOWER BOUNDS
•all lie on two line segmentsone connecting A to the vertical axisthe 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 ACTUAL LOCATIONS OF THE
PORTFOLIO•What if correlation coefficient (ij ) is
zero?
CONCAVITY OF THE EFFICIENT SET IMPLICATION:
•If ij < 0 line curves more to left
•If ij = 0 line curves to left
•If ij > 0 line 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
ri = return on security
rI = return on market index I
slope term
random error term
iIIiiIi rr 1
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
DIVERSIFICATION
PORTFOLIO RISK•TOTAL SECURITY RISK:
i
has two parts:
where = the market variance of index returns
= the unique variance of security i
returns
2222iiiIi
22 iI2i
DIVERSIFICATION
TOTAL PORTFOLIO RISK•also has two parts: market and
uniqueMarket Risk
– diversification leads to an averaging of market risk
Unique Risk– as a portfolio becomes more diversified, the
smaller will be its unique risk